Being Just about Adjustment in Clinical Trials

Estimation of the magnitude of effects and of the relevant precision in general needs inclusion of strata parameters even in balanced data.

Cox and Wermuth, 2003

Two Conditions

An interesting paper by David Cox and Nanny Wermuth[1] considers the case under which the 'marginal' regression of a random variable Y on another variable X will be the same as the conditional regression given a third variable W. A key well-known relationship to which they draw attention can be expressed in terms of the following equation

Here the first term (to the left of the = sign) is the marginal regression of Y on X, ignoring W, the second term (first on the RHS) is the conditional regression of Y on X given W and the third term is the product of the conditional regression of Y on W given X and the regression of W on X. (Note that there is a misprint in the original by C&W, which I have corrected.)

As Cox and Wermuth point out,

so that the requirement for the marginal and conditional estimates to be the same is that either the regression of Y on W is zero given X or that the regression of W on X is zero.

Clinical relevance

Now suppose that Y is an outcome of interest in a clinical trial in asthma and X is a treatment factor. For example Y might be forced expiratory volume in one second (FEV1) and X could be a factor that represents whether the patient was given a placebo or a beta-agonist. Now suppose that W is baseline FEV1, then the condition that the marginal and conditional effects of treatment are identical is either that the baseline is unpredictive of outcome given treatment or that the baseline is balanced between treatment groups. Cox and Wermuth give graphs representing possible relationships, which I have reproduced below.

Moving from left to right, the first picture in Figure 3 represents a situation in which the marginal and conditional estimates will not be the same, the middle one shows that W has no predictive value for Y given X and the third one shows that W and X are unrelated.

Means are not ends

Now consider the situation in terms of inferences as a whole, that is to say moving beyond estimands and estimates in terms of regression coefficients to the inferential support in terms of likelihood we would see in any given case. A possible situation is illustrated in Figure 4.

The situation is that a randomised clinical trial has been carried out and the difference in FEV1 at outcome between the two groups is observed to be 300ml. There are 50 patients per group and the standard deviation at outcome and baseline is 150ml with the correlation between the two being 0.8. Figure 4 gives the marginal unadjusted likelihood as a function of the posited treatment effect as a solid black line.

As regards the baseline differences between treatment groups, three cases are considered: differences = -30ml, 0ml or 30ml. Incorporating this information leads to the three adjusted likelihoods in the figure. The baseline is highly predictive, so the first of Cox and Wermuth's conditions does not apply. In consequence, the adjusted maximum likelihood estimator would only be the same as the unadjusted one in the case where the baseline difference is zero, satisfying the second of Cox and Wermuth's conditions.

So what?

So this. In my opinion, the degree of imbalance is irrelevant when deciding whether to adjust for a factor or not [2]. Some approaches to adjustment, for example propensity score matching, seem to regard it as important. I don't. In my opinion, this picks up the problem at the wrong end. If you look at Figure 4 you will see that none of the conditional likelihoods is similar to the unconditional one.

It is inferences that matter and that means that assessing uncertainty is important. Indeed, this is what Cox and Wermuth point out in their paper, hence the quote at the heading to this blog.

References

D. R. Cox and N. Wermuth (2003) A general condition for avoiding effect reversal after marginalization. Journal of the Royal Statistical Society Series B-Statistical Methodology, 937-941.

S. J. Senn (1994) Testing for baseline balance in clinical trials. Statistics in Medicine, 1715-1726.