Robust estimation of the causal effect of time-varying neighborhood factors on health outcomes

Abstract

The fundamental difficulty of establishing causal relationships between an exposure and an outcome in observational data involves disentangling causality from confounding factors. This problem underlies much of neighborhoods research, which abounds with studies that consider associations between neighborhood characteristics and health outcomes in longitudinal data. Such analyses are confounded by selection issues; individuals with above average health outcomes (or associated characteristics) may self-select into advantaged neighborhoods. Techniques commonly used to assess causal inferences in observational longitudinal data, such as inverse probability of treatment weighting (IPTW), may be inappropriate in neighborhoods data due to unique characteristics of such data. We advance the IPTW toolkit by introducing a procedure based on a multivariate kernel density function which is more appropriate for neighborhoods data. The proposed weighting method is applied in conjunction with a marginal structural model. Our empirical analyses use longitudinal data from the Health and Retirement Study; our exposure of interest is an index of neighborhood socioeconomic status (NSES), and we examine its influence on cognitive function. Our findings illustrate the importance of the choice of method for IPTW—the comparison weighting methods provide poor balance across the set of covariates (which is not the case for our preferred procedure) and yield misleading results when applied in the outcomes models. The utility of the multivariate kernel is also validated via simulation. In addition, our findings emphasize the importance of IPTW—controlling for covariates within a regression without IPTW indicates that NSES affects cognition, whereas IPTW-weighted models fail to show a statistically significant effect.

1 |. INTRODUCTION

One of the most difficult problems in statistical inference involves determination of whether associations observed in non-experimental data between an exposure (eg, a neighborhood characteristic) and an outcome (eg, health status) have causal implications or if they are instead explained by confounding factors. Longitudinal data offer a plethora of advantages when approaching this problem; however, those advantages induce a wealth of methodological challenges. Consequentially, the field of causal inference has expanded markedly over the past 15 years, producing a rich mixture of state-of-the-art methods for handling longitudinal observational data including methods to handle continuous exposures.^1–10 Available methods range from utilizing fully parametric approaches^11–14 to fully nonparametric,¹⁵ with a series on semiparametric methodologies as well.^16,17 Many of these papers shed light on how to utilize inverse probability of treatment weighting (IPTW) with time-varying exposures. IPTW, which captures the likelihood of an individual experiencing a particular exposure history, is used to weight individuals so that all the observed characteristics that are included in the model for IPTW are no longer correlated with the exposure, thereby allowing for robust estimation of the marginal impact of the exposure on outcomes.

Methods for causal inference with observational data such as IPTW are particularly important within neighborhoods research, which constitutes a substantial arm of modern epidemiologic science. Although randomized experiments in neighborhoods studies have been performed (see, for example, the Moving to Opportunity study^18–20), the vast majority of studies of neighborhood impacts are observational due to difficulties randomizing individuals into neighborhoods. The goal in neighborhoods research is often to discern whether a neighborhood level-exposure, such as socioeconomic status^21–24 or immigrant enclaves,²⁵ affects a health outcome, such as body mass index^26,27 or depressive symptoms.^27–29 However, the literature is replete with debate about whether causal links can be established in neighborhoods research.^30–34 In particular, individuals might knowingly or unknowingly select into neighborhoods based on a number of confounding factors including family poverty,³⁵ health conditions,^36–39 preferences for neighborhoods with greater walkability, healthy food access, etc,⁴⁰ and safety considerations.^41–43 These factors create neighborhoods in which residents differ on a number of underlying characteristics which are associated with outcomes, including age, gender, race/ethnicity, and general health status.^30,32,44,45 It is difficult to remove these imbalances when estimating associations between neighborhood factors and outcomes.

In our current study, we aim to assess if there is a causal link between neighborhood socioeconomic status (NSES) and cognition among a representative sample of older adults living in the United States. The public health and economic impact of cognitive impairment and dementia onset has been well-document in the US.⁴⁶ Yet, evidence is limited and inconsistent regarding the effectiveness of interventions targeting individual risk factors to slow cognitive decline.⁴⁷ As such, there is interest in gaining a better understanding of the contextual determinants of cognitive decline that may be a more fruitful for intervention for delaying the onset of cognitive impairment and dementia at a population level.⁴⁸ Recent studies have examined the role of neighborhood and other geographic level factors, collectively referred to hereafter as context, on cognitive aging, with the vast majority of studies focusing on measures of community socioeconomic status and disadvantage (for a review, see the work of Wu et al⁴⁹). It is of interest in this study to assess the potential role NSES might play on cognitive decline in older adults living in the United States (US) sample with a focus on how best to control for confounding effects. As noted above, it is pivotal that our analyses control for key confounders that involve the choice individuals in our sample make about which neighborhoods they want to live in.

In light of the challenges due to individual selection into neighborhoods, applications of IPTW in conjunction with marginal structural models (MSMs) in neighborhood research appear to be on the rise since such the methods are designed to handle confounding due to observed factors like neighborhood selection. Using MSMs with IPTW, causal links have been found between increased neighborhood poverty and increased alcohol consumption,⁵⁰ increased mortality risk,⁵¹ and increased odds of being obese.⁵² Furthermore, there has also been work that found a causal link between neighborhood context and epigenetic aging.⁵³ In addition, it has been shown that disadvantaged neighborhoods have a causal link to hindered verbal ability⁵⁴ and lower graduation rates.⁵⁵ The majority of these studies apply IPTW using MSMs on their outcomes following the methodology proposed by Robins and colleagues.^1–5,56 In addition, doubly robust methodology,^8,57–59 which uses regression modeling and IPTW in tandem, has grown in popularity recently with some implementation in studies that investigate links between neighborhood factors and health outcomes.^60,61

With the growing use of IPTW in neighborhoods research, it is critical to consider the special distributional characteristics of neighborhood-level data that may make the use of IPTW estimated via simple parametric models inappropriate. In particular, neighborhood-level exposures in longitudinal data tend to be static across most years of observation (ie, one’s neighborhood characteristics change meaningfully only if he or she relocates or moves). Traditional parametric approaches to estimate IPTW for longitudinal data are not built to handle such data characteristics. That is, the methods require that one evaluates residual densities for a conditional model that predicts the level of exposure at a given time point using prior values of the exposure. Commonly used Gaussian assumptions seem inappropriate since such models yield residuals with large masses near zero when applied to neighborhood-level exposures. Here, we propose a new method for estimating IPTW that is more appropriate for complex distributional structures seen in neighborhoods data. Specifically, we suggest a multivariate kernel density that circumvents the need for building conditional models of the exposure at one time period on the exposure at prior time periods.

In Section 2, we outline the outcomes models that will be used in our analyses as well as the candidate IPTW estimation methods that we will compare. Specifically, we consider an MSM that is a function of the cumulative value of the neighborhood exposure as suggested by Robins,^1,56 as well as doubly robust versions of this model. In addition, we outline our proposed method of estimating IPTW in Section 2, as well as five other methods that form a basis for comparison. We aim to compare the performance of the different estimation methods for the IPTW in terms of ability to balance across the set of observed covariates as well as inferences yielded in analyses involving outcomes. In Section 3, we present the findings from our motivating study using the proposed outcome models and IPTW estimators in an effort to estimate the effect of NSES on cognitive function among older adults living in the United States. Both the data application and simulations indicate that weights calculated via the multivariate kernel density estimator outperform all other methods considered. Section 4 presents findings from a detailed simulation study whose structure was inspired by our motivating case study. Lastly, discussion is provided in Section 5, and remnant content is given in supplementary materials.

2 |. STATISTICAL METHODS

Here, we describe the statistical methodology that will be used to draw causal inferences regarding the effect of neighborhood on cognitive function among older adults living in the US. We begin with some notation. Assume that we have data for n individuals at T time points (T = 6 in our data). Let Z_0,i denote a set of observed baseline covariates (eg, race, gender), also referred to as confounders, for individual i that are not time varying, and let Z_t,i denote a set of time-varying covariates or confounders (eg, stroke, blood pressure, diabetes) for t = 1, …, T. Furthermore, let X_t,i denote the value of the exposure variable (here NSES), and let Y_t,i denote the value of the outcome variable (here cognitive function) for individual i at time t for t = 1, …, T. Let ${\overline{\mathbf{X}}}_{t,i}={\left(X_{1,i},\dots ,X_{t,i}\right)}^{\prime }$ and ${\overline{\mathbf{Z}}}_{t,i}={\left({\mathbf{Z}}_{0,i}^{\prime },{\mathbf{Z}}_{1,i}^{\prime },\dots ,{\mathbf{Z}}_{t,i}^{\prime }\right)}^{\prime }$ denote the exposure and covariate histories up to time t. Thus, in our analysis, ${\overline{\mathbf{X}}}_{t,i}$ represents the individual’s neighborhood history of NSES values up to time t.

For our approach, we utilize an IPTW potential outcomes framework.^62,63 To elaborate, let ${\mathbf{Y}}_i\left({\overline{\mathbf{x}}}_{T,i}\right)={\left(Y_{1,i}\left({\overline{\mathbf{x}}}_{T,i}\right),\dots ,Y_{T,i}\left({\overline{\mathbf{x}}}_{T,i}\right)\right)}^{\prime }$ denote the potential value of the outcome trajectory that would have been observed at time t if individual i’s neighborhood exposure history had been ${\overline{\mathbf{x}}}_{T,i}$. The causal effect of experiencing an exposure history of ${\overline{\mathbf{x}}}_{T,i}$ in comparison to ${\overline{\mathbf{x}}}_{T,i}^{\prime }$ is defined as the difference in expected potential outcomes, which is given by

$$E\left[{\mathbf{Y}}_i\left({\overline{\mathbf{x}}}_{T,i}\right)-{\mathbf{Y}}_i\left({\overline{\mathbf{x}}}_{T,i}^{\prime }\right)\right]=E\left[{\mathbf{Y}}_i\left({\overline{\mathbf{x}}}_{T,i}\right)\right]-E\left[{\mathbf{Y}}_i\left({\overline{\mathbf{x}}}_{T,i}^{\prime }\right)\right].$$

Thus, in our example, this is the expected difference in two potential outcomes for an individual had he/she had two different histories in neighborhood exposures. This quantity is not observed directly for all possible exposure histories (since the same individual does not experience multiple treatment histories) and cannot be estimated using conditional models that control for covariates. Moreover, the true causal effects of interest might vary depending on the given interests of a study. Nonetheless, models that do not control for confounding covariates that predict the neighborhood exposure level and the outcome of interest are subject to confounding bias when trying to estimate the needed expected values. Therefore, it is prudent to reframe the analysis within a pseudo-population wherein the exposure is no longer confounded by the observed covariates—therein, the association between the outcome and exposure can be assessed marginally using marginal structural models (MSMs) as described in Section 2.1. IPTW is used to generate this pseudo-population. Weights are determined by calculating each individual’s likelihood of observing the treatment history that the individual observed given his or her observed confounding covariate values—residual densities are used to assess these likelihoods. Then, weighting based upon these densities balances the exposure history groups with respect to the observed covariate confounders by giving more weight to individuals with treatment histories that are underrepresented in the pseudo-population and less weight to exposure or neighborhood histories that are overrepresented. In the end, the IPTW serve to create comparable exposure history groups across the observed set of confounders that are used in the IPT model.

There are four assumptions that are required for robust inference from an MSM plus IPTW approach: consistency, exchangeability, positivity, and no misspecification of the model used to estimate weights. Consistency requires that a subject’s potential outcome under the exposure received (here, their observed neighborhood history) is equal to the subject’s observed outcome. That is, $Y_{t,i}=Y_{t,i}\left({\overline{\mathbf{X}}}_{T,i}\right)$. Exchangeability requires that there are no unobserved confounders that have not been included in the covariates used to estimate weights. In other terms, this implies

$$Y_{t,i}\left({\overline{\mathbf{X}}}_{T,i}\right)\perp {\overline{\mathbf{X}}}_{t,i}\mid {\overline{\mathbf{X}}}_{t-1,i},{\overline{\mathbf{Z}}}_{t,i},$$

where ⟂ indicates independence and thus that no important confounders were excluded from the IPT model when estimating the weights. Positivity requires that there is a nonzero probability of everyone in the sample receiving every level of the exposure. Consider, for example, that if a given level of the exposure were only observable among women, we could not consistently estimate the exposure effects among men. For the most part, the first two assumptions are generally determined as appropriate depending on the applied example at hand. The last assumption (also called the overlap assumption) is generally assessed using descriptive statistics to determine if there are any areas of the covariate distribution that might not be fully observed for all levels of the exposure (see the supplemental materials). Finally, the unobserved confounding assumption is typically assessed using sensitivity analyses (see, for example, other works^64–67).

Once the weights have been calculated, the causal parameter of interest from the pseudo-population may be extracted using a weighted regression model. When this regression does not control for covariates, it is an MSM in the vein of Robins.^1,56 However, doubly robust⁵⁷ versions of such a model can be employed by conditioning on covariates in addition to weighting the regression.

2.1 |. Calculation of weights for IPTW

Since IPTW with neighborhood data is complicated by the unique characteristics of such data (and therefore capturing the requisite densities is nontrivial), we consider a variety of methods for estimating the weights. We begin by outlining the traditional method for IPTW with longitudinal data and continuous exposures that was originally proposed by Robins^1,56 (this is manner in which IPTW is commonly applied within neighborhood research). We then discuss procedures which provide advancements, culminating in our preferred approach for IPTW when using neighborhood-level exposures, which is based on a multivariate kernel density.

In accordance with Robins,^1,56 IPTW is performed by first calculating a weight for individual i at time t for t = 1, …, T:

$$w_{t,i}=\frac{f\left(X_{t,i}\mid {\overline{\mathbf{X}}}_{t-1,i}\right)}{f\left(X_{t,i}\mid {\overline{\mathbf{X}}}_{t-1,i},{\overline{\mathbf{Z}}}_{t,i}\right)},$$ (1)

where f(·) denotes a density function. The numerator term $f\left(X_{t,i}\mid {\overline{\mathbf{X}}}_{t-1,i}\right)$ acts as a stabilizer. For analysis, one uses the cumulative product of weights

$$w_i=\prod _{t=1}^T{w}_{t,i},$$ (2)

which is the weight assigned to individual i for all time points.

An obstacle for implementation of IPTW is the fact that the true densities listed in the numerator and denominator of (1) are unknown. To approximate these densities, one could fit an appropriate density function to the residuals from the corresponding conditional models. However, the conditional densities are difficult to model in neighborhoods data. A neighborhood-level exposure (in our case, NSES) will be relatively stable over time for most study participants; however, when there is a change, it tends to be larger—this leads to residuals that have heavy tails. Therefore, we consider four methods of calculating weights for IPTW by modeling the residual densities needed in the calculation of w_t,i.

Let $r_{t,i}^{\left(1\right)}$ denote the residuals from a linear regression of X_t,i on X_1,i, …, X_t−1,i and an intercept term. Likewise, let $r_{t,i}^{\left(2\right)}$ denote the residuals from a linear regression of X_t,i on X_1,i, …, X_t−1,i and ${\overline{\mathbf{Z}}}_{t,i}$. If the density of $r_{t,i}^{(k)}$, which we represent by $f_t^{(k)}(\cdot )$ for j = 1, 2, were known, one could set $w_{t,i}=f_t^{(1)}\left(r_{t,i}^{(1)}\right)/f_t^{(2)}\left(r_{t,i}^{(2)}\right)$. We approximate $f_t^{(k)}(\cdot )$ for t = 1, …, T and k = 1, 2 via the following methods:

1. Univariate Normal – $f_t^{(k)}(\cdot )$ is represented by a normal density with mean zero and variance set to the sample variance of $\left\{r_{t,i}^{(k)}\right\}$.

2. Student’s t – $f_t^{(k)}(\cdot )$ is represented by a Student’s t distribution with 3 degrees of freedom. This mandates that $\left\{r_{t,i}^{(k)}\right\}$ first be scaled to have a variance of 3 prior to calculating the weights. We select 3 degrees of freedom because it is the smallest integer value that yields a density with finite variance. Finite variance is needed because we have to scale the residuals to have the same variance as the density that is being fit to them.

3. Mixture – We set $f_t^{(k)}(\cdot )=\rho g_t^{(k)}(\cdot )+(1-\rho )h_t^{(k)}(\cdot )$ for some mixing probability ρ ∈ (0, 1) where $g_t^{(k)}(\cdot )$ and $h_t^{(k)}(\cdot )$ are distinct normal densities. The parameters (ρ and the means and variances of the normal densities) are estimated by modeling $\left\{r_{t,i}^{(k)}\right\}$; we use the mixtools package in R for estimation.⁶⁸

4. Univariate Kernel – We estimate $f_t^{(k)}\left(r_{t,i}^{(k)}\right)$ by calculating

$${\widehat{f}}_t^{(k)}\left(r_{t,i}^{(k)}\right)=\frac{1}{nh}\sum _{j=1}^{n}K\left(\frac{r_{t,i}^{(k)}-r_{t,j}^{(k)}}{h}\right),$$ (3)

where K(·) is a kernel function and h is a bandwidth parameter. We use a standard normal density as the kernel function.

Distribution families not included above have been considered in the literature. For example, Do et al⁵¹ model an exposure that is bounded between 0 and 1 using a beta distribution which aims to more appropriately model the distribution of neighborhood-level exposures in a similar vain to our proposed work. However, the beta distribution does not work in this setting since we considering exposures that are unbounded.

Any inaccuracies in the estimation of the conditional densities will be exacerbated through the use of the product of weights that is seen in (2). Therefore, we propose a method for IPTW that circumvents the conditional models in (1). Note that w_i can be rewritten as a ratio of joint densities:

$$w_i=\frac{f\left({\overline{\mathbf{X}}}_{T,i}\right)}{f\left({\overline{\mathbf{X}}}_{T,i}\mid {\overline{\mathbf{Z}}}_{T,i}\right)},$$ (4)

which follows from the fact that joint densities may be expressed as a product of a sequence of conditional densities. This relationship assumes that the exposure at a given time is conditionally independent of future values. Therefore, we propose to calculate weights for IPTW by directly targeting the joint densities seen in (4). This should, in theory, help us bypass estimation problems that are encountered when building weights using residuals from a model that predicts the current exposure level from the prior year’s exposure.

Let ${\mathbf{r}}_i^{(1)}$ denote the (multivariate) residuals from a linear regression of ${\overline{\mathbf{X}}}_{T,i}$ on only an intercept term, and let ${\mathbf{r}}_i^{(2)}$ denote the (multivariate) residuals from a linear regression of ${\overline{\mathbf{X}}}_{T,i}$ on ${\overline{\mathbf{Z}}}_{T,i}$. We let f ^(k)(·) denote the multivariate density function of ${\mathbf{r}}_i^{(k)}$ for k = 1, 2– if f ^(k)(·) were known, we would set $w_i=f^{(1)}\left({\mathbf{r}}_i^{(1)}\right)/f^{(2)}\left({\mathbf{r}}_i^{(2)}\right)$. We outline two procedures for approximating the joint densities and compare their performance to the four methods list above.

1. Multivariate Normal – f ^(k)(·) is represented by a multivariate normal density with mean zero and variance set to the sample covariance matrix of $\left\{r_{t,i}^{(k)}\right\}$.

2. Multivariate Kernel – We estimate f ^(k)(·) via a multivariate kernel density function, which is denoted here as ${\widehat{f}}^{(k)}(\cdot )$.

Specifically, we set

$${\widehat{f}}^{(k)}(\mathbf{r})=\frac{1}{n}\sum _{i=1}^n{K}_{\mathbf{H}}\left(\mathbf{r}-{\mathbf{r}}_i^{(k)}\right)$$ (5)

for any length-T vector r, where H is a T × T matrix of bandwidth parameters, $K_{\mathbf{H}}(\mathbf{r})=|\mathbf{H}{|}^{-1/2}K\left({\mathbf{H}}^{-1/2}\mathbf{r}\right)$, and K(·) is a symmetric multivariate kernel function. It is common to let K_H(·) be represented by a multivariate normal density with a covariance matrix equivalent to the identity matrix. Lastly, the quantity in (4) is approximated with

$${\widehat{w}}_i=\frac{{\widehat{f}}^{(1)}\left({\mathbf{r}}_i^{(1)}\right)}{{\widehat{f}}^{(2)}\left({\mathbf{r}}_i^{(2)}\right)}.$$

In the empirical analyses provided in subsequent sections, the bandwidth parameter is calculated through Silverman’s rule of thumb.⁶⁹ When the multivariate kernel density is used, we set $\mathbf{\text{H}}=0.1\widehat{\Sigma }$, where $\widehat{\Sigma }$ is the sample correlation matrix of the appropriate multivariate residuals. Other more rigorous options were considered,^70–72 but these did not prove to be as effective at maintaining balance across the set of covariates. We use only Gaussian kernels.

To avoid distorting the underlying properties of the weights, we do not trim outlying weight values. In addition, note that weights calculated in accordance with (4) model an exposure on future values of the covariates. This functionality is not incorporated within (1). We argue that this facet of our methods enhances their robustness since weights calculated in the proposed manner will remove all associations between the exposure and the confounders. However, if needed, estimation of the weights in (4) can be performed in a manner that does not model current exposure on future covariate values.

2.1.1 |. Balance assessment

Imperative to all methods that invoke IPTW is the assumption that the weights obtain balance across the observed covariate set with respect to the exposure. Balance assessment with continuous treatment is more nuanced than with binary treatment. We follow guidance put forth by Zhu et al,¹⁷ wherein it is suggested to use sample correlations to assess balance. That is, if calculated correctly, the IPTW weights outlined here can be used to remove any correlation between the exposure/neighborhood history, ${\overline{\mathbf{X}}}_{T,i}$, and the set of covariates, ${\overline{\mathbf{Z}}}_{T,i}$.

2.1.2 |. Adjustments for survey weighting

Survey weights are often attached to microlevel data. For instance, the data may have been collected under a design that is not simple random sampling, and data are usually subject to nonresponse and/or attrition; survey weights inherently account for these complexities. In that vein, the Health and Retirement Survey (HRS) data set studied in Section 3 is subject to both nonresponse and design adjustments. The issue of performing propensity-score analysis (or IPTW analogously) using survey-weighted data has received attention in the literature. Ridgeway et al⁷³ and Griffin and Robbins⁷⁴ argue that when applying IPTW, survey weights should be included in the estimation of the weights for IPTW as well as in the estimation of the outcomes models. The latter is accomplished by using a final weight in outcomes analyses that is the product of the survey weight and the weight for IPTW.

Survey weights factor into the process of calculating IPTW weights as follows. Weighted least squares (with the survey weights) is used to approximate the residuals (ie, $r_{t,i}^{(1)}$, $r_{t,i}^{(2)}$, ${\mathbf{r}}_i^{(1)}$, and ${\mathbf{r}}_i^{(2)}$). Any parameters (eg, variances) that are calculated from the estimated residuals are calculated using weighted estimation techniques. Furthermore, the kernel density estimators of (3) and (5) are weighted; for example, we calculate ${\widehat{f}}^{(k)}(\mathbf{r})={\sum }_{i=1}^n{v}_i{K}_{\mathbf{H}}\left(\mathbf{r}-{\mathbf{r}}_i^{(k)}\right)/{\sum }_{i=1}^n{v}_i$, where {v_i} are the nonresponse weights.

2.2 |. Outcome models

As an alternative to structural nested models and related g-estimation,^75–78 Robins^1,56 presents MSMs for the potential outcome $Y_{t,i}\left({\overline{\mathbf{X}}}_{t,i}\right)$. To facilitate causal inferences with MSMs, he introduces IPTW in that context to remove the effects of the observed confounding covariates on the exposure effect estimates. Specifically, he considers MSMs of the form

$$E\left[Y_{t,i}\left({\overline{\mathbf{X}}}_{t,i}\right)\right]=g\left({\overline{\mathbf{X}}}_{t,i};\beta \right),$$ (6)

for some function g(·) and parameters β. This model is marginal because it examines the effect of the exposure histories (like our neighborhood histories) on the marginal distributions of the potential outcomes. Likewise, the term structural is invoked since counterfactual random variates are modeled.

Robins¹ and Hernán et al⁵ refine (6) by suggesting that the function g(·) depend primarily upon the cumulative exposure $\text{cum}\left({\overline{\mathbf{X}}}_{t,i}\right)={\sum }_{j=1}^t{X}_{j,i}$. We refine this model further in manner that is more appropriate for the neighborhood data in our study. Specifically, we consider

$$Y_{t,i}={\beta }_0+{\beta }_{0,i}+{\beta }_{1}t+{\beta }_2{t}^2+\alpha \text{cum}\left({\overline{\mathbf{X}}}_{t,i}\right)+U_{t,i},$$ (7)

for i = 1, …, n and t = 1, …, T, where β_0,i is an intercept random effect for individual i and where U_t,i is a mean zero error term. Here, 𝛼 is the causal parameter of interest and represents the average effect of a one-unit increase in NSES on cognition at the next time point. The sequence {U_t,i}_{t,i ≥ 1} is independent across t and i and is independent of the β_0,i. We orient this model in the paradigm of potential outcomes through the use of IPTW to weight the regression. That is, IPTW removes the correlation between the observed confounding covariates ${\overline{\mathbf{Z}}}_{T,i}$ and any function of the neighborhood exposure ${\overline{\mathbf{X}}}_{T,i}$.¹⁷ Ergo, IPTW is applicable for the general class of MSMs seen in (6), which includes (7). Recall that the average causal effect can be expressed is $E\left[Y_{t,i}\left({\overline{\mathbf{X}}}_{t,i}\right)\right]-E\left[Y_{t,i}\left({\overline{\mathbf{X}}}_{t,i}^{\prime }\right)\right]$, which under the MSM in (7) is $\alpha \text{cum}\left({\overline{\mathbf{X}}}_{t,i}\right)$; therefore, α has a causal interpretation.

IPTW has been applied previously in neighborhoods research,^50–55 and such applications tend to use a MSM that invokes cumulative sum (or moving average) of the exposure as in (7). Note that several authors, eg, Hernán et al⁵ and Cerdá et al,⁵⁰ consider MSMs that model $Y_{t,i}\left({\overline{\mathbf{X}}}_{t,i}\right)$ as a function of $\text{cum}\left({\overline{\mathbf{X}}}_{t-1,i}\right)$; this is to impose the assumption that the exposure does not have an instantaneous effect on the outcome. However, in the setting of our application, we feel it is reasonable to assume that current exposure levels can affect the outcome. (At the point when outcome measurements are taken, an individual has probably established him or herself in his or her respective neighborhood of a nonnegligible length of time.)

We also consider a version of (7) that controls for observed covariates within the regression. That is,

$$Y_{t,i}={\beta }_0+{\beta }_{0,i}+{\beta }_{1}t+{\beta }_2{t}^2+{\alpha }_1\text{cum}\left({\overline{\mathbf{X}}}_{t,i}\right)+{\gamma }_1^{\prime }{\mathbf{Z}}_{0,i}+{\gamma }_2^{\prime }{\mathbf{Z}}_{t,i}+U_{t,i}.$$ (8)

We apply the following outcome models in the ensuing sections:

• Model 1: A weighted version of the MSM (7) where covariates are not included and IPTW weights are used.

• Model 2: A weighted version of (8) where covariates are included and IPTW weights are used.

Note that Model 2 is considered doubly robust⁵⁷ because of the use of weights together with regression adjustment. As Robins and Rotnitzky⁷⁹ point out, so long as only one of the models, either that for the conditional mean of the outcome given covariates (the regression model) or that for the treatment indicator given covariates (the IPTW model) is correctly specified, the resulting estimator will be consistent. Robins and Rotnitzky⁷⁹ provide the needed proof of doubly robustness by showing that combining such IPTW with regression adjustment has consistency and demonstrated under the assumption that a parametric model applies to either the IPTW or the regression function, but not necessarily to both. Here, we use parametric modeling in our outcome model.

In the supplemental materials, we consider a separate outcome model that is in the vein of the disaggregation model presented by Jokela³⁴—this model is designed to disentangle between- and within-person effects of a neighborhood-level exposure. Therein, we provide results for IPTW-weighted versions of this model.

3 |. THE EFFECT OF NSES ON COGNITION

3.1 |. Study sample

To assess the influence of neighborhood on cognitive function, we use data from the Health and Retirement Study (HRS), which is a nationally representative, multicohort longitudinal panel of US adults aged 51 and older. The HRS provides one of the most comprehensive data sources on health in older Americans, with detailed data on health, residential status, and confounders such as economic and social well-being; further details are found in the work of Hauser and Willis.⁸⁰ Although some HRS data are publicly available (https://hrs.isr.umich.edu/data-products), the specific data used for this study are confidential.

HRS data are collected biennially. To focus on one decade of address data geocoded using the same 2000-based Census definitions for census tracts, we isolate to the six waves of survey data collected from 2000 to 2010. We examined the largest HRS cohort of individuals born between 1931 and 1941; as such, we avoid modeling NSES and covariate effects differently by cohort.

HRS data are subject to attrition (eg, only 58% of all individuals in our cohort are measured at each survey wave). Since our methodology mandates that only individuals with data observed at all time points for pertinent variables are used, attrition is handled by deleting cases with missing follow-up and using nonresponse weighting to account for nonresponse bias. The final cohort used in analyses contains 4253 individuals. Authors that have studied attrition in the HRS conclude that although there is some evidence of differential patterns of attrition across certain characteristics, differences are not particularly strong and can be captured by nonresponse weights.^81,82 Nonresponse weights corresponding to 2010 are factored into all analyses presented here, which ensures that our cohort is representative of the baseline sample (see the work of Heeringa and Connor⁸³ for further details on weighting in the HRS).

3.2 |. Measures

Outcome.

To assess cognitive function, we use a modified version of the telephone interview for cognitive status (TICS) (see the work of Crooks et al⁸⁴ for discussion of this instrument). We use the 27-point scale TICS that consisted of three tasks: an immediate and delayed word recall of a list of 10 nouns, a Serial Sevens subtraction test from 100, and a backwards count from 20 for 10 continuous numbers. Higher TICS means better cognitive status. For respondents who are unable to participate in the study, cognitive function is assessed through a series of questions to proxy respondents; however, only self-respondent data are included in the cognitive function score provided in the HRS and therefore utilized here.

Neighborhood socioeconomic status.

Neighborhood advantage is measured with an index of socioeconomic status that is a composite of tract-level median household income, percentage of female-headed households, percentage of adult workers who are unemployed, percentage of households below the poverty line, percentage of persons with a high school degree, and percentage of persons with a Bachelor’s degree.⁸⁵ This index is referred to as NSES and serves as our primary exposure (ie, treatment) variable.

Individual-level covariates.

Our analysis controls for several individual-level variables that could confound neighborhood associations. As noted below, our IPTW procedure is designed to balance our cohort on several covariates that may confound the association between cognitive function and NSES. Therefore, we include a number of individual-level covariates in our IPTW model including time invariant covariates such as gender, race/ethnicity, education, and age at baseline. Furthermore, we include time-varying values of sociodemographics including household wealth and retirement status, health behaviors including physical activity and alcohol use, and the presence of health conditions such as stroke, heart problems, high blood pressure, diabetes, chronic conditions, and depressive symptoms.

3.3 |. Descriptive statistics of the data

Table 1 provides descriptive statistics of covariates when calculated across the cohort used in analysis. It is also prudent to consider relevant characteristics of our cognitive outcome (TICS) and its unadjusted relationship with our exposure variable of interest (NSES) given the concerns we have about how this relationship might be biased by observed confounders in the HRS data set. First, there is a steady decline in cognitive function over time due to aging. The average TICS score across our analytic cohort in 2000 was 16.7; in 2010, the average score declined to 14.8. Figure 1 shows scatter plots of TICS vs NSES for each year from 2000 to 2010. It highlights that within each year there is an unadjusted positive association between TICS and NSES (that is relatively stable over time), suggesting that people living in higher NSES neighborhoods tend to have higher mean cognitive function. The correlation between these two variables is 0.269 in 2000 and is 0.274 in 2010. Note that the decline in mean TICS is small in comparison to the variability inherent in the outcome.

TABLE 1.

Characteristics of the Health and Retirement Study cohort (n = 4253) at baseline (2000) and the final follow-up (2010). ns are not weighted; percentages and means (with standard errors) are weighted to account for sample design and attrition

	Baseline (2000)		Last follow-up (2010)
	n/Mean	%/SE	n/Mean	%/SE
SOCIODEMOGRAPHICSs
Age (baseline)
< 62 Years	1546	35.2%	—	—
62–64 Years	1234	28.2%	—	—
65–67 Years	1038	25.4%	—	—
68+ Years	435	11.2%	—	—
Years (mean/se)	63.15	0.06	—	—
Gender
Male	1698	42.3%	—	—
Female	2555	57.8%	—	—
Race
African-American	587	7.2%	—	—
Hispanic	347	5.0%	—	—
Other	3319	87.8%	—	—
Education
0–11 Years	916	19.0%	—	—
12 Years	1589	37.7%	—	—
13–16 Years	1265	30.9%	—	—
17+ Years	483	12.5%	—	—
Years of education (mean/se)	12.8	0.05	—	—
Household wealth
Quintile 1 (poorest)	609	13.7%	477	10.5%
Quintile 2	771	16.2%	710	15.4%
Quintile 3	880	19.9%	881	19.7%
Quintile 4	950	23.6%	1007	24.1%
Quintile 5 (wealthiest)	1043	26.7%	1178	30.3%
Retired
No	2874	66.0%	1214	26.8%
Yes	1379	34.0%	3039	73.2%
HEALTH BEHAVIORS
Vigorous physical activity
No	2099	50.2%	3279	77.2%
Yes	2154	49.8%	974	22.9%
Alcohol use (no. of days/week with drinks)
0 Days	2909	66.0%	2822	64.5%
1–4 Days	904	22.2%	930	22.3%
5+ Days	440	11.9%	501	13.3%
HEALTH CONDITIONS
Stroke	137	3.7%	411	10.2%
Heart problems	579	15.6%	1341	33.6%
High blood pressure	1837	42.6%	2952	68.4%
Diabetes	510	12.5%	1156	27.0%
Chronic conditions (mean/se)	1.52	0.02	2.65	0.03
Depressive symptoms (CES-D ≥ 3)a	766	18.8%	786	19.5%

^aDepressive symptoms were measured using a Center for Epidemiologic Studies-Depression Scale score of 3 or greater.

FIGURE 1.

Scatter plots of telephone interview for cognitive status (TICS) versus neighborhood socioeconomic status (NSES) in 2000 (far left) through 2010 (far right) with best fitting lines in red [Colour figure can be viewed at wileyonlinelibrary.com]

Our doubly robust analysis is designed to assess whether this relationship continues to be maintained after we adjusted for confounding using IPTW and regression.

3.4 |. Assessing the quality of IPTW

Each of the six methods described in Section 2.1 are used to calculate weights for IPTW with the analytic HRS cohort. First, note that (as shown in Table A.10 of the supplemental materials) our cohort contains imbalance across the observed covariates considered here with respect to the exposure (NSES). For example, we see imbalance across race and wealth. However, the imbalance is not so strong as to suggest violations of the positivity assumption; that is, there is some degree of overlap across the categorizations of NSES with the various covariate categories.

Histograms of residuals from the sequence of conditional models may be used to assess the adequacy of the various methods of fitting univariate densities. As such, Figure 2 shows histograms of the studentized residuals from the conditional model for the NSES in 2010 when regressed on only the prior values of NSES (see the numerator model in the left panel) and then on the prior values of NSES along with the covariates (see the denominator model in the right panel). The fitted density curves from each of the four methods of univariate estimation are overlaid on both histograms. The supplementary materials contain analogous histograms that correspond to the other years. The histograms indicate that, as expected, the residuals have a large density near zero along with long tails (and excess kurtosis as a result). However, the mixture and univariate kernel methods appear to do the best job of capturing these characteristics.

FIGURE 2.

Histograms of the studentized residuals from the conditional models of the numerator and denominator seen in (1) for 2010 with density curves estimated from various methods [Colour figure can be viewed at wileyonlinelibrary.com]

Additionally, some brief descriptives of the weights calculated via all of the various methods for IPTW are provided in Table 2. Each set of weights is scaled to sum to the sample size prior to calculation of the descriptors. In the Table, the design effect, as approximated using the Kish approximation⁸⁶ $(\text{calculated} \text{as} \text{Kish} \text{DEFF} =n\sum w_i^2/{\left(\sum w_i\right)}^2)$, is listed for each set of weights. Note that a design effect quantifies the inflation in variance that results from weighting. The mixture and univariate kernel weights produce exorbitant design effects. The multivariate kernel yields the lowest design effect among the IPTW approaches.

TABLE 2.

Summary statistics of the various types of inverse probability of treatment weighting (IPTW) weights. All weights are scaled to sum to the sample size. Kish DEFF is the Kish approximation of the design effect

	IPTW
	Nonresp. Weights	Univariate Normal	t-dist.	Mixture	Uni. Kernel	Multi. Normal	Multi. Kernel
Mean	1.00	1.00	1.00	1.00	1.00	1.00	1.00
SD	0.52	8.22	3.90	39.97	45.69	3.93	1.84
Min	0.12	0.00	0.01	0.00	0.00	0.00	0.00
Max	5.20	334.42	159.90	2542.39	2951.76	199.36	57.75
DEFF	1.27	68.62	16.21	1598.03	2087.71	16.41	4.37

To assess whether IPTW provides balance for each estimation method considered, we calculate weighted correlations of NSES with covariates, which is in line with the suggestions of Zhu et al.¹⁷ Specifically, we calculate the sample correlation between the covariates and NSES with and without IPTW using the stacked data set (which has nT rows). If the weights are calculated correctly, the sample correlations will be close to zero. These correlations are shown in graphically in Figure 3, wherein box plots of the absolute value of the correlations when calculated using the various forms of IPTW are given. The correlations in Figure 3 are tabulated in the supplementary materials.

FIGURE 3.

Boxplots of absolute correlations with neighborhood socioeconomic status and covariates before and after applying inverse probability of treatment weighting (IPTW), where various methods are used for IPTW

Pre-IPTW weights are calculated using only nonresponse weights; moderate correlations with NSES existed prior to IPTW. For instance, NSES appears to be most heavily correlated with education (pre-IPTW correlation with NSES = 0.328), African American race (−0.313), and highest income category (0.312). IPTW with a multivariate kernel density appears to most effectively reduce the bulk of these correlations; therein, all weighted correlations are less than 0.129. The mixture and univariate kernel weights yield increased correlations; this may be due, in part, to the large design effects given by these weights. Note that balance assessment results are very similar when correlations are calculated using NSES-based terms that appear in our outcome models, (ie, $\text{cum}\left({\overline{\mathbf{X}}}_{t,i}\right)$).

3.5 |. Assessing the effect of NSES on cognitive function

We estimated the outcome models described earlier to determine the causal effects of NSES on cognitive function (TICS). We consider the two models outlined in Section 2.2: Model 1 (uses IPTW without controlling for covariates), and Model 2 (uses IPTW and controls for covariates). We also examine a version of Model 2 that does not use IPTW (but does use nonresponse weights)—an analogous version of Model 1 is not included since such a model would do nothing to account for confounding. Note that nonresponse weights factor into the calculation of IPTW weights as described in Section 2.1.2. That is, models used to calculate the IPTW weights are weighted via the nonresponse weights, and the IPTW-weighted models use the product of the nonresponse weights and the resulting IPTW weights as final weights. Results for the coefficients on cumulative NSES term are shown graphically in Figure 4 for each of the six methods of calculating IPTW. Complete results are provided in the supplementary materials for each method of IPTW.

FIGURE 4.

Coefficients for cumulative neighborhood socioeconomic status (with 95% confidence bands) for the six methods of inverse probability of treatment weighting (IPTW) and two outcome models

In the no-IPTW version of Model 2, we see a statistically significant positive effect cumulative NSES (regression coefficient = 0.049 and 95% CI = [0.024, 0.073]). This may suggest that living in more advantaged neighborhoods is beneficial for cognitive function. However, when IPTW with a multivariate kernel density is used (in either Model 1 or 2), this effect is reduced in magnitude and is no longer statistically significant (regression coefficient = 0.031 and 95% CI = [−.011, 0.072] for Model 2). The lack of statistical significance may be partly due to an increase in standard error that is caused by IPTW (ie, the weights induce a design effect).

Our assessments of the quality of the weights (which include balance assessments and simulations shown in the following section) indicated that the multivariate kernel weights perform the best. Therefore, the outcomes models for the other weights should not be used for inference and are provided only for the purposes of comparison. That said, Figure 4 illustrates that IPTW with poor weights can provide profoundly misleading results. Specifically, both Models 1 and 2 indicate a statistically significant negative effect of NSES on cognitive function when univariate normal weights are used. Additionally, mixture distribution weights indicate a strong positive effect in Models 1 and 2 that is more than six times as large as the effect seen in the version of Model 2 that excludes IPTW. We see an estimated effect that is nearly as large when the univariate kernel weights are used. Note that some of these large effects are not statistically significant due to the design effect induced by the respective kernel weights. Furthermore, the multivariate normal weights show a negative effect of NSES that is statistically significant in Model 1. In results presented here, weights found using a t-distribution provide findings that are fairly in line with those yielded by multivariate kernel weights. However, when the disaggregation model is considered in the supplemental materials, t-distribution weights yield potentially spurious findings.

To provide intuition behind the magnitude of the estimated effect of cumulative NSES on cognitive function, we present effect sizes as follows. We calculate the difference in the expected TICS in 2010 for an individual in the 10th percentile of cumulative NSES (in 2010) with the corresponding expected outcome for someone in the 90th percentile of NSES. This difference is divided by the standard deviation across our cohort of TICS in 2010 to yield the approximated effect size. In Model 1 with the multivariate kernel weights, we see a small effect size of 0.095. However, for other types of weights, this value is more substantial (eg, −0.564 for the univariate normal weights and 0.923 for the mixture weights).

It is important to revisit the key assumptions of IPTW analyses with MSMs. Diagnostics shown in the supplementary materials are consistent with the assumption of positivity holding among our cohort. A key limitation to our analyses and any analysis using IPTW is the assumption of no unobserved confounding. While our models control for a large set of key factors related to cognition and NSES (eg, age and health comorbidities), it is possible that our IPTW weights are missing a key factor. The best way to address the potential impacts of an unobserved confounder is through the use of sensitivity analyses. Typically, these sensitivity analyses aim to understand how strong an unobserved confounder would have to be to wipe away a significant finding.^66,87–89 In our case, we have no evidence of a significant impact of NSES on cognition. Thus, we would aim to understand how strong the relationship of an unobserved confounder would have to be to move our estimated effect into being a statistically significant finding. Such efforts are nontrivial in the continuous exposure setting and with repeated measures data. Future research should aim to develop such methods that allow for easier testing of the potential role unobserved confounding might have on these analyses.

4 |. SIMULATIONS

In this section, we present a simulation study that is used to evaluate the effectiveness of the methods proposed herein. We attempt to simulate data in the vein of the neighborhood data studied in Section 3. Specifically, the simulated values of a data unit’s exposure are the same as seen in the prior time period unless that individual was determined to have “moved.”

For each iteration of this simulation study, we generate synthetic versions of the exposure (NSES) and the outcome (TICS) as functions of the real values of a time invariant covariate and a time varying covariate from the HRS data. As such, our sample size (n = 4, 253) and number of time periods (T = 6) are the same as in our HRS cohort. For the time invariant covariate, which is denoted by Z_0,i for individual i, we use years of education. For the time-varying covariate (denoted by Z_t,i for individual i), we use household wealth (since household wealth has five categories, Z_t,i is four-dimensional after dropping a reference category).

The baseline value of synthetic NSES is generated using

$$X_{1,i}=0.15+{\lambda }_1{\mathbf{Z}}_{0,i}+{\mathbf{\lambda }}^{\prime }{\mathbf{Z}}_{1,i}+0.50V_{1,i}.$$ (9)

In (9), we set λ₁ = 0.17 and λ = (−0.11, −0.83, 0.07, 0.11)′, and V_t,i for t = 1, …, T is a t-distributed random variable with 3 degrees of freedom that has been rescaled to have unit variance. Then, for t > 1, we generate a binary random variable that indicates if individual i moves neighborhoods between time t − 1 and t. That is, a move occurs if R_i,t = 1, where P(R_i,t = 1) = 𝜋_i,t with

$$\log \left\{{\pi }_{i,t}/\left(1-{\pi }_{i,t}\right)\right\}=\varphi +0.18{\mathbf{Z}}_{0,i}+{\kappa }^{\prime }{\mathbf{Z}}_{t,i}+0.09\sum _{j=1}^{t-1}{(1/2)}^{t-j-1}{X}_{j,i}.$$ (10)

We use κ = (−0.08, −0.06, 0.05, 0.08)′ and ϕ = −2.6 in (10). Then, we generate the time varying exposure using

$$X_{t,i}=\left\{\begin{array}{ll}{X}_{t,i}=0.15+{\lambda }_1{\mathbf{Z}}_{0,i}+{\mathbf{\lambda }}^{\prime }{\mathbf{Z}}_{t,i}+0.3{\sum }_{j=1}^{t-1}{(1/2)}^{t-j-1}{X}_{j,i}+0.50V_{t,i},\hfill & \text{if} \hspace{1em}{R}_{i,t}=1,\hfill \\ X_{t,i}=X_{t-1,i}+0.02V_{t,i}\hfill & \text{otherwise}.\hfill \end{array}\right.$$ (11)

In (11), we also set λ₁ = 0.17 and λ = (−0.11, −0.83, 0.07, 0.11)′ . Note that the year-by-year variation in NSES for the neighborhood (indicated by the standard deviation of 0.02 in the above formula) is markedly less than the variation in NSES for movers (given by 0.50).

Lastly, the synthetic value of cognition (TICS) is generated with

$$Y_{t,i}=14.79+1.47{\mathbf{Z}}_{0,i}+{\xi }^{\prime }{\mathbf{Z}}_{t,i}+3.80U_{t,i},$$

for t = 1, …, T, where $\xi =(-0.36,-0.26,0.24,0.37{)}^{\prime }$ and where U_t,i has the same distribution as V_t,i. Note that all V_t,i and U_t,i are generated independently of one another for all t and i. The values of all regression coefficients used here are chosen so as to make the simulated data resemble as closely as possible the real HRS data subject to the above model structure. Note also that, under this setting, approximately 30% of individuals move at least once over the six time periods considered.

The data generation process described above is repeated independently for K = 1000 replications. Within each replication, the balance assessment process applied in the prior sections is used. That is, for all six weighting methods outlined in Section 2.1, we calculate the weighted correlation between the exposure (X_t,i) and the time-invariant covariate (Z_0,t) as well as the time-varying covariate (Z_t,i) using the tall (or stacked) version of the simulated data. For comparison, we also provide the unweighted correlations. When kernel densities are used, bandwidth parameters are determined in the same manner as described in the prior sections. Boxplots of the estimated correlations across all K replications are provided in Figure 5.

FIGURE 5.

Box plots of the simulated unweighted and weighted correlations between the exposure and the time-invariant covariate, education (top panel), in addition to the correlations between the exposure and a time-varying covariate (bottom panel), the highest wealth category, across K = 1000 replications [Colour figure can be viewed at wileyonlinelibrary.com]

Within each of the K replications, we estimate Models 1 and 2 (ie, weighted versions of (7) and (8), respectively) as described in Section 2.2. Individual-level random effects and temporal trend are, however, excluded here. For each coefficient that corresponds to a term involving the exposure variable within each model, we calculate bias and root-mean squared error as

$$\text{ bias }=\frac{1}{K}\sum _{k=1}^K{\widehat{\eta }}^{[k]}\hspace{1em}\text{and}\hspace{1em}\text{rMSE}=\frac{1}{K}\sum _{k=1}^K{\left({\widehat{\eta }}^{[k]}\right)}^2,$$

respectively, where ${\widehat{\eta }}^{[k]}$ is the value of a coefficient η as estimated within the kth iteration. Note that there should be no relationship between the outcome and the exposure that is not explained by covariates; ergo, for all relevant coefficients, η = 0. We calculate bias and rMSE for unweighted models and models calculated using the six weighting methods. Simulated bias and rMSE are shown in Table 3.

TABLE 3.

Bias and rMSE in estimators of a regression coefficient for various models and weighting methods across K = 1000 simulated data sets

	Biasa		rMSE
	Model 1	Model 2	Model 1	Model 2
Unweighted	0.3424	−0.0444	0.3429	0.0459
Normal	−0.0289	0.0070	0.1946	0.2023
t-dist.	−0.1016	−0.0028	0.1452	0.0569
Mixture	−0.2263	−0.0236	0.3754	0.3728
Uni. Kernel	−0.2450	−0.0275	0.3873	0.3692
M.V. Normal	−0.0284	0.0081	0.1893	0.2018
M.V. Kernel	0.0761	−0.0307	0.0814	0.0346

^aNote that all estimated biases for Models 1 and 2 are statistically significant at the 5% level

The results presented show that the weights calculated via a multivariate kernel outperform all other methods considered in the setting of this simulation. Specifically, Figure 5 illustrates that weights from a multivariate kernel reduce the correlation far more effectively than the other methods. Likewise, the multivariate kernel weights yield markedly smaller rMSE than all the other methods. In this simulation, controlling for covariates within the regression models (as in Model 2) appears to remove nearly all bias (which may not be the case in practice) with the multivariate kernel having rMSE that is a fraction of the rMSE of the other methods. Furthermore, note that the multivariate kernel tends to show slightly larger rMSE than the unweighted regression for Model 2. This is a consequence of variance inflation due to weighting and is an inherent trade-off in using IPTW to remove bias from observed confounders in estimation of causal effects.

Lastly, note that no method performs perfectly in this simulation study (all methods produce some degree of bias or excessive error as seen in Figure 5 and Table 3). This is, in part, because the data simulated here have very unique characteristics that no method is expected to capture perfectly. In fact, when comparing to the results in Section 3, we see that the simulated data may be harder for these methods to handle than the real data. For added rigor, the supplemental materials present an expanded version of this simulation study. In particular, simulation parameters are varied so as to alter the strength of the relationship between the exposure and covariates and to alter the frequency with which individuals move to new neighborhoods.

5 |. DISCUSSION

By proposing the use of a multivariate kernel density function in IPTW estimation, our work made contributions to statistical methods for causal inference utilizing IPTW and time-varying exposures. A multivariate kernel density allows one to circumvent the need to estimate a sequence of linear conditional models (one for each time point) as proposed by Robins.¹ This greatly reduces the degree to which parameterization is needed in estimation of the weights. The kernel removes the need for the treatment (or exposure) at one time point to be a linear function of treatment at prior time points. This type of procedure is particularly necessary for modeling neighborhood-level exposures since an individual’s neighborhood characteristics are mostly invariant over time (therefore, models that condition one time period’s exposure on the prior period’s will tend to have nonstandard probabilistic distributions). When competing methods (including traditional procedures) for IPTW with longitudinal data were applied to the cohort studied in our data analysis, the resulting weights provided poor balance. The use of a multivariate kernel density yielded dramatic improvement.

With more standard data, it is most likely that there will be some cases where employing a multivariate kernel density will be useful and others where it will not be (eg, when Gaussian assumptions do not hold). We believe that it is important for researchers to have multiple tools in their back pocket when trying to estimate the causal effects of continuous exposures or treatments. The continuous exposure problem is a fundamentally harder causal problem to tackle than the binary or multigroup settings. In practice, researchers should try their preferred approach and see if they have obtained good balance using the selected approach. If good balance is not obtained, the researcher should then utilize additional estimators to see if a better balance might be obtained. Thus, we believe it is useful for researchers to understand that both the more standard approach of using a sequence of linear conditional models and the multivariate kernel density method have merits for more standard applications not involving neighborhood data.

Furthermore, IPTW with a multivariate kernel removed the statistical significance of an estimated positive effect of neighborhood on cognitive function. However, when weights are calculated with competing methods, one observes markedly different effects (in some cases, we saw a strong positive effect of neighborhood; in others, the effect was negative and statistically significant).

Note that as is standard in other literature that uses IPTW within neighborhood analyses, eg, in the works of Cerdá et al⁵⁰ and Do et al,⁵¹ we do not consider neighborhood-level clustering within the estimation process for IPTW. This is reasonable in our applications since our data set is not large enough to contain substantial overlap of neighborhoods across cases but methodological adjustments akin to those suggested by Zubizarreta and Keele⁹⁰ to incorporate such data characteristics may be worth exploring in future work.

To summarize, our work emphasizes the need for doubly robust estimation in neighborhood research. In our application, we saw evidence that the traditional regression model alone was not sufficient for capturing the complex associations that exist in longitudinal neighborhood-based data. The use of IPTW via a multivariate kernel density shows promise as a means for capturing these associations. Therefore, we recommend that the methods outlined here be considered whenever an analyst aims to draw causal inferences within neighborhood research on health with longitudinal, observational data sets.

DATA AVAILABILITY STATEMENT

Data are not publicly available, but pseudo-data are available upon contacting the authors.