twoAssignment format: paper |
ANSWER SHEET (posted 4/10/08) |

**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]

Age Group (years) | Inmigrants on Jan 2, 2000 |
Population
in Year 2040 |

0 - 9 | ? |
? |

10 - 19 | ? |
? |

20 - 29 | ? |
? |

30 - 39 | ? |
? |

40 - 49 | ? |
? |

50 - 59 | ? |
? |

60 - 69 | ? |
? |

70 - 79 | ? |
? |

80 - 89 | ? |
? |

90 and above | ? |
? |

TOTAL | 50 |
? |

* 3. Optional Extra credit:* estimate the life expectancy (e

**Further comments / advice:**

**Cohort -Survival Method**

- Using matrix algebra is a mathematically powerful way to perform cohort
survival analysis (e.g., P
_{1}= C * P_{0}, or more generally, P_{n}= C^{n}* P_{0}, 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. - 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.
- 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).
- 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 (e _{0}) Estimation**

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

- 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}, T_{0}, and then finally e_{0}. - 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, L_{0}and L_{90}). Remember that l_{infinity}= 0 (i.e., everyone eventually dies). - 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!).