Cohort-Survival Analysis

600 people live in the mysterious country of Pangaea. The country is not found on any map, nor is it visited by ships or aircraft; as a result, there has been no migration to or from the country.

Use cohort-survival analysis to answer the following two questions.

1. SCENARIO ONE: Project the total population, and the number of people in each 10-year cohort, for the years 2010 and 2040 [the blue-shaded cells in the table below]. Show your work. [closed population scenario - population changes due only to natural increase/decrease  download this table in Excel format

Age Group (years)	Population in Year 2000	Survival Rate	Birth Rate	Population in Year 2010	Population in Year 2040
0 - 9	100	0.95	0	?	?
10 - 19	100	0.99	0.3	?	?
20 - 29	100	0.95	0.7	?	?
30 - 39	80	0.92	0.3	?	?
40 - 49	70	0.9	0.1	?	?
50 - 59	50	0.8	0	?	?
60 - 69	40	0.7	0	?	?
70 - 79	30	0.5	0	?	?
80 - 89	20	0.3	0	?	?
90 and above	10	0.1	0	?	?
TOTAL	600			?	?

Notes:
survival rate -- proportion of people in cohort who survive into next cohort.
birth rate -- the number of children born per person in cohort during the time interval (ten years)
Population for a given year is counted on January 1 (e.g., on January 1, 2000 the population was 600 people).

2. SCENARIO TWO (open population with migration): Pangaea is worried about insufficient population growth, and decides to use in-migration (rather than encouraging increased fertility) to stimulate faster population growth. The country decides to invite 50 outsiders to move to the island on January 2, 2000. To maximize the country's population in time for the nation's Centennial in the year 2040, what age group(s) should the 50 immigrants belong to in 2000? What would the total population and population by age group be in 2040? [the blue-shaded cells in the table below]

3. Optional Extra credit: estimate the life expectancy (e₀) of a Pangaean baby born in the year 2000. (Briefly explain and document your method.)

Further comments / advice:

Cohort -Survival Method

1. Using matrix algebra is a mathematically powerful way to perform cohort survival analysis (e.g., P₁ = C * P₀, or more generally, P_n = Cⁿ * P₀, where P_n is the age-specific population array for time period n and C is the components of change matrix, which includes both age-specific birth rates and survival rates). However, you do NOT need to answer this question using matrices. Instead, simply using Excel calculations may be an easier way to proceed.
2. Don't forget to include the ten percent of people who are already in the open cohort (90 years and older) who survive. These people survive into the same cohort.
3. There are several strategies for rounding numbers in your calculations. You can leave your numbers for cohort sizes in fractions, or you can round at each step. For this assignment, either approach is fine (but do note if you use a rounding strategy).
4. One important question (regarding how one calculates the numbers of births) is determining who is "at risk" of leading to a birth. Here there are three alternatives: (a) assume that anyone who enters a cohort (regardless of whether they die during that cohort period) is "at risk" (this is the assumption used in the in-class handwritten example, which will lead to a slight overestimate of children born); (b) assume that only those who survive the entire cohort period is "at risk" (which will lead to a slight underestimate of children born); or (c) use an average of the two (which would lead to a middle estimate). For this assignment, any of the three approaches is fine. Please do note what assumptions you make.

Life Expectancy (e₀) Estimation
(for a discussion on methodology, see CDC life tables, especially pp. 37 - 39)

1. Remember that calculating life expectancy is independent of both birth rates and the existing structure of the population (here, the allocation of the country's 600 residents by age group in the year 2000). In other words, the life expectancy of a baby at birth is NOT calculated based on either birth rates or the age structure of the existing population. To calculate life expectancies, you only need mortality rates (which is q_x = 1 - survival rates). Knowing q_x, you can then also calculate l_x, L_x, T₀, and then finally e₀.
2. Demographers usually estimate L_x differently for the youngest and oldest age cohorts as compared to the other cohorts (where we can assume that L_x = n * (l_x + l_x+1)/2. However, to keep this assignment simple, you may use the same assumption for these two cohorts (here, L₀ and L₉₀). Remember that l_infinity = 0 (i.e., everyone eventually dies).
3. Since the cohort sizes are n = 10 years (rather than n = 1 year), be sure to adjust your L_x to reflect this 10 year interval (otherwise you may get a life expectancy estimation of 5-8 years, which is a short life!).