Check Post-Weighting Balance for (A)IPW Estimators Using Generalized Additive Models

Description

This function calculates weighted means of covariates where weights in inverse propensity weights and then examines the differences in the weighted means across treated and control units as a diagnostic for covariate balance.

Usage

balance.IPW(pscore.formula, pscore.family,
             treatment.var, outcome.var, data = NULL, 
             divby0.action = c("fail", "truncate", "discard"), 
             divby0.tol = 1e-08, nboot = 501, 
             suppress.warnings = TRUE, ...)

Arguments

Details

This function provides diagnostic information that allows a user to judge whether the inverse propensity weights generated from a particular generalized additive model specification result in covariate balance across treated and control groups. The function is intended to be used before the estimate.ATE function in order to find a specification for the propensity score model that results in sufficient covariate balance.

The weighted mean differences between all variables in the dataset passed to balance.IPW are reported along with a z-statistics for these weighted differences. Univariate mean covariate balance is decreasing in the absolute value of the z-statistics (z-statistics closer to 0 imply better univariate mean balance).

Printing the output from balance.IPW will result in a table with k-2 rows (one for each variable other than the treatment and outcome variables) and 6 columns. The columns are (from left to right) the observed mean of the covariate among the treated units, the observed mean of the covariate among the control units, the weighted mean of the covariate among the treated units, the weighted mean of the covariate among the control units, the weighted mean difference, and the z-statistic for the difference.

It is often useful to include interactions and powers of the covariates in the dataset so that balance can be checked for these quantities as well.

Means, mean differences, and z-statistics are only reported for numeric covariates.

Value

An object of class balance with the following attributes:

Author(s)

Adam Glynn, Emory University

Kevin Quinn, University of Michigan

References

Adam N. Glynn and Kevin M. Quinn. 2010. "An Introduction to the Augmented Inverse Propensity Weighted Estimator." Political Analysis.

See Also

gam, estimate.ATE

Examples

## Not run: 
set.seed(1234)
## number of units in sample
n <- 2000


## measured potential confounders
z1 <- rnorm(n)
z2 <- rnorm(n)
z3 <- rnorm(n)
z4 <- rnorm(n)

## treatment assignment
prob.treated <-	pnorm(-0.5 + 0.75*z2)
x <- rbinom(n, 1, prob.treated)

## potential outcomes
y0 <- z4 + rnorm(n)
y1 <- z1 + z2 + z3 + cos(z3*2) + rnorm(n)

## observed outcomes
y <- y0
y[x==1] <- y1[x==1] 	     	


## put everything in a data frame
examp.data <- data.frame(z1, z2, z3, z4, x, y)

## augment data with interactions and powers of covariates
examp.data$z1z1 <- examp.data$z1^2
examp.data$z2z2 <- examp.data$z2^2
examp.data$z3z3 <- examp.data$z3^2
examp.data$z4z4 <- examp.data$z4^2

examp.data$z1z2 <- examp.data$z1 * examp.data$z2
examp.data$z1z3 <- examp.data$z1 * examp.data$z3
examp.data$z1z4 <- examp.data$z1 * examp.data$z4

examp.data$z2z3 <- examp.data$z2 * examp.data$z3
examp.data$z2z4 <- examp.data$z2 * examp.data$z4

examp.data$z3z4 <- examp.data$z3 * examp.data$z4



## check balance of a propensity score model that is not sufficient to
## control confounding bias

bal.1 <- balance.IPW(pscore.formula=x~s(z3)+s(z4),
                     pscore.family=binomial(probit),
                     treatment.var="x",
                     outcome.var="y",
                     data=examp.data,
                     nboot=250)

print(bal.1) ## some big z-statistics here indicating balance not so great


## try again
bal.2 <- balance.IPW(pscore.formula=x~z1+z2+z3+z4,
                     pscore.family=binomial(probit),
                     treatment.var="x",
                     outcome.var="y",
                     data=examp.data,
                     nboot=250)

print(bal.2) ## balance looks much better-- 
             ##    only 1 out of 14 zs > 2.0 in absval


## End(Not run)

Estimate Population Average Treatment Effects (ATE) Using Generalized Additive Models

Description

This function implements three estimators for the population ATE— a regression estimator, an inverse propensity weighted (IPW) estimator, and an augmented inverse propensity weighted (AIPW) estimator— using generalized additive models.

Usage

estimate.ATE(pscore.formula, pscore.family,
             outcome.formula.t, outcome.formula.c, outcome.family, 
             treatment.var, data = NULL, 
             divby0.action = c("fail", "truncate", "discard"), 
             divby0.tol = 1e-08, nboot = 501, 
             variance.smooth.deg = 1, variance.smooth.span = 0.75, 
             var.gam.plot = TRUE, suppress.warnings = TRUE, ...)

Arguments

Details

The three estimators implemented by this function are a regression estimator, an IPW estimator with weights normalized to sum to 1, and an AIPW estimator. Glynn and Quinn (2010) provides details regarding how each of these estimators are implemented. The AIPW estimator requires the specification of both a propensity score model governing treatment assignment and outcome models that describe the conditional expectation of the outcome variable given measured confounders and treatment status. The AIPW estimator has the so-called double robustness property. This means that if either the propensity score model or the outcomes models are correctly specified then the estimator is consistent for ATE.

Standard errors for the regression and IPW estimators can be calculated by either the bootstrap or by estimating the large sample standard errors. The latter approach requires estimation of the conditional variance of the disturbances in the outcome models given the propensity scores (see section IV of Imbens (2004) for details). The accuracy of these standard errors is only as good as one's estimates of these conditional variances.

Standard errors for the AIPW estimator can be calculated similarly. In addition, Lunceford and Davidian (2004) also discuss an empirical sandwich estimator of the sampling variance which is also implemented here.

Value

An object of class CausalGAM with the following attributes:

Author(s)

Adam Glynn, Emory University

Kevin Quinn, University of Michigan

References

Adam N. Glynn and Kevin M. Quinn. 2010. "An Introduction to the Augmented Inverse Propensity Weighted Estimator." Political Analysis.

Guido W. Imbens. 2004. "Nonparametric Estimation of Average Treatment Effects Under Exogeneity: A Review." The Review of Economics and Statistics. 86: 4-29.

Jared K. Lunceford and Marie Davidian. 2004. "Stratification and Weighting via the Propensity Score in Estimation of Causal Treatment Effects: A Comparative Study." Statistics in Medicine. 23: 2937-2960.

See Also

gam, balance.IPW

Examples

## Not run: 
## a simulated data example with Gaussian outcomes
## 

## number of units in sample
n <- 2000


## measured potential confounders
z1 <- rnorm(n)
z2 <- rnorm(n)
z3 <- rnorm(n)
z4 <- rnorm(n)

## treatment assignment
prob.treated <-	pnorm(-0.5 + 0.75*z2)
x <- rbinom(n, 1, prob.treated)

## potential outcomes
y0 <- z4 + rnorm(n)
y1 <- z1 + z2 + z3 + cos(z3*2) + rnorm(n)

## observed outcomes
y <- y0
y[x==1] <- y1[x==1] 	     	


## put everything in a data frame
examp.data <- data.frame(z1, z2, z3, z4, x, y)


## estimate ATE
##
## in a real example one would want to use a larger number of 
## bootstrap replications
##
ATE.out <- estimate.ATE(pscore.formula = x ~ s(z2),
                        pscore.family = binomial,
                        outcome.formula.t = y ~ s(z1) + s(z2) + s(z3) + s(z4),
                        outcome.formula.c = y ~ s(z1) + s(z2) + s(z3) + s(z4),
      			outcome.family = gaussian,
			treatment.var = "x",
                        data=examp.data,
                        divby0.action="t",
                        divby0.tol=0.001, 
                        var.gam.plot=FALSE,
			nboot=50)      	   	


## print summary of estimates
print(ATE.out)




## a simulated data example with Bernoulli outcomes
## 

## number of units in sample
n <- 2000


## measured potential confounders
z1 <- rnorm(n)
z2 <- rnorm(n)
z3 <- rnorm(n)
z4 <- rnorm(n)

## treatment assignment
prob.treated <-	pnorm(-0.5 + 0.75*z2)
x <- rbinom(n, 1, prob.treated)

## potential outcomes
p0 <- pnorm(z4)
p1 <- pnorm(z1 + z2 + z3 + cos(z3*2))
y0 <- rbinom(n, 1, p0)
y1 <- rbinom(n, 1, p1)

## observed outcomes
y <- y0
y[x==1] <- y1[x==1] 	     	


## put everything in a data frame
examp.data <- data.frame(z1, z2, z3, z4, x, y)


## estimate ATE
##
## in a real example one would want to use a larger number of 
## bootstrap replications
##
ATE.out <- estimate.ATE(pscore.formula = x ~ s(z2),
                        pscore.family = binomial,
                        outcome.formula.t = y ~ s(z1) + s(z2) + s(z3) + s(z4),
                        outcome.formula.c = y ~ s(z1) + s(z2) + s(z3) + s(z4),
      			outcome.family = binomial,
			treatment.var = "x",
                        data=examp.data,
                        divby0.action="t",
                        divby0.tol=0.001,
                        var.gam.plot=FALSE, 
			nboot=50)      	   	


## print summary of estimates
print(ATE.out)		     

## End(Not run)