ate.lik {iWeigReg}

Non-calibrated likelihood estimator for the causal-inference setup

Description

This function implements the non-calibrated (or non-doubly robust) likelihood estimator of the average treatment effect in causal inference in Tan (2006), JASA.

Usage

ate.lik(y, tr, p, g0,g1, X=NULL, evar=TRUE, inv="solve")

Arguments

y

A vector of observed outcomes.

tr

A vector of treatment indicators (=1 if treated or 0 if untreated).

p

A vector of known or fitted propensity scores.

g0

A matrix of calibration variables for treatment 0 (see the details).

g1

A matrix of calibration variables for treatment 1 (see the details).

X

The model matrix for the propensity score model, assumed to be logistic (set X=NULL if p is known or treated to be so).

evar

Logical; if FALSE, no variance estimation.

inv

Type of matrix inversion, set to "solve" (default) or "ginv" (which can be used in the case of computational singularity).

Details

The columns of g0 (or respectively g1) correspond to calibration variables for treatment 0 (or treatment 1), which can be specified to include a constant and the fitted outcome regression function for treatment 0 (or treatment 1). See the examples below. In general, a calibration variable is a function of measured covariates selected to exploit the fact that its weighted treatment-specific mean should equal to its unweighted population mean.

To estimate the propensity scores, a logistic regression model is assumed. The model matrix X does not need to be provided and can be set to NULL, in which case the estimated propensity scores are treated as known in the estimation. If the model matrix X is provided, then the "score," (tr-p)X, from the logistic regression is used to generate additional calibration constraints in the estimation. This may sometimes lead to unreliable estimates due to multicollinearity, as discussed in Tan (2006). Therefore, this option should be used with caution.

Variance estimation is based on asymptotic expansions in Tan (2013). Alternatively, resampling methods (e.g., bootstrap) can be used.

Value

mu

The estimated means for treatments 1 and 0.

diff

The estimated average treatment effect.

v

The estimated variances of mu, if evar=TRUE.

v.diff

The estimated variance of diff, if evar=TRUE.

w

The vector of calibrated weights.

lam

The vector of lambda maximizing the log-likelihood.

norm

The maximum norm (i.e., L_\infty norm) of the gradient of the log-likelihood at lam.

conv

Convergence status from trust.

References

Tan, Z. (2006) "A distributional approach for causal inference using propensity scores," Journal of the American Statistical Association, 101, 1619-1637.

Tan, Z. (2010) "Bounded, efficient and doubly robust estimation with inverse weighting," Biometrika, 97, 661-682.

Tan, Z. (2013) "Variance estimation under misspecified models," unpublished manuscript, http://www.stat.rutgers.edu/~ztan.

Examples

data(KS.data)
attach(KS.data)
z=cbind(z1,z2,z3,z4)
x=cbind(x1,x2,x3,x4)

#logistic propensity score model, correct
ppi.glm <- glm(tr~z, family=binomial(link=logit))

X <- model.matrix(ppi.glm)
ppi.hat <- ppi.glm$fitted

#outcome regression model, misspecified
y.fam <- gaussian(link=identity)

eta1.glm <- glm(y ~ x, subset=tr==1, 
               family=y.fam, control=glm.control(maxit=1000))
eta1.hat <- predict.glm(eta1.glm, 
               newdata=data.frame(x=x), type="response")

eta0.glm <- glm(y ~ x, subset=tr==0, 
               family=y.fam, control=glm.control(maxit=1000))
eta0.hat <- predict.glm(eta0.glm, 
               newdata=data.frame(x=x), type="response")

#ppi.hat treated as known
out.lik <- ate.lik(y, tr, ppi.hat, 
                     g0=cbind(1,eta0.hat),g1=cbind(1,eta1.hat))
out.lik$diff
out.lik$v.diff

#ppi.hat treated as estimated
out.lik <- ate.lik(y, tr, ppi.hat, 
                     g0=cbind(1,eta0.hat),g1=cbind(1,eta1.hat), X)
out.lik$diff
out.lik$v.diff