In science, we frequently see a comparison between two groups of people that differ on multiple demographic variables, say age, IQ, and income, investigating some dependent measures, say body mass index (BMI). Results are frequently reported as “the groups had substantially different BMIs, after controlling for X and Y” using ANCOVA or multiple regression. We are given the impression that this analysis shows that the groups would have different BMIs even if they had the same levels of X and Y.

Is this inference valid? Maybe. Extensive thought and scrutiny of the data would be required to determine whether this is a reasonable inference. I was thinking of discussing this a bit in my undergraduate teaching, so I asked about it on twitter. A bunch of people both provided helpful responses and asked me to report back.

An obvious problem is that the two groups may differ on other things besides X and Y, many of which you may not have even been measured. So the difference between the groups may be entirely attributable to those confounds. This post is about some less obvious problems. Here are some quick snippets from what people pointed me to.

First, from Miller & Chapman (2001), below. Thanks to @BrandesJanina for pointing me to this paper.

consider a data set in which two groups are older men and younger women, and gender is of interest as an independent variable, Grp. Using age as a covariate does indeed remove age variance. The problem is that, because age and gender are correlated in this data set, removing variance associated with Cov will also remove some (shared) variance due to Grp. Within this data set, there is no way to determine what values of DV men younger than those tested or women older than those tested would have provided. Far from “controlling for” age, the ANCOV A will systematically distort the gender variable. As in our presentation of Lord’s Paradox above, GrPres will not be a valid measure of the construct of gender….

Consider a data set consisting of childrens’ age, height, and weight. If we conduct an ANCOVA in which height is the covariate, age is the grouping variable, and weight is the dependent variable, we are attempting to ask whether younger and older children would differ in weight if they did not happen to differ in height. If the groups indeed do not differ on the covariate, this question can be asked. But if there is something about the construct of age in childhood that inherently involves differences in height, the question makes no sense, because then age with height partialed out would no longer be age. There is no way to “equate” older and younger children on height, because growth is an inherent (not chance or noise) differentiation of the two groups….

Cohen and Cohen (1983) provided the following extreme example: “Consider the fact that the difference in mean height between the mountains of the Himalayan and Catskill ranges, adjusting for differences in atmospheric pressure, is zero!” (p. 425), the point being that one has not in any sense “equated” the two mountain ranges by using atmospheric pressure as a covariate.

-Miller & Chapman (2001)

Let’s go back to my opening example of a BMI difference between two groups, after “controlling for” variables statistically. What if one of those variables controlled for was age? Well, if the two groups were people who exercise and people who don’t, there is very likely variance shared by age and level of exercise, and age likely has a causal influence on exercise (by various routes), so the meaning of the exercise factor is unclear after age has been removed.

### The problem of measurement (un)reliability

From Westfall & Yarkoni (submitted):

Suppose we are given city statistics covering a four-month summer period, and observe that swimming pool deaths tend to increase on days when more ice cream is sold. As astute analysts, we immediately identify average daily temperature as a confound: on hotter days, people are more likely to both buy ice cream and visit swimming pools. Using multiple regression, we can statistically control for this confound, thereby eliminating the direct relationship between ice cream sales and swimming pool deaths.

Now consider the following twist. Rather than directly observing recorded daily temperatures, suppose we obtain self-reported Likert ratings of subjectively perceived heat levels. A simulated batch of 120 such observations is illustrated in Figure 1, with the reliability of the subjective heat ratings set to 0.40—a fairly typical level of reliability for a single item in psychology1. Figure 2 illustrates what happens when the error-laden subjective heat ratings are used in place of the more precisely recorded daily temperatures. The simple relationship between ice cream sales and swimming pool deaths (Fig. 2A) is positive and substantial, r(118) = .49, p < .001. When controlling for the subjective heat ratings (Fig. 2B), the partial correlation between ice cream sales and swimming pool deaths is smaller, but remains positive and statistically significant, r(118) = .33, p < .001. Is the conclusion warranted that ice cream sales are a useful predictor of swimming pool deaths, over and above daily temperature? Obviously not. The problem is that subjective heat ratings are a noisy proxy for physical temperature, so controlling for the former does not equate observations on the latter. If we explicitly control for recorded daily temperatures (Fig. 2C), the spurious relationship is eliminated, as we would intuitively expect, r(118) = -.02, p = .81.

Given that most psychological measurements have considerable unreliability (lack of perfect correlation with the construct they are trying to get at), the problem is very general. And it can lead both to spurious conclusions of a relationship as well as spurious conclusions of a non-relationship.

I do not use ANCOVA or GLMs in this way so I may have given a misleading impression with some of what I have written or quoted above. If so, I would love to be corrected.