Dietary intake is a key variable in many epidemiological analyses. Nonetheless, most studies that have assessed dietary intake in childhood or adolescence have analyzed intake at just one age,^{1} ,^{2} within a narrow age range, or have used only a small number of repeat measurements,^{3} ,^{4} usually assuming that change in intake between ages is linear.^{5–} ^{9} These assumptions are likely to be incorrect, given the marked changes with age in characteristics that may plausibly influence dietary intake. Moreover, these approaches do not allow for the unbiased assessment of the determinants of intake and changes in intakes across different ages, or of whether different rates of change in intake over childhood and adolescence are associated with later outcomes. Linear-spline multilevel models (otherwise known as change-point, segmented, or piecewise linear multilevel models) are increasingly used to model childhood growth in life course epidemiology as they deal with many of the challenges associated with analyzing longitudinal data.^{10–} ^{13} We demonstrate how this approach can be applied to dietary data, using energy intake as an example, and how it can be used to combine data collected by 3-day food diaries at some ages and food frequency questionnaires (FFQs) at other ages, to produce individual dietary intake trajectories. We provide examples of how these trajectories can be used to assess: (1) determinants of dietary intake across childhood, using the association of maternal prepregnancy body mass index (BMI) with offspring energy intake from ages 3–13 years as an example, (2) associations of trajectories with later health outcomes, using the association of energy intake trajectories with BMI at 15 years as an example, and (3) the extent to which the trajectories might mediate associations between earlier exposures and later outcomes, with our example being the role of offspring energy intake trajectories in mediating the association between maternal prepregnancy BMI and offspring BMI at age 15. When the trajectories are an exposure for a later outcome (eg, 2 and 3 above), analyses can be performed either as a two-step process, in which individual-level intercepts and slopes are exported and used in standard regression models, or a single-step process using multivariate multilevel models. We discuss the advantages and disadvantages of each approach and compare the results from the two methods.

METHODS
Study Population
The Avon Longitudinal Study of Parents and Children (ALSPAC) is a prospective birth cohort from southwest England (full details in eAppendix; http://links.lww.com/EDE/A686 ).^{14} ,^{15} Our main study sample consists of 12,032 participants with at least one 3-day food diary or FFQ between ages 3 and 13 years. A subgroup of these (n = 4,197), who additionally had complete data for height and weight from the 15–16 year assessment, maternal prepregnancy BMI and all potential confounders, were included in the analyses to illustrate the use of trajectories as exposures, outcomes, and mediators. Ethical approval for the study was obtained from the ALSPAC Ethics and Law Committee and Local Research Ethics Committees.

Dietary Assessment
Three-day food diaries were mailed to parents when study children were age 3.5, 5, 7, 10, and 13 years. Parents recorded their child’s diet before the child reached age 10 years; from age 10 onward, children recorded their own diet with help from their parents. Diaries were requested for one weekend day and two weekdays. Similarly, FFQs were mailed to parents when the study children were age 3, 4, 7, and 9 years. Full details of FFQs and food diaries, and a detailed account of how nutrient and energy contents were calculated have been published previously,^{16} and are provided in the eAppendix (http://links.lww.com/EDE/A686 ).

Assessment of Maternal BMI, Offspring BMI, and Other Covariables
Mothers reported their age, parity, height, prepregnancy weight, and educational attainment in a questionnaire administered at enrolment. The highest parental occupation was used to allocate participants to family social class groups, using the 1991 British Office of Population and Census Statistics classification.^{17} The 15-year follow-up assessment at mean age 15.4 years was attended by 5515 children: weight was measured to the nearest 0.1 kg using Seca scales and height to the nearest 0.1 cm using a Harpenden stadiometer. Maternal and offspring BMIs were calculated as weight in kilograms divided by height in meters squared.

STATISTICAL ANALYSIS
Modeling Energy Intake Trajectories
Energy intake trajectories were estimated using linear-spline multilevel models with level-1 (measurement occasion) and level-2 (individual) random effects, allowing individuals to have different intercepts and slopes, and thus, their own trajectories. All multilevel models were fitted in MLwiN v.2.24^{18} using the Stata (StataCorp, College Station, TX) command “runmlwin.”^{19} All available energy intake measurements for a given child were included in the analyses under a missing-at-random (MAR) assumption. We decided a priori to place knot points at the target ages of follow-ups, to ensure availability of sufficient data at and around the knot points. Models with two knot points did not produce an appreciably better fit than models with a single knot point. The best model fit with a single knot point, judged by maximum-likelihood and differences between observed and predicted measurements, was obtained with the knot point at age 7 years, thus, estimating the intercept (energy intake at 3 years) and two different slopes (linear change in energy intake from 3 to 7 years and linear change in energy intake, with a different slope, from 7 to 13 years). Full details of knot point selection are in the eAppendix (http://links.lww.com/EDE/A686 ). Three occasion-level covariables were included in all multilevel models to account for variable measurement error: (1) an indicator for whether energy intake was recorded by food diary (reference category) or FFQ, (2) an indicator for whether the person reporting was a parent (reference category) or study child, and (3) a variable indicating how plausible a reported energy intake is. Plausibility of reported energy intake is the ratio of sex-, age-, and weight-specific expected energy requirements to reported energy intake, categorized as over-reporting, plausible reporting (reference category), or under-reporting;^{20} full details are in the eAppendix (http://links.lww.com/EDE/A686 ). We decided a priori to include these variables as fixed effects; measurement source (ie, food diary or FFQ) was also included as a level-1 random effect.

Analyzing boys and girls separately produced similar results to the combined model, albeit with different mean trajectories. Thus, we present results from the combined model with adjustment for sex. Three coefficients describe mean predicted intake for girls at age 3 years and average linear changes in intake from 3 to 7 and 7 to 13 years. A further three coefficients describe how the mean trajectory differed in boys. Individual-level residuals describe how each child’s intake at 3 years, and change in intake from 3 to 7 and 7 to 13 years differs from the average, and are used to create individual intercepts and slopes. All residuals were approximately normally distributed (eFigures 1–6; http://links.lww.com/EDE/A686 ). Energy intake trajectories were estimated using the following modeling equation: Kcal_{ij} = (β_{0} + u_{0j} + e_{0ij} ) + (β_{1} + u_{1j} )(S1_{ij} ) + (β_{2} + u_{2j} )(S2_{ij} ) + β_{3} (male_{j} ) + β_{4} (male_{j} * S1_{ij} ) + β_{5} (male_{j} * S2_{ij} ) + (β_{6} + e_{0ij} ) (source_{ij} ) + β_{7} (reporting individual_{ij} ) + β_{8} (under-reporter_{ij} ) + β_{9} (over-reporter_{ij} ) + e_{ij} (agewk_{ij} ) + e_{ij} (source_{ij} ) where, for individual _{j} at measurement occasion _{i} , β_{0} is the average predicted daily energy intake at 3 years in girls measured by a food diary; β_{1} is the average predicted linear change in daily energy intake per year in the first period (S1 = 3–7 years) in girls measured by a food diary; β_{2} is the average predicted linear change in daily energy intake per year in the second period (S2 = 7–13 years) in girls measured by a food diary; for energy intake at 3 years and changes in energy intake between 3–7 and 7–13 years, β_{3–5} are the differences between boys and girls; β_{6} is the average difference between FFQs and food diaries (coded 0 = food diary, 1 = FFQ); β_{7} is the average difference between child- and parent-reported (coded 0 = parent completion, 1 = child completion) energy intake; β_{8} is the average difference between under-reporters and plausible reporters (coded 0 = not under-reporter, 1 = under-reporter); β_{9} is the average difference between over-reporters and plausible reporters (coded 0 = not over-reporter, 1 = over-reporter); u represents individual-level random effects and e occasion-level random effects. “Agewk” (age in weeks) and "source" (measurement source) were fitted as level-1 random effects at the occasion level (1) to allow the variance of energy intake to change linearly with age and measurement source.

Energy Intake Trajectories as an Exposure, Outcome, or Mediator
Associations Between Maternal Prepregnancy BMI and Energy Intake Trajectories
To examine the association between prepregnancy BMI and various trajectories of offspring energy intake, interaction terms were fitted between maternal BMI and the intercept and two linear slopes in the multilevel models. Model 1 adjusts only for sex, measurement source, reporting individual, and plausibility of reporting. Model 2 additionally adjusts for potential confounding by maternal age, maternal education, social class, and parity. Coefficients from these models should be interpreted as the average difference in kilocalories at age 3 years, and the average differences in the rates of change in kilocalories per year from 3 to 7 and from 7 to 13 years, per unit increase in maternal BMI (fully adjusted modeling equation in eAppendix; http://links.lww.com/EDE/A686 ).

Associations Between Energy Intake Trajectories and Offspring BMI at 15 Years
Associations of energy intake from 3 to 13 years, with BMI at 15 years, were assessed in two ways. One was a two-step process in which individual intercepts and slopes were exported and related to BMI using linear regression in Stata version 11.^{21} The other was a single-step process using a bivariate multilevel model. In the single-step process, two equations are estimated simultaneously. The first dependent variable (equation 1) is energy intake, described by the linear-spline multilevel model described above. The second dependent variable (equation 2) is offspring BMI at age 15, described by a single individual-level random effect (bivariate equation in eAppendix; http://links.lww.com/EDE/A686 ). Regression coefficients are calculated from the covariance matrix of the individual-level random effects, using the Stata command reffadjust.^{22} ,^{23} Model 1 adjusts for sex, reporting individual, measurement source, and plausibility of reporting. Model 2 additionally adjusts for age at offspring BMI assessment, maternal prepregnancy BMI, maternal age, social class, maternal education, and parity. Model 3 additionally adjusts for all previous energy intakes. To compare results from the two-step process and bivariate model with a simpler statistical approach using the raw data, we assessed unadjusted associations of reported energy intake at 3 years, and change in reported intake from 3 to 7 and 7 to 13 years (restricting to food diary data) with later BMI, using linear regressions. To make a fair comparison between multilevel models and analysis of the raw data, we used an unadjusted multilevel model (excluding covariables outlined above), using only data from food diaries.

Mediation of Maternal BMI–Offspring BMI Associations by Offspring Energy Intake Trajectories
The role of offspring energy intake from 3 to 13 years in mediating the association between maternal prepregnancy BMI and offspring BMI at age 15 was assessed in both single- and two-step processes. In the two-step process, we conducted a path analysis in Mplus.^{24} ,^{25} Maternal prepregnancy BMI was the main exposure, offspring BMI at 15 years was the main outcome, and individual-level intercepts and slopes from the energy intake trajectories were included as potential mediators (Figure 1 ). Full details of how path analysis parameters are calculated and interpreted are in the eAppendix (http://links.lww.com/EDE/A686 ). The single-step process used a trivariate multilevel model. The dependent variables were maternal prepregnancy BMI in equation 1, described by a single individual-level random effect; energy intake in equation 2, described by the same linear-spline multilevel model, described previously; and offspring BMI in equation 3, described by a single individual-level random effect (trivariate equation provided in eAppendix; http://links.lww.com/EDE/A686 ). Regression coefficients are calculated from the covariance matrix of the individual-level random effects, using the Stata command “reffadjust.” For both the single- and two-step process, model 1 adjusts for sex, reporting individual, measurement source, and plausibility of reporting, and model 2 additionally adjusts for age at BMI assessment, maternal age, parity, maternal education, and social class.

FIGURE 1: A fully saturated path analysis model to assess the role of offspring energy intake from 3 to 13 years in mediating the association between maternal prepregnancy BMI and offspring BMI at mean age 15 years. Our path analysis model comprised four regression equations, estimated simultaneously using linear regressions: (1) offspring BMI was regressed on maternal prepregnancy BMI, energy intake at 3 years, change in energy intake from 3 to 7 years, and change in energy intake from 7 to 13 years; (2) change in energy intake from 7 to 13 years was regressed on maternal prepregnancy BMI, energy intake at 3 years, and change in energy intake from 3 to 7 years; (3) change in energy intake from 3 to 7 years was regressed on maternal prepregnancy BMI, energy intake at 3 years; and (4) energy intake at 3 years was regressed on maternal prepregnancy BMI. Arrow 1 represents the “direct effect” of maternal BMI on offspring BMI (not mediated by offspring energy intake from 3 to 13 years, but possibly, confounded and mediated by other factors not included in the model). All arrows except arrow 1 represent “specific indirect effects.” The total indirect effect estimate is the sum of all specific indirect effects. Arrows 2 and 4 (Path A) represent mediation by energy intake at 3 years only, and this path accounted for 46% of the total indirect effect (path model coefficient for this specific indirect effect 0.027; 95% CI = 0.011–0.043). Arrows 2, 3, and 5 (Path B) represent mediation by energy intake at 3 years and change in energy intake from 3 to 7 years, and this path accounted for 52% of the total indirect effect (path model coefficient for this specific indirect effect 0.032; 95% CI = 0.017–0.048). Paths A and B contributed the most to the total indirect effect in our model; all other pathways made minimal contribution.

Sensitivity Analyses
To assess potential bias caused by missing data, trajectories were estimated for participants with (1) at least two measures of reported energy intake from 3 to 13 years (n = 10,872), and (2) at least one measure of reported energy intake at 3 years and between 3–7 and 7–13 years (n = 7,954). To assess the implications of combining FFQ and food diary data in one multilevel model, we modeled energy intake trajectories for each measurement source separately and assessed the correlation of individual-level intercepts and slopes from the two separate models, using Pearson R correlation coefficients. We also repeated analyses for associations between energy intake from 3 to 13 years with offspring BMI, using the two-step process, separately for FFQ data (n = 11,086) and food diary data (n = 9,204). Models were the same as those in the main two-step process and bivariate multilevel model, except the model restricted to FFQ data made no adjustment for reporting individual as all FFQs were parent-completed.

RESULTS
Energy Intake Trajectories From Ages 3–13 Years
Dietary intake was assessed on a median of five occasions per child (interquartile range 3–7 occasions) between ages 3–13 years. Overall, energy intake increased from age 3–13 years, but the rate of increase was greater between ages 3–7 than 7–13 years (Table 1 , Figure 2 ). Individual variation in energy trajectories was smaller between 3–7 years compared with 7–13 years (eFigure A; http://links.lww.com/EDE/A686 ). Model fit was reasonable, given the measurement error inherent in dietary intake assessment; mean differences between observed and predicted energy intake were less than 4 kcal, with 95% of these differences lying between −271 kcal and +420 kcal (Table 1 ). The multilevel model predicted that compared with food diaries, FFQs overestimate energy intake by an average of 74 kcal (eTable A; http://links.lww.com/EDE/A686 ). At age 7 years, 5887 participants reported energy intake by both a food diary and an FFQ; reported energy intakes were similar by food diaries and FFQs at this age, although differences were considerable for some individuals (mean difference between FFQs and food diaries = 118 kcal, standard deviation = 473 kcal).

TABLE 1: A Summary of Measurements Included and Model Fit for Energy Intake Trajectories (n = 12,032)

FIGURE 2: Average energy intake trajectories in boys (n = 6,184) and girls (n = 5,848) from 3 to 13 years of age.

Energy Intake Trajectories as an Exposure, Outcome, and a Mediator
Predicted average energy intake at 3 years and change in intake from 3 to 7 years and 7–13 years were similar in the main analysis set (N = 12, 032) and the subgroup with data on maternal BMI, offspring BMI at 15 years, and all potential confounders (N = 4,197; eTable B, http://links.lww.com/EDE/A686 ).

Associations Between Maternal BMI and Energy Intake Trajectories
A greater maternal prepregnancy BMI was associated with greater offspring energy intake at age 3 years, a greater rate of increase in offspring energy intake between 3–7 years, and a slower rate of increase in energy intake between 7–13 years, after adjusting for maternal age, maternal education, parity, and social class (Table 2 ).

TABLE 2: Associations of Maternal Prepregnancy BMI with Energy Intake at 3 Years and Linear Changes in Energy Intake from Ages 3 to 7 Years and 7 to 13 Years (n = 4,197)

Associations Between Energy Intake Trajectories and Offspring BMI at 15 Years
In the two-step process, energy intake at 3 years and change in intake from 3 to 7 years were positively associated with BMI at age 15 in the unadjusted model. These associations remained after adjustment for age at offspring BMI assessment, maternal age, maternal education, parity, social class, and maternal prepregnancy BMI, and also after adjustment for previous energy intake. Energy intake from 7 to 13 years was inversely associated with offspring BMI in the unadjusted and confounder adjusted models; however, the association attenuated to the null after adjustment for previous energy intake (Table 3 ). Results were similar when assessed with a bivariate multilevel model, except that the standard errors for coefficients were slightly larger (Table 4 ). Using the simple statistical approach with raw observed food diary data, associations followed a similar pattern, but coefficients were substantially smaller compared with the coefficients from the main (fully adjusted) two-step process and bivariate model above (eTable C; http://links.lww.com/EDE/A686 ). However, the coefficients were similar in size to those from the unadjusted multilevel model of food diary data, which is a more appropriate comparison (eTable D; http://links.lww.com/EDE/A686 ).

TABLE 3: Associations of Energy Intake at 3 Years, and Linear Changes in Energy Intake from 3 to 7 Years and 7 to 13 Years with Body Mass Index at Age 15 Years Using the Two-Step Process (n = 4,197)

TABLE 4: Associations of Energy Intake at 3 Years, and Linear Changes in Energy Intake from 3 to 7 Years and 7 to 13 Years with Body Mass Index at Age 15 Years in Males and Females Using a Bivariate Multilevel Model (n = 4,197)

Mediation of Maternal BMI–Offspring BMI Associations by Offspring Energy Intake
In the two-step process, energy intake from 3 to 13 years accounted for 18% (95% confidence interval [CI] = 17–19%) of the association between maternal prepregnancy BMI and offspring BMI at 15 years, after adjustment for maternal education, parity, maternal age, and social class (Table 5 ). Results were similar when assessed with a trivariate multilevel model; the energy intake trajectories accounted for 25% of the total association between maternal BMI and offspring BMI at age 15, after adjustment for potential confounders (eTable E; http://links.lww.com/EDE/A686 ).

TABLE 5: Mediation of the Association Between Maternal BMI and Offspring BMI at Age 15 Years by Energy Intake at 3 Years and Linear Changes in Energy Intake from 3 to 7 and 7 to 13 Years, Using a Two-Step Path Analysis (n = 4,197)

Sensitivity Analysis
Sensitivity analyses confirmed that missing data did not bias the estimation of energy intake trajectories. Mean trajectories were similar when analyses were restricted to participants with (1) at least two measures of energy intake, and (2) energy intake at 3 years, and at least one measure of energy intake between 3 and 7 years, and between 7 and 13 years (eTables F and G; http://links.lww.com/EDE/A686 ). Mean energy intake trajectories were also similar when multilevel models were run separately for FFQs and food diaries (eTables H and I; http://links.lww.com/EDE/A686 ), although there was greater between-individual variation in the slopes of the trajectories (eFigures B and C; http://links.lww.com/EDE/A686 ) in the model restricted to FFQ data. Individual-level intercepts and the slope from 3 to 7 years from the model restricted to food diary data were moderately correlated with those from the model restricted to FFQ data (Pearson R = 0.36 for the intercepts and 0.34 for the first linear slopes from 3 to 7 years). Linear slopes from 7 to 13 years from the two separated models were weakly correlated (Pearson R = 0.04). Associations of energy intake from 3 to 13 years with offspring BMI at mean age 15 years were similar, when multilevel models used only FFQ data or only food diary data (eTables J and K; http://links.lww.com/EDE/A686 ), although coefficients were smaller for the model restricted to FFQ data. After adjustment for potential confounders and previous energy intake, the association between change in intake from 7 to 13 years and BMI was positive in the FFQ-only model (compared with null in the combined model and in the model restricted to food diary data).

DISCUSSION
We have shown how trajectories of dietary intake can be modeled using linear-spline multilevel models, and how data from food diaries and FFQs can successfully be combined into one model. We also demonstrated how trajectories can be used as exposures, outcomes and mediators in two-step processes and in multivariate multilevel models. This method of modeling change in dietary intake has important potential for accurately understanding the contribution of dietary intake to life course epidemiology. We chose to examine intake through childhood and adolescence because this is a key period for change in intake and for change in exposures that might influence intake. However, the methods could equally be applied to other stages of the life course. Furthermore, these methods can be extended to absolute and energy-adjusted micro-and macronutrients.

We are unaware of any other published studies that have used multilevel models to assess individual energy or macronutrient intake trajectories through childhood and adolescence. Other studies have assessed longitudinal changes in diet using generalized estimating equations.^{2} ,^{26} ,^{27} Although these models allow for aggregation of data and can account for repeat measures, they produce population averages rather than predicting individual trajectories, and hence cannot be used to explore associations with later outcomes at an individual level. Some studies have used multilevel modeling to assess associations of dietary intake through childhood and adolescence with later health outcomes.^{6–8} ,^{28} However, most models assume a linear rate of change in dietary intake over the measurement period. Our findings suggest that this may be an incorrect assumption, and that in a healthy population the rate of increase in energy intake is greater in earlier childhood (from 3 to 7 years) than in later childhood (7–13 years). Non-linearities may also be present at other ages.

Strengths
Multilevel modeling is useful for analyzing large numbers of repeat measures as it reduces the dimensionality of the data. We reduced up to nine measures per person, to just three summary measures. Because frequent repeat measurements are also likely to be collinear, summarizing trajectories with fewer coefficients reduces the chance of collinearity. Moreover, when there is a wide range of measurement times (eg, for an assessment planned for age 5 years, actual age varies from 4 to 6 years), these methods help decrease the associated measurement error. Trajectories can be modeled for each person with at least one diet measure under a MAR assumption, reducing potential for bias owing to missing data. The MAR assumption is likely to hold in our example because results were similar when analyses were repeated for children with two or more measures or at least one measure per linear-spline period.

Limitations
Linear-spline multilevel models are only an approximation of nonlinear relations. However, in our study, these models provided a reasonable description of the data. Furthermore, linear changes are easier to interpret than polynomial terms. Our adjustment for plausibility of reporting intake is subject to error: it is an approximation based on the ratio of estimated energy requirements (with an allowance for moderate physical activity levels) to reported energy intake. Although imperfect, some adjustment for the nondifferential measurement error in reporting is needed, and this method is widely used.^{29–33}

Combining FFQ and Food Diary Data
Similar to other studies, we found that FFQs measure energy intake with greater error than food diaries,^{34} ,^{35} but are considerably cheaper to administer. Thus, it would be advantageous if cohort studies could collect dietary intake data via a mixture of these. The FFQs had no portion size information; hence, the intakes derived from these may be inaccurate.^{16} The two sources of data are also conceptually different; FFQs capture usual intake, whereas, diaries represent a snapshot of foods eaten on particular days. However, both methods attempt to assess average energy intake, which is the quantity of interest in our analyses. On average, differences in reported energy intake at age 7 years (when both measures are available) were not large. Combining the two sources in a multilevel model has not only a financial advantage, but also the advantage of being able to account for the differential measurement error across the two data sources. We showed that multilevel modeling can be used to efficiently and validly combine data from FFQs and food diaries. On average, FFQs overestimated energy intake compared with food diaries in our study. However, mean trajectories were similar when they were modeled separately by measurement source. Individual-level intercepts and linear slopes from 3 to 7 years from the model restricted to food diary data were moderately correlated with those from the model restricted to FFQ data. However, linear slopes from 7 to 13 years from the two separated models were weakly correlated, most likely because there is only one FFQ after 7 years (age 9), compared with two food diaries (ages 10 and 13 years). Associations of energy intake from 3 to 13 years with later offspring BMI were similar, when the multilevel models were separated by measurement source, except that coefficients for the model restricted to FFQ data were smaller. These differences likely reflect regression dilution bias, because FFQs measure dietary intake with more error than food diaries.^{14} This would support the use of a combination of diet diaries and FFQs in main analyses in studies where both are available, as we have done here. The observed difference also highlights the potential for bias in studies that have access only to FFQ data.

One- Versus Two-Step Modeling
When dietary intake trajectories are an exposure for a later outcome, the analysis can be done as a two-step process, in which individual-level intercepts and slopes are exported and used in standard regression models, or as a single-step process, using multivariate multilevel models. These approaches have different advantages and disadvantages. In the two-step process, we assume the residuals are known, whereas, they are in fact estimated by the multilevel model. Therefore, standard errors could be underestimated. In the multivariate multilevel models, energy intake trajectories and their roles as exposures/mediators are estimated simultaneously, thus, the uncertainty in estimating individual-level intercepts and slopes is incorporated in their associations with outcomes. In our examples, results were similar across these methods, but CIs were slightly wider when a bivariate multilevel model was used. Therefore, any bias from assuming known residuals in the two-step process is fairly small in our examples. The potential bias of the two-step approach is balanced by the advantage that the individual intercepts and slopes can be generated by one analyst with knowledge of multilevel modeling software, and then, used by other analysts in standard regression models, which do not require expertise in multilevel modeling. A further disadvantage of the trivariate model, in assessing the role of energy intake trajectories in mediating the association between maternal and offspring BMI, is that the indirect effect is calculated by subtracting the direct effect from the total effect. Thus, no confidence interval for the indirect effect is calculated by the software (MLwiN), in contrast to the two-step path analysis using Mplus. Other techniques have been proposed to explore mediation analyses,^{36} ,^{37} which have advantages including the ability to assess exposure-mediator interactions. These methods, however, are not yet extendable to the multivariate multilevel framework.

Associations of Maternal BMI and Offspring BMI with Offspring Energy Intake Trajectories, and the Role of Energy Intake Trajectories in Mediating the Relationship Between Maternal and Offspring BMI
Greater energy intake at 3 years and a greater rate of increase in energy intake from 3 to 7 years was associated with higher BMI at age 15, whereas, a faster rate of increase in energy intake from 7 to 13 years was associated with lower BMI at age 15. Maternal BMI was also more strongly associated with offspring energy intake below age 7 years, and only weakly or inversely associated with changes in energy intake between 7 and 13 years. Measurement error is likely to partially explain these counter-intuitive findings. Measurement error in dietary intake increases strongly with age, perhaps, partly because children begin completing food diaries themselves and eating away from home more often. This is illustrated by the fact that the proportion of the cohort classified as “under-reporters” increases from 15% at age 3 years to 63% at age 13 years. We attempted to account for changing measurement error in our modeling structure, but it may still partially account for the unexpected direction of associations from 7 to 13 years. Furthermore, reporting of energy intake is often differential by weight status,^{38} and this is likely to worsen with age. Alternatively, overweight and obese children may stabilize their energy consumption at a high level at an early age, and not undergo the growth-related increases in energy consumption necessary for normal or underweight children. This is supported by the fact that individual-level slopes for energy intake from 3 to 7 years were negatively correlated with slopes from 7 to 13 years (Pearson R correlation coefficient = −0.45). With the simpler statistical approach using the raw data, patterns of associations were similar to the two-step process and bivariate model, but coefficients were substantially smaller. This is likely to reflect the different covariables included in the models because the results from the analysis of the raw data were similar to those generated from the two-step process, using an unadjusted multilevel model. However, regression dilution bias is also likely to be important because parameter estimates in the standard regression model using raw data are likely to be biased by the measurement error in the dietary data. Although analyses using the raw data have the advantage of being simpler to perform in standard statistical software, they require all participants to have all measurements, assume that measurements were taken at the same ages for all participants, and cannot account for variation in measurement error across various data sources, ages, etc; these concerns are addressed by multilevel models.

Linear-spline multilevel models are useful for summarizing trajectories of dietary intake. The trajectories can be used as exposures, outcomes, and mediators to assess potential determinants and consequences of variation in dietary intake throughout childhood and adolescence. Changes in dietary intake throughout these periods may be importantly related to later health, thus, it is crucial that dietary intake is modeled correctly. Analysing FFQs and food diaries separately produced similar results to a combined model, so we conclude that they can reasonably be combined in a multilevel model with allowance for the differential measurement error between them. We obtained similar results from a two-step approach using individual-level intercepts and slopes exported from the multilevel model in standard regression models, and single-step multivariate multilevel models.

ACKNOWLEDGMENTS
We thank the families who took part in this study, the midwives for their help in recruiting them, and the whole ALSPAC team.

REFERENCES
1. Shultis WA, Leary SD, Ness AR, Bain CJ, Emmett PMALSPAC Study Team. . Does birth weight predict childhood diet in the Avon longitudinal study of parents and children? J Epidemiol Community Health. 2005;59:955–960

2. Magarey AM, Daniels LA, Boulton TJ, Cockington RA. Does fat intake predict adiposity in healthy children and adolescents aged 2–15 y? A longitudinal analysis. Eur J Clin Nutr. 2001;55:471–481

3. Mikkilä V, Räsänen L, Raitakari OT, Pietinen P, Viikari J. Longitudinal changes in diet from childhood into adulthood with respect to risk of cardiovascular diseases: The Cardiovascular Risk in Young Finns Study. Eur J Clin Nutr. 2004;58:1038–1045

4. Gillman MW, Rifas-Shiman SL, Frazier AL, et al. Family dinner and diet quality among older children and adolescents. Arch Fam Med. 2000;9:235–240

5. Grillenberger M, Neumann CG, Murphy SP, et al. Intake of micronutrients high in animal-source foods is associated with better growth in rural Kenyan school children. Br J Nutr. 2006;95:379–390

6. Phillips SM, Bandini LG, Cyr H, Colclough-Douglas S, Naumova E, Must A. Dairy food consumption and body weight and fatness studied longitudinally over the adolescent period. Int J Obes Relat Metab Disord. 2003;27:1106–1113

7. Phillips SM, Bandini LG, Naumova EN, et al. Energy-dense snack food intake in adolescence: longitudinal relationship to weight and fatness. Obes Res. 2004;12:461–472

8. Burns JS, Dockery DW, Neas LM, et al. Low dietary nutrient intakes and respiratory health in adolescents. Chest. 2007;132:238–245

9. Berkey CS, Gardner JD, Frazier AL, Colditz GA. Relation of childhood diet and body size to menarche and adolescent growth in girls. Am J Epidemiol. 2000;152:446–452

10. Howe LD, Tilling K, Galobardes B, Smith GD, Gunnell D, Lawlor DA. Socioeconomic differences in childhood growth trajectories: at what age do height inequalities emerge? J Epidemiol Community Health. 2012;66:143–148

11. Tilling K, Davies NM, Nicoli E, et al. Associations of growth trajectories in infancy and early childhood with later childhood outcomes. Am J Clin Nutr. 2011;94(suppl 6):1808S–1813S

12. Howe LD, Tilling K, Benfield L, et al. Changes in ponderal index and body mass index across childhood and their associations with fat mass and cardiovascular risk factors at age 15. PLoS One. 2010;5:e15186

13. Howe LD, Matijasevich A, Tilling K, et al. Maternal smoking during pregnancy and offspring trajectories of height and adiposity: comparing maternal and paternal associations. Int J Epidemiol. 2012;41:722–732

14. Boyd A, Golding J, Macleod J, et al. Cohort Profile: The ‘Children of the 90s’ - the index offspring of the Avon Longitudinal Study of Parents and Children. Int J Epidemiol. 2013;42:111–127

15. Fraser A, Macdonald-Wallis C, Tilling K, et al. Cohort Profile: The Avon Longitudinal Study of Parents and Children: ALSPAC mothers cohort. Int J Epidemiol. 2013;42:97–110

16. Emmett P. Dietary assessment in the Avon Longitudinal Study of Parents and Children. Eur J Clin Nutr. 2009;63(suppl 1):S38–S44

17. OPCS. Standard Occupational Classification Volume 3. 1991 London HMSO

18. Rasbash J, Charlton C, Browne WJ, Healy M, Cameron B MLwiN Version 2.02. 2005 Bristol, United Kingdom Centre for Multilevel Modelling, University of Bristol

19. Leckie G, Charlton C runmlwin: Stata Module for Fitting Multilevel Models in the MLwiN Software Package. 2011 Bristol, United Kingdom Centre for Multilevel Modelling, University of Bristol

20. Torun B. Energy requirements of children and adolescents. Public Health Nutr. 2005;8:968–993

21. StataCorp. Stata Statistical Software: Release 11. 2009 College Station, TX StataCorp LP

22. Macdonald-Wallis C, Lawlor DA, Palmer T, Tilling K. Multivariate multilevel spline models for parallel growth processes: application to weight and mean arterial pressure in pregnancy. Stat Med. 2012;31:3147–3164

23. Palmer T, Macdonald-Wallis CREFFADJUST: Stata module to estimate adjusted regression coefficients for the association between two random effects variables. Available at:

http://ideas.repec.org/c/boc/bocode/s457403.html . Accessed 26 July 2012

24. Gamborg M, Andersen PK, Baker JL, et al. Life course path analysis of birth weight, childhood growth, and adult systolic blood pressure. Am J Epidemiol. 2009;169:1167–1178

25. Muthén LK, Muthén BO Mplus User’s Guide. 19986th ed Los Angeles, CA Muthén & Muthén

26. Kronsberg SS, Obarzanek E, Affenito SG, et al. Macronutrient intake of black and white adolescent girls over 10 years: the NHLBI Growth and Health Study. J Am Diet Assoc. 2003;103:852–860

27. Post GB, Vente Wd, Kemper HCG, Twisk JWR. Longitudinal trends in and tracking of energy and nutrient intake over 20 years in a Dutch cohort of men and women between 13 and 33 years of age: The Amsterdam growth and health longitudinal study. Br J Nutr. 2001;85:375–85

28. Grillenberger M, Neumann CG, Murphy SP, et al. Intake of micronutrients high in animal-source foods is associated with better growth in rural Kenyan school children. Br J Nutr. 2006;95:379–390

29. Johnson L, Mander AP, Jones LR, Emmett PM, Jebb SA. A prospective analysis of dietary energy density at age 5 and 7 years and fatness at 9 years among UK children. Int J Obes (Lond). 2008;32:586–593

30. Johnson L, van Jaarsveld CHM, Emmett PM, et al. Dietary energy density affects fat mass in early adolescence and is not modified by FTO variants. PLoS ONE. 2009;4

31. O’Connor J, Ball EJ, Steinbeck KS, et al. Comparison of total energy expenditure and energy intake in children aged 6-û9 y. Am J Clin Nutr. 2001;74:643–649

32. Cribb VL, Jones LR, Rogers IS, Ness AR, Emmett PM. Is maternal education level associated with diet in 10-year-old children? Public Health Nutr. 2011;14:2037–2048

33. Börnhorst C, Huybrechts I, Hebestreit A, et al.IDEFICS consortium. Diet-obesity associations in children: approaches to counteract attenuation caused by misreporting. Public Health Nutr. 2013;16:256–266

34. Day N, McKeown N, Wong M, Welch A, Bingham S. Epidemiological assessment of diet: a comparison of a 7-day diary with a food frequency questionnaire using urinary markers of nitrogen, potassium and sodium. Int J Epidemiol. 2001;30:309–317

35. Bingham SA, Gill C, Welch A, et al. Validation of dietary assessment methods in the UK arm of EPIC using weighed records, and 24-hour urinary nitrogen and potassium and serum vitamin C and carotenoids as biomarkers. Int J Epidemiol. 1997;26(suppl 1):S137–S151

36. VanderWeele TJ. Mediation and mechanism. Eur J Epidemiol. 2009;24:217–224

37. VanderWeele TJ. Bias formulas for sensitivity analysis for direct and indirect effects. Epidemiology. 2010;21:540–551

38. Fisher JO, Johnson RK, Lindquist C, Birch LL, Goran MI. Influence of body composition on the accuracy of reported energy intake in children. Obes Res. 2000;8:597–603