Estimating and Using Propensity Score in Presence of Missing Background Data. An Application to Assess the Impact of Childbearing on Wellbeing Alessandra Mattei Dipartimento di Statistica “G. Parenti” Universit` a degli Studi di Firenze mattei@ds.unifi.it
Outline 1. Motivation of the study 2. Estimating causal effects through a quasi-experimental approach 3. Estimating propensity scores with incomplete data 4. Estimating the causal effects of a childbearing on economic wellbeing in Indonesia using the Indonesia Family Life Survey (IFLS) 5. Concluding remarks
Motivation of the Study • We compare three different approaches of handling missing background data in the estimation and use of propensity scores: 1. A complete-case analysis 2. A pattern-mixture model based approach developed by Rosenbaum and Rubin (1984) 3. A multiple imputation approach • We make explicit the assumptions underlying each approach by illustrating the interaction between the treatment assignment mechanism and the missing data mechanism • We apply these methods to assess the impact of childbearing events on individuals’ wellbeing in Indonesia, using a sample of women from the Indonesia Family Life Survey
The Quasi-Experimental Approach • We use appropriate econometric techniques based on longitudinal micro data in order to identify the causal effects of childbearing events on poverty • We consider the endogenous variable of interest, here change in fertility, as treatment variable Z , and divides individuals into two groups: – those who experienced a childbirth - the treatment group, indicated by Z = T , and – those who did not - the control group, indicated by Z = C • The outcome variable, say Y , is a measure of wellbeing • Strong Ignorability Assumption (Rosenbaum and Rubin, 1983) (i) Z is independent of the potential outcomes ( Y ( C ) , Y ( T )) conditional on X = x ( Unconfoundedness Assumption ) (ii) η < Pr ( Z = 1 | X = x ) < 1 − η , for some η > 0
The Unconfoundedness Assumption The unconfoundedness assumption requires that all variables that affect both outcome and the likelihood of receiving the treatment are observed or that all the others are perfectly collinear with the observed ones • This assumption is not testable, it is a very strong assumption, and one that need not generally be applicable • Selection may also take place on the basis of unobservable characteristics We view it as a useful starting point for two reasons 1. In our study, we have carefully investigated which variables are most likely to confound any comparison between treated and control units 2. Any alternative assumptions that not rely on unconfoundedness, while allowing for consistent estimation of the causal effects of interest, must make alternative untestable assumptions
The Propensity Score • The propensity score is the conditional probability of receiving a particular treatment ( Z = T ) versus control ( Z = C ) given a vector of observed covariates, X , e = e ( X ) = Pr ( Z = T | X ) • Balancing of pre-treatment variables given the propensity score If e ( X ) is the propensity score, then Z ⊥ X | e ( X ) • Unconfoundedness given the propensity score Z ⊥ ( Y ( C ) , Y ( T )) | X = ⇒ Z ⊥ ( Y ( C ) , Y ( T )) | e ( X )
Notation Let the response indicator be 1 , if the value of the k covariate for the i th subject is observed R ik = 0 , if the value of the k covariate for the i th subject is missing for i = 1 , . . . , N and k = 1 , . . . , K . Let X = ( X obs , X mis ), where X obs = { X ik : R ik = 1 } X mis = { X ik : R ik = 0 } e
Estimating Propensity Score with Incomplete Data • It is not clear how the propensity score should be estimated when some covariate values are missing • The missingness itself may be predictive about which treatment is received • Any technique for estimating propensity score in the presence of covariate missing data will have to either make a stronger assumption regarding ignorability of the assignment mechanism or will have to make an assumption about the missing data mechanism • In order to have ignorability of the assignment mechanism, for all of the techniques here described, we will maintain the following assumption: Pr ( Z | X , R, Y ( C ) , Y ( T )) = Pr ( Z | X , R )
Complete-Data Analysis • A complete-data analysis uses only observations where all variables are observed • To make valid causal inferences with this approach we require that data is Missing Completely At Random (MCAR, Little and Rubin): 1987): Pr( R | X , Z ) = Pr( R ) – This means that the units removed from the data set, those with missing data, are just a simple random sample of the other Note that Pr ( Z | X , R, Y ( C ) , Y ( T )) = Pr ( Z | X , R ) and Pr( R | X , Z ) = Pr( R ) ⇓ Pr ( Z | X , R, Y ( C ) , Y ( T )) = Pr ( Z | X )
Rosenbaum - Rubin Approach The Propensity Scores with Incomplete Data The generalized propensity score, which conditions on all of the observed covariate information, is � � e ∗ = e ∗ ( X obs , R ) = Pr Z = T | X obs , R • Balancing of pre-treatment variables given the generalized propensity score � � X obs , R | e ∗ ( X obs , R ) Z ⊥ • Unconfoundedness given the generalized propensity score � � X obs , R ⇒ Z ⊥ ( Y ( C ) , Y ( T )) | e ∗ ( X obs , R ) Z ⊥ ( Y ( C ) , Y ( T )) | =
Rosenbaum - Rubin Approach Assumptions The Rosenbaum-Rubin method relies on either one of the following assumptions: Pr( Z | X , R ) ≡ Pr( Z | X obs , X mis , R ) = Pr( Z | X obs , R ) or Pr( Y ( C ) , Y ( T ) | X obs , X mis , R ) Pr( Y ( C ) , Y ( T ) | X , R ) ≡ Pr( Y ( C ) , Y ( C ) | X obs , R ) = • The Rosenbaum - Rubin method does not make any assumption about the missing data mechanism
Rosenbaum - Rubin Approach Drawbacks • The Rosenbaum - Rubin method does assume that either – all missing covariate values are independent of the the assignment mechanism conditional on the missing data patterns or – or that they are independent of the potential outcomes conditional on observed covariate values and the missing data patterns • Since the Rosenbaum - Rubin method specifies one model for both handling missing data and estimating propensity scores, it is not possible to incorporate the outcome variable Y into this model even though it might provide useful information about missing values
Multiple Imputation and Propensity Score Methods The latent ignorability of the assignment mechanism Using Multiple Imputation (MI) to handling incomplete data covariates, we essentially assume the latent ignorability of the assignment mechanism Pr( Z | X , R, Y ( C ) , Y ( T )) = Pr( Z | X ) . • In our case, the assignment mechanism is ignorable only conditional on complete covariate data (which includes, of course, values that in practice are missing) • Computationally, this is achieved by filling in the missing covariate values using MI
Multiple Imputation and Propensity Score Methods Assumptions on the assignment mechanism • Imputations may in principle be created under any kind of model for the missing data mechanism, and the resulting inferences will be valid under that mechanism (Rubin, 1987) • In our application, MI was performed assuming that the missing observations are Missing At Random (MAR), that is, Pr( R | X , Z, Y ( C ) , Y ( T )) = Pr( R | X obs , Z, Y obs ) , � � n where Y obs = Y obs i =1 , Y obs = I { Z i = T } Y i ( T ) + I { Z i = C } Y i ( C ) i i – This MAR assumption involves all the observed variables – In our application, we perform MI in two way: ∗ including Y in the model, and ∗ not including Y in the imputation model
Multiple Imputation and Propensity Score Methods Estimators Let � ATT l and se 2 l denote the point estimate and variance respectively from the l th ( l = 1 , . . . , m ) dataset. Then, � m 1 � � ATT = ATT l m l =1 � � � � 1 + 1 � se 2 W + se 2 ATT = V ar B m where � m se 2 1 l =1 se 2 = Within-imputation variance W l m � � 2 � m ATT l − � � se 2 1 = ATT Between-imputation variance B m − 1 l =1 • In our application, MI was performed using the mvis module in STATA (Patrick Royston, 2004), which is based on MICE method of multiple multivariate imputation (van Buuren et al., 1999)
Matching Estimators of the ATT Effect based on the Propensity Score • The Nearest Neighbor Matching Estimator • The Kernel Matching Estimator • The Stratification Matching Estimator Irrespective of the method of handling missing data, the propensity score analysis is implemented by the use of the pscore module in STATA written by Becker and Ichino (2000)
The Indonesia Family Life Survey Data • The IFLS consists of three waves (1993, 1997, 2000) plus a special wave (1998), which we will not use in our study • We will use a subsample of panel ever-married women age 15-49 • In our study the outcome variable is a measure of monetary wellbeing, given by the annual value of the total household consumption expenditures adjusted for price variability across space and time and household heterogeneity – Adjustment for price variability ∗ We divided the nominal consumption expenditures by the national consumption price index (IFS, 2002) – Adjustment for household heterogeneity ∗ We adjust our income-based measure of wellbeing for household heterogeneity by applying the following equivalence scale: � Total number of persons in the household
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