Title: Testing models empirically: Estimating effect sizes
Course: UGQuants (SSPS) > Section: Causality

Testing models empirically: Estimating effect sizes

The final step in considering the association between income and health is to calculate the values for the three causal paths between the variables in the model, corresponding to the proportions in our three way crosstabulation and expressing the probability that someone in each age and income group will enjoy good health.

First we can calculate the direct effect of income upon health by calculating the differences in proportions for those on high and low incomes in good health, doing so both for our old group and for the younger group i.e. controlling for age. For the young the difference is .122. For the old it is .071. Next we take the average of these differences, which is .097. Thus, as we saw from simply looking at the two partial tables in the crosstab, taking age into account has reduced the effect of high income upon the likeliness of being in very good health overall (remember that before controlling for age the difference in proportions was .128).

Now we can calculate the direct effect of age upon health, controlling for income. Among the low income group .188 of the older and .304 of the younger people report very good health: a difference of .116. In the above median income group .260 of the old and .427 of the young report good health: a difference of .167. The average difference in proportions is therefore .142.

We can also calculate the direct effect of age upon income, represented by the arrow running from the age variable to the income variable (using table 3). While .604 of young (up to 50) respondents had above median incomes, this was true of only .402 of older respondents, so that the difference in proportions is .202.

To obtain the size of an indirect effect, we multiply the effects of the relevant causal path together. Thus the indirect effect of age on health, through the effect of age upon income is .202 * .097 = 0.020. Indirect effects are by their nature usually small.

Finally we can summarise the effects of each independent variable by summing the effects along the relevant causal paths. The total effect of age is .202 + .020 = .222. This means that being in the younger age group increases the proportion reporting very good health by this amount. (We can check how good an estimate this is by looking at the marginals in table 3. There the effect is .379-..218 = .161. Given our rough and ready procedure of averaging proportion differences, this is a good estimate.)