Methodological challenges in studying the causal determinants of child growth

Abstract

Background: Previous studies of early life influences on later growth in childhood have varied in their analytical approaches, particularly with respect to ‘adjustment’ for differences in size at the beginning of the growth period examined.

Methods: We compared three commonly used statistical models to assess the effect of maternal body mass index (BMI) on growth between 6.5 and 11.5 years in a large cohort of Belarusian children, as follows: (Model 1) analysis of the difference in anthropometric measurements between the two ages; (Model 2) analysis of the measurement at 11.5 years after adjustment for the same measurement at 6.5 years; and (Model 3) analysis of the difference in measurements after adjustment for the measurement at 6.5 years (mathematically identical to Model 2).

Results: Among PROBIT children of obese mothers (BMI ≥ 30 kg/m²) vs those of mothers with normal BMI (18.5 to < 25 kg/m²), Model 1 yielded larger increases in most weight and adiposity outcomes than did Model 2. We show that these larger effects arise because Model 2 parameterizes the effect of maternal BMI twice in same model: once for its effect on size at 6.5 years, and a second time for its effect on growth over the 5-year period between 6.5 and 11.5 years. Similar results were obtained in analogous analyses from cohorts in Boston, MA, and Singapore.

Conclusion: Analysing the effect of exposure on change in outcome between two ages (Model 1) is clearly preferable to ‘adjustment’ for the outcome at the earlier age whenever the exposure under study affects the outcome at the earlier age.

Introduction

With the recent focus on the developmental origins of health and disease (DOHaD), studies of growth during fetal life, infancy and childhood have increased in frequency and importance. Early growth has been studied as an exposure causing long-term adult outcomes like high blood pressure, coronary artery disease and diabetes¹; as an outcome caused by maternal and familial influences, including infant feeding, later diet and physical activity^2–9; and as a causal intermediate between early life exposures and later chronic disease.^10–13

By definition, growth is increase in size over time. It can be characterized by its overall trajectory, using either parametric or nonparametric longitudinal models. Other attempts have focused on age period-specific effects. Growth during a specific age period is likely to be predicted by growth at earlier periods, owing to persistent genetic, nutritional or other influences that caused earlier growth, and may even be ‘caused’ by earlier growth (e.g. catch-up or catch-down growth). Investigators of childhood growth have therefore analysed growth in specific age periods by employing a variety of statistical adjustment methods for size attained at earlier ages and even, in some studies, at later ages as well.¹⁴

We have previously reported on problems that arise when adjusting for current body size (height, weight, or BMI) as a causal intermediate between earlier growth and current adiposity.^15,16 In this paper, we use the same study cohort derived from the Promotion of Breastfeeding Intervention Trial (PROBIT) to compare and contrast three commonly used analytical approaches when childhood growth is itself the study outcome of interest, here studied as an effect of maternal BMI. Using the same dataset, we compare the results of the different analytical approaches, discuss the potential biases associated with each, and recommend strategies for future research.

Methods

This is an observational analysis of children who participated in the Promotion of Breastfeeding Intervention Trial (PROBIT)—a cluster-randomized trial of a breastfeeding promotion intervention in the Republic of Belarus. The design of PROBIT,¹⁷ as well as the anthropometric methods and results at ages 6.5 and 11.5 years,^6,18 have been previously published. Briefly, the clusters randomized were maternity hospitals and one affiliated polyclinic (outpatient clinic where children receive routine health care) per hospital. Clusters were randomized to a control intervention (continuation of the breastfeeding practices and policies in effect at the time of randomization) or an experimental intervention based on the Baby-Friendly Hospital Initiative.

The trial recruited 17 046 newborn infants from 31 maternity hospitals and polyclinics. All infants were born in 1996-97 and were healthy, singleton, ≥ 37 weeks gestational age, weighed ≥ 2500 g at birth, had a 5-min Apgar score ≥ 5 and were enrolled during their postpartum stay. The two randomized groups were similar in baseline sociodemographic and clinical variables, including maternal age and occupation, number of other children at home, the proportion of mothers who had breastfed a previous child ≥ 3 months, caesarean delivery, maternal smoking during pregnancy, birthweight, gestational age and 5-min Apgar score.

For simplicity and clarity, the current analysis is limited to comparisons of children born to obese others (BMI ≥ 30 kg/m²) and those born to mothers of normal BMI (≥ 18.5 but < 25 kg/m²). Maternal BMI was based on the paediatrician’s measurement of the mother’s height and weight at the 11.5-year follow-up visit in 9491 mothers, with an additional 2260 based on self-report.¹⁹ Paternal height and weight were based on the mother’s report at the 6.5-year visit.

Follow-up interviews and examinations at 6.5 and 11.5 years of age were performed by one or two paediatricians (depending on volume) at each of the 31 polyclinics. The training and quality assurance procedures at both the 6.5- and 11.5-year follow-up visits have been described in detail previously.^6,15 The children eligible for the present study were the 7582 children born to obese or normal-BMI mothers who attended the follow-up visits at both 6.5 and 11.5 years. To exclude implausible measurements, we eliminated a priori all those < −4 standard deviations (n = 0–15) or > +4 standard deviations (n = 7–133) from the mean at either age; those for whom the change from the 6.5- to the 11.5-year measurement was negative for height (n = 4) or weight (n = 17); those for whom the decrease exceeded 10% of the 6.5-year value for waist (n = 27) or hip (n = 4) circumference; and those for whom the decrease exceeded 20% of the 6.5-year value for triceps (n = 633) or subscapular (n = 389) skinfold thickness.

We considered three statistical analytical approaches to estimate the causal effect of maternal obesity on growth in height, weight and adiposity between the ages of 6.5 and 11.5 years among PROBIT children. For ease of interpretation and comparison, all outcome trajectories are represented by straight lines. As graphically depicted in Figure 1, the simplest method (Model 1) of analysis is to compare the mean change in the anthropometric measure between 6.5 and 11.5 years of age in children in the two maternal BMI groups (a−c) − (b−d).

Figure 1.

Graphical depiction of trajectory (growth) in anthropometric outcomes to ages 6.5 and 11.5 years in children born to mothers with obese vs normal BMI. Growth is assumed to be linear from birth to age 11.5. According to Model 1, the differences in outcome at age 6.5 and 11.5 in those born to obese and normal-BMI mothers is denoted by a − c and b − d, respectively. If growth in the normal group between 6.5 and 11.5 years were the same as that in the obese group (i.e. if the lines were parallel, rather than diverging), the difference in the normal group would be ${\text{b}}^{\prime }-\text{d}$, rather than b − d, where $\text{a}-\text{c }={\text{b}}^{\prime }-\text{d}$.

For Model 1, we can write the following simple linear regression equation:

$${\text{difference}}_{\text{11}.\text{5}-\text{6}.\text{5 y}}={\alpha }_{\text{1}}+{\beta }_{\text{1}}*\text{obesity}+\epsilon$$ (Equation 1)

In other words, the difference in any of the measures between 6.5 and 11.5 years is equal to α₁ (the intercept denoting the mean difference in the normal maternal BMI group, i.e. when obesity = 0), plus the effect (β) of obesity times the value of obesity (0 or 1), plus a random error term (ε) for each child. Referring to Figure 1: α₁ = b − d and ${\beta }_{\text{1}}={\text{b}}^{\prime }-\text{b}$.

A second approach is to analyse the size at age 11.5 years after adjusting for the same measurement at age 6.5 years using linear regression (Model 2). For Model 2, the corresponding equation is:

$${\text{size}}_{\text{11}.\text{5 y}}={\alpha }_{\text{2}}+{\beta }_{\text{2}}*\text{obesity}+{\gamma }_2*{\text{size}}_{\text{6}.\text{5 y}}+\epsilon$$ (Equation 2)

In Equation 2, parameter α₂ does not denote the mean of the normal maternal BMI group, but the mean of the normal group when the size at age 6.5 years is 0. Equation 2 indicates that size (any of the anthropometric measures) at age 11.5 years is a function of both maternal obesity and the same anthropometric measure at age 6.5 years.

A ‘hybrid’ approach (Model 3) analyses the change in the value of the anthropometric measurement between 6.5 and 11.5 years, while also adjusting for the measurement at 6.5 years. Its corresponding equation is:

$${\text{difference}}_{\text{11}.\text{5}-\text{6}.\text{5 y}}={\alpha }_{\text{3}}+{\beta }_3*\text{obesity}+{\gamma }_{\text{3}}*{\text{size}}_{\text{6}.\text{5 y}}+\epsilon$$ (Equation 3)

Equation 3 looks very much like Equation 2. The only difference is in γ₃ vs γ₂, because the size at age 11.5 (Model 2) must add the difference in size between ages 6.5 and 11.5 years to the size at age 6.5 years, as written in Equation 1. We can re-write Equation 3 as follows:

$${\text{size}}_{\text{11}.\text{5 y}}={\alpha }_{\text{3}}+{\beta }_{\text{3}}*\text{obesity}+{\gamma }_{\text{3}}*{\text{size}}_{\text{6}.\text{5 y}}+{\text{size}}_{\text{6}.5\text{y}}+\epsilon$$

Therefore, ${\alpha }_{\text{3}}={\alpha }_{\text{2}},{\beta }_{\text{3}}={\beta }_{\text{2}}$, and ${\gamma }_{\text{3}}={\gamma }_{\text{2}}-\text{ 1}$, or equivalently, ${\gamma }_{\text{2}}={\gamma }_{\text{3}}+\text{ 1}$. Thus Model 3 and Model 2 are mathematically equivalent, and we show only the output for Model 2 in the Results section.

All analyses were based on the MIXED procedure in SAS (Version 9.3; SAS Institute, Cary, NC), which accounts for the clustered measurement of the outcomes, in addition to the exposure and covariates included in the model. For both models, we show the results both with and without adjustment for other covariates, including maternal height and paternal height and BMI, geographical region, urban vs rural residence, maternal education and the child’s exact age at the two follow-up visits. We also used multiple imputation to account for missing values of maternal and (especially) paternal height and BMI, with 20 imputations based on the Markov-chain Monte Carlo technique, using SAS procedures PROC MI to impute the missing values and PROC MIANALYZE to analyse the imputed datasets.

To assess the robustness of our findings based on PROBIT, we carried out similar analyses (including the multiple imputation for parental height and BMI) in two smaller cohorts: the Project Viva cohort from Boston, MA,²⁰ and the GUSTO cohort from Singapore.^21,22 For Viva, we used the mother’s self-reported pre-pregnancy height and weight to calculate BMI, with anthropometric outcomes assessed in early (range 3-5 years) and middle (range 6-10 years) childhood. For GUSTO, maternal BMI was based on the mother’s measured height and weight at 4 years, with anthropometric outcomes assessed at ages 2 and 4 years.

Results

Table 1 compares the two maternal BMI groups studied in PROBIT according to potentially confounding baseline characteristics. The mean BMI was 22.3 kg/m² in the normal BMI group and 34.7 kg/m² in the obese group.

Table 1.

Comparison of baseline characteristics in obese and normal maternal BMI groups

Characteristic	Normal maternal BMI	Obese maternal BMI	P-value
Place of residence (%)			<0.0001
East/urban	33.6	29.3
East/rural	13.2	18.5
West/urban	27.3	20.2
West/rural	25.9	32.0
Maternal education (%)			0.006
Completed university	13.7	11.1
Partial university	51.9	52.8
Completed secondary school	30.9	33.0
Incomplete secondary school	3.6	3.2
Maternal height (cm)	164.4	163.3	<0.0001
Paternal height (cm)	176.4	175.5	<0.0001
Paternal BMI (kg/m2)	25.4	26.3	<0.0001

Table 2 shows the PROBIT results for Model 1, comparing the differences in anthropometric measure increases between 6.5 and 11.5 years in children born to obese mothers vs those born to mothers of normal BMI. For all of the measures, the increase (growth) in the measure over the 5-year interval was significantly larger in children born to obese mothers than in those born to mothers with normal BMI, with similar results for the crude and adjusted models, and in adjusted models with and without multiple imputation.

Table 2.

Comparison of mean (95% CI) differences in anthropometric measurements between 6.5 and 11.5 years of age in children born to obese mothers (n = 2239-2528)^a vs those with normal BMI (n = 4353-5043)^a (Model 1)

Anthropometric measure	Crudeb	Adjustedc	Adjustedc with MI
Height (cm)	+ 0.26 (+ 0.003 to + 0.52)	+ 0.57 (+ 0.33 to + 0.81)	+ 0.58 (+ 0.35 to + 0.81)
Weight (kg)	+ 2.80 (+ 2.47 to + 3.13)	+ 2.84 (+ 2.51 to + 3.17)	+ 2.92 (+ 2.60 to + 3.23)
BMI (kg/m2)	+ 0.94 (+ 0.84 to + 1.04)	+ 0.89 (+ 0.78 to + 1.00)	+ 0.91 (+ 0.81 to + 1.01)
Waist circumference (cm)	+ 2.22 (+ 1.93 to + 2.51)	+ 2.22 (+ 1.92 to + 2.51)	+ 2.21 (+ 1.93 to + 2.50)
Hip circumference (cm)	+ 2.03 (+ 1.75 to + 2.31)	+ 2.00 (+ 1.73 to + 2.28)	+ 2.08 (+ 1.81 to + 2.35)
Triceps SF (mm)	+ 1.71 (+ 1.48 to + 1.95)	+ 1.63 (+ 1.38 to + 1.88)	+ 1.65 (+ 1.41 to + 1.88)
Subscapular SF (mm)	+ 1.53 (+ 1.33 to + 1.73)	+ 1.49 (+ 1.28 to + 1.71)	+ 1.48 (+ 1.28 to + 1.69)
Sum of SF (mm)	+ 3.19 (+ 2.78 to + 3.61)	+ 3.11 (+ 2.67 to + 3.54)	+ 3.11 (+ 2.69 to + 3.52)

MI = multiple imputation; SF, skinfold.

^aRanges shown reflect different numbers of children with missing values; the lower end of the range for sum of SF, the upper end for height.

^bCrude: adjusted for clustering of measurement by polyclinic.

^cAdjusted: also adjusted for covariates in Table 1.

The results for Model 2 (Table 3), comparing the differences between the two groups of children at 11.5 years, after adjustment for the same measure at age 6.5 years, differ from those for Model 1. Except for the triceps skinfold thickness, the differences are smaller than those for Model 1, again with similar findings in the crude and adjusted models, and in adjusted models with and without multiple imputation.

Table 3.

Comparison of mean (95% CI) anthropometric measurements at 11.5 years in children born to obese mothers (n = 2239-2528)^a vs those with normal BMI (n = 4353-5043),^a adjusted for same measurement at 6.5 years (Model 2)

Anthropometric	Crudeb	Adjustedc	Adjustedc with MI
Height (cm)	+ 0.24 (− 0.02 to + 0.49)	+ 0.55 (+ 0.31 to + 0.79)	+ 0.57 (+ 0.34 to + 0.80)
Weight (kg)	+ 2.00 (+ 1.70 to + 2.30)	+ 2.07 (+ 1.77 to + 2.37)	+ 2.10 (+ 1.81 to + 2.39)
BMI (kg/m2)	+ 0.82 (+ 0.72 to + 0.93)	+ 0.80 (+ 0.69 to + 0.91)	+ 0.81 (+ 0.71 to + 0.91)
Waist circumference (cm)	+ 2.07 (+ 1.78 to + 2.36)	+ 2.12 (+ 1.82 to + 2.42)	+ 2.10 (+ 1.81 to + 2.39)
Hip circumference (cm)	+ 1.91 (+ 1.64 to + 2.19)	+ 1.92 (+ 1.64 to + 2.20)	+ 2.00 (+ 1.73 to + 2.27)
Triceps SF (mm)	+ 1.72 (+ 1.48 to + 1.96)	+ 1.65 (+ 1.40 to + 1.90)	+ 1.67 (+ 1.43 to + 1.91)
Subscapular SF (mm)	+ 1.31 (+ 1.11 to + 1.51)	+ 1.30 (+ 1.09 to + 1.50)	+ 1.29 (+ 1.09 to + 1.49)
Sum of SF (mm)	+ 2.86 (+ 2.45 to + 3.27)	+ 2.82 (+ 2.39 to + 3.26)	+ 2.82 (+ 2.41 to + 3.23)

MI, multiple imputation; SF, skinfold.

^aRanges shown reflect different numbers of children with missing values; the lower end of the range for sum of SF, the upper end for height.

^bCrude: adjusted for clustering of measurement by polyclinic.

^cAdjusted: also adjusted for covariates in Table 1.

For the Viva cohort, Model 1 yielded modestly larger adjusted effects of maternal obesity than did Model 2 for change in BMI (1.00 vs 0.96 kg/m²), height (0.73 vs 0.58 cm), weight (2.32 vs 1.71 kg) and waist circumference (3.20 vs 3.15 cm), the identical effect for sum (triceps + subscapular) of skinfolds (3.89 mm), but a smaller effect for hip circumference (1.66 vs 2.05 cm).

The GUSTO cohort results were also similar to those from PROBIT and Viva. Model 1 adjusted effect estimates were larger for height (0.24 vs 0.23 cm), weight (0.27 vs 0.22 kg), BMI (0.15 vs 0.13 kg/m²) and sum (triceps + subscapular) of skinfolds (0.91 vs 0.50 mm), but smaller for waist circumference (0.27 vs 0.48 cm). Hip circumference was not measured in GUSTO.

Discussion

We have compared three commonly used approaches to analysis of growth in anthropometric measures in the PROBIT cohort between two time points (t₁ and t₂) in relation to maternal BMI. One approach is based on the t₂ − t₁ difference alone, another is based on the t₂ measure adjusted for the t₁ measure and the third (‘hybrid’) approach is based on the t₂ − t₁ difference adjusted for the t₁ measure. The second and third approaches yielded identical results, with (generally) smaller effects of maternal BMI on growth between ages 6.5 and 11.5 years than those obtained using the first approach.

We chose to compare these three different analytical approaches because all three are in common use and because we are unaware of any consensus or of previous discussion about which approach is preferable and why. Our analyses indicate that the results and the inferences derived from them differ between the first approach and the other two. The direction and magnitude of the differences we observed in the three cohorts that we analysed were not sufficient to mask or reverse the overall effect of maternal obesity. But for other exposures, and especially those with smaller effects, the differences in results could result in major differences in the causal inferences derived therefrom.

Our goal in the remainder of this paper is to explain the reasons for the observed differences in results between Model 1 and Model 2 (or Model 3), to suggest which approach is preferable for causal inference and to provide arguments for that preference. Referring to Figure 1, it is clear that size at age 6.5 years also depends on maternal obesity; the average difference between the obese and normal maternal BMI groups is c − d. The important point here is that because maternal BMI affects size at age 6.5, the effect of maternal obesity appears twice on the right side of Equation 2 (representing Model 2). The net result is a systematic underestimate of the effect of maternal obesity. Size at age 6.5 can thus be thought of as a mediator of the effect of maternal obesity on size at age 11.5. Model 2 essentially ‘adjusts’ for size at age 6.5 as if it were a confounder, rather than a mediator, resulting in over-adjustment and a systematic underestimate of the effect of maternal obesity on growth between 6.5 and 11.5 years.^15,16,23,24

This ‘double counting’ for the effect of maternal obesity can be seen clearly by re-writing Equation 1 for size_{6.5 y} instead of difference_{11.5-6.5 y}:

$${\text{size}}_{\text{6}.\text{5 y}}={{\alpha }^{\prime }}_1+{{\beta }^{\prime }}_1*\text{obesity}+\epsilon \text{(Equation 1′)}$$

where ${\alpha }_{\text{1}}{}^{\prime }$ is the average size of children in the normal maternal BMI group at 6.5 years (${\alpha }_{\text{1}}{}^{\prime }=\text{ d}$ in Figure 1) and ${\beta }_{\text{1}}{}^{\prime }=\text{ c }-\text{ d}$. Then Equation 2 can be re-written as:

$$\begin{array}{l}{\text{size}}_{\text{11}.\text{5 y}}={\alpha }_{\text{2}}+{\beta }_2*\text{obesity}+{\gamma }_2*{\text{size}}_{\text{6}.\text{5 y}}\\ ={\alpha }_{\text{2}}+{\beta }_2*\text{obesity}+{\gamma }_2*\left({\alpha }_{\text{1}}{}^{\prime }+{\beta }_{\text{1}}\prime *\text{obesity}\right) \\ ={\alpha }_{\text{2}}+{\gamma }_2*{\alpha }_{\text{1}}{}^{\prime }+[{\beta }_{\text{2}}+({\gamma }_2*{\beta }_{\text{1}}{}^{\prime })]*\text{obesity}+\epsilon \end{array}$$

 Because Model 2 tends to underestimate the effect of maternal obesity, it is incorrect, at least from a causal perspective (maternal obesity as a cause of growth between ages 6.5 and 11.5 years). In contrast, Model 1 adequately captures the maternal obesity effect on growth between 6.5 and 11.5 years and, as long as the multivariable model captures all of the relevant confounders (i.e. in the absence of residual confounding), should reflect the causal effect of maternal obesity on growth.

The preference for Model 1 over Model 2 (or Model 3) hinges on previous exposure to a cause (here, maternal obesity) that affects the outcome at t_1. Had the exposure begun at or after t₁, Model 2 might be equivalent or even superior to Model 1, because the γ coefficients would control for possible confounding (or random error) by differences in size at t₁ and thus for regression to the mean (larger children at t₁ would tend to be smaller at t₂ in the absence of exposure). This might even be an advantage for randomized trials, where imbalance in outcome at t₁ at the time of randomization could bias the outcome measured at t₂.²⁵ Model 1 can be shown to yield effect estimates identical to those of the interaction term between maternal obesity and the age difference (t₂ − t₁) in a repeated-measures analysis of variance (ANOVA) analysis (data not shown but available on request).

The over-adjustment bias we have demonstrated has been referred to as Lord’s paradox in some recent publications.^26,27 Lord’s paradox refers not to a bias, however, but to a difference in the causal effect of interest.^28,29 In Lord’s paradox, a comparison of two groups’ estimated changes in outcome between two time points t₁ and t₂ is very different if the analysis of change is unconditional on the outcome at t₁ (our Model 1) or conditional on (i.e. adjusted for) the outcome at t₁ (our Model 3). This ‘paradox’ is a mathematical truism, and neither analysis is incorrect if exposure has no effect on outcome at time 1. The ‘paradox’ is merely one of interpretation, depending on the analyst’s preference for a conditional or unconditional effect; but the conditional analysis is clearly incorrect (biased) whenever the exposure begins prior to and influences the outcome at t₁, rather than beginning at t₁.

The parallel analysis of data from the Viva and the GUSTO cohorts yielded very similar results to those from PROBIT. Estimates from Model 2 were generally smaller than those from Model 1. For several of the measurements presented (triceps skinfold thickness for PROBIT, hip circumference for Viva, waist circumference for GUSTO), however, a reverse pattern was observed. The algebraic arguments presented above provide no explanation for such an occurrence. The anomalous results probably stem from the regression to the mean mentioned above. For those measurements resulting in larger effect estimates from Model 2 than from Model 1, adjustment for regression to the mean appears to have increased those effect estimates to an extent greater than the over-adjustment bias (systematic underestimate). The balance of opposing adjustments for those outcomes was a small net increase in the estimate of the maternal obesity effect.

The net result is an unpredictable combination of the two adjustments: a systematic underestimate (bias) due to over-adjustment and an increase due to adjustment for regression to the mean. In all three of the study cohorts we examined, however, the net result of the combination was a biased underestimate for most of the outcomes. The modest differences in magnitude and direction of the two adjustments observed among the three cohorts for each specific outcome probably reflect differences in timing and type (measured vs self-reported) of the exposure measurement, i.e. differences in measurement error, as well as differences in the ages and populations studied.

Several limitations of our study should be acknowledged. First, maternal BMI does not remain constant over time, but is a time-varying exposure for which we used the measure obtained at a single time only: at the 11.5-year follow-up in PROBIT, pre-pregnancy in Viva and at the 4-year follow-up in GUSTO. Moreover, maternal obesity during pregnancy increases fetal growth³⁰ and can have both genetic and behavioural influences (through diet and physical activity) during infancy and childhood. Nonetheless, BMI in adults is known to track strongly over time. PROBIT mothers who were obese when their child was 11.5 years old were likely to have been obese during pregnancy and during the child’s earlier life. We used the mother’s BMI at the child’s 11.5-year visit because it was based on the paediatrician’s measurement of her height and weight in the majority of cases, rather than on the self-reported heights and weights we obtained at the 6.5-year visit. Maternal height and weight at the 11.5-year follow-up was self-reported in some (< 20%) mothers and was missing in 15.3% of children who were followed up at 11.5 years. The similar results we observed based on multiple imputation, however, provide some reassurance with regard to the latter issue.

We believe that the choice among the analytical approaches we have compared, and the impact of that choice on the resulting findings and inferences, have broad applicability for life-course epidemiology. This field is characterized by longitudinal follow-up (cohort studies) with repeated measures of outcomes that are expected to change with advancing age of the participating cohort members. The optimal analytical approach depends on biological understanding and well-defined hypotheses about the causal relationships between the study exposure (putative cause) and its timing relative to the age or age period at which the outcomes (putative effects) are measured. Moreover, the approaches compared in this paper are limited to analysis of growth between two ages (time points). Analysis of trajectories over longer periods with multiple measurements may require more complex methods, including parametric trajectory models, splines and latent class models.

Our focus here has been on growth (increases in anthropometric measures) in childhood, but the same principles should apply in other areas. For example, Glymour et al. examined declines in cognitive ability among the elderly.³¹ Those authors focused on measurement error and risk of collider stratification bias, rather than the effect of exposure (years of schooling completed) on cognitive ability at the earlier age. They came to the same conclusion as ours: a clear preference for analysis of the difference in outcome between t₁ and t₂ without adjustment for the outcome measured at t₁. Our results suggest that even in the absence of measurement error and collider bias, however, analyses of outcome at t₂ that ‘adjust’ for the same outcome at t₁ will be biased whenever the baseline (t₁) measure of the outcome is affected by the exposure.

Funding

This study was supported by a grant from the Canadian Institutes of Health Research. The 11.5-year follow-up of PROBIT was also supported by grants from the European Union, Early Nutrition Programming Long-term Efficacy and Safety (FOOD-DT-2005-007036) and the USA National Institutes of Health (R01 HD050758). The NIHR Bristol Nutrition Biomedical Research Unit is funded by the National Institute for Health Research (NIHR) and is a partnership between the University Hospitals Bristol NHS Foundation Trust and the University of Bristol.

Conflict of interest: None.

KEY MESSAGES

• Many cohort studies assess the effects of one or more early exposures on the same outcome over successive ages (‘waves’) at follow-up, but analyses of these effects have been highly variable.

• We compared three statistical models commonly used in such studies: (Model 1) analysis of the difference in outcomes from time 1 to time 2; (Model 2) analysis of the time 2 outcome adjusted for the time 1 outcome; and (Model 3) analysis of the difference adjusted for the time 1 outcome.

• We show that Models 2 and 3 are equivalent but that both systematically underestimate the effect of exposure, owing to its effect on the time 1 outcome.

• We also explain how Model 2 (or Model 3) can yield a higher estimate than Model 1 when the impact of adjustment for regression to the mean from time 1 to time 2 exceeds the bias due to over-adjustment.