# R Macros Tutorial

### Introduction

The Toolkit for Weighting and Analysis of Nonequivalent Groups, twang, contains a set of functions and procedures to support causal modeling of observational data through the estimation and evaluation of propensity scores and associated weights. This package was developed in 2004. After extensive use, it received a major update in 2012. This tutorial provides an introduction to twang and demonstrates its use through illustrative examples.

The foundation to the methods supported by twang is the propensity score. The propensity score is the probability that a particular case would be assigned or exposed to a treatment condition. Rosenbaum & Rubin (1983) showed that knowing the propensity score is sufficient to separate the effect of a treatment on an outcome from observed confounding factors that influence both treatment assignment and outcomes, provided the necessary conditions hold. The propensity score has the balancing property that given the propensity score the distribution of features for the treatment cases is the same as that for the control cases. While the treatment selection probabilities are generally not known, good estimates of them can be effective at diminishing or eliminating confounds between pretreatment group differences and treatment outcomes in the estimation of treatment effects.

There are now numerous propensity scoring methods in the literature. They differ in how they estimate the propensity score (e.g. logistic regression, CART), the target estimand (e.g. treatment effect on the treated, population treatment effect), and how they utilize the resulting estimated propensity scores (e.g. stratification, matching, weighting, doubly robust estimators). We originally developed the twang package with a particular process in mind, namely, generalized boosted regression to estimate the propensity scores and weighting of the comparison cases to estimate the average treatment effect on the treated (ATT). However, we have updated the package to also meaningfully handle the case where interest lies in using the population weights (e.g., weighting of comparison and treatment cases to estimate the population average treatment effect, ATE). The main workhorse of twang is the ps() function which implements generalized boosted regression modeling to estimate the propensity scores. However, the framework of the package is flexible enough to allow the user to use propensity score estimates from other methods and to assess the usefulness of those estimates for ensuring equivalence (or “balance”) in the pretreatment covariate distributions of treatment and control groups using tools from the package. The same set of functions is also useful for other tasks such as non-response weighting, as discussed in Section 4.

The twang package aims to (i) compute from the data estimates of the propensity scores which yield accurate causal effect estimates, (ii) check the quality of the resulting propensity score weights by assessing whether or not they have the balancing properties that we expect in theory, and (iii) use them in computing treatment effect estimates.

The twang package and this tutorial were developed under NIDA grants R01 DA017507 and R01 DA015697-03

### An ATT example to start

If you have not already done so, install twang by typing install.packages("twang"). twang relies on other R packages, especially gbm, survey, and lattice. You may have to run install.packages() for these as well if they are not already installed. You will only need to do this step once. In the future running update.packages() regularly will ensure that you have the latest versions of the packages, including bug fixes and new features.

To start using twang, first load the package. You will have to do this step once for each R session that you run. We also set the seed of R’s pseudo random number generator so that the results are exactly replicable. (There is a stochastic element to the fitting of the propensity score models.)

> library(twang)

> set.seed(1)

To demonstrate the package we utilize data from Lalonde’s National Supported Work Demonstration analysis (Lalonde 1986, Dehejia & Wahba 1999, http://users.nber.org/~rdehejia/nswdata2.html). This dataset is provided with the twang package.

> data(lalonde)

R can read data from many other sources. The manual "R Data Import/Export," available at http://cran.r-project.org/doc/manuals/R-data.pdf, describes that process in detail.

For the lalonde dataset, the variable treat is the 0/1 treatment indicator, 1 indicates "treatment" by being part of the National Supported Work Demonstration and 0 indicates "comparison" cases drawn from the Current Population Survey. In order to estimate a treatment effect for this demonstration program that is unbiased by pretreatment group differences on other observed covariates, we include these covariates in a propensity score model of treatment assignment: age, education, black, Hispanic, having no degree, married, earnings in 1974 (pretreatment), and earnings in 1975 (pretreatment). Note that we specify no outcome variables at this time. The ps() function is the primary method in twang for estimating propensity scores. This step is computationally intensive and can take a few minutes.

> ps.lalonde <- ps(treat ~ age + educ + black + hispan + nodegree +

+ married + re74 + re75,

+ data = lalonde,

+ n.trees=5000,

+ interaction.depth=2,

+ shrinkage=0.01,

+ perm.test.iters=0,

+ stop.method=c("es.mean","ks.max"),

+ estimand = "ATT",

+ verbose=FALSE)

The arguments to ps() require some discussion. The first argument specifies a formula indicating that treat is the 0/1 treatment indicator and that the propensity score model should predict treat from the eight covariates listed there separated by "+". As when fitting linear models via the lm() function, "+" does *not *mean that these variables are being summed; in contrast to the lm() function, neither does it mean that the model is linear. This is just R’s notation for including predictor variables in the model. There is no need to specify interaction terms in the formula. There is also no need — and it can be counterproductive — to create indicator, or "dummy coded," variables to represent categorical covariates, provided the categorical variables are stored as a factor or as ordered (see help(factor) for more details).

The next argument, data, indicates the dataset. n.trees, interaction.depth, and shrinkage are parameters for the gbm model that ps() computes and stores. The resulting gbm object describes a family of candidate propensity score models indexed by the number of GBM iterations from one to n.trees. The argument n.trees is the maximum number of iterations that gbm will run. ps() will issue a warning if the estimated optimal number of iterations is too close to the bound selected in this argument because it indicates that balance may improve if more complex models (i.e., those with more trees) are considered. The user should increase n.trees or decrease shrinkage if this warning appears. The argument interaction.depth controls the level of interactions allowed in the GBM. The default is 3 and we typically use the default in our analyses. The GBM estimation algorithm uses shrinkage to enhance the smoothness of resulting model. The shrinkage argument controls the amount of shrinkage. Small values such as 0.005 or 0.001 yield smooth fits but require greater values of n.trees to achieve adequate fits. Computational time increases inversely with shrinkage argument. Additional details on n.trees, interaction.depth, and shrinkage can be found McCaffrey, Ridgeway, and Morral (2004).

perm.test.iters specifies whether *p*-values for KS statistics should be calculated using Monte Carlo methods, which is slow but can be accurate, or estimated using an analytic approximation that is fast, but produces poor estimates in the presence of many ties. If perm.test.iters=0 is called, then analytic approximations are used. If perm.test.iters=500 is called, then 500 Monte Carlo trials are run to establish the reference distribution of KS statistics for each covariate. Higher numbers of trials will produce more precise *p*-values. Specifying permtestiters greater than zero can greatly slow down the twang computations. We tend to rely on the approximations (permtestiters=0) when using twang in practice.

The estimand argument is used to indicate whether the analyst is interested in estimating the average treatment effect (ATE) or the average treatment effect on the treated (ATT), as we do above. ATE addresses the question of how outcomes would differ if everyone in the sample were given the treatment versus everyone being given the control (Wooldridge, 2002). ATT, on the other hand, estimates the analogous quantity averaging only over the subjects who were actually treated. The estimand argument was added to the 2012 revision of the package which integrated ATE weighting into the package and the ps function estimate of the propensity score.

The stop.method argument specifies a set (or sets) of rules and measures for assessing the balance, or equivalence, established on the pretreatment covariates of the weighted treatment and control groups. The ps function selects the optimal number of GBM iterations to minimize the differences between the treatment and control groups as measured by the rules of the given stop.method object. The package includes four built-in stop.method objects. They are es.mean, es.max, ks.mean, and ks.max. The four stopping rules are defined by two components: a balance metric for covariates and rule for summarizing across covariates. The balance metric summarizes the difference between two univariate distributions of a single pre-treatment variable (e.g., age). The default stopping rules in twang use two balance metrics: absolute standardized bias (also referred to as the absolute standardized mean difference or the *E *ffect *S *ize) and the Kolmogorov-Smirnov (KS) statistic. The stopping rule use two different rules for summarizing across covariates: the mean of the covariate balance metrics ("mean") or the maximum of the

balance metrics ("max"). The first piece of the stopping rule name identifies the balance metric (ES or KS) and the second piece specifies the method for summarizing across balance metrics. For instance, es.mean uses the effect size or the absolute standardized bias and summarizes across variables with the mean and the ks.max uses the KS statistics to assess balances and summarizes using the maximum across variables and the other two stopping rules use the remaining two combinations of balance metrics and summary statistics. The variable distributions used in the balance metrics depend on whether we are interested in estimating the ATT or ATE, and correct specification of these distributions is set automatically by the specification of the estimand in the ps() function.

Having fit the ps object, the analyst should perform several diagnostic checks before estimating the causal effect in question. The first of these diagnostic checks makes sure that the specified value of n.trees allowed GBM to explore sufficiently complicated models. We can do this quickly with the plot() function.1 As a default, the plot() function applied to a ps object gives the balance measures as a function of the number of iterations in the GBM algorithm, with higher iterations corresponding to more complicated fitted models. In the example below, 2127 iterations minimized the average effect size difference and 1756 iterations minimized the largest of the eight Kolmogorov-Smirnov (KS) statistics computed for the covariates. If it appears that additional iterations would be likely to result in lower values of the balance statistic, n.trees should be increased. However, after a point, additional complexity typically makes the balance worse, as in the example below. This figure also gives information on how compatible two or more stopping rules are: if the minima for multiple stopping rules under consideration are near one another, the results should not be sensitive to which stopping rule one uses for the final analysis. See Section 5.3 for a discussion of these and other balance measures.

> plot(ps.lalonde)

If we wish to focus on only one stopping rule, the plotting commands also take a subset argument.

> plot(ps.lalonde, subset = 2)

The gbm package has various tools for exploring the relationship between the covariates and the treatment assignment indicator if these are of interest. summary() computes the relative influence of each variable for estimating the probability of treatment assignment. The gbm estimates depend on the number of iterations at which the gbm model is evaluated, which is specified by the n.trees argument in the summary method for gbm. In this example, we choose the number of iterations to be the optimal number for minimizing the largest of the KS statistics. This value can be found in the ps.lalonde$desc$ks.max.ATT$n.trees. Figure 1 shows the barchart of the relative influence and is produced when plot=TRUE in the call to summary().

> summary(ps.lalonde$gbm.obj,

+ n.trees=ps.lalonde$desc$ks.max.ATT$n.trees,

+ plot=FALSE)

var | rel.inf | |

black | black | 57.83094183 |

age | age | 16.51591133 |

re74 | re74 | 15.61359263 |

re75 | re75 | 3.58011684 |

married | married | 3.02141880 |

educ | educ | 2.91846328 |

nodegree | nodegree | 0.43884552 |

hispan | hispan | 0.08070978 |

### 2.1 Assessing "balance" using balance tables

Having estimated the propensity scores, bal.table() produces a table that shows how well the resulting weights succeed in manipulating the control group so that its weighted pretreatment characteristics match, or balance, those of the unweighted treatment group if estimand = "ATT" or the control and treatment groups so that the weighted pretreatment characteristics match, or balance, with one another if estimand = "ATE". By default, the bal.table() function uses the value of estimand set with the ps() function call. For example, in the analysis we set estimand = "ATT" when calling ps() to estimate the propensity scores and the resulting ps.object, ps.lalonde, contains an element "estimand" which takes the value "ATT". The function bal.table() checks this value and automatically uses ATT weights when checking balance and comparing the distributions of pre-treatment variables for the weighted control group with those from the unweighted treatment group.

Figure 1: Relative influence of the covariates on the estimated propensity score.

> lalonde.balance <- bal.table(ps.lalonde)

> lalonde.balance

$unw

tx.mn | tx.sd | ct.mn | ct.sd | std.eff.sz | stat | p | ks | ks ks.pval | |

age | 25.816 | 7.155 | 28.030 | 10.787 | -0.309 | -2.994 | 0.003 | 0.158 | 0.003 |

educ | 10.346 | 2.011 | 10.235 | 2.855 | 0.055 | 0.547 | 0.584 | 0.111 | 0.074 |

black | 0.843 | 0.365 | 0.203 | 0.403 | 1.757 | 19.371 | 0.000 | 0.640 | 0.000 |

hispan | 0.059 | 0.237 | 0.142 | 0.350 | -0.349 | -3.413 | 0.001 | 0.083 | 0.317 |

nodegree | 0.708 | 0.456 | 0.597 | 0.491 | 0.244 | 2.716 | 0.007 | 0.111 | 0.074 |

married | 0.189 | 0.393 | 0.513 | 0.500 | -0.824 | -8.607 | 0.000 | 0.324 | 0.000 |

re74 | 2095.574 | 4886.620 | 5619.237 | 6788.751 | -0.721 | -7.254 | 0.000 | 0.447 | 0.000 |

re75 | 1532.055 | 3219.251 | 2466.484 | 3291.996 | -0.290 | -3.282 | 0.001 | 0.288 | 0.000 |

$es.mean.ATT

tx.mn | tx.sd | ct.mn | ct.sd | std.eff.sz | stat | p | ks | ks.pval | |

age | 25.816 | 7.155 | 25.802 | 7.279 | 0.002 | 0.015 | 0.988 | 0.122 | 0.892 |

educ | 10.346 | 2.011 | 10.573 | 2.089 | -0.113 | -0.706 | 0.480 | 0.099 | 0.977 |

black | 0.843 | 0.365 | 0.842 | 0.365 | 0.003 | 0.027 | 0.978 | 0.001 | 1.000 |

hispan | 0.059 | 0.237 | 0.042 | 0.202 | 0.072 | 0.804 | 0.421 | 0.017 | 1.000 |

nodegree | 0.708 | 0.456 | 0.609 | 0.489 | 0.218 | 0.967 | 0.334 | 0.099 | 0.977 |

married | 0.189 | 0.393 | 0.189 | 0.392 | 0.002 | 0.012 | 0.990 | 0.001 | 1.000 |

re74 | 2095.574 | 4886.620 | 1556.930 | 3801.566 | 0.110 | 1.027 | 0.305 | 0.066 | 1.000 |

re75 | 1532.055 | 3219.251 | 1211.575 | 2647.615 | 0.100 | 0.833 | 0.405 | 0.103 | 0.969 |

$ks.max.ATT

tx.mn | tx.sd | ct.mn | ct.sd | std.eff.sz | stat | p | ks | ks.pval | |

age | 25.816 | 7.155 | 25.764 | 7.408 | 0.007 | 0.055 | 0.956 | 0.107 | 0.919 |

educ | 10.346 | 2.011 | 10.572 | 2.140 | -0.113 | -0.712 | 0.477 | 0.107 | 0.919 |

black | 0.843 | 0.365 | 0.835 | 0.371 | 0.022 | 0.187 | 0.852 | 0.008 | 1.000 |

hispan | 0.059 | 0.237 | 0.043 | 0.203 | 0.069 | 0.779 | 0.436 | 0.016 | 1.000 |

nodegree | 0.708 | 0.456 | 0.601 | 0.490 | 0.235 | 1.100 | 0.272 | 0.107 | 0.919 |

married | 0.189 | 0.393 | 0.199 | 0.400 | -0.024 | -0.169 | 0.866 | 0.010 | 1.000 |

re74 | 2095.574 | 4886.620 | 1673.666 | 3944.600 | 0.086 | 0.800 | 0.424 | 0.054 | 1.000 |

re75 | 1532.055 | 3219.251 | 1257.242 | 2674.922 | 0.085 | 0.722 | 0.471 | 0.094 | 0.971 |

bal.table() returns information on the pretreatment covariates before and after weighting. The object is a list with named components, one for an unweighted analysis (named unw) and one for each stop.method specified, here es.mean and ks.max. McCaffrey et al (2004) essentially used es.mean for the analyses, but our more recent work has sometimes used ks.max. See McCaffrey et al. (2013) for a more detailed description of these choices.

If there are missing values (represented as NA) in the covariates, twang will attempt to construct weights that also balance rates of missingness in the treatment and control arms. In this case, the bal.table() will have an extra row for each variable that has missing entries.

The columns of the table consist of the following items:

**tx.mn**, **ct.mn** The treatment means and the control means for each of the variables. The unweighted table (unw) shows the unweighted means. For each stopping rule the means are weighted using weights corresponding to the gbm model selected by ps() using the stopping rule. When estimand = "ATT" the weights for the treatment group always equal 1 for all cases and there is no difference between unweighted and propensity score weighted tx.mn.

**tx.sd**, **ct.sd** The propensity score weighted treatment and control groups’ standard deviations for each of the variables. The unweighted table (unw) shows the unweighted standard deviations

**std.eff.sz** The standardized effect size, defined as the treatment group mean minus the control group mean divided by the treatment group standard deviation if estimand = "ATT" or divided by the pooled sample (treatment and control) standard deviation if estimand = "ATE". (In discussions of propensity scores this value is sometimes referred to as "standardized bias".) Occasionally, lack of treatment group or pooled sample variance on a covariate results in very large (or infinite) standardized effect sizes. For purposes of analyzing mean effect sizes across multiple covariates, we set all standardized effect sizes larger than 500 to NA (missing values).

**stat**, **p** Depending on whether the variable is continuous or categorical, stat is a t-statistic or a *χ* 2 statistic. p is the associated p-value

**ks**, **ks.pval** The Kolmogorov-Smirnov test statistic and its associated p-value. P-values for the KS statistics are either derived from Monte Carlo simulations or analytic approximations, depending on the specifications made in the perm.test.iters argument of the ps function. For categorical variables this is just the *χ* 2 test p-value

Components of these tables are useful for demonstrating that pretreatment differences between groups on observed variables have been eliminated using the weights. The xtable package aids in formatting for LATEX and Word documents. Table 1 shows the results for ks.max reformatted for a LATEX document. For Word documents, paste the LATEX description of the table into a Word document, highlight it and use Word tools to convert the text to a table using "&" as the separator.

> library(xtable)

> pretty.tab <- lalonde.balance$ks.max.ATT[,c("tx.mn","ct.mn","ks")]

> pretty.tab <- cbind(pretty.tab, lalonde.balance$unw[,"ct.mn"])

> names(pretty.tab) <- c("E(Y1|t=1)","E(Y0|t=1)","KS","E(Y0|t=0)")

> xtable(pretty.tab,

+ caption = "Balance of the treatment and comparison groups",

+ label = "tab:balance",

+ digits = c(0, 2, 2, 2, 2),

+ align=c("l","r","r","r","r"))

Table 1: Balance of the treatment and comparison groups

The summary() method for ps objects offers a compact summary of the sample sizes of the groups and the balance measures. If perm.test.iters>0 was used to create the ps object, then

Monte Carlo simulation is used to estimate p-values for the maximum KS statistic that would be expected across the covariates, had individuals with the same covariate values been assigned to groups randomly. Thus, a p-value of 0.04 for max.ks.p indicates that the largest KS statistic found across the covariates is larger than would be expected in 96% of trials in which the same cases were randomly assigned to groups.

> summary(ps.lalonde)

In general, weighted means can have greater sampling variance than unweighted means from a sample of equal size. The effective sample size (ESS) of the weighted comparison group captures this increase in variance as

The ESS is approximately the number of observations from a simple random sample that yields an estimate with sampling variation equal to the sampling variation obtained with the weighted comparison observations. Therefore, the ESS will give an estimate of the number of comparison participants that are comparable to the treatment group when estimand = "ATT". The ESS is an accurate measure of the relative size of the variance of means when the weights are fixed or they are uncorrelated with outcomes. Otherwise the ESS underestimates the effective sample size (Little & Vartivarian, 2004). With propensity score weights, it is rare that weights are uncorrelated with outcomes. Hence the ESS typically gives a lower bound on the effective sample size, but it still serves as a useful measure for choosing among alternative models and assessing the overall quality of a model, even if it provides a possibly conservative picture of the loss in precision due to weighting.

The ess.treat and ess.ctrl columns in the summary results shows the ESS for the estimated propensity scores. Note that although the original comparison group had 429 cases, the propensity score estimates effectively utilize only 23 or 27.1 of the comparison cases, depending on the rules and measures used to estimate the propensity scores. While this may seem like a large loss of sample size, this indicates that many of the original cases were unlike the treatment cases and, hence, were not useful for isolating the treatment effect. Moreover, similar or even greater reductions in ESS would be expected from alternative approaches to using propensity scores, such as matching or stratification. Since the estimand of interest in this example is ATT, ess.treat = n.treat throughout (i.e., all treatment cases have a weight of 1).

### 2.2 Graphical assessments of balance

The plot() function can generate useful diagnostic plots from the propensity score objects. The full set of plots available in twang and the argument value of plot to produce each one are given in Table 2. The convergence plot — the default — was discussed above.

The plot() function takes a plots argument in order to produce other diagnostic plots. For example, specifying plots = 2 or plots = "boxplot" produces boxplots illustrating the spread of the estimated propensity scores in the treatment and comparison groups. Whereas propensity score stratification requires considerable overlap in these spreads, excellent covariate balance can often be achieved with weights, even when the propensity scores estimated for the treatment and control groups show little overlap.

> plot(ps.lalonde, plots=2)

The effect size plot illustrates the effect of weights on the magnitude of differences between groups on each pretreatment covariate. These magnitudes are standardized using the standardized effect size described earlier. In these plots, substantial reductions in effect sizes are observed for most variables (blue lines), with only one variable showing an increase in effect size (red lines), but only a seemingly trivial increase. Closed red circles indicate a statistically significant difference, many of which occur before weighting, none after. In some analyses variables can have very little variance in the treatment group sample or the entire sample and group differences can be very large relative to the standard deviations. In these situations, the user is warned that some effect sizes are too large to plot.

> plot(ps.lalonde, plots=3)

When many of the p-values testing individual covariates for balance are very small, the groups are clearly imbalanced and inconsistent with what we would expect had the groups been formed by random assignment. After weighting we would expect the p-values to be larger if balance had been achieved. We use a QQ plot comparing the quantiles of the observed p-values to the quantiles of the uniform distribution (45 degree line) to conduct this check of balance. Ideally, the p-values from independent tests in which the null hypothesis is true will have a uniform distribution. Although the ideal is unlikely to hold even if we had random assignment (Bland, 2013), severe deviation of the p-values below the diagonal suggests lack of balance and p-values running at or above the diagonal suggests balance might have been achieved. The p-value plot (plots=4 or plots="t") allows users to visually to inspect the p-values of the t-tests for group differences in the covariate means.

> plot(ps.lalonde, plots = 4)

Before weighting (closed circles), the groups have statistically significant differences on many variables (i.e., p-values are near zero). After weighting (open circles) the p-values are generally above the 45-degree line, which represents the cumulative distribution of a uniform variable on [0,1]. This indicates that the p-values are even larger than would be expected in a randomized study.

One can inspect similar plots for the KS statistic with the argument plots = "ks" or

> plot(ps.lalonde, plots = 5)

In all cases, the subset argument can be used if we wish to fous on results from one stopping rule.

> plot(ps.lalonde, plots = 3, subset = 2)

### 2.3 Analysis of outcomes

A separate R package, the survey package, is useful for performing the outcomes analyses using weights. Its statistical methods account for the weights when computing standard error estimates. It is not a part of the standard R installation but installing twang should automatically install survey as well.

> library(survey)

The get.weights() function extracts the propensity score weights from a ps object. Those weights may then be used as case weights in a svydesign object. By default, it returns weights corresponding to the estimand (ATE or ATT) that was specified in the original call to ps(). If needed, the user can override the default via the optional estimand argument.

> lalonde$w <- get.weights(ps.lalonde, stop.method="es.mean")

> design.ps <- svydesign(ids=~1, weights=~w, data=lalonde)

The stop.method argument specifies which GBM model, and consequently which weights, to utilize.

The svydesign function from the survey package creates an object that stores the dataset along with design information needed for analyses. See help(svydesign) for more details on setting up svydesign objects.

The aim of the National Supported Work Demonstration analysis is to determine whether the program was effective at increasing earnings in 1978. The propensity score adjusted test can be computed with svyglm.

> glm1 <- svyglm(re78 ~ treat, design=design.ps)

> summary(glm1)

The analysis estimates an increase in earnings of $733 for those that participated in the

NSW compared with similarly situated people observed in the CPS. The effect, however, does not appear to be statistically significant.

Some authors have recommended utilizing both propensity score adjustment and additional covariate adjustment to minimize mean square error or to obtain "doubly robust" estimates of the treatment effect (Huppler-Hullsiek & Louis 2002, Bang & Robins 2005). These estimators are consistent if either the propensity scores are estimated correctly *or *the regression model is specified correctly. For example, note that the balance table for ks.max.ATT made the two groups more similar on nodegree, but still some differences remained, 70.8% of the treatment group had no degree while 60.1% of the comparison group had no degree. While linear regression is sensitive to model misspecification when the treatment and comparison groups are dissimilar, the propensity score weighting has made them more similar, perhaps enough so that additional modeling with covariates can adjust for any remaining differences. In addition to potential bias reduction, the inclusion of additional covariates can reduce the standard error of the treatment effect if some of the covariates are strongly related to the outcome.

> glm2 <- svyglm(re78 ~ treat + nodegree, design=design.ps)

> summary(glm2)

Adjusting for the remaining group difference in the nodegree variable slightly increased the

estimate of the program’s effect to $920, but the difference is still not statistically significant. We can further adjust for the other covariates, but that too in this case has little effect on the estimated program effect.

> glm3 <- svyglm(re78 ~ treat + age + educ + black + hispan + nodegree +

+ married + re74 + re75,

+ design=design.ps)

> summary(glm3)

### 2.4 Estimating the program effect using linear regression

The more traditional regression approach to estimating the program effect would fit a linear model with a treatment indicator and linear terms for each of the covariates.

> glm4 <- lm(re78 ~ treat + age + educ + black + hispan + nodegree +

+ married + re74 + re75,

+ data=lalonde)

> summary(glm4)

This model estimates a rather strong treatment effect, estimating a program effect of $1548 with a p-value=0.048. Several variations of this regression approach also estimate strong program effects. For example using square root transforms on the earnings variables yields a p- value=0.016. These estimates, however, are very sensitive to the model structure since the treatment and control subjects differ greatly as seen in the unweighted balance comparison ($unw) from bal.table(ps.lalonde).

### 2.5 Propensity scores estimated from logistic regression

Propensity score analysis is intended to avoid problems associated with the misspecification of covariate adjusted models of outcomes, but the quality of the balance and the treatment effect estimates can be sensitive to the method used to estimate the propensity scores. Consider estimating the propensity scores using logistic regression instead of ps().

> ps.logit <- glm(treat ~ age + educ + black + hispan + nodegree +

+ married + re74 + re75,

+ data = lalonde,

+ family = binomial)

> lalonde$w.logit <- rep(1,nrow(lalonde))

> lalonde$w.logit[lalonde$treat==0] <- exp(predict(ps.logit,subset(lalonde,treat==0)))

**predict()** for logistic regression model produces estimates on the log-odds scale by default. Exponentiating those predictions for the comparison subjects gives the ATT weights *p/*(1 *− p*).

**dx.wts()** from the twang package diagnoses the balance for an arbitrary set of weights producing a balance table. This function requires the user to specify the estimand argument in order to perform the appropriate calculations relative to the target group on which we are drawing inferences.

> bal.logit <- dx.wts(x = lalonde$w.logit,

+ data=lalonde,

+ vars=c("age","educ","black","hispan","nodegree",

+ "married","re74","re75"),

+ treat.var="treat",

+ perm.test.iters=0, estimand = "ATT")

> bal.logit

type n.treat n.ctrl ess.treat ess.ctrl max.es

1 unw 185 429 185 429.00000 1.7567745

2 185 429 185 99.81539 0.1188496 mean.es max.ks mean.ks iter

1 0.5687259 0.6404460 0.27024507 NA

2 0.0318841 0.3078039 0.09302319 NA

Applying the bal.table() function to this object returns a variable-by-variable summary of balance, just like it did for the ps object.

> bal.logit <- dx.wts(x = lalonde$w.logit,

+ data=lalonde,

+ vars=c("age","educ","black","hispan","nodegree",

+ "married","re74","re75"),

+ treat.var="treat",

+ perm.test.iters=0, estimand = "ATT")

> bal.logit

type n.treat n.ctrl ess.treat ess.ctrl max.es

1 unw 185 429 185 429.00000 1.7567745

2 185 429 185 99.81539 0.1188496 mean.es max.ks mean.ks iter

1 0.5687259 0.6404460 0.27024507 NA

2 0.0318841 0.3078039 0.09302319 NA

For weights estimated with logistic regression, the largest KS statistic was reduced from the unweighted sample’s largest KS of 0.64 to 0.31, which is still quite a large KS statistic. Table 3 shows the details of the balance of the treatment and comparison groups. The means of the two groups appear to be quite similar while the KS statistic shows substantial differences in their distributions.

> pretty.tab <- bal.table(bal.logit)[[2]][,c("tx.mn","ct.mn","ks")]

> pretty.tab <- cbind(pretty.tab, bal.table(bal.logit)[[1]]$ct.mn)

> names(pretty.tab) <- c("E(Y1|t=1)","E(Y0|t=1)","KS","E(Y0|t=0)")

> xtable(pretty.tab,

+ caption = "Logistic regression estimates of the propensity scores",

+ label = "tab:balancelogit",

+ digits = c(0, 2, 2, 2, 2),

+ align=c("l","r","r","r","r"))

> glm6 <- svyglm(re78 ~ treat, design=design.logit)

> summary(glm6)

The analysis estimates an increase in earnings of $1214 for those that participated in the NSW compared with similarly situated people observed in the CPS. Table 5 compares all of the treatment effect estimates.

### 3 An ATE example

In the analysis of Section 2, we focused on estimating ATT for the lalonde dataset. In this situation, the ATE is not of great substantive interest because not all people who are offered entrance into the program could be expected to take advantage of the opportunity. Further, there is some evidence that the treated subjects were drawn from a subset of the covariate space. In particular, in an ATE analysis, we see that we are unable to achieve balance, especially for the "black" indicator.

We now turn to an ATE analysis that is feasible and meaningful. We focus on the lindner dataset, which was included in the USPS package (Obenchain 2011), and is now included in twang for convenience. A tutorial by Helmreich and Pruzek (2009; HP) for the PSAgraphics package also uses propensity scores to analyze a portion of these data. HP describe the data as follows on p. 3 with our minor recodings in square braces:

The lindner data contain data on 996 patients treated at the Lindner Center, Christ Hospital, Cincinnati in 1997. Patients received a Percutaneous Coronary Intervention (PCI). The data consists of 10 variables. Two are outcomes: [sixMonthSurvive] ranges over two values... depending on whether patients surved to six months post treatment [denoted by TRUE] or did not survive to six months [FALSE]... Secondly, cardbill contains the costs in 1998 dollars for the first six months (or less if the patient did not survive) after treatment... The treatment variable is abcix, where 0 indicates PCI treatment and 1 indicates standard PCI treatment and additional treatment in some form with abciximab. Covariates include acutemi, 1 indicating a recent acute myocardial infarction and 0 not; ejecfrac for the left ventricle ejection fraction, a percentage from 0 to 90; ves1proc giving the number of vessels (0 to 5) involved in the initial PCI; stent with 1 indicating coronary stent inserted, 0 not; diabetic where 1 indicates that the patient has been diagnosed with diabetes, 0 not; height in centimeters and female coding the sex of the patent, 1 for female, 0 for male.

HP focus on cardbill — the cost for the first months after treatment — as their outcome of interest. However, since not all patients survived to six months, it is not clear whether a lower value of cardbill is good or not. For this reason, we choose six-month survival (sixMonthSurvive) as our outcome of interest.

Ignoring pre-treatment variables, we see that abcix is associated with lower rates of 6-month mortality:

> data(lindner)

> table(lindner$sixMonthSurvive, lindner$abcix)

0 | 1 |

FALSE 15 | 11 |

TRUE 283 | 687 |

> chisq.test(table(lindner$sixMonthSurvive, lindner$abcix))

Pearson*s Chi-squared test with Yates* continuity correction

data: table(lindner$sixMonthSurvive, lindner$abcix)

X-squared = 8.5077, df = 1, p-value = 0.003536

The question is whether this association is causal. If health care policies were to be made on the basis of these data, we would wish to elicit expert opinion as to whether there are likely to be other confounding pretreatment variables. For this tutorial, we simply follow HP in choosing the pre-treatment covariates. The twang model is fit as follows

> set.seed(1)

> ps.lindner <- ps(abcix ~ stent + height + female + diabetic +

+ acutemi + ejecfrac + ves1proc, data = lindner,

+ verbose = FALSE, estimand = "ATE")

We set estimand = "ATE" because we are interested in the effects of abciximab on everyone in the population. We do not specify the stopping rules. Consequently ps() uses the defaults: es.mean and ks.mean. We then inspect pre- and post-weighting balance with the command

> bal.table(ps.lindner)

$unw | |||||||||

tx.mn | tx.sd | ct.mn | ct.sd | std.eff.sz | stat | p | ks | ks.pval | |

stent | 0.705 | 0.456 | 0.584 | 0.494 | 0.257 | 3.624 | 0.000 | 0.121 | 0.004 |

height | 171.443 | 10.695 | 171.446 | 10.589 | 0.000 | -0.005 | 0.996 | 0.025 | 0.999 |

female | 0.331 | 0.471 | 0.386 | 0.488 | -0.115 | -1.647 | 0.100 | 0.055 | 0.531 |

diabetic | 0.205 | 0.404 | 0.268 | 0.444 | -0.152 | -2.127 | 0.034 | 0.064 | 0.349 |

acutemi | 0.179 | 0.384 | 0.060 | 0.239 | 0.338 | 5.923 | 0.000 | 0.119 | 0.005 |

ejecfrac | 50.403 | 10.419 | 52.289 | 10.297 | -0.181 | -2.640 | 0.008 | 0.114 | 0.008 |

ves1proc | 1.463 | 0.706 | 1.205 | 0.480 | 0.393 | 6.693 | 0.000 | 0.188 | 0.000 |

$ks.mean.ATE

tx.mn | tx.sd | ct.mn | ct.sd | std.eff.sz | stat | p | ks | ks.pval | |

stent | 0.683 | 0.466 | 0.657 | 0.475 | 0.054 | 0.718 | 0.473 | 0.025 | 1.000 |

height | 171.470 | 10.550 | 171.591 | 10.589 | -0.011 | -0.155 | 0.877 | 0.015 | 1.000 |

female | 0.338 | 0.473 | 0.345 | 0.476 | -0.015 | -0.200 | 0.841 | 0.007 | 1.000 |

diabetic | 0.215 | 0.411 | 0.229 | 0.421 | -0.033 | -0.432 | 0.666 | 0.014 | 1.000 |

acutemi | 0.148 | 0.355 | 0.107 | 0.310 | 0.120 | 1.331 | 0.183 | 0.040 | 0.935 |

ejecfrac | 51.052 | 10.333 | 51.604 | 9.110 | -0.056 | -0.798 | 0.425 | 0.027 | 0.999 |

ves1proc | 1.395 | 0.666 | 1.337 | 0.573 | 0.094 | 1.203 | 0.229 | 0.028 | 0.999 |

$es.mean.ATE

20

tx.mn | tx.sd | ct.mn | ct.sd | std.eff.sz | stat | p | ks | ks.pval | |

stent | 0.683 | 0.466 | 0.653 | 0.477 | 0.063 | 0.849 | 0.396 | 0.030 | 0.996 |

height | 171.468 | 10.534 | 171.501 | 11.040 | -0.003 | -0.039 | 0.969 | 0.018 | 1.000 |

female | 0.337 | 0.473 | 0.350 | 0.478 | -0.028 | -0.380 | 0.704 | 0.013 | 1.000 |

diabetic | 0.214 | 0.411 | 0.237 | 0.426 | -0.055 | -0.719 | 0.473 | 0.023 | 1.000 |

acutemi | 0.148 | 0.355 | 0.110 | 0.313 | 0.114 | 1.303 | 0.193 | 0.038 | 0.950 |

ejecfrac | 51.008 | 10.389 | 51.386 | 9.396 | -0.038 | -0.532 | 0.595 | 0.027 | 0.999 |

ves1proc | 1.398 | 0.670 | 1.350 | 0.589 | 0.076 | 0.951 | 0.342 | 0.023 | 1.000 |

This balance table shows that stent, acutemi, and ves1proc were all significantly imbalanced before weighting. After weighting (using either stop.method considered) we do not see problems in this regard. Examining plot(ps.lindner, plots = x) for x running from 1 to 5 does not reveal problems, either. In regard to the optimize plot, we note that the scales of the KS and ES statistics presented in the optimize plots are not necessarily comparable. The fact that the KS values are lower than the ES values in the optimize plot does not suggest that the KS stopping rule is finding superior models. Each panel of the optimize plot indicates the gbm model that minimizes each stopping rule. The panels should not be compared other than to compare the number of iterations selected by each rule.

> plot(ps.lindner, plots = 1)

> plot(ps.lindner, plots = 2)

> plot(ps.lindner, plots = 3)

> plot(ps.lindner, plots = 4)

> plot(ps.lindner, plots = 5)

From a call to summary(), we see that the es.mean.ATE stopping rule results in a slightly higher ESS with comparable balance measures, so we proceed with those weights. Also, we note that ess.treat is no longer equal to n.treat since we are focusing on ATE rather than ATT.

> summary(ps.lindner)

n.treat | n.ctrl | ess.treat | ess.ctrl | max.es |

unw | 698 | 298 | 663.8400 298.0000 | 0.3925637 |

ks.mean.ATE | 698 | 298 | 655.6754 228.8501 | 0.1197287 |

es.mean.ATE | 698 | 298 | 663.8400 237.1444 | 0.1135056 |

As before, we use the survey package to reweight our sample and perform the analysis.

> lindner$w <- get.weights(ps.lindner, stop.method = "es.mean")

> design.ps <- svydesign(ids=~1, weights = ~w, data = lindner)

> svychisq(~sixMonthSurvive + abcix, design = design.ps)

Pearson's X^2: Rao & Scott adjustment

data: svychisq(~sixMonthSurvive + abcix, design = design.ps) F = 11.378, ndf = 1, ddf = 995, p-value = 0.0007718

The reweighting does not diminish the association between the treatment and the outcome. Indeed, it is marginally more significant after the reweighting.

### 4 Non-response weights

The twang package was designed to estimate propensity score weights for the evaluation of treatment effects in observational or quasi-experimental studies. However, we find that the package includes functions and diagnostic tools that are highly valuable for other applications, such as for generating and diagnosing nonresponse weights for survey nonresponse or study attrition. We now present an example that uses the tools in twang. This example uses the subset of the US Sustaining Effects Study data distributed with the HLM software (Bryk, Raudenbush, Congdon, 1996) and also available in the R package mlmRev. The data include mathematics test scores for 1721 students in kindergarten to fourth grade. They also include student race (black, Hispanic, or other), gender, an indicator for whether or not the student had been retained in grade, the percent low income students at the school, the school size, the percent of mobile students, the students’ grade-levels, student and school IDs, and grades converted to year by centering. The study analysis plans to analyze growth in math achievement from grade 1 to grade

4 using only students with complete data. However, the students with complete data differ from other students. To reduce bias that could potentially result from excluding incomplete cases, our analysis plan is to weight complete cases with nonresponse weights.

The goal of nonresponse weighting is to develop weights for the respondents that make them look like the entire sample — both the respondents and nonrespondents. Since the respondents already look like themselves, the hard part is to figure out how well each respondent represents the nonrespondents. Nonresponse weights equal the reciprocal of the probability of response and are applied only to respondents.

Note that the the probability of response is equivalent to the propensity score if we consider subjects with an observed outcome to be the "treated" group, and those with an unobserved outcome to be the "controls". We wish to reweight the sample to make it equivalent to the population from which the sample was drawn, so ATE weights are more appropriate in this case. Further, recall that the weights for the treated subjects are 1*/p *in an ATE analysis. Therefore we can reweight the sample of respondents using the get.weights() function.

Before we can generate nonresponse weights, we need to prepare the data using the following commands. First we load the data.

> data(egsingle)

Next we create a response indicator variable that we can merge onto the student by test score level data. We want to include only students with scores from all of grades 1 to 4. The data include scores from kindergarten (grade = 0) to grade 5 with some students having multiple scores from the same grade. First we keep the unique grades for each student:

> tmp <- tapply(egsingle$grade, egsingle$childid, unique)

Because students do not all have the same number of score, tapply() returns a list with one element per student. Each element contains the unique set of grades observed for each student. We now check for grades in 1 to 4:

> tmp <- lapply(tmp, function(x){return(x %in% 1:4)})

The list tmp now contains a boolean vector for each student, where "TRUE" indicates the grade took on a value in 1 to 4. The sum of this vector for each student determines how many of grades 1 to 4 we observed for him or her.

> tmp <- lapply(tmp, sum)

A student is a respondent if he or she has scores from all four of grades 1 to 4 or if the value of tmp is 4.

> tmp <- sapply(tmp, function(x){as.numeric(x == 4)})

We create a data frame of response indicators so we can merge them onto the test scores data:

> tmp <- data.frame(tmp)

> names(tmp) <- "resp"

> tmp$childid <- row.names(tmp)

and merge this back to create a single data frame

> egsingle <- merge(egsingle, tmp)

Because nonresponse is a student-level variable rather than a student-by-year-level variable we create one record per student.

> egsingle.one <-unique(egsingle[,-c(3:6)])

We also create a race variable

> egsingle.one$race <- as.factor(race <- ifelse(egsingle.one$black==1, 1,

+ ifelse(egsingle.one$hispanic==1, 2, 3)))

As discussed above, to use ps() to estimate nonresponse, we need to let respondents be the treatment group by modeling an indicator of response.

> egsingle.ps <- ps(resp ~ race + female + size + lowinc + mobility,

+ data=egsingle.one,

+ stop.method=c("es.mean","ks.max"),

+ n.trees=5000,

+ verbose=FALSE,

+ estimand = "ATE")

As in standard propensity score applications, we should check that n.trees was set large enough, so that balance would seem not to be improved if more complex models were considered. Recall that for es.mean.ATE the measure is the average effect size difference between the two groups and for ks.max.ATE the measure is the largest of the KS statistics.

> plot(egsingle.ps)

By default the balance table generated by ps() compares the weighted treatment group (respondents) to the weighted comparison group (nonresponders) – both groups weighted to equal the overall population. However, the goal is to weight the respondents to match the population not to compare the weighted respondents and nonrespondents. The default balance table may be useful for evaluating the propensity scores, but it does not directly assess the quality of the weights for balancing the weighted respondents with the overall population.

We can "trick" the dx.wts() function in the twang package into making the desired comparison. We want to compare the weighted respondents to the unweighted full sample. When evaluating ATT weights we compare the weighted comparison group with the unweighted treatment group. If we apply dx.wts() to a data set where the "treatment" group is the entire esingle.one sample and the "control" group is the esingle.one respondents and the weights equal one for every student in the pseudo-treatment group and equal the weights from ps() for every student in the pseudo-control group, we can obtain the balance statistics we want.

We begin by setting up the data with the pseudo-treatment and control groups. We add

ATE weights from the "ks.max" stopping rule as our nonresponse weights.

> egsingle.one$wgt <- get.weights(egsingle.ps, stop.method="ks.max")

We now stack the full sample and the respondents. The variable "nr2" is the pseudo-treatment indicator. We set it equal to one for the full sample and 0 for the respondents. Similarly, "wgt2" is the pseudo-ATT weight which is set equal to one for the full sample and equal to the nonresponse weights for the respondents.

> egtmp <- rbind(data.frame(egsingle.one, nr2=1, wgt2=1),

+ data.frame(egsingle.one, nr2=0, wgt2=egsingle.one$wgt)[egsingle.one$resp==1,])

We now run dx.wts() to obtain the balance statistics. Switching to ATT from ATE yields a warning that can be ignored in this case.

> egdxwts <- dx.wts(x=egtmp$wgt2,

+ data=egtmp,

+ estimand="ATT",

+ vars=c("race", "female", "size", "lowinc", "mobility"),

+ treat.var="nr2")

> pretty.tab<-bal.table(egdxwts)[[2]][,c("tx.mn","ct.mn","std.eff.sz","ks")]

> names(pretty.tab) <- c("OverallS Sample","Weighted responders","Std ES","KS")

> xtable(pretty.tab,

+ caption = "Balance of the nonrespondents and respondents",

+ label = "tab:balance2",

+ digits = c(0, 2, 2, 2, 2),

+ align=c("l","r","r","r","r"))

The balance is very good. We can use the weighted respondents for analysis. We select only the records with an observed outcome. This will be our analysis sample and the variable "wgt" will contains the nonreponse weights.

> egsinge.resp <- merge(subset(egsingle, subset=resp==1),

+ subset(egsingle.one, subset=resp==1,

+ select=c(childid, wgt)) )

### The details of twang

### Propensity scores and weighting

Propensity scores can be used to reweight comparison cases so that the distribution of their features match the distribution of features of the treatment cases, for ATT, or cases from both treatment and control groups to match each other, for ATE (Rosenbaum 1987, Wooldridge 2002, Hirano and Imbens 2001, McCaffrey *et al.* 2004) Let *f*(**x**∣*t* = 1) be the distribution of features for the treatment cases and *f*(**x**∣*t* = 0) be the distribution of features for the comparison cases. If treatments were randomized then we would expect these two distributions to be similar. When they differ for ATT we will construct a weight, *w*(**x**), so that

*f*(**x**∣*t* = 1) = *w*(**x**)*f*(**x**∣*t* = 0).

For example, if *f*(age=65, sex=F∣*t* = 1) = 0.10 and *f*(age=65, sex=F∣*t* = 0) = 0.05 (i.e. 10% of the treatment cases and 5% of the comparison cases are 65 year old females) then we need to give a weight of 2.0 to every 65 year old female in the comparison group so that they have the same representation as in the treatment group. More generally, we can solve Equation 3, below, for *w*(**x**) and apply Bayes Theorem to the numerator and the denominator to give an expression for the propensity score weight for comparison cases,

where *K* is a normalization constant that will cancel out in the outcomes analysis. Equation ([eq:weight]) indicates that if we assign a weight to comparison case *i* equal to the odds that a case with features **x**_{i} would be exposed to the treatment, then the distribution of their features would balance. Note that for comparison cases with features that are atypical of treatment cases, the propensity score *P*(*t* = 1∣**x**) would be near 0 and would produce a weight near 0. On the other hand, comparison cases with features typical of the treatment cases would receive larger weights.

For ATE, each group is weighted to match the population. The weight must satisfy:

*f*(**x**∣*t* = 1) = *w*(**x**)*f*(**x**), and

*f*(**x**∣*t* = 0) = *w*(**x**)*f*(**x**), and

Again using Bayes Theorem we obtain *w*(**x**) = 1/*f*(**x**∣*t* = 1) for the treatment group and *w*(**x**) = 1/*f*(**x**∣*t* = 0) for the control group.

### 5.2. Estimating the propensity score

In randomized studies *P*(*t* = 1∣**x**) is known and fixed in the study design. In observational studies the propensity score is unknown and must be estimated, but poor estimation of the propensity scores can cause just as much of a problem for estimating treatment effects as poor regression modeling of the outcome. Linear logistic regression is the common method for estimating propensity scores, and can suffice for many problems. Linear logistic regression for propensity scores estimates the log-odds of a case being in the treatment given **x**as

Usually, *β* is selected to maximize the logistic log-likelihood

Maximizing the above provides the maximum likelihood estimates of *β*. However, in an attempt to remove as much confounding as possible, observational studies often record data on a large number of potential confounders, many of which can be correlated with one another. Standard methods for fitting logistic regression models to such data with the iteratively reweighted least squares algorithm can be statistically and numerically unstable. To improve the propensity score estimates we might also wish to include non-linear effects and interactions in **x**. The inclusion of such terms only increases the instability of the models.

One increasingly popular method for fitting models with numerous correlated variables is the lasso (least absolute subset selection and shrinkage operator) introduced in statistics in Tibshirani (1996). For logistic regression, lasso estimation replaces ([eq:loglikelihood]) with a version that penalizes the absolute magnitude of the coefficients

The second term on the right-hand side of the equation is the penalty term since it decreases the overall of ℓ*β* when there are coefficients that are large in absolute value. Setting *λ* = 0 returns the standard (and potentially unstable) logistic regression estimates of *β*. Setting *λ* to be very large essentially forces all of the *β*_{j} to be equal to 0 (the penalty excludes *β*_{0}). For a fixed value of *λ* the estimated $\hat\beta$ can have many coefficients exactly equal to 0, not just extremely small but precisely 0, and only the most powerful predictors of *t* will be non-zero. As a result the absolute penalty operates as a variable selection penalty. In practice, if we have several predictors of *t* that are highly correlated with each other, the lasso tends to include all of them in the model, shrink their coefficients toward 0, and produce a predictive model that utilizes all of the information in the covariates, producing a model with greater out-of-sample predictive performance than models fit using variable subset selection methods.

Our aim is to include as covariates all piecewise constant functions of the potential confounders and their interactions. That is, in **x** we will include indicator functions for continuous variables like *I*(age < 15), *I*(age < 16), …, *I*(age < 90), etc., for categorical variables like *I*(sex = male), *I*(prior MI = TRUE), and interactions among them like *I*(age < 16)*I*(sex = male)*I*(prior MI = TRUE). This collection of basis functions spans a plausible set of propensity score functions, are computationally efficient, and are flat at the extremes of **x** reducing the likelihood of propensity score estimates near 0 and 1 that can occur with linear basis functions of **x**. Theoretically with the lasso we can estimate the model in ([eq:lasso]), selecting a *λ* small enough so that it will eliminate most of the irrelevant terms and yield a sparse model with only the most important main effects and interactions. Boosting (Friedman 2001, 2003, Ridgeway 1999) effectively implements this strategy using a computationally efficient method that Efron *et al.* (2004) showed is equivalent to optimizing ([eq:lasso]). With boosting it is possible to maximize ([eq:lasso]) for a range of values of *λ* with no additional computational effort than for a specific value of *λ*. We use boosted logistic regression as implemented in the generalized boosted modeling (gbm) package in R (Ridgeway 2005).

## Evaluating the weights

As with regression analyses, propensity score methods cannot adjust for unmeasured covariates that are uncorrelated with the observed covariates. Nonetheless, the quality of the adjustment for the observed covariates achieved by propensity score weighting is easy to evaluate. The estimated propensity score weights should equalize the distributions of the cases’ features as in ([eq:balance]). This implies that weighted statistics of the covariates of the comparison group should equal the same statistics for the treatment group. For example, the weighted average of the age of comparison cases should equal the average age of the treatment cases. To assess the quality of the propensity score weights one could compare a variety of statistics such as means, medians, variances, and Kolmogorov-Smirnov statistics for each covariate as well as interactions. The twang package provides both the standardized effect sizes and KS statistics and p-values testing for differences in the means and distributions of the covariates for analysts to use in assessing balance.

## Analysis of outcomes

With propensity score analyses the final outcomes analysis is generally straightforward, while the propensity score estimation may require complex modeling. Once we have weights that equalize the distribution of features of treatment and control cases by reweighting. For ATT, we give each treatment case a weight of 1 and each comparison case a weight *w*_{i} = *p*(**x**_{i})/(1 − *p*(**x**_{i})). To estimate the ATE, we give control cases weight *w*_{i} = 1/*p*(**x**_{i}) and we give the treatment cases *w*_{i} = 1/(1 − *p*(**x**_{i}). We then estimate the treatment effect estimate with a weighted regression model that contains only a treatment indicator. No additional covariates are needed if the weights account for differences in **x**.

A combination of propensity score weighting and covariate adjustment can be useful for several reasons. First, the propensity scores may not have been able to completely balance all of the covariates. The inclusion of these covariates in addition to the treatment indicator in a weighted regression model may correct this if the imbalance is relatively small. Second, in addition to exposure, the relationship between some of the covariates and the outcome may also be of interest. Their inclusion can provide coefficients that can estimate the direction and magnitude of the relationship. Third, as with randomized trials, stratifying on covariates that are highly correlated with the outcome can improve the precision of estimates. Lastly, the some treatment effect estimators that utilize an outcomes regression model and propensity scores are "doubly robust" in the sense that if either the propensity score model is correct or the regression model is correct then the treatment effect estimator will be unbiased (Bang & Robins 2005).

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