Technical Appendix for Modeling Individual Heterogeneity

First, a linear regression model assuming homoscedasticity and adjusting for a vector () of all the pre-specified covariates (age, type of residence, education, wealth, and marital status) was constructed to serve as the base model for comparison with subsequent models.

(Model 1)

In this ordinary least squares model, represents an individual-specific residual that captures the difference between BMI value for each observation and the population average, conditional on the adjusted variables. Assuming that these homoscedastic residuals are normally distributed around a mean of 0, Model 1 estimates the level-1 (between-individual) variance in BMI as .

Next we modeled BMI variance as a function of each of the explanatory variables, while adjusting for the same set of variables in the fixed part. For example, to test whether between-individual variation in BMI is truly constant across all age groups, we derived separate level-1 variance for each of the seven age groups in Model 2a (where = age 15-19, = age 20-24, , = age45-49):

(Model 2a)

At level-1, the residuals can no longer be summarized with one variance, and instead a variance covariance matrix is estimated.

For categorical variables, not all the terms in the matrix get estimated since individuals can only be in one category on a single dimension. Due to the dummy/indicator coding approach, only the base variance () and the associated covariance terms (highlighted) that represent the differential in the variation between the reference group and each of the respective categories can be estimated (Goldstein, 2005). In this case, the total between-individual variance in BMI for age 15-19 years (intercept) is estimated directly as , and a linear model can be estimated to derive the between-individual variation for all the other age categories. For instance, the variation for individuals aged 20-24 years is derived as , and the variation for age 25-29 years is derived as .

We repeated this procedure of modeling the difference in variability between categories and then deriving the variances indirectly for type of residency (Model 2b), wealth quintiles (Model 2c), education (Model 2d), and marital status (Model 2e). Based on the estimates presented in Supplementary Table 1, the between-individual variance in BMI for each model was calculated as:

Relevant terms (functional forms) / Total variation
Model 2a: heterogeneity by age
15-19 yrs / / 9.84
20-24 yrs / / 12.95
25-29 yrs / / 16.34
30-34 yrs / / 18.81
35-39 yrs / / 20.50
40-44 yrs / / 22.79
45-49 yrs / / 23.21
Model 2b: heterogeneity by place of residence
Urban / / 20.21
Rural / / 13.88
Model 2c: heterogeneity by education
No education / / 14.20
Primary education / / 16.57
Secondary education / / 17.51
Higher education / / 19.69
Model 2d: heterogeneity by wealth quintiles
Q1 / / 13.62
Q2 / / 14.85
Q3 / / 15.92
Q4 / / 17.93
Q5 / / 20.10
Model 2e: heterogeneity by marital status
Never married / / 12.50
Married/ Living together / / 17.81
Widowed/ Divorced/ Separated / / 20.84

To explore potential interaction between age and socioeconomic factors, we first included the interaction terms between age and education in the fixed part (Model 3a). Then, we further modeled BMI variance as a function of all possible cross-classification of age groups and education levels (Model 3b).To obtain the total variance for each of the 28 social groups (7 age categories x 4 education levels) directly from the model, we used separate coding in the random part. Assuming = age 15-19yrs with no education, …, = age 45-49yrs with higher education, the total between-individual variance in BMI for the youngest group of women with no education is directly estimated as and the variance for oldest and the most educated women is given as .

Due to the separate coding approach, only the variance terms in the diagonal (highlighted) are estimated. Similarly, we estimated interaction effect between age and wealth quintiles (Model 4a) and further expanded the model to estimate individual variance as a function of age and wealth (Model 4b).