Recently, Glynn and Quinn (2010) applied generalized additive models (GAM) [e. g., Hastie & Tibshi-rani, 1990] for the estimation of average total effects based on predictions of the (group-specific) covariate regression (see the accompanyingR-package for details, Glynn & Quinn, 2009).
Statistical Inference Although the resulting estimator of the average total effect for linear parameterized covariate-treatment regressions is equivalent to the estimator based on moderated regressions, the regres-sion estimates discussed in Schafer and Kang (2008) are interesting with respect to the statistical inference.
The authors provide adjusted, robust standard errors, which do not assume that the variance of the out-come is modeled correctly (see subsection 3.2.2.1 on page 58).
Note that even though Wooldridge (2001) suggested flexible nonparametric techniques for the estima-tion of the condiestima-tional regressions E (Y |Z , X = j ) [for example kernel estimators, Härdle & Linton, 1994], in order to derive valid standard errors, he recommended flexible parametric models based on low-order polynomials (see, e. g., R. J. A. Little et al., 2000, and subsection 2.2.2). Glynn and Quinn (2009) provide bootstrap standard errors for the average total effect.
Example For an application of Schafer and Kang’s approach to the quasi-experimental example, the group-specific regressions [Equation (2.6)] could be extended for multiple covariates Z ≡ (Z1,..., ZK) to EX =j(Y |Z ) = β0j+PK
k=1
¡βk jZk
¢, for example, under a linearity assumption conditional on X = j .
After-wards, a predicted score would be computed for each student under the math treatment and under the language treatment as the weighted sum of the observed values of the covariates zik and the regression coefficients ˆβk j, estimated by ordinary least-squares:
b
yi j= ˆβ0j+ XK k=1
¡ˆβk jzik
¢. (2.10)
Finally, the average of the difference between the predicted scores, i. e., N1PN
i
¡ybi1− byi0
¢is computed as an
estimator of the average total effect, AT E10.
2.3 Methods based on Propensity Score
The adjustment methods described so far are based on the regression of the outcome variable Y on the ob-served covariates Z and the treatment variable X . These methods are criticized mainly for the following two reasons: The parametrization of the outcome model (i. e., the functional form of the covariate-treatment regression) is not fixed when the outcome variable Y is analyzed for the first time (Rubin, 2001, 2008b).
Therefore, there is a risk that the final result of the analysis (e. g., the estimated average total effect) is taken into account for constructing and specifying the regression model of the (parametric) covariate-treatment regression. Typical post hoc modifications are the adaptation of the functional form of the regression, but
2.3 Methods based on Propensity Score 31
these modifications might also comprise the inclusion or exclusion of covariates, or functions of covari-ates (for instance, the product of two covaricovari-ates). Furthermore, although this is of course not specific to the estimation of causal effects, it is often noted, that typical model-based analyses provide no warning that comparisons may be based on extreme extrapolations (Rubin, 1997, p. 762, see also, e. g., King & Zeng, 2006;
Stürmer, Joshia, Glynn, Avorna, & Rothman, 2006; Zanutto, 2006). We will summarize and discuss these practical issues regarding the adjustment procedures in section 2.5 subsequent to the following presenta-tion of alternative adjustment methods.
Over the last two decades, a set of alternative methods have been developed and become widely ac-cepted and extensively used. These methods are based on the conditional probability of assignment to treatment given the covariates, popularized as propensity score by Rosenbaum and Rubin (1983b) [see also Rosenbaum, 2002b, 2002c].6As described in section 1.1.8, the theoretical foundation for the identification of the average total effect based on the propensity π is the assumption of Z -conditonal independence of X and the true outcomes, which implies unbiasedness of the regression E (Y |X ,π).
In quasi-experiments the Z -conditional propensity functions are unknown and have to be estimated.
Therefore, adjustment methods based on the propensity score are sometimes called two-step approaches (Dehejia & Wahba, 1999). The estimation of the propensity score for a multivariate vector Z ≡ (Z1,..., ZK) can be performed, for example, by using a parametric logistic regression model (see Rosenbaum & Rubin, 1984),
P (X = j |Z ) = exp¡ α0+PK
k=1αk· Zk
¢ 1 + exp¡
α0+PK
k=1αk· Zk
¢ . (2.11)
The parameterization of Equation (2.11) represents a functional form assumption, used for the assignment model (as introduced in section 1.1.8). The estimated propensity score bπi for unit i is obtained as the pre-dicted score based on the estimated regression coefficients bαkand the observed values zkifor each covariate Zk. The regression parameters αkfor a logistic regression model are usually estimated by maximum likeli-hood (see, e. g., Agresti, 2007). It is known that if this assignment model is not specified correctly the average total effect, estimated by conditioning on the estimated propensity score, can be biased (Drake, 1993). That means for the simple logistic model in Equation (2.11), the estimate of the average total effect might be bi-ased if linearity and additivity of the logit ln£
P (X = j |Z )/¡
1 − P(X = j |Z )¢¤
= α0+PK
k=1(αk· Zk) do not hold.
There are several alternative strategies for the estimation of propensity scores beyond the simple lo-gistic regression (see, e. g., King & Zeng, 2001, for neuronal networks, McCaffrey, Ridgeway, & Morral, 2004, for a propensity score estimation with boosted regressions, B. K. Lee, Lessler, & Stuart, 2009, for the
appli-6Note that in line with Steyer et al. (in press) we distinguish the conditional probability P (X = j |Z = z) [the propensity score for X = j and Z = z] and the function πX =j≡ P (X = j |Z ) [the Z -conditional propensity for X = j ] from the true treatment probability functions P (X = j |C x) [the true propensity functions, see section 1.1.6].
2.3 Methods based on Propensity Score 32
cation of classification trees, and Keele, 2008; Woo, Reiter, & Karr, 2008, for a smoothing splines generalized additive model, to name at least a few of them).
For a large number of covariates, the estimation of the regression of the outcome variable Y on all co-variates Z = (Z1,..., ZK), the treatment variable X , as well as additional interaction terms (e. g., Z1· X ,..., ZK· X ) might become complicated due to the problem commonly known as curse of dimension-ality (see, e. g., Kotz, Read, Balakrishnan, & Vidakovic, 2005, and Hirano & Imbens, 2001). This is especially the case with small samples (for an application of propensity score based adjustment methods for small samples see, e. g., Cepeda, Boston, Farrar, & Strom, 2003). Even if the estimation can be performed with-out difficulties, redundant regressors might increase the estimated standard errors unnecessarily (because of multicollinearity). Hence, under certain conditions the two-step adjustment based on πj= P(X = j |Z ) and E (Y |X ,πj) might be easier to model and might give more reliable results than the outcome modeling approach based on the covariate-treatment regression E (Y |X , Z ).
The estimated propensity scores can be used in a variety of different data analysis techniques (see, e. g., Guo & Fraser, 2010, for a textbook presentation of different propensity score techniques). Basically, all methods make use of the property that if the treatment assignment is strongly ignorable given the co-variates, then the observed difference in the means of the outcome variable between treatment and control group conditional on the propensity score is an unbiased estimator of the conditional total effect given that particular value of the propensity score (see subsection 1.1.8). Note that if the expectation of the conditional total effect given a value of the (estimated) propensity score is taken over the (distribution of the propen-sity score in the) population of treated units, an estimator of the average treatment effect of the treated is constructed (see subsection 1.1.3), whereas the average total effect is obtained with respect to the (distri-bution of the propensity score in the) total population. The various methods summarized in the following paragraphs differ mainly technically in the way the continuous and uni-dimensional propensity scores are treated for the computation or approximation of the expectation.
Example In the examples of Shadish et al. (2008a) and Pohl et al. (submitted), the propensity scores were estimated by using logistic regressions. The procedures were designed and applied in order to achieve acceptable balance of the considered covariates Z ≡ (Z1,..., ZK), a property we will discuss in subsection 2.5. Furthermore, Luellen, Shadish, and Clark (2005) compared three approaches for the construction of propensity scores for data from the same within-study comparison: classification tree analysis, bagging classification trees and simple logistic regression.
2.3 Methods based on Propensity Score 33
2.3.1 Propensity Score Subclassification
A very fundamental approach for the adjustment based on the estimated propensity scores is the creation of S strata, in which the units of the treatment group and the control group have identical (or at least “com-parable”) propensity scores. In order to apply the propensity score subclassification approach7as an ad-justment method, a new variable πs is created, for instance with approximately equal sized classes based on the percentiles of the estimated propensity score distribution. For this purpose, a corresponding value 1,...,S — according to the estimated propensity score percentile — is assigned to the variable πs for each unit U = u, regardless of the observed value of the treatment variable X . The underlying idea can be de-scribed as conditional mean adjustment, where the adjusted means E¡
EX =j(Y |π)¢
are identified (or at least approximated) by the expectation of EX =j(Y |πs= s), for each of the S = s strata and each treatment group X = j , over the marginal distribution of πs. The difference between the two adjusted means for group X = j and X = k yields an estimator of the average total effect ATEj k(see, Steyer et al., in press).
Rosenbaum and Rubin (1984) suggest forming S = 5 strata, because Cochran (1968a) reported a 90 % bias reduction for the adjustment by subclassification for a single confounding covariate with 5 strata. This finding was replicated for propensity score subclassification by Drake (1993), who also pointed out that
“there was some bias left in estimating treatment effect when using the propensity score approach in the con-tinuous model”. To remove this (remaining) bias, which is sometimes called residual confounding (e. g., Austin & Mamdani, 2006), the combination of different adjustment procedures is discussed in the literature (see subsection 2.4).
For the estimation of the conditional expectations EX =j(Y |πS= s) it is necessary that at least one unit be observed with an estimated propensity score in each stratum s for each treatment group j . In other words, the number of strata has to be small enough that the conditional mean differences P F E10;πs=s— used as estimators of CTE10;πs=s— are empirically identified. Hence, the optimal number of strata depends on the data. For a small dataset it may not be possible to use five strata (Rubin, 1997), whereas for a large dataset more than five propensity score subclasses may be desirable (Huppler Hullsiek & Louis, 2002). For πs as introduced above, the conditional total effect of treatment X = j compared to treatment X = k for each stratum is expressed as:
CTEj k;πs=s= EX =j(Y |πs= s) − EX =k(Y |πs= s). (2.12)
7Note that this approach is described under different names, i. e., beside propensity score subclassification, for instance, propensity score stratification (e. g., Senn, Graf, & Caputo, 2007), and blocking on the propensity score (e. g., Imbens, 2004) are commonly used.
2.3 Methods based on Propensity Score 34
The average total effect is identified as the weighted sum over the strata-specific conditional total effects,
ATEj k= XS s=1
µNs
N CTEj k;πs=s
¶
, (2.13)
with Nsas the number of units within each stratum and N =PS
s=1(Ns) as the total sample size. FromNNs it follows that the weighting is proportional to the number of observations falling in each stratum.
Statistical Inference Standard errors for the propensity score subclassification ATE–estimator are com-monly calculated as
V ar¡ ATEd10¢
= XS s=1
·µNs
N
¶2
Var¡CTE10;πs=s
¢¸
, (2.14)
i. e., based on the variance for the conditional total effect estimator given πs= s (see, e. g., Benjamin, 2003, and Guo & Fraser, 2010, p. 66). As discussed by Zanutto (2006), these standard errors are only approxi-mations, because they do not account for the fact that subclassification is based on estimated propensity scores. Alternative standard errors exist, for example, based on bootstrapping confidence intervals (see, e. g., Tu & Zhou, 2003).
Example For the quasi-experimental study used as an illustration of different adjustment methods, Shadish et al. (2008a) applied the subclassification approach on the estimated propensity score as described in this paragraph and reported standard errors from N = 1000 bootstrap samples. Due to a smaller sample size, Pohl et al. (submitted) did not perform the propensity score subclassification as described here. In-stead, a weighted regression analysis was performed with individual weights obtained from the stratum membership for each unit. This is a special version of inverse-propensity weighting with coarsened weights as described in the next paragraph (see, e. g., Freedman & Berk, 2008, for the similarity of simple weighted regression and inverse-propensity weighting).
2.3.2 Inverse-Propensity Weighting
Propensity score subclassification can be understood as a special case of a more general weighting approach (Lunceford & Davidian, 2004),8and the relation of both approaches can be described as bias-variance trade-off (Tan, 2007): On the one hand the bias of the (approximated) ATE–estimator decreases with an increas-ing number of strata, but on the other hand the variance of the ATE–estimator (i. e., the standard error) decreases with an increasing sample size per stratum. A similar approach to inverse-propensity weighting is known in survey sampling (see, e. g., Kish, 1965), and the estimator of the average total effect can be
ex-8Also the reverse was argued, that “... weighting can also be viewed as the limit of subclassification as the number of observations and subclasses tend to infinity.” (see Rubin, 2001, p. 173).
2.3 Methods based on Propensity Score 35
pressed as the difference of two Horvitz-Thompson estimators of the adjusted means (Horvitz & Thompson, 1952). The average total effect based on the weighted outcome is defined as
ATE10 = E (τ1) − E(τ0)
´[see Steyer et al., in press]. Estimates of the average total
effect based on weighting are obtained by replacing the true Z -conditional probabilities P(X = j |Z ) with the estimated propensity scores ˆπ. The propensity scores are estimated under the assumption that the assignment model is correctly specified (see above).9 DifferentRimplementations for various weighting estimators are available (see, e. g., Ridgeway, McCaffrey, & Morral, 2006; Glynn & Quinn, 2009).
It is well known in the literature that inverse-propensity weighting is susceptible for extreme propen-sity scores: “This process [inverse-propenpropen-sity weighting] can generate unrealistically extreme weights when an estimated propensity score is near zero or one, something that is avoided in the subclassification approach”
(Rubin, 2001, p. 184). This makes inverse-propensity weighting prone to misspecified propensity score models (Kang & Schafer, 2007a). Propensity score subclassification does not incorporate the raw estimated treatment probabilities for the computation of the conditional average total effect CTE10;πs=sand is there-fore, compared to the weighting approach, more robust against misspecifications of the assignment model (see also R. J. A. Little & Rubin, 2002).
Statistical Inference In order to obtain a valid standard error for the estimated average total effect based on inverse-propensity weighting, it is necessary to account for the fact that the weights are based on the propensity scores bπ estimated from the sample. Unfortunately “weights are routinely treated as known by nearly all researchers who use them” (Morgan & Todd, 2008, p. 278). Nevertheless, correct standard errors for a specific ATE–estimator obtained from inverse-propensity weighting with propensity scores estimated by a logistic regression [see Equation (2.11)] were recently derived by Schafer and Kang (2008).
9Note that two versions of the inverse-propensity weighting estimator exist in the literature which differ with respect to the normal-ization of the weights
(see, e. g., Rosenbaum, 1987, 1998) or d
(see, e. g., Hirano & Imbens, 2001; Tan, 2007). Both procedures are assumed to be equal for large sample sizes, because the two correction terms have an expectations of one (a formal argumentation can be found, e. g., in Lunceford & Davidian, 2004).
2.3 Methods based on Propensity Score 36
Example Shadish et al. (2008a) applied inverse-propensity weighting as well as weighting in combination with adjustment for additional covariates as described by Rubin (2001). In order to obtain standard errors for the ATE–estimator, the authors used a bootstrapping approach again.
2.3.3 Propensity Score Matching
Propensity score matching describes a class of methods dealing with pairing units from the treatment group and the control group with equal (or at least “similar”) values of the estimated propensity scores. Subclassi-fication and propensity score matching share the same idea of comparing treated and untreated units con-ditional on the estimated propensity score, i. e., both approaches are based on the assumption of unbiased-ness of E (Y |X ,π). Unlike the propensity subclassification approach, propensity score matching is applied by the selection of a matched sample of treated and untreated units with similar propensity score distribu-tions instead of dividing the sample into S distinct strata with similar values of the propensity scores. Orig-inally, matching was described as “sampling from a large reservoir of potential controls to produce a control group of modest size in which the distribution of covariates is similar to the distribution in the treated group”
(Rosenbaum & Rubin, 1983b, p. 48). Hence, propensity score matching is asymmetric in the sense that unmatched units (from the larger group) are typically discarded for the estimation of adjusted treatment effects based on propensity score matched samples (see, e. g., Schafer & Kang, 2008, for some comments about the consequences). This is an implicit restriction of the analyzed sample to the region of common support (see, e. g., Guo & Fraser, 2010, and subsection 2.5).
Nonparametric matching on the observed covariates (as described above) and matching on the (esti-mated) propensity score is often performed based on the same matching algorithms. Different matching techniques can be described again as a trade-off between incomplete matching (i. e., failing to match all treated subjects to control units) versus inexact matching (i. e., the matching of dissimilar, not comparable subjects).
Matching on propensity scores and matching on observed covariates are often combined, a strategy described as the observational study analog of blocking in a randomized experiment (Rubin & Thomas, 2000). For instance, one of the most common techniques is the one-one matching of units based on the minimal Mahalanobis distance of the observed covariates Z = (Z1,..., ZK), where the matching is restricted to units within a range of the estimated propensity score logit [ ˆπl n= ln¡
π/(1 − ˆπ)ˆ ¢
, see Schafer & Kang, 2008, for a short summary of this matching procedure]. For this matching strategy, c is called the caliper, a pre-specified tolerance in terms of the propensity, and observed covariate matching is performed only in the range ˆπl n± c (see Rosenbaum & Rubin, 1985, for the selection of value for the caliper). A survey of all different matching estimators is beyond the scope of this review (see, e. g., Rubin & Thomas, 1996,
2.3 Methods based on Propensity Score 37
D’Agostino, 1998, Rosenbaum, 2002b, Imbens, 2004, and Baser, 2006, for a survey of techniques). Gu and Rosenbaum (1993) compare different multivariate matching methods and Froelich (2004) gives a review of the small sample properties of various matching estimators. The choice between different propensity score matching estimators can be made based on the distribution of the propensity scores in the treatment group and the control group (see Dehejia & Wahba, 2002, for details). Finally, see Ho, Imai, King, and Stuart (2005) for an implementation of the Mahalanobis matching inR, Leuven and Sianesi (2003) for aStatapackage and Parsons (2004) for an implementation of different matching estimators inSAS.
Statistical Inference The ATE–estimator based on the propensity score matched samples can be com-puted as a simple mean difference. A simple t-test for unpaired samples is suggested to test the hypothesis of no average total effect (see, e. g., Schafer & Kang, 2008). This procedure is valid only for known propensity scores (Imbens, 2004), a limitation which is often ignored in applied research.10
Bootstrapping approaches are implemented for the computation of standard errors for the average total effect based on matching estimators (see, e. g., Becker & Ichino, 2002), but Abadie and Imbens (2006) showed that bootstrapping yields biased variance estimates for matching estimators. As discussed, e. g., by Stuart (2008), with respect to valid standard errors for matching estimators there is no consensus in the statistical literature (see also Hill & Reiter, 2006).
Finally, note that an interesting conclusion was drawn by Ho, Imai, King, and Stuart (2007, p. 224, nomenclature changed), who described matching as nonparametric preprocessing to reduce model depen-dence (i. e., to make the adjustment more robust against misspecified functional form assumptions for the outcome model or the assignment model): “We thus take advantage of a common feature of all the methods of computing uncertainty estimates associated with regression-type parametric methods: They are all con-ditional on the pretreatment variables Z (and X ), which are therefore treated as fixed and exogenous. Since our preprocessing procedures modify the raw data only in ways that are solely a function of Z , a reasonable method for defining uncertainty is to continue to treat Z , and thus our entire preprocessing procedures, as fixed”. In section 3.2.5 we will derive under which conditions the mentioned assumption of fixed regressors (i. e., the treatment of covariates and the treatment variable as non-stochastic regressors) yields unbiased standard errors of the average total effect.
10See, for instance, Gelman and Hill (2007, p. 212): “The standard errors presented for the analyses fitted to matched samples are not technically correct. First, matching induces correlation among the matched observations. The regression model, however, if correctly specified, should account for this by including the variables used to match. Second, our uncertainty about the true propensity score is not reflected in our calculations. This issue has no perfect solution to date and is currently under investigation by researchers in this field.”