A Novel Criterion for Selecting Covariates
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STATISTICS
ates, though common in practice, has been criticized (U-15) because treatment effect estimates may be biased if they are perfectly balanced yet prognostic variables are omitted (16). A compelling approach, then, would be to adjust for all prognostic covariates, balanced or not (15). This approach results in the most precise estimates of treatment effects (17). However, given two covariates, only one of which can be included in a model, could the fact that one is much more unbalanced compensate for the better predictive ability of the other? We consider this question in the remainder of this article.
0 M l T T l N G IM P O RTA N T COVA R IATES FROM THE MODEL We preface our development by noting that stratifying for every potentially prognostic covariate could cause each stratum to consist of but one patient. In the interest of parsimony, choices often need to be made: they need to be guided by an assessment of how well each option meets the objectives. This raises the question of why one would adjust for a covariate. Koch et al. (18) offer five benefits in adjusting an analysis for one or several covariates, including: Bias reduction, Better power through variance reduction, Creating comparable comparison groups, Clarifying the predictive ability of one or several covariates,and 5. Clarifying the uniformity of treatment effects over subpopulations defined by particular values or ranges of covariates.
1. 2. 3. 4.
In a subsequent article (19), some of the same authors (and others) stated that the first benefit does not apply to randomized trials, because "randomization provides statistically equivalent groups at baseline (that is, any departures from equivalence are random)." Yet, systematic baseline imbalances (selection bias) can, in fact, occur even in properly randomized trials (1,2),and so even the first benefit applies to randomized trials. To demonstrate how omitting a relevant covariate can lead to unreliable results, consider a hypothetical study
comparing two drugs, A and B, with a binary outcome. Suppose that overall U6 of 400 (34%) patients respond to A while only 100 of 400 (25%) respond to B ( p = O . O 0 3 by one-sided Fisher's exact test), yet, only 8 of 80 males (10%) responded to A but 64 of 320 (20%)responded to B, making B better for males (p = 0.02). If A is better overall and B is better for males, then to compensate, it would appear that A should be much better for females. Yet, 128 of 320 females (40%)responded to A and 36 of 80 (45%) responded to B, making B better again (although not statisticallysignificantly,but certainly by retaining the proportions and increasingthe sample size one could make the p-value arbitrarily low, and the stratified analysis is significant, with p = 0.034 one-sided). So, while B is better for each gender, omitting gender and looking at overall results would be misleading, and would lead to the reversed conclusion that A is better. The salient features that enable omitting a covariate from the analysis to lead to Simpson's paradox (8,20) are the asso
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