## February 24, 2009

### Stationary population models and graduation

In odd moments in the last few weeks, I've been playing around with a standard demographic concept, the stationary population model. This is one of those things that don't really exist in reality, a population with constant mortality and fertility rates, with no migration in or out, and where the population is the same every year (no natural increase). In essence, a stationary population model is like a stripped-down car, something with all the extras out of the way so you can look at the engine while it's running. The question I've had is, if one looks at a stationary population model of high school, what can one say about a high school if one observes the total enrollment, the ninth-grade enrollment, the number of graduates, and the distribution of graduates by years in high school?

A few minutes of scribbling shows that the crude graduation rate (or the number of graduates divided by the total enrollment) is equal to the probability of graduating times the rate of new ninth graders entering every year. The probability of graduating and the number of new ninth graders are both interesting and unobserved quantities. Unfortunately, they're also dependent on a crucial third unobserved quantity, the difference between the entering-ninth-grade rate and the proportion of the high school in ninth grade. (One way of interpreting this is the overestimate of entering 9th graders. Another interpretation is the proportion of total school life experienced in repeating ninth grade.)

Because my life is now booked, I've only spent odd moments away from a computer on this exercise, but the obvious next step is to generate some simulated stationary populations (e.g., bootstrap samples of NELS:88, constrained to confirm to a range of graduation probabilities) and then look for regularities in the relationships between the underlying population measures and what would normally be observed from published data. Given the inherent constraints of the true value for the entering-ninth-grade rate (between 0.25 and the observed ninth-grade proportion), and a few other things, I suspect that regularities exist. **Update**: the problems of writing when sick is that one forgets obvious things like, *NELS data sets do not have year-by-year information on enrollment*. On the other hand, the Bureau of Labor Statistics longitudinal surveys (starting in 1979 and 1997) are every year...

Then the next step is to move on to a stable population model, where you relax the zero-growth assumption and assume a constant growth rate. That's important because school populations do not remain constant. (Neither does growth remain constant, but a stable population model introduces one level of complexity, and it's loads easier to understand than the full-blown, "let the population do what it wants to" model.) The problem here is that one crucial number in a stable population model is a term that normally corresponds to the mean length of a generation. This has no clear interpretation in a model of high school enrollment, so that's an interesting hurdle.

Incidentally, if anyone wants to jump ahead of me on this research program, feel free to dive in. The water's fine, I'm not likely to follow up for some months, and there are some interesting payoffs. Among other things, in a stationary population model, the product of life expectancy at birth and the birth rate is always one. In the school parallel with a stationary population model, if you multiply the entering-ninth-grade rate by the average time spent in high school, you will always get one. From there and the data on graduates, it's simple to calculate the average time spent in high school by those who eventually drop out.

Tags: graduation measures

Posted in Research on February 24, 2009 12:39 AM |