i UNIVERSITY OF GHANA DEPARTMENT OF STATISTICS DETERMINANTS OF THE PROMOTION OF UNIVERSITY OF GHANA LECTURERS: A SURVIVAL ANALYSIS APPROACH BY CRYSTAL BUBUNE LETSA 10441866 THESIS SUBMITTED TO THE DEPARTMENT OF STATISTICS, UNIVERSITY OF GHANA IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF A MASTER OF PHILOSOPHY DEGREE IN STATISTICS JUNE 2015 University of Ghana http://ugspace.ug.edu.gh i ABSTRACT The desire to reach the peak of one’s chosen career makes us curious and hardworking enough to attain positions and ranks. This is no different in the academia where lecturers look forward for promotion. However, there are perceived and actual factors contributing to promotion among University of Ghana lecturers which is examined. The factors contributing to promotion of University of Ghana lecturer and how long it takes a lecturer to earn a first promotion, were the objectives of this study. The Kaplan-Meier (KM) method was used in descriptive analyses. Proportional Hazards (PH); Cox, Exponential, Weibull and Gompertz and Acceleration Failure Time (AFT); Exponential, Weibull, Log-logistic and Lognormal modelling were techniques employed in further analyses. According to the KM estimate, the average time a lecturer first promotion is 8.09 years. Based on AIC values of 518.20076 and 3459.829, the best-fitting PH and AFT models were the Gompertz and Weibull distributions respectively. Married lecturers have the same chance to earn a first promotion as compared to their single counterparts. Again, male lecturers and female lecturers in the University of Ghana do not have different time to first promotion. University of Ghana http://ugspace.ug.edu.gh ii DECLARATION Candidate’s Declaration I hereby declare that this submission is my work in partial fulfilment of MPhil and that, to the best of my knowledge, it contains no material previously published by another person nor material which has been accepted for the award of any other degree of the University, except where due acknowledgement has been made in the text. CRYSTAL BUBUNE LETSA ____________________ __________ (10441866) SIGNATURE DATE Certified by: DR. KWABENA DOKU-AMPONSAH ____________________ __________ (SUPERVISOR) SIGNATURE DATE Certified by: DR. SAMUEL IDDI ____________________ __________ (SUPERVISOR) SIGNATURE DATE University of Ghana http://ugspace.ug.edu.gh iii ACKNOWLEDGEMENTS Firstly, I wish to Thank Dr. Kwabena Doku-Amponsah and Dr. Samuel Iddi for being very supportive supervisors. Their constant encouragement, patience and support, made this work possible. I am very grateful to the entire staff at the Department of Statistics, University of Ghana for their regular support, advice and friendship. I also express my gratitude to Dr. Mettle, the Head of Department for his regular support in my research. My sincere thanks also go to Mr Daniel Baidoo, the Director of HRODD, University of Ghana, Mr. Torto and Aunty Mary all of HRODD, for helping me obtain the data for this work. Again, I thank Mr. Sarpong of the School of Physical Sciences; who gave some ideas which helped shaping this work. Special thanks to Mr. M.K. Vowotor for being of enormous help and support throughout my thesis. Special thanks to all my friends especially Hannah, Sarah, Kassim, Shalom and Priscilla who encouraged and helped me with a lot of great ideas for this project work. Finally, I would like to thank my dearest siblings and Aunt Comfort Mensah for their undying love and support. University of Ghana http://ugspace.ug.edu.gh iv DEDICATION To My loving parents; Fautine Enyonam Letsa and Kefas Mawuli Letsa University of Ghana http://ugspace.ug.edu.gh v TABLE OF CONTENTS UNIVERSITY OF GHANA .................................................................................................. i ABSTRACT ....................................................................................................................... i DECLARATION ............................................................................................................... ii ACKNOWLEDGEMENTS ................................................................................................ iii DEDICATION ................................................................................................................. iv TABLE OF CONTENTS .................................................................................................... v APPENDICES ................................................................................................................ vii LIST OF FIGURES ......................................................................................................... viii LIST OF TABLE ............................................................................................................... ix LIST OF SYMBOLS ......................................................................................................... xi CHAPTER ONE ............................................................................................................... 1 INTRODUCTION ............................................................................................................. 1 1.1 Introduction .................................................................................................. 1 1.2 A Lecturer and Promotion at the University of Ghana ................................. 1 1.3 Research Problem Statement ....................................................................... 3 1.4 Research Questions ...................................................................................... 3 1.5 Statistical Methodology ................................................................................ 4 1.6 Objectives of Study ....................................................................................... 4 1.7 Significance of This Study .............................................................................. 5 1.8 Scope of Study .............................................................................................. 5 1.9 Limitations .................................................................................................... 6 1.10 Organisation of Work .................................................................................... 6 CHAPTER TWO .............................................................................................................. 7 LITERATURE REVIEW ..................................................................................................... 7 2.1 Introduction .................................................................................................. 7 2.2 Job Mobility................................................................................................... 7 2.3 Promotion of Workers .................................................................................. 8 2.4 Promotion of Lecturers in the University of Ghana ...................................... 9 2.5 Applications of Survival Analysis ................................................................... 9 2.6 Survival Analysis and Promotion of Workers .............................................. 10 CHAPTER THREE .......................................................................................................... 12 METHODOLOGY .......................................................................................................... 12 3.1 Introduction .................................................................................................. 12 University of Ghana http://ugspace.ug.edu.gh vi 3.2 Study Area ................................................................................................... 12 3.3 Experimental Design and Subjects .............................................................. 13 3.4 Ethics ........................................................................................................... 14 3.5 Statistical Techniques ................................................................................. 14 3.5.1 Overview of Survival Analysis ............................................................. 14 3.5.2 Censoring and Truncation ................................................................... 17 3.5.3 Estimation of Survival Data ................................................................. 19 3.5.4 Proportional Hazard Modelling ........................................................... 31 3.5.5 AFT Modelling ..................................................................................... 38 3.5.6 Diagnostic Tests .................................................................................. 41 3.5.7 Modelling and Model Selection .......................................................... 44 3.6 Statistical Software ..................................................................................... 44 3.7 Dissemination of Results ............................................................................. 45 CHAPTER FOUR ........................................................................................................... 46 RESULTS AND DISCUSSION ......................................................................................... 46 4.1 Introduction ................................................................................................ 46 4.3 Variable Coding ........................................................................................... 47 4.4 Preliminary Analysis .................................................................................... 48 4.5 Further Analysis .......................................................................................... 57 CHAPTER FIVE ............................................................................................................. 64 CONCLUSION AND RECOMMENDATIONS .................................................................. 64 5.1 Introduction ................................................................................................ 64 5.2 Conclusion ................................................................................................... 64 5.3 Recommendation ........................................................................................ 65 APPENDICES ................................................................................................................ 69 Appendix A .................................................................................................................. 69 FORM FOR APPLICATION FOR PROMOTION ........................................................... 69 Appendix B .................................................................................................................. 72 Procedure for the Appointment and Promotion of Senior Members .................... 72 Appendix C .................................................................................................................. 78 Some Tables and Graphs for Chapter 4 .................................................................. 78 Appendix D .................................................................................................................. 92 Codes Used In Data Analysis ................................................................................... 92 Appendix E ................................................................................................................ 101 Data Used In Research .......................................................................................... 101 Data Used In Research (continuation) .................................................................. 112 University of Ghana http://ugspace.ug.edu.gh vii APPENDICES Appendix A: Application for Promotion Form 69 Appendix B: Schedule F: Procedure for the Appointment and Promotion of Senior Members 72 Appendix C: Some Tables and Graphs for Chapter 4 78 Appendix D: Codes Used In Data Analysis 91 Appendix E: Data Used In Research 100 University of Ghana http://ugspace.ug.edu.gh viii LIST OF FIGURES Figure 4.1 A Kaplan-Meier Survival Plot of time to first promotion in Days 51 Figure 4.2 A plot of survival of Male and Female Lecturers 53 Figure C1 Comparison of Survivorship across groups for the categorical covariates: Marital Status, Colleges and Region of Origin. 80 Figure C2 Plots of log negative log Kaplan-Meier Survival Function against time (Days) for the covariates: Gender, Marital Status, Colleges and Region of Origin. 82 Figure C3 Plots of log negative log Kaplan-Meier Survival Function against log of time for the covariates: Gender, Marital Status, Colleges and Region of Origin. 83 University of Ghana http://ugspace.ug.edu.gh ix LIST OF TABLE Table 4.1 Frequency Distribution of Lecturers by Ranks 49 Table 4.2 Frequency Distribution of Lecturers by College of Work 49 Table 4.3 A table showing the Cox PH test for College as a Covariate 55 Table 4.4 The Test of College and Interaction 56 Table 4.5 Comparison of Hazard Ratios for Cox, Exponential, Weibull and Gompertz Full Models 58 Table 4.6 PH distribution models, log-likelihood and AIC values. 56 Table 4.7 The Gompertz PH Full Model Table of Hazard Ratios, Standard Errors and 95% Confidence Interval. 59 Table 4.8 Comparison of standard errors of Exponential, Weibull, Log-logistic and Lognormal AFT models 61 Table 4.9 Comparison of AIC values 61 Table 4.10 AFT Model of the Weibull Distribution 62 Table C1 Mean Age of Lecturers at Present and at the Time of First Promotion 78 Table C2 Frequency distribution of Region of Origin of Lecturers 78 Table C3 Median Time to first promotion Based on Marital Status 79 Table C4 Median Time to first promotion Based on College of Affiliation 79 Table C5 Median Time to first promotion Based on Region of Origin 79 Table C6 Log Rank Test for Gender of Lecturers 80 Table C7 Log Rank Test for Marital Status of Lecturers 81 Table C8 Log Rank Test for Colleges of Affiliation of Lecturers 81 Table C9 Log Rank Test for Region of Origin of Lecturers 81 Table C10 Cox PH Assumption Test for Gender 83 Table C11 Cox PH Assumption Test for Age of Lecturers 83 Table C12 Cox PH Assumption Test for Region of Origin of 84 University of Ghana http://ugspace.ug.edu.gh x Lecturers Table C13 Cox PH Assumption Test for Marital Status 84 Table C14 Interaction of Marital Status of Lecturers with Time 84 Table C15 The Exponential PH Full Model Table of Hazard Ratios, Standard Errors and 95% Confidence Interval 85 Table C16 The Weibull PH Full Model Table of Hazard Ratios, Standard Errors and 95% Confidence Interval 86 Table C17 Cox PH Full Model Table of Hazard Ratios, Standard Errors and 95% Confidence Interval 87 Table C18 Full AFT model Log-logistic distribution 88 Table C19 AFT model of the Lognormal Distribution 99 Table C20 AFT model of the Exponential Distribution 90 University of Ghana http://ugspace.ug.edu.gh xi LIST OF SYMBOLS AFT Accelerated Failure Time AIC Akaike Informative Criterion CI Confidence Interval KM Kaplan-Meier MLE Maximum Likelihood Estimator PH Proportional Hazards HR Hazard Ratio University of Ghana http://ugspace.ug.edu.gh 1 CHAPTER ONE INTRODUCTION 1.1 Introduction This study uses Survival Analysis to determine factors that contribute to the time a lecturer in the University of Ghana is promoted. The format of this chapter is as follows: it starts with a brief overview of the research topic, and then proceeds to take a look at the general profile of the study area, the problem statement, research questions and objectives. Research methodology, justification of the study as well as scope and limitations of the study are also discussed. 1.2 A Lecturer and Promotion at the University of Ghana A lecturer at the University of Ghana is a senior member who researches, teaches and may be responsible for some administrative and management role. The balance between teaching and research will depend on whether his/her department in the university focuses on research and/or teaching as well as the individual’s interest and goals. Lecturing as a job can be a highly rewarding career, with the freedom to research in one’s chosen area of interest, and the possibility of making a high impact in scientific applications for example. Lecturers may research as individuals or work in a group, which may be multidisciplinary and/or involve collaboration across universities and even countries. Sometimes they work individually on problems that interest them in University of Ghana http://ugspace.ug.edu.gh 2 particular. Often they supervise research students or postdoctoral researchers. (University lecturer, 2014). The major plus point of an academic job, is the relative freedom to set your own working hours, and your research agenda. The job can be very rewarding, with the opportunity to influence the direction of research in your field. It can also be varied, with time split between a variety of teaching, research and extracurricular activities. There is the opportunity (and expectation) to undertake international travel to attend conferences and workshops. Benefits, such as pension schemes, are usually excellent in comparison to the private sector. Down point includes workload that can often be high. In more senior positions, the time available for research may be diminished due to administrative and supervisory responsibilities. Marking exams and assignments can be tedious especially for larger classes and some courses especially those involving more practical sessions. And the pressure to produce research output and attract funding can be high; popularly known among the University community as to either ‘publish or perish.’ Presentations about researches at seminars and conferences, and write ups in recognised journal articles are well-known factors which enable lecturers to be promoted. Being promoted is an elevation in status or position. However there may be other contributing factors which may influence a lecturer’s promotion, which this research seeks to do. University of Ghana http://ugspace.ug.edu.gh 3 1.3 Research Problem Statement There is nothing as refreshing as someone seeing his or her efforts being recognised. Promotions are therefore of importance to every worker since it is a sign of acknowledgement for hard-work and gives the worker some level of prestige, privileges and sometimes, added responsibilities. Institutions may either have unique statutes or be using another institution’s statutes as reference. The promotion of lecturers in each institution is dependent on the governing statutes. There are however some requirements common to all the institutions. There may be fixed time to promotion or variable timing. There are contributing factors to a person’s promotion whether the time is fixed or variable. Promotion in The University of Ghana may be an issue on the mind of already employed lecturers, a debate among aspiring lecturers and a reference to other higher institutions of learning. The disambiguation of the issues bothering on University of Ghana lecturers is what this project seeks to do, laying bare the problem of how long it will take a lecturer to earn his/her first promotion, subsequent promotions and the factors that contribute to earning a promotion. 1.4 Research Questions Questions that would be answered in this research are: 1. When does a lecturer earn his/her first promotion? 2. What factors determine the promotion of a University of Ghana lecturer? University of Ghana http://ugspace.ug.edu.gh 4 1.5 Statistical Methodology Survival analysis is a set of statistical methods for studying the timing and occurrence of event(Allison, 1995). Survival Analysis typically focuses on time to event data. In the most general sense, it consists of techniques for nonnegative valued random variables, such as: time to death, time to onset (or relapse) of a disease, length of stay in a hospital, duration of a strike, money paid by health insurance, viral load measurements, time to finishing a doctoral dissertation and many others. Areas of survival studies include: clinical trials, prospective cohort studies, retrospective cohort studies, retrospective correlative studies (Ibrahim, 2014). 1.6 Objectives of Study This study seeks to focus on the technique of Survival Analysis to determine factors that influence the time to which a University of Ghana lecturer earns a promotion. That is, to know the median time to first promotion and subsequent ones if applicable and also to investigate the major contributing factors to a lecturer’s promotion. The focus will be to obtain a theoretical basis for solution to problems relating to promotion of lecturers, as well as obtaining a model for a given set of determinants, it can aid in predicting the mean or median time to first promotion of any newly employed lecturer. University of Ghana http://ugspace.ug.edu.gh 5 1.7 Significance of This Study This study is of great significance because: * It will give upcoming lecturers an insight on how to balance, set priorities, their responsibilities and where their focus should lie in order to climb the next ladder in academia. * It will serve as a reference point for other higher institutions of learning on where their targets should be, especially during recruitments. * It will enable the University review her policy on promotion if results show that the existing one is not effective. 1.8 Scope of Study The study involves currently employed full time University of Ghana lecturers. A secondary data collected from The Human Resource Directorate of The University of Ghana is used for the work. However a sample is used, the sampling of lecturers is done randomly with three lecturers selected from each Department. The main technique is the Survival Analysis because it incorporates censored observations. Various techniques of Survival Analysis are considered specifically, Proportional Hazard (PH) modelling and the Accelerated Failure Time (AFT) modelling. The PH models include: Cox Regression, Weibull, Gompertz and Exponential models while the AFT models are: Exponential, Weibull, Log-logistic and Lognormal models. The R and STATA were the statistical software for the analyses even though the data was first obtained in a Microsoft Excel spreadsheet. University of Ghana http://ugspace.ug.edu.gh 6 1.9 Limitations The main limitation of this work is the time frame. Also, since the data secondary, it is not 100% accurate. It may also have some missing information since the database for lecturers is still under construction. Much literature is also not available especially on the use of Survival Analysis in the study of promotion of workers, especially lecturers. 1.10 Organisation of Work The research work has been arranged in this Thesis as follows: Chapter One provides the introduction to the entire study. It examines the general background of the basic factors that determine the promotion of University of Ghana lecturers, the problem statement, research questions and objectives, research methodology, significance of the study, scope and limitations of the study. Chapter Two reviews related literature based on the thesis objectives and preferred models to be used in achieving these objectives. Expected outcome of the study and other comparative results of similar studies are also discussed. Chapter Three describes the theory of the model to be used, formulations and methods of obtaining solution. In Chapter Four, the data collection method is discussed, the data is analysed and displayed. This chapter also contains the interpretation and discussion of the results. Finally, Chapter 5 concludes the entire study and states specific recommendations to stakeholders based on the major findings made in the study. University of Ghana http://ugspace.ug.edu.gh 7 CHAPTER TWO LITERATURE REVIEW 2.1 Introduction The study of job mobility is usually done in fields of Business, Economics and Psychology. These fields will therefore explore relatively less sophisticated statistical techniques. However in this chapter, review is done on job mobility and the superior technique of Survival analysis. 2.2 Job Mobility Job mobility is the concept used to describe any movement on a job. The following types of job mobility can be distinguished: i. Occupational/career mobility: when an employee carries out a different job (occupational category) for the same employer (e.g. moving from junior to senior manager). ii. Intra-sectorial mobility: when an employee carries out the same job for another employer in the same sector (e.g. moving as a post-doc researcher from one university to another). iii. Inter-sectorial mobility: when a researcher carries out the same job for another employer in another sector (e.g. moving from university to industry). Job-to-job mobility refers to job mobility within or between sectors and implies a movement from one employer to another (IDEA Consult et al, 2010). University of Ghana http://ugspace.ug.edu.gh 8 Internal labour markets are clusters of jobs that have three basic structural features: job ladders, entry at the bottom, and internal promotion based on knowledge or skill development. This research is solely going to be on transition of a worker from one level to another defined above as occupational or career mobility (Althauser, R. & Kalleberg, 1981). 2.3 Promotion of Workers The determinants of mobility may depend on the definition adopted; promotions are differentiated by whether or not they are associated with a change in the tasks performed. There are existing economic theories which have addressed mechanisms generating labour mobility within and between firms. One model assumes that in the labour market, employers determine wages and employment (vacancies) and predicts that workers change jobs in response to differences in wage rates. If workers were the same, then firms will easily find it easy to replace a worker. While productivity is increased marginally by human capital, some firms need specific skills in order to increase productivity. Hence, employee turnover becomes a matter of importance since some firms train some employees in order to have the desired skill for the firm’s progress. The firms, mostly private ones may increase wages of highly skilled workers in order to retain them. In order to increase profitability of workers, employers may also create a promotion scheme which will have an associated wage and privileges to motivate workers in firm-specific human capital (Ferreira, 2009). Promotions can also be interpreted in the context of tournaments. A promotion is a prize that is allocated to workers who rank higher than all other workers in a group over a given period. The probability of promotion provides University of Ghana http://ugspace.ug.edu.gh 9 incentive to exert effort, and winners are moved to higher positions that involve e.g. higher prestige, higher responsibility, or higher earnings (Bognamo, 2001; Lazear, 1981) . Findings that clearly emerged relates to gender differentials in promotions. Their existence depends heavily on the definition of promotion adopted. Duration models’ results suggest that older individuals experience longer survival times to promotion than their younger counterparts. The magnitude of this effect is even stronger when considering just the time to first promotion. Older employees are left for last in promotion (Ferreira, 2009; Portela & Machado, 2013). 2.4 Promotion of Lecturers in the University of Ghana The University of Ghana statutes provides the procedure for promotion of lecturers. And no matter the level one is qualified to go in for; an aspiring lecturer must download and fill an online application form, found in Appendix A; this form gives information on what factors are scrutinized in order to grant promotions. The application goes through a series of processes. Appendix B has some information on the criteria for promotion in the University of Ghana. (Basic Laws of The University of Ghana, 2012). 2.5 Applications of Survival Analysis All the cases in the above examples have data consist of censored observations in which the end event has not happened in every observation or when information on a case is only known for a limited duration. The censoring time is the main information to find cumulative survival probability in the survival analysis. In other words, the great advantage of using survival University of Ghana http://ugspace.ug.edu.gh 10 analysis is to analyse censored cases in analysis. Another strength of using survival analysis is the use of covariates and the ability to assess the magnitude of specific influences which can be analysed (e.g., age, level of education) (Wang, Little and DelHomme-Little, 2012). 2.6 Survival Analysis and Promotion of Workers Survival analysis is a technique that has been used in the management literature to examine phenomena such as turnover, absenteeism, and population ecology (Lee et al., 2008; Morita, Lee, & Mowday, 1989, 1993). This approach is chosen to examine the impact of different types of boundary crossings on the probability of career advancement because this analytical technique is designed to model “time to event” data and is well suited for data at different time points with dependent variables (Morita et al., 1989). In addition, compared to other regression techniques (e.g., logistic and OLS regression), survival analysis is particularly appropriate for the current study because the probabilities of the occurrence of an event (career advancement) are described as a function of time, and it allows the probabilities to differ from one time point to another (Morita et al., 1993). The response variable is the time to advancement in career. The predictor variables included sex (0=female, 1=male), marital status (0=not married, 1=married), and level of education as control variables because prior research suggests that these variables may play substantive roles in career advancement. Meta-analysis found that sex is a key socio-demographic variable that predicts career success, with men achieving greater objective career success than women. Marital status has been shown to influence career success, with married individuals achieving more favourable career outcomes University of Ghana http://ugspace.ug.edu.gh 11 than unmarried individuals (Judge & Bretz, 1994; Ng et al., 2005; Pfeffer & Ross, 1982). Level of education, as a form of human capital, has also been found to be positively related to career outcomes (Ng et al., 2005; Pfeffer & Ross, 1982). University of Ghana http://ugspace.ug.edu.gh 12 CHAPTER THREE METHODOLOGY 3.1 Introduction This chapter sets to describe in detail, the procedure of the research. The main statistical technique used in this work is the survival analysis. The chapter also describes the study area, the design of the experiment. Survival analysis is used to analyze data corresponding to survival time. Thus survival time is the time to occurrence of an event, it is also known as failure-time data or end point (Hoon, 2008). Time to events include survival time in promotional and non- promotional events. An example of survival time in a promotional event is the survival time until promoted or until being promoted in an institution. On the other hand, an example of survival time in non-promotional event is the time to graduation until one gets a job. A famous example of the application of survival analysis for non-promotional event is finding the determinants of the survival of a network in franchising, where time is an important variable for the development of franchising (Perrigot et al, 2004). 3.2 Study Area The study incudes presently employed full time lecturers in the University of Ghana. University of Ghana http://ugspace.ug.edu.gh 13 3.3 Experimental Design and Subjects A secondary data was collected from the Human Resource Directorate of the University of Ghana. Four lecturers are sampled from each department. The data points to be collected from the secondary data will be as follows: Data points Description 1. Nationality 2. Date of employment The date a lecturer is employed 3. **Level of employment The level a lecturer was employed whether as an assistant lecturer, lecturer, senior lecturer, etc. 4. Qualification The qualification a lecturer had as at time of employment. 5. Gender 6. Religion 7. Region of origin 8. Age The lecturer’s date of birth or age as at time of employment. 9. **Schools attended Names of tertiary institutions a lecturer attended and countries where the schools are located. 10. Promotions and dates The date a lecturer gets promoted. 11. **Publications The number of publications a lecture submits in order to earn a promotion. 12. **Programmes read The programmes a lecturer read for undergraduate study and postgraduate studies 13. Marital status The marital status is preferred with two levels single or married with date of marriage specified. 14. Children Number of children and dates of births if available. The names of children are to be excluded. ** indicates that data point was not provided. University of Ghana http://ugspace.ug.edu.gh 14 3.4 Ethics There is the need for high ethical standards in any research. The rights of the lecturers whose data were extracted were respected during the course of the research. In addition, consent was sought from the University of Ghana before the data extraction commenced. The University was also assured that the information gathered was confidential. 3.5 Statistical Techniques 3.5.1 Overview of Survival Analysis Survival analysis is an analysis involving the study of time to occurrence of an event of interest. The study of time-to-event has data with special features which makes this method unique. Some of these features are censoring and truncation. And these require special treatments when analysing the data. There are four functions under survival analysis, they are: a. Survival function b. Hazard function c. Cumulative hazard function d. Mean residual life Survival function Survival function ( )(tS ) is the probability of an individual surviving at least to time t. It is written mathematically as; University of Ghana http://ugspace.ug.edu.gh 15 (3) )( )( (2) )()( (1) )()( dt tdS tf dttftS tTPtS t −= = >= ∫ ∞ The survival function is a monotonically decreasing function; no matter the distribution, as time increases, the chance of survival decreases i.e. 0)()(lim)( and 1)0( t ===∞= ∞→ tStSSS . And )(tf is the density function of the given distribution. Hazard Function The hazard function is the instantaneous rate at which an event occurs given that no previous event has occurred. The hazard rate in itself is not a probability. It happens with small changes and may have values greater than 1. However, it is nonnegative. Hazard rates are not directly observable quantities, they are estimates and vary from one individual to the other unless in a homogenous population. The hazard function is given as [ ] t tT lim)( 0 ∆ ≥∆+<< = →∆ ttTtP th t (4) The hazard function gives more information about mechanisms underlying failure than the survival function. Thus for a continuous random variable, the hazard function is obtained by; University of Ghana http://ugspace.ug.edu.gh 16 [ ] dt tsd tS tf th )(ln )( )( )( − == (5) The cumulative hazard H(t) is the sum of hazards to a time of interest say, t, which is actually integration for a continuous random variable and summation in discrete random variables. )](ln[ )()( 0 tSduuhtH t −== ∫ (6) Hence the survival function can be deduced from the relations above.    −=−= ∫ t duuhtHtS 0 )(exp)](exp[)( (7) The Density Function Another function under function of the survival analysis is the probability density function; for a continuous random variable and probability mass function; for a discrete random variable. The continuous distribution T has a cumulative distribution function given a 0 t],[)( ≥≤= tTPtF which yields the density function when differentiated duuftF dt tF tf t )()( )( )( 0∫=⇒= (8) The density function is related to the survival function by [ ] )(1)( tFtTPtS −=>= . Where T is the survival time; )(tS is the probability that a randomly selected person will survive to at least time t (the survival function). University of Ghana http://ugspace.ug.edu.gh 17 Mean Residual Life Functions The mean residual life for an individual at age t measures the individual’s average remaining life. It is the average time left for an individual who is yet to experience an event. )( )()( )( )()( 0 00 0 000 0 tS dttftt tmrl tTtTEtmrl t∫ ∞ − = ≥−= (9) 3.5.2 Censoring and Truncation Censoring occurs when an observation is incomplete due to a random occurrence. The random occurrence must be independent of the event of interest. Censoring usually occurs because a person does not experience the event before the study time set by researcher elapses, the subject drops out as study progresses (attrition) or the subject is not available for a follow-up test. There are three types of censoring: left, interval and right censoring. Left censoring: When the event interest has already occurred at the time of observation but the exact time is not known. Interval censoring: This type of censoring occurs during observation such that, even though the precise time of occurrence is unknown, there is an interval bound around the event occurring which is known. Right censoring: The right censoring has three types of censoring. The fixed types 1 and 2 and the random censoring. University of Ghana http://ugspace.ug.edu.gh 18 • Fixed type 1 censoring: in this type of censoring, a censored subject does not experience the event of interest because the experiment is set to end after “C” years of follow-up. Hence any subject who does not experience the event before the end of the experiment and follow-up time elapses is censored. • Random censoring: This type of censoring occurs when subjects have different censoring time even though the experimental design has a fixed study time. • Fixed type 2 censoring: In this type of censoring, the experimental design is such that, there are a pre-specified number of events of interest. Hence any subject that does not experience the event before the required number is obtained is censored. Truncation on the other hand, is an incomplete observation pertaining to a subject but its occurrence is due to the form of selection inherent in the design of experiment employed. The types of truncation include left and right truncation. Unlike censoring which has partial information on the subject, in truncation the researcher has no information on the subject. Left truncation: Individuals whose event time is greater than some truncation threshold are observed, otherwise they are truncated. Right truncation: The inclusion of subjects with time-to-event lesser than a truncation threshold in a study In order to have an effective methodology, the cause of both truncation and censoring should be independent of the failure of event of interest. And if present in a dataset, should be included in the analyses. University of Ghana http://ugspace.ug.edu.gh 19 3.5.3 Estimation of Survival Data Survival data can be estimated in three main ways; the parametric, semi-parametric and non-parametric approaches. The Kaplan-Meier (KM) estimation is an example of a non- parametric modelling of the survival function. The Cox Proportional Hazard is an example of the semi-parametric model and the Gamma, Exponential and Lognormal distributions are some examples of parametric models. In the parametric approach, the survival time is assumed to follow a particular parametric distribution. And the data must initially be tested to follow certain assumption of the particular distribution. These tests are powerful but may lead to inaccurate conclusions if they are used inappropriately. However, in the nonparametric modelling, assumptions are not stringent and therefore are more flexible and used for convenience. The semi-parametric tests are flexible which make some assumptions about the time-to-event which can be tested and model the effects of covariates. These survival estimation techniques are however able to estimate censored or non- censored data. The two main procedures employed in this work are estimation using semi-parametric and parametric models. The semi-parametric model is the Cox Proportional Hazard (PH) regression model. There are three parametric models; the best-fitting parametric model is in turn compared to the Cox PH model in order to know which method of modelling the data produces a better fitting model. Thus it is proper to use parametric models which use the same method of modelling as the Cox PH regression model. University of Ghana http://ugspace.ug.edu.gh 20 Parametric models are modelled using two main procedures; Accelerated Failure Time (AFT) and Proportional Hazard methods. The AFT’s are usually linked to the survival function. They measure the covariates’ effects which are multiplicative on the survival times. The covariates either accelerate or decelerate the occurrence of an event. And AFT’s are also assumed to be constant for any survival percentile. The PH models are associated with the hazard function; they measure the risk of the occurrence of an event of interest. The determinants are multiplicative to hazards and either increase or decrease hazards. The AFT and PH models have their unique limitations. In the PH model for instance, the model tends to be restrictive and it is not easy to interpret due to its non-linear nature. This non-linearity makes it not easy to analyse the factors on especially long-term effects. Log-logistic, Lognormal, Gamma and Inverse Gaussian distributions are parametric distributions which use the AFT method of estimating survival data. The Weibull and exponential distributions are distributions which use both AFT and PH modelling technique. The Gompertz distribution is one of the main parametric models used in PH modelling. The succeeding section discuss the types of models; parametric, semi- parametric and non-parametric and the techniques of modelling; PH and AFT. 3.5.3.1 Non Parametric Model: Kaplan-Meier (KM) A preliminary analysis is conducted for the survival data like any other data undergoing analysis. Most of the descriptive tests are non-parametric tests because they are fairly robust, efficient compared to parametric tests and are University of Ghana http://ugspace.ug.edu.gh 21 simple to understand and interpret. The first step is to obtain the survival times for the data. Then the survival function using the Kaplan-Meier (KM) also known as the Product Limit estimator, a non-parametric modelling approach is used. Let )(td denote number of events (promotions) at time t . )(td is either 1 if a lecturer has earned a promotion or 0 otherwise, but there is an allowance for the possibility of ties in which the number of promotions is greater than 1. The number of individuals at risk at time t is denoted by )(tn . The KM survival function is given as ∏ ≤       −= itt tn td KM )( )( 1 (10) Right censoring is accommodated in the KM method which is a right continuous step function which has a jump at the “death” times. After obtaining the survival functions, a graph of the survival function against time is plotted to enhance understanding of the function. Again, the survivorship of categorical variables can be assessed by various tests, a number of statistical tests have been proposed to answer this question, and most software packages provide results from at least two of these tests. However, comparison of the results obtained by different packages can become confusing due to small but annoying differences in terminology and methods used to calculate the tests (Hosmer, Lemeshow & May, 2008). Some of the tests include, Wilcoxon test weights, Log Rank Tests, Tarone- Ware weights, Peto-Prentice weights and other tests peculiar to certain software packages. However, in this work, the Log Rank test is used to assess differences in survivorship across categories because of its availability in the R package and the ease in result interpretation. University of Ghana http://ugspace.ug.edu.gh 22 The Log Rank test can also be extended to covariates with more than two categories. The null hypothesis is generally given by; :0H There are no differences between survival curves. Like the KM test, this test has same assumptions. These are; a. the survival probabilities are the same for subjects recruited early and late in the study, b. censoring is unrelated to chances of survival, c. the events happened at the times specified. Test statistic for the Log rank test is obtained in similar to the chi-square test procedure. The observed survival and the expected survival cell counts over categories. )var( )( statisticrank Log 22 2 22 EO EO − − = for a two category covariate, where O is the observed survival count and E is the expected survival count. Hence the test statistic can be approximated as ∑ = − ≈ n i i ii E EO 1 2)( statisticrank Log (11) This approximation formula applies to covariates with more than two categories. The hypothesis for this test, given for instance for the covariate gender is given as; 0H : There is no statistically significant difference between the survival of males and females. University of Ghana http://ugspace.ug.edu.gh 23 :1H There is a statistically significant difference in the survival of males and females. Tested at a 0.05 level of significance, a resulting p-value lesser than 0.05, implies that, survival is different for males and females. The hypothesis is set in like fashion for the test of survivorship for the rest of the covariates. 3.5.3.2 Semi-parametric Regression Model: Cox Regression Cox Regression modelling is usually used for data which have subjects in groups with additional characteristics that may affect outcomes. The variables may be used as covariates which may be explanatory variables, confounders, risk factors or independent variables. The Cox PH model is not a fully parametric model because even though its survival time has a parametric regression structure, its dependence on time is left unspecified (Hosmer, Lemeshow & May, 2008). It is a semi-parametric model because even if the regression parameters (the betas) are known, the distribution of the outcome is unknown. The Cox regression model is one of the most used modelling methods in survival analysis because it is robust; the results from using the Cox model will closely approximate the results for the correct parametric model. Also, Cox model always yields a nonnegative hazard function. The exponential part of the hazard function is appealing because it ensures that the fitted model will always give estimated hazards that are non-negative. Another appealing property of the Cox model is that, even though the baseline hazard part of the model is unspecified, it is still possible to estimate the β’s in the exponential part of the model. The measure of effect, which is called a hazard ratio, is University of Ghana http://ugspace.ug.edu.gh 24 calculated without having to estimate the baseline hazard function. The final reason for the popularity of the Cox model is that, it uses more information— the survival times—than the logistic model, which considers a (0,1) outcome and ignores survival times and censoring.(Kleinbaum & Klein, 2005). The general model form of the Cox is given by; pis th k i i X i thXth 1,2,3,..., with parameters theare ' and hazard, baseline theis )( where; (12) 1 exp)( 0 ),( 0 =         ∑ = = β β The parameters in the equation are estimated using the procedure of partial likelihood; the procedure takes into consideration censored data. The likelihood function is the product of the number of failure times say k . ∏ ∑ ≥ = d Uncensore )exp( )exp( )( i ij Y YY j i X X L β β β (13) The log partial likelihood is then given by: ( ){ } )exp(log)(log)( d Uncensore ∑ ∑ ≥−== i ij Y YY ji XXLl ββββ (14) After forming the likelihood function, it is maximised (partial derivatives of log L are obtained with respect to the parameters in the equation) then the resulting equation is solved in order to derive valid partial MLE’s of β . The partial likelihood is effectual if no two subjects experience the event simultaneously (presence of ties). If this occurs some permutations are done to cater for the situation and hence the use of approximations like the Breslow or Effron’s approximation to partial log-likelihood. University of Ghana http://ugspace.ug.edu.gh 25 In Cox models, the basic assumption is that the hazard rates for varying values of the covariates are proportional. The Cox model generates regression coefficients that can be used to determine the percent change in the hazard rate (i.e., the increase or decrease in the probability of a promotion) given changes in the independent variable (e.g., Age, marital status, number of publications), (Zheng, Veiga & Powell, 2011). The hazard ratio is the ratio of hazard of one individual in a study to that of another individual comparing some covariates. It is written as exp)(ˆ exp)(ˆ ),(ˆ ),(ˆ 1 0 1 * 0*      Χ      Χ = Χ Χ = ∑ ∑ = = ∧ k i ii k i ii th th th th HR β β (15) which becomes ( )    Χ−Χ= ∑ = k i iiiHR 1 *exp β because the baseline hazards cancel out each other. 3.5.3.3 Parametric Models i. Exponential The exponential distribution is one of the most commonly used parametric models; it has one parameter and a constant hazard. The probability density function is given by 0, where )( >= − λλ λ xexf x (16) The survival function and the hazard function of the exponential distribution are University of Ghana http://ugspace.ug.edu.gh 26 λ λ = −= )( )exp()( th ttS (17) The exponential distribution is mathematically tractable due to its memory-less property given by )()( zTPtTztTP ≥=≥+≥ , (18) This property makes the use of the exponential distribution limited in application fields like in industrial and health research. The no memory property simply means that, the occurrence of a future event does not depend on an event that has already occurred. Hence the mean residual life is constant and given by λ1)()( ==>− TEtTtTE . ii. Weibull The Weibull distribution is used in parametric modelling of survival data analysis and has two parameters. This distribution can reduce to an exponential distribution and it is very flexible. The parameters are αλ and , the scale and shape parameter respectively. The shape parameter α can take different values which in turn affects the shape of the hazard function. When 1>α the hazard function is increasing, 1=α implies the hazard is constant and similar to the hazard in an exponential model and when 1<α the hazard is a decreasing function. Thus the parameter α is referred to as the shape parameter. The product of the survival function and hazard function gives the density function. University of Ghana http://ugspace.ug.edu.gh 27 (19) 0, tand 0, )exp( )()()( )( )exp()( 1 1 ≥>−= ×= = −= − − λαλλα λα λ αα α α tt thtStf tth ttS There are some assumptions that must be tested to ascertain the appropriateness of Weibull distribution. Even though the assumptions in other distributions differ depending on whether it is AFT or PH, for the Weibull distribution, if the PH assumption holds then the AFT assumption holds and vice versa. iii. Gompertz The Gompertz model like the Weibull is also a two-parameter model; it is a parametric model that uses PH modelling but not AFT. The Gompertz is popularly used in modelling mortality curves. The hazard function of the model is given by teth αθ=)(0 (20) and survival function     −= )1(exp)( tetS α α θ (21) thus the density function is; 0 and 0, where )1(exp)( ≥>     −= t etetf t αθ α θαθ α (22) University of Ghana http://ugspace.ug.edu.gh 28 iv. Log-Logistic The log- logistic is a parametric modelling distribution that strictly models using the AFT approach of modelling in Survival analysis. The hazard function is not monotonic, it is given by: p ppt th λ λ + = − 1 )( 1 (23) where p is the shape parameter and λ the scale parameter. For 1≤p , the hazard decreases and for 1>p , the hazard is unimodal. As the name suggests, the log-logistic is a proportional odds survival model and therefore has the underlying assumption that, proportional odds (PO) have odds ratio constant over time. The definitions are thence given in terms of odds. Survival Odds, is the odds of surviving beyond a time t p p p t t tS t tS tTP tTP tS tS λ λ λ + =− + = ≤ > = − 1 )(1 and 1 1 )( where )( )( )(1 )( (24) Failure Odds, is the odds of experiencing the event by time t; this is given by the reciprocal of equation (39) as: University of Ghana http://ugspace.ug.edu.gh 29 p p p p t t t t tTP tTP tS tS λ λ λ λ =       +       + = > ≤ = − 1 1 1 )( )( )( )(1 (25) The log-logistic has its survival function as ( )p p t t tS p 1 1 1 1 1 )( λ λ + = + = (26) v. Lognormal The lognormal is also a parametric distribution that is AFT, not proportional odds and its survival and hazard can just be expressed in terms of integrals. Lognormal and log-logistic distributions are similar and result in similar models. The survival time t is such that, tlog is normally distributed with mean µ and variance 2σ . The popularity of the lognormal distribution is due in part to the fact that the cumulative values of y=log t can be obtained from the tables of the standard normal distribution and the corresponding values of t are then found by taking antilogs. Thus, the percentiles of the lognormal distribution are easy to find (Lee & Wang, 2003). University of Ghana http://ugspace.ug.edu.gh 30 The survival and probability density are given by ( )∫ ∞     −−= t dxx x tS 2 2 log 2 1 exp 1 2 1 )( µ σπσ (27) ( ) 2,0 log 2 1 exp 2 1 )( 2 2 >>    −−= σµ σπσ tx t tf (28) Let aa log- then ),exp( =−= µµ the equations (27) and (27) become: ( )∫ ∞    −= t dxax x tS 2 2 log 2 1 exp 1 2 1 )( σπσ (29) (30) log1)(    −= σ at GtS G(y) is the cumulative distribution function of a normal standard variable and given by the equation below: ( ) (31) 2 1 )( 0 22 dueyG y u∫ −= ( ) log 2 1 exp 2 1 )( 2 2    −= at t tf σπσ (32) The lognormal distribution is specified completely by the two parameters µ and 2σ , which are scale parameters not location and scale parameters as in the normal distribution. Time t cannot assume zero values since log t is not defined for t = 0. The distribution is flexible and positively skewed and the greater the value of 2σ , the greater the skewness (Lee & University of Ghana http://ugspace.ug.edu.gh 31 Wang, 2003). The hazard function associated with the lognormal distribution is given by; ( ) ( )[ ] ( )σ σπσ atG att th log1 2logexp21 )( 22 − − = (33) 3.5.4 Proportional Hazard Modelling Proportional hazard models are either parametric or semi-parametric. Parametric models aid in the interpretation of the parameters and some functions, especially the hazard rate. The semi-parametric model does not require the choice of a particular probability distribution to represent the survival time. The semi-parametric models have unspecified baseline hazard while the parametric models have the baseline hazard specified to follow a particular distribution. They assume that there is an underlying hazard rate over time, differences in the relative hazard rate at a point in time, is as a result differences in covariates. In other words, they assume no interaction between time and covariates. Mathematically, kk xxxethth βββ +++= ...0 2211)()( (35) where, )(0 th is some time function of the hazard rate known as the baseline hazard. They are called the proportional hazards model because; the hazard for any individual is a fixed proportion of the hazard for any other individual. (Klein, Moeschberger, 2003). To see this, take the ratio of the hazards for two individuals i and j, with covariates University of Ghana http://ugspace.ug.edu.gh 32 ( ) ( ) [ ] [ ] ( )[ ]∑∑ ∑ = = = Χ−Χ= Χ Χ = Ζ Ζ p k kkkp k kk p k kk th th h h 1 10 10 *exp *exp)( exp)( * β β β (34) What is important about this equation is that )(0 th cancels out of the numerator and denominator. As a result, the ratio of the hazards is constant over time. Proportional hazards models have a number of advantages. First, proportional hazards modelling uses information on survival time (i.e., promotion), rather than relying solely on a simple dichotomous dependent variable. Second, the proportional hazards model allowed us to differentiate between an employee who gets promoted in year one and an employee who gets promoted in another year. This is an important distinction, as some have noted that failure to consider the timing of employee job movements may result in biased findings (Morita et al., 1993). Third, proportional hazards modelling can control for censored data. In some cases, the exact survival time is unknown, although it is known to be greater than the specified value. Censoring occurs when the study ends without all the employees getting promoted. If the effect of censoring was ignored, the sample size and power will reduce and in effect render the results biased. There are tests available to check data suitability for PH modelling. Tests of PH assumptions for parametric and semi-parametric tests take similar formats. If we graph the log hazards for any two individuals for instance, the proportional hazards property implies that the hazard functions should be strictly parallel (Allison, 1995). However, to test the PH assumption, a plot of ( )( ) ( )ttS logagainst loglog − is an appropriate graphical test. The Survival estimates used for this plot are KM survival estimates of two or more levels of University of Ghana http://ugspace.ug.edu.gh 33 covariates and parallel and linear lines indicates that the PH assumption hence AFT assumption holds. And therefore the use of a Weibull distribution on the survival data is reasonable. If the lines in the plot is straight and the slope is equal to 1, then either an AFT and PH exponential modelling is appropriate. When the lines are parallel but not linear, a Cox Model can be used because it is PH but not Weibull hence not AFT. If the lines are straight but not parallel, then neither PH or AFT holds, but Weibull with a different shape parameter can be used. Finally, if the lines are neither straight nor parallel, the PH assumption is violated and hence the data does not follow a Weibull distribution. The points above are summarised as: Summary of possible results for plot of ln[−ln ˆS(t)] against ln(t) 1. Parallel straight lines ⇒Weibull, PH, and AFT assumptions hold 2. Parallel straight lines with slope of 1⇒Exponential. PH and AFT 3. Parallel but not straight lines⇒ PH but not Weibull, not AFT (can use Cox model) 4. Not parallel and not straight⇒ Not Weibull, PH violated 5. Not parallel but straight lines⇒ Weibull holds, but PH and AFT violated, different (Klein & Kleinbaum,2005) The PH assumption requires that the Hazard Ratio (HR) is constant over time, or equivalently, that the hazard for one individual is proportional to the hazard for any other individual, where the proportionality constant is independent of time denote the set of X’s for two individuals. Hence if the test University of Ghana http://ugspace.ug.edu.gh 34 of assumption is violated for a given PH modelling technique, that method will not be executed. In investigating a group, the length of survival is related to various characteristics. And therefore an underlying hazard for the “average” subject denoted by )(0 th is used to specify the hazard function. This is written as (36) ),()()( 0 Xththth = where ),( Xth is a function that may change with time . Equation can be rewritten as (37) )( )( ),( 0 th th Xth = If both )(th and )(0 th change with time, their ratio can remain constant thus a PH; this implies that, at any given time t the hazard rate applying to a subject will be times that of an average subject for a constant hXth =),( not changing with time. The logarithm of hazard and of the HR are often used and written as [ ] (38) )( )( loglog),(log 0 β=      == th th hXth Such that β and h do not depend on time. The strength of the PH model developed by Cox (1972) is not only that it allows survival data arising from a non-constant hazard rate to be modelled, but it does so without making any assumption about the underlying distribution of the hazards in different University of Ghana http://ugspace.ug.edu.gh 35 groups, except that the hazards in the groups remain proportional over time (Machin, Cheung & Parmar). The PH and HR can be extended to compare two groups; one group is treated as a control or reference group )(thC . And the other group used in comparison is the treatment group )(thN . The HR for comparing two groups is derived as )exp( )( )exp()( )( )( )exp()()( )()( 0 0 0 0 β β β = == = = HR th th th th HR thth thth C N N C (39) In PH modelling for a given coefficient say iβ , the coefficient and hazard ratios are interpreted as follows; a. For ,0iβ there is a higher hazard and therefore a poorer survival associated with the coefficient because 1)exp( >iβ . Parametric PH Models The parametric distributions which can be modelled using the PH procedure include the Exponential, Weibull and Gompertz distributions. While the Exponential and Weibull employ both AFT and PH modelling University of Ghana http://ugspace.ug.edu.gh 36 techniques, the Gompertz uses solely PH modelling approach to analyse survival data. In estimating parametric survival models, time is assumed to follow some distribution whose probability density function )(tf . The survival function can be modelled for both censored and non-censored data which makes it a better option for modelling as compared to the density function. Also, the hazard function is also more interesting than the survival function because it examines an individual’s risk of experiencing an event of interest as compared to the survival which looks at the experience of the event by every subject in the long run. Survival estimates obtained from parametric survival models typically yield plots more consistent with a theoretical survival curve. If the investigator is comfortable with the underlying distributional assumption, then parameters can be estimated that completely specify the survival and hazard functions. This simplicity and completeness are the main appeals of using a parametric approach (Kleinbaum & Klein, 2005). The exponential distribution has a constant hazard of λ=)(th hence this is substituted into (35) as: ∑ = Χ= n i iiXth 1 )(exp),( βλ (40) and the log of the hazard is also deduced in order to linearize the equation, thus equation (17) becomes University of Ghana http://ugspace.ug.edu.gh 37 ∑ = Χ+= n i iiXth 1 )()log(),(log βλ (41) However, the hazard of the exponential function is re-parametrized as )...exp( , )( 110 kk XX where th βββλ λ +++= = (42) Written as: [ ] (43) exp)(),( 10 ∑= Χ+=Χ k i ii tth ββλ The Weibull and exponential distribution are reparametrized and specified as the baseline hazard for the equation to become a parametric PH model. For the Weibull with a hazard function 1)( −= αλαtth is reparametrized as exp 1 0       += ∑ = k i ii Xββλ (44) Hence the parametric form of the equation is substituted as (45) expexp)( 1 0 1 0 1       +×      += ∑∑ == − k i ii k i ii XXtth ββββα α Thus equation 23 is the Weibull proportional hazard model for the covariates investigated under the time to first promotion of lecturers. The exponential proportional hazard model is similar to the Weibull and shown below in equation 46. University of Ghana http://ugspace.ug.edu.gh 38 (46) expexp)( 1 0 1 0       +×      += ∑∑ == k i ii k i ii XXth ββββ For the hazard of the exponential λ reparametrized as .exp 1 0       += ∑ = k i ii Xββλ Again, a Gompertz PH model is obtained when the baseline hazard of the PH model is specified as a Gompertz distribution )][exp(0 th γ= is given by 3.5.5 AFT Modelling Hazard functions are described as functions of explanatory variables but it is possible to let the explanatory variables act via a scale factor directly on time. (Hougaard, 2000). The AFT model can be expressed on a log scale as an additive term as: ∑ = ++= k i iii XT 1 0)log( σεαα , (48) But multiplicative on T and given as: ( )i k i ii XT σεαα exp.exp).exp( 1 0       = ∑ = (49) where iα are the unknown parameters, iX are covariates of interest for i=1,2,…,k and iε is the random error of the equation which follows a particular distribution. The σ scales the error term and is reparametrized in R as: p 1 =σ therefore equations (28) and (29) become, (47) )Covariateexp()][exp(),( 10 ββγ +×= tXth University of Ghana http://ugspace.ug.edu.gh 39 ∑ = ++= k i iii p XT 1 0 1 )log( εαα and             = ∑ = i k i ii p XT εαα 1exp.exp).exp( 1 0 respectively. Therefore, for the weibull model, t is obtained by [ ]         −=⇒ =−⇔−= α α αα λ λλ 1 1 1 ))(log( ))(log()exp()( tSt ttSttS (50) When it is reparametrized, [ ] )Covariateexp())(ln( )Covariateexp( 1 10 1 101 αα αα λ α α +−= += tSt (51) Hence the acceleration factor for a fixed value S(t)=q is calculated for instance for Gender with levels Female=0 and Male=1, the acceleration factorγ is [ ] [ ] )exp( )-(1)exp( )(0)exp()ln( )(1)exp()ln( 1 010 10 1 10 1 αγ αααγ αα αα γ α α = += +− +− = q q (52) The exponential model like the weibull is reparametrized in similar fashion. University of Ghana http://ugspace.ug.edu.gh 40 [ ] 1))(ln( )exp()( −×−=⇒ −= λ λ tSt ttS (53) let )Covariateexp( 10 1 ααλ +=− , then equation becomes; [ ] )Covariateexp())(ln( 10 αα +×−= tSt (54) Thus, the acceleration factor is obtained by; [ ] [ ] )exp( )-(1)exp( )(0)exp()ln( )(1)exp()ln( 1 010 10 10 αγ αααγ αα αα γ = += +− +− = q q (55) For a covariate with two levels 0 1nd 1 and S(t)=q. The survival function of the log-logistic model is given by, pt tS λ+ = 1 1 )( and can be rewritten as ppt tS )(1 1 )( 1 λ+ = . The equation is written in terms of t as 11 1 1 )( 1 −      ×      −= p p tS t λ , where the shape parameter is reparametrized as )Covariateexp( 10 1 ααλ +=p . The acceleration factor is deduced for qtS =)( as; [ ] )Covariateexp(1 10 1 1 αα +×−= − pqt (56) University of Ghana http://ugspace.ug.edu.gh 41 Therefore for a dummy variable with two levels, the acceleration factor for the log-logistic model is; [ ] [ ] )exp( )-(1)exp( )(0)exp(1 )(1)exp(1 1 010 10 1 1 10 1 1 αγ αααγ αα αα γ = += +− +− = − − p p q q (57) 3.5.6 Diagnostic Tests In order to use PH and AFT models, it is prudent to check for data suitability. There various methods of achieving this aim, some of which are discussed below and employed in this work. Proportional hazard models are tested for suitability by using analytical or graphical tests. The test is based on chi-square tests or large sample Z tests are used to assess whether the hazard ratio of two observations is constant. Each covariate is tested individually, for a p-value, say greater than 0.10, the PH assumption is reasonable, whereas a small p-value, say less than 0.05, suggests that the variable being tested does not satisfy this assumption. The analytical test is a more powerful test because decisions are based on p-value unlike the graphical approach which is concluded based on a researcher’s observation. Plots available in using the graphical approach include: (a) Plots of survival estimates for two groups, using KM estimates (b) Plots of log negative log estimated survival function versus log of time for two or more subgroups, using KM estimates University of Ghana http://ugspace.ug.edu.gh 42 (c) Plots of observed survival probabilities versus expected under PH model (d) Plots of weighted Schoenfeld residuals from Cox models verses time Schoenfeld residuals are defined for every subject who has an event, for each independent variable in the model. Although it may be usual to study residuals the plots may be difficult to interpret and any patterns observed may be a consequence of the censoring as much as a ‘true’ reflection of the data. Thus care needs to be taken not to over-interpret any residual plots displayed (Machin, Cheung & Parmar, 2006). To interpret plots of log negative log estimated survival function versus log of time for two or more subgroups, using KM estimates for categories, parallel lines indicate that PH assumptions hold. In a situation where the PH assumptions do not hold, the covariates interaction with time is suspected. However, there are some interventions that can make the data suitable or the methodology in analysis is switched. Some of the interventions are: 1. to use a parametric model in which the PH assumption is not important, 2. Examine the structural form of the predictors in the model. Perhaps they have a more complicated relationship with the log hazard. 3. Repeat the analysis by stratifying on the exposure variable; if there are no other covariates of interest then do not fit any model, just obtain Kaplan–Meier curves for each exposure group separately. If there are University of Ghana http://ugspace.ug.edu.gh 43 additional control variables fit a Cox model in which the levels of the exposure variable are treated as strata. 4. Start the analysis at a time when the PH assumption appears to hold, and use a Cox PH model for only those individuals that survived that long. 5. Fit one Cox model to the early data and a different Cox model for the later data to get two different hazard ratio estimates, one for each of these two time periods. 6. Fit a modified Cox model that includes a time-dependent variable that incorporates the interaction of exposure with time. Such a model is called an extended Cox model. (Weiss, 2010). There are available tests in order to ascertain whether the models have been correctly chosen, these tests are referred to as goodness-of-fit tests. There are tests of subsets of parameters in a distribution, all parameters in a distribution, appropriateness of distribution family, among several other tests. The generalized gamma model is commonly used as a tool for picking an appropriate parametric model for survival data but, rarely, as the final parametric model. Wald or likelihood ratio tests of the hypotheses that 1=θ or 0=θ provide a means of checking the assumption of a Weibull or log normal regression model, respectively. With the exception of the Weibull and log normal distribution, it is difficult to use a formal statistical test to discriminate between parametric models because the models are not nested in a larger model which includes all the regression models discussed in this chapter. University of Ghana http://ugspace.ug.edu.gh 44 One way of selecting an appropriate parametric model is to base the decision on minimum Akaike information criterion (AIC). AIC is useful in selecting the best model in the set; however, if all the models are very poor, AIC will still select the one estimated to be best, but even that relatively best model might be poor in an absolute sense. Thus, every effort must be made to ensure that the set of models is well founded (Burnham & Anderson). For the parametric models discussed, the AIC is given by AIC=-2*log (Likelihood) + 2(p + k ), where p is the number of regression parameters and k is the number of parameters (Klein & Moeschberger, 2003). The Wald Test, Log-Likelihood and Score tests however, are tests of parameters, while the AIC and Bayesian Information Criterion (BIC) are also procedures for selecting best-fitting model. 3.5.7 Modelling and Model Selection As discussed above, there are two main modelling techniques in survival analysis; PH and AFT modelling and both approaches are employed in this work. The AIC values of the models were used to choose the best fitting model. The best model ideally, should have the smallest AIC as possible. The AIC values of parametric models for the PH model are compared separately and turn compared to the semi-parametric model. Then, the best-fitting AFT model is also selected based on the AIC values. 3.6 Statistical Software There are various statistical software available to analyse survival data, these include SPSS, SAS, STATA, and R; to mention a few. In this research however, the data was obtained in the form of a MS excel spreadsheet. And University of Ghana http://ugspace.ug.edu.gh 45 analyses were done using R and STATA. The AFT models were built in R while the PH models were built using STATA, because, the package survreg in R does not conveniently execute the Gompertz modelling. The package flexsurvreg could execute the Gompertz modelling analysis but also had a limitation of sample size of less than 170, hence the switch to STATA. 3.7 Dissemination of Results The results of this study will be published as a dissertation for the award of the M.Phil Statistics. A copy of this dissertation will be placed in University of Ghana library. University of Ghana http://ugspace.ug.edu.gh 46 CHAPTER FOUR RESULTS AND DISCUSSION 4.1 Introduction In this chapter, the results of the analyses are presented and discussed. The preliminary analyses include the general description of the data. A further analyses is performed with results of the various models used presented. 4.2 Choice of Explanatory Variables The event of interest in this research is the time to first promotion. This may occur due to so many factors; however the covariates explored in this work include sex, marital status, the number of children, the qualification as at time of employment, nationality, religion, date of employment, the college of affiliation of the lecturer. Other important covariates are the number of publications, the number of teaching hours and whether a lecturer has other administrative roles to play or not, but these data were not made available and hence unaccounted for in this work. Men and women have different attitudes in relation to work that is why the variable gender is included. According to Portela & Machado (2013), Judge & Bretz (1994), Ng et al. (2005) and Pfeffer & Ross (1982), younger people and married people tend to excel at work as compared to older people and unmarried people. There are also opinions that some disciplines of study are more difficult than others. People generally deem the sciences as a more challenging disciple as compared to the arts and therefore will expect those in the field of science to have a flexible rule with regards to promotion. The University of Ghana http://ugspace.ug.edu.gh 47 reason for choosing nationality as a covariate is to ascertain or dismiss the role of nepotism in promotion, and also to see if one’s background plays much role in his/her performance at work. Descriptive statistics is conducted. This includes the construction of frequency tables which indicate; the number of males and females lecturers, the number of lecturers per college, the mean age of lecturers, among others. 4.3 Variable Coding In this work, the response variable is time to first promotion in days with predictor variables as status, gender, age at first promotion, marital status, region of origin and college to which a lecturer is affiliated. The censoring status is coded as    = censored isLecturer 0 promotionfirst hadLecturer 1 status The gender is categorised as males (M) and females (F), marital status as married and single. There are four colleges in the University and these are coded as;        = Humanities of College CH SciencesHealth of College CHS Education of College CE Sciences Applied And Basic of College CBAS College University of Ghana http://ugspace.ug.edu.gh 48 And the Region of origin was divided into 11 regions and is coded as shown below:                  = Regionn WesterWES Region Volta VOL Region Upper WestUW RegionEast pper UUE NationalsOther ON RegionNorthern NOR Region AccraGreater GA RegionEastern EAS Region Central CEN Region Ahafo Brong BA Region Ashanti ASH Origin ofRegion 4.4 Preliminary Analysis The data used was a secondary data with 218 respondents; the respondents are currently employed full time lecturers in the University of Ghana. The data generally spans between the years 1979 to 2015; the longest serving lecturer for the sample used in this work, was employed in 1979 and the least serving lecturer was appointed in 2015. Out of the sample 60 were females and 158 males. There are 181 married lecturers and 37 are single. Table 4.1 shows the distribution of lecturers by ranks whiles table 4.2 gives the distribution of lecturers per the four main colleges in the University of Ghana. University of Ghana http://ugspace.ug.edu.gh 49 Table 4.1 Frequency Distribution of Lecturers by Ranks Qualification Number of Lecturers Professor 15 Associate 35 Senior Lecturer 75 Lecturer 31 Research Fellow 10 Others 39 Table 4.2: Frequency Distribution of Lecturers by College of Work College Number of Lecturers College of Basic and Applied Sciences 65 College of Education 9 College of Health Sciences 70 College of Humanities 74 The College of Education had the smallest sample of lecturers which may be as a result of it being composed of fewer Departments as compared to the others. The youngest lecturer at date of appointment was 20 years, which may be that the person at the time of employment was not employed to the rank of a lecturer. The oldest lecturer was employed at the age of 57 years who may have worked elsewhere before being employed in this institution. The mean age of employment of lecturers was 35.46 years. The mean age of the University of Ghana lecturers is 49.75 years while the mean age as at the time of first promotion is 43.64 years. University of Ghana http://ugspace.ug.edu.gh 50 Tables and figures not displayed in this chapter are displayed in Appendix C, and the names preceded with C. Table C1 is a table showing the mean age of lecturers both in present and at the time of first promotion with respect to their various Colleges. Table C2 also shows the frequency distribution of lecturers by their region of origin. As mentioned above, the KM survival function and curve was used as a descriptive tool to explore the general form of the data. The median survival time is 2952 days which has a 95% lower confidence limit as 2459 days and an upper confidence limit as 3318 day. This means that, on average a lecturer earns a promotion after 2952 which is approximately 8.09 years. The time to first promotion of a University of Ghana lecturer on average, can be as small as 2459 days (6.74years) or as much as 3318 days (9.09 years). However, according to the estimate, the shortest time a lecturer had his or her first promotion was after 72 days which is probable because, a lecturer may be due promotion in another University and transfer to the University of Ghana. And by virtue of his or her teaching experience and other qualifications, is given a relatively shorter probation period. And the longest time was 10866 days which is approximately 29.77years. The time to first promotion was however obtained by taking the difference between the date a lecturer earns his or her first promotion and the date of first appointment. University of Ghana http://ugspace.ug.edu.gh 51 0 2000 4000 6000 8000 10000 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 t(Days) S (t) Figure 4.1: A Kaplan-Meier Survival Plot of time to first promotion in Days Figure 4.1 shows the survival function with 95% confidence band. The “+” on the curve indicates censoring; lecturers did not earn a first promotion. And from the curve above, more lecturers experienced censoring at earlier times; between 0 to 2000 days. The censored lecturers may be lecturers still on probation or those who have not tendered in any application for promotion as the time of this study. Also, the KM survival curve was fitted with univariate covariates; the curve of each categorical covariate was fitted in order to test for differences in survival across groups. Gender was the first covariate explored; the median survival time across gender was 3223 days for females with Confidence Interval (CI) as [2459,4139] . This means that, for female lecturers, the average time to earn a first promotion was 3223days (8.83 years) and this average time can be as low as 2459 days (6.774 years) and as high as 4139 University of Ghana http://ugspace.ug.edu.gh 52 days (11.34 years). 2898 days (7.94 years) is the average time to promotion for males with CI [2349, 3318]. The CI signifies that, at a confidence level of 95%, male lecturers earned a first promotion with the shortest time being 2349 days (6.44 years) and the longest time on average as 3318 days (9.09 years). It is clear that on average, male lecturers earned first promotions earlier than the female lecturers. Out of the 60 female lecturers, 53 had their first promotions and 139 of the 158 male lecturers earned their first promotions. The median time to promotion is also calculated for the covariates marital status, college and region of origin, these are displayed in tables C4 to C6. Figure 4.2 is the survival plot for gender; it gives an insight into the time to first promotion of male and female lecturers. Similar plots for the rest of the categorical covariates; marital status, college and region of origin are exhibited in Figure C1. In Figure C2 are plots of cumulative hazards across categories for the covariates. These plots also give an idea as to whether the PH assumption holds or not. University of Ghana http://ugspace.ug.edu.gh 53 0 2000 4000 6000 8000 10000 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 t (Days) S (t) Figure 4.2: A plot of survival of Male and Female Lecturers The next exploratory test is log-rank test. This test is used to check if there are any differences in survival across groups say marital status. The hypothesis sets out to test if the survival of married lecturers significantly differs from their single counterparts. Hence for the log rank test, the general null hypothesis states that, the chance of survival across a categorical group is the same. The first test tested survivorship of gender, the null hypothesis and alternate hypothesis were set as; :0H Time to first promotion for male and female lecturers is the same. :1H Time to first promotion of male and female lecturers differed. University of Ghana http://ugspace.ug.edu.gh 54 It was observed that, at 5% significance level the p-value was 0.841. Thus, the null hypothesis is not rejected and hence concluded that, there is no statistically significant difference in the survival of males and females. Which implies that, the time to first promotion among male and female lecturers statistically, were not different. In like manner, the log rank test is conducted to test differences in marital status; if there is any difference in the time to first promotion of married lecturers as compared to their single counterparts, test differences in colleges; if there are differences in time to first promotion in the four colleges. And the last test was to find out if there exist differences in the time to first promotion in the region of origin of lecturers. Results of the stated log rank tests are displayed in Table C7 through to Table C10 all tested at 5% significance. From the results, p-value was 0.221 for the test of survivorship for marital status. The time to first promotion statistically, was not different for married and single lecturers. This value was not significant so the null hypothesis was not rejected. Log rank tests for Colleges and Region of origin were however significant with p-values of 0.0183 and 0.00128 respectively. This means that, there were significant differences in survivorship across those categories. And this signifies that, time to first promotion differs for the four Colleges and Regions of origin are different. That is, for instance, the time a lecturer in the College of Basic and Applied Sciences earned a first promotion is not the same as the time to first promotion of a lecturer in the other Colleges. In order to be able to use Proportional Hazard (PH) Regression method, the underlying assumption for data suitability is tested. The PH University of Ghana http://ugspace.ug.edu.gh 55 assumption like the log rank test is conducted for one covariate at a time. The graphical approaches used were the plot of survival using KM estimates, the plot of ( )[ ])(ˆloglog tS− versus time and ( )[ ])(ˆloglog tS− versus )log(time . Figure C2 and Figure C3 are plots of ( )[ ])(ˆloglog tS− versus time and ( )[ ])(ˆloglog tS− versus )log(time respectively. However, the outputs of these plots were not very clear for interpretation; especially for the covariates with more than two categories. The graphs showed that the Weibull and Exponential PH assumptions were not likely to hold but this is not a formal test. Since the parametric tests are more powerful as compared to the semi- parametric and nonparametric models, the regression is still carried out after testing the PH assumption. Table 4.3 shows the results for the univariate test of College. Table 4.3: A table showing the Cox PH test for College as a Covariate College Rho chi-square p-value Education -0.0261 0.13 0.7100 Health Sci. - 0.1042 2.09 0.1478 Humanities -0.1723 5.67 0.0173 GLOBAL NA 5.85 0.1191 From table 4.3, the Global result indicates that the null hypothesis is not to be rejected because the test was not statistically significant. But, the test of individual categories shows that the College of Humanities (CH) was statistically significant. Hence a further test was conducted to validate the University of Ghana http://ugspace.ug.edu.gh 56 acceptance of proportionality. The test was to verify if College had an interaction with time. The result is as shown in Table 4.4. Table 4.4: The Test of College and Interaction The results show that, there is no interaction with time and hence the test of constant hazard is true, thus the global test result validated. This means that the baseline hazard is constant and hence the use of Cox PH modelling is appropriate. The Cox PH assumption goodness-of-fit test shown in Table C11, indicates a p-value of 0.0675 which is greater than 0.05, so the conclusion is made that the covariate Gender, satisfies the Cox PH assumption. In Table C12 of Appendix C, the p-value of Age of a lecture at first promotion and marital status were 0.345 and 0.695 which means that the PH assumption is not violated by these covariates. Therefore, these covariates can be used in the Cox PH regression modelling. The global test for the region of origin of lecturers is not statistically significant; has p-value of 0.259, so the PH College βˆ ) ˆexp(β Standard error ( βˆ ) z-score p-value CE 1.25e-01 1.133 0.938120 0.134 0.89 CHS 2.63e-01 1.301 0.390181 0.673 0.50 CH -1.06e-01 0.899 0.367433 -0.289 0.77 Time -2.39e-02 0.976 0.001952 -12.239 0.00 CE*time -9.53e-05 1.000 0.000343 -0.278 0.78 CHS*time -1.19e-04 1.000 0.000110 -1.074 0.28 CH*time 1.29e-05 1.000 0.000119 0.108 0.91 University of Ghana http://ugspace.ug.edu.gh 57 assumption is not violated. Marital status of an individual is susceptible to change as time goes on, even though the PHA is not violated because the p- value as shown in Table 14 is 0.695, the test with time interaction was still carried out. This further test, in Table C15, confirms that the Cox PHA was not violated and marital status of lecturers does not necessarily change over time. Cox PH regression is therefore conducted since the covariates tested satisfy the PH assumption. 4.5 Further Analysis One part of this research was to use PH models of parametric and semi-parametric nature to verify which of these tests was superior in terms of fit, and the other part is to model using parametric distributions; Exponential, Weibull, Log-logistic and Lognormal to choose the best fitting model for the data based on Akaike Information Criterion (AIC). Results for three full PH models are displayed in Tables C16 through to C18. The tables display the Hazard ratios, Standard error and 95% confidence intervals. However, the best-fitting PH model is selected based on the AIC values of the models. It can be observed from Table 4.5 that, the hazard ratios for the Gompertz model are generally smaller. University of Ghana http://ugspace.ug.edu.gh 58 Table 4.5: Comparison of Hazard Ratios for Cox, Exponential, Weibull and Gompertz Full Models Name Covariate Hazard Ratios Cox Exponential Weibull Gompertz Age AgeFP 0.9282514 0.9533675 0.9343619 0.9300939 Gender Male 1.2103270 1.1977120 1.2686390 1.1927680 College CE 1.1624450 1.0678080 1.0834140 1.0943490 CHS 0.5756674 0.7014479 0.6150247 0.5680341 CH 0.8451392 1.0270810 0.9121821 0.8058880 Regions BA 0.5441232 0.8483473 0.7617721 0.6080597 CEN 1.6224440 1.3490160 1.5632730 1.5101810 EAS 1.0427200 1.1551080 1.0387160 0.9802927 GA 0.6942323 0.7453483 0.6865262 0.6589495 NOR 4.3526660 2.2735330 4.0015840 4.0471570 ON 1.7228250 1.3549550 1.6160680 1.7264010 UE 0.8827092 0.8406996 0.8804941 0.9411511 UW 0.6570495 0.7422572 0.6596121 0.6602135 VOL 1.1616830 1.1108670 1.1504800 1.1498150 WES 0.5669474 0.7131871 0.6500310 0.5586578 Marital SINGLE 0.8616368 0.8991458 0.8916579 0.9049309 PH models were compared using AIC values. Table 4.6 is comparing log-likelihood and AIC values the full regression models on degrees of freedom equal to 18. Table 4.6: PH models, log-likelihood and AIC values. Model Number of Parameters Log Likelihood AIC Cox N/A -796.19523 1624.39046 Exponential 1 -266.76882 567.53764 Weibull 2 -246.79702 529.59404 Gompertz 2 -241.10038 518.20076 University of Ghana http://ugspace.ug.edu.gh 59 In the Table 4.6, it is observed that, the Gompertz model had the lowest AIC value of 518.20076. Comparing the parametric PH models, the Gompertz model has the smallest AIC value and when it is compared further to the semi-parametric model (Cox PH model), it still has a lesser value. Thus, using a PH modelling technique on this data implies that, the Gompertz model is the best-fitting model. Modelling with the Gompertz distribution, Table 4.7 is composed of the following: hazard ratios, the standard errors, p-values and confidence intervals. Table 4.7: The Gompertz PH Full Model Table of Hazard Ratios, Standard Errors and 95% Confidence Interval. Covariate Category Hazard Ratio Std. Err. Z P>|z| Age AgeFP 0.9300939 0.0111667 -6.04 0.000 Gender G2 1.1927680 0.2149385 0.98 0.328 College of Affiliation CE 1.0943490 0.4175831 0.24 0.813 CHS 0.5680341 0.1139611 -2.82 0.005 CH 0.8058880 0.1597714 -1.09 0.276 Region of Origin BA 0.6080597 0.2966976 -1.02 0.308 CEN 1.5101810 0.4604371 1.35 0.176 EAS 0.9802927 0.2755335 -0.07 0.944 GA 0.6589495 0.1881989 -1.46 0.144 NOR 4.0471570 1.8546270 3.05 0.002 ON 1.7264010 0.7638916 1.23 0.217 UE 0.9411511 0.5911121 -0.10 0.923 UW 0.6602135 0.2972374 -0.92 0.356 VOL 1.1498150 0.3185507 0.50 0.614 WES 0.5586578 0.2258614 -1.44 0.150 Marital SINGLE 0.9049309 0.1936178 -0.47 0.641 Intercept 0.0044480 0.0024228 -9.94 0.000 Gamma 0.0002554 0.0000342 7.47 0.000 University of Ghana http://ugspace.ug.edu.gh 60 From the results, the chance of a lecturer earning a first promotion decreases by 6.99% for a unit increase in age. (This was calculated by 100[1- (HR)]%). A lecturer affiliated to the College of Health Sciences had a hazard rate 0.43197(43.197%) lesser than a lecturer from the College of Basic and Applied Sciences. Meaning that, a lecturer from the College of Health Sciences, is 0.43197 times less likely to earn a first promotion compared to a colleague from the College of Basic and Applied Sciences. The HR was 4.0471570 for Northern region. This means a lecturer from this region is 3.0471570 more likely to earn a first promotion compared to a lecturer from the Ashanti region. The rest of the HR may be increasing or decreasing but are not significant; hence they are statistically not different from their reference categories. The next approach was to model the data using the AFT approach with the Exponential, Weibull, Log-logistic and Lognormal distributions. Table 4.8 is a comparison of standard errors of parameter estimates of the full models. It was realised that the weibull model has generally smaller standard errors. This signifies that its parameter estimation is more accurate as compared to the other models. University of Ghana http://ugspace.ug.edu.gh 61 Table 4.8: Comparison of standard errors of Exponential, Weibull, Log-logistic and Lognormal AFT models Covariate Category Parametric Models Log-logistic Lognormal Weibull Exponential Intercept 0.4433 0.5010 0.37048 0.5539 Age AgeFP 0.0093 0.0105 0.00798 0.0118 College of Affiliation CE 0.3679 0.4362 0.25550 0.3821 CHS 0.2392 0.2766 0.13137 0.1956 CH 0.2207 0.2465 0.12801 0.1876 Region of Origin BA 0.2148 0.2570 0.32321 0.4826 CEN 0.3341 0.4023 0.20225 0.3023 EAS 0.3636 0.3949 0.18534 0.2727 GA 0.4732 0.5927 0.18854 0.2819 NOR 0.3275 0.4111 0.29903 0.4457 ON 0.2160 0.2488 0.29366 0.4402 UE 0.2894 0.3675 0.41865 0.6293 UW 0.2957 0.3517 0.29843 0.4449 VOL 0.1522 0.1764 0.18422 0.2743 WES 0.1487 0.1712 0.26635 0.3987 Marital Status SINGLE 0.1652 0.1900 0.14086 0.2093 Gender MALE 0.1316 0.1526 0.11922 0.1736 The models were compared using their AIC values; the weibull distribution had the smallest AIC and the lognormal the highest AIC value. Thus, the Weibull is the best-fitting AFT model (displayed in Table 4.9). Table 4.9: Comparison of AIC values Distribution Number of Parameters Value of Scale Parameter Log- likelihood AIC Value Exponential 1 1.000 -1732.7 3499.451 Weibull 2 0.666 -1711.9 3459.829 Log-logistic 2 0.481 -1727.0 3490.042 Lognormal 2 0.942 -1738.5 3513.096 University of Ghana http://ugspace.ug.edu.gh 62 Table 4.10: AFT Model of the Weibull Distribution Covariate iβˆ )ˆexp( iβ Std. Error Z P (Intercept) 5.9712 0.37048 16.117 <0.001 Age AgeFP 0.0497 1.0510 0.00798 6.225 <0.001 College of Affiliation CE -0.0431 0.9578 0.25550 -0.169 0.866 CHS 0.2868 1.3322 0.13137 2.183 0.029 CH 0.0673 1.0696 0.12801 0.526 0.623 Region of Origin BA 0.1588 1.1721 0.32321 0.491 0.133 CEN -0.3041 0.7378 0.20225 -1.504 0.768 EAS -0.0546 0.9469 0.18534 -0.294 0.211 GA 0.2357 1.2658 0.18854 1.250 0.002 NOR -0.9490 0.3871 0.29903 -1.161 0.648 ON -0.3408 0.7112 0.29366 -0.188 0.246 UE 0.0789 1.0821 0.41865 0.202 0.851 UW 0.2547 1.2901 0.29843 0.854 0.393 VOL -0.0841 0.9193 0.18422 -0.456 0.648 WES 0.2670 1.3060 0.26635 1.002 0.316 Marital Status Single 0.0357 1.0363 0.14086 0.253 0.800 Gender Male -0.1707 0.8431 0.11922 -1.432 0.152 Log(scale) -0.4068 0.05759 -7.063 <0.001 In Table 4.10, for a unit increase in age, the acceleration factor is 1.0510. The percentage acceleration factor is obtained by the ]% )ˆexp( -100[1 β . It means that, a unit increase in age of a lecturer will imply the decrease in his or her chance of earning a first promotion by 5.10% provided he or she has not already earned a promotion. A lecturer from the College of Health Sciences is also less likely to earn a first promotion compared to a lecturer from the College of basic and Applied Sciences. Because the chance of not getting a first promotion is accelerated by a factor of 1.3322, which means a first promotion by a lecturer from this college is delayed approximately 1.3 times a lecturer from the College of basic and University of Ghana http://ugspace.ug.edu.gh 63 Applied Sciences. Again, using Ashanti region as a reference category for regions of origin, a lecturer from the Greater Accra region has an acceleration factor of 1.2658. It means that the chance of earning a first promotion is prolonged and hence, compared to a lecturer from the Ashanti region, a lecturer from the Greater Accra region will take a longer time to earn a first promotion. Even though it is observed from the Table 4.10 that other categories have decreasing and increasing acceleration factors, they are not statistically significant and therefore not different from their respective reference categories. The shape parameter for the model was 0.666 means the hazard rate; the chance of earning a first promotion, for the distribution in general, is decreasing. The model had a p-value less than 0.001(5.6e-09) and therefore significant. Therefore the weibull model indicates that, the chance of promotion in general, decreases with time when predicted with the factors age, region of origin, marital status, gender and college of affiliation. University of Ghana http://ugspace.ug.edu.gh 64 CHAPTER FIVE CONCLUSION AND RECOMMENDATIONS 5.1 Introduction This chapter is the final chapter of this work; it will comprise the conclusion and recommendation. The conclusion will provide the summary and the decision of the findings. And the recommendations will advocate for areas of this work that need further investigations. 5.2 Conclusion The median time to first promotion is 8.09 years, which agrees with the University Statutes that, at the end of the ninth year the appointment shall terminate unless the lecturer or research fellow can be promoted to a grade above that of a lecturer. The Gompertz model is the best model if modelling using a PH method, while the Weibull is best-fitting in AFT modelling. However, other models may be more suitable in the presence of other covariates like the number of publications a lecturer had, the notable contributions he or she made to the university or community, the number of children he or she has, among other variables that are likely to affect his or her progress in academia. Using either the PH or AFT best-fitting model; the chance of getting a first promotion decreases as a lecturer grows older. Chances of married lecturers earning a first promotion are the same as unmarried lecturers. Again, University of Ghana http://ugspace.ug.edu.gh 65 male lecturers do not earn first promotion earlier than female lecturers in the University of Ghana. 5.3 Recommendation This study was limited to the study of time until first promotion, a further study of time until subsequent promotions of University of Ghana lecturers which will be categorized as recurrent events survival analysis may also help arrive at an in-depth understanding of issues pertaining to promotion in the University. Also the use of a primary data may also go a long way to boost the information obtained from the secondary data. University of Ghana http://ugspace.ug.edu.gh 66 REFERENCES Allison, P. D. (1995). Survival Analysis Using SAS. North Carolina: SAS Institute Inc. Althauser, R. & Kalleberg, A. L. (1981). Firms, Occupations and the Structure of Labour Markets: A Conceptual Analysis. Sociological Perspectives on Labour Markets, In I. Berg, 119–149. Basic Laws of University of Ghana. 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Kaplan, E. L., & Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53, 457-481. Klein, J. P. & Moeschberger, M.L.,(2003) Survival analysis : techniques for censored and truncated data. (2nd Edition). Springer, New York. Lazear, E. P. & R. S. (1981). Rank-order Tournaments as Optimum Labour Contracts. Journal of Political Economy, 89(5), 841–864. Lee, T. H., Gerhart, B., Weller, I., & Trevor, C. O. (2008). Understanding voluntary turnover: Path-specific job satisfaction effects and the importance of unsolicited job offers. Academy of Management Journal, 51, 651−671 Machin, D., Cheung, Y.B. & Parmar, M. K. B. (2006). Survival Analysis: A Practical Approach (2nd Edition). Wiley and Sons, Ltd. Maindonald, J. & Braun, W. J. (2006). Data Analysis and Graphics Using R – an Example-Based Approach (2nd Edition). Cambridge University Press. Morita, J. G., Lee, T. W., & Mowday, R. T. (1989). 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A handbook of statistical analyses using Stata (3rd Edition). CRC Press LLC. Royston, P. (2004). Flexible parametric alternatives to the Cox model: update. The Stata Journal, 4(1), 98–101. The New International Webster’s Pocket Dictionary of the English Language, (2002). Promotion, Trident Press International, 2002 Ed. pg354 University lecturer, (2014). Statslife, Royal Statistical Society. http://www.statslife.org.uk/careers/types-of-job /66-careers/job- types/1132-university-lecturer. Assessed on 12/10/2014 Wang E., Little B.B. & DelHomme-Little B.A. (2012). Factors contributing to tourists' length of stay in Dalian north-eastern China —A survival model analysis. Tourism Management Perspectives. 4, 67–72. Venables, W. N., & Ripley, B. D. (2002). Modern Applied Statistics with S (4th Ed.). Springer Science & Business Media. Zheng C., Veiga J. F. & Powell G. N.(2011). A survival analysis of the impact of boundary crossings on managerial career advancement up to midcareer. Journal of Vocational Behaviour. 79, 230–240. University of Ghana http://ugspace.ug.edu.gh 69 APPENDICES Appendix A FORM FOR APPLICATION FOR PROMOTION UNIVERSITY OF GHANA CONFIDENTIAL APPLICATION FOR PROMOTION SENIOR STAFF GRADE [STRICTLY FOR SENIOR STAFF] (Two copies to be completed, one of which is to be submitted to the Registrar) SECTION A PERSONAL RECORD: (To be completed by the candidate) 1. Full name of candidate:……………………………………………………… 2. Date of Birth: ………………………………………………………………… 3. Department: …………………………………………………………………... 4. Year and Grade of First Appointment in the University: ………………………. …………………………………………………………………………………… 5. Qualification on your first Appointment in the University: ………………… …………………………………………………………………………… 6. Present Grade/Post: ………………………………………………………... 7. Application for Promotion to: …………………………………………………... 8. Date of last promotion (if applicable): ………………………………………….. 9. Qualification as at last promotion: …………………………………………… University of Ghana http://ugspace.ug.edu.gh 70 10. Additional qualifincations obtained since last Appointment/Promotion (List subjects studied & grades obtained where applicable) ……………………..…………………………………………………………… 11. Duties performed in present grade: ………………………………..…….… …………………………………………….……………………………………….. …………………………………………………………………………………… 12. Do you have any suggestions for improvement of your performance? …………………………………………………………………………………… …………………………………………………………………………………… 13. Justification for promotion: ………………………………………………… ………………………………………………………………………………… 14. Do you supervise other staff? ………………………………………………… 15. If yes, give details: …………………………………………………………… …………………………………………………………………………………… …………..……………………… …………………………. (Signature of Applicant) (Date) ………………………………………………. Contact Number University of Ghana http://ugspace.ug.edu.gh 71 SECTION B CONFIDENTIAL ASSESSMENT (To be completed personally by Head of Department/Organisation) 1. KNOWLEDGE OF WORK i. To what extent is he/she conversant with his/her work? (Where necessary refer for Technical/Professional assessment) ……………………………………………………………………………………… ii. Is he/she well-informed as to the rules and regulations relevant to his/her duties? …………………………………………………………………………………… iii. Any other comments?………………………………………………………….. …………………………………………………………………………………….... 2. SENSE OF RESPONSIBILITY AND DEGREE OF OUTPUT i. To what extent is the Officer conscientious in the performance of his/her duties? (Please tick as appropriate) Very Good Good Satisfactory Indifferent University of Ghana http://ugspace.ug.edu.gh 72 Appendix B Schedule F Procedure for the Appointment and Promotion of Senior Members Vacancies 1. (1) Vacancies shall be announced by internal or external advertisement as appropriate. (2) The vacancies may be filled through: (a) secondment from other universities under a scheme of staff exchange; (b) technical assistance; (c) a recommendation to the Vice-Chancellor by the Dean in consultation with the Director or Head of Department, as appropriate; or (d) application by individuals on their own initiative. (3) Despite subsection (1), a senior member of the University may apply for promotion at any time. Appointment by promotion 2. (1) Promotion shall normally proceed from one rank to the immediate next rank that is, from lecturer to senior lecturer to associate professor to professor. (2) Despite the normal progression as stated in subsection (1) a senior member of the University may apply at any time to be promoted to a rank for which that member feels qualified. University of Ghana http://ugspace.ug.edu.gh 73 (3) In the case of an application for promotion from lecturer to senior lecturer the applicant shall have completed the first two years of probation before becoming eligible to apply. Assistant Lecturer or Assistant Research Fellow 10. An applicant who does not hold a minimum researched master’s degree may be appointed to the grade of assistant lecturer for two years, and exceptionally for a third year, but the applicant must have registered for a researched higher degree or shall be expected to do so on appointment. Lecturer or Research Fellow 11. (1) For the appointment of a lecturer or research fellow, training in research as evidenced by a higher researched degree, preferably a doctorate degree or its equivalent or higher professional qualification is required. (2) The appointment shall normally be for six years, the first two years of which shall be regarded as a period of probation. (3) The appointment shall be reviewed before the end of the sixth year and may normally be renewed for no longer than three more years. (4) At the end of the ninth year the appointment shall terminate unless the lecturer or research fellow can be promoted to a grade above that of a lecturer. (5) In exceptional circumstances, the Appointments Board may, on the recommendation of the Faculty Appointments Review Committee, extend the appointment for a further period not exceeding two more years, at the end of University of Ghana http://ugspace.ug.edu.gh 74 which the appointment shall terminate unless the lecturer or research fellow can be promoted to a grade above that of a lecturer. Senior Lecturer or Senior Research Fellow 12. (1) Appointment or promotion to the grade of senior lecturer or senior research fellow shall be considered on the basis of significant performance in: (a) scholarship as exemplified through research or contribution to knowledge through publications, (b) teaching, and (c) extension work or service. (2) The Head of Department, Dean or Director shall provide an assessment on teaching taking into account student assessment and external examiners’ comments. (3) Extension work or service shall include matters described in subsection (3) of section 7. (4) Applications for promotion based solely on teaching and extension work or service, or any other contributions that do not normally result in publications, shall not be considered during the first regular Six-year contract. (5) Two assessors as described in subsections (6) and (7) of section 5 shall be required. Associate Professor 13. (1) Appointment or promotion to the grade of associate professor shall be on the basis of outstanding scholarship in the candidate’s field of teaching and University of Ghana http://ugspace.ug.edu.gh 75 research and contribution to the intellectual life of the University and the development of the country and on teaching or extension service which shall be treated as described in subsection (1) of section 12. (2) Two external assessors as described in 5.6. and 5.7 shall be required. (3) Appointment is tenured. Professor 14. (1) Appointment or promotion to the grade of professor shall be on the basis of internationally acknowledged scholarship in the candidate’s field of teaching and research and contribution to the intellectual life of the University and the development of the country and on teaching or extension work and service which shall be treated as described in subsection (1) of section 12. (2) Two external assessors as described in subsections (6) and (7) of section 5 shall be required. (3) Appointment is tenured. Promotion based on long service: 32. A lecturer, in exceptional cases, may be promoted on the basis of objective assessment of teaching, extension service and at least one publication, in which case an interview as provided in subsection (2) of section 6 shall be administered. Appointment or Promotion of Lecturer to Senior Lecturer University of Ghana http://ugspace.ug.edu.gh 76 Publication 33. (1) Wherever possible assessment shall be by peer review within Ghana. (2) In all cases, there shall be two assessors one of whom shall, wherever possible, be from a cognate discipline. (3) The assessors shall be appointed or nominated by the Dean on the advice of the Head of Department or Director. Teaching or Departmental Work 34. (1) There shall be student assessment in all cases. (2) The Head of Department shall provide an assessment on teaching or departmental work taking into account student assessment and external examiner’s comments. Appointment or Promotion from Senior Lecturer to Associate Professor 35. (1) Appointment or promotion from senior lecturer to associate professor shall be based on work of outstanding scholarship in the candidate’s field. (2) Mode of assessment shall be by two external assessors who shall be nominated or appointed by the Dean or Director in consultation with the Head of Department. (3) For all professorial appointments, in order to avoid delays, it may be advisable to request three assessments any two of which may be used to arrive at a decision. University of Ghana http://ugspace.ug.edu.gh 77 Appointment or Promotion from Associate Professor to Professor. 36. (1) Appointments or promotion from associate professor to professor grade shall be based on work of internationally acclaimed scholarship. (2) There shall be two external assessors who shall be appointed or nominated by the Dean or Director in consultation with the Head of Department. University of Ghana http://ugspace.ug.edu.gh 78 Appendix C Some Tables and Graphs for Chapter 4 Table C1: Mean Age of Lecturers at Present and at the Time of First Promotion College Mean Age of Lecturers (Presently) Mean Age of Lecturers (At Appointment) Mean Age of Lecturers (At first promotion) CBAS 49.98462 35.53846 43.44615 CE 54.77778 34.66667 44.22222 CHS 48.50000 35.32857 44.94286 CH 49.85135 35.2027 42.36486 Table C2: Frequency distribution of Region of Origin of Lecturers Region of Origin Number of Lecturers Ashanti Region 26 Brong Ahafo Region 7 Central Region 25 Eastern Region 44 Greater Accra Region 37 Northern Region 9 Upper East Region 3 Upper West Region 7 Volta 41 Western Region 10 Other Nationals 9 University of Ghana http://ugspace.ug.edu.gh 79 Table C3: Median Time to first promotion Based on Marital Status College Number of Lecturers number at start events median 0.95LCL 0.95UCL Married 181 181 161 3060 2579 3399 Single 37 37 31 2435 1705 4550 Table C4: Median Time to first promotion Based on College of Affiliation College Number of Lecturers number at start events median 0.95LCL 0.95UCL CBAS 65 65 58 3335 2708 3832 CE 9 9 9 2459 1705 NA CHS 70 70 55 3587 2951 4993 CH 74 74 70 2247 1673 2983 CBAS=COLLEGE OF BASIC AND APPLIED SCIENCE CE=COLLEGE OF EDUCATION CHS=COLLEGE OF HEALTH SCIENCES CH=COLLEGE OF HUMANITIES Table C5: Median Time to first promotion Based on Region of Origin Region Number of lecturers Number at start Events Median 0.95% LCL 0.95% UCL ASHANTI 26 26 22 2839 1321 4656 BRONG AHAFO 7 7 6 6727 2100 NA CENTRAL 25 25 22 1924 1053 3571 EASTERN 44 44 41 2983 1280 3578 GREATER 37 37 33 4077 3653 5368 NORTHERN 9 9 7 1248 962 NA OTHERS 9 9 7 2434 1550 NA UPPER EAST 3 3 3 3223 744 NA UPPER WEST 7 7 7 5110 1953 NA VOLTA 41 41 35 2435 1485 NA WESTERN 10 10 9 3441 3198 3832 University of Ghana http://ugspace.ug.edu.gh 80 Figure C1: Comparison of Survivorship across groups for the categorical covariates: Marital Status, Colleges and Region of Origin. Log Rank Tests Table C6: Log Rank Test for Gender of Lecturers Gender Number of Lecturers Observed Expected ( ) E EO 2^− ( ) V EO 2^− Female 60 53 51.8 0.0290 0.0403 Male 158 139 140.2 0.0107 0.0403 Chi-square value = 0 on 1 degrees of freedom, p-value= 0.841 0 2000 4000 6000 8000 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 Survivorship Based o t S (t ) Married Single 0 2000 4000 6000 8000 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 Survivorship Based o t S (t ) CBAS EDU HS HUM 0 2000 4000 6000 8000 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 Survivorship Based o t S (t ) ASH BA CEN EAS GA NOR OTHERS UE UW VOL WES University of Ghana http://ugspace.ug.edu.gh 81 Table C7: Log Rank Test for Marital Status of Lecturers Marital Status Number of Lecturers Observed Expected ( ) E EO 2^− ( ) V EO 2^− Married 181 161 166.7 0.193 1.5 Single 37 31 25.3 1.272 1.5 Chi-square = 1.5 on 1 degrees of freedom, p= 0.221 Table C8: Log Rank Test for Colleges of Affiliation of Lecturers College Number of Lecturers Observed Expected ( ) E EO 2^− ( ) V EO 2^− CBAS 65 58 52.51 0.5742 0.81129 CE 9 9 8.78 0.0054 0.00572 CHS 70 55 75.18 5.4187 9.20195 CH 74 70 55.52 3.7738 5.36816 Chi-square = 10 on 3 degrees of freedom, p-value= 0.0183 Table C9: Log Rank Test for Region of Origin of Lecturers Region of Origin Number of Lecturers Observed Expected ( ) E EO 2^− ( ) V EO 2^− ASHANTI 26 22 20.22 0.157 0.178 BRONG AHAFO 7 6 11.35 2.518 3.079 CENTRAL 25 22 15.73 2.499 2.751 EASTERN 44 41 34.31 1.306 1.605 GREATER ACCRA 37 33 47.81 4.588 6.186 NORTHERN 9 7 2.31 9.522 9.859 OTHERS 9 7 4.63 1.217 1.258 UPPER EAST 3 3 2.15 0.340 0.346 UPPER WEST 7 7 9.27 0.557 0.590 VOLTA 41 35 28.97 1.256 1.497 WESTERN 10 9 15.27 2.573 2.966 Chi-square = 28.9 on 10 degrees of freedom, p-value = 0.00128 University of Ghana http://ugspace.ug.edu.gh 82 0 2000 6000 10000 -5 -4 -3 -2 -1 0 1 log(-log(S(t))) versus t (Days) lo g  lo g S t  Gender female male 0 2000 6000 10000 -5 -4 -3 -2 -1 0 1 log(-log(S(t))) versus t(Days) lo g  lo g S t  Marital Status 1 2 Figure C2: Plots of log negative log Kaplan-Meier Survival Function against time (Days) for the covariates: Gender, Marital Status, Colleges and Region of Origin. 0 2000 4000 6000 8000 -4 -3 -2 -1 0 1 log(-log(S(t))) versus t t (Days) lo g  lo g S t  Colleg 1 2 3 4 0 2000 4000 6000 8000 -3 -2 -1 0 1 log(-log(S(t))) versus t t (Days) lo g  lo g S t  ASH BA CEN EAS GA NOR OTHERS UE UW VOL WES University of Ghana http://ugspace.ug.edu.gh 83 Figure C3: Plots of log negative log Kaplan-Meier Survival Function against log of time for the covariates: Gender, Marital Status, Colleges and Region of Origin. Table C10: Cox PH Assumption Test for Gender Gender Rho chi-square p-value Male -0.132 3.34 0.0675 4 5 6 7 8 9 -5 -4 -3 -2 -1 0 1 A Graph of log(-log S(t)) again log(t) lo g  lo g S t  female male 4 5 6 7 8 9 -5 -4 -3 -2 -1 0 1 A Graph of log(-log S(t)) again log (t) lo g  lo g S t  Married Single 4 5 6 7 8 9 -4 -3 -2 -1 0 1 A Graph of log(-log S(t)) agains logt lo g  lo g S t  CBAS CE CHS CH 4 5 6 7 8 9 -3 -2 -1 0 1 A Graph of log(-log S(t)) again logt lo g  lo g S t  ASH BA CEN EAS GA NOR OTHERS UE UW VOL WES University of Ghana http://ugspace.ug.edu.gh 84 Table C11: Cox PH Assumption Test for Age of Lecturers Age Rho chi-square p-value At First Promotion 0.0719 0.891 0.345 Table C12: Cox PH Assumption Test for Region of Origin of Lecturers Region Rho Chisq p- value BRONG AHAFO -0.09825 2.06542 0.151 CENTRAL -0.09813 1.83210 0.176 EASTERN -0.08835 1.48218 0.223 GREATER ACCRA 0.05508 0.56962 0.450 NORTHERN 0.04501 0.40071 0.527 OTHER NATIONALS 0.00628 0.00746 0.931 UPPER EAST 0.03214 0.19604 0.658 UPPER WEST 0.05284 0.52073 0.471 VOLTA -0.05004 0.47350 0.491 WESTERN -0.02639 0.13224 0.716 GLOBAL NA 12.39986 0.259 Table C13: Cox PH Assumption Test for Marital Status Marital Status Rho Chi-square p-value Single 0.0282 0.153 0.695 University of Ghana http://ugspace.ug.edu.gh 85 Table C14: Interaction of Marital Status of Lecturers with Time Covariate βˆ )ˆexp(β Standard error ( βˆ ) z-score p-value Marital Status -0.383994 0.681 0.405047 -0.948 0.34 time -0.023896 0.976 0.001944 -12.292 0.00 Marital status*time 0.000123 1.000 0.000148 0.828 0.41 University of Ghana http://ugspace.ug.edu.gh 86 Table C15: The Exponential PH Full Model Table of Hazard Ratios, Standard Errors and 95% Confidence Interval. Covariate Hazard Ratio Standard Error Z P>|z| [95% Confidence Interval] Lower Upper Age AgeFP 0.9533675 0.0108653 -4.19 0.000 .932308 .9749027 Gender Male 1.1977120 0.2081914 1.04 0.299 .8519087 1.683883 College of Affiliation CE 1.0678080 0.4084444 0.17 0.864 .5045478 2.259873 CHS 0.7014479 0.1364792 -1.82 0.068 .4790479 1.027098 CH 1.0270810 0.1925983 0.14 0.887 .7111932 1.483277 Region of Origin BA 0.8483473 0.4094402 -0.34 0.733 .3294246 2.184698 CEN 1.3490160 0.4077698 0.99 0.322 .7459721 2.43956 EAS 1.1551080 0.3146588 0.53 0.597 .6772513 1.970132 GA 0.7453483 0.2099057 -1.04 0.297 .4291836 1.294421 NOR 2.2735330 1.0131400 1.84 0.065 .9492614 5.445238 ON 1.3549550 0.5963919 0.69 0.490 .5718252 3.210602 UE 0.8406996 0.5292819 -0.28 0.783 .2447638 2.887583 UW 0.7422572 0.3299757 -0.67 0.503 .3105608 1.774035 VOL 1.1108670 0.3045821 0.38 0.701 .6490468 1.901288 WES 0.7131871 0.2842012 -0.85 0.396 .3265911 1.557409 Marital Status SINGLE 0.8991458 0.1898228 -0.50 0.615 .5944688 1.359976 Intercept 0.0024709 0.0013402 -11.07 0.000 .0008534 .0071538 University of Ghana http://ugspace.ug.edu.gh 87 Table C16: The Weibull PH Full Model Table of Hazard Ratios, Standard Errors and 95% Confidence Interval. Covariate Covariate Hazard Ratio Standard Error Z P>|z| [95% Confidence Interval] Lower Upper Age AgeFP 0.9343619 0.0110583 -5.74 0.000 0.9129375 0.9562891 Gender Male 1.2686390 0.2278055 1.33 0.185 0.8922577 1.8037880 College of Affiliation CE 1.0834140 0.4167688 0.21 0.835 0.5097455 2.3026900 CHS 0.6150247 0.1211820 -2.47 0.014 0.4179989 0.9049195 CH 0.9121821 0.1754412 -0.48 0.633 0.6257039 1.3298240 Region of Origin BA 0.7617721 0.3703488 -0.56 0.576 0.2937640 1.9753840 CEN 1.5632730 0.4766266 1.47 0.143 0.8600263 2.8415660 EAS 1.0387160 0.2892000 0.14 0.891 0.6018725 1.7926230 GA 0.6865262 0.1943322 -1.33 0.184 0.3941949 1.1956480 NOR 4.0015840 1.8407330 3.01 0.003 1.6243450 9.8579270 ON 1.6160680 0.7146779 1.09 0.278 0.6792514 3.8449320 UE 0.8804941 0.5537426 -0.20 0.840 0.2566882 3.0202790 UW 0.6596121 0.2955444 -0.93 0.353 0.2740949 1.5873630 VOL 1.1504800 0.3180377 0.51 0.612 0.6692277 1.9778100 WES 0.6500310 0.2598678 -1.08 0.281 0.2969223 1.4230670 Marital Status SINGLE 0.8916579 0.1902021 -0.54 0.591 0.5869837 1.3544730 Intercept 0.0001105 0.0000859 -11.7 0.000 0.0000241 0.0005071 /ln_p 0.3978201 0.0576391 6.90 0.000 0.2848495 0.5107907 P 1.4885760 0.0858002 1.3295621 0.6666090 1/p 0.6717829 0.0387210 0.6000209 0.7521274 University of Ghana http://ugspace.ug.edu.gh 88 Table C17: Cox PH Full Model Table of Hazard Ratios, Standard Errors and 95% Confidence Interval. Covariate Hazard Ratio Std. Err. Z P>|z| [95% Confidence Interval] Lower Upper Age AgeFP 0.9282514 0.0113343 -6.10 0.000 0.9063003 0.9507342 Gender Male 1.2103270 0.2211094 1.04 0.296 0.8460581 1.7314300 College of Affiliatio n CE 1.1624450 0.4474603 0.39 0.696 0.5466626 2.4718700 CHS 0.5756674 0.1157999 -2.75 0.006 0.3881021 0.8538809 CH 0.8451392 0.1681261 -0.85 0.398 0.5722637 1.2481310 BA 0.5441232 0.2892902 -1.14 0.252 0.1919309 1.5425870 Region of Origin CEN 1.6224440 0.4996184 1.57 0.116 0.8872563 2.9668130 EAS 1.0427200 0.2943607 0.15 0.882 0.5996150 1.8132700 GA 0.6942323 0.1985741 -1.28 0.202 0.3963074 1.2161230 NOR 4.3526660 2.0221120 3.17 0.002 1.7511120 10.8192400 ON 1.7228250 0.7655158 1.22 0.221 0.7211410 4.1158730 UE 0.8827092 0.5559866 -0.20 0.843 0.2568482 3.0336030 UW 0.6570495 0.2970691 -0.93 0.353 0.2708618 1.5938530 VOL 1.1616830 0.3229749 0.54 0.590 0.6736489 2.0032790 WES 0.5669474 0.2341850 -1.37 0.169 0.2523149 1.2739210 Marital Status SINGLE 0.8616368 0.1866897 -0.69 0.492 0.5635020 1.3175070 University of Ghana http://ugspace.ug.edu.gh 89 LOG LOGISTIC AFT MODELLING Table C18: AFT model of the Log-logistic Distribution Scale= 0.481 Loglik(model)= -1727 Loglik(intercept only)= -1765.6 Chisq= 77.24 on 16 degrees of freedom, p= 5.2e-10 Number of Newton-Raphson Iterations: 5 for n= 218 Covariate Coefficient Exp(Coef.) Standard Error Z P (Intercept) 5.2559 0.4433 11.855 <0.001 Age AgeFP 0.0630 1.0650 0.0093 6.773 <0.001 Region of Origin BA 0.2791 1.3219 0.3679 0.759 0.448 CEN -0.2811 0.7550 0.2392 -1.176 0.240 EAS -0.3640 0.6949 0.2207 -1.650 0.099 GA 0.3220 1.3799 0.2148 1.499 0.134 NOR -0.7256 0.4840 0.3341 -2.172 0.029 ON -0.4043 0.6674 0.3636 -1.112 0.266 UE 0.4521 1.5716 0.4732 0.955 0.339 UW 0.3911 1.4786 0.3275 1.178 0.239 VOL -0.0696 0.9327 0.2160 -0.322 0.747 WES 0.4627 1.5883 0.2894 1.599 0.110 College of Affiliation CE -0.1248 0.8826 0.2957 -0.422 0.673 CHS 0.2261 1.2537 0.1522 1.486 0.137 CH -0.1643 0.8484 0.1487 -1.105 0.269 Marital Status Single 0.1317 1.1408 0.1652 0.797 0.426 Gender Male -0.2401 0.7865 0.1316 -1.825 0.068 Log(scale) -0.7324 0.0617 -11.876 <0.001 University of Ghana http://ugspace.ug.edu.gh 90 1. LOGNORMAL AFT MODELLING Table C19: AFT Model of the Lognormal Distribution Covariates Of the equation Value Std. Error Z P (Intercept) 5.5543 0.5010 11.0871 <0.001 Age AgeFP 0.0562 1.0578 0.0105 5.3712 <0.001 Region of Origin BA -0.1069 0.8986 0.4362 -0.2451 0.806 CEN -0.2909 0.7476 0.2766 -1.0519 0.293 EA -0.5386 0.5835 0.2465 -2.1855 0.029 GA 0.3004 1.3504 0.2570 1.1688 0.242 NOR -0.6372 0.5288 0.4023 -1.5839 0.113 ON -0.5593 0.5716 0.3949 -1.4161 0.157 UE 0.4265 1.5316 0.5927 0.7195 0.472 UW 0.4256 1.5305 0.4111 1.0351 0.301 VOL -0.1690 0.8445 0.2488 -0.6793 0.497 WES 0.5031 1.6538 0.3675 1.3690 0.171 College of Affiliation CE 0.0114 1.0114 0.3517 0.0325 0.974 CHS 0.1978 1.2187 0.1764 1.1213 0.262 CH -0.2078 0.8123 0.1712 -1.2136 0.225 Marital Status Single 0.0965 1.1013 0.1900 0.5078 0.612 Gender Male -0.2606 0.7706 0.1526 -1.7076 0.088 Log(scale) - 0.0596 0.0506 -1.1778 0.239 Scale= 0.942 Log likelihood (model) = -1738.5 Log likelihood (intercept only)= -1770. Chi-square = 63.09on 16 degrees of freedom, p= 1.6e-07 Number of Newton-Raphson Iterations: 4 for a sample size n= 218 University of Ghana http://ugspace.ug.edu.gh 91 EXPONENTIAL AFT MODELLING Table C20: AFT Model of the Exponential Distribution Coefficient Exp (Coefficient) Std. Error Z p- value (Intercept) 5.8695 0.5539 10.596 <0.001 Age AgeFP 0.0514 1.0527 0.0118 4.339 <0.001 College of Affiliation CE -0.0607 0.9411 0.3821 -0.159 0.874 CHS 0.3228 1.3810 0.1956 1.650 0.099 CH -0.0250 0.9753 0.1876 -0.133 0.894 Region of Origin BA 0.1446 1.1556 0.4826 0.300 0.764 CEN -0.3027 0.7388 0.3023 -1.002 0.322 EAS -0.1744 0.8399 0.2727 -0.639 0.612 GA 0.2793 1.3222 0.2819 0.991 0.468 NOR -0.8344 0.4341 0.4457 -1.872 0.065 ON -0.3193 0.7267 0.4402 -0.725 0.468 UE 0.1734 1.1893 0.6293 0.276 0.783 UW 0.2786 1.3213 0.4449 0.626 0.531 VOL -0.0986 0.9061 0.2743 -0.359 0.719 WES 0.3220 1.3799 0.3987 0.808 0.419 Marital Status Single 0.0771 1.0801 0.2093 0.369 0.712 Gender Male -0.1904 0.8266 0.1736 -1.096 0.273 Scale fixed at 1 Log likelihood (model) = -1732.7 Log likelihood (intercept only) = -1754.4 Chi-square = 43.34 on 16 degrees of freedom, p= 0.00025 Number of Newton-Raphson Iterations: 4 for a sample size n= 218 University of Ghana http://ugspace.ug.edu.gh 92 Appendix D Codes Used In Data Analysis D1: Analyses in R 1. Packages used >library(MASS) >library(splines) >library(survival) 2. Cox PH Modelling >Mycox<-read.csv("C:/Users/Crystal/Desktop/The +Sis/Data/mycox.csv",head=TRUE) >attach(Mycox) ## Descriptives >summary(Mycox) # Kaplan-Meier Estimation of Survival Function >(KM<-survfit(Surv(time, status)~1, data=Mycox)) > summary(KM) >plot(KM, main="Estimated Survival function plot", +ylab="S(t)",xlab="t(Days)") # survival function (where "+" denotes right censoring) S<-with(Mycox, Surv(time,status==1)) >par(mfrow=c(1,3)) ##Univariate Test of Covariate Survival Time #U1 Gender as a covariate >(U1 <- survfit(S~factor(Gender), data=Mycox)) #Survival function plot >plot(U1, col=4:7,main= "Survivorship of Males and Female Lecturers", +ylab="S(t)",xlab="t") >legend("bottomleft", c("females", "males"),col=4:5, lty=1) #U2 Marital_Status as a covariate >(U2 <- survfit(S~Marital_Status, data=Mycox)) #Survival function plot >plot(U2, col=8:9,main= "Survivorship Based on Marital Status", +ylab="S(t)",xlab="t") >legend("topright", c("Married", "Single"),col=8:9, lty=1) University of Ghana http://ugspace.ug.edu.gh 93 #U3 College as a covariate >(U3 <- survfit(S~factor(College), data=Mycox)) #Survival function plot >plot(U3, col=11:14,main= "Survivorship Based on College", +ylab="S(t)",xlab="t") >legend("topright", c("CBAS", "EDU","HS","HUM"),col=11:14, lty=1) #U4 Ethnicity as a covariate >(U4 <- survfit(S~factor(Ethnicity), data=Mycox)) #Survival funtion plot >plot(U4, col=1:13,main= "Survivorship Based on Region of Origin", +ylab="S(t)",xlab="t") >legend("topright", c( +"ASH","BA","CEN","EAS","GA","NOR","OTHERS","UE","UW","VOL", +"WES"),col=1:13, cex=0.85,lty=1) ## Log- Rank Tests for Categorical Predictors >(UU1<-survdiff(S~factor(Gender), data=Mycox, rho=1)) >(UU2<-survdiff(S~factor(Marital_Status), data=Mycox, rho=0)) >(UU3<-survdiff(S~factor(College), data=Mycox)) >(UU4<-survdiff(S~factor(Ethnicity), data=Mycox, rho=1)) ##Test for proportional hazard assumption for a Cox Regression Model fit >model1 <- coxph(S~ Ethnicity, data= Mycox, method="breslow") >model2 <- coxph(S~ Marital_Status,data=Mycox,method="breslow") >model3 <- coxph(S~ Gender, data= Mycox, method="breslow") >model4 <- coxph(S~College, data= Mycox, method="breslow") >model5 <- coxph(S~AgeFP, data= Mycox, method="breslow") >model6tplot<- function(x){ +cox.zph(x) +} >(tplot1<-tplot(model1)) >(tplot2<-tplot(model2)) >(tplot3<-tplot(model3)) >(tplot4<-tplot(model4)) >(tplot15<-tplot(model5)) >(tplot1<-tplot(model6)) ##Graphical test for proportional hazard assumption >par(mfrow=c(1,2)) University of Ghana http://ugspace.ug.edu.gh 94 # Plot of log negative log Kaplan-Meier survival function versus t #Gender >Gender<-rep(1:2, U1$strata) >plot((U1$time),log(-log(U1$surv)), type='n', main="log(- +log(S(t))) versus t for Gender" ,ylab=expression(log(- +log(S(t)))), xlab="t (Days)") >lines(U1$time[Gender==1], log(-log(U1$surv[Gender==1])), +col="purple",type='s') >lines(U1$time[Gender==2], log(-log(U1$surv[Gender==2])), +col="darkgreen",type='s') >legend('bottomright', legend=c("female","male"), +title="Gender",col=c("purple", "darkgreen"), +lty=1,cex=.9,bty='n') # Marital_Status >Marital_Status<-rep(1:2, U2$strata) >plot((U2$time),log(-log(U2$surv)), type='n', main="log(- +log(S(t))) versus t for Marital Status" ,ylab=expression(log(- +log(S(t)))), xlab="t(Days)") >lines(U2$time[Marital_Status==1], log(- +log(U2$surv[Marital_Status==1])), +col="violetred", type='s') >lines(U2$time[Marital_Status==2], log(- +log(U2$surv[Marital_Status==2])), col="darkblue", type='s') >legend('bottomright', legend=1:2, title="Marital Status", +col=c("violetred","darkblue" ), lty=1 ,cex=.9,bty='n') #College >College<-rep(1:4, U3$strata) >plot((U3$time),log(-log(U3$surv)), type='n',main="log(- +log(S(t))) versus t for College" , ylab=expression(log(- +log(S(t)))), xlab="t (Days)") >lines(U3$time[College==1], log(-log(U3$surv[College==1])), +col=3,type='s') >lines(U3$time[College==2], log(-log(U3$surv[College==2])), +col=4, type='s') >lines(U3$time[College==3], log(-log(U3$surv[College==3])), +col=5, type='s') >lines(U3$time[College==4], log(-log(U3$surv[College==4])), +col=6,type='s') >legend('bottomright', legend=1:4, title='College', col=3:6, +lty=1,cex=.9,bty='n') #Ethnicity >Ethnicity<-rep(1:11, U4$strata) >plot((U4$time),log(-log(U4$surv)), type='n',main="log(- +log(S(t))) versus t for Region of Origin" , +ylab=expression(log(-log(S(t)))), xlab="t (Days)") >lines(U4$time[Ethnicity==1], log(-log(U4$surv[Ethnicity==1])), +col=11, type='s') >lines(U4$time[Ethnicity==2], log(-log(U4$surv[Ethnicity==2])), +col=12, type='s') University of Ghana http://ugspace.ug.edu.gh 95 >lines(U4$time[Ethnicity==3], log(-log(U4$surv[Ethnicity==3])), +col=13,type='s') >lines(U4$time[Ethnicity==4], log(-log(U4$surv[Ethnicity==4])), +col=14, type='s') >lines(U4$time[Ethnicity==5], log(-log(U4$surv[Ethnicity==5])), +col=15,type='s') >lines(U4$time[Ethnicity==6], log(-log(U4$surv[Ethnicity==6])), +col=16,type='s') >lines(U4$time[Ethnicity==7], log(-log(U4$surv[Ethnicity==7])), +col=17,type='s') >lines(U4$time[Ethnicity==8], log(-log(U4$surv[Ethnicity==8])), +col=18,type='s') >lines(U4$time[Ethnicity==9], log(-log(U4$surv[Ethnicity==9])), +col=19,type='s') >lines(U4$time[Ethnicity==10], log(- +log(U4$surv[Ethnicity==10])), col=20, type='s') >lines(U4$time[Ethnicity==11], log(- +log(U4$surv[Ethnicity==11])), col=21, type='s') >legend("bottomright", c( +"ASH","BA","CEN","EAS","GA","NOR","OTHERS","UE","UW","VOL", +"WES"),col=1:13+, cex=0.7,lty=1) ##Testing PH assumption with interaction with time >(PH3 <-coxph(S~Marital_Status*time, +data=Mycox,method="breslow")) >(PH4 <- coxph(S~ College*time, data= Mycox, method="breslow")) #Plots of log negative log of Kaplan-Meier Survival functions against log t # data named Mypar for parametric analyses par(mfrow=c(2,2)) #Gender as a covariate >(fit1<-survfit(Surv(time, status)~factor(Gender), data=Mypar)) >Gender<-rep(1:2 , fit1$strata) >plot(log(fit1$time),log(-log(fit1$surv)), type='n',main="A +Graph of log(-log S(t)) against log (t) for Gender", +ylab=expression(log(-log(S(t)))), xlab="t") >lines(lowess(log(-log(fit1$surv[Gender==1]))~ +log(fit1$time)[Gender==1]), lty=5,col=1) >lines(lowess(log(-log(fit1$surv[Gender==2]))~ +log(fit1$time)[Gender==2]), +lty=5,col=2) >legend('bottomright', paste(c("female","male")), col=3:4, +lty=1, cex=.9, +bty='n') #Marital_Status >fit2<-survfit(Surv(time, status)~factor(Marital_Status), +data=Mypar) >Marital_Status<-rep(1:2 , fit2$strata) University of Ghana http://ugspace.ug.edu.gh 96 >plot(log(fit2$time),log(-log(fit2$surv)), type='n',main="A +Graph of log(-+log S(t)) against log (t) for Marital Status", +ylab=expression(log(-log(S(t)))), xlab="t") >lines(lowess(log(log(fit2$surv[Marital_Status==1]))~log(fit2$ +time)[Marital+_Status==1]), lty=5,col=3) >lines(lowess(log(- +log(fit2$surv[Marital_Status==2]))~log(fit2$time)[Marital_ + Status==2]), lty=5,col=4) >legend('bottomright', paste(c("Married", "Single")), col=5:6, lty=1, +cex=.9, bty='n') #College as a covariate >(fit3<-survfit(Surv(time, status)~factor(College), data=Mypar)) >College<-rep(1:4 , fit3$strata) >plot(log(fit3$time),log(-log(fit3$surv)), type='n',main="A +Graph of log(-+log S(t)) against log (t) for Colleges", +xlab=expression(log(t)), +ylab=expression(log(-log(S(t))))) >lines(lowess(log(-log(fit3$surv[College==1]))~ +log(fit3$time)[College==1]), lty=5,col=18) >lines(lowess(log(-log(fit3$surv[College==2]))~ +log(fit3$time)[College==2]), lty=5, col=19) >lines(lowess(log(-log(fit3$surv[College==3]))~ +log(fit3$time)[College==3]), lty=2, col=20) >lines(lowess(log(-log(fit3$surv[College==4]))~ +log(fit3$time)[College==4]), lty=2, col=22) >legend('topleft', paste(c("CBAS","CE","CHS", "CH")), col=7:10, +lty=1, cex=.9, bty='n') #Ethnicity as a covariate >(fit4<-survfit(Surv(time, status)~factor(Ethnicity), +data=Mypar)) >Ethnicity<-rep(1:11 , fit4$strata) >plot(log(fit4$time),log(-log(fit4$surv)), type='n',main="A Graph of log(-+log S(t)) against log (t) for Region of Origin", +xlab=expression(log(t)), ylab=expression(log(-log(S(t))))) >lines(lowess(log(log(fit4$surv[Ethnicity==1]))~log(fit4$time)[ +Ethnicity==1]), lty=5,col=18) >lines(lowess(log(log(fit4$surv[Ethnicity==2]))~log(fit4$time)[ +Ethnicity==2]), lty=5, col=19) >lines(lowess(log(log(fit4$surv[Ethnicity==3]))~log(fit4$time)[ +Ethnicity==3]), lty=2, col=20) >lines(lowess(log(log(fit4$surv[Ethnicity==4]))~log(fit4$time)[ +Ethnicity==4]), lty=2, col=21) >lines(lowess(log(log(fit4$surv[Ethnicity==5]))~log(fit4$time)[ +Ethnicity==5]), lty=2, col=22) >lines(lowess(log(log(fit4$surv[Ethnicity==6]))~log(fit4$time)[ +Ethnicity==6]), lty=2, col=23) >lines(lowess(log(log(fit4$surv[Ethnicity==7]))~log(fit4$time)[ +Ethnicity==7]), lty=2, col=24) >lines(lowess(log(log(fit4$surv[Ethnicity==8]))~log(fit4$time)[ +Ethnicity==8]), lty=2, col=25) >lines(lowess(log(log(fit4$surv[Ethnicity==9]))~log(fit4$time)[ +Ethnicity==9]), lty=2, col=26) University of Ghana http://ugspace.ug.edu.gh 97 >lines(lowess(log(log(fit4$surv[Ethnicity==10]))~log(fit4$time) + [Ethnicity==10]), lty=2, col=27) >lines(lowess(log(log(fit4$surv[Ethnicity==11]))~log(fit4$time) + [Ethnicity==11]), lty=2, col=28) >legend('topleft',paste(c("ASH","BA","CEN","EAS","GA","NOR", +"OTHERS","UE","UW","VOL","WES")), col=18:28, lty=1, cex=.4, +bty='n') ## Cox PH modelling >(FC <- coxph(S~ AgeFP + Gender + factor(College) + factor(Ethnicity)+ + Marital_Status, data= Mycox, method="breslow")) >summary(FC) # Extraction of AIC values >extractAIC(FC) # Log-likelihood of a model >(ll <-FC$loglik) #Central Tendencies(mean,median) of covariates # mean time to earn a promotion based on gender with(Mycox, tapply(time[status==1],Gender[status==1], mean)) # median time to earn a promotion based on gender with(Mycox, tapply(time,Gender, median)) # median time to earn a promotion based on gender with(Mycox, tapply(time[status==0],Gender[status==0], median)) # mean time to earn a promotion based on marital status with(Mycox, tapply(time[status==1],Marital_Status[status==1], mean)) median time to earn a promotion based on marital status with(Mycox, tapply(time[status==1],Marital_Status[status==1], median)) mean age to earn a promotion based by gender with(Mycox, tapply(AgeFP,Gender, mean)) with(Mycox, tapply(time[status==1],College[status==1], mean)) # mean time to promotion University of Ghana http://ugspace.ug.edu.gh 98 3. AFT Modelling ##Loading data and some descriptives >Mypar<-read.csv("C:/Users/Crystal/Desktop/The +Sis/Data/mycox.csv", head=TRUE) >attach(Mypar) ##Exponential Regression # Full model for exponential regression >fitFE<-survreg(formula=Surv(time, status)~AgeFP + College + +Ethnicity+Marital_Status +Gender, data=Mypar, +dist="exponential") >confint(fitFE, level=0.95) >summary(fitFE) ##Weibull Regression # Full model >fitFW<-survreg(formula=Surv(time, status)~AgeFP + College + +Ethnicity + Marital_Status+ Gender, data=Mypar, +dist="weibull") >confint(fitFW, level=0.95) >summary(fitFW) ##Log-logistic regression # full model >fitFL<-survreg(formula=Surv(time, status)~AgeFP + Ethnicity + +College + Marital_Status + Gender , data=Mypar, +dist="loglogistic") >confint(fitFL, level=0.95) >summary(fitFL) ##Log-normal regression # Full model >fitLN<-survreg(formula=Surv(time, status)~AgeFP + Ethnicity + +College + Marital_Status + Gender , data=Mypar, +dist="lognorm") >confint(fitLN, level=0.95) >summary(fitLN) ## model comparison #full model comparison >anova(fitFL,fitFE,fitFW, fitLN) University of Ghana http://ugspace.ug.edu.gh 99 #AIC values for full and reduced models >AIC(fitFE,fitFW, fitFL, fitLN) D2 Analyses in STATA PH modelling in Stata # Declare data as survival data. stset time,(failure==1)scale(1) # Variable coding -For Gender tabulate Gender, gen(G) -For Marital Status tabulate MaritalStatus, gen(MS) -For Ethnicity tabulate Ethnicity, gen(E) -For Colleges tabulate College, gen(C) # Models a. Cox Stcox AgeFP E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E1 C2 C3 C4 C1 MS2 MS1 G2 G1 -Store model Estimates store C - AIC value Estat ic b. Exponential streg AgeFP E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E1 C2 C3 C4 C1 MS2 MS1 G2 G1, dist(exponential) time -Store model Estimates store E - AIC value Estat ic c. Weibull streg AgeFP E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E1 C2 C3 C4 C1 MS2 MS1 G2 G1, dist(weibull) time -Store model Estimates store W - AIC value Estat ic d. Gompertz University of Ghana http://ugspace.ug.edu.gh 100 streg AgeFP E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E1 C2 C3 C4 C1 MS2 MS1 G2 G1, dist(gompertz) time -Store model Estimates store G - AIC value Estat ic University of Ghana http://ugspace.ug.edu.gh 101 Appendix E Data Used In Research Gender Appointment Date Post Name Appointment Code Prog/Course Department Faculty School Name 1. M 1-Mar-00 ASSO. PROFESSOR 1 ZOOLOGY DEPT. OF ANIMAL BIO&CONS. SCI. SCH OF BIOLOGICAL SCIENCES 2. F 15-May-89 SNR. LECTURER 1 RELIGION DEPARTMENT OF RELIGIONS SCHOOL OF ARTS 3. M 3-Oct-84 SENIOR RESEARCH FELLOW 1 LIPREC LIVESTOCK&POULTRY RES. CENTRE SCHOOL OF AGRICULTURE 4. M 5-Nov-02 PROFESSOR 1 LINGUISTICS DEPT. OF LINGUISTICS SCHOOL OF LANGUAGES 5. F 3-Oct-83 ASSO. PROFESSOR 1 SOCIOLOGY DEPT. OF SOCIOLOGY SCHOOL OF SOCIAL SCIENCES 6. F 2-May-84 ASSO. PROFESSOR 1 GEO & RES DEV DEPT. OF GEOGRAPHY & RES. DEV. SCHOOL OF SOCIAL SCIENCES 7. M 4-Jan-93 SNR. LECTURER 1 AGRIC. ENGINEERING AGRICULTURAL ENGINEERING SCHOOL OF ENGINEERING 8. F 14-Mar-86 PROFESSOR 1 NUTRITION & FOOD SCIENCE DEPT. OF NUTRITION & FOOD SCI. SCH OF BIOLOGICAL SCIENCES 9. M 1-Sep-86 SNR. LECTURER 1 DANCE STUDIES DEPT. OF DANCE STUDIES SCHOOL OF PERFORMING ARTS 10. M 1-Sep-90 ASSO. PROFESSOR 1 GEOLOGY DEPT. OF EARTH SCIENCES SCH. OF PHYSICAL & MATH. SC. 11. M 10-Feb-87 SNR. LECTURER 1 AGRIC EXTENSION DEPT. OF AGRIC. EXTENSION SCHOOL OF AGRICULTURE 12. M 1-Nov-87 PROFESSOR 1 CROP SCIENCE DEPT. OF CROP SCIENCE SCHOOL OF AGRICULTURE 13. M 1-Aug-90 SNR. LECTURER 1 ACCOUNTING ACCOUNTING STUDIES UNIV. OF GHANA BUSINESS SCHOOL 14. M 1-Oct-87 SNR RESIDENT TUTOR 1 ADULT EDUC. LEGON DEPT. OF DISTANCE. EDUCATION. SCHOOL OF CONT. & DIST. EDUC. 15. M 1-Oct-87 SNR. LECTURER 1 SCDE DEPT. OF DISTANCE. EDUCATION. SCHOOL OF CONT. & DIST. EDUC. University of Ghana http://ugspace.ug.edu.gh 102 16. M 23-Mar-88 PROFESSOR 1 BOTANY DEPT. OF BOTANY SCH OF BIOLOGICAL SCIENCES 17. M 9-Feb-93 SNR. LECTURER 1 AGRIC. ENGINEERING AGRICULTURAL ENGINEERING SCHOOL OF ENGINEERING 18. M 1-Aug-03 SNR. LECTURER 1 MODERN LANGUAGE MODERN LANGUAGES SCHOOL OF LANGUAGES 19. F 1-Nov-06 SNR. LECTURER 1 ECONOMICS DEPT. OF ECONOMICS SCHOOL OF SOCIAL SCIENCES 20. F 20-May-89 PROFESSOR 1 BIOCHEMISTRY DEPT. OF BIOCHEM, CELL&MOL.BIO SCH OF BIOLOGICAL SCIENCES 21. F 22-May-89 PROFESSOR 1 AFRICAN STUDIES AFRICAN STUDIES INSTITUTE OF AFRICAN STUDIES 22. M 22-Aug-89 ASSO. PROFESSOR 1 CROP SCIENCE DEPT. 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FOR MED.RES 211 F 1-Jul-00 RESEARCH FELLOW 1 NMIMR PARASITOLOGY UNIT NOGUCHI MEM. INST. FOR MED.RES 212 M 1-Jul-97 RESEARCH FELLOW 1 NUTRITION UNIT (NMIMR) NUTRITION UNIT NOGUCHI MEM. INST. FOR MED.RES 213 M 1-Apr-02 LECTURER 1 MED LABORATORY (SAHS) DEAN'S OFFICE, SBAHS SCH. OF BIOMED & ALLIED HEALTH University of Ghana http://ugspace.ug.edu.gh 111 214 M 1-May-02 LECTURER 1 BIOMATERIAL SCIENCE DEPT. OF BIOMATERIAL SCIENCE SCH. OF MEDICINE & DENTISTRY 215 F 1-Jun-07 LECTURER 1 MEDICAL BIOCHEMISTRY (UGMS) DEAN'S OFF. - MED & DENTISTRY SCH. OF MEDICINE & DENTISTRY 216 F 15-Jun-09 SNR. LECTURER 1 PUBLIC HEALTH PUBLIC HEALTH SCHOOL OF PUBLIC HEALTH 217 F 4-Aug-08 LECTURER 1 PHARMACY & MICROBIOLOGY DEAN'S OFFICE, SCH OF PHARMACY SCHOOL OF PHARMACY 218 M 1-Mar-04 LECTURER 1 DEPT OF ANATOMY DEPARTMENT OF ANATOMY SCH. OF BIOMED & ALLIED HEALTH University of Ghana http://ugspace.ug.edu.gh 112 Data Used In Research (continuation) College Birth Date AgeFP Marital Status Ethnic Group Name Country Name Censoring Date of first promotion Appointment Date time to promotion in days 1. COLLEGE OF BASIC AND APPLIED SCIENCE 8-Mar-56 43 Married GREATER ACCRA GHANA 1 3/1/2000 3/1/1990 3653 2. COLLEGE OF HUMANITIES 17-Apr-60 37 Married UPPER EAST GHANA 1 3/12/1998 5/15/1989 3223 3. COLLEGE OF BASIC AND APPLIED SCIENCE 3-Jun-56 43 Married UPPER WEST GHANA 1 7/17/1999 10/3/1984 5400 4. COLLEGE OF HUMANITIES 3-Aug-55 47 Married CENTRAL GHANA 1 11/5/2002 6/1/1996 2348 5. COLLEGE OF HUMANITIES 10-Sep-54 40 Single GREATER ACCRA GHANA 1 12/1/1994 10/3/1983 4077 6. COLLEGE OF HUMANITIES 12-Oct-55 39 Married UPPER EAST GHANA 1 9/1/1995 5/2/1984 4139 7. COLLEGE OF BASIC AND APPLIED SCIENCE 9-Mar-55 49 Married GREATER ACCRA GHANA 1 10/1/2004 1/4/1993 4288 8. COLLEGE OF BASIC AND APPLIED SCIENCE 21-Oct-57 42 Married GREATER ACCRA GHANA 1 11/4/1999 3/14/1986 4983 9. COLLEGE OF HUMANITIES 4-Aug-57 55 Married GREATER ACCRA GHANA 1 7/1/2013 9/1/1986 9800 10. COLLEGE OF BASIC AND APPLIED SCIENCE 20-Jul-57 41 Married GREATER ACCRA GHANA 1 1/17/1999 9/1/1990 3060 11. COLLEGE OF BASIC AND APPLIED SCIENCE 16-May-56 50 Married GREATER ACCRA GHANA 1 9/15/2006 2/10/1987 7157 12. COLLEGE OF BASIC AND APPLIED SCIENCE 29-Jan-58 43 Single EASTERN GHANA 1 10/25/2001 11/1/1987 5107 13. COLLEGE OF HUMANITIES 23-Mar-58 48 Married GREATER ACCRA GHANA 1 5/1/2006 8/1/1990 5752 14. COLLEGE OF EDUCATION 18-Oct-58 33 Single GREATER ACCRA GHANA 1 6/1/1992 10/1/1987 1705 15. COLLEGE OF EDUCATION 28-Feb-55 38 Married BRONG AHAFO GHANA 1 7/1/1993 10/1/1987 2100 16. COLLEGE OF BASIC AND APPLIED SCIENCE 24-Feb-58 41 Married VOLTA GHANA 1 6/2/1999 3/23/1988 4088 17. COLLEGE OF BASIC AND APPLIED SCIENCE 26-Jul-62 47 Single VOLTA GHANA 1 9/1/2009 2/9/1993 6048 University of Ghana http://ugspace.ug.edu.gh 113 18. COLLEGE OF HUMANITIES 3-Dec-56 53 Single NORTHERN GHANA 1 1/5/2010 8/1/2003 2349 19. COLLEGE OF HUMANITIES 10-Feb-59 47 Married EASTERN GHANA 1 11/1/2006 11/1/1997 3287 20. COLLEGE OF BASIC AND APPLIED SCIENCE 8-Jan-56 40 Single GREATER ACCRA GHANA 1 11/1/1996 5/20/1989 2722 21. COLLEGE OF HUMANITIES 27-Mar-60 36 Married CENTRAL GHANA 1 10/1/1996 5/22/1989 2689 22. COLLEGE OF BASIC AND APPLIED SCIENCE 23-Apr-56 43 Married EASTERN GHANA 1 6/9/1999 8/22/1989 3578 23. COLLEGE OF BASIC AND APPLIED SCIENCE 25-Jan-59 38 Married VOLTA GHANA 1 4/1/1997 11/1/1989 2708 24. COLLEGE OF HUMANITIES 22-Sep-56 49 Married EASTERN GHANA 1 12/1/2005 5/31/2002 1280 25. COLLEGE OF HUMANITIES 18-Jul-63 38 Married VOLTA GHANA 1 10/23/2001 10/1/1994 2579 26. COLLEGE OF BASIC AND APPLIED SCIENCE 3-Oct-57 53 Married UPPER WEST GHANA 1 1/1/2011 9/1/1990 7427 27. COLLEGE OF HUMANITIES 27-Jun-57 47 Married WESTERN GHANA 1 8/1/2004 3/1/1995 3441 28. COLLEGE OF HUMANITIES 10-Nov-58 37 Married VOLTA GHANA 1 11/17/1995 10/4/1995 44 29. COLLEGE OF HUMANITIES 13-May-58 49 Married UPPER WEST GHANA 1 4/15/2008 1/15/1992 5935 30. COLLEGE OF BASIC AND APPLIED SCIENCE 4-Jun-65 43 Married WESTERN GHANA 1 3/1/2009 10/2/1995 4899 31. COLLEGE OF HUMANITIES 28-May-56 44 Married GREATER ACCRA GHANA 1 2/16/2001 9/13/1993 2713 32. COLLEGE OF EDUCATION 17-Jul-60 50 Married BRONG AHAFO GHANA 1 2/1/2011 9/1/1992 6727 33. COLLEGE OF EDUCATION 10-Feb-63 47 Single EASTERN GHANA 1 11/1/2010 1/13/1993 6501 34. COLLEGE OF HUMANITIES 15-Mar-57 42 Married CENTRAL GHANA 1 5/19/1999 8/2/1993 2116 35. COLLEGE OF BASIC AND APPLIED SCIENCE 19-Jun-63 47 Married UPPER WEST GHANA 1 10/12/2010 10/15/1996 5110 36. COLLEGE OF EDUCATION 29-Mar-61 39 Single VOLTA GHANA 1 3/29/2000 7/5/1993 2459 37. COLLEGE OF BASIC AND APPLIED SCIENCE 14-Sep-58 46 Married ASHANTI GHANA 1 7/6/2005 12/3/1993 4233 38. COLLEGE OF HUMANITIES 13-Aug-57 40 Married UPPER WEST GHANA 1 7/1/1998 10/1/1993 1734 39. COLLEGE OF BASIC AND APPLIED SCIENCE 20-Aug-57 54 Married EASTERN GHANA 1 10/1/2011 9/5/1994 6235 40. COLLEGE OF BASIC AND APPLIED SCIENCE 3-Mar-61 43 Married GREATER ACCRA GHANA 1 10/1/2004 9/1/1994 3683 University of Ghana http://ugspace.ug.edu.gh 114 41. COLLEGE OF HUMANITIES 17-Nov-56 55 Single ASHANTI GHANA 1 3/12/2012 9/1/1994 6402 42. COLLEGE OF BASIC AND APPLIED SCIENCE 26-Apr-55 50 Married GREATER ACCRA GHANA 1 6/1/2005 4/3/1995 3712 43. COLLEGE OF HUMANITIES 15-Jul-62 47 Married GREATER ACCRA GHANA 1 5/1/2010 10/1/2005 1673 44. COLLEGE OF HEALTH SCIENCES 22-Aug-58 42 Married GREATER ACCRA GHANA 1 3/23/2001 5/1/1995 2153 45. COLLEGE OF HUMANITIES 24-Mar-60 39 Married VOLTA GHANA 1 8/1/1999 10/1/1995 1400 46. COLLEGE OF HEALTH SCIENCES 12-Apr-60 49 Married EASTERN GHANA 1 1/1/2010 5/1/1996 4993 47. COLLEGE OF BASIC AND APPLIED SCIENCE 20-Feb-66 42 Married GREATER ACCRA GHANA 1 4/16/2008 6/2/1997 3971 48. COLLEGE OF EDUCATION 22-Feb-56 42 Married VOLTA GHANA 1 3/13/1998 11/1/1996 497 49. COLLEGE OF HUMANITIES 10-Oct-67 45 Married EASTERN GHANA 1 6/1/2013 10/1/1996 6087 50. COLLEGE OF EDUCATION 26-Jul-56 57 Married GREATER ACCRA GHANA 1 8/1/2013 3/10/1997 5988 51. COLLEGE OF HUMANITIES 1-Dec-55 47 Married VOLTA GHANA 1 6/16/2003 4/21/1997 2247 52. COLLEGE OF HUMANITIES 2-Jul-57 41 Married ASHANTI GHANA 1 10/10/1998 10/1/1997 374 53. COLLEGE OF BASIC AND APPLIED SCIENCE 2-Mar-55 54 Single OTHER AFRICAN AUSTRALIA 1 2/1/2010 1/26/1998 4389 54. COLLEGE OF BASIC AND APPLIED SCIENCE 22-Nov-60 52 Married ASHANTI GHANA 1 3/18/2013 9/14/1998 5299 55. COLLEGE OF HEALTH SCIENCES 21-Oct-56 49 Married EASTERN GHANA 1 6/15/2006 1/5/1998 3083 56. COLLEGE OF HUMANITIES 8-Oct-58 41 Married CENTRAL GHANA 1 8/14/2000 12/17/1997 971 57. COLLEGE OF BASIC AND APPLIED SCIENCE 20-Dec-60 44 Married EASTERN GHANA 1 10/1/2005 1/11/1999 2455 58. COLLEGE OF BASIC AND APPLIED SCIENCE 23-Feb-64 39 Single UPPER WEST GHANA 1 10/1/2003 5/27/1998 1953 59. COLLEGE OF BASIC AND APPLIED SCIENCE 5-Jun-61 45 Married EASTERN GHANA 1 7/1/2006 6/1/1998 2952 60. COLLEGE OF BASIC AND APPLIED SCIENCE 17-Nov-67 38 Married CENTRAL GHANA 1 12/22/2005 6/6/1997 3121 61. COLLEGE OF HUMANITIES 18-Jan-55 58 Married VOLTA GHANA 1 6/11/2013 11/20/1998 5317 62. COLLEGE OF HUMANITIES 22-Sep-54 56 Married GREATER ACCRA GHANA 1 6/13/2011 10/1/1996 5368 63. COLLEGE OF HUMANITIES 17-May-57 53 Married VOLTA GHANA 1 7/1/2010 6/18/2009 378 University of Ghana http://ugspace.ug.edu.gh 115 64. COLLEGE OF BASIC AND APPLIED SCIENCE 1-Apr-58 51 Married VOLTA GHANA 1 9/10/2009 3/15/1999 3832 65. COLLEGE OF HUMANITIES 14-Nov-66 41 Married GREATER ACCRA GHANA 1 12/1/2007 2/1/1999 3225 66. COLLEGE OF HUMANITIES 12-Feb-65 43 Married WESTERN GHANA 1 6/1/2008 8/30/1999 3198 67. COLLEGE OF BASIC AND APPLIED SCIENCE 23-Aug-55 55 Married GREATER ACCRA GHANA 1 3/1/2011 2/7/2000 4040 68. COLLEGE OF HUMANITIES 6-Feb-57 48 Married CENTRAL GHANA 1 3/1/2005 1/3/2000 1884 69. COLLEGE OF BASIC AND APPLIED SCIENCE 28-Oct-61 48 Married EASTERN GHANA 1 4/1/2010 3/1/2001 3318 70. COLLEGE OF BASIC AND APPLIED SCIENCE 11-May-62 41 Married VOLTA GHANA 1 10/1/2003 3/3/2000 1307 71. COLLEGE OF HUMANITIES 17-Aug-62 44 Married BRONG AHAFO GHANA 1 7/23/2007 4/17/2000 2653 72. COLLEGE OF HUMANITIES 18-Nov-66 39 Married OTHER GHANAIANS GHANA 1 7/1/2006 11/1/1999 2434 73. COLLEGE OF BASIC AND APPLIED SCIENCE 13-Dec-68 34 Single CENTRAL GHANA 1 8/1/2003 9/12/2000 1053 74. COLLEGE OF BASIC AND APPLIED SCIENCE 5-Nov-68 44 Single ASHANTI GHANA 1 6/1/2013 9/1/2000 4656 75. COLLEGE OF BASIC AND APPLIED SCIENCE 7-Nov-67 39 Married ASHANTI GHANA 1 2/12/2007 9/1/2000 2355 76. COLLEGE OF HUMANITIES 18-Feb-58 45 Married VOLTA GHANA 1 3/1/2003 9/1/2000 911 77. COLLEGE OF HEALTH SCIENCES 26-Nov-67 39 Married VOLTA GHANA 1 7/1/2007 6/2/1999 2951 78. COLLEGE OF HUMANITIES 7-Sep-70 30 Single EASTERN GHANA 1 8/1/2001 4/24/2001 99 79. COLLEGE OF HUMANITIES 29-Apr-63 43 Married NORTHERN GHANA 1 6/13/2006 1/1/2002 1624 80. COLLEGE OF HUMANITIES 19-Mar-62 48 Married EASTERN GHANA 1 4/1/2010 6/12/2001 3215 81. COLLEGE OF BASIC AND APPLIED SCIENCE 7-Sep-69 37 Married CENTRAL GHANA 1 11/15/2006 7/1/2001 1963 82. COLLEGE OF HUMANITIES 28-Jul-72 41 Married OTHER AFRICAN GHANA 1 11/15/2013 10/15/2001 4414 83. COLLEGE OF HUMANITIES 30-Sep-71 30 Married CENTRAL GHANA 1 8/1/2002 11/6/2001 268 84. COLLEGE OF EDUCATION 15-Oct-58 52 Married CENTRAL GHANA 1 1/1/2011 8/1/2006 1614 85. COLLEGE OF EDUCATION 11-May-70 40 Married ASHANTI GHANA 1 8/1/2010 4/7/2003 2673 86. COLLEGE OF HUMANITIES 21-Aug-72 30 Married UPPER EAST GHANA 1 2/1/2003 1/18/2001 744 University of Ghana http://ugspace.ug.edu.gh 116 87. COLLEGE OF BASIC AND APPLIED SCIENCE 14-Sep-61 53 Married GREATER ACCRA GHANA 1 10/1/2014 6/10/2003 4131 88. COLLEGE OF HUMANITIES 19-Oct-67 46 Married WESTERN GHANA 1 12/1/2013 9/1/1997 5935 89. COLLEGE OF HUMANITIES 2-Jul-64 38 Married OTHER GHANAIANS GHANA 1 4/14/2003 4/12/2002 367 90. COLLEGE OF BASIC AND APPLIED SCIENCE 1-Oct-71 41 Married VOLTA GHANA 1 4/1/2013 8/25/2003 3507 91. COLLEGE OF HUMANITIES 17-Mar-75 32 Married ASHANTI GHANA 1 4/12/2007 3/1/2004 1137 92. COLLEGE OF HUMANITIES 1-Jan-58 46 Married NORTHERN GHANA 1 11/26/2004 4/1/2003 605 93. COLLEGE OF HUMANITIES 4-Mar-69 43 Married GREATER ACCRA GHANA 1 2/1/2013 11/23/2004 2992 94. COLLEGE OF HUMANITIES 25-Feb-70 35 Single CENTRAL GHANA 1 2/25/2005 6/1/2003 635 95. COLLEGE OF HUMANITIES 14-Aug-72 36 Married ASHANTI GHANA 1 2/11/2009 7/1/2005 1321 96. COLLEGE OF HUMANITIES 30-Aug-72 39 Single VOLTA GHANA 1 4/1/2012 11/1/2005 2343 97. COLLEGE OF HUMANITIES 17-Feb-70 36 Married ASHANTI GHANA 1 3/29/2006 6/1/2004 666 98. COLLEGE OF BASIC AND APPLIED SCIENCE 28-Dec-61 47 Married VOLTA GHANA 1 8/2/2009 2/1/2007 913 99. COLLEGE OF HUMANITIES 16-Nov-78 28 Single CENTRAL GHANA 1 11/1/2007 7/1/2006 488 100 COLLEGE OF HUMANITIES 1-Oct-73 33 Single EASTERN GHANA 1 12/1/2006 8/1/2006 122 101 COLLEGE OF BASIC AND APPLIED SCIENCE 1-May-71 39 Married VOLTA GHANA 1 9/1/2010 11/2/2006 1399 102 COLLEGE OF BASIC AND APPLIED SCIENCE 20-Sep-70 36 Married VOLTA GHANA 1 2/1/2007 1/1/2006 396 103 COLLEGE OF HUMANITIES 11-Dec-65 46 Married NORTHERN GHANA 1 3/30/2012 8/7/2007 1697 104 COLLEGE OF HUMANITIES 7-Nov-79 33 Married ASHANTI GHANA 1 7/1/2013 8/7/2007 2155 105 COLLEGE OF HEALTH SCIENCES 26-Mar-65 46 Married VOLTA GHANA 1 2/1/2012 1/8/2008 1485 106 COLLEGE OF BASIC AND APPLIED SCIENCE 20-Jun-67 41 Single GREATER ACCRA GHANA 1 6/23/2008 11/1/2007 235 107 COLLEGE OF HUMANITIES 22-Feb-72 39 Married NORTHERN GHANA 1 12/1/2011 7/1/2008 1248 108 COLLEGE OF BASIC AND APPLIED SCIENCE 14-Aug-68 40 Married EASTERN GHANA 1 1/19/2009 9/1/2005 1236 109 COLLEGE OF BASIC AND APPLIED SCIENCE 22-Jul-71 37 Single EASTERN GHANA 1 2/18/2009 8/2/2005 1296 University of Ghana http://ugspace.ug.edu.gh 117 110 COLLEGE OF HUMANITIES 23-Mar-72 39 Married VOLTA GHANA 1 9/1/2011 6/9/2009 814 111 COLLEGE OF HUMANITIES 23-Feb-79 30 Single EASTERN GHANA 1 8/18/2009 11/1/2007 656 112 COLLEGE OF HUMANITIES 1-Dec-69 41 Single EASTERN GHANA 1 1/1/2011 8/3/2009 516 113 COLLEGE OF HUMANITIES 23-Oct-71 37 Single CENTRAL GHANA 1 10/1/2009 9/11/2008 385 114 COLLEGE OF BASIC AND APPLIED SCIENCE 28-Aug-67 42 Married ASHANTI GHANA 1 8/2/2010 1/11/2008 934 115 COLLEGE OF HUMANITIES 3-Sep-67 42 Married EASTERN GHANA 1 8/1/2010 6/1/2010 61 116 COLLEGE OF HUMANITIES 24-Aug-75 34 Married CENTRAL GHANA 1 8/1/2010 8/1/2005 1826 117 COLLEGE OF HEALTH SCIENCES 2-Apr-60 47 Married GREATER ACCRA GHANA 1 9/1/2007 1/1/1998 3530 118 COLLEGE OF HEALTH SCIENCES 7-Jun-68 41 Married VOLTA GHANA 1 5/1/2010 6/1/2003 2526 119 COLLEGE OF HEALTH SCIENCES 17-Aug-56 54 Married GREATER ACCRA GHANA 1 3/1/2011 1/1/1997 5172 120 COLLEGE OF HEALTH SCIENCES 11-Jul-62 50 Married NORTHERN GHANA 1 12/1/2012 8/1/2009 1218 121 COLLEGE OF HEALTH SCIENCES 25-May-58 48 Married EASTERN GHANA 1 5/1/2007 4/1/1997 3682 122 COLLEGE OF HEALTH SCIENCES 28-Feb-59 54 Married CENTRAL GHANA 1 12/1/2013 9/21/1999 5185 123 COLLEGE OF HEALTH SCIENCES 8-Jan-61 48 Married EASTERN GHANA 1 9/10/2009 1/3/2001 3172 124 COLLEGE OF HEALTH SCIENCES 15-Sep-59 54 Married CENTRAL GHANA 1 1/1/2014 1/11/1987 9852 125 COLLEGE OF HEALTH SCIENCES 1-Jul-55 56 Married GREATER ACCRA GHANA 1 2/1/2012 6/6/2005 2431 126 COLLEGE OF HEALTH SCIENCES 7-Mar-60 48 Married ASHANTI GHANA 1 4/9/2008 7/1/2000 2839 127 COLLEGE OF HEALTH SCIENCES 28-Jul-69 40 Married EASTERN GHANA 1 11/1/2009 4/1/2008 579 128 COLLEGE OF HEALTH SCIENCES 14-Apr-56 54 Married VOLTA GHANA 1 3/1/2011 10/15/1990 7442 129 COLLEGE OF HEALTH SCIENCES 24-Oct-57 50 Married ASHANTI GHANA 1 4/1/2008 7/1/1996 4292 130 COLLEGE OF HEALTH SCIENCES 20-Jun-57 51 Married CENTRAL GHANA 1 12/16/2008 12/1/1996 4398 131 COLLEGE OF HEALTH SCIENCES 15-Aug-55 56 Married BRONG AHAFO GHANA 1 9/1/2011 1/4/2011 240 132 COLLEGE OF HEALTH SCIENCES 2-Nov-70 40 Married OTHER AFRICAN GHANA 1 8/1/2011 5/4/2007 1550 University of Ghana http://ugspace.ug.edu.gh 118 133 COLLEGE OF HEALTH SCIENCES 11-Nov-60 53 Married EASTERN GHANA 1 2/1/2014 8/1/1987 9681 134 COLLEGE OF HEALTH SCIENCES 21-May-56 52 Married WESTERN GHANA 1 10/1/2008 1/1/1979 10866 135 COLLEGE OF HEALTH SCIENCES 28-May-58 55 Married BRONG AHAFO GHANA 1 3/1/2014 1/1/1983 11382 136 COLLEGE OF HUMANITIES 26-Mar-57 56 Single EASTERN GHANA 1 9/1/2013 8/1/2012 396 137 COLLEGE OF HEALTH SCIENCES 11-Jun-67 42 Married ASHANTI GHANA 1 9/11/2009 9/1/1998 4028 138 COLLEGE OF HEALTH SCIENCES 4-Aug-65 47 Married ASHANTI GHANA 1 7/3/2013 4/1/1993 7398 139 COLLEGE OF HEALTH SCIENCES 27-Nov-66 45 Married BRONG AHAFO GHANA 1 8/1/2012 4/1/1993 7062 140 COLLEGE OF HEALTH SCIENCES 17-Apr-68 41 Married VOLTA GHANA 1 4/13/2010 1/1/1996 5216 141 COLLEGE OF HEALTH SCIENCES 25-Dec-66 45 Married NORTHERN GHANA 1 2/7/2012 6/20/2009 962 142 COLLEGE OF HUMANITIES 15-Dec-61 49 Married GREATER ACCRA GHANA 1 10/1/2011 4/1/1982 10775 143 COLLEGE OF BASIC AND APPLIED SCIENCE 3-Feb-56 57 Married WESTERN GHANA 1 10/1/2013 10/6/1986 9857 144 COLLEGE OF HUMANITIES 27-Feb-65 47 Married VOLTA GHANA 1 10/1/2012 2/3/1992 7546 145 COLLEGE OF HUMANITIES 12-Dec-64 46 Married GREATER ACCRA GHANA 1 11/1/2011 10/1/1991 7336 146 COLLEGE OF BASIC AND APPLIED SCIENCE 26-Jan-68 46 Single EASTERN GHANA 1 10/1/2014 10/1/2001 4748 147 COLLEGE OF HEALTH SCIENCES 10-Nov-62 49 Single VOLTA GHANA 1 10/1/2012 4/17/2000 4550 148 COLLEGE OF HUMANITIES 12-Jun-73 39 Single GREATER ACCRA GHANA 1 4/1/2013 8/1/2001 4261 149 COLLEGE OF BASIC AND APPLIED SCIENCE 3-Apr-73 38 Married WESTERN GHANA 1 10/1/2011 8/14/2002 3335 150 COLLEGE OF BASIC AND APPLIED SCIENCE 15-May-67 45 Married CENTRAL GHANA 1 3/1/2013 5/22/2003 3571 151 COLLEGE OF HUMANITIES 23-Sep-57 55 Married OTHER GHANAIANS 1 3/1/2013 11/10/2003 3399 152 COLLEGE OF HEALTH SCIENCES 22-Aug-69 43 Married UPPER WEST GHANA 1 5/1/2013 9/1/2004 3164 153 COLLEGE OF HUMANITIES 14-Aug-64 49 Married EASTERN GHANA 1 12/1/2013 10/1/2005 2983 154 COLLEGE OF BASIC AND APPLIED SCIENCE 9-Jun-55 52 Married GREATER ACCRA GHANA 1 7/1/2007 2/10/2006 506 155 COLLEGE OF BASIC AND APPLIED SCIENCE 14-Oct-74 34 Married WESTERN GHANA 1 5/4/2009 4/26/2006 1104 University of Ghana http://ugspace.ug.edu.gh 119 156 COLLEGE OF HUMANITIES 15-Aug-72 39 Married EASTERN GHANA 1 3/1/2012 10/11/2006 1968 157 COLLEGE OF BASIC AND APPLIED SCIENCE 2-Sep-76 30 Married EASTERN GHANA 1 11/13/2006 10/26/2005 383 158 COLLEGE OF HEALTH SCIENCES 13-Sep-66 40 Married VOLTA GHANA 1 10/18/2006 11/1/2005 351 159 COLLEGE OF BASIC AND APPLIED SCIENCE 29-Apr-73 36 Married VOLTA GHANA 1 11/1/2009 8/29/2007 795 160 COLLEGE OF BASIC AND APPLIED SCIENCE 7-Jan-79 32 Single VOLTA GHANA 1 8/1/2011 9/1/2007 1430 161 COLLEGE OF BASIC AND APPLIED SCIENCE 5-Dec-61 48 Married CENTRAL GHANA 1 8/1/2010 1/8/2008 936 162 COLLEGE OF BASIC AND APPLIED SCIENCE 13-Apr-73 37 Married ASHANTI GHANA 1 8/1/2010 1/8/2001 3492 163 COLLEGE OF BASIC AND APPLIED SCIENCE 3-Jul-76 35 Married CENTRAL GHANA 1 4/8/2012 10/1/2010 555 164 COLLEGE OF HEALTH SCIENCES 22-Sep-71 38 Married ASHANTI GHANA 1 2/1/2010 9/1/2006 1249 165 COLLEGE OF HEALTH SCIENCES 19-Jul-78 34 Single VOLTA GHANA 1 6/11/2013 10/11/2006 2435 166 COLLEGE OF HEALTH SCIENCES 1-Nov-70 36 Married WESTERN GHANA 1 11/1/2006 3/1/2000 2436 167 COLLEGE OF HEALTH SCIENCES 1-Jul-55 57 Married EASTERN GHANA 1 6/11/2013 4/1/1993 7376 168 COLLEGE OF HEALTH SCIENCES 21-Jan-60 48 Married EASTERN GHANA 1 4/29/2008 9/21/1999 3143 169 COLLEGE OF HEALTH SCIENCES 15-May-64 46 Married VOLTA GHANA 1 4/18/2011 11/21/1999 4166 170 COLLEGE OF HEALTH SCIENCES 28-Jan-64 48 Married GREATER ACCRA GHANA 1 10/15/2012 10/1/1991 7685 171 COLLEGE OF HEALTH SCIENCES 24-Nov-61 46 Married CENTRAL GHANA 1 8/1/2008 5/2/2007 457 172 COLLEGE OF HEALTH SCIENCES 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