Title: Testing models empirically: Assessing relationships
Course: UGQuants (SSPS) > Section: Causality

Testing models empirically: Assessing relationships

If we consider first the relationship between self-assessed health and income using the data in ESS 2010 as shown in table 1, it seems fairly clear that there is one. Those in households with higher incomes report better health. This is a good example to consider possible theories of cause and effect. Our data cannot tell us if money buys health or whether better health facilitates greater earning capacity or whether there is another uncontrolled variable that determines both health and income. Older, especially retired people, may have both lower incomes and poorer health, but as a result of their age rather than any direct relationship between health and income as such. We can test this idea by controlling for age (table 2).

The relation between X (income) and Y (health) can be expressed in terms of whatever measure of association we prefer. Let us use the difference in proportions. Table 1 shows that those on median incomes or below are less likely to describe their health as good and about three times as likely to describe it as bad or very bad. Once we control for age, by splitting our analysis into two groups, the under 50s and 50+'s (table 2), a rather different picture emerges. There is still a relationship between health and income for the younger group: but it is weaker than before. The proportion of the low income group describing their health as very bad, bad, or fair has smaller, and there are therefore much more low income receivers with good or very good health. In the ‘good’ category, the direction even switches: the number of those with an above average household income reporting good health is slightly smaller than that of the other group. By contrast, in the 50+ age group differences widen: almost twice as many in the lower income group describe their health as bad or very bad compared to the higher income group. Whatever the nature of the relationship between income and health, age seems to play a substantial part in it.

We can describe this relationship more precisely using the difference in proportions method, and focusing on those who say their health is very good. We will take low income and very bad health as our base categories. First looking at all respondents of all ages and taking those on low (up to median) incomes as our base category for our independent variable, we can see that  .239 of them said they had very good health. In addition, .367 of those in high (above median) incomes said they had very good health. The difference in proportions between the two categories is .128. The total proportion in very good health is therefore the proportion in the 'base' category (low income) in very good health plus the extra proportion in the non-base category (high income) who were in very good health multiplied by the proportion of respondents in the non-base category. This is shown in the following diagram.