Matching with Sampling Weights
In MatchIt: Nonparametric Preprocessing for Parametric Causal Inference

```{=html}

```r
knitr::opts_chunk$set(echo = TRUE, eval=T)
options(width = 200, digits = 4)

#Generating data similar to Austin (2009) for demonstrating treatment effect estimation with sampling weights
gen_X <- function(n) {
  X <- matrix(rnorm(9 * n), nrow = n, ncol = 9)
  X[,5] <- as.numeric(X[,5] < .5)
  X
}

#~20% treated
gen_A <- function(X) {
  LP_A <- - 1.2 + log(2)*X[,1] - log(1.5)*X[,2] + log(2)*X[,4] - log(2.4)*X[,5] + log(2)*X[,7] - log(1.5)*X[,8]
  P_A <- plogis(LP_A)
  rbinom(nrow(X), 1, P_A)
}

# Continuous outcome
gen_Y_C <- function(A, X) {
  2*A + 2*X[,1] + 2*X[,2] + 2*X[,3] + 1*X[,4] + 2*X[,5] + 1*X[,6] + rnorm(length(A), 0, 5)
}
#Conditional:
#  MD: 2
#Marginal:
#  MD: 2

gen_SW <- function(X) {
  e <- rbinom(nrow(X), 1, .3)
  1/plogis(log(1.4)*X[,2] + log(.7)*X[,4] + log(.9)*X[,6] + log(1.5)*X[,8] + log(.9)*e +
             -log(.5)*e*X[,2] + log(.6)*e*X[,4])
}

set.seed(19599)

n <- 2000
X <- gen_X(n)
A <- gen_A(X)
SW <- gen_SW(X)

Y_C <- gen_Y_C(A, X)

d <- data.frame(A, X, Y_C, SW)

Introduction

Sampling weights (also known as survey weights) frequently appear when using large, representative datasets. They are required to ensure any estimated quantities generalize to a target population defined by the weights. Evidence suggests that sampling weights need to be incorporated into a propensity score matching analysis to obtain valid and unbiased estimates of the treatment effect in the sampling weighted population [@dugoff2014; @austin2016; @lenis2019]. In this guide, we demonstrate how to use sampling weights with MatchIt for propensity score estimation, balance assessment, and effect estimation. Fortunately, doing so is not complicated, but some care must be taken to ensure sampling weights are incorporated correctly. It is assumed one has read the other vignettes explaining matching (vignette("matching-methods")), balance assessment (vignette("assessing-balance")), and effect estimation (vignette("estimating-effects").

We will use the same simulated toy dataset used in vignette("estimating-effects") except with the addition of a sampling weights variable, SW, which is used to generalize the sample to a specific target population with a distribution of covariates different from that of the sample. Code to generate the covariates, treatment, and outcome is at the bottom of vignette("estimating-effects") and code to generate the sampling weights is at the end of this document. We will consider the effect of binary treatment A on continuous outcome Y_C, adjusting for confounders X1-X9.

head(d)

library("MatchIt")

Matching

When using sampling weights with propensity score matching, one has the option of including the sampling weights in the model used to estimate the propensity scores. Although evidence is mixed on whether this is required [@austin2016; @lenis2019], it can be a good idea. The choice should depend on whether including the sampling weights improves the quality of the matches. Specifications including and excluding sampling weights should be tried to determine which is preferred.

To supply sampling weights to the propensity score-estimating function in matchit(), the sampling weights variable should be supplied to the s.weights argument. It can be supplied either as a numerical vector containing the sampling weights, or a string or one-sided formula with the name of the sampling weights variable in the supplied dataset. Below we demonstrate including sampling weights into propensity scores estimated using logistic regression for optimal full matching for the average treatment effect in the population (ATE) (note that all methods and steps apply the same way to all forms of matching and all estimands). ```{asis, echo = !requireNamespace("optmatch", quietly = TRUE)}

Note: if the optmatch, marginaleffects, or sandwich packages are not available, the subsequent lines will not run.

```r
#In case packages goes offline, don't run lines below
if (!requireNamespace("optmatch", quietly = TRUE) ||
    !requireNamespace("marginaleffects", quietly = TRUE) ||
    !requireNamespace("sandwich", quietly = TRUE)) knitr::opts_chunk$set(eval = FALSE)

mF_s <- matchit(A ~ X1 + X2 + X3 + X4 + X5 + 
                  X6 + X7 + X8 + X9, data = d,
                method = "full", distance = "glm",
                estimand = "ATE", s.weights = ~SW)
mF_s

Notice that the description of the matching specification when the matchit object is printed includes lines indicating that the sampling weights were included in the estimation of the propensity score and that they are present in the matchit object. It is stored in the s.weights component of the matchit object. Note that at this stage, the matching weights (stored in the weights component of the matchit object) do not incorporate the sampling weights; they are calculated simply as a result of the matching.

Now let's perform full matching on a propensity score that does not include the sampling weights in its estimation. Here we use the same specification as was used in vignette("estimating-effects").

mF <- matchit(A ~ X1 + X2 + X3 + X4 + X5 + 
                X6 + X7 + X8 + X9, data = d,
              method = "full", distance = "glm",
              estimand = "ATE")
mF

Notice that there is no mention of sampling weights in the description of the matching specification. However, to properly assess balance and estimate effects, we need the sampling weights to be included in the matchit object, even if they were not used at all in the matching. To do so, we use the function add_s.weights(), which adds sampling weights to the supplied matchit objects.

mF <- add_s.weights(mF, ~SW)

mF

Now when we print the matchit object, we can see lines have been added identifying that sampling weights are present but they were not used in the estimation of the propensity score used in the matching.

Note that not all methods can involve sampling weights in the estimation. Only methods that use the propensity score will be affected by sampling weights; coarsened exact matching or Mahalanobis distance optimal pair matching, for example, ignore the sampling weights, and some propensity score estimation methods, like randomForest and bart (as presently implemented), cannot incorporate sampling weights. Sampling weights should still be supplied to matchit() even when using these methods to avoid having to use add_s.weights() and remembering which methods do or do not involve sampling weights.

Assessing Balance

Now we need to decide which matching specification is the best to use for effect estimation. We do this by selecting the one that yields the best balance without sacrificing remaining effective sample size. Because the sampling weights are incorporated into the matchit object, the balance assessment tools in plot.matchit() and summary.matchit() incorporate them into their output.

We'll use summary() to examine balance on the two matching specifications. With sampling weights included, the balance statistics for the unmatched data are weighted by the sampling weights. The balance statistics for the matched data are weighted by the product of the sampling weights and the matching weights. It is the product of these weights that will be used in estimating the treatment effect. Below we use summary() to display balance for the two matching specifications. No additional arguments to summary() are required for it to use the sampling weights; as long as they are in the matchit object (either due to being supplied with the s.weights argument in the call to matchit() or to being added afterward by add_s.weights()), they will be correctly incorporated into the balance statistics.

#Balance before matching and for the SW propensity score full matching
summary(mF_s)

#Balance for the non-SW propensity score full matching
summary(mF, un = FALSE)

The results of the two matching specifications are similar. Balance appears to be slightly better when using the sampling weight-estimated propensity scores than when using the unweighted propensity scores. However, the effective sample size for the control group is larger when using the unweighted propensity scores. Neither propensity score specification achieves excellent balance, and more fiddling with the matching specification (e.g., by changing the method of estimating propensity scores, the type of matching, or the options used with the matching) might yield a better matched set. For the purposes of this analysis, we will move forward with the matching that used the sampling weight-estimated propensity scores (mF_s) because of its superior balance. Some of the remaining imbalance may be eliminated by adjusting for the covariates in the outcome model.

Note that had we not added sampling weights to mF, the matching specification that did not include the sampling weights, our balance assessment would be inaccurate because the balance statistics would not include the sampling weights. In this case, in fact, assessing balance on mF without incorporated the sampling weights would have yielded radically different results and a different conclusion. It is critical to incorporate sampling weights into the matchit object using add_s.weights() even if they are not included in the propensity score estimation.

Estimating the Effect

Estimating the treatment effect after matching is straightforward when using sampling weights. Effects are estimated in the same way as when sampling weights are excluded, except that the matching weights must be multiplied by the sampling weights to yield accurate, generalizable estimates. match.data() and get_matches() do this automatically, so the weights produced by these functions already are a product of the matching weights and the sampling weights. Note this will only be true if sampling weights are incorporated into the matchit object.

Below we estimate the effect of A on Y_C in the matched and sampling weighted sample, adjusting for the covariates to improve precision and decrease bias.

md_F_s <- match.data(mF_s)

fit <- lm(Y_C ~ A * (X1 + X2 + X3 + X4 + X5 + 
             X6 + X7 + X8 + X9), data = md_F_s,
          weights = weights)

library("marginaleffects")
avg_comparisons(fit,
                variables = "A",
                vcov = ~subclass,
                newdata = subset(md_F_s, A == 1),
                wts = "weights")

Note that match.data() and get_weights() have the option include.s.weights, which, when set to FALSE, makes it so the returned weights do not incorporate the sampling weights and are simply the matching weights. Because one might to forget to multiply the two sets of weights together, it is easier to just use the default of include.s.weights = TRUE and ignore the sampling weights in the rest of the analysis (because they are already included in the returned weights). avg_comparisons() also works more smoothly when the weights supplied to weights is a single variable rather than the product of two.

knitr::opts_chunk$set(eval = TRUE)

Code to Generate Data used in Examples

#Generatng data similar to Austin (2009) for demonstrating 
#treatment effect estimation with sampling weights
gen_X <- function(n) {
  X <- matrix(rnorm(9 * n), nrow = n, ncol = 9)
  X[,5] <- as.numeric(X[,5] < .5)
  X
}

#~20% treated
gen_A <- function(X) {
  LP_A <- - 1.2 + log(2)*X[,1] - log(1.5)*X[,2] + log(2)*X[,4] - log(2.4)*X[,5] + 
    log(2)*X[,7] - log(1.5)*X[,8]
  P_A <- plogis(LP_A)
  rbinom(nrow(X), 1, P_A)
}

# Continuous outcome
gen_Y_C <- function(A, X) {
  2*A + 2*X[,1] + 2*X[,2] + 2*X[,3] + 1*X[,4] + 2*X[,5] + 1*X[,6] + rnorm(length(A), 0, 5)
}
#Conditional:
#  MD: 2
#Marginal:
#  MD: 2

gen_SW <- function(X) {
  e <- rbinom(nrow(X), 1, .3)
  1/plogis(log(1.4)*X[,2] + log(.7)*X[,4] + log(.9)*X[,6] + log(1.5)*X[,8] + log(.9)*e +
             -log(.5)*e*X[,2] + log(.6)*e*X[,4])
}

set.seed(19599)

n <- 2000
X <- gen_X(n)
A <- gen_A(X)
SW <- gen_SW(X)

Y_C <- gen_Y_C(A, X)

d <- data.frame(A, X, Y_C, SW)