To match or not to match when 2 exposures are inextricably confounded with other variables?

Question

I want to study the effect of two Groups of patients ($X_1$) on $y$ (a test performance score), in a GLM framework. Age ($X_2$) and Education ($X_3$) are potential confounders on $y$.

However its not possible to match these two groups for age, as they are illnesses that occur in different age groups-one group is younger than the other. Hence the mean ages are significantly different between these groups.

Adding age as a covariate could potentially cause multicollinearity problem as age is significantly different between groups, and make the estimation of group effect ($β_1$) erroneous.

Is recruiting a control group with age distribution comparable to the pooled patient groups, hence of a mean age mid-way between the two patient groups a good idea to improve the statistical power of the study? In this case my group factor $X_1$ will have three levels. Can this reduce the multicollinearity problem to an extent as the ages of patients in the two patient groups are approximately represented in the control group also..? Should I add an interaction term of Age*Group in the GLM to account for the age difference between groups?

Answer 1

(Nitpick: I wouldn't say age and education are "confounders on $y$", I would say they are "confounded with group".)

Many people seem to think matching is necessary when working within the strategy of controlling for confounding statistically. It isn't. Statistically controlling for confounders by including them in a multiple regression model does the same job. Matching can be more efficient (which is a real benefit, to be sure). But matching won't be possible, if the groups cannot exist in the same regions of a confounder (your situation, as I understand it). You could match each group against a separate group of controls, but those two control groups wouldn't be matched to each other.

I would skip matching as the preferred approach in this case. Recruit a set of controls who span the ranges of the confounders for both groups. That is, if disease 1 only occurs between ages 30 and 40, and disease 2 only occurs between 50 and 60, recruit disease-free controls between 30 and 60. The controls should also span education similarly. You'll want the controls to be fairly uniformly distributed, not just some 30 year olds and some 60 year olds. Then you can fit a function to, say, age, and control for that effect.

The question of whether you need an age by group interaction is a different issue. If you think the relationship between age and $y$ differs depending on disease status, then you need an interaction to control for that.

Be aware that all of this is likely to increase the required sample size, maybe substantially.