1. The data in data(bangladesh) are 1934 women from the 1989 Bangladesh Fertility Survey. For each woman, we know which district she lived in, her number of living.children, her age.centered, whether she lived in an urban center, and finally whether or not she used contraception (use.contraception).
In this first problem, I only want you to investigate the proportion of women using contraception in each district. Use partial pooling (varying effects). Then compare the varying effect estimates to the raw empirical proportion in each district. Explain the differences between the estimates and the data.
Note that district number 54 is absent in the data. This causes some problems in indexing the parameters. The simplest fix is just to tell ulam manually how long the vector should be, like this: vector[61]:a ~ normal(abar,sigma). Pay special attention to district number 54’s estimate.
2. First, draw a DAG that includes all five variables: (1) Uses contraception C (2) Age A (3) Children K (4) Urban U (5) District D. You don’t have to be an expert on fertility. But do think about which variables can influence which other variables.
Second, design an estimation strategy to identify both the total and direct causal effects of living in an urban center on contraceptive use. Those are your estimands. Consider causal relationships among the variables. Then use your DAG to justify an adjustment set that will yield the estimate of the causal effect of urban living on contraceptive use.
Do not run a statistical model (yet). I just want you to try to design an analysis. There is no firm right answer. Just apply the backdoor criterion and rules of d-separation (the elemental confounds) correctly to the DAG you design.
4-OPTIONAL CHALLENGE. Can you also go beyond the parameter estimates from problem 3 and compute a marginal causal effect of urban living for each district, using a standard age distribution? Use any population age distribution you like. The important thing is to project the estimates from the sample of each district (which does not have representative age distributions) to the population. If you think the different districts should have different age distributions, that would be even more interesting.
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