Institute of Child Health, Centre for Paediatric Epidemiology and Biostatistics, University College London, London, United Kingdom WC1N 1EH
In a recent Journal article, Tu et al. (1) showed how adjustment for current weight tends to increase the inverse association between birth weight and blood pressure. They assumed that current weight is positively correlated with blood pressure and birth weight, and that birth weight and blood pressure are only weakly correlated. They tested the effect of adjustment by simulating different correlation scenarios.
Their argument focuses on how adjusting for current weight alters the association. This depends only on the correlation structure, so the case can be made explicitly, more elegantly, and without loss of generality by expressing the three variables as z scores, that is, with mean zero and standard deviation one. Doing so avoids the need to specify means and standard deviations for the variables.
Assume that the correlations of birth weight and current weight with blood pressure are r1 and r2, respectively, while the correlation for birth weight versus current weight is r12. In the z-score regression of blood pressure on birth weight, the unadjusted birth-weight coefficient is equal to the correlation r1; adjusted for current weight, the birth-weight coefficient is .
The adjusted coefficient is more negative than the unadjusted coefficient (i.e., ß1 < r1) if r1 < r2/r12, although, if r1 is sufficiently large, ß1 may still be positive. The stronger condition r1 < r2r12 ensures that ß1 < 0. Thus, under weak conditions, the adjustment increases the inverse association between birth weight and blood pressure, confirming the findings of Tu et al. (1).
Note that r2 and r12 need not be positive. As long as they are of the same sign, either positive or negative, the two conditions hold. So, the findings apply quite generally to situations in which a covariate correlates in the same direction with birth weight and outcome.
Tu et al. (1) argue that adjusting for a confounder on the causal pathway, that is, current weight, invalidates the inference that birth weight is inversely associated with blood pressure. The algebra above shows that such an adjustment is very likely to make the association more negative, but it does not help to decide whether such an adjustment is valid. Colleagues and I have argued elsewhere that the change in sign of the weight-versus-outcome correlation over time indicates the importance of weight change as opposed to birth weight (2
, 3
).
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