Life Table and Population Projection Using the Leslie Matrix A PROJECT SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Amber Koslucher IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE Dr. Richard Green July, 2016 Acknowledgements I would like to give a special thanks to my advisor, Dr. Richard Green. His help and knowledge have been a tremendous help to me during my undergraduate and graduate career at the University of Minnesota Duluth. All of his time and help have been greatly appreciated. I would also like to thank my parents for supporting me throughout my academic career. Their constant support and encouragement have helped me along the way. i Abstract A life table is a table that shows different values pertaining to the mortality rate for humans. The table is broken up by age and includes the following values: the probability that a person of age x will die within the next year, the number of people surviving from the original cohort to age x, the number of people of age x that die within the year, and the expected lifetime remaining for a person of age x. These values are often used to compare the health of different countries. This can be done by comparing their life expectancy at birth, the age-adjusted mortality rates, or by looking at their population pyramids. An age-distribution is the number of people alive for each age x from the original cohort. Using mortality rates along with fertility rates, future age-distributions can be determined. The Leslie Matrix is a tool used to make such projections. The Leslie Matrix uses fertility and survival rates to project the age- distribution to the following year. The Leslie Matrix will be used to project the United States population from the year 2010 to the years 2011, 2016, and 2050. ii Contents Acknowledgements i Abstract ii List of Tables v List of Figures vii 1 Life Table 1 1.1 Definitions and Assumptions . .1 1.2 History of Life Table . .2 1.3 Life Table Calculations . .2 1.4 Life Table Applications . .4 1.5 Life Table Example . .5 2 Life Expectancy 10 2.1 Life Expectancy . 10 2.2 Importance of Life Expectancy . 10 2.3 Comparison . 11 3 Mortality Rates 13 3.1 Crude Mortality Rate . 13 3.2 Age-Specific Mortality Rate . 14 3.3 Age-Adjusted Mortality Rate . 16 3.4 Comparison . 19 iii 4 Life Table Functions 21 4.1 Density Function . 21 4.2 Cumulative Distribution Function . 22 4.3 Survival Function . 23 4.4 Hazard Function . 26 5 Population Pyramids 27 5.1 Definition . 27 5.2 Types of PopulationPyramids . 27 6 Population Projections 32 6.1 Population Projection Definition . 32 6.2 Leslie Matrix . 32 6.3 Stable Age-Distribution . 34 7 My Findings and Projections 37 7.1 Data Being Used . 37 7.2 United States Projections . 43 7.3 United States Projections Using Different Mortality and Fertility Rates 50 7.4 Stable Age Vector . 56 7.5 Conclusion . 58 References 60 iv List of Tables 1.1 Graunt's Life Table . .3 1.2 United States Life Table 2010 . .5 1.2 United States Life Table 2010 . .6 1.2 United States Life Table 2010 . .7 1.2 United States Life Table 2010 . .8 1.2 United States Life Table 2010 . .9 2.1 Comparing Life Expectancy, 2010 - 2014 . 11 3.1 Comparing Crude Mortality Rates 2010 - 2014 . 14 3.2 United States 2000 and 1940 Standard Population . 18 3.3 Comparing Crude and Age-Specific Mortality Rates 2010 - 2014 . 19 7.1 Birth Rates by Age of Mother. United States, 2010 . 38 7.2 Survival and Fertility Rates for Women in the United States, 2010 . 39 7.2 Survival and Fertility Rates for Women in the United States, 2010 . 40 7.2 Survival and Fertility Rates for Women in the United States, 2010 . 41 7.2 Survival and Fertility Rates for Women in the United States, 2010 . 42 7.3 Comparing Female Age-Distributions of the United States: 2010, 2011, 2016 and 2050 . 44 7.3 Comparing Female Age-Distributions of the United States: 2010, 2011, 2016 and 2050 . 45 7.3 Comparing Female Age-Distributions of the United States: 2010, 2011, 2016 and 2050 . 46 7.3 Comparing Female Age-Distributions of the United States: 2010, 2011, 2016 and 2050 . 47 v 7.4 Total Number of Women and People for United States: 2010, 2011, 2016, and 2050 . 49 7.5 Total Number of Women and People for United States: 2010, 2011, 2016, and 2050, with Changes in Mortality Rates . 53 7.6 Total Number of Women and People for United States: 2010, 2011, 2016, and 2050, with Changes in Fertility Rates . 56 7.7 Rate of Population Change for Leslie Matrices Used . 57 vi List of Figures 3.1 Age-Specific Mortality Rates for the United States, 2010 . 15 3.2 Log(Age-Specific Mortality Rates)+5 for the United States, 2010 . 16 3.3 Comparing Crude and Age-Adjusted Mortality, United States . 19 4.1 Density Function for the United States, 2010 . 22 4.2 Cumulative Function for the United States, 2010 . 23 4.3 Three Types of Survivorship Curves . 24 4.4 Survival Function for the United States, 2010 . 25 5.1 Three Types of Population Pyramids . 28 5.2 Population Pyramid of United States . 29 5.3 Population Pyramid of Nigeria . 29 5.4 Population Pyramid of Japan . 30 5.5 Population Pyramid of the World . 30 7.1 Birth Rates for the United States 2010 . 38 7.2 United States Age-Distributions: 2010, 2011, 2016, 2050 . 48 7.3 United States Projection to 2016 with Differing Mortality Rates . 51 7.4 United States Projection to 2050 with Differing Mortality Rates . 51 7.5 United States Projection to 2016 with Differing Fertility Rates . 54 7.6 United States Projection to 2050 with Differing Fertility Rates . 54 7.7 Stable Age-Distributions with Different Mortality and Fertility Rates . 58 vii Chapter 1 Life Table 1.1 Definitions and Assumptions A life table is a table that shows, for each age, what the probability is that a person will die before his or her next birthday, the number of the original cohort that survived, the number of people from the cohort dying, and the life expectancy remaining. There are two types of life tables, the cohort life table and the period life table. The cohort life table, also called the generation life table, presents the mortality experience of a particular cohort from the moment of birth until death. Therefore the cohort life table will follow the original cohort until no one from the cohort remains. Humans tend to live for a long time, and thus it will take a long time to complete a cohort life table. Since it generally takes longer than 100+ years to complete a cohort life table, this form of a life table is not very practical and therefore is not often used. The period life table, also called the current life table, presents what would happen to a hypothetical cohort if it experienced throughout its entire lifetime the mortality conditions that are current for each age group. Newborns experience mortality rates currently present for newborns. Then next year the newborns that survived their first year of life will be a year old and they will experience mortality rates that one-year olds are currently experiencing, and so on. Therefore the period life table is complete immediately, and this makes this form of the life table much more applicable. The period life table may thus be thought of as rendering a snapshot of current mortality experiences[1]. From here on we will be referring to the period life table when mentioning 1 2 life table. There are some assumptions that are made in constructing a life table. It is assumed that the cohort being considered is closed to migration. Therefore no one is entering or leaving the population. This assumption of course holds when looking at the World population, but not when looking at the United States population. We assume that everyone in the cohort experiences the same mortality rates. Thus there is no inequality in health; higher class people experience the same rates as lower class people. This assumption also does not hold in the United States. Lastly, it is assumed that deaths are evenly distributed throughout the year[2]. This is not always the case. For example, newborns are much more likely to die right after birth than they are months later. 1.2 History of Life Table Data on fertility and mortality rates have been around for a long time. John Graunt is credited with producing the first life table in 1662. The life table that he made closely resembles the life tables used today. His observations were based on the bills of mortality of the city of London. They were used to create a system to warn of the onset and spread of bubonic plague in the city. His work resulted in the first statistically-based estimation of the population of London[3]. 1.3 Life Table Calculations A life table starts with a cohort of some given number of newborns. The cohort can be of any size. In the United States it is common to use 100,000 newborns when con- structing a life table. The World Health Organization, an agency of the United Nations that is concerned with public health, also uses 100,000 newborns when constructing a life table for the World.