Z-scores and the birthweight paradox

Abstract

Investigators have long puzzled over the observation that low-birthweight babies of smokers tend to fare better than low-birthweight babies of non-smokers. Similar observations have been made with regard to factors other than smoking status, including socio-economic status, race and parity. Use of standardised birthweights, or birthweight z-scores, has been proposed as an approach to resolve the crossing of the curves that is the hallmark of the so-called birthweight paradox. In this paper, we utilise directed acyclic graphs, analytical proofs and an extensive simulation study to consider the use of z-scores of birthweight and their effect on statistical analysis. We illustrate the causal questions implied by inclusion of birthweight in statistical models, and illustrate the utility of models that include birthweight or z-scores to address those questions. Both analytically and through a simulation study we show that neither birthweight nor z-score adjustment may be used for effect decomposition. The z-score approach yields an unbiased estimate of the total effect, even when collider-stratification would adversely impact estimates from birthweight-adjusted models; however, the total effect could have been estimated more directly with an unadjusted model. The use of z-scores does not add additional information beyond the use of unadjusted models. Thus, the ability of z-scores to successfully resolve the paradoxical crossing of mortality curves is due to an alteration in the causal parameter being estimated (total effect), rather than adjustment for confounding or effect decomposition or other factors.

Figure 1

Birthweight-specific infant mortality curves for smokers and non-smokers.

Figure 2

Directed acyclic graphs (DAG) representing causal relationships between a risk factor, birthweight, unmeasured variables $\left(\mathcal{U}\right)$ and neonatal mortality. (a) DAG representing both direct and indirect effects of birthweight on neonatal mortality. (b) DAG representing collider-stratification bias introduced by adjusting for birthweight in the presence of an unmeasured confounder, $\mathcal{U}$. (c) DAG representing collider-stratification bias introduced by adjusting for birthweight in the presence of an unmeasured confounder, $\mathcal{U}$, under alternative causal assumptions. (d) DAG depicting deterministic relationship between birthweight, smoking and individual z-scores. (e) DAG representing relationships among variables under z-score adjustment.

Figure 3

Relative bias of the direct effect of smoking in a model adjusted for birthweight in the presence of an unmeasured confounder. The effect of smoking on birthweight is e = -200, the direct effect is Θ_D = c = 0.2, the total effect is Θ_T = c + d*e = 0.3, d = -0.0005, and b corresponds to the effect of the unmeasured confounder on neonatal mortality.

Figure 4

Z-score-specific mortality curves for smokers and nonsmokers. Effect of smoking on birthweight represented by e, direct effect Θ_D = c, total effect Θ_T = c + d*e, d = -0.0005. (a) e = -100, c = 0.2; (b) e = -100, c = 0.4; (c) e = -500, c = 0.2; (d) e = -500, c = 0.4.