Population Projection: Definitions & Mathematical Basis


In its basic form, the cohort-survival method is the essence of simplicity:

  • Population[t+1] = Population[t] + Natural Increase + Net Migration

It states that the population at the next time interval (interval "t + 1") is the population at the beginning time interval ("t") plus the net natural increase(or decrease) plus the net migration. This is calculated for men and women for each age-group. Population at the beginning interval can be obtained from the U.S. Bureau of Census (1990), which presents the information for 5-year age cohorts, divided by sex. Other census publications (1995, 1994, 1975) present population data for coarser age groupings.

The results of this sort of analysis are usually presented in both a numerical and a graphic form. In the ideal case, survival rates are fairly constant and the net migration is not significant. A graphic presentation with male cohorts on one side and female cohorts on the other will take the shape of a pyramid; there are many people on the bottom (the "birth cohort") and the number of people in each group declines with age (Figure 3.3). From this ideal form, a graphic presentation of population estimation using the cohort survival method is called a "population pyramid." See Page & Patton (1991) for further description of the population pyramid.

Time interval is determined by the age cohorts. The smallest time interval for which an estimate can be made is the length of time it takes all the members of an age cohort (say, everyone age 10 - 14) to pass on to the next age grouping (the 15 - 19 year-old group). All of the cohorts must be the same dimension (5-year increments, or 7-year increments, or whatever), since over the course of the analysis each group must pass from one cohort to the next with no one left behind. It is not possible to estimate the population 3 years from now using 5-year cohorts, nor is it possible to estimate the population 7 years from now using 5-year cohorts. All estimates must be use time-intervals which are multiples of the cohort size.

Natural increase is the difference between the number of children born and the number of people who die during one time interval. The analysis, however, is being done in terms of age-cohorts for each sex. Children can only be born into the first cohort (if you are using 5-year cohorts, no one is ever born into the 5 - 9 year-old cohort, for example). But people die in all of the cohorts (including the birth cohort). Further, the number of males has no direct effect on the number of children born (this is not the same as saying they have nothing to do with it). Children are born only to women of childbearing age; girls and post-menopausal women also have no direct effect on the number of children born.

In symbolic form, these relations can be expressed as follows:

  • Birthrate[cohort x] = Births / Female population

This is called the "surviving birthrate," and is expressed as a proportion (usually "per 1000") of the number of women in the childbearing age groups. "Childbearing age" is, by convention, considered to be between the ages of 10 and 49, although the birth rates for the 10 -14 and the 45 - 49 age-groups are rather low. It is possible (although a bit tedious) to calculate the birth rate for the most recent interval, or even over several intervals. Fortunately, birth rates are published by State in the Vital Statistics of the United States (U.S. National Center for Health Statistics, 1995). State demographer's offices often have similar information.

Survival rate is calculated in a similar fashion:

  • Survival Rate[cohort x] = 1 - (Deaths[cohort x] / Population[cohort x])

The survival rate for each cohort is 100%, less the "crude death rate." The crude death rate is the ratio of the number of deaths in an age-group for one time interval, divided by the population of that age group at the mid-point of the time interval. Again, these rates are published in Vital Statistics.

There is one convention which you must be careful to observe: Any time-rate (survival rate, birth rate, migration rate, etc.) is calculated for a specific time interval. If you are using a different time interval, you must adjust the rate accordingly. As an example, most of the rates in Vital Statistics are given for five-year age-groups; suppose you are working with ten-year age groups. Your age-groups are twice the size of the standard; what is the proper calculation for the rate of change? It is the standard rate multiplied by itself (i.e., squared), not twice the standard rate (i.e., multiplied by 2). In a more general form,

  • Survival Rate[t+n] = Survival Rate[t]**n

This says that the survival rate for the "nth" interval is the survival rate for one interval raised to the "nth" power.

If you think about it, this makes sense. Suppose a survival rate of 50% (0.50--a very low survival rate). If you want to carry this rate over two standard intervals (i.e., over 10 years instead of 5), then in the second interval the standard survival rate should be applied to the survivors of the first interval; or, symbolically,

  • Population[t+2] = Survival Rate * Population[t+1]

where:

  • Population[t+1] = Survival Rate * Population[t]

therefore,

  • Population[t+2] = Survival Rate * (Survival Rate * Population[t] )

or, more simply,

  • Population[t+2] = (Survival Rate * *2)*(Population[t] )

Besides, the other way (doubling the survival rate) results in nonsense: 50% survive one time period, but if you jump ahead to the second period you find 100% surviving; what happened to all those people who died a few years before?

Net Migration is the difference between the number of people moving in and the number of people moving out. People may "move in" or "move out" by a variety of means: Some may choose to come or go, and do so under their own power. Others will have the decision made for them (children and convicted felons go where they are told). Keep in mind, also, that migration is independent of survival rate. A local population may experience more growth from natural increase than from migration. It is even possible for one to counteract the other. Throughout much of the Middle Ages, the natural increase in cities was negative (people died faster than others were being born; infant mortality was high and plague was common) but cities grew rapidly as changing social and technological conditions pulled people in from the countryside. On the other hand, in many Northern cities in the United States, growth (what there is of it) comes from natural increase; the decline in birthrate in the 1970s was a particularly heavy blow for these cities.

There are many ways to calculate net migration. It is possible to construct quite complex linear models to predict migration for each cohort. The model in this chapter takes a simpler route, basing the net migration rate on past performance. The assumption is that the rate of migration for the next time interval will be the same as the rate of migration for the last time interval for each cohort. Symbolically,

  • Migration Rate[t-1/t] = {(Popltn[t] - Popltn[t-1])-Natural Increase} / Population[t]

In other words, the migration rate from the previous period (t-1) to the present (t) is the change in population from the previous period to the present, less the natural increase. This net change due to migration is expressed as a proportion of the present population. Migration rates may be calculated for each cohort; in Minnesota, the State Demographer's Office also has migration rates available by County, based on the decennial census.

Migration rates and survival rates "point forward." They are attached to the current age group, and represent the "transition rate" to the next age-group. In other words, the product of the migration rate and the population of a current cohort (say, ages 5 - 9) will result in the population of the next cohort (ages 10 -14) in the next time period.


 

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© 1996 A.J.Filipovitch
Revised 11 March 2005