CBPS: Covariate Balancing Propensity Score (CBPS) Estimation

Description

CBPS estimates propensity scores such that both covariate balance and prediction of treatment assignment are maximized. The method, therefore, avoids an iterative process between model fitting and balance checking and implements both simultaneously. For cross-sectional data, the method can take continuous treatments and treatments with a control (baseline) condition and either 1, 2, or 3 distinct treatment conditions.

Usage

CBPS(formula, data, na.action, ATT = 1, iterations = 1000, 
	       standardize = TRUE, method = "over", twostep = TRUE, 
		   sample.weights = NULL, baseline.formula = NULL, 
		   diff.formula = NULL, ...)
	  CBPS.fit(treat, X, baselineX, diffX, ATT, method, 
			   iterations, standardize, twostep, 
			   sample.weights, ...)

Arguments

formula

An object of class formula (or one that can be coerced to that class): a symbolic description of the model to be fitted.

data

An optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which CBPS is called.

na.action

A function which indicates what should happen when the data contain NAs. The default is set by the na.action setting of options, and is na.fail if that is unset.

ATT

Default is 1, which finds the average treatment effect on the treated interpreting the second level of the treatment factor as the treatment. Set to 2 to find the ATT interpreting the first level of the treatment factor as the treatment. Set to 0 to find the average treatment effect. For non-binary treatments, only the ATE is available.

iterations

An optional parameter for the maximum number of iterations for the optimization. Default is 1000.

standardize

Default is TRUE, which normalizes weights to sum to 1 within each treatment group. For continuous treatments, normalizes weights to sum up to 1 for the entire sample. Set to FALSE to return Horvitz-Thompson weights.

method

Choose "over" to fit an over-identified model that combines the propensity score and covariate balancing conditions; choose "exact" to fit a model that only contains the covariate balancing conditions.

twostep

Default is TRUE for a two-step estimator, which will run substantially faster than continuous-updating. Set to FALSE to use the continuous-updating estimator described by Imai and Ratkovic (2014).

treat

A vector of treatment assignments. Binary or multi-valued treatments should be factors. Continuous treatments should be numeric.

A covariate matrix.

sample.weights

Survey sampling weights for the observations, if applicable. When left NULL, defaults to a sampling weight of 1 for each observation.

baseline.formula

Used only to fit iCBPS (see Fan et al). Currently only works with binary treatments. A formula specifying the balancing covariates in the baseline outcome model, i.e., E(Y(0)|X).

diff.formula

Used only to fit iCBPS (see Fan et al). Currently only works with binary treatments. A formula specifying the balancing covariates in the difference between the treatment and baseline outcome model, i.e., E(Y(1)-Y(0)|X).

baselineX

Similar to baseline.formula, but in matrix form.

diffX

Similar to diff.formula, but in matrix form.

...

Other parameters to be passed through to optim().

Value

fitted.values

The fitted propensity score

deviance

Minus twice the log-likelihood of the CBPS fit

weights

The optimal weights. Let \(\pi_i = f(T_i | X_i)\). For binary ATE, these are given by \(\frac{T_i}{\pi_i} + \frac{(1 - T_i)}{(1 - \pi_i)}\). For binary ATT, these are given by \(\frac{n}{n_t} * \frac{T_i - \pi_i}{1 - \pi_i}\). For multi_valued treatments, these are given by \(\sum_{j=0}^{J-1} T_{i,j} / \pi_{i,j}\). For continuous treatments, these are given by \(\frac{f(T_i)}{f(T_i | X_i)}\). These expressions for weights are all before standardization (i.e. with standardize=FALSE). Standardization will make weights sum to 1 within each treatment group. For continuous treatment, standardization will make all weights sum to 1. If sampling weights are used, the weight for each observation is multiplied by the survey sampling weight.

The treatment vector used

The covariate matrix

model

The model frame

converged

Convergence value. Returned from the call to optim().

call

The matched call

formula

The formula supplied

data

The data argument

coefficients

A named vector of coefficients

sigmasq

The sigma-squared value, for continuous treatments only

The J-statistic at convergence

mle.J

The J-statistic for the parameters from maximum likelihood estimation

var

The covariance matrix for the coefficients.

Ttilde

For internal use only.

Xtilde

For internal use only.

beta.tilde

For internal use only.

simgasq.tilde

For internal use only.

Details

Fits covariate balancing propensity scores.

References

Imai, Kosuke and Marc Ratkovic. 2014. ``Covariate Balancing Propensity Score.'' Journal of the Royal Statistical Society, Series B (Statistical Methodology). http://imai.princeton.edu/research/CBPS.html Fong, Christian, Chad Hazlett, and Kosuke Imai. ``Parametric and Nonparametric Covariate Balancing Propensity Score for General Treatment Regimes.'' Unpublished Manuscript. http://imai.princeton.edu/research/files/CBGPS.pdf Fan, Jianqing and Imai, Kosuke and Liu, Han and Ning, Yang and Yang, Xiaolin. ``Improving Covariate Balancing Propensity Score: A Doubly Robust and Efficient Approach.'' Unpublished Manuscript. http://imai.princeton.edu/research/CBPStheory.html

Examples

Run this code

# NOT RUN {
###
### Example: propensity score matching
###

##Load the LaLonde data
data(LaLonde)
## Estimate CBPS
fit <- CBPS(treat ~ age + educ + re75 + re74 + 
			I(re75==0) + I(re74==0), 
			data = LaLonde, ATT = TRUE)
summary(fit)
# }
# NOT RUN {
## matching via MatchIt: one to one nearest neighbor with replacement
library(MatchIt)
m.out <- matchit(treat ~ fitted(fit), method = "nearest", 
				 data = LaLonde, replace = TRUE)

### Example: propensity score weighting 
###
## Simulation from Kang and Shafer (2007).
set.seed(123456)
n <- 500
X <- mvrnorm(n, mu = rep(0, 4), Sigma = diag(4))
prop <- 1 / (1 + exp(X[,1] - 0.5 * X[,2] + 
			 0.25*X[,3] + 0.1 * X[,4]))
treat <- rbinom(n, 1, prop)
y <- 210 + 27.4*X[,1] + 13.7*X[,2] + 13.7*X[,3] + 13.7*X[,4] + rnorm(n)

##Estimate CBPS with a misspecified model
X.mis <- cbind(exp(X[,1]/2), X[,2]*(1+exp(X[,1]))^(-1)+10, 
			  (X[,1]*X[,3]/25+.6)^3, (X[,2]+X[,4]+20)^2)
fit1 <- CBPS(treat ~ X.mis, ATT = 0)
summary(fit1)
	
## Horwitz-Thompson estimate
mean(treat*y/fit1$fitted.values)
## Inverse propensity score weighting
sum(treat*y/fit1$fitted.values)/sum(treat/fit1$fitted.values)

rm(list=c("y","X","prop","treat","n","X.mis","fit1"))

### Example: Continuous Treatment
set.seed(123456)
n <- 1000
X <- mvrnorm(n, mu = rep(0,2), Sigma = diag(2))
beta <- rnorm(ncol(X)+1, sd = 1)
treat <- cbind(1,X)%*%beta + rnorm(n, sd = 5)

treat.effect <- 1
effect.beta <- rnorm(ncol(X))
y <- rbinom(n, 1, (1 + exp(-treat.effect*treat - 
				   X%*%effect.beta))^-1)

fit2 <- CBPS(treat ~ X)
summary(fit2)
summary(glm(y ~ treat + X, weights = fit2$weights, 
			family = "quasibinomial"))

rm(list=c("n", "X", "beta", "treat", "treat.effect",
		  "effect.beta", "y", "fit2"))

### Simulation example: Improved CBPS (or iCBPS) from Fan et al
set.seed(123456)
n <- 500
X <- mvrnorm(n, mu = rep(0, 4), Sigma = diag(4))
prop <- 1 / (1 + exp(X[,1] - 0.5 * X[,2] + 0.25*X[,3] + 0.1 * X[,4]))
treat <- rbinom(n, 1, prop)
y1 <- 210 + 27.4*X[,1] + 13.7*X[,2] + 13.7*X[,3] + 13.7*X[,4] + rnorm(n)
y0 <- 210 + 13.7*X[,2] + 13.7*X[,3] + 13.7*X[,4] + rnorm(n)
##Estimate iCBPS with a misspecificied model
X.mis <- cbind(exp(X[,1]/2), X[,2]*(1+exp(X[,1]))^(-1)+10, 
			   (X[,1]*X[,3]/25+.6)^3, (X[,2]+X[,4]+20)^2)
fit1 <- CBPS(treat ~ X.mis, baseline.formula=~X.mis[,2:4], 
			 diff.formula=~X.mis[,1], ATT = FALSE)
summary(fit1)
# }

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Description

Usage

Arguments

Value

Details

References

See Also

Examples