University of Ghana http://ugspace.ug.edu.gh UNIVERSITY OF GHANA ERROR ASSESSMENT IN THE APPLICATION OF FOREIGN LIFE TABLES IN GHANA AND COMPETING RISK ANALYSIS BY EMMANUEL KOJO AIDOO (10336154) THIS THESIS IS SUBMITTED TO UNIVERSITY OF GHANA, LEGON IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE AWARD OF MPHIL STATISTICS DEGREE JULY, 2017 University of Ghana http://ugspace.ug.edu.gh DECLARATION Candidate’s Declaration I, Emmanuel Kojo Aidoo hereby declare that apart from references to other people’s publications, which have been duly acknowledged, this thesis is a result of my independent ideas, thought, deliberations and has not been submitted for the award of any degree at this institution and other universities elsewhere. SIGNATURE: …………………………………… DATE: ……………… EMMANUEL KOJO AIDOO (10336154) Supervisors’ Declaration We hereby certify that this thesis was prepared from the candidate’s own work and supervised in accordance with guidelines on supervision of thesis laid down by the University of Ghana. SIGNATURE: …………………………………… DATE: ……………… DR. ISAAC BAIDOO (PRINCIPAL SUPERVISOR) SIGNATURE: …………………………………… DATE: ……………… DR. KWABENA DOKU-AMPONSAH (CO-SUPERVISOR) i University of Ghana http://ugspace.ug.edu.gh ABSTRACT The application of foreign life table in Sub-Sahara Africa had been a major concern all over the world. However, assessing the error for applying the foreign life tables as well as analysing the impact of the leading causes of death on Ghanaian longevity had not been studied. This study aimed at assessing the error of applying Foreign Life Tables in Ghana and analysing the impacts of the leading causes of death in Ghana. A cohort data of 3260 were collected from the University of Ghana Hospital and the causes of death and the gender of each respondent were recorded. The study revealed that Ghanaians life expectancy at birth is 51.25 (combined sex), 49.51 years for male and 53.38 for females. Using UK, USA and South African life tables, the study found out the error obtained from applying South African life table is small relative to other foreign life tables but were significantly different from the Ghana life table. The mortality of males was greater than the females in most ages. The study also found out that cardiovascular disease and HIV/AIDS were the leading causes of death in Ghana. It was revealed that the life expectancy of Ghanaians will increase by 16.021% and 17.587% for males and females respectively, if cardiovascular disease is eliminated in the population. The study recommended that there is a need to construct a life table for Ghana and the Government of Ghana and other agencies should create awareness on cardiovascular diseases and HIV/AIDS. ii University of Ghana http://ugspace.ug.edu.gh DEDICATION This work is dedicated to my parents and all the lecturers in Department of Statistics, University of Ghana. iii University of Ghana http://ugspace.ug.edu.gh ACKNOWLEDGEMENT My first appreciation goes to the Almighty God for his Mercy and Grace upon my life throughout my study in this University. My profound appreciation also goes to my supervisors, Dr. Isaac Baidoo and Dr. K. Doku Amponsah whose suggestions and criticisms helped to enrich my work. I also thank all the lecturers of Statistics Department, especially Dr. F.O. Mettle, Dr. E. N. N. Nortey, Dr. L. Asiedu, Mr. E. N. B. Quaye and Mr. A.A. Asare-Kumi for their services and pieces of advice throughout my two years of study in this University. I wish to thank my friends, Mr. Enoch Sakyi-Yeboah, Mr. Steven Nkrumah, Mr. Obu-Amoah, Mr. Agyarko, Mr. Alidu Uzeru, Miss Millicent Narh, Mr. Prince Owusu Agyeman and Mr. Armachie Joseph for their words of encouragement. My special and sincere gratitude goes to my parents Rev. and Mrs. Aidoo and my Uncle Mr. Appiah Boateng for their financial support and advice, May the good Lord continue to bless you all. iv University of Ghana http://ugspace.ug.edu.gh TABLE OF CONTENTS Content Page DECLARATION ................................................................................................................... i ABSTRACT .......................................................................................................................... ii DEDICATION .................................................................................................................... iii ACKNOWLEDGEMENT ................................................................................................... iv TABLE OF CONTENTS ...................................................................................................... v LIST OF FIGURES........................................................................................................... viii LIST OF TABLES ............................................................................................................... ix CHAPTER ONE ................................................................................................................... 1 INTRODUCTION................................................................................................................. 1 1.1 Background of the study ............................................................................................. 1 1.2 Statement of the Problem ............................................................................................ 3 1.3 Research Questions ..................................................................................................... 4 1.4 Objectives of the Study ............................................................................................... 4 1.5 Hypotheses .................................................................................................................. 4 1.6 Significance of the study ............................................................................................. 5 1.7 Scope of the study ....................................................................................................... 5 1.8 Limitations of the Research ........................................................................................ 5 1.9 Organization of the study ............................................................................................ 6 CHAPTER TWO .................................................................................................................. 7 LITERATURE REVIEW...................................................................................................... 7 2.0 Introduction ................................................................................................................. 7 2.1 Definition and Concept of life table ............................................................................ 7 2.2 Methods of Constructing Life Table ........................................................................... 8 2.3 Importance of Life Table ............................................................................................ 8 2.4 Definitions and Concepts of Competing risk .............................................................. 8 2.4.1 Crude probability ................................................................................................. 9 2.4.2 Net probability ..................................................................................................... 9 2.4.3 Partial crude probability ....................................................................................... 9 2.5 Previous Studies on Life Table ................................................................................. 10 2.6 Previous Studies that used multiple Decrement life table and Competing risk ........ 12 2.7 Summary of Literature Reviewed ............................................................................. 20 CHAPTER THREE ............................................................................................................. 21 METHODOLOGY .............................................................................................................. 21 3.0 Introduction ............................................................................................................... 21 3.1 Setting ....................................................................................................................... 21 3.2 Study population ....................................................................................................... 22 3.3 The Data .................................................................................................................... 22 v University of Ghana http://ugspace.ug.edu.gh 3.4 Data Analysis ............................................................................................................ 22 3.5 Construction of Life Table ........................................................................................ 23 3.6 Error Assessment ...................................................................................................... 24 3.6.1 Absolute Error Analyses .................................................................................... 24 3.6.2 Univariate Analysis of Variance (ANOVA) ...................................................... 24 3.6.3 Paired t-test......................................................................................................... 25 3.7 Force of Mortality Model .......................................................................................... 27 3.8 Computation of competing risks ............................................................................... 28 3.8.1 Crude Probability ............................................................................................... 29 3.8.2 Net probability ................................................................................................... 30 3.8.3 Partial Crude Probability .................................................................................... 31 3.8.4 Assumptions of Competing Risks ...................................................................... 33 3.9 The Log Rank Test .................................................................................................... 34 3.10 Summary of Methodology ...................................................................................... 34 CHAPTER FOUR ............................................................................................................... 35 DATA ANALYSIS AND DISCUSSION ........................................................................... 35 4.0 Introduction ............................................................................................................... 35 A CONSTRUCTION OF LIFE TABLES ...................................................................... 35 4.1 Abridge Life Table .................................................................................................... 35 4.1.1 Abridged Life Table for Males .......................................................................... 35 4.1.2 Abridged Life Table for Female ........................................................................ 37 4.1.3 Life Expectancy for Combined Sex ................................................................... 39 4.2 Comparing the Study Life Table with Other Foreign Life Tables ............................ 40 4.2.1 Life Expectancy for males in the various Countries .......................................... 40 4.2.2 Life Expectancy for females in the various Countries ....................................... 41 4.2.3 Testing for the Univariate Normality ................................................................. 42 4.2.4 Univariate Analysis of Variance (ANOVA) ...................................................... 42 4.2.5 Testing for the difference between South Africa Life Table and the Study Life Table ............................................................................................................................ 43 4.3 Modelling he Force of Mortality ............................................................................... 44 4.3.1 Force of Mortality Functions for Males ............................................................. 44 4.3.2 Force of Mortality Functions for Females ......................................................... 45 4.3.3 Force of Mortality Functions for Combine Sex ................................................. 46 4.3.4 Graphical Representation ................................................................................... 47 4.4 Testing for the equality of survivor functions among Gender .................................. 47 B COMPETING RISK ANALYSIS ............................................................................... 49 4.5 Multiple Decrement Table ........................................................................................ 49 4.6 Crude Probabilities .................................................................................................... 51 4.6.1 Crude Probabilities of Cardiovascular Diseases ................................................ 51 4.6.2 Crude Probabilities of HIV/AIDS ...................................................................... 54 4.6.3 Crude Probabilities of Lower Respiration Infections......................................... 55 4.6.4 Crude Probabilities of Malaria ........................................................................... 57 4.6.5 Graphical Presentation Crude Probabilities ....................................................... 58 vi University of Ghana http://ugspace.ug.edu.gh 4.7 Net Probabilities ........................................................................................................ 59 4.7.1 Net Probabilities for Males ................................................................................ 59 4.7.2 Graphical Presentation the Female Net Probabilities......................................... 60 4.8 The Impact of Eliminating the Major Causes of Diseases ........................................ 60 4.8.1 The Impact of Eliminating Cardiovascular Diseases ......................................... 61 4.8.1.1 Percentage Decrease in Probability of Dying Given that Cardiovascular Diseases is eliminated .............................................................................................. 61 4.8.1.2 Life Expectancy Gain from Eliminating Cardiovascular Diseases .............. 62 4.8.2 The Impact of Eliminating HIV/AIDS ............................................................... 64 4.8.2.1 Percentage Decrease in Probability of Dying Given that HIV/AIDS is eliminated ................................................................................................................. 64 4.8.2.2 Life Expectancy Gain from Eliminating HIV/AIDS ................................... 65 4.8.3 The Effect of Eliminating Lower Respiration .................................................... 66 4.8.3.1 Percentage Decrease in Probability of Dying Given that Lower Respiration (LIR) is eliminated ................................................................................................... 66 4.8.3.2 Life Expectancy Gain from Eliminating Lower Respiration ....................... 67 4.8.4 The Effect of Eliminating Malaria ..................................................................... 69 4.8.4.1 Percentage Decrease in Probability of Dying Given that Eliminating Malaria Diseases .................................................................................................................... 69 4.8.4.2 Life Expectancy Gain from Eliminating Malaria......................................... 70 CHAPTER FIVE ................................................................................................................. 73 SUMMARY, CONCLUSION AND RECOMMENDATION ........................................... 73 5.1 Introduction ............................................................................................................... 73 5.2 Summary ................................................................................................................... 73 5.3 Summary of findings ................................................................................................. 74 5.3 Conclusions ............................................................................................................... 75 5.4 Recommendations ..................................................................................................... 75 REFERENCES .................................................................................................................... 76 APPENDICES..................................................................................................................... 80 Appendix I: Complete Life Table ................................................................................... 80 Appendix II Abridge Life Table ..................................................................................... 85 Appendix III Kaplan-Meier Survival Curve ................................................................... 87 Appendix IV Multiple Decrement Table for Mortality................................................... 88 Appendix V: Net Probabilities (When there is Only that Risk) ...................................... 92 Appendix VI: Net Probabilities (When The Risk Is Eliminated) ................................. 100 vii University of Ghana http://ugspace.ug.edu.gh LIST OF FIGURES Figure 4.1: Life Expectancy plot for Combined sex ........................................................... 39 Figure 4.2: Life Expectancy for males in the various Countries......................................... 40 Figure 4.3: Life Expectancy for females in the various Countries ..................................... 41 Figure 4.4: Force of Mortality Plot for Males, Female and Combines Sex. ....................... 47 Figure 4.5: Crude Probabilities ........................................................................................... 58 Figure 4.6: Net Probabilities for Males ............................................................................... 59 Figure 4.7: Net Probabilities for Females ........................................................................... 60 viii University of Ghana http://ugspace.ug.edu.gh LIST OF TABLES Table 3.1: ANOVA Table ................................................................................................... 25 Table 3.2: Critical Region of Paired T-test ......................................................................... 27 Table 3.3: Distribution and its Force of Mortality Function ............................................... 28 Table 4.1A: Abridged Life Table for Males ....................................................................... 36 Table 4.1B: Abridged Life Table for Males........................................................................ 37 Table 4.2A: Female Abridge Life Table ............................................................................. 38 Table 4.2B: Female Abridge Life Table Cont. ................................................................... 39 Table 4.3: Shapiro-Wilk normality test ............................................................................. 42 Table 4.4: One-way ANOVA ............................................................................................. 42 Table 4.5: Independent Paired T-test .................................................................................. 43 Table 4.6: Modelling the force of Mortality for Males ....................................................... 44 Table 4.7: Modelling the force of Mortality for Females ................................................... 45 Table 4.8: Modelling the force of Mortality for Combine Sex ........................................... 46 Table 4.9: Log-rank test ...................................................................................................... 48 Table 4.10A: Multiple Decrement Table for Combined Sex .............................................. 50 Table 4.10B: Multiple Decrement Table for (Combined Sex) ........................................... 51 Table 4.11: Crude Probabilities of Cardiovascular Diseases .............................................. 53 Table 4.12: Crude Probabilities of HIV/AIDS .................................................................... 54 Table 4.13: Crude Probabilities of Lower Respiration Infections ...................................... 56 Table 4.14: Crude Probabilities of Malaria ......................................................................... 57 Table 4.15: Percentage Decrease in Probability of Dying Given that Cardiovascular Diseases is eliminated .................................................................................... 62 Table 4.16: Life Expectancy Gain from Eliminating Cardiovascular Diseases.................. 63 ix University of Ghana http://ugspace.ug.edu.gh Table 4.17: Percentage Decrease in Probability of Dying Given that HIV/AIDS is eliminated ........................................................................................................................ 64 Table 4.18: Percentage Decrease in Probability of Dying Given that HIV/AIDS is eliminated ........................................................................................................................ 65 Table 4.19: Percentage Decrease in Probability of Dying Given that Lower Respiration is eliminated ....................................................................................................... 67 Table 4.20 Life Expectancy Gain from Eliminating Lower Respiration ............................ 68 Table 4.21: Percentage Decrease in Probability of Dying Given that Eliminating Malaria Diseases .......................................................................................................... 69 Table 4.21: Percentage Decrease in Probability of Dying Given that Eliminating Malaria Diseases Cont. ................................................................................................ 70 Table 4.22: Life Expectancy Gain from Eliminating Malaria ............................................ 71 Table 4.22 Life Expectancy Gain from Eliminating Malaria Cont. .................................... 72 x University of Ghana http://ugspace.ug.edu.gh CHAPTER ONE INTRODUCTION 1.1 Background of the study A life table is a useful tool for studying the implications of observed mortality rates in a population (Coale, Demeny, & Vaughan, 2013). Life tables provide the most complete description of mortality and serves as a key indicator of the health and wellbeing of any population. It has many applications in various areas of research where birth, death and illness may take place. There are two forms of the life table in general, we have the cohort life table and the period or current life table. A cohort life table records the actual mortality or death till the last member of the group. A period life table on the other hand, is the most common form. It is based on the mortality rates for a particular year, or averages over a few consecutive years (Denton & Spencer, 2011). The period life table draws out the implications for survivorship and life expectancy of the observed age-specific mortality probabilities of a given period. This is done under the assumption that the probabilities remain constant. Cohort and period life tables may either be complete or abridged. In Complete life tables, the functions or columns of life table are calculated for each year of life. For abridged life table, the ages of the year interval are greater than one, , taking the initial year as 0 to 1 year (Livingstone et al., 2015). Life tables are mostly used by life insurance companies in determining the accurate premium for their clients. The required data needed for construction of a life table are obtained from vital registration and population censuses. In Sub-Saharan Africa, these accurate basic data do not exist due to lack of functioning vital registration systems and incompleteness of coverage and errors in reporting (Mathers, Ma Fat, Inoue, Rao, & Lopez, 2005). As a result of that actuaries and various insurance companies in Sub-Saharan Africa had adopted foreign life tables in setting their policy rates. The actuaries applied the foreign 1 University of Ghana http://ugspace.ug.edu.gh life table to determine the amount of premium to be paid by a person falling in a specific age group. However, this application of the foreign life table on sub-Saharan Africa also have shortcomings. According to Hunter (2001), a life table can also have a multiple mode of decrement (death) instead of only one mode as mention earlier. In the multiple decrements, the causes of the death of the population is not grouped into one but are grouped in many mutually exclusive causes. Every human is continuously exposed to many risks of death such as cancer, heart disease, accidents, stroke, diabetes. Hence there are various risks competing for the life of an individual and death is attributed to one cause. According to Lin, So, and Johnston (2012), competing risks arise in studies in which individuals are subjected to a number of potential failure events and the occurrence of one event might impede the occurrence of other events. Competing risk analyses enable one to determine the impact of a disease in a particular population through the calculation of life expectancy gain when that disease is eliminated. In Ghana, few people had constructed life table. Kpedekpo (1969), constructed a working life for males in Ghana and estimated the average number of years of working life, remaining to those of a given age and also compared the working life tables for Ghana to other industrialized countries such as the United States, England and Wales are made. Katara, Mohammed, Osman, and Faisal (2014), constructed an abridged life table in Ghana and estimated the life expectancy at birth to be 50.32 years for both male and female. Moreover, the World Health Organization (WHO) had been computing a period life table in Ghana from 2000, using the country’s vital statistics. However, there are lots of limitation of the nation is vital statistics since not all deaths and births are reported. Hence the actuaries, the insurance companies and policy makers in Sub-Saharan Africa adjusted the foreign life tables in determining their policy rates. 2 University of Ghana http://ugspace.ug.edu.gh 1.2 Statement of the Problem The construction of a life table requires the gathering of good empirical demographic data, which are inaccuracies in Sub-Saharan African countries like Ghana. Therefore, Sub- Saharan African countries adjust the foreign countries’ life tables to study the longevity of its citizens (Katara et al., 2014), Clearly, these tables would not reflect Sub-Saharan African mortality rates in general and Ghana in particular. Mortality experiences generally vary greatly from place to place due to differences in occupation, place of residence and causes of death. Moreover, accessibility to standard health care may be higher in the developed countries than developing countries like Ghana. This absence of standard health care in the developing countries affects mortality experiences in the country. Death trends, patterns and causes, in developed countries where these tables are developed are different from that of Ghana. For example, according to World Health Organization (2000), malaria is the third leading cause of death in Africa, whiles it is not among the 50 leading causes of death in the world. However, life tables are expected to take into account mortality experiences in the environment in which it is to be applied. Therefore, using foreign life tables to study the mortality rate in Ghana have its own errors. However, assessing the error of applying the foreign life table had not been taken into consideration. Moreover, the impact of the leading cause of death has not been evaluated in Ghana. The theory of competing risk provides convenient methods of analysis of such problems. It is in this light that, this research will assess the errors of applying the various foreign life tables in Ghana and also to evaluate the impact of leading causes of death. 3 University of Ghana http://ugspace.ug.edu.gh 1.3 Research Questions The study is guided by the following research questions;  Are the foreign adjusted life tables significantly different from the computed life table?  What is the expectation of life at birth in Ghana?  What is the force of mortality function of Ghana?  What are the mortality rates by Gender?  What are the life expectancy gain when eliminating the leading cause of death? 1.4 Objectives of the Study The main objective is to assess the error in the application of foreign life table in Ghana and perform competing risk analysis on the leading causes of death. The specific objectives are;  To determine the expectation of life at birth in Ghana.  To assess the error generated from adjusted foreign life tables.  To derive the survival function of Ghana.  To compare the mortality experience by males and females.  To determine the life expectancy gain when eliminating the leading cause of death. 1.5 Hypotheses The study tested two hypotheses namely; 1. 𝐻1 :The foreign life tables are not significantly different from the computed life table 𝐻0: The foreign life tables are significantly different from the computed life table 2. 𝐻0 : The mortality experiences by gender are the same in Ghana. 𝐻1: The mortality experiences by gender are significantly different in Ghana. 4 University of Ghana http://ugspace.ug.edu.gh 1.6 Significance of the study Life expectancy is a key characteristic of human longevity and aids in formulating important policies. However, due to inaccurate data, Ghanaians use foreign life tables to make their policies. The significances of this research are as follows:  Through the assessment of error of applying foreign life table in Ghana, the actuaries and the insurance companies will know the deviations, in calculating premiums and pension liabilities.  The life table that will be constructed will serve as an indicator of the health of the nation which will help the government to make good policies.  This research will serve as literature in research in the areas of health, population, pension and insurance. 1.7 Scope of the study This research constructed a life table from a cohort of 3,260 people from birth to death and highlights the cause of death of each person for the period 1927 to 2016. The life table was compared to other foreign life tables that were used in the country in order to assess the error of applying foreign life tables in Ghana. 1.8 Limitations of the Research Despite the efforts to minimise all limitations, there were certain constraints within which the research was done. The main limitation was on the data integrity; that is the geographical concentration of the sample (Greater Accra Region) which was used to represent the mortality in Ghana. 5 University of Ghana http://ugspace.ug.edu.gh 1.9 Organization of the study The first chapter is the introduction of the study which focuses on the background, problem statement, research questions, objectives and significance of the study. The second chapter provides a review of relevant literature on life table and competing risk. Chapter three discusses the research methods which include the research design, the setting, the target population, and data collections. Chapter four presents the results and chapter five presents the conclusion and recommendations of the study. 6 University of Ghana http://ugspace.ug.edu.gh CHAPTER TWO LITERATURE REVIEW 2.0 Introduction This chapter presents a review of related literature of the study. It also reviews previous studies on life table and competing risks. 2.1 Definition and Concept of life table A life table is a rectangular matrix, showing changes in a standard set of life table functions across ages (Grigoriev et al., 2014). The life table tells us the probability of surviving any particular year of age and the remaining life expectancy for people at different ages. According to Lambert, Dickman, Nelson, and Royston (2010) “The life table is a mathematical model that portrays mortality condition at a particular time among a population and provides a basis for measuring longevity”. Life tables are usually constructed separately for men and for women because of their different in mortality rates (Lambert et al., 2010). There are two types of life table namely; Cohort or Generation Life Table, and Period or Current Life Table. The Cohort or Generation Life Table: It presents the age specific mortality experience of a given birth of a group of persons all born at the same time over many years. Period Life Table presents the age specific mortality conditions pertaining to a given or other short time period (Oeppen & Vaupel, 2002). A life table can have many modes of decrement instead of only one mode of decrement. This is known as multiple modes of decrement. 7 University of Ghana http://ugspace.ug.edu.gh 2.2 Methods of Constructing Life Table Cohort life tables are constructed on the basis of a single cross-sectional time data for a generation. There is also a longitudinal life table method which takes a real cohort of persons that starts life at a specific age interval and follow it throughout life until they all die. Moreover, there are two ways of constructing life table, namely; complete and abridge. A complete life table is constructed on the basis of single years of ages and abridged life table is constructed wherein ages are grouped in 5 or 10 years of interval, taking the initial year as 0 to 1. 2.3 Importance of Life Table 1. Life table is used to project future population on the basis of the present death rate. 2. It helps in determining the average expectation of life based on age specific death rates. 3. The method of constructing a life table can be followed to estimate the cause of specific death rates, male and female death rates, etc. 4. The survival rates in a life table can be used to calculate the net migration rate on the basis of age distribution at 5 or 10 years interval. 5. Life tables can be used to compare population trends at national and international levels. 6. Life tables are applied in the insurance field to calculate the amount of premium that is to be paid by a person falling in a specific age group. 2.4 Definitions and Concepts of Competing risk Gichangi and Vach (2005), defined competing risk as a situation in which an individual is subjected to many distinct groups which are mutually exclusive. Putter, Fiocco, and Geskus (2007) also defined competing risks as a situation in which an individual is exposed to a risk of death from more than one mutually exclusive causes such that the occurrence of one of these will prevent any other event from ever happening. 8 University of Ghana http://ugspace.ug.edu.gh These mutually exclusive causes of failure are considered as competing events and the problems resulting from such data are commonly referred to as competing risk problems. In analysing competing risk, the following probabilities must be clearly defined; Crude probability, Net probability and Partial probability. 2.4.1 Crude probability Ratib, Fleming, Crooks, Walker, and West (2015), defined crude probability as the probability of dying from a specific cause in the present of all other causes acting in the population. This implies that with regards to crude probability, an individual will get to the likelihood of dying from that particular risk even though the person is exposed to many risks of death in the population. 2.4.2 Net probability Pietersen et al. (2014), defined net probability in two ways; The first definition is net probability is the probability that an individual will die if there is only one particular risk acting in the population. Therefore, in this definition, there is an assumption that there is only one cause or risk of death acting in the population. The second definition is that, net probability is the probability that a person will die if a specific risk of death is eliminated in the population. For instance, net probability will enable one to know the probability that an individual at age x will die if there is no HIV/AIDS in the population. 2.4.3 Partial crude probability Partial crude probability as the probability that a person will die from a specific cause when another risk(s) is eliminated from the population (Gerds, Scheike, & Andersen, 2012). 9 University of Ghana http://ugspace.ug.edu.gh Therefore, partial crude probability is the combination of both net probability and crude probability. For example, partial probability answers the probability of dying from HIV/AIDS when tuberculosis is eliminated in the population. 2.5 Previous Studies on Life Table This chapter reviews some articles of life tables constructed in Ghana and other countries. Katara et al. (2014), constructed an abridge life table in the Tamale metropolis. Information regarding to death experiences and number of inhabitants in a house was gathered from 100 selected houses distributed in 10 selected suburbs in the Tamale metropolis. The hypothesis of the study was tested using chi-square test and Wilcoxon signed ranked test. It was found that mortality among males was higher than their female counterparts in most age groups. The study also found out that the expectation of life at birth for Ghana was 50.32 (combined sex). Kpedekpo (1969), constructed a working life for males in Ghana and provides information on the expected average number of years of working life, remaining to those of a given age and many other aspects of working life. People of the working age were categorised into two broad groups, namely economically active and those who were economically inactive during a period of one month preceding the Census midnight. Kpedekpo also compared the working life tables of Ghana to other industrialized countries such as the United States of America (USA), United Kingdom (UK) and Wales. He found out that the working life table for Ghana was significantly different from that of United States, England and Wales. Arias (2014), constructed a complete life table in 2009 for the United States by gender and race in 2009. The causes of deaths as well as the population per gender were obtained. She found out that changes in mortality levels by age and cause of death have a major effect on changes in life expectancy. She found out that the life expectancy at birth increase in 2009 10 University of Ghana http://ugspace.ug.edu.gh was higher than 2008 and the reason was attributed to decreases in mortality from heart disease, cancer, unintentional injuries, stroke, and lower respiratory infections (LRI). She also found out that the life expectancy for females were higher than that of males and also the life expectancy for the white population were higher than that of the black population. Scherbov and Ediev (2011), performed a research on “Significance of life table estimates for small populations”. For one particular life table, they simulated various population size ranges from one thousand, five thousand, ten thousand, twenty-five thousand and one million people. From their analysis they found out that, for all small population less than ten thousand there was an upwards bias in life expectancy estimated. They concluded that higher population leads to lower life expectancies. They observed that to estimate a life expectancy at 60 years with a standard error of about 0.25 years, the population size should be above 100,000. They also revealed that abridged life table calculations can lead to strong biases when the population size are not evenly spread. Denton and Spencer (2011), performed a dynamic extension of the period life table. They addressed the problem of period life table that their life expectancy typically ignores the fact that the expectancies make no allowance for future declines in mortality rates. They found out that the dynamic extension provides an informative supplement to standard life table calculations. Therefore, they concluded that the use of period life expectancies as if they represented the future usage is not appropriate and hence recommended that the use of dynamic life expectancies over period life expectancies. Zeng, Morgan, Wang, Gu, and Yang (2012), discussed the method for calculating healthy life expectancy by considering the dynamic changes of mortality and health of people in United States. The data were obtained from the National Health Interview Survey in the United States. They constructed a cohort life table to estimate and forecast healthy life 11 University of Ghana http://ugspace.ug.edu.gh expectancy. They modelled the dynamic changes of both the mortality and health processes by using the Lee Carter model and constructs their stochastic projections. 2.6 Previous Studies that used multiple Decrement life table and Competing risk This section, reviewed articles that used multiple decrement life table and competing risk in its analyses. According to Haile (2008), the study of competing risks can be traced to a paper in 1958 where Mendenhall and Hader (1958) presented a model for the analysis of failure-time distributions when there are two, or more, types of failure. They illustrate their theory by analysing some data on the failure times of radio transmitter receivers. The failures were classed into two types, those confirmed on arrival at the maintenance centre and those unconfirmed. Due to restrictions on time available for testing, the sample were done through censoring in which the experimenter frequently desires to conclude the life test after a predetermined length of time has elapsed or after a predetermined number of units have failed. Cox (1959), also illustrated and distinguished between a number of models that can be used for competing risk data. Using data that had previously been presented by Mendenhall and Hader (1958), Cox presented and discussed a number of models that could be used to analyse data where the failures were classified into two types. Cox suggested that, in modelling the parametric forms of competing risk data, Gompertz, exponential and Weibull parametric models should be used. Chiang (1970), later studied competing risk in heath prospective by evaluating the impact of Cardiovascular-renal diseases on human longevity in USA. He obtained the data through, US Census of Population 1960. He calculated the net probability for eliminating cardiovascular diseases from the population and also calculated the probability that an 12 University of Ghana http://ugspace.ug.edu.gh individual alive will die without specifying any cause of death. The expectancies were derived from these probabilities and the difference between the two life expectancies as well as the probabilities were computed. Chiang found out that the difference in net probabilities of cardiovascular diseases was lower for females than that of the males and the life expectancy gain for the females were also higher than that of the males. Luptáková and Bilíková (2014), performed a research on actuarial modelling of life insurance using many modes of decrement. The decrement models were death, voluntary withdrawal from the course and expulsion from the course due to poor results. The value of life insurance was computed by following a group of students on a two year post high school course. They first dealt with mortality table as a single-decrement model, where the students only leave the course because of death. They later extend the single decrement model to multiple decrements where students leave the course for reasons other than death such as voluntary withdrawal from the course and expulsion from the course due to poor results. With the assumptions that decrement intensities in both the multiple decrement model and single-decrement models are independent of time and hence are not influenced by the operation of the other decrements. . Lai and Hardy (1999), measured the impact of premature deaths on US population of the working age (15-64 years). They compared two indicators and found out the appropriate one for measuring the impact. The indicators were the life expectancy by elimination of deaths from HIV/AIDS, diseases of the heart and malignant neoplasms and the years of potential life lost due to these causes. They used a monthly vital statistics report and data from National Centre for health Statistics. Multiple decrement life table technique was used to compute the life expectancy due to elimination of death from diseases such as heart diseases, malignant neoplasms and HIV/AIDS. They found out that the life expectancy gains for eliminating cardiovascular diseases were higher in ages above 60 years. They revealed 13 University of Ghana http://ugspace.ug.edu.gh that out that the life expectancy gains for eliminating HIV/AIDS from US population was higher for females than the males. Schwartländer et al. (2011), assessed the human cost of mortality due to intentional violence in over 90 countries. Mortality data collected from international organizations and country statistical offices for the year 2004 were used. All homicides reported to the WHO were included. They employed multiple decrement life table analysis to estimate the potential gains in life expectancy that could be achieved by reducing the risk of intentional injury to deaths to a proposed “regular” level of 1.27 deaths per 100,000 persons. They found out that, Regional potential gains in life expectancy ranges from 0.44 years for men in the Americas to 0.02 years for women in the Western Pacific. They also noticed that violence prevention programs are likely to have the highest overall impact in countries such as Jamaica, Colombia, and Brazil characterized as they had high life expectancies and high levels of homicides. Katzmarzyk, Church, Craig, and Bouchard (2009), obtained a measured that can be used to explain the mortality trends of a population. The two measures they used for analysing the effect of premature deaths were years of potential life lost and potential gains in life expectancy. Since the effect of premature deaths cannot be quantified using general mortality rates they are dominated by chronic diseases among the elderly. They used these measures enable to examine the premature mortality patterns of a population in terms of causes of death. The effects of premature deaths cannot be quantified by using general mortality rates since they are dominated by chronic diseases among the elderly. They obtained their data from Demographic and Health Survey 1998, 2003 and 2008 and the province and district death statistics derived by Turkish Statistical Institute (TURKSTAT) for the years 2000 and 2008. The data were analysed using cause specific mortality analyses, single and multiple decrement life tables. The results of the potential gains in life expectancy 14 University of Ghana http://ugspace.ug.edu.gh analyses were represented by complete and partial elimination of causes of death. Years of potential life lost results were estimated as lifetime years of potential life lost. From their findings, they suggested that the overall effect of premature mortality shows a decreasing trend during the period 2000 to 2008 in Turkey. Cardiovascular diseases and cancers are the leading causes of death affecting premature mortality. They observed that the impact of cancers and injuries on premature mortality are greater for the younger age groups in Turkey. Rosamond et al. (2008) in his articles “Bounds in Competing Risks Models and the War on Cancer” derived a framework to estimate competing risk models with interval outcome data and discrete explanatory variables. They released that competing risks models used mortality from multiple causes and it was difficult to get a clear picture of the trends in cancer. They estimated changes in cancer and cardiovascular mortality from 1970 to 2000. They found out that find that males die from cardiovascular disease than the females. Wolbers et al. (2014) studied the objectives and approaches of competing risks analyses. They gave a non-technical overview of competing risks concepts for descriptive and regression analyses. For descriptive statistics, they concluded that the cumulative incidence function (CIF) was the most important tool. For regression analyses, they suggested regression models for the cumulative incidence function and the cause-specific hazard function as the appropriate models. They stressed the importance of choosing the various statistical methods that were appropriate, if competing risks were present. The data used for their study were obtained from the implantable cardioverter-defibrillator registry of the Department of Cardiology, University Hospital Basel, Switzerland. The study included 442 subjects with dilated cardiomyopathy with an implantable cardioverter-defibrillator implanted for primary or secondary prevention, a median age of 63.4 years, and a median follow-up duration of 3.3 years. The study quantified the benefit of implantable 15 University of Ghana http://ugspace.ug.edu.gh cardioverter-defibrillator implantation in an unselected routine-care population by analysing the time from implantable cardioverter-defibrillator implantation to the first appropriate implantable cardioverter-defibrillator therapy or death without prior appropriate implantable cardioverter-defibrillator therapy. Implantable cardioverter-defibrillator therapy that failed to save the patient’s life at the time of the arrhythmia was classified as death, not as appropriate implantable cardioverter-defibrillator therapy. They found out that for the composite endpoint, the standard survival analysis without competing risks was appropriate, and the effect estimates of covariates on the hazard function and the CIF are identical. They also found that both advanced age and an implantable cardioverter-defibrillator implantation for secondary prevention show a highly significant association with an increased rate (and risk) of the combined outcome of prior implantable cardioverter-defibrillator therapy or death. Albertsen, Hanley, Gleason, and Barry (1998) performed a follow up study with consideration of competing risks. The data was collected by the Tumor Registry of the California State Department of Public Health which consists of 5982 patients admitted to certain California hospitals and clinics between January 1, 1982, and December 31, 1994, with a diagnosis of cancer of the cervix uteri. The date of entrance to follow-up for each patient is the date of hospital admission. The survival experience of the patients grouped according to their withdrawal status was constructed as well as the survival experience of non-withdrawals. The deaths were further divided by cause where 1105 deaths were due to cancer of the cervix uteri and 182 deaths were from all other causes. The survival status of the 1954 admissions was determined at the close of the study, as it is for patients’ due for withdrawal in any interval. In this study, 576 patients withdrew alive in the first interval, and 89 patients died before the closing date. The crude and net probabilities of cancer of the cervix uteri and other causes were estimated. They found out that since only two risks were 16 University of Ghana http://ugspace.ug.edu.gh studied, the probability of dying from cancer of the cervix uteri was equal to the net probability of death when cancer of the cervix uteri is eliminated as a risk of death from the population. For each age interval the estimated net probability was always greater than the corresponding crude probability. Grude (2011), studied risk factors for breast, uterine and ovarian cancer. Theory of competing risks was used to identify possible risk factors for breast, uterine and ovarian cancer. This was done by performing regression on the cause specific hazard functions, the sub distribution hazard functions and two approximate methods. Cox regression was used for a complete analysis of the medical data. By following 61457 women over approximately 50 years, he noticed 3407 cases of breast cancer, 934 of uterine cancer and 843 of ovarian cancer. The data used in the analysis was selected from a screening program organized by the Norwegian Cancer Society for early diagnosis of breast cancer. His study found out that each additional birth decreases the risk of getting breast, uterine or ovarian cancer by 10 %, 10 % and 16%, respectively. He also found out that age at first birth, the risk of getting breast or uterine cancer whiles age at last birth affect the risk of uterine and breast cancer because early last birth is protective against breast cancer compared to late last birth. Age at menarche affects the risk of getting uterine and breast cancer as each year increase, in age at menarche decreases the risk of uterine and breast cancer with approximately 12% and 5%, respectively. But ovarian cancer risk was not affected by age at menarche. It was seen that obesity affects the risk of getting breast, uterine and ovarian cancer for postmenopausal women. Oeppen and Vaupel (2002), did a research on competing risks analysis of end stage renal disease and mortality among adults with diabetes in Canada. The objective was to determine whether there are significant disparities in the risk of end stage renal disease and mortality without end stage renal disease between diabetic First Nations and other Saskatchewan 17 University of Ghana http://ugspace.ug.edu.gh people in Canada. The data was drawn from the Saskatchewan Ministry of Health administrative databases from 1980 to 2000. The competing risks survival analysis that was used were a Cox cause-specific model, Weibull proportional hazards (PH) model and piece- wise exponential PH hazards model. System Dynamics modelling (SDM) and agent-based modelling (ABM) methods were also used to build dynamic models of diabetic patients’ progression to end stage renal disease. There was a total of 90,429 diabetic people in the study cohort, from 1980 to 2005. Among them, 8,254 (9%) of them were First Nations people. After adjusting for diabetes diagnosis age, sex, interaction between age and sex and interaction between age and ethnicity, Oeppen and Vaupel (2002) found out that First Nations had higher risk of death than other Saskatchewan given the same sex and diabetes diagnosis age (younger than 81 years old). Using the same hazard rate estimations from competing risks survival analysis, the agent-based modelling model demonstrated a better match between historical data and model predicted data compared to the System Dynamics model. They concluded that a much younger age of diabetes diagnosis among First Nations compared to other Saskatchewan likely contributes to higher rates of end stage renal disease because of a differential mortality effect First Nations with diabetes are more likely to live long enough to develop end stage renal disease. Lin et al. (2012), performed a quantitative evaluation of competing risks in occupational studies. They noticed that adjustment for competing risks allowed more meaningful comparisons of cause-specific mortality of two populations, especially if dying from all other causes is significantly different between the two populations. In their study, they identified a method for adjusting competing causes of death in the calculation of relative risk. This method identifies three factors, namely; magnitude of the overall mortality risk of the study population, differential risk or adjustment factor for all other causes between two populations and age interval used in mortality calculation. Hence, the impact of competing 18 University of Ghana http://ugspace.ug.edu.gh risks is increased if the mortality risk of the study population is high. They used a refinery cohort data for which there was a certain age groups unadjusted for competing risks. They found out that the relative risk for groups unadjusted for competing risks was overestimated by 9%. They concluded that the impact of competing risks in their study was relatively small. Higgins, Hoffman, and Dworkin (2010), did competing risks analysis in epidemiology by reviewing the concepts of rate and risk. They also introduced the analogous concepts of rate and risk in the context of competing risks. They used data from the European Group for Blood and Marrow Transplantation (EBMT). The data consist of all chronic myeloid leukemia (CML) patients, having received an allogeneic stem cell transplantation from an Human Leukocyte Antigen (HLA) during the years 2000–2008. They did a follow up of 8.5 years of 3,982 patients who were Philadelphia chromosome positive, transplanted with bone marrow or peripheral blood, and were above 18 years of age. All-cause mortality analyses were performed and they noticed they are equivalent due to their one-to-one correspondence. They realized that in the presence of competing risk, when rates are now cause-specific hazards and risks are cumulative incidences a one-to-one correspondence between a single rate and the corresponding risk no longer exists. Therefore, any given cumulative incidence depends on all cause-specific hazards and vice versa. Also, covariates may affect the cause 𝑖 specific hazard and the cause 𝑖 cumulative incidence differently. They found out that the Kaplan–Meier estimator provides a biased estimate of the cumulative incidence in the presence of competing risks. In fitting a regression model, they found out that, Cox regression models for cause-specific hazards are easy to fit and it provides parameter estimates which possess simple rate ratio interpretations. 19 University of Ghana http://ugspace.ug.edu.gh 2.7 Summary of Literature Reviewed From the literature reviewed, there are limited research on the impact of cardiovascular disease and other killer disease such as HIV/AIDS, Malaria on human longevity in Ghana. Moreover, there are few literatures on life table in Ghana. The literatures above give in- depth knowledge on how, the researcher could construct a life in Ghana and also analyses the impact of the major causes of death in the country. 20 University of Ghana http://ugspace.ug.edu.gh CHAPTER THREE METHODOLOGY 3.0 Introduction This chapter sets out the methodology that was used to achieve the study. The succeeding section presents the research approach and highlights on the setting, population of the study, sample techniques and size, data collection instrument, as well as methods of data analysis. 3.1 Setting The study was done at University of Ghana Hospital. The hospital is located at Legon in La- Nkwantanang district in Greater Accra Region. The hospital is often attended by students, staff of the university and people living around the institution. The hospital was established in 1957 and is owned by the University of Ghana. It was under the charge of Dr. A.B. Boyd (a Scottish Doctor) and was assisted by one nursing staff. The hospital started as a clinic, sharing all facilities in common with the Achimota Hospital. In 1959, five (5) heath personnel consisting of a doctor and four nurses moved from Achimota to start work at the then University College Hospital. The facilities of the hospital grew over time to include Maternity Ward and Staff Quarters. The hospital is a quasi-government hospital with a bed capacity of 130 comprising of General Wards, Maternity Wing, Casualty and Emergency Ward, Pediatric Unit, Dental Unit and Operating Theatre. The mission of the hospital is to enhance the health status of all employees and students of the University of Ghana by providing them with world class Medical Services of the highest quality. Their mission also includes enhancing the health status of the communities in University of Ghana’s immediate environs by providing affordable client focused quality health services. The hospital has established a Primary Health Care outreach programme aimed at teaching and advising students, pregnant women, nursing mothers and the general public about personal hygiene, 21 University of Ghana http://ugspace.ug.edu.gh good diet, child care, including immunization against childhood communicable diseases, family planning and school health services. The main referral point for the hospital is the Korle-Bu Teaching Hospital and 37 Military Hospital. The hospital has recently introduced specialist consultancy services. 3.2 Study population According to Rothman, Greenland, and Lash (2008) ,a population is defined as a group of individuals, persons, objects, or items from which samples are obtained for measurement. Study population is a population from which the sample actually was drawn and about which a conclusion can be made (Degu & Yigzaw, 2006). The study is targeting the population of Ghana and hence conclusion made from this research will be generalized to the entire population of Ghana. 3.3 The Data Secondary data were used for the study. The data were extracted from the administrative records unit of the hospital. The data include the causes of death by gender and age from the year the hospital started to 2015. The data were then converted into a cohort of 3250 people from birth to death where the researcher assumed that the individuals selected were born in the same year. The causes of death were grouped into five, namely; death due to cardiovascular diseases, HIV/AIDs, lower respiratory infections, malaria and other causes of death.’ 3.4 Data Analysis To ensure accuracy in the data processing, data editing and clearing of the data were done before analysing the data. The data were coded in order to make possible inputting into the data processor. The codes were transformed into units to facilitate their description and 22 University of Ghana http://ugspace.ug.edu.gh analyses. Diagrammatic presentation by means of tables and graphs were done. Although there will be several electronic means of analysing data, STATA and R were used for the analysis. 3.5 Construction of Life Table In a cohort life table, given the number of people from the cohort 𝑙𝑜 , 𝑙𝑗 , is the number of people at the beginning of exact age 𝑗 and 𝑑𝑗, the number of death at each age. The following represent the columns that can be found in life tables: i. ?̂?𝑗 , Proportion dying (𝑗, 𝑗 + 1): The probability that an individual alive at age will die during the age interval (𝑗, 𝑗 + 1) is calculated directly as 𝑑𝑗 ?̂?𝑗 = … … … . (3.1) 𝑙𝑗 ii. ?̂?𝑗 , Proportion dying (𝑗, 𝑗 + 1): The probability that an individual alive at age will survive during the age interval (𝑗, 𝑗 + 1) is calculated directly as 𝑙𝑗+1 ?̂?𝑗 = … … … . (3.2) 𝑙𝑗 iii. 𝑎𝑥, Average fraction of the last year of life for age 𝑥 . Each of the people who die during the interval (𝑥, 𝑥 + 1) has lived 𝑥 complete years plus some fraction of the year (𝑥, 𝑥 + 1). Usually at the beginning of the year it is 0.10 and after 5 years it is taken as 0.5. iv. 𝐿𝑥 The number of years lived in the interval (𝑥, 𝑥 + 1). Each member of the cohort who survives the year (𝑥, 𝑥 + 1) contributes one year to 𝐿𝑥, while each member who dies during the year(𝑥, 𝑥 + 1) contributes to an average fraction 𝑎𝑥. Hence the formula is 𝐿𝑥 = 𝑙𝑥 − (1 − 𝑎𝑥)𝑑𝑥 … … … (3.3) 23 University of Ghana http://ugspace.ug.edu.gh v. 𝑇𝑥 The total number of people lived beyond age 𝑥. This is equal to the sum of the number of years lived in each interval. This implies 𝑇𝑥 = 𝐿𝑥 + 𝐿𝑥+1 + ⋯ + 𝐿𝑛 … … (3.4) vi. ?̂?𝑥 Observed expectation of life at age x. This is the average number of years yet to be lived by an individual at exact age x. 𝑇𝑥 ?̂?𝑥 = … … . . (3.5) 𝑙𝑥 3.6 Error Assessment 3.6.1 Absolute Error Analyses Absolute error measures how a measurement deviate from the true value. It is also an indication of the uncertainty in a measurement. Let 𝑒𝑎𝑏 be absolute error 𝑒𝑎𝑏 = |𝑇𝑟𝑢𝑒 𝑣𝑎𝑙𝑢𝑒 − 𝐴𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒 𝑣𝑎𝑙𝑢𝑒| … … . (3.6) = |𝑋 − 𝑋′| … … . (3.7) 3.6.2 Univariate Analysis of Variance (ANOVA) The univariate Analysis of Variance (ANOVA) was done test whether the absolute error obtained from the three foreign life tables differ. Suppose the absolute error of observations are as follows; Absolute error from UK life table: 𝑋11,𝑋12, … , 𝑋1𝑛 Absolute error from USA life table:𝑋21,𝑋22, … , 𝑋2𝑛 Absolute error from South Africa (SA) life table:𝑋21,𝑋22, … , 𝑋2𝑛 µ1 = Absolute error from UK life table µ2 = Absolute error from USA life table 24 University of Ghana http://ugspace.ug.edu.gh µ3 = Absolute error from SA life table HYPOTHESES 𝐻0: µ1 = µ2 = µ3 𝐻1: µ1 ≠ µ2 ≠ µ3 Table 3.1: ANOVA Table SOV Df SS MS F 𝑆𝑆𝑅 Regression 𝑝 − 1 SSR 𝑀𝑆𝑅 = 𝑃 − 1 𝑆𝑆𝐸 𝑀𝑆𝑅 Error 𝑛 − 𝑝 SSE 𝑀𝑆𝐸 = 𝑃 − 1 𝑀𝑆𝐸 Total 𝑛 − 1 SST 𝑛 = 101 𝑝 = 3 𝑆𝑂𝑉 = 𝑆𝑜𝑢𝑟𝑐𝑒 𝑜𝑓 𝑉𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 𝑑𝑓 = 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 𝑜𝑓 𝑓𝑟𝑒𝑒𝑑𝑜𝑚 𝑆𝑆 = 𝑆𝑢𝑚 𝑜𝑓 𝑆𝑞𝑢𝑎𝑟𝑒𝑠 𝑀𝑆 = 𝑀𝑒𝑎𝑛 𝑆𝑢𝑚 𝑜𝑓 𝑆𝑞𝑢𝑎𝑟𝑒𝑠 3.6.3 Paired t-test Pared t-test was constructed to determine whether the minimum error obtained from the three foreign life tables significantly different from zero. This will tell us whether there is a significant need to construct a life table for Ghana or we can rely on of the foreign life table that exhibit the minimum error. 25 University of Ghana http://ugspace.ug.edu.gh Specifying the model and simple testing for µ𝑫 Suppose the life expectancy observations are as follows; The life expectancy of the study life table: 𝑋11,𝑋12, … , 𝑋1𝑛 The foreign life table with the minimum error:𝑋21,𝑋22, … , 𝑋2𝑛 Since the data is paired, we can study the differences 𝐷𝑖 = 𝑋1𝑖 − 𝑋2𝑖 , 𝑖 = 1, 2, … … . . , 𝑛 ?̅?−µ𝐷 Assume that 𝐷 is normally distributed, 𝑡 = ~ 𝑡 … … … (3.8) 𝑠𝐷/√𝑛 𝑛−1 Where µ𝐷 = µ1 − µ2 µ1 = The life expectancy of the study life table µ2 = The foreign life table with the minimum error ∑ 𝐷 Sample mean = ?̅? = 𝑖 where D is the difference between the study life table and the 𝑛 foreign life table with the minimum error 𝑁 = 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑟𝑡𝑖𝑜𝑛 (𝑖𝑛 𝑡ℎ𝑖𝑠 𝑐𝑎𝑠𝑒 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑒𝑛𝑡 𝑤𝑎𝑠 101) Sample standard deviation 2 ∑(𝐷−?̅?)2 ∑ 𝐷2 (∑ 𝐷) − 𝑠 = √ = √ 𝑛𝐷 … … . . (3.9) 𝑛−1 𝑛−1 HYPOTHESES 𝐻0: µ𝐷 = 0 , µ𝐷 = µ1 − µ2 = 0 𝐻1: µ𝐷 ≠ 0(𝑇𝑤𝑜 − 𝑡𝑎𝑖𝑙𝑒𝑑) 𝐻1: µ𝐷˂0 (𝑙𝑒𝑓𝑡 − 𝑡𝑎𝑖𝑙𝑒𝑑), 26 University of Ghana http://ugspace.ug.edu.gh 𝐻1: µ𝐷˃0 (𝑟𝑖𝑔ℎ𝑡 − 𝑡𝑎𝑖𝑙𝑒𝑑) TEST STATISTIC ?̅? T = ~ 𝑇𝑛−1 … … . . (3.10) 𝑠𝐷/√𝑛 CRITICAL REGION The critical value is obtained from t-distribution table with n-1 degrees of freedom. Table 3.2: Critical Region of Paired T-test LEFT-TAILED RIGHT-TAILED TWO-TAILED Critical Value(s) −𝑡𝛼,𝑛−1 𝑡𝛼,𝑛−1 −𝑡𝑎 𝑎𝑛𝑑 𝑡𝑎 ,𝑛−1 ,𝑛−1 2 2 Critical Region(s) T ≤ −𝑡𝛼,𝑛−1 T ≥ 𝑡𝛼,𝑛−1 T ≤ −𝑡𝛼,𝑛−1 𝑜𝑟 T ≥ 𝑡𝛼,𝑛−1 3.7 Force of Mortality Model According to Lee and Wang (2003), force of mortality measures the instantaneous rate of mortality of at a specific age. Supposed 𝑙𝑥 is the number of people living at aged 𝑥. Then the force of mortality which is the instantaneous rate of mortality at age 𝑥 is given as 1 𝑑 𝜇𝑥 = − 𝑙𝑥 … … … (3.11) 𝑙𝑥 𝑑𝑥 𝑑 𝜇𝑥 = − 𝑙𝑜𝑔𝑙𝑥 … … … … (3.12) 𝑑𝑥 27 University of Ghana http://ugspace.ug.edu.gh There are many force of mortality models, this study used 4 survival models and the one with the least R-square was taken. The survival models considered were; Weibull model, Gompertz model and logistic and log-logistic. Table 3.3: Distribution and its Force of Mortality Function Force of Mortality Function Distribution 𝜇𝑥 Gompertz 𝛽𝑒𝜃𝑡 𝛽 𝑥 𝛽−1 Weibull ( ) 𝜃 𝜃 𝛽𝑒𝜃𝑡 Logistic 𝛾𝛽 [1 + (𝑒𝜃𝑡−1)] 𝜃 𝛽𝜃𝑥𝛽−1 Log-logistic (1 + 𝜃𝑥𝛽) 3.8 Computation of competing risks Suppose that five risks of death are acting mutually exclusively on the population of patient in University of Ghana Hospital. Let the risk of cardiovascular diseases, HIV/AIDs, lower respiratory infections, malaria and other risk of death be denoted as 𝑅1, 𝑅2, 𝑅3, 𝑅4 𝑎𝑛𝑑 𝑅5 respectively. Causes of Deaths Cardiovascular diseases HIV/AIDs Risk of Death Lower respiratory infections malaria Other risk of death Figure 3.1 Competing risk Model: Death Attributed to 5 Mutually Exclusive Causes 28 University of Ghana http://ugspace.ug.edu.gh For each risk, 𝑅𝛾 there is a corresponding force of mortality, which is presented by 𝜇(𝜏; 𝛾). Therefore ( ; )0()  P{anindividual aliveat t will dieintheinterval (t, t ) fromcause R }.....(3.13) 𝑤ℎ𝑒𝑟𝑒 𝛾 = 1,2,3, … ,5 Let the sum for each force of mortality of a risk be denoted by 𝜇(𝜏). 6 ∑ 𝜇(𝜏; 𝛾) = 𝜇(𝑡) … … … (3.14) 𝛾=1 and each risk 𝑅𝛾 the ratio 𝜇(𝜏; 𝛾)⁄𝜇(𝑡) depends only on (𝑥𝑗 , 𝑥𝑗+1) and not on 𝑡. The probability that an individual alive at 𝑥𝑗 will live in the interval (𝑥𝑗 , 𝑥𝑗+1) without specify the cause of death is given as 𝑥𝑗+1 𝑝𝑗 = 𝑒𝑥𝑝 {− ∫ 𝜇(𝑡)𝑑𝑡}, … … … (3.15) 𝑥𝑗 and the probability of dying within the interval (𝑥𝑗 , 𝑥𝑗+1) is given as 𝑥𝑗+1 𝑞𝑗 = 1 − 𝑒𝑥𝑝 {− ∫ 𝜇(𝑡)𝑑𝑡} … … … (3.16) 𝑥𝑗 3.8.1 Crude Probability Crude Probability is the probability of dying from a specific risk (𝑅𝛾) in the present of all other risks acting in the population. Let 𝑄𝑗𝛾 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑐𝑟𝑢𝑑𝑒 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 , to derive the crude probability let us consider the point 𝑡 within the interval (𝑥𝑗 , 𝑥𝑗+1) . That is 𝑥𝑗 < 𝑡 ≤ 𝑥𝑗+1 29 University of Ghana http://ugspace.ug.edu.gh 𝑥𝑗+1 𝑡 𝑄𝑗𝛾 = ∫ 𝑒𝑥𝑝 {− ∫ 𝜇(𝜏)𝑑𝜏} 𝜇(𝑡; 𝛾)𝑑𝑡 … … … (3.17) 𝑥𝑗 𝑥𝑗 𝑑𝑗𝛾 𝑄𝑗𝛾 = … … … (3.18) 𝑙𝑗 Variance of Crude Probability The variance of the crude probability can be calculated as 𝑄𝑗𝛾(1 − 𝑄𝑗𝛾) 𝑉𝑎𝑟(𝑄𝑗𝛾) = … … … (3.19) 𝑙0𝑝0𝑗 𝑙0 𝑖𝑠 𝑡ℎ𝑒 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑠𝑖𝑧𝑒 (𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑜𝑝𝑙𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑐𝑜ℎ𝑜𝑟𝑡) 𝑝0𝑗 = 𝑝0𝑝1 … 𝑝𝑗−1 𝑄𝑗𝛾 = 𝑐𝑟𝑢𝑑𝑒 𝑝𝑟𝑜𝑏𝑎𝑏𝑙𝑖𝑡𝑦 𝑎𝑛𝑑 𝛾 = 1,2,3,4,5 3.8.2 Net probability The net probability of death in the interval (𝑥𝑗 , 𝑥𝑗+1) when 𝑅𝛾 is the only risk acting in the population is denoted as 𝑞𝑗𝛾 and defined as 𝑥𝑗+1 𝑞𝑗𝛾 = 1 − 𝑒𝑥𝑝 {− ∫ 𝜇(𝑡; 𝛾)𝑑𝑡} … … … (3.20) 𝑥𝑗 𝑄𝑗𝛾 ( ⁄𝑞 )𝑗 𝑞𝑗𝛾 = 1 − 𝑝 … … … (3.21) 𝑗 The net probability of death when a specific risk 𝑅𝛾 is eliminated is denoted by 𝑞𝑗.𝛾 and the formula is calculated as 𝑥𝑗+1 𝑞𝑗.𝛾 = 1 − 𝑒𝑥𝑝 {− ∫ 𝜇(𝑡) − 𝜇(𝑡; 𝛾)𝑑𝑡} … … … (3.22) 𝑥𝑗 30 University of Ghana http://ugspace.ug.edu.gh 𝜇(𝑡)−𝜇(𝑡;𝛾) 𝜇(𝑡) 𝑞𝑗.𝛾 = 1 − 𝑝 … … … (3.23) 𝑗 (𝑞 −𝑄 )⁄𝑞 𝑞𝑗.𝛾 = 1 − 𝑝 𝑗 𝑗𝛾 𝑗 … … … (3.24) 𝑗 Variance of Net Probability The variance of the net probability of death when 𝑅𝛾 is the only risk acting in the population is given as 2 (1 − 𝑞𝑗𝛾) 𝑣𝑎𝑟( 𝑞𝑗𝛾) = {𝑝𝑗𝑙𝑜𝑔(1 − 𝑞𝑗𝛾) 𝑙𝑜𝑔(1 − 𝑞 2 𝑗.𝛾) + 𝑄𝑗𝛾} … … … (3.25) 𝑙0𝑝0𝑗𝑝𝑗𝑞𝑗 The variance of the net probability of death if 𝑅𝛾 is eliminated from the population is given as 2 (1 − 𝑞𝑗.𝛾) 𝑣𝑎𝑟( 𝑞𝑗.𝛾) = {𝑝𝑗𝑙𝑜𝑔(1 − 𝑞𝑗𝛾) 𝑙𝑜𝑔(1 − 𝑞𝑙 𝑝 𝑝 𝑞 𝑗.𝛾 ) 0 0𝑗 𝑗 𝑗 2 + (𝑞𝑗 − 𝑄𝑗𝛾) } … … … (3.26) 3.8.3 Partial Crude Probability Let an assumed specific risk 𝑅1 is eliminated from the population, in the presence of all other risks. Now the probability of death from a specific cause when a risk 𝑅1 is eliminated from the population within the interval (𝑥𝑗 , 𝑥𝑗+1) is denoted by 𝑄𝑗𝛾.1. 𝑥𝑗+1 𝑡 𝑄𝑗𝛾.1 = ∫ 𝑒𝑥𝑝 {− ∫ (𝜇(𝜏) − 𝜇(𝜏; 1))𝑑𝜏} 𝜇(𝑡; 𝛾)𝑑𝑡 … … … (3.27) 𝑥𝑗 𝑥𝑗 𝜇(𝑡; 𝛾) 𝑥𝑗+1 𝑡 𝑄𝑗𝛾.1 = ∫ 𝑒𝑥𝑝 {− ∫ (𝜇(𝜏) − 𝜇(𝜏; 1))𝑑𝜏} [𝜇(𝑡)𝜇(𝜏) − 𝜇(𝜏; 1) 𝑥𝑗 𝑥𝑗 − 𝜇(𝑡; 1)]𝑑𝑡 … … (3.28) 31 University of Ghana http://ugspace.ug.edu.gh 𝑄 𝑥𝑗+1𝑗𝛾 𝑄𝑗𝛾.1 = = (1 − 𝑒𝑥𝑝 {− ∫ [𝜇(𝑡) − 𝜇(𝑡; 1)]𝑑𝑡}) … … … (3.29) 𝑞𝑗 − 𝑄𝑗1 𝑥𝑗 𝑄𝑗𝛾 𝑄𝑗𝛾.1 = 𝑞𝑗.1 … … … (3.30) 𝑞𝑗 − 𝑄𝑗1 𝑄𝑗𝛾 (𝑞𝑗−𝑄𝑗1)⁄𝑞𝑄𝑗𝛾.1 = (1 − 𝑝 𝑗) 𝛾 = 2,3, … ,5 … … … (3.31) 𝑞 − 𝑄 𝑗𝑗 𝑗1 Now if 𝑅1 𝑎𝑛𝑑 𝑅2 are eliminated, the partial probability of death from a specific cause when risks 𝑅1 𝑎𝑛𝑑 𝑅2 are eliminated from the population within the interval (𝑥𝑗 , 𝑥𝑗+1) is denoted by 𝑄𝑗𝛾.12. 𝑥𝑗+1 𝑡 𝑄𝑗𝛾.1 = ∫ 𝑒𝑥𝑝 {− ∫ (𝜇(𝜏) − 𝜇(𝜏; 1) 𝑥𝑗 𝑥𝑗 − 𝜇(𝜏; 2))𝑑𝜏} 𝜇(𝑡; 𝛾)𝑑𝑡 … … … (3.32) 𝜇(𝑡; 𝛾) 𝑥𝑗+1 𝑡 𝑄𝑗𝛾.12 = ∫ 𝑒𝑥𝑝 {− ∫ (𝜇(𝜏) − 𝜇(𝜏; 1) − 𝜇(𝜏; 2))𝑑𝜏} [𝜇(𝑡)𝜇(𝜏) − 𝜇(𝜏; 1) − 𝜇(𝜏; 2) 𝑥𝑗 𝑥𝑗 − 𝜇(𝑡; 1)]𝑑𝑡 … … … (3.33) 𝑄 𝑥𝑗+1𝑗𝛾 𝑄𝑗𝛾.12 = = (1 − 𝑒𝑥𝑝 {− ∫ [𝜇(𝑡) − 𝜇(𝑡; 1) − 𝜇(𝜏; 2)]𝑑𝑡}) … … (3.34) 𝑞𝑗 − 𝑄𝑗1 − 𝑄𝑗2 𝑥𝑗 𝑄𝑗𝛾 𝑄𝑗𝛾.12 = (1𝑞𝑗 − 𝑄𝑗1 − 𝑄𝑗2 (𝑞𝑗−𝑄𝑗1−𝑄𝑗2)⁄𝑞− 𝑝 𝑗) … … … (3.35) 𝑗 32 University of Ghana http://ugspace.ug.edu.gh Variance of Partial Probability The variance of the partial probability of death from a specific cause when a risk 𝑅1 is eliminated from the population within the interval (𝑥𝑗 , 𝑥𝑗+1) is given by 𝑞𝑗 − 𝑄𝑗1 − 𝑄𝑗𝛾 𝑉𝑎𝑟(𝑄 2𝑗𝛾.1) = 𝑄𝑗𝛾.1 𝑙0𝑝0𝑗(𝑞𝑗 − 𝑄𝑗1)𝑄𝑗𝛾 2 2 (𝑄𝑗𝛾.1(𝑞𝑗 − 𝑄𝑗1) − 𝑄𝑗𝛾) 𝑙𝑜𝑔𝑝𝑗 + [(𝑞𝑗 − 𝑄𝑗1) + 𝑄𝑗1𝑝𝑗 ( ) ] … (3.36) 𝑙0𝑝0𝑗(𝑞𝑗 − 𝑄𝑗1) 𝑞𝑗 The variance of the partial probability of death from a specific cause when the risks 𝑅1 𝑎𝑛𝑑 𝑅2 are eliminated from the population within the interval (𝑥𝑗 , 𝑥𝑗+1) is given by 𝑞𝑗 − 𝑄𝑗1 − 𝑄𝑗2 − 𝑄𝑗𝛾 𝑉𝑎𝑟(𝑄𝑗𝛾.12) = 𝑄 2 𝑗𝛾.12 𝑙0𝑝0𝑗(𝑞𝑗 − 𝑄𝑗1 − 𝑄𝑗2)𝑄𝑗𝛾 2 (𝑄𝑗𝛾.12(𝑞𝑗 − 𝑄𝑗1 − 𝑄𝑗2) − 𝑄𝑗𝛾) + [(𝑞𝑗 − 𝑄𝑗1 − 𝑄𝑗2) 𝑙0𝑝0𝑗(𝑞𝑗 − 𝑄𝑗1 − 𝑄𝑗2) 2 𝑙𝑜𝑔𝑝𝑗 + 𝑄𝑗12𝑝𝑗 ( ) ] … … … (3.37) 𝑞𝑗 3.8.4 Assumptions of Competing Risks 1. We assume that the set of 𝑟𝑖𝑠𝑘𝑠 𝛾 are mutually exclusive and exhaustive. 2. At any particular death the cause of death is only associated with a single cause 33 University of Ghana http://ugspace.ug.edu.gh 3.9 The Log Rank Test Let e be the expected cell counts for male and e be the expected cell counts for female 1 j 1 j  n  j e =     1j  × n +n   m +m  ........(3.38)  1j 2j  1j 2j      n  j Where   represent the proportional in the risk set and   n +n   1j 2j    m +m   represent the death for both gender, If  1j 2j  O E  m e  i i   ij ij  then the Log-Rank can be computed as j1  2 O E2 2  .........(3.39) var(O E ) 2 2 3.10 Summary of Methodology The researcher used absolute error analyses, univariate analysis of variance and paired t-test to assess the error of applying the foreign life table in Ghana. In order to derive the force of mortality function, a complete single decrement life table was constructed. The mortality rate was used to fit existing survival models such as Gomertz model, Weibull model, logistic and log-logistic and the most appropriate model that fit with the highest R-square was taken. Moreover, crude and net probabilities of the leading cause of death was computed and with the help of multiple decrement life table, the various life expectancy gains from the elimination of the leading cause of death was estimated. 34 University of Ghana http://ugspace.ug.edu.gh CHAPTER FOUR DATA ANALYSIS AND DISCUSSION 4.0 Introduction This chapter presents the analysis and discussion pertaining to error assessment in the applications of Foreign Life Tables in Ghana and competing risk analysis. The analysis includes construction of Abridge Life Table, Force of mortality function and competing risks analysis. A CONSTRUCTION OF LIFE TABLES The construction of life table was done in complete and abridge method. The complete life table can be seen in Appendix I and the abridge life table are presented in Table 4.1 to 4.4. 4.1 Abridge Life Table An abridge life table was constructed for male and female. This was done in order to determine the probability of dying and the expected life for each age interval and gender. The results were presented in Table 4.1 to 4.3. 4.1.1 Abridged Life Table for Males Table 4.1 represent the abridge life able for males, the life expectancy ( e j ) was computed with reference to equation (3.5). For example, in calculating for the life expectancy for ages between 0 and 1, 𝑇𝑗⁄𝑙𝑗 = 9360.54⁄1890 = 49.53. It was revealed that the life expectancy at birth was 49.53. The highest life expectancy record was 51.7417 and it was in the age group of 5 to 10 years. Then the life expectancy begins to decrease as the age group increase. According to Oeppen and Vaupel (2002) life expectancy had a positive relation with age. They explained the immune system of humans becomes weak as ones’ years 35 University of Ghana http://ugspace.ug.edu.gh increases hence are not able to fight diseases. From Katara et al. (2014) , also constructed a local life able in Tamale and had the life expectancy at birth to be for male was 45 years. With regards to probability of dying ( q j ), at the initial stage or at birth q j was 0.0312 signifying that the probability that a male child at birth will not survive his/her next birthday was 0.0312. The probability of dying at the age of 100 and over interval is 1. This means that eventually every human being will die (a sure event). Table 4.1A: Abridged Life Table for Males d j Tj [x j  x j1) q j  l j d j L j Tj e j  l j l j [0-1) 0.0312 1890 59 1836.9 93605.39 49.5267 [1-5) 0.1038 1831 190 6860.4 91768.49 50.1193 [5-10) 0.036 1641 59 8057.5 84908.09 51.7417 [10-15) 0.0076 1582 12 7880 76850.59 48.5781 [15-20) 0.0248 1570 39 7752.5 68970.59 43.9303 [20-25) 0.0222 1531 34 7570 61218.09 39.9857 [25-30) 0.0301 1497 45 7372.5 53648.09 35.8371 [30-35) 0.0482 1452 70 7085 46275.59 31.8702 [35-40) 0.0658 1382 91 6682.5 39190.59 28.3579 [40-45) 0.0837 1291 108 6185 32508.09 25.1805 [45-50) 0.1014 1183 120 5615 26323.09 22.2511 [50-55) 0.1232 1063 131 4987.5 20708.09 19.4808 36 University of Ghana http://ugspace.ug.edu.gh Table 4.1B: Abridged Life Table for Males [x j  x j1) d j l j d j L j Tj Tj q j  e j  l j l j [60-65) 0.1944 792 154 3575 11410.59 14.4073 [65-70) 0.2586 638 165 2777.5 7835.59 12.2815 [70-75) 0.3044 473 144 2005 5058.09 10.6936 [75-80) 0.3252 329 107 1377.5 3053.09 9.2799 [80-85) 0.3964 222 88 890 1675.59 7.5477 [85-90) 0.5448 134 73 487.5 785.59 5.8626 [90-95) 0.5902 61 36 215 298.09 4.8867 [95-100) 0.76 25 19 77.5 83.09 3.3236 100+ 1 6 6 5.59 5.59 0.9317 4.1.2 Abridged Life Table for Female An abridge life tables for female population was constructed and the results is seen in Table 4.2. It can be seen that the life expected at birth was 53.6256 years and in the age group of 5 to 10 years, the life expectancy increases to 56.2354 years. However, from 10 years onwards the life expectancy started decreasing. With regards to the probability of dying ( q ), from the Table 4.2, the probability that a female child at birth will not survive to the j next year was 0.02920. Then in the year group of 1 to 5 years the probability of the child dying increases to 0.10150. However, in the age group of 5 to 10 years the probability of the female child dying decreases. This implies that at early stage of life the probability a child will die in the age of 1 to 5 years is very high but when she passed that stage the probability that he will survive increases. According to Marmot (2005) children under 5 37 University of Ghana http://ugspace.ug.edu.gh years venerable to many death due to because their immune system are well adapted to the environment. Observing the probability of the people of the old age, it was found that, the probability of dying of them increases rapidly and their life expectancy also decreases. For instance, at age group 90 to 95 years the number of years expected for a female to live is 5.25 years and the probability that she will survive from the age group is 0.573. The standard error and the main functions of the female life table can be seen in Appendix II. Table 4.2A: Female Abridge Life Table [x j  x j1) q j l j d j L j Tj e j [0-1) 0.029 1370 40 1334.00 73467.10 53.63 [1-5) 0.102 1330 135 4990.60 72133.10 54.24 [5-10) 0.022 1195 26 5910.00 67142.50 56.19 [10-15) 0.013 1169 15 5807.50 61232.50 52.38 [15-20) 0.022 1154 25 5707.50 55425.00 48.03 [20-25) 0.024 1129 27 5577.50 49717.50 44.04 [25-30) 0.049 1102 54 5375.00 44140.00 40.05 [30-35) 0.059 1048 62 5085.00 38765.00 36.99 [35-40) 0.056 986 55 4792.50 33680.00 34.16 [40-45) 0.061 931 57 4512.50 28887.50 31.03 [45-50) 0.065 874 57 4227.50 24375.00 27.89 [50-55) 0.060 817 49 3962.50 20147.50 24.66 38 University of Ghana http://ugspace.ug.edu.gh Table 4.2B: Female Abridge Life Table Cont. [x j  x j1) q j l j d j L T e j j j [55-60) 0.090 768 69 3667.50 16185.00 21.07 [60-65) 0.122 699 85 3282.50 12517.50 17.91 [65-70) 0.168 614 103 2812.50 9235.00 15.04 [70-75) 0.235 511 120 2255.00 6422.50 12.57 [75-80) 0.263 391 103 1697.50 4167.50 10.66 [80-85) 0.337 288 97 1197.50 2470.00 8.58 [85-90) 0.461 191 88 735.00 1272.50 6.66 [90-95) 0.573 103 59 367.50 537.50 5.22 [95-100) 0.727 44 32 140.00 170.00 3.86 100+ 1.000 12 12 30.00 30.00 2.50 4.1.3 Life Expectancy for Combined Sex The complete and abridged life table for combined sex was computed and they can be seen in Appendix II. The life expectancy of the complete life table was plotted to study the trend of the life expectancy at each age. 60 50 40 30 20 10 0 0 20 40 60 80 100 120 Age Figure 4.1: Life Expectancy plot for Combined sex Combing the sex, it was revealed that the life expectancy at birth for Ghana was 51.1359. The life expectancy increases 53.342 at age 5 when it reaches its maximum and then slopes 39 Life Expectancy in years University of Ghana http://ugspace.ug.edu.gh downwards from left to right. This is because children under five years’ experience many risk of death. Hence their probability of dying becomes higher thereby reducing the life expectancy. However, at 5 years their probability of dying decreases drastically as a result of immunization and others health practices hence shoot their life expectancy high. 4.2 Comparing the Study Life Table with Other Foreign Life Tables In order to assess the errors of applying the foreign life tables to Ghanaian mortalities, the life expectancy of complete life table constructed was used to compare three foreign life tables. The foreign life tables were USA Period Life Table, United Kingdom Life Table and South Africa Life Table. The life expectancies of the countries and that of this study were presented in Figure 4.2 and 4.3. 4.2.1 Life Expectancy for males in the various Countries Male Life Expectancy of UK, USA, SA, GHANA 80 Variable UK USA 70 SA GH 60 50 40 30 20 10 0 1 10 20 30 40 50 60 70 80 90 100 Age Figure 4.2: Life Expectancy for males in the various Countries 40 Years University of Ghana http://ugspace.ug.edu.gh From Figure 4.2, comparing the life expectancy of the various countries, the life expectancy at birth for UK, USA, South Africa and Ghana (the study life table) was 79.09, 76.53, 59.30 and 49.51 respectively. The graph revealed that the life expectancy of USA and UK behave in almost the same manner and they far away from the Ghana life expectancy. The South Africa one is closed to the Ghana life expectancy. 4.2.2 Life Expectancy for females in the various Countries Female life Expectancy plot of UK, USA, SA, GH 90 Variable UK 80 USA SA 70 GH 60 50 40 30 20 10 0 1 10 20 30 40 50 60 70 80 90 100 Age Figure 4.3: Life Expectancy for females in the various Countries From Figure 4.3, it can be seen that the life expectancies for UK and that of US were far away from the life expectancy of Ghana. However, the South Africa one was closest to Ghana. 41 Year University of Ghana http://ugspace.ug.edu.gh 4.2.3 Testing for the Univariate Normality A univariate normality test was done to test whether the errors obtained from the three foreign life tables are normally distributed. When the data is univariate normal then Univariate Analysis of Variance (ANOVA) can be carried out. Table 4.3: Shapiro-Wilk normality test W-Statistic P-value 0.82725 0.1814 The null hypothesis of the Shapiro-Wilk’s test states that the data is normal. From Table 4.3, the results revealed that the p-value of Shapiro-Wilk’s statistic is greater than 0.05. This implies that we reject the null hypothesis and conclude that they are normal. Hence one-way univariate ANOVA was performed to compare for the differences between the errors obtained from the three foreign life tables and identify the one with the minimum average. 4.2.4 Univariate Analysis of Variance (ANOVA) Table 4.4: One-way ANOVA P- Gender Country Mean Mean Square F-value value UK 11.3137 Male USA 10.0085 2477.977 56.793 0.0000 SA 2.1560 UK 10.7598 Female USA 9.9932 1019.291 23.490 0.0000 SA 4.914 42 University of Ghana http://ugspace.ug.edu.gh From Table 4.4, it can be seen that the error obtained from the males’ life tables from the foreign life tables were significantly different with the South Africa exhibiting the minimum absolute error on the average. The United Kingdom error exhibit the highest error, followed by the USA foreign Life Table. This means that applying the UK and USA life tables on Ghana population is not appropriate. However, applying the South Africa Life table on Ghana population will yield a relative small error compare with the other two. 4.2.5 Testing for the difference between South Africa Life Table and the Study Life Table From Table 4.4, it was established that the South Africa Life table had the minimum error, but is that error significantly different from zero? To answer this question paired t-test was performed to determine if the error is significantly different from zero. The results were presented in Table 4.5 Table 4.5: Independent Paired T-test Gender t-statistics p-value Male 4.0039 0.000 Female 19.2074 0.000 The null hypothesis of the paired t-test state that the paired difference (d) is equal to zero, whiles the alternative state that it is not different from zero. From Table 4.8, it can be seen that the p-value for paired difference of the south Africa and that of the Ghana are greater than 0.05 (𝑡 = 4.0039, 𝑝 = 0.000 𝑚𝑎𝑙𝑒 ; 𝑡 = 19.2074, 𝑝 = 0.000 𝑓𝑒𝑚𝑎𝑙𝑒). This implies that the life expectancies of the ages of South Africa life table and that of the study Life Table are significantly difference. This reject the study first hypothesis which state that foreign life tables are not significantly different from the study life table. Hence applying 43 University of Ghana http://ugspace.ug.edu.gh South Africa life table and other foreign life table to Ghana population will also exhibit some significant errors and hence there is a need to construct our own life table. 4.3 Modelling he Force of Mortality The second objective of the study was to derive the force of mortality function for Ghana. The studies used four parametric force of motility models namely Gompertz, Weibull, Logistic and Log-logistic. Choosing R-square as the test of goodness fit, the model that come out with the highest R-square was taken. The ages were group into two “5 years and below” and “above 5 years”. 4.3.1 Force of Mortality Functions for Males Table 4.6: Modelling the force of Mortality for Males x 5 x 5 R-square R-square Model Gompertz 0.896 0.904 Weibull 0.905 0.881 Logistic 0.896 0.865 Log-logistic 0.854 0.865 Table 4.6 displayed the R-square of the various mortality models, Weibull model obtained the highest R-square for age 5 and below, whiles the Gompertz model obtained the highest R-Square for ages above 5 years. This implies that for ages of 5 years and below the force of mortality follows a Weibull, whiles after 5 years the force of mortality follows a Gompertz model. The function of the force of mortality can be seen in equation 4.1. 44 University of Ghana http://ugspace.ug.edu.gh  0.96935 x  0.006134     x  5 x   4.99715  … … … … … . (4.1)  0.0509x  0.0017e x  5 4.3.2 Force of Mortality Functions for Females Table 4.7: Modelling the force of Mortality for Females x 5 x 5 R-square R-square Model Gompertz 0.895 0.984 Weibull 0.909 0.881 Logistic 0.726 0.855 Log-logistic 0.754 0.865 From Table 4.7, the model that obtained the highest R-square was Weibull and Gompertz for age 5 and below and above 5 years respectively. Hence Gompertz force of mortality model was appropriate for ages above 5 years and Weibull force of mortality model was appropriate for ages 5 and below. The force of mortality functions for the females can be seen in equation 4.2.  0.971168 x  0.003666 x  5 x      8.10423  … … … … … … . . (4.2)  0.0016e0.0485x x  5 45 University of Ghana http://ugspace.ug.edu.gh 4.3.3 Force of Mortality Functions for Combine Sex Table 4.8: Modelling the force of Mortality for Combine Sex x 5 x 5 R-square R-square Model Gompertz 0.796 0.984 Weibull 0.899 0.881 Logistic 0.792 0.825 Log-logistic 0.754 0.865 With regards to combining the sex, the Weibull force of mortality were appropriate for ages 5 years and below and Gompertz model was also appropriate for ages 5 years and above. The force of mortality functions can be seen in equation 4.3.  0.971168 x  0.003666  x  5x    8.10423  … … … … … … . . (4.3)   0.0016e 0.0485x x  5 46 University of Ghana http://ugspace.ug.edu.gh 4.3.4 Graphical Representation 0.30 Variables Male (ux) Female (ux) 0.25 Combine (ux) 0.20 0.15 0.10 0.05 0.00 1 10 20 30 40 50 60 70 80 90 100 Age Figure 4.4: Force of Mortality Plot for Males, Female and Combines Sex. The functions obtained in equation 4.1 to 4.3 were used to plot the force of mortality curve and graph were displayed in Figure 4.4. From Figure 4.4, it can be seen that the males exhibit the highest force of mortality in almost all the ages than the females. This implies that instantaneous rate of mortality of the males were higher than that of females. 4.4 Testing for the equality of survivor functions among Gender From Figure 4.4, the force of mortality for males were higher than that on the females most of the ages. The study therefore employed the log rank equality of survivor to test if in deed the mortality or the survival for the gender significantly differ. This was done to meet the third objectives of the study. 47 Force of Mortality University of Ghana http://ugspace.ug.edu.gh Table 4.9: Log-rank test Gender Death observed Death expected Male 1890 1702.54 Female 1370 1557.46 Total 3260 3260.00 Chi-square (1) = 46.22 P-value = 0.0000 𝐻𝑜: The survival curves for male and female are the same 𝐻𝑎: The survival curves for male and female are significantly different From Table 4.9, it can be seen that the survival curves for male and female are significantly different (𝑝 = 0.00). This implies that the mortality of the male is really greater than the females. According to Austad (2006), the males mortality are always higher than the females this is because the metabolism system of male developed faster than the females and hence males metabolism breaks down faster than the females causing the mortality in the males to be higher than the females. Kaplan-Meier survival curve was also done to analysis the survival trend per gender and it can be seen in appendix III. 48 University of Ghana http://ugspace.ug.edu.gh B COMPETING RISK ANALYSIS This section presents the competing risk analysis, in which the causes of death were grouped into 5 main causes, namely; Cardiovascular, HIV/AIDS, Lower respiratory infections, Malaria and Other causes. 4.5 Multiple Decrement Table The multiple decrement table displayed the ages and causes of death for each age interval. The males multiple decrement table and females multiple decrement table can be seen in Appendix IV, whiles Table 4.10 displayed the multiple decrement table for combined sex. From Table 4.10, it was revealed that among the four leading causes of death clearly specified, malaria recorded the highest death for children under one year (5 deaths were recorded), whiles for ages between 1 to 5 years’ death due to Lower respiratory infections was the highest recording a death of 42. Many research such as had confirmed that malaria is one of leading causes of death for children in Africa. The age group that recorded the highest death of the cardiovascular diseases were 65 to 70 years. For ages, less than 15 years death due to cardiovascular diseases were few. According to Katzmarzyk et al. (2009) cardiovascular diseases are caused by poor blood circulatory due to lack of exercise, overweight, smoking, diabetes, stress and alcohol. Therefore, people in these age group lack exercise and bad blood circulation. With regards to HIV/AIDs the age group of 45 to 50 years recorded the highest death whiles for ages of 80 years and above recorded few deaths. According to Schwartländer et al. (2011), the major causes of HIV/AIDS is by sexual intercourse and most people becomes sexual inactive at age 65 that time hence reduced the risk of being infected by the diseases, But for the youth they are sexually active and therefor stand at a higher risks of being infected by the HIV/AIDs. However, the virus stays longer a little before it kills the person 49 University of Ghana http://ugspace.ug.edu.gh this explained the reason why ages of 20 to 30 years did not record the highest death of HIV/AIDS but rather from 45 years to 50 years. Table 4.10A: Multiple Decrement Table for Combined Sex Deaths Deaths Deaths by No. of by Deaths by Deaths by by Total Age lower people Cardio HIV/AIDs Malaria other death Group respiratory living vascular causes 𝛼𝑙 𝛽 𝛽 𝛽 𝛽𝑥𝑗 (𝛼𝑑) 1 𝑥 (𝛼𝑑) 2 3 𝑥 (𝛼𝑑)𝑥 (𝛼𝑑) 4 𝛽 (𝛼𝑑) 5𝑥 [xj - x 𝑗 𝑗 𝑗 𝑗 𝑥𝑗 j+1) (𝛼𝑑)𝑥 𝑗 [0-1) 3260 0 2 7 13 77 99 [1-5) 3161 1 2 82 28 212 325 [5-10) 2836 0 1 17 11 56 85 [10-15) 2751 1 5 5 2 14 27 [15-20) 2724 2 2 6 2 52 64 [20-25) 2660 3 3 2 1 52 61 [25-30) 2599 5 28 3 4 59 99 [30-35) 2500 14 33 3 11 71 132 [35-40) 2368 17 34 4 9 82 146 [40-45) 2222 37 40 1 3 84 165 [45-50) 2057 55 31 0 7 84 177 [50-55) 1880 63 24 0 4 89 180 50 University of Ghana http://ugspace.ug.edu.gh Table 4.10B: Multiple Decrement Table for (Combined Sex) Deaths Deaths by Deaths by No. of Deaths by Deaths by by Cardio lower Total death Age people HIV/AIDs Malaria other Group vascular respiratory living causes [55-60) 1700 78 15 0 5 111 209 [60-65) 1491 103 8 1 6 121 239 [65-70) 1252 156 4 5 9 94 268 [70-75) 984 113 1 1 13 136 264 [75-80) 720 102 1 1 11 95 210 [80-85) 510 85 0 1 19 80 185 [85-90) 325 67 2 4 17 71 161 [90-95) 164 36 0 0 2 57 95 [95-100) 69 17 0 4 12 18 51 100+ 18 7 0 1 2 8 18 4.6 Crude Probabilities The crude probability obtained present the probability of dying from a specific disease in the present of other disease. The crude probabilities of all the leading causes of diseases were computed with respect to age and gender and the results were presented in Table 4.11 to Table 4.15. 4.6.1 Crude Probabilities of Cardiovascular Diseases The crude probabilities of dying from Cardiovascular Diseases and as well as its standard error with respect to age and gender were computed with referenced to equation (3.18) and (3.19) respectively and it can be seen in Table 4.11. 51 University of Ghana http://ugspace.ug.edu.gh From Table 4.11, it was observed that the probability that a less than 5 years will die from cardiovascular diseases is negligible. Probability of 0 was recorded for males whiles 0.00075 was recorded for females. This means that children under 5 years are not at risk of being death from Cardiovascular Diseases. However, for both male and female the aged people (above 65 years) recorded high crude probabilities from dying from cardiovascular diseases and hence implies that death due to cardiovascular diseases are often experienced by this age groups. Comparing with gender, in all the ages except 100 years and above, the probability of a male dying from Cardiovascular Diseases were higher than that of the females one. According to Graham et al. (2007) males suffered from death of cardiovascular disease because of stress and activities such as smoking and alcoholism. Rosamond et al. (2008) in his research also confirmed that the males the probability that a male will die from cardiovascular diseases are higher than that of a female. 52 University of Ghana http://ugspace.ug.edu.gh Table 4.11: Crude Probabilities of Cardiovascular Diseases Age Group Male Female [x - x ) ˆ Q S Q ˆ S j j+1 jCD Q̂ jCD Q̂jCD jCD [0-1) 0.00000 0.00000 0.00000 0.00000 [1-5) 0.00000 0.00000 0.00075 0.00064 [5-10) 0.00000 0.00000 0.00000 0.00000 [10-15) 0.00000 0.00000 0.00086 0.00073 [15-20) 0.00064 0.00064 0.00087 0.00074 [20-25) 0.00065 0.00065 0.00177 0.00107 [25-30) 0.00267 0.00133 0.00091 0.00077 [30-35) 0.00551 0.00194 0.00573 0.00198 [35-40) 0.00796 0.00239 0.00609 0.00211 [40-45) 0.01782 0.00368 0.01504 0.00340 [45-50) 0.03128 0.00506 0.02059 0.00409 [50-55) 0.04045 0.00604 0.02448 0.00460 [55-60) 0.04936 0.00710 0.04167 0.00614 [60-65) 0.08207 0.00975 0.05436 0.00730 [65-70) 0.13950 0.01372 0.10912 0.01071 [70-75) 0.11205 0.01450 0.11742 0.01212 [75-80) 0.14894 0.01963 0.13555 0.01474 [80-85) 0.15766 0.02446 0.17361 0.01900 [85-90) 0.18657 0.03365 0.21990 0.02552 [90-95) 0.13115 0.04322 0.27184 0.03732 [95-100) 0.28000 0.08980 0.22727 0.05379 100+ 0.333333 0.19245 0.416667 0.121169 53 University of Ghana http://ugspace.ug.edu.gh 4.6.2 Crude Probabilities of HIV/AIDS The probability of dying from HIV/AIDS (crude probability) and its standard error were computed and the results were displayed in Table 4.12. Table 4.12: Crude Probabilities of HIV/AIDS Age Group Male Female [x - x ) Q ˆ S Qˆ S j j+1 jHIV Q̂ jHIV Q̂jHIV jHIV [0-1) 0.00053 0.00053 0.00073 0.00062 [1-5) 0.00000 0.00000 0.00150 0.00090 [5-10) 0.00061 0.00058 0.00000 0.00000 [10-15) 0.00190 0.00109 0.00171 0.00103 [15-20) 0.00127 0.00090 0.00000 0.00000 [20-25) 0.00000 0.00000 0.00266 0.00130 [25-30) 0.00267 0.00133 0.02178 0.00374 [30-35) 0.00689 0.00217 0.02195 0.00385 [35-40) 0.01302 0.00305 0.01623 0.00343 [40-45) 0.01782 0.00368 0.01826 0.00374 [45-50) 0.01775 0.00384 0.01144 0.00306 [50-55) 0.01129 0.00324 0.01469 0.00358 [55-60) 0.00858 0.00302 0.00911 0.00292 [60-65) 0.00505 0.00252 0.00572 0.00243 [65-70) 0.00313 0.00221 0.00326 0.00196 [70-75) 0.00211 0.00211 0.00000 0.00000 [75-80) 0.00304 0.00303 0.00000 0.00000 [80-85) 0.00000 0.00000 0.00000 0.00000 [85-90) 0.00746 0.00743 0.00524 0.00445 [90-95) 0.00000 0.00000 0.00000 0.00000 [95-100) 0.00000 0.00000 0.00000 0.00000 100+ 0.00000 0.00000 0.00000 0.00000 From the Table 4.13, it can be seen that the probability of dying from HIV/AIDS among people of ages below 20 years and above 90 years were negligible ( 𝑝𝑟𝑜𝑝 < 0.005). This 54 University of Ghana http://ugspace.ug.edu.gh means that the probability that a person below 20 years and above 90 years dying from HIV/AIDS is very small. Hence people less than 20 years and above 90 years have a very low risk of dying from HIV/AIDS. For ages of 34 to 55 years in both male and female groups, the probability of a person dying from HIV.AIDS are more than 0.005. This implies that people in the age groups of 34 to 55 years have a higher chance of dying from HIV/AIDS. Leclerc‐Madlala (1997) explained that youth are sexually active and a person can stay for a minimum of 10 years before dying from HIV/AIDS and thus answer why the probability of dying from HIV/AIDS are higher from 45 to 60 years. 4.6.3 Crude Probabilities of Lower Respiration Infections The Lower Respiration Infections that were recorded in the study were bronchitis and pneumonia. The probability of dying from Lower Respiration Infections (LIR), in the present of all the other causes of death were computed as well as its standard error with respect to gender and age. From Table 4.13, highest probability of dying from LIR were in the age group of 95 to 100 years and children from 1 to 5 years. According to Lodha, Kabra, and Pandey (2013), Pneumonia is the leading causes of for children under five years in developing countries such as Ghana. They explained that children under five years cannot withstand to excessive cold and are likely to have pneumonia diseases. Comparing with the gender, the probability that a female child will die from pneumonia is higher than the male. The probability of dying from LIR was also higher in 95 to 100 years. World Health Organization (2009) also explained that very old people cannot also stand cold weather and are at a higher risk of being infected with pneumonia. 55 University of Ghana http://ugspace.ug.edu.gh Table 4.13: Crude Probabilities of Lower Respiration Infections Age Group Male Female [x - x ) Qˆ S Q ˆ S j j+1 jLR Q̂ jLR Q̂jLR jLR [0-1) 0.00106 0.00075 0.00365 0.00139 [1-5) 0.02294 0.00350 0.03008 0.00399 [5-10) 0.00366 0.00141 0.00921 0.00223 [10-15) 0.00063 0.00063 0.00342 0.00145 [15-20) 0.00318 0.00142 0.00087 0.00074 [20-25) 0.00065 0.00065 0.00089 0.00075 [25-30) 0.00134 0.00094 0.00091 0.00077 [30-35) 0.00207 0.00119 0.00000 0.00000 [35-40) 0.00289 0.00145 0.00000 0.00000 [40-45) 0.00077 0.00077 0.00000 0.00000 [45-50) 0.00000 0.00000 0.00000 0.00000 [50-55) 0.00000 0.00000 0.00000 0.00000 [55-60) 0.00000 0.00000 0.00000 0.00000 [60-65) 0.00126 0.00126 0.00000 0.00000 [65-70) 0.00470 0.00271 0.00326 0.00196 [70-75) 0.00211 0.00211 0.00000 0.00000 [75-80) 0.00304 0.00303 0.00000 0.00000 [80-85) 0.00000 0.00000 0.00347 0.00295 [85-90) 0.00746 0.00743 0.01571 0.00766 [90-95) 0.00000 0.00000 0.00000 0.00000 [95-100) 0.04000 0.03919 0.06818 0.03235 100+ 0.00000 0.00000 0.08333 0.06793 56 University of Ghana http://ugspace.ug.edu.gh 4.6.4 Crude Probabilities of Malaria Table 4.14 displayed the crude probability and the standard error of dying from malaria. Table 4.14: Crude Probabilities of Malaria Age Group Male Female [x - x ) Qˆ S Qˆ S j j+1 jm Q̂ jm Q̂jm jm [0-1) 0.00265 0.00118 0.00584 0.00175 [1-5) 0.00874 0.00218 0.00902 0.00221 [5-10) 0.00609 0.00182 0.00084 0.00068 [10-15) 0.00063 0.00063 0.00086 0.00073 [15-20) 0.00064 0.00064 0.00087 0.00074 [20-25) 0.00065 0.00065 0.00000 0.00000 [25-30) 0.00067 0.00067 0.00272 0.00134 [30-35) 0.00482 0.00182 0.00382 0.00162 [35-40) 0.00434 0.00177 0.00304 0.00149 [40-45) 0.00232 0.00134 0.00000 0.00000 [45-50) 0.00423 0.00189 0.00229 0.00138 [50-55) 0.00188 0.00133 0.00245 0.00147 [55-60) 0.00536 0.00239 0.00000 0.00000 [60-65) 0.00631 0.00281 0.00143 0.00122 [65-70) 0.01254 0.00441 0.00163 0.00139 [70-75) 0.01480 0.00555 0.01174 0.00406 [75-80) 0.01824 0.00738 0.01279 0.00484 [80-85) 0.05856 0.01576 0.02083 0.00717 [85-90) 0.05224 0.01922 0.05236 0.01372 [90-95) 0.01639 0.01626 0.00971 0.00823 [95-100) 0.16000 0.07332 0.18182 0.04950 100+ 0.00000 0.00000 0.16667 0.09160 From Table 4.14, it revealed that children from 1 to 5 years as well as the aged people from 85 above are the age groups that recorded a significant crude probabilities. In African in 57 University of Ghana http://ugspace.ug.edu.gh every 30 seconds a child in Africa dies from malaria (Sachs & Malaney, 2002). According to Mermin et al. (2006) aged people are more likely to die from malaria than the adult people, this is because the elderly people have a weaker immune. 4.6.5 Graphical Presentation Crude Probabilities The crude probabilities of all the leading causes of diseases were displayed in a graphical form in Figure 4.4. 0.40000 0.35000 0.30000 0.25000 0.20000 0.15000 0.10000 0.05000 0.00000 0 20 40 60 80 100 120 -0.05000 AGE GROUP CD HIV LR M Figure 4.5: Crude Probabilities From Figure 4.5, it can be seen that in the ages of 55 years and above the crude probability of cardiovascular diseases were higher than all the other diseases. This implies that that probability that a 55 year above person will die from cardiovascular diseases is high. From literature Katzmarzyk et al. (2009) found out that the leading causes of death for old people and the entire population was cardiovascular diseases. For children with 1 to 5 years, the 58 CRUDE PROB. University of Ghana http://ugspace.ug.edu.gh probability that a person will die from Lower Respiratory Infections were higher, followed by malaria. 4.7 Net Probabilities The net probability with respect to the probability of dying when only a particular risk is acting in the population for all the leading causes of death and their standard error for each age group and gender were computed with referenced to equation (3.24) and (2.26) respectively. The graphic presentation of these net probabilities per age gender were displayed in Figure 4.6 and 4.7. 4.7.1 Net Probabilities for Males 1.20000 1.00000 0.80000 0.60000 0.40000 0.20000 0.00000 0 20 40 60 80 100 120 -0.20000 AGE GROUP CD HIV LR m Figure 4.6: Net Probabilities for Males 59 NET PROB. University of Ghana http://ugspace.ug.edu.gh 4.7.2 Graphical Presentation the Female Net Probabilities 1.20000 1.00000 0.80000 0.60000 0.40000 0.20000 0.00000 0 20 40 60 80 100 120 -0.20000 AGE GROUP CD HIV LR m Figure 4.7: Net Probabilities for Females From the Figure 4.6 and Figure 4.7, the net probability of all the leading causes of death for age 40 and below were less than 0.0005. This implies if the death is only caused cardiovascular, HIV/AIDS, almost everyone in the age of 40 years and below will not die but will experience mortality from ages above 40 years. 4.8 The Impact of Eliminating the Major Causes of Diseases The objective of the study was to determine the impact of the leading causes of death. As a result of that the net probability of the various causes of death probability ( qˆ j. ) were calculated and the results were used to compute for the life expectancy for eliminating the leading Causes of death for each age group and gender. The impacts were examined in two ways, they are by calculating the percentage decrease in probability of dying given that a 60 NET PROB. University of Ghana http://ugspace.ug.edu.gh particular risk of death is eliminated and the life expectancy gain when the cause of death is eliminated. 4.8.1 The Impact of Eliminating Cardiovascular Diseases 4.8.1.1 Percentage Decrease in Probability of Dying Given that Cardiovascular Diseases is eliminated The probability of dying at a specific age is represented by qˆ j , whiles qˆ j.CD represent the probability of dying given that the cardiovascular disease is eliminated. From Table 4.15, it was seen that the percentage decrease in probability of dying given that cardiovascular disease is eliminated are less than 1% for age 15 and below. This implies that at age 15 and below when the cardiovascular disease is eliminated the impact will not felt. On the contrary, in aged years especially 95 to 100 years for males and 65 to 70 years for the females the impact of eliminating the leading cause of death will felt massively since they recorded a percentage decreases above 15%. This massive reduction implies that when there are no cardiovascular diseases the probability that a male will die within 95 years 100 will decrease by 64% and 62% for female aged of 65 to 70 years. 61 University of Ghana http://ugspace.ug.edu.gh Table 4.15: Percentage Decrease in Probability of Dying Given that Cardiovascular Diseases is eliminated Male Female Age Prob. of CD Percentage Prob. of CD Percentage Group Dying eliminated Decrease Dying eliminated Decrease [xj - xj+1) qˆ j - qˆ ˆ ˆqˆ qˆ j.CD q - q qˆ qˆ j j.CDj j.CD j j.CD qˆ ˆj q j [0-1) 0.03122 0.03122 0.000% 0.02920 0.02920 0.000% [1-5) 0.10377 0.10377 0.000% 0.10150 0.10079 0.702% [5-10) 0.03595 0.03595 0.000% 0.02176 0.02176 0.000% [10-15) 0.00759 0.00759 0.000% 0.01283 0.01198 6.627% [15-20) 0.02484 0.02421 2.533% 0.02166 0.02081 3.958% [20-25) 0.02221 0.02156 2.909% 0.02391 0.02216 7.325% [25-30) 0.03006 0.02743 8.766% 0.04900 0.04812 1.807% [30-35) 0.04821 0.04282 11.180% 0.05916 0.05359 9.413% [35-40) 0.06585 0.05812 11.729% 0.05578 0.04985 10.632% [40-45) 0.08366 0.06645 20.570% 0.06122 0.04654 23.979% [45-50) 0.10144 0.07131 29.701% 0.06522 0.04510 30.853% [50-55) 0.12324 0.08456 31.386% 0.05998 0.03594 40.071% [55-60) 0.15021 0.10353 31.079% 0.08984 0.04923 45.208% [60-65) 0.19444 0.11747 39.588% 0.12160 0.06918 43.108% [65-70) 0.25862 0.12876 50.214% 0.16775 0.06216 62.943% [70-75) 0.30444 0.20501 32.662% 0.23483 0.12526 46.659% [75-80) 0.32523 0.19203 40.954% 0.26343 0.13793 47.639% [80-85) 0.39640 0.26218 33.860% 0.33681 0.18044 46.425% [85-90) 0.54478 0.40397 25.847% 0.46073 0.27589 40.120% [90-95) 0.59016 0.50031 15.224% 0.57282 0.36039 37.085% [95-100) 0.76000 0.27177 64.241% 0.72727 0.59068 18.782% 100+ 1.00000 1.00000 0.000% 1.00000 1.00000 0.000% 4.8.1.2 Life Expectancy Gain from Eliminating Cardiovascular Diseases Table 4.16 represent the life expectancy gain from Eliminating Cardiovascular Diseases. From the Table, it was recorded that when Cardiovascular Diseases.is eliminated Ghanaians life expectancy at birth will increase by 16.021% when the person is a male and 17.587% when the person is a female. Comparing the gender, the impact it felt more in females than the males. Lai and Hardy (1999) in their study also found out that in US, the life expectancy gain for eliminating cardiovascular diseases were higher in ages above 60 years. 62 University of Ghana http://ugspace.ug.edu.gh Table 4.16: Life Expectancy Gain from Eliminating Cardiovascular Diseases Life CD Expectancy Life CD Expectancy Age Expectancy eliminated Gain Expectancy eliminated Gain Group eˆ j.CDeˆ j eˆ j.CDeˆ j [xj - xj+1) eˆ j.CD eˆ j eˆ eˆ ˆj j.CD e j eˆ j [0-1) 58.97507 49.52666 16.021% 65.06901 53.62562 17.587% [1-5) 59.87875 50.11933 16.299% 66.02296 54.23541 17.854% [5-10) 62.63107 51.74168 17.387% 69.24540 56.18619 18.859% [10-15) 59.87363 48.57812 18.866% 65.72863 52.38024 20.308% [15-20) 55.31216 43.93031 20.577% 61.49438 48.02860 21.898% [20-25) 51.62212 39.98569 22.542% 57.74514 44.03676 23.739% [25-30) 47.70353 35.83707 24.875% 53.99277 40.05445 25.815% [30-35) 43.97468 31.87024 27.526% 51.58207 36.98950 28.290% [35-40) 40.81668 28.35788 30.524% 49.33617 34.15822 30.764% [40-45) 38.14597 25.18055 33.989% 46.75448 31.02846 33.635% [45-50) 35.60730 22.25113 37.510% 43.85486 27.88902 36.406% [50-55) 33.00095 19.48080 40.969% 40.71511 24.66034 39.432% [55-60) 30.53310 16.86759 44.756% 37.03759 21.07422 43.100% [60-65) 28.26096 14.40731 49.020% 33.63296 17.90773 46.755% [65-70) 25.87525 12.28149 52.536% 30.57361 15.04072 50.805% [70-75) 23.09312 10.69364 53.693% 26.98120 12.56849 53.418% [75-80) 20.73252 9.27991 55.240% 24.21353 10.65857 55.981% [80-85) 17.35443 7.54770 56.509% 20.99918 8.57639 59.158% [85-90) 13.66876 5.86261 57.109% 17.54902 6.66230 62.036% [90-95) 9.64172 4.88672 49.317% 13.89320 5.21845 62.439% [95-100) 5.02407 3.32360 33.846% 9.62293 3.86364 59.850% 100+ 1.03376 0.93167 9.876% 4.96212 2.50000 49.618% 63 University of Ghana http://ugspace.ug.edu.gh 4.8.2 The Impact of Eliminating HIV/AIDS 4.8.2.1 Percentage Decrease in Probability of Dying Given that HIV/AIDS is eliminated Table 4.17: Percentage Decrease in Probability of Dying Given that HIV/AIDS is eliminated Male Female Prob. of HIV/AIDS Percentage Prob. of HIV/AIDS Percentage Dying eliminated Decrease Dying eliminated Decrease Age Group qˆ - qˆj j.HIV qˆ j - qˆqˆ qˆ j.HIV qˆ qˆ [x - x j j.HIV j j.HIVj j+1) qˆ qˆj j [0-1) 0.03122 0.03070 1.669% 0.02920 0.02848 2.464% [1-5) 0.10377 0.10377 0.000% 0.10150 0.10008 1.405% [5-10) 0.03595 0.03536 1.665% 0.02176 0.02176 0.000% [10-15) 0.00759 0.00569 24.929% 0.01283 0.01113 13.259% [15-20) 0.02484 0.02358 5.067% 0.02166 0.02166 0.000% [20-25) 0.02221 0.02221 0.000% 0.02391 0.02129 10.992% [25-30) 0.03006 0.02743 8.766% 0.04900 0.02753 43.825% [30-35) 0.04821 0.04147 13.985% 0.05916 0.03763 36.387% [35-40) 0.06585 0.05318 19.244% 0.05578 0.03988 28.501% [40-45) 0.08366 0.06645 20.570% 0.06122 0.04337 29.166% [45-50) 0.10144 0.08446 16.737% 0.06522 0.05409 17.060% [50-55) 0.12324 0.11261 8.623% 0.05998 0.04563 23.921% [55-60) 0.15021 0.14227 5.286% 0.08984 0.08111 9.721% [60-65) 0.19444 0.18991 2.333% 0.12160 0.11623 4.421% [65-70) 0.25862 0.25593 1.042% 0.16775 0.16478 1.772% [70-75) 0.30444 0.30268 0.577% 0.23483 0.23483 0.000% [75-80) 0.32523 0.32274 0.764% 0.26343 0.26343 0.000% [80-85) 0.39640 0.39640 0.000% 0.33681 0.33681 0.000% [85-90) 0.54478 0.53984 0.906% 0.46073 0.45694 0.824% [90-95) 0.59016 0.59016 0.000% 0.57282 0.57282 0.000% [95-100) 0.76000 0.76000 0.000% 0.72727 0.72727 0.000% 100+ 1.00000 1.00000 0.000% 1.00000 1.00000 0.000% From Table 4.17, the percentage decrease in probability of dying given HIV/AIDS is eliminated in the population are higher in most of the ages in female that that of the males. This implies that when there are no HIV/AIDs in the population the females will survive 64 University of Ghana http://ugspace.ug.edu.gh longer than the males. According to Higgins et al. (2010), the probability a male will contract the virus from females is less than a female contracting the virus from the male. His explained that the females the probability a female will contract AIDS are higher than that of the males. 4.8.2.2 Life Expectancy Gain from Eliminating HIV/AIDS Table 4.18: Percentage Decrease in Probability of Dying Given that HIV/AIDS is eliminated Male Female Life HIV/AIDS Expectancy Life HIV/AIDS Expectancy Expectancy eliminated Gain Expectancy eliminated Gain Age Group eˆ j.HIV eˆ j eˆ j.HIV eˆ j [xj - xj+1) eˆ j.HIV eˆ ˆ ˆj eˆ ej j.HIV e j eˆ j [0-1) 52.88507 49.52666 6.350% 59.61999 53.62562 10.054% [1-5) 53.55622 50.11933 6.417% 60.36400 54.23541 10.153% [5-10) 55.57283 51.74168 6.894% 62.89232 56.18619 10.663% [10-15) 52.51616 48.57812 7.499% 59.23208 52.38024 11.568% [15-20) 47.80196 43.93031 8.099% 54.86842 48.02860 12.466% [20-25) 43.89180 39.98569 8.899% 51.02293 44.03676 13.692% [25-30) 39.82752 35.83707 10.019% 47.07207 40.05445 14.908% [30-35) 35.87361 31.87024 11.160% 43.31019 36.98950 14.594% [35-40) 32.30085 28.35788 12.207% 39.83910 34.15822 14.260% [40-45) 28.93653 25.18055 12.980% 36.29242 31.02846 14.504% [45-50) 25.74144 22.25113 13.559% 32.69751 27.88902 14.706% [50-55) 22.74921 19.48080 14.367% 29.24863 24.66034 15.687% [55-60) 20.09551 16.86759 16.063% 25.37585 21.07422 16.952% [60-65) 17.69060 14.40731 18.560% 22.11397 17.90773 19.021% [65-70) 15.74911 12.28149 22.018% 19.27551 15.04072 21.970% [70-75) 14.45978 10.69364 26.046% 16.94795 12.56849 25.841% [75-80) 13.35556 9.27991 30.517% 15.32101 10.65857 30.432% [80-85) 11.81108 7.54770 36.096% 13.53406 8.57639 36.631% [85-90) 10.15547 5.86261 42.271% 11.94268 6.66230 44.214% [90-95) 8.34156 4.88672 41.417% 10.56167 5.21845 50.591% [95-100) 4.87657 3.32360 31.846% 8.39286 3.86364 53.965% 100+ 1.04367 0.93167 10.732% 4.77941 2.50000 47.692% 65 University of Ghana http://ugspace.ug.edu.gh From Table 4.18 the life expectancy gain from Eliminating AIDS at birth for the females was 10.054% and that of the males were 6.350. This implies that comparing by gender, the impact it eliminating AIDS will be higher in females’ mortality that the males’ mortality. This confirm to, Lai and Hardy (1999) who found out that the life expectancy for eliminating HIV/AIDS from US population was higher for females than the males. 4.8.3 The Effect of Eliminating Lower Respiration 4.8.3.1 Percentage Decrease in Probability of Dying Given that Lower Respiration (LIR) is eliminated From Table 4.19, it can be seen that at age 10 and below there are a huge decrease in the probability of dying when in LIR is eliminated in the population. For instance, assuming that there is no LIR, the probability that a child at aged 5 to 10 years given she is a female will decrease by 42.039%. With respect to the male the probability that a child will die when LIR is eliminated will decrease by 21.172%. This implies that when LIR is eliminated in the population, the children mortality rate will reduce. 66 University of Ghana http://ugspace.ug.edu.gh Table 4.19: Percentage Decrease in Probability of Dying Given that Lower Respiration is eliminated Male Female Age Group Prob. of LR Percentage Prob. of LR Percentage [x Dying eliminated Decrease Dying eliminated Decrease j - xj+1) qˆ - qˆ qˆ ˆ qˆ ˆ j j.LR - q ˆ ˆ j j.LR j q j.LR q qqˆ j j.LR ˆ j q j [0-1) 0.03122 0.03017 3.338% 0.02920 0.02559 12.339% [1-5) 0.10377 0.08180 21.172% 0.10150 0.07255 28.522% [5-10) 0.03595 0.03236 10.003% 0.02176 0.01261 42.039% [10-15) 0.00759 0.00696 8.304% 0.01283 0.00943 26.541% [15-20) 0.02484 0.02169 12.680% 0.02166 0.02081 3.958% [20-25) 0.02221 0.02156 2.909% 0.02391 0.02304 3.661% [25-30) 0.03006 0.02874 4.380% 0.04900 0.04812 1.807% [30-35) 0.04821 0.04619 4.185% 0.05916 0.05916 0.000% [35-40) 0.06585 0.06305 4.254% 0.05578 0.05578 0.000% [40-45) 0.08366 0.08291 0.886% 0.06122 0.06122 0.000% [45-50) 0.10144 0.10144 0.000% 0.06522 0.06522 0.000% [50-55) 0.12324 0.12324 0.000% 0.05998 0.05998 0.000% [55-60) 0.15021 0.15021 0.000% 0.08984 0.08984 0.000% [60-65) 0.19444 0.19331 0.582% 0.12160 0.12160 0.000% [65-70) 0.25862 0.25458 1.564% 0.16775 0.16478 1.772% [70-75) 0.30444 0.30268 0.577% 0.23483 0.23483 0.000% [75-80) 0.32523 0.32274 0.764% 0.26343 0.26343 0.000% [80-85) 0.39640 0.39640 0.000% 0.33681 0.33399 0.835% [85-90) 0.54478 0.53984 0.906% 0.46073 0.44926 2.490% [90-95) 0.59016 0.59016 0.000% 0.57282 0.57282 0.000% [95-100) 0.76000 0.71876 5.426% 0.72727 0.69194 4.858% 100+ 1.00000 1.00000 0.000% 1.00000 1.00000 0.000% 4.8.3.2 Life Expectancy Gain from Eliminating Lower Respiration From Table 4.20, it can be seen that the life expectancy for females increased by 8.114% when LIR is eliminated from the population, whiles for males the life expectancy increases by 6.310%. This implies on the average eliminating LIR in the population will have a 67 University of Ghana http://ugspace.ug.edu.gh significant increase in our life expectancy and females impact will be higher than that of the males. Table 4.20 Life Expectancy Gain from Eliminating Lower Respiration Male Female Age Group Life LR Expectancy Life LR Expectancy Expectancy eliminated Gain Expectancy eliminated Gain [xj - x ˆj+1) e j.LReˆ j eˆ j.LReˆ j eˆ ˆ j.LR e j eˆ eˆ ˆj j.LR e j eˆ j [0-1) 52.86252 49.52666 6.310% 58.36073 53.62562 8.114% [1-5) 53.50326 50.11933 6.325% 58.88816 54.23541 7.901% [5-10) 54.06563 51.74168 4.298% 59.28024 56.18619 5.219% [10-15) 50.74025 48.57812 4.261% 54.97551 52.38024 4.721% [15-20) 46.06761 43.93031 4.639% 50.45099 48.02860 4.801% [20-25) 42.00100 39.98569 4.798% 46.41807 44.03676 5.130% [25-30) 37.83983 35.83707 5.293% 42.39905 40.05445 5.530% [30-35) 33.84553 31.87024 5.836% 39.30243 36.98950 5.885% [35-40) 30.29992 28.35788 6.409% 36.47760 34.15822 6.358% [40-45) 27.08223 25.18055 7.022% 33.35765 31.02846 6.982% [45-50) 24.18647 22.25113 8.002% 30.23479 27.88902 7.759% [50-55) 21.48946 19.48080 9.347% 27.03125 24.66034 8.771% [55-60) 18.97957 16.86759 11.128% 23.47774 21.07422 10.237% [60-65) 16.66541 14.40731 13.550% 20.37730 17.90773 12.119% [65-70) 14.73525 12.28149 16.652% 17.62186 15.04072 14.647% [70-75) 13.37884 10.69364 20.071% 15.27344 12.56849 17.710% [75-80) 12.25833 9.27991 24.297% 13.63487 10.65857 21.829% [80-85) 10.76471 7.54770 29.885% 11.88385 8.57639 27.832% [85-90) 9.27821 5.86261 36.813% 10.38911 6.66230 35.872% [90-95) 7.89326 4.88672 38.090% 9.28779 5.21845 43.814% [95-100) 4.85444 3.32360 31.535% 7.83186 3.86364 50.668% 100+ 1.06143 0.93167 12.225% 4.67262 2.50000 46.497% 68 University of Ghana http://ugspace.ug.edu.gh 4.8.4 The Effect of Eliminating Malaria 4.8.4.1 Percentage Decrease in Probability of Dying Given that Eliminating Malaria Diseases The probability of dying given that malaria is eliminated from the population presented as qˆ j.CD From Table 4.21, it was seen that the percentage decrease in probability of dying when malaria is eliminated are high in almost all the age group in the population. This implies in Ghana everyone is at risk of dying from malaria. Hence when the disease is eliminated the mortality rate experience in the country will reduced a crossed the age groups. Table 4.21: Percentage Decrease in Probability of Dying Given that Eliminating Malaria Diseases Male Female Age Prob. of LR Percentage Prob. of LR Percentage Group Dying eliminated Decrease Dying eliminated Decrease [xj - xj+1) qˆ j - qˆ ˆ ˆqˆ qˆ j.MR q qˆ j - q j.MR j j.m j qˆqˆ j.m j qˆ j [0-1) 0.03122 0.02861 8.352% 0.029197 0.023427 19.764% [1-5) 0.10377 0.09546 8.005% 0.101504 0.092915 8.462% [5-10) 0.03595 0.02995 16.692% 0.021757 0.020929 3.806% [10-15) 0.00759 0.00696 8.304% 0.012831 0.011981 6.627% [15-20) 0.02484 0.02421 2.533% 0.021664 0.020806 3.958% [20-25) 0.02221 0.02156 2.909% 0.023915 0.023915 0.000% [25-30) 0.03006 0.02940 2.189% 0.049002 0.046344 5.425% [30-35) 0.04821 0.04349 9.779% 0.05916 0.055451 6.269% [35-40) 0.06585 0.06164 6.386% 0.055781 0.05282 5.308% [40-45) 0.08366 0.08143 2.661% 0.061224 0.061224 0.000% [45-50) 0.10144 0.09742 3.957% 0.065217 0.063003 3.396% [50-55) 0.12324 0.12147 1.430% 0.059976 0.057599 3.962% 69 University of Ghana http://ugspace.ug.edu.gh Table 4.21: Percentage Decrease in Probability of Dying Given that Eliminating Malaria Diseases Cont. Male Female Age Prob. of LR Percentage Prob. of LR Percentage Group Dying eliminated Decrease Dying eliminated Decrease qˆ qˆ qˆ - qˆ qˆ qˆ qˆ - qˆ [x - x ) j j.m j j.MR j j.m j j.MRj j+1 qˆ qˆj j [55-60) 0.15021 0.14526 3.298% 0.089844 0.089844 0.000% [60-65) 0.19444 0.18877 2.919% 0.121602 0.120261 1.103% [65-70) 0.25862 0.24779 4.189% 0.167752 0.166267 0.885% [70-75) 0.30444 0.29206 4.068% 0.234834 0.224525 4.390% [75-80) 0.32523 0.31018 4.628% 0.263427 0.252413 4.181% [80-85) 0.39640 0.34966 11.790% 0.336806 0.319742 5.066% [85-90) 0.54478 0.50909 6.550% 0.460733 0.42153 8.509% [90-95) 0.59016 0.57988 1.742% 0.572816 0.566613 1.083% [95-100) 0.76000 0.54744 27.968% 0.727273 0.622605 14.392% 100+ 1.00000 1.00000 0.000% 1 1 0.000% 4.8.4.2 Life Expectancy Gain from Eliminating Malaria Table 4.22 represent the life expectancy gain from malaria. it be seen that when malaria is eliminated in Ghana, the life expectancy at birth will increase by 6.048% for males and 5.275% for females. This means that when the country succeeds in preventing death caused by malaria, the life expectancy of the citizens will increase and hence will have an impact in the life of the people. 70 University of Ghana http://ugspace.ug.edu.gh Table 4.22: Life Expectancy Gain from Eliminating Malaria Male Female Age Group Life malaria Expectanc Life malaria Expectancy Expectancy eliminated y Gain Expectancy eliminated Gain [xj - xj+1) eˆ j.MReˆ j eˆ j.MR eˆ j eˆ ˆ ˆ ˆ j.m e j eˆ j e j.m e j eˆ j [0-1) 52.71504 49.52666 6.048% 56.61174 53.62562 5.275% [1-5) 53.26254 50.11933 5.901% 56.96329 54.23541 4.789% [5-10) 54.67545 51.74168 5.366% 58.57202 56.18619 4.073% [10-15) 51.26044 48.57812 5.233% 54.75000 52.38024 4.328% [15-20) 46.59525 43.93031 5.719% 50.37202 48.02860 4.652% [20-25) 42.66662 39.98569 6.283% 46.36936 44.03676 5.030% [25-30) 38.53239 35.83707 6.995% 42.42222 40.05445 5.581% [30-35) 34.59858 31.87024 7.886% 39.31797 36.98950 5.922% [35-40) 31.01867 28.35788 8.578% 36.41978 34.15822 6.210% [40-45) 27.82904 25.18055 9.517% 33.24948 31.02846 6.680% [45-50) 24.98427 22.25113 10.939% 30.18192 27.88902 7.597% [50-55) 22.29499 19.48080 12.623% 26.96890 24.66034 8.560% [55-60) 19.87433 16.86759 15.129% 23.39752 21.07422 9.930% [60-65) 17.61765 14.40731 18.222% 20.35666 17.90773 12.030% 71 University of Ghana http://ugspace.ug.edu.gh Table 4.22 Life Expectancy Gain from Eliminating Malaria Cont. Male Female Age Group Life malaria Expectan Life malaria Expectancy Expectancy eliminated cy Gain Expectancy eliminated Gain [x - x ) eˆ eˆ eˆ j.MReˆ j eˆ eˆ eˆ j.MReˆj j+1 j.m j j j.meˆ j j eˆ j [65-70) 15.80820 12.28149 22.309% 17.65721 15.04072 14.818% [70-75) 14.61401 10.69364 26.826% 15.46818 12.56849 18.746% [75-80) 13.64221 9.27991 31.976% 13.85894 10.65857 23.093% [80-85) 12.26074 7.54770 38.440% 12.15237 8.57639 29.426% [85-90) 10.36267 5.86261 43.426% 10.70850 6.66230 37.785% [90-95) 8.41524 4.88672 41.930% 9.49704 5.21845 45.052% [95-100) 4.94881 3.32360 32.840% 8.15315 3.86364 52.612% 100+ 1.04300 0.93167 10.674% 4.71264 2.50000 46.951% 72 University of Ghana http://ugspace.ug.edu.gh CHAPTER FIVE SUMMARY, CONCLUSION AND RECOMMENDATION 5.1 Introduction This chapter presents the summary, conclusions and recommendations of errors assessment of applying the foreign life tables in Ghana and competing risk analysis. 5.2 Summary The application of foreign life table in Sub-Sahara Africa had been a major concern all over the world. However, assessing the error for applying the foreign life tables as well as analysing the impact of the leading causes of death on Ghanaian longevity had not been studied. This study aimed at assessing the error of applying Foreign Life Table in Ghana and analysing the impacts of the leading causes of death on he lives of Ghanaians. The specific objectives were to construct a life table for Ghana and to determine the expectation of life at birth in Ghana. The other specific objectives were to assess the error generated from applying foreign life table, to derive the survival function of Ghana, to compare the mortality experience by male and female and to determine the life expectancy, gain when eliminating the leading cause of death. A cohort data was extracted from the administrative records unit of University of Ghana Hospital. The gender, age and the age which each cohort member die was recorded. The causes of death were grouped into five main causes namely; death due to cardiovascular diseases, HIV/AIDs, lower respiratory infections, malaria and other causes of death. 73 University of Ghana http://ugspace.ug.edu.gh 5.3 Summary of findings The study revealed that the highest life expectancy record was at age 5. Then the life expectancy begins to decrease as the ages increases, leading to a negative relationship between age and life expectancy. It can be seen that the life expected at birth was for females was 53.6256 years, that of males was 49.5267 and 51.1359 was record for combine sex. The complete life table was compare with some selected foreign life tables to assess the error of applying these foreign life tables in Ghana. Plotting a graph in Figure 4.2, it was revealed that graph life expectancy of USA and UK behave in almost the same manner and they were far away from the study life expectancy. The South Africa table was the closest to the study life table. The United Kingdom error exhibit the highest error, followed by the USA foreign Life Table whiles South Africa Life table yielded the lowest error. The force of mortality of males and females were compared and it was found that the instantaneous rate of mortality of the males were higher than that of females. The log-rank test performed also confirms that the survival curves for male and female were significantly different. In calculating the crude probabilities for each cause of death it was realised that for children under 5 years, the probability of them dying from Lower Respiratory Infections and malaria were very high, whiles for the aged people 60 years and above, their chances of dying from cardiovascular diseases were very high. Examine the impact of the leading causes it was revealed that the life expectancy at birth for males in Ghana will increase by 16.021% and that of the female by 17.587% when death caused by Cardiovascular Diseases are eliminated. It was also found out that Ghana life expectancy will increase by 10.054% and 6.350% for male and female respectively, if the death of no HIV/AIDs diseases in the population the life expectancy at birth will increase by 6.048% for males and 5.275% for females. For malaria, it was released that if the country eliminates the death of malaria Ghanaians life expectancies will across all the ages. 74 University of Ghana http://ugspace.ug.edu.gh 5.3 Conclusions The following conclusions were drawn from the study: The life expectation at birth for Ghana is about 51.1359 (combined sex), 53.63 years for females and 49.5267 years for males. Mortality among males is higher than their females’ counterparts in most age groups. The leading causes of death in Ghana is Cardiovascular disease followed by HIV/AIDS. The life expectancy of males will increase by 16.021% and 6.35% if cardiovascular diseases and HIV/AIDS is eliminated respectively in Ghana. Also, life expectancy of females will increase by 17.587% and 10.054% if cardiovascular diseases and HIV/AIDS is eliminated in Ghana. 5.4 Recommendations Base on the findings and conclusion the following recommendation was made;  There is a need for Ghana to have its own life table since the foreign life tables differ from Ghanaian mortality.  The foreign life table that is similar to Ghana mortality is the South Africa Life table and therefore institutions and agencies such as SSNIT and insurance companies continue applying South African life table than other foreign life table.  The government and other agencies should create awareness on cardiovascular diseases since it is the leading causes of death in the country.  The Ghana AIDS commission should continue to create public awareness since the cause of death due to HIV/AIDS is still prevalence 75 University of Ghana http://ugspace.ug.edu.gh REFERENCES Albertsen, P. C., Hanley, J. A., Gleason, D. F., & Barry, M. J. (1998). 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Population research and policy review, 31(2), 207-234. 79 University of Ghana http://ugspace.ug.edu.gh APPENDICES Appendix I: Complete Life Table Combined Sex Age 𝑙𝑖 𝑞𝑗 𝑑𝑗 𝑎𝑗 𝐿𝑗 𝑇 𝑒 Group 𝑗 𝑗 0 0.0304 3260 99 0.1 3170.90 166703.19 51.1359 1 0.0310 3161 98 0.43 3105.14 163532.29 51.7344 2 0.0284 3063 87 0.45 3015.15 160427.15 52.3758 3 0.0269 2976 80 0.47 2933.60 157412.00 52.8938 4 0.0207 2896 60 0.49 2865.40 154478.40 53.3420 5 0.0109 2836 31 0.5 2820.50 151613.00 53.4602 6 0.0089 2805 25 0.5 2792.50 148792.50 53.0455 7 0.0050 2780 14 0.5 2773.00 146000.00 52.5180 8 0.0011 2766 3 0.5 2764.50 143227.00 51.7813 9 0.0043 2763 12 0.5 2757.00 140462.50 50.8370 10 0.0036 2751 10 0.5 2746.00 137705.50 50.0565 11 0.0011 2741 3 0.5 2739.50 134959.50 49.2373 12 0.0018 2738 5 0.5 2735.50 132220.00 48.2907 13 0.0015 2733 4 0.5 2731.00 129484.50 47.3782 14 0.0018 2729 5 0.5 2726.50 126753.50 46.4469 15 0.0026 2724 7 0.5 2720.50 124027.00 45.5312 16 0.0055 2717 15 0.5 2709.50 121306.50 44.6472 17 0.0048 2702 13 0.5 2695.50 118597.00 43.8923 18 0.0052 2689 14 0.5 2682.00 115901.50 43.1021 19 0.0056 2675 15 0.5 2667.50 113219.50 42.3250 20 0.0041 2660 11 0.5 2654.50 110552.00 41.5609 80 University of Ghana http://ugspace.ug.edu.gh Complete Life Table Combined Sex Cont. Age 𝑙𝑖 𝑞𝑗 𝑑𝑗 𝑎𝑗 𝐿𝑗 𝑇𝑗 𝑒 Group 𝑗 21 0.0042 2649 11 0.5 2643.50 107897.50 40.7314 22 0.0049 2638 13 0.5 2631.50 105254.00 39.8992 23 0.0050 2625 13 0.5 2618.50 102622.50 39.0943 24 0.0050 2612 13 0.5 2605.50 100004.00 38.2864 25 0.0065 2599 17 0.5 2590.50 97398.50 37.4754 26 0.0074 2582 19 0.5 2572.50 94808.00 36.7188 27 0.0074 2563 19 0.5 2553.50 92235.50 35.9873 28 0.0102 2544 26 0.5 2531.00 89682.00 35.2524 29 0.0075 2518 19 0.5 2508.50 87151.00 34.6112 30 0.0108 2499 27 0.5 2485.50 84642.50 33.8705 31 0.0065 2472 16 0.5 2464.00 82157.00 33.2350 32 0.0134 2456 33 0.5 2439.50 79693.00 32.4483 33 0.0132 2423 32 0.5 2407.00 77253.50 31.8834 34 0.0105 2391 25 0.5 2378.50 74846.50 31.3034 35 0.0152 2366 36 0.5 2348.00 72468.00 30.6289 36 0.0129 2330 30 0.5 2315.00 70120.00 30.0944 37 0.0091 2300 21 0.5 2289.50 67805.00 29.4804 38 0.0145 2279 33 0.5 2262.50 65515.50 28.7475 39 0.0120 2246 27 0.5 2232.50 63253.00 28.1625 40 0.0149 2219 33 0.5 2202.50 61020.50 27.4991 81 University of Ghana http://ugspace.ug.edu.gh Complete Life Table Combined Sex Cont. Age 𝑙𝑖 𝑞𝑗 𝑑𝑗 𝑎𝑗 𝐿𝑗 𝑇𝑗 𝑒Group 𝑗 41 0.0151 2186 33 0.5 2169.50 58818.00 26.9067 42 0.0172 2153 37 0.5 2134.50 56648.50 26.3114 43 0.0165 2116 35 0.5 2098.50 54514.00 25.7628 44 0.0135 2081 28 0.5 2067.00 52415.50 25.1877 45 0.0205 2053 42 0.5 2032.00 50348.50 24.5244 46 0.0164 2011 33 0.5 1994.50 48316.50 24.0261 47 0.0172 1978 34 0.5 1961.00 46322.00 23.4186 48 0.0175 1944 34 0.5 1927.00 44361.00 22.8194 49 0.0183 1910 35 0.5 1892.50 42434.00 22.2168 50 0.0181 1875 34 0.5 1858.00 40541.50 21.6221 51 0.0190 1841 35 0.5 1823.50 38683.50 21.0122 52 0.0194 1806 35 0.5 1788.50 36860.00 20.4097 53 0.0209 1771 37 0.5 1752.50 35071.50 19.8032 54 0.0219 1734 38 0.5 1715.00 33319.00 19.2151 55 0.0236 1696 40 0.5 1676.00 31604.00 18.6344 56 0.0242 1656 40 0.5 1636.00 29928.00 18.0725 57 0.0260 1616 42 0.5 1595.00 28292.00 17.5074 58 0.0273 1574 43 0.5 1552.50 26697.00 16.9612 59 0.0287 1531 44 0.5 1509.00 25144.50 16.4236 82 University of Ghana http://ugspace.ug.edu.gh Complete Life Table Combined Sex Cont. Age 𝑙𝑖 𝑞𝑗 𝑑𝑗 𝑎𝑗 𝐿𝑗 𝑇𝑗 𝑒𝑗 Group 60 0.0296 1487 44 0.5 1465.00 23635.50 15.8948 61 0.0319 1443 46 0.5 1420.00 22170.50 15.3642 62 0.0344 1397 48 0.5 1373.00 20750.50 14.8536 63 0.0371 1349 50 0.5 1324.00 19377.50 14.3643 64 0.0393 1299 51 0.5 1273.50 18053.50 13.8980 65 0.0409 1248 51 0.5 1222.50 16780.00 13.4455 66 0.0434 1197 52 0.5 1171.00 15557.50 12.9971 67 0.0472 1145 54 0.5 1118.00 14386.50 12.5646 68 0.0504 1091 55 0.5 1063.50 13268.50 12.1618 69 0.0550 1036 57 0.5 1007.50 12205.00 11.7809 70 0.0582 979 57 0.5 950.50 11197.50 11.4377 71 0.0597 922 55 0.5 894.50 10247.00 11.1139 72 0.0611 867 53 0.5 840.50 9352.50 10.7872 73 0.0627 814 51 0.5 788.50 8512.00 10.4570 74 0.0642 763 49 0.5 738.50 7723.50 10.1225 75 0.0644 714 46 0.5 691.00 6985.00 9.7829 76 0.0659 668 44 0.5 646.00 6294.00 9.4222 77 0.0657 624 41 0.5 603.50 5648.00 9.0513 78 0.0669 583 39 0.5 563.50 5044.50 8.6527 79 0.0717 544 39 0.5 524.50 4481.00 8.2371 83 University of Ghana http://ugspace.ug.edu.gh Complete Life Table Combined Sex Cont. Age 𝑙𝑖 𝑞𝑗 𝑑𝑗 𝑎𝑗 𝐿𝑗 𝑇Group 𝑗 𝑒𝑗 80 0.0733 505 37 0.5 486.50 3956.50 7.8347 81 0.0791 468 37 0.5 449.50 3470.00 7.4145 82 0.0882 431 38 0.5 412.00 3020.50 7.0081 83 0.0941 393 37 0.5 374.50 2608.50 6.6374 84 0.1011 356 36 0.5 338.00 2234.00 6.2753 85 0.1156 320 37 0.5 301.50 1896.00 5.9250 86 0.1201 283 34 0.5 266.00 1594.50 5.6343 87 0.1285 249 32 0.5 233.00 1328.50 5.3353 88 0.1382 217 30 0.5 202.00 1095.50 5.0484 89 0.1497 187 28 0.5 173.00 893.50 4.7781 90 0.1572 159 25 0.5 146.50 720.50 4.5314 91 0.1567 134 21 0.5 123.50 574.00 4.2836 92 0.1681 113 19 0.5 103.50 450.50 3.9867 93 0.1702 94 16 0.5 86.00 347.00 3.6915 94 0.1795 78 14 0.5 71.00 261.00 3.3462 95 0.2031 64 13 0.5 57.50 190.00 2.9688 96 0.2157 51 11 0.5 45.50 132.50 2.5980 97 0.2500 40 10 0.5 35.00 87.00 2.1750 98 0.2667 30 8 0.5 26.00 52.00 1.7333 99 0.3182 22 7 0.5 18.50 26.00 1.1818 100+ 1.0000 15 15 0.5 7.50 7.50 0.5000 84 University of Ghana http://ugspace.ug.edu.gh Appendix II Abridge Life Table Abridge Life Table for Combined Sex Age 𝑙𝑖 𝑞𝑗 𝑑𝑗 𝑎Group 𝑗 𝑛𝑗 𝐿𝑗 𝑇𝑗 𝑒𝑗 [0-1) 0.030 3260 99 0.1 1 3170.9 167081.9 51.252 [1-5) 0.103 3161 325 0.39 4 11851 163911 51.854 [5-10) 0.030 2836 85 0.5 5 13967.5 152060 53.618 [10-15) 0.010 2751 27 0.5 5 13687.5 138092.5 50.197 [15-20) 0.023 2724 64 0.5 5 13460 124405 45.670 [20-25) 0.023 2660 61 0.5 5 13147.5 110945 41.709 [25-30) 0.038 2599 99 0.5 5 12747.5 97797.5 37.629 [30-35) 0.053 2500 132 0.5 5 12170 85050 34.020 [35-40) 0.062 2368 146 0.5 5 11475 72880 30.777 [40-45) 0.074 2222 165 0.5 5 10697.5 61405 27.635 [45-50) 0.086 2057 177 0.5 5 9842.5 50707.5 24.651 [50-55) 0.096 1880 180 0.5 5 8950 40865 21.737 85 University of Ghana http://ugspace.ug.edu.gh Abridge Life Table for Combined Sex Cont. [55-60) 0.123 1700 209 0.5 5 7977.5 31915 18.774 [60-65) 0.160 1491 239 0.5 5 6857.5 23937.5 16.055 [65-70) 0.214 1252 268 0.5 5 5590 17080 13.642 [70-75) 0.268 984 264 0.5 5 4260 11490 11.677 [75-80) 0.292 720 210 0.5 5 3075 7230 10.042 [80-85) 0.363 510 185 0.5 5 2087.5 4155 8.147 [85-90) 0.495 325 161 0.5 5 1222.5 2067.5 6.362 [90-95) 0.579 164 95 0.5 5 582.5 845 5.152 [95-100) 0.739 69 51 0.5 5 217.5 262.5 3.804 100+ 1.000 18 18 0.5 5 45 45 2.500 86 University of Ghana http://ugspace.ug.edu.gh Appendix III Kaplan-Meier Survival Curve Kaplan-Meier survival estimates 0 20 40 60 80 100 analysis time gender = Male gender = Female 87 0.00 0.25 0.50 0.75 1.00 University of Ghana http://ugspace.ug.edu.gh Appendix IV Multiple Decrement Table for Mortality Multiple Decrement Table for Male Mortality Deaths Deaths Deaths by Deaths No. of by Deaths by by by Total Age people Cardio lower HIV/AIDs other death Group respiratory Malaria living vascular causes 𝛼𝑙 𝛽 𝛽(𝛼𝑑) 1 (𝛼𝑑) 2 𝛽 𝛽 𝛽 [x - x ) 𝑥𝑗 𝑥 𝑥 (𝛼𝑑) 3 𝑥 (𝛼𝑑) 4 𝑥 (𝛼𝑑) 5 𝑥 j j+1 𝑗 𝑗 𝑗 𝑗 𝑗 (𝛼𝑑)𝑥 𝑗 [0-1) 1890 0 1 2 5 51 59 [1-5) 1831 0 0 42 16 132 190 [5-10) 1641 0 1 6 10 42 59 [10-15) 1582 0 3 1 1 7 12 [15-20) 1570 1 2 5 1 30 39 [20-25) 1531 1 0 1 1 31 34 [25-30) 1497 4 4 2 1 34 45 [30-35) 1452 8 10 3 7 42 70 [35-40) 1382 11 18 4 6 52 91 [40-45) 1291 23 23 1 3 58 108 [45-50) 1183 37 21 0 5 57 120 [50-55) 1063 43 12 0 2 74 131 88 University of Ghana http://ugspace.ug.edu.gh Multiple Decrement Table for Male Mortality Cont. Deaths Deaths Deaths by Deaths No. of by Deaths by by by Total Age people Cardio lower HIV/AIDs other death Group respiratory Malaria living vascular causes 𝛼𝑙𝑥 𝛽 𝛽 𝛽 𝛽 𝛽 [x - x ) 𝑗 (𝛼𝑑) 1 (𝛼𝑑) 2𝑥 𝑥 (𝛼𝑑) 3 𝑥 (𝛼𝑑) 4 (𝛼𝑑) 5𝑥 𝑥 j j+1 𝑗 𝑗 𝑗 𝑗 𝑗 (𝛼𝑑)𝑥 𝑗 [50-55) 1063 43 12 0 2 74 131 [55-60) 932 46 8 0 5 81 140 [60-65) 792 65 4 1 5 79 154 [65-70) 638 89 2 3 8 63 165 [70-75) 473 53 1 1 7 82 144 [75-80) 329 49 1 1 6 50 107 [80-85) 222 35 0 0 13 40 88 [85-90) 134 25 1 1 7 39 73 [90-95) 61 8 0 0 1 27 36 [95-100) 25 7 0 1 4 7 19 100+ 6 2 0 0 0 4 6 89 University of Ghana http://ugspace.ug.edu.gh Multiple Decrement Table for Female Mortality Deaths Deaths by Deaths by Deaths No. of Deaths by by Cardio by Total Age Group people lower HIV/AIDs other death vascular respiratory Malaria living causes [xj - xj+1) 𝛼𝑙 𝛽1 𝛽 𝛽 𝛽 𝛽𝑥𝑗 (𝛼𝑑)𝑥 (𝛼𝑑) 2 ( ) 𝑥 (𝛼𝑑) 3 4 5 𝛼𝑑 𝑥 𝑗 𝑗 𝑥 (𝛼𝑑)𝑥 (𝛼𝑑)𝑗 𝑗 𝑥 𝑗𝑗 [0-1) 1370 0 1 5 8 26 40 [1-5) 1330 1 2 40 12 80 135 [5-10) 1195 0 0 11 1 14 26 [10-15) 1169 1 2 4 1 7 15 [15-20) 1154 1 0 1 1 22 25 [20-25) 1129 2 3 1 0 21 27 [25-30) 1102 1 24 1 3 25 54 [30-35) 1048 6 23 0 4 29 62 [35-40) 986 6 16 0 3 30 55 [40-45) 931 14 17 0 0 26 57 [45-50) 874 18 10 0 2 27 57 [50-55) 817 20 12 0 2 15 49 90 University of Ghana http://ugspace.ug.edu.gh Multiple Decrement Table for Female Mortality Cont. Deaths Deaths by Deaths by Deaths No. of Deaths by by Cardio by Total Age people lower HIV/AIDs other death Group vascular respiratory Malaria living causes [xj - xj+1) 𝛼𝑙 𝛽1 𝛽2 𝛽3 𝛽4 𝛽𝑥𝑗 (𝛼𝑑)𝑥 (𝛼𝑑)𝑥 (𝛼𝑑)𝑥 (𝛼𝑑) (𝛼𝑑) 5 ( ) 𝛼𝑑 𝑥 𝑗 𝑗 𝑗 𝑥𝑗 𝑥 𝑗 𝑗 [55-60) 768 32 7 0 0 30 69 [60-65) 699 38 4 0 1 42 85 [65-70) 614 67 2 2 1 31 103 [70-75) 511 60 0 0 6 54 120 [75-80) 391 53 0 0 5 45 103 [80-85) 288 50 0 1 6 40 97 [85-90) 191 42 1 3 10 32 88 [90-95) 103 28 0 0 1 30 59 [95-100) 44 10 0 3 8 11 32 100+ 12 5 0 1 2 4 12 91 University of Ghana http://ugspace.ug.edu.gh Appendix V: Net Probabilities (When there is Only that Risk) Net Probabilities for Cardiovascular Diseases Age Group Male Female [x - x ) qˆ S Sj j+1 ˆ jCD q̂ q q̂jCD jCD jCD [0-1) 0.00000 0.00000 0.00000 0.00000 [1-5) 0.00000 0.00000 0.00079 0.00030 [5-10) 0.00000 0.00000 0.00000 0.00000 [10-15) 0.00000 0.00000 0.00086 0.00036 [15-20) 0.00064 0.00029 0.00088 0.00035 [20-25) 0.00066 0.00030 0.00179 0.00054 [25-30) 0.00271 0.00069 0.00093 0.00036 [30-35) 0.00563 0.00105 0.00588 0.00105 [35-40) 0.00820 0.00132 0.00624 0.00113 [40-45) 0.01843 0.00229 0.01540 0.00216 [45-50) 0.03244 0.00348 0.02107 0.00282 [50-55) 0.04225 0.00426 0.02493 0.00348 92 University of Ghana http://ugspace.ug.edu.gh Net Probabilities for Cardiovascular Diseases Cont. Age Group Female Male [xj - xj+1) qˆ S qˆ SjCD q̂ q̂ jCD jCD jCD [55-60) 0.05208 0.00505 0.04272 0.00504 [60-65) 0.08722 0.00754 0.05632 0.00619 [65-70) 0.14906 0.01156 0.11259 0.01077 [70-75) 0.12508 0.01126 0.12526 0.01214 [75-80) 0.16485 0.01617 0.14557 0.01572 [80-85) 0.18191 0.02002 0.19079 0.02162 [85-90) 0.23625 0.02866 0.25527 0.03109 [90-95) 0.17981 0.03562 0.33212 0.05141 [95-100) 0.67043 0.04740 0.33371 0.07934 100+ 1.00000 1.00000 93 University of Ghana http://ugspace.ug.edu.gh Net Probabilities for HIV/AIDS Age Male Female Group [xj - xj+1) qˆ jHIV S q̂ qˆ S jHIV jHIV q̂ jHIV [0-1) 0.00054 0.00024 0.00074 0.00029 [1-5) 0.00000 0.00000 0.00158 0.00043 [5-10) 0.00062 0.00026 0.00000 0.00000 [10-15) 0.00190 0.00069 0.00172 0.00057 [15-20) 0.00129 0.00044 0.00000 0.00000 [20-25) 0.00000 0.00000 0.00269 0.00070 [25-30) 0.00271 0.00069 0.02208 0.00283 [30-35) 0.00703 0.00122 0.02237 0.00274 [35-40) 0.01338 0.00185 0.01656 0.00226 [40-45) 0.01843 0.00229 0.01867 0.00250 [45-50) 0.01854 0.00231 0.01176 0.00183 [50-55) 0.01198 0.00176 0.01503 0.00235 94 University of Ghana http://ugspace.ug.edu.gh Net Probabilities for HIV/AIDS Cont. Age Female Male Group [xj - xj+1) qˆ jHIV S q̂ qˆ S jHIV jHIV q̂ jHIV [55-60) 0.00926 0.00158 0.00950 0.00169 [60-65) 0.00560 0.00128 0.00608 0.00136 [65-70) 0.00362 0.00114 0.00356 0.00111 [70-75) 0.00252 0.00111 0.00000 0.00000 [75-80) 0.00367 0.00162 0.00000 0.00000 [80-85) 0.00000 0.00000 0.00000 0.00000 [85-90) 0.01072 0.00477 0.00699 0.00370 [90-95) 0.00000 0.00000 0.00000 0.00000 [95-100) 0.00000 0.00000 0.00000 0.00000 100+ 1.00000 1.00000 95 University of Ghana http://ugspace.ug.edu.gh Net Probabilities for Lower Respiratory Disease Age Male Female Group [x S Sj - xj+1) qˆ jLR q̂ qˆ jLR q̂ jLR jLR [0-1) 0.00107 0.00035 0.00370 0.00076 [1-5) 0.02393 0.00221 0.03122 0.00271 [5-10) 0.00372 0.00075 0.00926 0.00164 [10-15) 0.00063 0.00032 0.00344 0.00094 [15-20) 0.00322 0.00078 0.00088 0.00035 [20-25) 0.00066 0.00030 0.00090 0.00036 [25-30) 0.00136 0.00045 0.00093 0.00036 [30-35) 0.00212 0.00058 0.00000 0.00000 [35-40) 0.00299 0.00071 0.00000 0.00000 [40-45) 0.00081 0.00036 0.00000 0.00000 [45-50) 0.00000 0.00000 0.00000 0.00000 [50-55) 0.00000 0.00000 0.00000 0.00000 96 University of Ghana http://ugspace.ug.edu.gh Net Probabilities for Lower Respiratory Disease Cont. Age Female Male Group [x S Sj - xj+1) qˆ ˆjLR q̂ q q̂ jLR jLR jLR [55-60) 0.00000 0.00000 0.00000 0.00000 [60-65) 0.00140 0.00062 0.00000 0.00000 [65-70) 0.00543 0.00141 0.00356 0.00111 [70-75) 0.00252 0.00111 0.00000 0.00000 [75-80) 0.00367 0.00162 0.00000 0.00000 [80-85) 0.00000 0.00000 0.00422 0.00213 [85-90) 0.01072 0.00477 0.02083 0.00662 [90-95) 0.00000 0.00000 0.00000 0.00000 [95-100) 0.14664 0.04135 0.11468 0.04360 100+ 1.00000 0.00000 1.00000 0.00000 97 University of Ghana http://ugspace.ug.edu.gh Net Probabilities for Malaria Age Group Male Female [xj - xj+1) qˆ jMR S q̂ qˆ Sq̂ jMR jMR jMR [0-1) 0.00268 0.00061 0.00591 0.00105 [1-5) 0.00918 0.00116 0.00947 0.00118 [5-10) 0.00619 0.00105 0.00085 0.00032 [10-15) 0.00063 0.00032 0.00086 0.00036 [15-20) 0.00064 0.00029 0.00088 0.00035 [20-25) 0.00066 0.00030 0.00000 0.00000 [25-30) 0.00068 0.00031 0.00279 0.00067 [30-35) 0.00493 0.00097 0.00393 0.00082 [35-40) 0.00448 0.00090 0.00313 0.00073 [40-45) 0.00242 0.00064 0.00000 0.00000 [45-50) 0.00445 0.00094 0.00236 0.00067 98 University of Ghana http://ugspace.ug.edu.gh Net Probabilities for Malaria Cont. Age Group Female Male [xj - xj+1) qˆ S ˆ S jMR q̂ q q̂jMR jMR jMR [50-55) 0.00201 0.00064 0.00252 0.00074 [55-60) 0.00580 0.00121 0.00000 0.00000 [60-65) 0.00700 0.00145 0.00152 0.00064 [65-70) 0.01440 0.00242 0.00178 0.00077 [70-75) 0.01749 0.00313 0.01329 0.00268 [75-80) 0.02182 0.00427 0.01473 0.00341 [80-85) 0.07187 0.01072 0.02508 0.00564 [85-90) 0.07269 0.01378 0.06777 0.01317 [90-95) 0.02447 0.01116 0.01431 0.00833 [95-100) 0.46968 0.05462 0.27734 0.07209 100+ 1.00000 1.00000 99 University of Ghana http://ugspace.ug.edu.gh Appendix VI: Net Probabilities (When The Risk Is Eliminated) Net Probabilities of Cardiovascular Disease Age Group Male Female [x - x S Sj j+1) qˆ ˆ j.CD q̂ q q̂j.CD j.CD j.CD [0-1) 0.03122 0.00000 0.02920 0.00000 [1-5) 0.10377 0.00000 0.10079 0.00029 [5-10) 0.03595 0.00000 0.02176 0.00000 [10-15) 0.00759 0.00000 0.01198 0.00031 [15-20) 0.02421 0.00028 0.02081 0.00032 [20-25) 0.02156 0.00028 0.02216 0.00045 [25-30) 0.02743 0.00056 0.04812 0.00034 [30-35) 0.04282 0.00081 0.05359 0.00084 [35-40) 0.05812 0.00100 0.04985 0.00088 [40-45) 0.06645 0.00147 0.04654 0.00131 [45-50) 0.07131 0.00190 0.04510 0.00152 [50-55) 0.08456 0.00225 0.03594 0.00161 100 University of Ghana http://ugspace.ug.edu.gh Net Probabilities of Cardiovascular Disease Cont. Age Group Male Female [xj - xj+1) qˆ j.CD S q̂ qˆ Sq̂ j.CD j.CD j.CD [55-60) 0.10353 0.00267 0.04923 0.00214 [60-65) 0.11747 0.00344 0.06918 0.00270 [65-70) 0.12876 0.00435 0.06216 0.00328 [70-75) 0.20501 0.00566 0.12526 0.00488 [75-80) 0.19203 0.00705 0.13793 0.00618 [80-85) 0.26218 0.00975 0.18044 0.00857 [85-90) 0.40397 0.01592 0.27589 0.01349 [90-95) 0.50031 0.02434 0.36039 0.02330 [95-100) 0.27177 0.06333 0.59068 0.05181 100+ 1.00000 1.00000 101 University of Ghana http://ugspace.ug.edu.gh Net Probabilities of HIV/AIDS Age Group Male Female [xj - xj+1) qˆ S Sq̂ qˆ j.HIV j.HIV j.HIV q̂ j.HIV [0-1) 0.03070 0.00023 0.02848 0.00027 [1-5) 0.10377 0.00000 0.10008 0.00041 [5-10) 0.03536 0.00025 0.02176 0.00000 [10-15) 0.00569 0.00041 0.01113 0.00042 [15-20) 0.02358 0.00039 0.02166 0.00000 [20-25) 0.02221 0.00000 0.02129 0.00054 [25-30) 0.02743 0.00056 0.02753 0.00124 [30-35) 0.04147 0.00089 0.03763 0.00135 [35-40) 0.05318 0.00122 0.03988 0.00127 [40-45) 0.06645 0.00147 0.04337 0.00139 [45-50) 0.08446 0.00158 0.05409 0.00125 [50-55) 0.11261 0.00142 0.04563 0.00143 102 University of Ghana http://ugspace.ug.edu.gh Net Probabilities of HIV/AIDS cont’d Age Group Female Male [x - x ) qˆ S qˆ Sj j+1 j.HIV q̂ q̂j.HIV j.HIV j.HIV [55-60) 0.14227 0.00138 0.08111 0.00134 [60-65) 0.18991 0.00120 0.11623 0.00121 [65-70) 0.25593 0.00110 0.16478 0.00106 [70-75) 0.30268 0.00109 0.23483 0.00000 [75-80) 0.32274 0.00159 0.26343 0.00000 [80-85) 0.39640 0.00000 0.33681 0.00000 [85-90) 0.53984 0.00463 0.45694 0.00361 [90-95) 0.59016 0.00000 0.57282 0.00000 [95-100) 0.76000 0.00000 0.72727 0.00000 100+ 1.00000 1.00000 103 University of Ghana http://ugspace.ug.edu.gh Net Probabilities of Lower Respiration Age Male Female Group [xj - xj+1) qˆ S ˆ S j.LR q̂ q q̂j.LR j.LR j.LR [0-1) 0.03017 0.00032 0.02559 0.00057 [1-5) 0.08180 0.00140 0.07255 0.00151 [5-10) 0.03236 0.00059 0.01261 0.00074 [10-15) 0.00696 0.00026 0.00943 0.00055 [15-20) 0.02169 0.00058 0.02081 0.00032 [20-25) 0.02156 0.00028 0.02304 0.00033 [25-30) 0.02874 0.00041 0.04812 0.00034 [30-35) 0.04619 0.00052 0.05916 0.00000 [35-40) 0.06305 0.00063 0.05578 0.00000 [40-45) 0.08291 0.00035 0.06122 0.00000 [45-50) 0.10144 0.00000 0.06522 0.00000 [50-55) 0.12324 0.00000 0.05998 0.00000 104 University of Ghana http://ugspace.ug.edu.gh Net Probabilities of Lower Respiration Cont. Age Female Male Group [xj - x Sj+1) qˆ q̂ j.LR qˆ S j.LR q̂ j.LR j.LR [55-60) 0.15021 0.00000 0.08984 0.00000 [60-65) 0.19331 0.00061 0.12160 0.00000 [65-70) 0.25458 0.00135 0.16478 0.00106 [70-75) 0.30268 0.00109 0.23483 0.00000 [75-80) 0.32274 0.00159 0.26343 0.00000 [80-85) 0.39640 0.00000 0.33399 0.00208 [85-90) 0.53984 0.00463 0.44926 0.00613 [90-95) 0.59016 0.00000 0.57282 0.00000 [95-100) 0.71876 0.05718 0.69194 0.03883 100+ 1.00000 1.00000 105 University of Ghana http://ugspace.ug.edu.gh Net Probabilities of Malaria Age Male Female Group [x S Sj - xj+1) qˆ j.m q̂ qˆ j.m j.m q̂ j.m [0-1) 0.02861 0.00050 0.02343 0.00069 [1-5) 0.09546 0.00095 0.09291 0.00096 [5-10) 0.02995 0.00073 0.02093 0.00029 [10-15) 0.00696 0.00026 0.01198 0.00031 [15-20) 0.02421 0.00028 0.02081 0.00032 [20-25) 0.02156 0.00028 0.02391 0.00000 [25-30) 0.02940 0.00029 0.04634 0.00058 [30-35) 0.04349 0.00077 0.05545 0.00070 [35-40) 0.06164 0.00077 0.05282 0.00064 [40-45) 0.08143 0.00060 0.06122 0.00000 [45-50) 0.09742 0.00084 0.06300 0.00061 [50-55) 0.12147 0.00061 0.05760 0.00066 106 University of Ghana http://ugspace.ug.edu.gh Net Probabilities of Malaria Cont. Age Female Male Group [xj - xj+1) qˆ j.m S q̂ qˆ S j.m j.m q̂ j.m [55-60) 0.14526 0.00110 0.08984 0.00000 [60-65) 0.18877 0.00133 0.12026 0.00062 [65-70) 0.24779 0.00215 0.16627 0.00075 [70-75) 0.29206 0.00279 0.22452 0.00238 [75-80) 0.31018 0.00374 0.25241 0.00303 [80-85) 0.34966 0.00788 0.31974 0.00488 [85-90) 0.50909 0.01141 0.42153 0.01040 [90-95) 0.57988 0.01055 0.56661 0.00804 [95-100) 0.54744 0.08555 0.62260 0.05173 100+ 1.00000 1.00000 107