# Residual Confounding

The two age categories for the data in Table 10-3 may not be sufficient to control all of the age confounding in the data. More strata with narrower boundaries usually can control confounding more effectively than fewer strata with broader boundaries. If age strata (or strata by any continuously measured stratification factor) are broad, there may be confounding within them. A stratified analysis controls only between-stratum confounding, not within-stratum confounding. Within-stratum confounding is often referred to as residual confounding. The same term is used to describe confounding from factors that are not controlled at all in a study or from factors that are controlled but are measured inaccurately.

To avoid within-stratum residual confounding, it is desirable to carve the data into more strata and to avoid open-ended strata (eg, age 55+) when possible. On the other hand, stratifying too finely may stretch the data unreasonably, producing small frequencies of events within cells and leading to imprecise results. Finding the best number of strata to use in a given analysis often requires balancing the need to control confounding against the need to avoid random error in the estimation and ends up being a compromise.

The unconfounded estimate of the risk difference, 3.5%, is unconfounded only to the extent that stratification into these two broad age categories removes age confounding. It is likely that some residual confounding remains (see box) and that the risk difference that is fully unconfounded by age is smaller than 3.5%.

We can also calculate a pooled estimate of the risk ratio from the data in Table 10-3 using Equation 10-2:

This result, like that for the risk difference, is closer to the null value than the crude risk ratio of 1.44, indicating that some age confounding has been removed by the stratification. The pooled estimate is within the range of the stratum- specific estimates, as it must be mathematically. Note, however, that for the risk ratio, the stratum-specific estimates for the data in Table 10-3, 1.81 and 1.19, differ considerably from one another. The wide range between them includes the pooled estimate and the estimate of effect from the crude data. When the stratum-specific estimates of effect are almost identical, as they are for the risk differences in the data in Table 10-3, we have a good idea of what the pooled estimate will be just from inspecting the stratum-specific data. When the stratum-specific estimates vary, it is not clear on inspection what the pooled estimate will be.

As stated earlier, the equations used to obtain pooled estimates are premised on the assumption that the effect is constant across strata. The pooled risk ratio of 1.33 for the previous example is premised on the assumption that there is a single value for the risk ratio that applies to both the young and the old stratum. This assumption seems reasonable for the risk difference calculation, for which the two strata gave almost the same estimate of risk difference, but how can we use this assumption to estimate the risk ratio when the two age strata give such different risk ratio estimates? The assumption does not imply that the estimates of effect will be the same or even almost the same in each stratum. It allows for statistical variation over the strata. It is possible to conduct a statistical evaluation, called a test of heterogeneity or a test of homogeneity, to determine whether the variation in estimates from one stratum to another is compatible with the assumption that the effect is uniform.4 In any event, it is helpful to bear in mind that the assumption that the effect is uniform is probably wrong in most situations. It is asking too much to have the effect be absolutely constant over the categories of some stratification factor. It is more realistic to consider the assumption as a fictional convenience, one that facilitates the computation of a pooled estimate. Unless the data demonstrate some clear pattern of variation that undermines the assumption that the effect is uniform over the strata, it is usually reasonable to use a pooled approach, despite the fiction of the assumption. In Table 10-3, the variation of the risk ratio estimates for the two age strata is not striking enough to warrant concern about the assumption that the risk ratio is uniform. If a more formal statistical evaluation of the assumption of uniformity were undertaken for these data (calculating a P value to test the assumption), it would support the view that the assumption of a uniform risk ratio for the data in Table 10-3 is reasonable.