Title: Testing models empirically: Path diagram with control variable
Course: UGQuants (SSPS) > Section: Causality

Testing models empirically: Path diagram with control variable

Again we will take 'very good health' as our non-base category for the dependent variable (because it is very good health that we are interested in explaining). The base categories for our independent variable income and control variable age will be low income and 50+ (we have already seen that the younger and richer report better health, so this will keep the differences positive in our calculations).

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.

Above is our causal path diagram for the variables age, income and health.

Arrow a represents the direct effect of age (being under 50) upon income.

Arrow b shows the direct effect of (high) income upon health, controlling for age.

Arrow c shows the direct effect of age on health controlling for income.

We also have to add arrows leading to the income and health variables which show the proportions unpredicted by our model.

Arrow d shows the proportion of those aged 50 and over  who are nevertheless in the high income group (reminding us that not all older respondents are poorer).

Arrow e shows the proportion of those in our two base categories (older and poorer respondents) who nevertheless are in the non-base category of the dependent variable. This represents the effect of all the other variables that are not in our model, reminding us that age and income are not the only determinants of good health.

Explanation

To facilitate your understanding of the diagram, here is how we arrive at arrow figures a-e:

a: We can 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.

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 count | 997 | 843 | 1840

Data source: ESS 2010 (UK)

b: We can calculate the direct effect of income upon health by calculating the differences in proportions for those on high and low incomes in very good health, doing so both for our old group and for the younger group i.e. controlling for age as we do in table 4 below. For the young the difference is .122. For the old it is .071. Next we take the average of these differences, which is .097.

Table 4: Health by household income, controlling for age

Age | Health | Household income
Up to median | Above median | Total
Up to 49 | Very good | 30.5% | 42.7% | 37.9%
Very bad to good | 69.6% | 57.3% | 62.2%
Total count | 394 | 599 | 993
50+ | Very good | 18.9% | 26.0% | 21.8%
Very bad to good | 81.2% | 74.0% | 78.3%
Total count | 503 | 338 | 841

Data source: ESS 2010 (UK). 24.2% missing.

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)

c: Now we can calculate the direct effect of age upon health, controlling for income (table 5 below). 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.

Table 5: Health by age, controlling for household income

HH Income | Health | Age
Up to 49 | 50+ | Total
Up to median | Very good | 30.4% | 18.8% | 23.9%
Very bad to good | 69.6% | 81.2% | 76.1%
Total count | 395 | 504 | 899
Above median | Very good | 42.7% | 26.0% | 36.7%
Very bad to good | 57.3% | 74.0% | 63.3%
Total count | 599 | 338 | 937

d: Look again at table 3 on income above. Seek out the 50+ age group and see which proportion of it have high incomes (0.402).

e: Look at table 2. What proportion of the 50+ age group who have below median income say they are in very good health?(18.9% = 0.189)

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.)