estimate.ATE {CausalGAM}

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

pscore.formula

A formula expression for the propensity score model. See the documentation of gam for details.

pscore.family

A description of the error distribution and link function to be used for the propensity score model. See the documentation of gam for details.

outcome.formula.t

A formula expression for the outcome model under active treatment. See the documentation of gam for details.

outcome.formula.c

A formula expression for the outcome model under control. See the documentation of gam for details.

outcome.family

A description of the error distribution and link function to be used for the outcome models. See the documentation of gam for details.

treatment.var

A character variable giving the name of the binary treatment variable in data. If treatment.var is a numeric variable, it is assumed that control corresponds to sort(unique(treatment.values))[1] and treatment corresponds to sort(unique(treatment.values))[2]. If treatment.var is a factor, it is assumed that control corresponds to levels(treatment.values)[1] and treatment corresponds to levels(treatment.values)[2].

data

A non-optional data frame containing the variables in the model. data cannot contain any missing values.

divby0.action

A character variable describing what action to take when some estimated propensity scores are less than divby0.tol or greater than 1 - \code{divby0.tol}. Options include: ‘⁠fail⁠’ (abort the call to estimate.ATE), ‘⁠truncate⁠’ (set all estimated propensity scores less than divby0.tol equal to divby0.tol and all estimated propensity scores greater than 1 - \code{divby0.tol} equal to 1 - \code{divby0.tol}), and ‘⁠discard⁠’ (discard units that have estimate propensity scores less than divby0.tol or greater than 1 - \code{divby0.tol}). Note that discarding units will change the estimand.

divby0.tol

A scalar in [0,0.5) giving the tolerance level for extreme propensity scores. Defaults to 1e-8. See divby0.action for details.

nboot

Number of bootrap replications used for calculating bootstrap standard errors. If nboot is less than or equal to 0 then bootstrap standard errors are not calculated. Defaults to 501.

variance.smooth.deg

The degree of the loess smooth used to calculate the conditional error variance of the outcome models given the estimated propensity scores. Possible values are 0, 1, or 2. Defaults to 1. If set to a value less than 0 than the conditional error variance will not be calculated and the estimated asymptotic standard errors will not be reported. See lo for details.

variance.smooth.span

The span of the loess smooth used to calculate the conditional error variance of the outcome models given the estimated propensity scores. Defaults to 10.75. If set to a value less than or equal to 0 than the conditional error variance will not be calculated and the estimated asymptotic standard errors will not be reported.See lo for details.

var.gam.plot

Logical value indicating whether the estimated conditional variances should be plotted against the estimated propensity scores. Setting var.gam.plot to TRUE is useful for judging whether variance.smooth.deg and variance.smooth.span were set appropriately. Defaults to TRUE.

suppress.warnings

Logical value indicating whether warnings from the gam fitting procedures should be suppressed from printing to the screen. Defaults to TRUE.

...

Further arguments to be passed.

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:

ATE.AIPW.hat

AIPW estimate of ATE.

ATE.reg.hat

Regression estimate of ATE.

ATE.IPW.hat

IPW estimate of ATE.

ATE.AIPWsand.SE

Empirical sandwich standard error for ATE.AIPW.hat.

ATE.AIPW.asymp.SE

Estimated asymptotic standard error for ATE.AIPW.hat.

ATE.reg.asymp.SE

Estimated asymptotic standard error for ATE.reg.hat.

ATE.IPW.asymp.SE

Estimated asymptotic standard error for ATE.IPW.hat.

ATE.AIPW.bs.SE

Estimated bootstrap standard error for ATE.AIPW.hat.

ATE.reg.bs.SE

Estimated bootstrap standard error for ATE.reg.hat.

ATE.IPW.bs.SE

Estimated bootstrap standard error for ATE.IPW.hat.

ATE.AIPW.bs

Vector of bootstrap replications of ATE.AIPW.hat.

ATE.reg.bs

Vector of bootstrap replications of ATE.reg.hat.

ATE.IPW.bs

Vector of bootstrap replications of ATE.IPW.hat.

gam.t

gam object from fitted outcome model under treatment.

gam.c

gam object from the fitted outcome model under control.

gam.ps

gam object from the fitted propensity score model.

truncated.indic

Logical vector indicating which rows of data had extreme propensity scores truncated.

discarded.indic

Logical vector indicating which rows of data were discarded because of extreme propensity scores.

treated.value

Value of treatment.var that corresponds to active treatment.

control.value

Value of treatment.var that corresponds to control.

treatment.var

treatment.var

n.treated.prediscard

Number of treated units before any truncations or discards.

n.control.prediscard

Number of control units before any truncations or discards.

n.treated.postdiscard

Number of treated units after truncations or discards.

n.control.postdiscard

Number of control units after truncations or discards.

pscores.prediscard

Estimated propensity scores before any truncations or discards.

pscores.postdiscard

Estimated propensity scores after truncations or discards.

cond.var.t

Vector of conditional error variances for the outcome for each unit under treatment given the unit's estimated propensity score.

cond.var.c

Vector of conditional error variances for the outcome for each unit under control given the unit's estimated propensity score.

call

The initial call to estimate.ATE.

data

The data frame sent to estimate.ATE.

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)