4.1 Conditional measures versus adjusting for exposure
An ‘unexplained residuals’ (UR) modeling framework allows examination of the association of several measurements of an exposure and their relative importance over time with the outcome (23). These measures are often operationalized as residuals from a linear regression. However, one could use non-linear approaches to estimate conditional measures while compromising interpretability as residuals absent of correlations with all previous measures of wealth. The below equation (Eq. 5) describes the mathematical quantity of conditional measure (ci,t) for an exposure (wi,t) as the difference beyond what is predicted by previous measures of the exposure from time 1 to t -1.
ci,t = \({\text{w}}_{\text{i},\text{t}}- {\widehat{\text{w}}}_{\text{i},\text{t}}\) = \({\text{w}}_{\text{i},\text{t}}- \text{f}({\text{w}}_{\text{i},1},{\text{w}}_{\text{i},2},{\text{w}}_{\text{i},3},\dots ,{\text{w}}_{\text{i},\text{t}-1})\) [5]
We demonstrate the use of conditional measures with a simple example using two time points (t = 1, 2) and exposure wi,t measured at time ‘t’ for individual ‘i'. A fixed effects approach for repeated measures of the exposure is provided in Eq. 6. The conditional measures approach is provided in Eq. 7. Previous research has demonstrated how both models are equivalent for w2 such that a2 = a'2 (23). However, a0 ≠ a'0 and a1 ≠ a'1, leading to debates (see Section 8. Limitations of the approach) over the relevance of the anchor measure (w1 in our case) that is used as the predictor for other measures (such as w2), rendering it different from the fixed effects approach with repeated measures of the exposure.
E[yi] = a0 + a1 E[wi,1] + a2 E[wi,2] + Covariates [6]
E[yi] = a'0 + a'1 E[wi,1] + a'2 E[ci,2] + Covariates [7]
4.3 Conditional wealth
We extend the conditional growth model to a measure derived from asset-based indices, which we call a conditional asset index. We henceforth refer to it as conditional wealth given that asset-based indices are a proxy for wealth in LMICs. Similar to conditional growth, conditional wealth would allow us to identify stages in the life course at which changes in wealth beyond that predicted by past measures of wealth are differentially associated with health outcomes. This is especially important as LMICs experience slow economic growth, high or rising wealth inequality, and intergenerational social persistence to identify sensitive periods when relative social mobility (positional mobility) is important. The importance of positional mobility assumes that relative position in the wealth hierarchy matters.
We propose that conditional wealth (ci,t) for a life stage ‘t’ and individual ‘i’ is the difference in wealth in that life stage from that which could be predicted by all prior individual measures of wealth and the overall wealth trajectory of the population under study (Eq. 8). For our previous example with two time points, which could be extended to more than two time points, where \(\text{g}\left({\text{w}}_{\text{i},1}\right)= {\widehat{\text{w}}}_{\text{i},2}\), we propose:
wi,2 = g(wi,1) + ci,2 [8]
Conditional wealth is the unexplained residual of the regression of wealth at time 2 as the dependent variable on all previous measures of wealth (in this case time 1) as linearly associated independent variables (Eq. 9).
wi,2 = b0 + b1 wi,1 + ci,2 [9]
such that E[ci,2] = 0; Var[ci,2] = σ2t=2
Re-writing Eq. 9, we define conditional wealth as the magnitude of change in relative position for an individual: ci,2 = wi,2 – [b0 + b1 wi,1]. The above example could be extended to more than two time points easily and we demonstrate the same empirically in Section 7.4.
A previous study by Arnold et. al. has clearly demonstrated appropriate confounder adjustment mathematically and using causal diagrams for conditional measures (or unexplained residuals) (23). Arnold et. al. recommend adjustment for confounders at both stages, i.e. during construction of conditional measures and during estimation of association with outcomes (Eq. 10). The conditional wealth derived using the Arnold et. al. approach would then be uncorrelated with previous measures of wealth and with the confounders (23). Assume X1 is a predictor of wealth at time 1 (e.g. maternal schooling), and X2 is a predictor of conditional wealth, and is partly predicted by w1 (e.g. attained schooling). A directed acyclic graph (DAG) for how wealth, conditional wealth, schooling and outcome are related is provided in Fig. 1. The outcome regression as per Arnold et. al. (Eq. 11) is fit with the anchor measure, confounders, and conditional wealth.
wi,2 = b'0 + b'1 wi,1 + b'2 Xi,1 + b'3 Xi,2 + c'i,2 [10]
E[yi] = a'0 + a'1 E[wi,1] + a'2 E[Xi,1] + a'3 E[Xi,2] + a'3 E[c'i,2] [11]
We deviate from the approach by Arnold et. al. during the first stage (using Eq. 9) since we are interested in understanding the ‘absorbed effect’ of omitted predictors (Eq. 12) on the conditional measures (23). In this case, absorbed effect refers to the variability in conditional wealth explained by predictors of conditional measures, where ui,2 is the error term when predicting conditional wealth.
ci,2 = d0 + d1 Xi,1 + d2 Xi,2 + ui,2 [12]
A path analysis approach (Fig. 1A), where all past measures of the exposure and other covariates are predictors of the exposure at time t, is equivalent to the conditional measures approach (Fig. 1B). Studies estimating the association of conditional wealth with health outcomes should adjust for the first (or anchor) measure of wealth, life course covariates and past measures of conditional wealth, but not for any other wealth measure. Our final outcome regression (Eq. 13) would yield equivalent regression coefficients (i.e. a1 = a'1, a2 = a'2 etc.) as Arnold et.al (Eq. 11).
E[yi] = a0 + a1 E[wi,1] + a2 E[Xi,1] + a3 E[Xi,2] + a3 E[ci,2] [13]
Given these estimates, our approach is intended to understand how one could intervene on conditional wealth, while at the same time get unbiased estimates of relative position effect on the outcome after appropriate confounder/covariate adjustment. Studies estimating the association of early life variables on conditional wealth should not adjust for the anchor measure since it is assumed to be uncorrelated with the conditional wealth measure (Supplementary Fig. 1). There is no covariance between the anchor measure and a predictor of conditional wealth (say X2) that also covaries with conditional wealth. For example, only the component of attained schooling, say from an intervention such as mandatory schooling, which doesn’t depend on early life wealth predicts conditional wealth. Conditional wealth is, in effect, a decomposition of current wealth into explained and unexplained components that are uncorrelated with each other. Conditional wealth is therefore the magnitude of change in relative position for an individual.