Z-scores and the birthweight paradox

Enrique F Schisterman; Brian W Whitcomb; Sunni L Mumford; Robert W Platt

doi:10.1111/j.1365-3016.2009.01054.x

. Author manuscript; available in PMC: 2009 Sep 14.

Published in final edited form as: Paediatr Perinat Epidemiol. 2009 Sep;23(5):403–413. doi: 10.1111/j.1365-3016.2009.01054.x

Z-scores and the birthweight paradox

Enrique F Schisterman ^a, Brian W Whitcomb ^a,^b, Sunni L Mumford ^a,^c, Robert W Platt ^d

PMCID: PMC2742985 NIHMSID: NIHMS135844 PMID: 19689489

Abstract

Summary

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.

Keywords: birthweight, birthweight paradox, simulation, directed acyclic graphs, z-scores

Introduction

Birthweight is a robust predictor of neonatal and infant mortality.¹^-³ In the United States in 2004, infants born weighing <2500 g were reported to have 25-fold higher mortality rates than those weighing ≥2500 g.⁴^,⁵ As a result, investigators of associations of perinatal mortality with various factors also linked to birthweight often consider birthweight as a confounder and utilise traditional epidemiological approaches to modelling, despite the fact that birthweight often lies on the causal pathway and should not be adjusted for.⁶ Assessments of associations between maternal smoking,⁷ multiple pregnancies,⁸ placenta praevia,⁹ black race¹⁰^-¹² and infant mortality, have employed analyses stratified on (or adjusted for) birthweight. In the context of numerous risk factors, stratification on birthweight yields a puzzling result that has long been a source of controversy: in the lower stratum of birthweight, babies born to mothers with the risk factor of interest tend to have better outcomes than babies born to mothers without the risk factor.² The ‘birthweight paradox’ as described by Wilcox² is shown graphically in Fig. 1,¹³ and is characterised by the crossover of the stratum-specific mortality curves.

Birthweight-specific infant mortality curves for smokers and non-smokers.¹³

Various approaches have been suggested for addressing the paradox,¹⁴^-¹⁶ but no consensus has been reached in the literature.³^,¹⁷^,¹⁸ Recognising that adjustment for birthweight itself may have inadvertent effects on estimation, some have adopted a strategy dependent upon z-scores of birthweight that is more akin to standardisation.²^,¹⁴ Z-scores - birthweight in units of standard deviations within strata of the other risk factors of interest - are generated and then used for adjustment, for example as a regression covariate.¹⁴ Proponents of z-scores have suggested the paradox to be an artifact of adjustment for unstandardised birthweight, and claim that the substitution of z-scores for birthweight eliminates the crossover of the curves and produces unbiased results.¹⁴ Greenland and others have cautioned against the use of standardised regression coefficients on the grounds that standardisation of coefficients may result in confounding and also because there is a loss of interpretability. The issues regarding use of standardised regression coefficients and z-scores are very much related because adjusting for z-scores influences interpretation of the effect of interest.¹⁹^,²⁰

We have previously utilised directed acyclic graphs (DAG) to show that adjustment for birthweight in the estimation of total effects of an exposure of interest (e.g. smoking on infant mortality) introduces bias under certain plausible causal assumptions.²¹^,²² The paradoxical crossing of curves may be explained as the product of stratification on a common effect in the presence of unmeasured confounders.²¹^,²³^-²⁶ In this paper, we expand this finding to consideration of standardised birthweight - birthweight z-scores - and provide estimates for the potential magnitude of bias. We use DAGs to illustrate causal questions in the context of direct, indirect and total effects on an outcome of interest, and to describe how z-scores fit into causal systems relating risk factors with birthweight and neonatal mortality. We present analytical results, as well as a simulation study to evaluate the use of z-scores as a regression term on estimates under a range of scenarios.

The causal question and causal diagrams

Causal diagrams are useful tools for displaying the causal structure underlying relations in epidemiological studies.²⁷^,²⁸ DAGs may be used to show how standard adjustment (stratification or regression) for variables affected by exposure may create bias by introducing a spurious (non-causal) association between the exposure and the outcome.²⁴^,²⁸ A simple DAG representing the factors under consideration is shown in Fig. 2a.

Directed acyclic graphs (DAG) representing causal relationships between a risk factor, birthweight, unmeasured variables $(U)$ 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, $U$ . (c) DAG representing collider-stratification bias introduced by adjusting for birthweight in the presence of an unmeasured confounder, $U$ , 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.

Direct and indirect effects

The DAGs may be used to illustrate the notion of effect decomposition from total into direct and indirect effects.²⁹^-³³ This circumstance arises when a causal network resembles that shown in Fig. 2a, in which there are two paths from the exposure to neonatal mortality: one direct effect of exposure and one indirect effect that passes through birthweight. Effect decomposition is of particular relevance if interventions are possible that act on one of the separate effects. In the context of birthweight and neonatal mortality, those interested in assessing risk factors such as smoking have often been concerned about the effects on mortality which are separate from those mediated through birthweight.²¹^,²² It should be noted that decomposition is only valid under certain very strong assumptions. Specifically, validity requires absence of confounding, monotonicity, no unit-level interaction, use of linear contrasts as measures of effects and homogeneity of causal effects across strata of birthweight.²⁵^,³⁰^,³¹ It is doubtful that these assumptions would be met in most real world applications, putting into question the utility of effect decomposition under most circumstances. However, for illustration we assume that these assumptions are met, and assume the causal structure displayed in Fig. 2a. Therefore, regression models for mortality may be considered as follows:Total effects model:

l o g (p (N M = 1)) = α_{1} + (β_{R F_{1}} \times R F);

(1)

Effect decomposition model:

l o g (p (N M = 1)) = α_{2} + (β_{R F_{2}} \times R F) + (β_{B W T} \times B W T);

(2)

where p is the probability of neonatal mortality (NM), RF represents the risk factor of interest and BWT represents birthweight. According to the causal structure represented in Fig. 2a, we assume that the risk factor is the only cause of differences in birthweight. If we then assume that the association between the risk factor and birthweight follows a linear function such that:

B W T = γ + ({\overset{‒}{ω}}_{R F} \times R F),

(3)

then the effect decomposition model is equal to:

\begin{matrix} l o g (p (N M = 1)) = & α_{2} + (β_{R F_{2}} \times R F) + (β_{B W T} \times B W T)) \\ = & α_{2} + (β_{R F_{2}} \times R F) + β_{B W T} (γ + ({\overset{‒}{ω}}_{R F} \times R F)) \\ = & α_{2} + β_{B W T} \times γ + (β_{R F_{2}} \times R F) + \\ (β_{B W T} \times {\overset{‒}{ω}}_{R F} \times R F) . \end{matrix}

(4)

It then follows that

β_{R F_{1}} = β_{R F_{2}} + β_{B W T} \times {\overset{‒}{ω}}_{R F} .

(5)

The direct effect is represented by β_RF₂, and the indirect effect as $β_{B W T} \times {\overset{‒}{ω}}_{E}$ . The sum of these effects is the total effect of exposure on the log rate ratio of neonatal mortality.

Unmeasured factors - confounders and colliders

The presence of additional factors in causal networks when one is interested in total effect decomposition is potentially problematic. As shown in Fig. 2b, adjustment for birthweight, in the presence of an unmeasured variable that affects both birthweight and mortality, introduces collider-stratification bias by creating a spurious association between the unmeasured variable $U$ and exposure (the back door pathway is shown as a dotted line). Additionally, the indirect effect estimate is confounded by the unmeasured factor, $U$ . The direct effect is also biased and cannot be estimated without bias using the available data. Collider-stratification bias is a result of adjusting for birthweight whenever the relation between birthweight and mortality is confounded by an unmeasured factor and there is a link between the exposure of interest and birthweight (Fig. 2b). This is true even when there is no direct effect of the exposure on mortality or between birthweight and neonatal mortality (as represented in Fig. 2c).

Application of z-scores for adjustment

Use of z-scores within strata of the other risk factors of interest has been suggested as an approach to remove the crossover of the curves and produce unbiased results.¹⁰^,¹¹^,¹⁴^,²⁶ This approach requires calculation of the z-score for each individual birthweight (bw_ij) within strata such that:

Z_{i} = \frac{b w_{i j} - {\overset{‒}{b w}}_{j}}{S D (b w_{j})},

(6)

where i represents individuals 1 through n; j represents strata of another risk factor 1 through k, bw_ij is the birthweight of the ith individual in the jth stratum, ${\overset{‒}{b w}}_{j}$ is the estimated mean birthweight for the jth stratum, and SD(bw_j) is the standard deviation of birthweight in the jth stratum. These methods have been applied extensively in the analysis of various risk factors on neonatal mortality when birthweight is considered as a covariate. A modified approach has also been proposed, where instead of the z-score, the within-stratum percentile is utilised to ensure the curves do not cross. These claims have not been evaluated either analytically or numerically. Nevertheless, the effect of use of z-scores in regression models has not been explicitly stated; it is unclear if the estimators for the main exposure effect adjusted by z-scores that yield the ‘uncrossing’ of the curves yield unbiased estimators for the direct effect, indirect effect, both or neither.

Smoking, birthweight z-scores and neonatal mortality

When the interest is estimation of the total effect of an exposure, both that acting through its effect on birthweight and directly on neonatal mortality, the question of whether to use birthweight or z-scores is an academic one, as neither will be in the model. Thus, we limit our focus to whether the inclusion of a regression term for birthweight z-scores in models of neonatal mortality allows for decomposition of total effects into direct and indirect effects.

Z-scores may be seen as a variable transformation approach. Although it is clear that both birthweight and exposure have a deterministic relation with individual z-scores (Fig. 2d), the effect of the transformation is to create a birthweight variable that is independent of exposure status. It may be demonstrated in datasets with information on smoking (yes/no), birthweight and neonatal mortality, and in which a relationship between birthweight and neonatal mortality is observed, that z-scores retain an association with birthweight and neonatal mortality but are independent of smoking status. One way to think about the independence of z-scores and smoking is to consider the simple case where smoking shifts the birthweight distribution. In this manner, the counterfactual smoking birthweight for non-smokers could be calculated by subtracting a given value and the counterfactual non-smoking birthweight for smokers could be calculated by adding the same value. A given birth would then be the same distance from the mean smoking birthweight as it would be from the mean non-smoking birthweight if all smokers had been non-smokers. Thus, because the z-score is a measure of distance from the mean, it remains independent of the smoking stratum the birth is in. The DAG in Fig. 2e shows the resultant relationships among variables in the system. This DAG suggests that the z-score, because of its retained relationship with birthweight, will be related to outcome whenever birthweight is related to outcome. Effect estimates for z-scores will differ from those of birthweight because of the different scale due to transformation. The exact relationship between regression coefficients of the relationship between z-score and outcome and regression coefficients of birthweight and outcome is demonstrated in an analytical proof (see Appendix 1).

Note that Fig. 2e represents a situation that has been described as ‘unfaithfulness’.³⁴ Unfaithfulness is the situation where the DAG does not represent reality and the true causal structure. In this case, the smoking variable was used to perform the transformation thereby implying the presence of an arrow between smoking and z-score, as shown in Fig. 2d; however, the transformed variable, z-score, is independent of smoking, thereby implying no arrow between the two nodes as shown in Fig. 2e. There are important implications of the fact that smoking is uncorrelated with birthweight z-scores. First, the independence of smoking and birthweight z-scores suggests that estimates from smoking-only models and those adjusted for z-score should be equal. That is:

\begin{matrix} smoking ⊥ ⊥ z - score & \Rightarrow \\ E (Y ∣ smoking) = E & (Y ∣ smoking, z - score) \end{matrix}

Accordingly, the DAG suggests that the use of z-scores should be effective for avoiding collider-stratification bias; the backdoor pathway opened with stratification on birthweight is no longer so with the z-score. This result also is demonstrated in an analytic proof (see Appendix 2).

Simulation study

An extensive simulation study was conducted to compare estimates of exposure effects from the unadjusted (crude) model, the birthweight-adjusted model and the adjusted model utilising birthweight z-scores based on the causal relationships illustrated in Fig. 2e. These simulations were run in the absence (Table 1) and presence (Table 2) of unmeasured factors that confound the relationship between birthweight and neonatal mortality. Birthweight was simulated as:

B W T = 3500 + (e \times smoking) + (a \times U_{B W T}),

(7)

where e is the effect of smoking on birthweight with values (-100, -150, -200, -500),³⁵ $U_{B W T}$ represents the unmeasured variable affecting birthweight with values (0, 1), and a is the effect of the unmeasured factor with values (-100, -200, -400, -500, -1000).

Table 1.

Simulation study results: true total and direct effects of smoking (S) on neonatal mortality (NM) and parameter estimates under the unadjusted (β̂_S), birthweight (BWT) adjusted (β̂_S|BWT) and z-score (Z) adjusted (β̂_S|Z) regression models^a

True effects			Effect estimated in regression models
Smoking on BWT (e)	Direct effect Θ_D = c	Total effect Θ_T = c + d*e	Smoking			Smoking\|BWT			Smoking\|z-score
Smoking on BWT (e)	Direct effect Θ_D = c	Total effect Θ_T = c + d*e	β̂_S	Bias	MSE	β̂_S\|BWT	Bias	MSE	β̂_S\|Z	Bias	MSE
-100	0.2	0.250	0.24057	-0.00943	0.07955	0.19065	-0.00935	0.08051	0.24060	-0.00940	0.07954
	0.3	0.350	0.34278	-0.00722	0.07631	0.29266	-0.00734	0.07734	0.34279	-0.00721	0.07632
	0.4	0.450	0.44356	-0.00644	0.06938	0.39306	-0.00694	0.07049	0.44355	-0.00645	0.06938
-150	0.2	0.275	0.26422	-0.01078	0.07724	0.18798	-0.01202	0.07942	0.26424	-0.01076	0.07724
	0.3	0.375	0.36341	-0.01159	0.07372	0.28901	-0.01099	0.07611	0.36341	-0.01159	0.07371
	0.4	0.475	0.47021	-0.00479	0.06767	0.39538	-0.00462	0.06988	0.47020	-0.00480	0.06767
-200	0.2	0.300	0.28665	-0.01335	0.07690	0.18608	-0.01392	0.08098	0.28666	-0.01334	0.07689
	0.3	0.400	0.39026	-0.00974	0.07307	0.29001	-0.00999	0.07771	0.39028	-0.00972	0.07307
	0.4	0.500	0.49566	-0.00434	0.06788	0.39602	-0.00398	0.07125	0.49567	-0.00433	0.06788
-500	0.2	0.450	0.43903	-0.01097	0.06884	0.19157	-0.00843	0.09362	0.43903	-0.01097	0.06884
	0.3	0.550	0.54691	-0.00309	0.06478	0.29914	-0.00086	0.08890	0.54691	-0.00309	0.06478
	0.4	0.650	0.64553	-0.00447	0.06083	0.39743	-0.00257	0.08451	0.64554	-0.00446	0.06083

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BWT = 3500 + (e × smoking) + (a × $U_{B W T}$ ), $U_{B W T}$ = 0; log(p(NM = 1))=-3.5+(c×smoking)+(d×BWT)+(b× $U_{N M}$ ), $U_{N M}$ = 0, d=-0.0005.

MSE, Mean squared error.

Table 2.

Simulation study results: true total and direct effects of smoking (S) on neonatal mortality (NM) and parameter estimates in the presence of an unmeasured confounder $U$ of the birthweight-mortality relation under the unadjusted (β̂_S), birthweight (BWT) adjusted (β̂_S|BWT) and z-score (Z) adjusted (β̂_S|Z) regression models^a

True effects		Effect estimated in regression models
		Smoking			Smoking\|BWT			Smoking\|z-score
$U$ on BWT (a)	$U$ on NM (b)	β̂_S	Bias	MSE	β̂_S\|BWT	Bias	MSE	β̂_S\|Z	Bias	MSE
-100	0.2	0.28653	-0.01347	0.07311	0.18396	-0.01604	0.07717	0.28654	-0.01346	0.07311
	0.4	0.28688	-0.01312	0.07157	0.18146	-0.01854	0.07591	0.28688	-0.01312	0.07157
	1.0	0.28871	-0.01129	0.06411	0.17204	-0.02796	0.06858	0.28872	-0.01128	0.06411
	1.6	0.29167	-0.00833	0.05383	0.15925	-0.04075	0.05891	0.29167	-0.00833	0.05383
-200	0.2	0.28634	-0.01366	0.07282	0.18121	-0.01879	0.07687	0.28635	-0.01365	0.07282
	0.4	0.28708	-0.01292	0.07128	0.17596	-0.02404	0.07570	0.28709	-0.01291	0.07128
	1.0	0.28912	-0.01088	0.06298	0.15490	-0.04510	0.06870	0.28914	-0.01086	0.06298
	1.6	0.29176	-0.00824	0.05263	0.12618	-0.07382	0.06135	0.29177	-0.00823	0.05264
-400	0.2	0.28669	-0.01331	0.07184	0.17648	-0.02352	0.07583	0.28670	-0.01330	0.07184
	0.4	0.28718	-0.01282	0.06993	0.16492	-0.03508	0.07459	0.28720	-0.01280	0.06993
	1.0	0.28951	-0.01049	0.06172	0.12224	-0.07776	0.07127	0.28955	-0.01045	0.06173
	1.6	0.29108	-0.00892	0.05024	0.06430	-0.13570	0.07165	0.29115	-0.00885	0.05026
-1000	0.2	0.28642	-0.01358	0.06883	0.16859	-0.03141	0.07175	0.28645	-0.01355	0.06884
	0.4	0.28790	-0.01210	0.06617	0.14923	-0.05077	0.07071	0.28795	-0.01205	0.06618
	1.0	0.28986	-0.01014	0.05695	0.07869	-0.12131	0.07331	0.29000	-0.01000	0.05699
	1.6	0.29097	-0.00903	0.04367	0.00086	-0.19914	0.08418	0.29095	-0.00905	0.04368

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BWT = 3500 + (e × smoking) + (a × $U_{B W T}$ ), $U_{B W T}$ = 1, e=-200; log(p(NM = 1))=-3.5+(c×smoking)+(d×BWT)+(b× $U_{N M}$ ), $U_{N M}$ = 1, c=0.2, d=-0.0005. Direct effect Θ_D=c=0.2; Total effect Θ_T=c+d*e=0.3.

MSE, Mean squared error.

Neonatal mortality risk was simulated as:

\begin{matrix} \log (p (N M = 1)) = & - 3.5 + (c \times smoking) + \\ (d \times B W T) + (b \times U_{N M}), \end{matrix}

(8)

where c is the direct effect of smoking with effects (0.2, 0.3, 0.4), and $U_{N M}$ is an unmeasured factor affecting neonatal mortality with values (0, 1), b is the effect of the unmeasured factor with values (0.2, 0.4, 1.0, 1.6), and d is the effect of birthweight on neonatal mortality (-0.0005). The coefficients a, b, c, d and e represent direct effects of one variable on another and are displayed in Fig. 2e. The choice of parameters was based on realistic scenarios; where there is no effect of the unmeasured factors $U_{B W T}$ and $U_{N M}$ the values of $U$ are set to 0. Using these equations, the true direct effect of smoking is equal to Θ_D = c, the true indirect effect is Θ_I = d*e, and therefore the true total effect of smoking on neonatal mortality is Θ_T = c + d*e.

We utilised three regression models of neonatal mortality for estimation of direct, indirect and total effects of the binomial factor smoking (S), accounting for birthweight in grams (BWT), or birthweight z-scores (Z) as follows:Unadjusted Model:

\log (p (N M = 1)) = α_{1} + β_{S} S,

(9)

Adjusted model by birthweight:

\log (p (N M = 1)) = α_{2} + β_{S ∣ B W T} S + β_{B W T} B W T,

(10)

Adjusted model by birthweight z-score:

\log (p (N M = 1)) = α_{3} + β_{S ∣ Z} S + β_{Z} Z,

(11)

where β_S is the total effect, β_S|BWT is the direct effect, and β_S|Z is the standardised effect. For each scenario, 5000 datasets were generated with 10 000 observations each and the mean and variance of the estimates, β_S, β_S|BWT and β_S|Z were calculated.

Simulation study results

No unmeasured confounding factors of the birthweight-neonatal mortality relation

The results of simulations where smoking, birthweight and neonatal mortality are sufficient to describe the causal system (Fig. 2a) can be found in Table 1. Examining these results, we find that the unadjusted and the z-score-adjusted model calculate the same point estimates and variances for the total effect of smoking on neonatal mortality. In other words, inclusion of birthweight z-scores does not alter the estimate of the total effect from the basic unadjusted model. Conversely, we observe that the birthweight-adjusted model correctly estimates the direct effect of smoking on neonatal mortality. When smoking and birthweight are sufficient to predict neonatal mortality and there are no unmeasured confounders or other causes of mortality, the effect estimate for birthweight itself is equivalent to the indirect effect of smoking on neonatal mortality. However, sufficiency of the DAG, and the absence of unmeasured confounding are strong assumptions. Violation of these assumptions can substantially impair the effect decomposition approach.²⁵^,³⁰

Unmeasured confounding factors of the birthweight-neonatal mortality relation

Table 2 displays the results of simulations run under the addition of a factor $U$ that confounds the relationship between birthweight and neonatal mortality, reflecting a potential for collider-stratification (Fig. 2b). The factor $U$ was used to create the birthweight variable as described above, but these models did not include $U$ in order to simulate the effects of an unmeasured confounder. The unadjusted model yielded unbiased estimates of the total effect, under various assumptions regarding the strength of the associations between $U$ and birthweight and neonatal mortality. Adjustment for birthweight resulted in unbiased estimates of the direct effect when the effect of $U$ on birthweight was small. As the effect of $U$ on birthweight and the effect of $U$ on neonatal mortality increased, estimation of the direct effect was negatively biased, and the relative bias was also a function of the strength of these effects (Fig. 3). Z-score-adjusted models yielded nearly identical estimates as unadjusted models. Z-score-specific mortality curves for smokers and nonsmokers, as shown in Fig. 4, further demonstrate that while adjusting for z-scores removes the paradoxical crossing of the curves, this adjustment is not an issue of confounding. Rather, at issue is the effect being estimated, as the distance between the curves for smokers and nonsmokers is equal to the total effect.

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.

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.

As previously discussed, z-score-adjusted models estimated the total effects of smoking, with bias equivalent to the unadjusted model. When simulations do not include an unmeasured factor affecting birthweight and neonatal mortality, z-score parameter estimates can be transformed to recapture the unstandardised birthweight-neonatal mortality relationship, as described in Appendix 1. When $U$ was included in the causal system, neither the birthweight-adjusted nor the z-score adjusted model were effective for effect decomposition, with estimates of the direct smoking effect being biased (birthweight adjustment), or unbiased estimates of the total effect (z-score adjustment).

Discussion

Attempts to resolve the birthweight paradox have led investigators to a variety of practices regarding the handling of birthweight, in models of outcomes and risk factors of interest, such as inclusion of birthweight in regression models.¹⁴ Others have proposed various approaches to the use of z-scores towards this end, leading to questions about the effect of modelling standardised birthweight.²^,³^,¹⁸ In this paper, we have evaluated the effect of inclusion of the z-score as a regression term both in the absence and presence of unmeasured factors using a simulation study and analytical proofs. We have demonstrated that adjustment of z-scores in regression models yields unbiased estimates for the total effect of the primary factor of interest when birthweight does and does not represent a collider. It should be pointed out that these results match results from the unadjusted model, and that z-score adjustment offers no advantage over unadjusted models. However, in the presence of confounding of the risk factor-outcome relationship, graphical representation of the z-scores (which removes the crossing of the curves, and gives an estimate of the total effect) does not allow for multivariable confounding adjustment. Only a model-based approach using the unadjusted model as the base model can be used to find unconfounded estimates of the total effect.

It is not uncommon practice to adjust for an intermediate variable (results of birthweight) when estimating the effect of a risk factor on an outcome (e.g. smoking on neonatal mortality). However, birthweight adjustment is not recommended for effect decomposition. The potential pitfalls of effect decomposition have been described in the literature and require several strong assumptions, which are generally not met in practice.²³^,²⁵^,³⁰ The results presented here show that models that use adjustment for z-scores are not effective for separating direct and indirect effects, adding to the concerns regarding effect decomposition. Models adjusting for z-scores can be used to estimate total effects, but not direct effects. We have demonstrated that a scaled version of the indirect effects can be evaluated through consideration of the estimate for birthweight z-scores, which may be rescaled to reflect unstandardised birthweight. However, this approach is not recommended because it also requires strong assumptions, including equal variances, and is contingent upon additional computation to address potential differences in birthweight distributions between strata of the standardising factor (e.g. smoking) (see Appendix 1). Regardless, the interpretation of unadjusted, birthweight-adjusted and z-score-adjusted estimates is important to understand for models to reflect investigator hypotheses correctly.

In this paper, we have used causal diagrams to illustrate the causal relationships among risk factors, birthweight, z-scores of birthweight and neonatal mortality both in the absence and presence of unmeasured common causes of birthweight and neonatal mortality. Importantly, many such factors have been postulated in the literature, including genetic, environmental and behavioural factors. Both analytically and through an extensive simulation study, we have shown that neither birthweight nor z-scores may be used for effect decomposition, and that the true utility of z-scores is in estimating unbiased total effects of exposures, even when collider-stratification would adversely influence estimates from birthweight-adjusted models. However, the use of z-scores does not add any additional information beyond the use of unadjusted models.

While the work presented here has only focused on the relationship between smoking, birthweight and neonatal mortality, the implications are more farreaching. The issue of the birthweight paradox and intersecting mortality curves is a more general phenomenon and applies to other outcomes, such as stillbirth,¹⁶ other exposures, such as race, parity, altitude and infant sex,⁷ and other standardised variables, such as gestational age.¹⁷ It should be acknowledged that the ability of z-scores to successfully resolve the paradoxical crossing of the curves (in any of these situations), is due to an alteration in the causal parameter being estimated (total effect), rather than adjustment for confounding or effect decomposition.

Acknowledgements

The authors thank Dr Miguel Hernán and Dr Sonia Hernández-Diaz for their valuable comments. We are also grateful to the editor and referees for their helpful comments which improved this paper. This research was supported by the Intramural Research Program of the National Institutes of Health, Eunice Kennedy Shriver National Institute of Child Health and Human Development. Robert W. Platt is a Chercheur-boursier of the Fonds de la Recherche en Santé du Québec (FRSQ), and is a member of the Research Institute of the McGill University Health Centre, which is supported in part by the FRSQ.

Appendix 1

In the case where there are no unmeasured factors affecting birthweight and neonatal mortality, z-score parameter estimates can be transformed to recapture the unstandardised birthweight-neonatal mortality relation. To this end, we define:

S: Binomial factor indicating smoking status (1 = Yes, 0 = No),
BWT: Birthweight in grams,
Z: Standardised birthweight (z-score),
NM: Neonatal mortality rate.

We consider the following three models where α₁, α₂, and α₃ correspond to the intercept in models 1 through 3, respectively:

log(p(NM = 1)) = α₁ + β_SS + ε₁;
log(p(NM = 1)) = α₂ + β_S|_BWTS + β₂BWT + ε₂;
log(p(NM = 1)) = α₃ + β_S|ZS + β₃Z + ε₃.

Z_S and Z_NS are the z-scores or standardised birthweight for smokers and nonsmokers, respectively.

For smokers, Z_{S} = \frac{B W T - (\overset{‒}{B W T} ∣ (S = 1))}{s d (\overset{‒}{B W T} ∣ (S = 1))} .

For nonsmokers, Z_{N S} = \frac{B W T - (\overset{‒}{B W T} ∣ (S = 0))}{s d (\overset{‒}{B W T} ∣ (S = 0))} .

For smokers the models become:

\begin{matrix} \log (p (N M = 1)) & = α_{1} + β_{S} + ε_{1}, \\ \log (p (N M = 1)) & = α_{2} + β_{S ∣ B W T} + β_{2} (B W T ∣ (S = 1)) + ε_{2}, \\ \log (p (N M = 1)) & = α_{3} + β_{S ∣ Z} + β_{3} Z + ε_{3}, \\ = α_{3} + β_{S ∣ Z} + β_{3} \frac{B W T - (\overset{‒}{B W T} ∣ (S = 1))}{s d (B W T ∣ (S = 1))} + ε_{3} \\ = (α_{3} - \frac{β_{3} (\overset{‒}{B W T} ∣ (S = 1))}{s d (B W T ∣ (S = 1))}) + β_{S ∣ Z} + \\ \frac{β_{3}}{s d (B W T ∣ (S = 1))} B W T + ε_{3} . \end{matrix}

For nonsmokers the models become:

\begin{matrix} \log (p (N M = 1)) & = α_{1} + ε_{1}, \\ \log (p (N M = 1)) & = α_{2} + β_{2} (B W T ∣ (S = 0)) + ε_{2}, \\ \log (p (N M = 1)) & = α_{3} + β_{3} Z + ε_{3} \\ = α_{3} + β_{3} \frac{B W T - (\overset{‒}{B W T} ∣ (S = 0))}{s d (B W T ∣ (S = 0))} + ε_{3} \\ = (α_{3} - \frac{β_{3} (\overset{‒}{B W T} ∣ (S = 0))}{s d (\overset{‒}{B W T} ∣ (S = 0))}) + \\ \frac{β_{3}}{s d (\overset{‒}{B W T} ∣ (S = 0))} B W T + ε_{3} . \end{matrix}

Consider the expectation of log(p(NM = 1)) in Model 2 for smokers and nonsmokers, respectively:

α_{1} + β_{S} = α_{2} + β_{S ∣ B W T} + β_{2} (\overset{‒}{B W T} ∣ (S = 1)),

α_{1} = α_{2} + β_{2} (\overset{‒}{B W T} ∣ (S = 0)) .

Therefore, we have

β_{S} = β_{S ∣ B W T} + β_{2} {(\overset{‒}{B W T} ∣ (S = 1)) - (\overset{‒}{B W T} ∣ (S = 0))} .

If sd(BWT|(S = 1)) = sd(BWT|(S = 0)), we have

β_{2} = \frac{β_{3}}{s d (B W T ∣ (S = 0))} .

Appendix 2

In order to show that conditioning on z-scores removes the association between smoking status and an unmeasured variable associated with birthweight and infant mortality which is created by conditioning on the collider birthweight (Fig. 2e), we define:

W: Birthweight,
S: Indicator of smoking status (1 = Yes, 0 = No),
$U$ : An unmeasured Bernoulli random variable with $P (U = 1) = p$ , and $P (U = 0) = 1 - p = q$ .

The linear model is: $W = W_{0} + a U + b S$ .For smokers, S = 1, the birthweight can be written as:

W_{S} = W_{0} + a U + b,

For nonsmokers, S = 0, the birthweight can be written

as : W_{N S} = W_{0} + a U .

It follows that:

E (W_{S}) = W_{0} + b + a p, V a r (W_{S}) = a^{2} p (1 - p)

E (W_{N S}) = W_{0} + a p, V a r (W_{N S}) = a^{2} p (1 - p)

Application of the standardisation procedure Z = (X - μ_W)/σ_W on smokers and nonsmokers separately, yields the z-scores:

\begin{matrix} Z_{S} & = \frac{W_{0} + a U + b - (W_{0} + a p + b)}{\sqrt{a^{2} p (1 - p)}} = \frac{a (U - p)}{\sqrt{a^{2} p (1 - p)}} \\ = \frac{s i g n (a)}{\sqrt{p (1 - p)}} \times (U - p), \end{matrix}

\begin{matrix} Z_{N S} & = \frac{W_{0} + a U - (W_{0} + a p)}{\sqrt{a^{2} p (1 - p)}} = \frac{a (U - p)}{\sqrt{a^{2} p (1 - p)}} \\ = \frac{s i g n (a)}{\sqrt{p (1 - p)}} \times (U - p), \end{matrix}

thus Z_S = Z_NS = Z.It then follows that,

\begin{matrix} P (Z = z ∣ S = s) = P (Z = z), \\ P (U = u ∣ S = s) = P (U = u), \\ and P (U = u ∣ S = 1, Z_{S} = z) = P (U = u ∣ S = 0, Z_{N S} = z) . \end{matrix}

Consequently, $P (U = u ∣ S = 1, Z_{S} = z) = P (U = u ∣ S = 0, Z_{N S} = z) = P (U = u ∣ Z = z)$ , by

\begin{matrix} P (U = u ∣ S = s, Z = z) & = \frac{P (U = u, S = s, Z = z)}{P (Z = z, S = s)} \\ = \frac{P (U = u, S = s, Z = z)}{P (Z = z ∣ S = s) \cdot P (S = s)} \\ = \frac{P (U = u, S = s, Z = z)}{P (Z = z) \cdot P (S = s)} \\ = \frac{P (U = u, Z = z ∣ S = s) \cdot P (S = s)}{P (Z = z) \cdot P (S = s)} \\ = \frac{P (U = u, Z = z)}{P (Z = z)} \\ = \frac{P (U = u ∣ Z = z) \cdot P (Z = z)}{P (Z = z)} \\ = P (U = u ∣ Z = z) \end{matrix}

This is equivalent to: $U ⊥ ⊥ S ∣ Z$ .

References

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16.Joyce R, Peacock J. A comparison of methods of adjusting stillbirth and neonatal mortality rates for birthweight in hospital and geographical populations. Paediatric and Perinatal Epidemiology. 2003;17:119–124. doi: 10.1046/j.1365-3016.2003.00486.x. [DOI] [PubMed] [Google Scholar]
17.Wilcox AJ, Weinberg CR. Invited commentary: analysis of gestational-age-specific mortality - on what biologic foundations? American Journal of Epidemiology. 2004;160:213–214. doi: 10.1093/aje/kwh204. [DOI] [PubMed] [Google Scholar]
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21.Hernandez-Diaz S, Schisterman EF, Hernan MA. The birth weight ‘paradox’ uncovered? American Journal of Epidemiology. 2006;164:1115–1120. doi: 10.1093/aje/kwj275. [DOI] [PubMed] [Google Scholar]
22.Hernandez-Diaz S, Wilcox AJ, Schisterman EF, Hernan MA. From causal diagrams to birth weight-specific curves of infant mortality. European Journal of Epidemiology. 2008;23:163–166. doi: 10.1007/s10654-007-9220-4. [DOI] [PMC free article] [PubMed] [Google Scholar]
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26.Wilcox AJ, Russell IT. Perinatal mortality: standardizing for birthweight is biased. American Journal of Epidemiology. 1983;118:857–864. doi: 10.1093/oxfordjournals.aje.a113704. [DOI] [PubMed] [Google Scholar]
27.Greenland S, Pearl J, Robins JM. Causal diagrams for epidemiologic research. Epidemiology. 1999;10:37–48. [PubMed] [Google Scholar]
28.Hernan MA, Hernandez-Diaz S, Werler MM, Mitchell AA. Causal knowledge as a prerequisite for confounding evaluation: an application to birth defects epidemiology. American Journal of Epidemiology. 2002;155:176–184. doi: 10.1093/aje/155.2.176. [DOI] [PubMed] [Google Scholar]
29.Greenland S, Robins JM. Identifiability, exchangeability, and epidemiological confounding. International Journal of Epidemiology. 1986;15:413–419. doi: 10.1093/ije/15.3.413. [DOI] [PubMed] [Google Scholar]
30.Kaufman JS, MacLehose RF, Kaufman S. A further critique of the analytic strategy of adjusting for covariates to identify biologic mediation. Epidemiologic Perspectives and Innovations. 2004;1:4. doi: 10.1186/1742-5573-1-4. [DOI] [PMC free article] [PubMed] [Google Scholar]
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32.Kaufman S, Kaufman JS, MacLehose RF, Greenland S, Poole C. Improved estimation of controlled direct effects in the presence of unmeasured confounding of intermediate variables. Statistics in Medicine. 2005;24:1683–1702. doi: 10.1002/sim.2057. [DOI] [PubMed] [Google Scholar]
33.Petersen ML, Sinisi SE, van der Laan MJ. Estimation of direct causal effects. Epidemiology. 2006;17:276–284. doi: 10.1097/01.ede.0000208475.99429.2d. [DOI] [PubMed] [Google Scholar]
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35.Kramer MS. Determinants of low birth weight: methodological assessment and meta-analysis. Bulletin of the World Health Organization. 1987;65:663–737. [PMC free article] [PubMed] [Google Scholar]

[R1] 1.Basso O, Wilcox AJ, Weinberg CR. Birth weight and mortality: causality or confounding? American Journal of Epidemiology. 2006;164:303–311. doi: 10.1093/aje/kwj237. [DOI] [PubMed] [Google Scholar]

[R2] 2.Wilcox AJ. On the importance - and the unimportance - of birthweight. International Journal of Epidemiology. 2001;30:1233–1241. doi: 10.1093/ije/30.6.1233. [DOI] [PubMed] [Google Scholar]

[R3] 3.Wilcox AJ. Commentary: On the paradoxes of birthweight. International Journal of Epidemiology. 2003;32:632–633. doi: 10.1093/ije/dyg193. [DOI] [PubMed] [Google Scholar]

[R4] 4.MacDorman MF, Callaghan WM, Mathews TJ, Hoyert DL, Kochanek KD. Trends in preterm-related infant mortality by race and ethnicity, United States, 1999-2004. International Journal of Health Services. 2007;37:635–641. doi: 10.2190/HS.37.4.c. [DOI] [PubMed] [Google Scholar]

[R5] 5.Mathews TJ, MacDorman MF. Infant mortality statistics from the 2004 period linked birth/infant death data set. National Vital Statistics Reports. 2007;55:1–32. [PubMed] [Google Scholar]

[R6] 6.Schisterman EF, Cole SR, Platt RW. Over-adjustment bias and unnecessary adjustment in epidemiologic studies. Epidemiology. 2009;20:488–495. doi: 10.1097/EDE.0b013e3181a819a1. [DOI] [PMC free article] [PubMed] [Google Scholar]

[R7] 7.Wilcox AJ. Birth weight and perinatal mortality: the effect of maternal smoking. American Journal of Epidemiology. 1993;137:1098–1104. doi: 10.1093/oxfordjournals.aje.a116613. [DOI] [PubMed] [Google Scholar]

[R8] 8.Buekens P, Wilcox A. Why do small twins have a lower mortality rate than small singletons? American Journal of Obstetrics and Gynecology. 1993;168:937–941. doi: 10.1016/s0002-9378(12)90849-2. [DOI] [PubMed] [Google Scholar]

[R9] 9.Ananth CV, Wilcox AJ. Placental abruption and perinatal mortality in the United States. American Journal of Epidemiology. 2001;153:332–337. doi: 10.1093/aje/153.4.332. [DOI] [PubMed] [Google Scholar]

[R10] 10.Wilcox AJ, Russell IT. Birthweight and perinatal mortality: II. On weight-specific mortality. International Journal of Epidemiology. 1983;12:319–325. doi: 10.1093/ije/12.3.319. [DOI] [PubMed] [Google Scholar]

[R11] 11.Wilcox AJ, Russell IT. Birthweight and perinatal mortality: I. On the frequency distribution of birthweight. International Journal of Epidemiology. 1983;12:314–318. doi: 10.1093/ije/12.3.314. [DOI] [PubMed] [Google Scholar]

[R12] 12.Wilcox AJ, Russell IT. Birthweight and perinatal mortality: III. Towards a new method of analysis. International Journal of Epidemiology. 1986;15:188–196. doi: 10.1093/ije/15.2.188. [DOI] [PubMed] [Google Scholar]

[R13] 13.MacDorman MF, Atkinson JO. Infant mortality statistics from the linked birth/infant death data set - 1995 period data. Monthly Vital Statistics Reports. 1998;46:1–22. [PubMed] [Google Scholar]

[R14] 14.Hertz-Picciotto I, Din-Dzietham R. Comparisons of infant mortality using a percentile-based method of standardization for birthweight or gestational age. Epidemiology. 1998;9:61–67. doi: 10.1097/00001648-199801000-00009. [DOI] [PubMed] [Google Scholar]

[R15] 15.Hertz-Picciotto I. Is it time to abandon adjustment for birth weight in studies of infant mortality? Paediatric and Perinatal Epidemiology. 2003;17:114–116. doi: 10.1046/j.1365-3016.2003.00483.x. [DOI] [PubMed] [Google Scholar]

[R16] 16.Joyce R, Peacock J. A comparison of methods of adjusting stillbirth and neonatal mortality rates for birthweight in hospital and geographical populations. Paediatric and Perinatal Epidemiology. 2003;17:119–124. doi: 10.1046/j.1365-3016.2003.00486.x. [DOI] [PubMed] [Google Scholar]

[R17] 17.Wilcox AJ, Weinberg CR. Invited commentary: analysis of gestational-age-specific mortality - on what biologic foundations? American Journal of Epidemiology. 2004;160:213–214. doi: 10.1093/aje/kwh204. [DOI] [PubMed] [Google Scholar]

[R18] 18.Wilcox AJ. Invited commentary: the perils of birth weight - a lesson from directed acyclic graphs. American Journal of Epidemiology. 2006;164:1121–1123. doi: 10.1093/aje/kwj276. [DOI] [PubMed] [Google Scholar]

[R19] 19.Greenland S, Schlesselman JJ, Criqui MH. The fallacy of employing standardized regression coefficients and correlations as measures of effect. American Journal of Epidemiology. 1986;123:203–208. doi: 10.1093/oxfordjournals.aje.a114229. [DOI] [PubMed] [Google Scholar]

[R20] 20.Greenland S, Maclure M, Schlesselman JJ, Poole C, Morgenstern H. Standardized regression coefficients: a further critique and review of some alternatives. Epidemiology. 1991;2:387–392. [PubMed] [Google Scholar]

[R21] 21.Hernandez-Diaz S, Schisterman EF, Hernan MA. The birth weight ‘paradox’ uncovered? American Journal of Epidemiology. 2006;164:1115–1120. doi: 10.1093/aje/kwj275. [DOI] [PubMed] [Google Scholar]

[R22] 22.Hernandez-Diaz S, Wilcox AJ, Schisterman EF, Hernan MA. From causal diagrams to birth weight-specific curves of infant mortality. European Journal of Epidemiology. 2008;23:163–166. doi: 10.1007/s10654-007-9220-4. [DOI] [PMC free article] [PubMed] [Google Scholar]

[R23] 23.Cole SR, Hernan MA. Fallibility in estimating direct effects. International Journal of Epidemiology. 2002;31:163–165. doi: 10.1093/ije/31.1.163. [DOI] [PubMed] [Google Scholar]

[R24] 24.Hernan MA, Hernandez-Diaz S, Robins JM. A structural approach to selection bias. Epidemiology. 2004;15:615–625. doi: 10.1097/01.ede.0000135174.63482.43. [DOI] [PubMed] [Google Scholar]

[R25] 25.Robins JM, Greenland S. Identifiability and exchangeability for direct and indirect effects. Epidemiology. 1992;3:143–155. doi: 10.1097/00001648-199203000-00013. [DOI] [PubMed] [Google Scholar]

[R26] 26.Wilcox AJ, Russell IT. Perinatal mortality: standardizing for birthweight is biased. American Journal of Epidemiology. 1983;118:857–864. doi: 10.1093/oxfordjournals.aje.a113704. [DOI] [PubMed] [Google Scholar]

[R27] 27.Greenland S, Pearl J, Robins JM. Causal diagrams for epidemiologic research. Epidemiology. 1999;10:37–48. [PubMed] [Google Scholar]

[R28] 28.Hernan MA, Hernandez-Diaz S, Werler MM, Mitchell AA. Causal knowledge as a prerequisite for confounding evaluation: an application to birth defects epidemiology. American Journal of Epidemiology. 2002;155:176–184. doi: 10.1093/aje/155.2.176. [DOI] [PubMed] [Google Scholar]

[R29] 29.Greenland S, Robins JM. Identifiability, exchangeability, and epidemiological confounding. International Journal of Epidemiology. 1986;15:413–419. doi: 10.1093/ije/15.3.413. [DOI] [PubMed] [Google Scholar]

[R30] 30.Kaufman JS, MacLehose RF, Kaufman S. A further critique of the analytic strategy of adjusting for covariates to identify biologic mediation. Epidemiologic Perspectives and Innovations. 2004;1:4. doi: 10.1186/1742-5573-1-4. [DOI] [PMC free article] [PubMed] [Google Scholar]

[R31] 31.Kaufman JS, MacLehose RF, Kaufman S, Greenland S. The mediation proportion. Epidemiology. 2005;16:710. doi: 10.1097/01.ede.0000171282.54664.71. [DOI] [PubMed] [Google Scholar]

[R32] 32.Kaufman S, Kaufman JS, MacLehose RF, Greenland S, Poole C. Improved estimation of controlled direct effects in the presence of unmeasured confounding of intermediate variables. Statistics in Medicine. 2005;24:1683–1702. doi: 10.1002/sim.2057. [DOI] [PubMed] [Google Scholar]

[R33] 33.Petersen ML, Sinisi SE, van der Laan MJ. Estimation of direct causal effects. Epidemiology. 2006;17:276–284. doi: 10.1097/01.ede.0000208475.99429.2d. [DOI] [PubMed] [Google Scholar]

[R34] 34.Rothman KJ, Greenland S, Lash TL. Modern Epidemiology. Lippincott Williams and Wilkins; Philadelphia, PA: 2008. [Google Scholar]

[R35] 35.Kramer MS. Determinants of low birth weight: methodological assessment and meta-analysis. Bulletin of the World Health Organization. 1987;65:663–737. [PMC free article] [PubMed] [Google Scholar]

PERMALINK

Z-scores and the birthweight paradox

Enrique F Schisterman

Brian W Whitcomb

Sunni L Mumford

Robert W Platt