## Causal inference with general treatment regimes: Generalizing the propensity score (2004)

Venue: | Journal of the American Statistical Association |

Citations: | 32 - 7 self |

### BibTeX

@ARTICLE{Imai04causalinference,

author = {Kosuke Imai and David A. Van Dyk},

title = {Causal inference with general treatment regimes: Generalizing the propensity score},

journal = {Journal of the American Statistical Association},

year = {2004},

volume = {99},

pages = {854--866}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this article we develop the theoretical properties of the propensity function, which is a generalization of the propensity score of Rosenbaum and Rubin. Methods based on the propensity score have long been used for causal inference in observational studies; they are easy to use and can effectively reduce the bias caused by nonrandom treatment assignment. Although treatment regimes need not be binary in practice, the propensity score methods are generally confined to binary treatment scenarios. Two possible exceptions have been suggested for ordinal and categorical treatments. In this article we develop theory and methods that encompass all of these techniques and widen their applicability by allowing for arbitrary treatment regimes. We illustrate our propensity function methods by applying them to two datasets; we estimate the effect of smoking on medical expenditure and the effect of schooling on wages. We also conduct simulation studies to investigate the performance of our methods.

### Citations

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Citation Context ...dinal treatment is also a special case of the propensity function. We can use the same setup as in the example with a categorical treatment, except that we model π(X) using an ordinal logistic model (=-=McCullagh and Nelder 1989-=-). In this case π(X) is determined by the scalar X ⊤ β,whereβ is a (p × 1) parameter vector; in the general framework ψ = β and θψ(X) = X ⊤ β.Lu et al. (2001) mentioned the possibility of using Gaussi... |

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Citation Context ...oth coefficient model by letting the causal effect as well as an intercept vary smoothly as a function of ˆθ. In parallel with the aforementioned twopart model, we use the generalized additive model (=-=Hastie and Tibshirani 1990-=-) with the binomial family and logistic link to model the probability of positive medical costs, and use the Gaussian family and identity link to model the conditional distribution of log(Y ). We fit ... |

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Citation Context ...tional studies. In fact, there exists empirical evidence that in certain situations the propensity score method produces more reliable estimates of causal effects than other estimation methods (e.g., =-=Dehejia and Wahba 1999-=-; Imai 2004). The propensity score is called a balancing score because, conditional on the propensity score, the binary treatment assignment and the observed covariates are independent (Rosenbaum and ... |

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Citation Context ...ysis. The effect of education on income has long been an important topic in economics; researchers have quantified the effect by comparing years of education and individual wage in IV analyses (e.g., =-=Angrist and Krueger 1991-=-, 1992; Card 1995; Kling 2001). But the use of IV estimation in observational studies is vulnerable to criticism concerning the validity of the instrument (e.g., Bound, Jaeger, and Baker 1995). Thus i... |

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Citation Context ...onstraints, then θψ(X) = π(X) is a tmax-dimensional parameter that corresponds to the set of tmax propensity scores proposed by Imbens (2000). We might use nested logistic regression (as suggested in =-=Imbens 2000-=-) or a multinomial probit model (e.g., Imai and van Dyk 2004) to model the dependence of π(X) on X; in either case, ψ represents the regression coefficients. Example With an Ordinal Treatment. The pro... |

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Citation Context ...al outcomes and potential treatment assignments given X, (b) be monotonically predictive of the treatment assignment given X, and (c) affect only the outcome variable through the treatment variables (=-=Angrist and Imbens 1995-=-; Angrist, Imbens, and Rubin 1996). As Angrist and Krueger (1995) pointed out, the key here is that the assignment mechanism only for the lottery code (and not that for education level) needs to be st... |

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Citation Context ...of Rosenbaum and Rubin (1983b) has found wide applicability in empirical research; in particular, the method has rapidly become popular in the social sciences (e.g., Heckman, Ichimura, and Todd 1998; =-=Lechner 1999-=-; Imai 2004). The propensity score aims to control for differences between the treatment groups when the treatment is binary; it is defined as the conditional probability of assignment to the treatmen... |

65 |
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Citation Context ...and Wahba 1999; Imai 2004). The propensity score is called a balancing score because, conditional on the propensity score, the binary treatment assignment and the observed covariates are independent (=-=Rosenbaum and Rubin 1983-=-b). If we further assume the conditional independence between treatment assignment and potential outcomes given the observed covariates, then it is possible to obtain unbiased estimates of treatment e... |

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Citation Context ...of pretreatment covariates, Xi, the possibly multivariate value of the treatment received, TA i ,andthe value of the outcome variable associated with this treatment, Yi. Using the Rubin causal model (=-=Holland 1986-=-) as a framework for causal inference, we define a set of potential outcomes, Y ={Yi(tP ), tP ∈ T for i = 1,...,n}, whereTisa set of potential treatment values and Yi(tP ) is a random variable that ma... |

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Citation Context ...iple factors and their interactions. In political science, one may be interested in the combined effects of different voter mobilization strategies, such as phone calls and door-to-door visits (e.g., =-=Gerber and Green 2000-=-). Treatment can also be measured in terms of frequency and duration, for example, the health effects of smoking. These examples illustrate the need to extend the propensity score, a prominent methodo... |

40 |
Interpreting Instrumental Variable Estimates of the Returns to Schooling”, Working Paper 415
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Citation Context ... long been an important topic in economics; researchers have quantified the effect by comparing years of education and individual wage in IV analyses (e.g., Angrist and Krueger 1991, 1992; Card 1995; =-=Kling 2001-=-). But the use of IV estimation in observational studies is vulnerable to criticism concerning the validity of the instrument (e.g., Bound, Jaeger, and Baker 1995). Thus improving the performance of I... |

37 | Jackknife Instrumental Variables Estimation - ANGRIST, IMBENS, et al. - 1999 |

36 |
Nonparametric regression techniques in economics. Journal of Economic Literature 36, no. 2: 669–721. Merlin Mack Hanauer was born July 22, 1978 in Forks of Salmon
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Citation Context ...03). In the context of linear regression, we can allow the regression coefficients to vary with θ, in what is known as a smooth coefficient model (DiNardo and Tobias 2001; Li, Huang, Li, and Fu 2002; =-=Yatchew 1998-=-). We illustrate this strategy with a continuous treatment in Section 3.3 and with a bivariate treatment in Section 4.3. Known Propensity Functions. Even if the true propensity function is known, adju... |

32 |
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Citation Context ...ed by logistic regression (Rosenbaum and Rubin 1984, 1985). The advantage of using estimated propensity scores in place of true propensity scores has been discussed at length in the literature (e.g., =-=Rosenbaum 1987-=-; Robins, Rotnitzky, and Zhao 1995; Rubin and Thomas 1996; Heckmen et al. 1998; Hirano, Imbens, and Ridder 2003); see also Section 5.3. Indeed, even in randomized experiments where the randomization s... |

31 |
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Citation Context ...ent may not be binary or even categorical. For example, in clinical trials, one may be interested in estimating the dose-response function where the drug dose may take on a continuum of values (e.g., =-=Efron and Feldman 1991-=-). Alternatively, the treatment may be ordinal. In economics, an important quantity of interest is the effect of schooling on wages, where schooling is measured as years of education in school (e.g., ... |

31 |
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Citation Context ...hen model the conditional distribution of log(Y ), given T A and X, for those individuals who reported positive medical expenditure, p(log(Y )|Y >0,TA , X), using Gaussian linear regression (see also =-=Olsen and Schafer 2001-=-; Javaras and van Dyk 2003). Using this two-part model, we estimate the effects of smoking on medical costs within each of the 10 subclasses. Finally, we compute the weighted average of the 10 within-... |

29 | Nonparametric density and regression estimation - DINARDO, TOBIAS - 2001 |

29 |
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Citation Context ...nd to attain higher levels of education and might be expected to earn higher wages for any given level of education they might have attained. Without controlling for a richer set of covariates (e.g., =-=Rouse 1995-=-), Assumption 2 is unjustifiable. Our criticism of the ignorability assumption is substantive in nature; Rosenbaum and Rubin (1983a) described a method for quantifying the sensitivity of results to As... |

29 |
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Citation Context ... Rather than fixing φ in each of several subclasses, we can allow φ to vary smoothly as a function of θ; that is, by computing φ(θ) using a flexible model, such as penalized regression splines (e.g., =-=Wood 2003-=-). In the context of linear regression, we can allow the regression coefficients to vary with θ, in what is known as a smooth coefficient model (DiNardo and Tobias 2001; Li, Huang, Li, and Fu 2002; Ya... |

28 | Do Get-Out-The-Vote Calls Reduce Turnout? The Importance of Statistical Methods for Field Experiments
- Imai
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Citation Context ...nd Rubin (1983b) has found wide applicability in empirical research; in particular, the method has rapidly become popular in the social sciences (e.g., Heckman, Ichimura, and Todd 1998; Lechner 1999; =-=Imai 2004-=-). The propensity score aims to control for differences between the treatment groups when the treatment is binary; it is defined as the conditional probability of assignment to the treatment group giv... |

28 | A Bayesian analysis for the multinomial probit model using marginal data augmentation - Imai, Dyk - 2005 |

27 | Matching with doses in an observational study of a media campaign against drug abuse - Lu, Zanutto, et al. - 2001 |

25 |
Effects of Misspecification of the Propensity Score on Estimators of Treatment Effect
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Citation Context ... measure. In practice, ignorability is a nontrivial assumption that should be made only with great care; omitting covariates can seriously bias estimates of causal effects (Rosenbaum and Rubin 1983a; =-=Drake 1993-=-); see also Section 5. For clarity, we maintain Assumptions 1 and 2 and discuss generalization of the propensity score method under these assumptions. When making causal inference, the distribution p{... |

23 |
Matching to Remove Bias
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(Show Context)
Citation Context ... as described earlier. To assess our methods, however, we generate the outcome variable using various known functions of the covariates and treatment variables. In particular, we follow others (e.g., =-=Rubin 1973-=-, 1979; Rubin and Thomas 2000) and use an exponential function to create models with varying degrees of nonlinearity and nonadditivity. Specifically, we closely follow the simulation studies described... |

19 | Estimating and using propensity scores with partially missing data - D’Agostino, Rubin - 2000 |

19 |
Af�nely invariant matching methods with ellipsoidal distributions. Annals of Statistics 1992; 20: 1079–93. 28 Rubin DB, Thomas N. Characterizing the effect of matching using linear propensity score methods with normal covariates
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Citation Context ...large literature on the advantage of adjusting for the estimated propensity score rather than the true propensity score in both observational studies and randomized experiments (e.g., Rosenbaum 1987; =-=Rubin and Thomas 1992-=-, 1996; Hill, Rubin, and Thomas 1999). The advantage of the estimated propensity score can be understood by identifying two types of errors that can occur when estimating treatment effects. First, the... |

14 | Semiparametric Smooth Coefficient Models - Li, Huang, et al. - 2002 |

12 | Propensity Scores - Joffe, Rosenbaum - 1999 |

9 |
Estimating Outcome Distributions for
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Citation Context ...elative weight of subclass j. Equation (2) shows how we can approximate the marginal distributions of the potential outcomes. Although these distributions are sometimes appropriate in practice (e.g., =-=Imbens and Rubin 1997-=-), more often they are summarized by the relevant causal effect. This causal effect is generally a function of φ, for instance, the regression coefficient of Y(t P ) on t P . In practice, additional a... |

8 |
Comment on Inference for semiparametric models: some questions and an answer, by
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Citation Context ...n t P . In practice, additional adjustment within each subclass is desirable to further reduce bias. For example, some authors suggest adjusting for the covariates in the within-subclass model (e.g., =-=Robins and Rotnitzky 2001-=-). We believe that this is generally a useful strategy for accounting for the within-subclass variability of θ, and thus we include available covariates when fitting the within-subclass models in our ... |

5 | Disease cases and their medical costs attributable to smoking: An analysis of the national medical expenditure survey - Johnson, Dominici, et al. - 2003 |

3 | On the application of probability theory to agricultural experiments. Essay on principles. Section 9 - Dabrowska, Speed - 1990 |

2 | Multiple imputation for incomplete data with semicontinuous variables - Javaras, Dyk - 2003 |

1 |
An Analysis of Survey Data on Smoking Using Propensity Scores
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(Show Context)
Citation Context ... scores. Because this method is confined to a binary treatment, the focus has been on the comparison of smokers and nonsmokers without distinguishing among smokers based on how much they smoke (e.g., =-=Larsen 1999-=-; Rubin 2001). In contrast, our proposed method can estimate the causal effects of the frequency and duration of smoking. We use the data that Johnson, Dominici, Griswold, and Zeger (2003) extracted f... |