Perils and prospects of using aggregate area level socioeconomic information as a proxy for individual level socioeconomic confounders in instrumental variables regression

Abstract

A frequent concern in making statistical inference for causal effects of a policy or treatment based on observational studies is that there are unmeasured confounding variables. The instrumental variable method is an approach to estimating a causal relationship in the presence of unmeasured confounding variables. A valid instrumental variable needs to be independent of the unmeasured confounding variables. It is important to control for the confounding variable if it is correlated with the instrument. In health services research, socioeconomic status variables are often considered as confounding variables. In recent studies, distance to a specialty care center has been used as an instrument for the effect of specialty care vs. general care. Because the instrument may be correlated with socioeconomic status variables, it is important that socioeconomic status variables are controlled for in the instrumental variables regression. However, health data sets often lack individual socioeconomic information but contain area average socioeconomic information from the US Census, e.g., average income or education level in a county. We study the effects on the bias of the two stage least squares estimates in instrumental variables regression when using an area-level variable as a controlled confounding variable that may be correlated with the instrument. We propose the aggregated instrumental variables regression using the concept of Wald’s method of grouping, provided the assumption that the grouping is independent of the errors. We present simulation results and an application to a study of perinatal care for premature infants.

Appendix: Consistency of the TSLS estimator from the aggregate IV regression and its variance estimate

In this section, we will show the show the consistency of $\hat{\beta}^{agg}$ obtained from the aggregate IV regression in Sect. 4 We will provide an estimate for the variance of $\hat{\beta}^{agg}$.

Let (Y ^agg, X ^agg, D ^agg, Z ^agg) denote vectors of aggregated (Y, X, D, Z), and $\underline{{\bf W}}^{agg} = [{\bf D}^{agg}, {\bf X}^{agg}]$ and $\underline{{\bf A}}^{agg} = [{\bf Z}^{agg}, {\bf X}^{agg}]$. We use $(\underline{{\bf W}}^{agg}, \underline{{\bf A}}^{agg})$ to distinguish (W ^agg, A ^agg) used in Sect. 3 in which W ^agg = [D, X ^agg] and A ^agg = [Z, X ^agg]. Let (H _Y , H _A , H _W ) be aggregation errors for (Y, A, W), where ${\bf H}_{Y} = {\bf Y} - {\bf Y}^{agg},\, {\bf H}_{A} = {\bf A} - \underline{{\bf A}}^{agg}$, and ${\bf H}_{W} = {\bf W} - \underline{{\bf W}}^{agg}$. The TSLS estimator $\hat{\beta}^{agg}$ obtained from all aggregate variables is

$$ \begin{aligned} \hat{\beta}^{agg} &= (\underline{{\bf A}}^{agg^{T}} \underline{{\bf W}}^{agg})^{-1} \underline{{\bf A}}^{agg^{T}}{\bf Y}^{agg} \\ &= (\underline{{\bf A}}^{agg^{T}} \underline{{\bf W}}^{agg})^{-1} \underline{{\bf A}}^{agg^{T}}\{(\underline{{\bf W}}^{agg} + {\bf H}_{W}) \beta - {\bf H}_{Y} + \varepsilon\} \\ &= \beta + (\underline{{\bf A}}^{agg^{T}} \underline{{\bf W}}^{agg})^{-1} \underline{{\bf A}}^{agg^{T}}({\bf H}_{W} \beta - {\bf H}_{Y} + \varepsilon) \end{aligned} $$

(10)

If we can show that $\underline{{\bf A}}^{agg^{T}}({\bf H}_{W} \beta - {\bf H}_{Y} + \varepsilon) \xrightarrow{p}{\bf 0}$ in (10), then $\hat{\beta}^{agg}$ is a consistent estimator for β. We could write the aggregate matrices $({\bf Y}^{agg}, \underline{{\bf A}}^{agg}, \underline{{\bf W}}^{agg})$ as (G Y, G A, G W). The matrix G is a diagonal grouping matrix, where

$$ {\bf G} = \left[\begin{array}{llll} {\bf G}_{1} & {\bf 0} & \cdots & {\bf 0} \\ {\bf 0} & {\bf G}_{2} & \cdots & {\bf 0} \\ \vdots & \vdots & \ddots & \vdots \\ {\bf 0} & {\bf 0} & \cdots & {\bf G}_{j}\\ \end{array}\right] \quad \hbox{and} \quad {\bf G}_{j} = \left[\begin{array}{lll} 1/n_{j} & \cdots & 1/n_{j} \\ \vdots & \ddots & \vdots \\ 1/n_{j} & \cdots & 1/n_{j}\\ \end{array}\right]. $$

Thus, $\underline{{\bf A}}^{agg^{T}}({\bf H}_{W}\beta - {\bf H}_{Y} + \varepsilon)$ can be written as ${\bf A}^{T}{\bf G}^{T}\{({\bf W} - {\bf G}{\bf W})\beta - ({\bf Y} - {\bf G}{\bf Y}) + \varepsilon\}$. Since G is a symmetric and idempotent matrix, G ^T(W − G W) and G ^T(Y − G Y) are zero. Also, ${\bf A}^{T}{\bf G}^{T}\varepsilon$ converges in probability to zero because of the assumption of independence between G and $\varepsilon$. The variance of $\hat{\beta}^{agg}$ is

$$ \begin{aligned} Var\left(\hat{\beta}^{agg}\right) &= Var\left\{(\underline{{\bf A}}^{agg^{T}} \underline{{\bf W}}^{agg})^{-1} \underline{{\bf A}}^{agg^{T}} ({\bf H}_{W} \beta - {\bf H}_{Y} + \varepsilon)\right\} \\ & = ({\bf A}^{T}{\bf G}{\bf W})^{-1}{\bf A}^{T}{\bf G} \times Var\left\{({\bf W} - {\bf G}{\bf W})\beta - ({\bf Y} - {\bf G}{\bf Y}) + \varepsilon\right\} \times {\bf G}{\bf A}({\bf W}^{T}{\bf G}{\bf A})^{-1} \\ &= ({\bf A}^{T}{\bf G}{\bf W})^{-1}{\bf A}^{T}{\bf G} \times Var\left\{({\bf G}{\bf Y} - {\bf G}{\bf W} \beta)\right\} \times {\bf G}{\bf A}({\bf W}^{T}{\bf G}{\bf A})^{-1} \\ &= ({\bf A}^{T}{\bf G}{\bf W})^{-1}{\bf A}^{T}{\bf G} \times Var({\bf G}\varepsilon) \times {\bf G}{\bf A}({\bf W}^{T}{\bf G}{\bf A})^{-1}, \end{aligned} $$

(11)

where $Var({\bf G} \varepsilon)$ can be estimated by

$$ \begin{aligned} \widehat{Var}({\bf G}\varepsilon) &= \left[\begin{array}{llll} \hat{\Upsigma}_{1} & {\bf 0} & \cdots & {\bf 0} \\ {\bf 0} & \hat{\Upsigma}_{2} & \cdots & {\bf 0} \\ \vdots & \vdots & \ddots & \vdots \\ {\bf 0} & {\bf 0} & \cdots & \hat{\Upsigma}_{j}\\ \end{array}\right] \quad \hbox{and} \quad \\ \hat{\Upsigma}_{j} &= \left[\begin{array}{ccc} (n_{j}-1)^{-1}\sum_{i=1}^{n_{j}}(y_{j}^{agg}-[z_{j}^{agg}, x_{j}^{agg}]\hat{\beta}^{agg})^{2} & \cdots & (n_{j}-1)^{-1}\sum_{i=1}^{n_{j}}(y_{j}^{agg}-[z_{j}^{agg}, x_{j}^{agg}]\hat{\beta}^{agg})^{2} \\ \vdots & \ddots & \vdots \\ (n_{j}-1)^{-1}\sum_{i=1}^{n_{j}}(y_{j}^{agg}-[z_{j}^{agg}, x_{j}^{agg}]\hat{\beta}^{agg})^{2} & \cdots & (n_{j}-1)^{-1}\sum_{i=1}^{n_{j}}(y_{j}^{agg}-[z_{j}^{agg}, x_{j}^{agg}]\hat{\beta}^{agg})^{2}\\ \end{array}\right]. \end{aligned} $$