Title: Testing models empirically: Adding the control variable
Course: UGQuants (SSPS) > Section: Causality

Testing models empirically: Adding the control variable

In considering the relationship between income and health we should have a clear idea of what each of the models we wanted to test for would look like when we add our control variable age.

For us to accept Model 1 as accurate we would need evidence that there is no relationship between age and income. However, from our earlier crosstab on age and income we can see that this is not the case. About two fifth of younger respondents have incomes up to the median, compared to almost three fifth of older respondents. We can, therefore, safely discount model 1. |

 | If Model 2 is accurate, and age only has an effect through income, we would expect to find no relationship between age and health, once we control for income, since any apparent relationship between age and health would in fact be due to the way age varies with income, not any independent effect of age.  From our crosstab of age and health (and income), again we find evidence to refute our hypothesis. Even if we control for income, our younger age group still reports better health. We can safely discount model 2.

If Model 3 is accurate, if we control for age, we ought to find that the relationship between income and health disappears. The correlation between health and income would be due simply to the fact that age causes people to have lower incomes and also causes people to have poorer health. In such a situation we would define the relationship between income and health as spurious: it was produced by the prior relationship of both variables to age, which is the real explanatory variable. If we run a crosstab between health and income controlling for age we do find that the relationship is weaker for younger respondents, but it is if anything stronger for older respondents. We might conclude that the relationship has a spurious component. |

 | This takes us to Model 4. It does indeed look as if both age and income affect health, and that part of this is because age affects income too. By examining the crosstab we can see that for younger respondents, income has an effect, but it is not substantial. For older respondents the effect of income is much greater. However, how about the combined effect of age and income? Our next step is to estimate the size of the different relationships. We’ll use the difference in proportions method.