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# Impact of Data Order for Propensity Score Matching

## Jason M. Bryer [jason@bryer.org](mailto:jason@bryer.org)
#### Project website: [jason.bryer.org/PSAboot](http://jason.bryer.org/PSAboot)
#### Bug Reports: [github.com/jbryer/PSAboot/issues](https://github.com/jbryer/PSAboot/issues)

```{r, echo=FALSE, results='hide'}
set.seed(2112)
```


Propensity score matching (PSM; Rosenbaum & Rubin, 1983) has become a popular approach for adjuting for selection bias in observational studies. However, recent studies have shown that under certain circumstances PSM may increase bias. In particular, Lunt (2014) showed that the order in which treated units are matched, along with caliper specifications, can impact the estimates. This paper explores the effects of matching order using the `PSAboot` package for bootstrapping propensity score analysis using two of the more popular R packages for matching, `MatchIt` (Ho, Imai, King, & Stuart, 2011) and `Matching` (Sekhon, 2011).

Set the number of bootstrap samples. This should be set to at least 100 but kept small to reduce the execution time for CRAN submissions.

```{r}
boot.M = 10
```

### Matching Procedures


```{r functions}
boot.matchit.random <- function(Tr, Y, X, X.trans, formu, ...) {
	boot.matchit(Tr=Tr, Y=Y, X=X, X.trans=X.trans, formu=formu, m.order='random', ...)
}

boot.matching.random <- function(Tr, Y, X, X.trans, formu, ...) {
	boot.matching(Tr=Tr, Y=Y, X=X, X.trans=X.trans, formu=formu, replace=FALSE)
}

SimpleMatch <- function(Tr, Y, X, X.trans, formu, caliper=0.25, ...) {
	if(!is.logical(Tr)) {
		Tr <- as.logical(Tr)
	}
	formu <- update.formula(formu, 'treat ~ .')
	ps <- fitted(glm(formu, data=cbind(treat=Tr, X), family=binomial(logit)))
	matches <- data.frame(Treat=which(Tr), Treat.Y=Y[Tr], Treat.ps=ps[Tr],
						  Control=as.integer(NA), Control.Y=as.numeric(NA), 
						  Control.ps=as.numeric(NA))
	available.Control <- !Tr
	for(i in which(Tr)) {
		d <- abs(ps[i] - ps[!Tr & available.Control])
		if((min(d) / sd(ps)) < caliper)
			m <- which(!Tr & available.Control)[which(d == min(d))]
		if(length(m) > 1) {
			m <- m[1]
		}
		if(length(m) > 0) {
			matches[matches$Treat == i,]$Control <- m
			matches[matches$Treat == i,]$Control.Y <- Y[m]
			matches[matches$Treat == i,]$Control.ps <- ps[m]
			available.Control[m] <- FALSE
		}
	}
	match.t <- t.test(matches$Treat.Y, matches$Control.Y, paired=TRUE)

	return(list(
		summary=c(estimate=unname(match.t$estimate),
				  ci.min=match.t$conf.int[1],
				  ci.max=match.t$conf.int[2],
				  p=match.t$p.value,
				  t=unname(match.t$statistic)),
		details=c(Matches=matches, t.test=match.t),
		balance=balance.matching(matches$Treat, matches$Control, X.trans) ))
}
```



```{r setup, echo=FALSE, results='hide', message=FALSE}
library(PSAboot)
library(multilevelPSA)
library(reshape2)
library(ggplot2)
```

### Example 1: Lalonde

First example will use the classic `lalonde` data (LaLonde, 1986; Dehejia & Wahba, 1999).

```{r laonde}
data("lalonde", package='Matching')
```

Typically, bootstrapping draws `M` random samples with replacement. However, by setting the `control.replace` and `treated.replace` parameters to `FALSE` and the bootstrap sample sizes equal to the number of observations we can evaluate the impact of ordering.

```{r lalonde.psaboot, cache=FALSE, warning=FALSE}
lalonde.boot <- PSAboot(Tr=lalonde$treat,
						Y=lalonde$re78,
						X=lalonde[,c(1:8)],
						seed=2112,
						M=boot.M,
						control.sample.size = 260, control.replace = FALSE,
						treated.sample.size = 185, treated.replace = FALSE,
						methods=c(getPSAbootMethods()[c('Matching','MatchIt')],
								  'MatchingRandom'=boot.matching.random,
								  'MatchItRandom'=boot.matchit.random,
								  'NearestNeighbor'=SimpleMatch))
```

Boxplot of estimated effect sizes.

```{r lalonde-boxplot, fig.width=12, fig.height=4.0, warning=FALSE, message=FALSE}
boxplot(lalonde.boot)
```

```{r lalonde-balance, fig.width=12, fig.height=4, warning=FALSE}
lalonde.bal <- balance(lalonde.boot)
tmp.bal <- melt(lalonde.bal$pooled)
tmp.est <- lalonde.boot$pooled.summary[,c('iter','method','estimate')]
tmp <- merge(tmp.bal, tmp.est, by.x=c('Var1','Var2'), by.y=c('iter','method'))
ggplot(tmp, aes(x=value, y=estimate, group=Var2)) + geom_point(alpha=.5) + 
	facet_wrap(~ Var2, nrow=1) + xlab('Balance') + ylab('Estimate')
```

### Exmaple 2: Tutoring

```{r tutoring}
data(tutoring, package='TriMatch')
tutoring$treatbool <- tutoring$treat != 'Control'
```


```{r tutoring-psaboot, cache=FALSE, warning=FALSE}
tutoring.boot <- PSAboot(Tr=tutoring$treatbool, 
						 Y=tutoring$Grade, 
						 X=tutoring[,c('Gender', 'Ethnicity', 'Military', 'ESL',
						 			  'EdMother', 'EdFather', 'Age', 'Employment',
						 			  'Income', 'Transfer', 'GPA')], 
						 seed=2112,
						 M=boot.M,
						 control.sample.size=918, control.replace=FALSE,
						 treated.sample.size=224, treated.replace=FALSE,
						 methods=c(getPSAbootMethods()[c('Matching','MatchIt')],
						 		  'MatchingRandom'=boot.matching.random,
						 		  'MatchItRandom'=boot.matchit.random,
						 		  'NearestNeighbor'=SimpleMatch))
```


```{r tutoring-boxplot, fig.width=12, fig.height=4.0, warning=FALSE, message=FALSE}
boxplot(tutoring.boot)
```

```{r tutoring-balance, fig.width=12, fig.height=4, warning=FALSE}
tutoring.bal <- balance(tutoring.boot)
tmp.bal <- melt(tutoring.bal$pooled)
tmp.est <- tutoring.boot$pooled.summary[,c('iter','method','estimate')]
tmp <- merge(tmp.bal, tmp.est, by.x=c('Var1','Var2'), by.y=c('iter','method'))
ggplot(tmp, aes(x=value, y=estimate, group=Var2)) + geom_point(alpha=.5) + 
	facet_wrap(~ Var2, nrow=1) + xlab('Balance') + ylab('Estimate')
```

### References

Daniel E. Ho, Kosuke Imai, Gary King, Elizabeth A. Stuart (2011). MatchIt: Nonparametric Preprocessing for Parametric Causal Inference. Journal of Statistical Software, Vol. 42, No. 8, pp. 1-28. URL http://www.jstatsoft.org/v42/i08/
  
Jasjeet S. Sekhon (2011). Multivariate and Propensity Score Matching Software with Automated Balance Optimization: The Matching Package for R. Journal of Statistical Software, 42(7), 1-52. URL http://www.jstatsoft.org/v42/i07/.

Lunt, M. (2014). Selecting an appropriate caliper can be essential for achieving good balance with propensity score matching. Practice of Epidemiology, 179(2), 226-235.


