Title: Testing models empirically: An example
Course: UGQuants (SSPS) > Section: Causality

Testing models empirically: An example

Let’s investigate this empirically with data from the European Social Survey (2010).

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 such a relationship. Those in households with higher incomes report better health:

Table 1: Self-assessed health by household income

Health | Household income
Up to median | Above median | Total
Very good | 23.9% | 36.7% | 30.7%
Good | 39.8% | 41.5% | 40.7%
Fair | 24.8% | 18.3% | 21.5%
Bad | 9.1% | 2.7% | 5.8%
Very bad | 2.3% | .9% | 1.6%
Total count | 899 | 937 | 1836

Data source: ESS 2010 (UK)

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.

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 these may be the 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):

Table 2: Self-assessed health by household income by age

Age | Health | Household income
Up to median | Above median | Total
Up to 49 | Very good | 30.5% | 42.7% | 37.9%
Good | 41.1% | 39.7% | 40.3%
Fair | 20.1% | 16.0% | 17.6%
Bad | 6.6% | 1.0% | 3.2%
Very bad | 1.8% | .5% | 1.0%
Total | 394 | 599 | 993
50+ | Very good | 18.9% | 26.0% | 21.8%
Good | 38.8% | 44.7% | 41.7%
Fair | 28.4% | 22.2% | 25.9%
Bad | 11.1% | 5.6% | 8.9%
Very bad | 2.8% | 1.5% | 2.3%
Total | 503 | 338 | 841

Data source: ESS 2010 (UK)

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 who describe their health as very bad, bad, or fair is smaller, and there are therefore many more low income receivers with good or very good health.

In the ‘good’ health category, the direction 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.

Likewise we find, in table 3, an association between income and age, with the younger group much more likely to have above median household incomes:

Table 3: Age and income

Household income | Age
Up to 49 | 50+ | Total
Up to median | 39.6% | 59.8% | 48.9%
Above median | 60.4% | 40.2% | 51.1%
Total | 997 | 843 | 1840

Data source: ESS 2010 (UK)