method_cbps {WeightIt}

Covariate Balancing Propensity Score Weighting

Description

This page explains the details of estimating weights from covariate balancing propensity scores by setting method = "cbps" in the call to weightit() or weightitMSM(). This method can be used with binary, multi-category, and continuous treatments.

In general, this method relies on estimating propensity scores using generalized method of moments and then converting those propensity scores into weights using a formula that depends on the desired estimand. This method relies on code written for WeightIt using optim().

Binary Treatments

For binary treatments, this method estimates the propensity scores and weights using optim() using formulas described by Imai and Ratkovic (2014). The following estimands are allowed: ATE, ATT, and ATC.

Multi-Category Treatments

For multi-category treatments, this method estimates the generalized propensity scores and weights using optim() using formulas described by Imai and Ratkovic (2014). The following estimands are allowed: ATE and ATT.

Continuous Treatments

For continuous treatments, this method estimates the generalized propensity scores and weights using optim() using a modification of the formulas described by Fong, Hazlett, and Imai (2018). See Details.

Longitudinal Treatments

For longitudinal treatments, the weights are computed using methods similar to those described by Huffman and van Gameren (2018). This involves specifying moment conditions for the models at each time point as with single-time point treatments but using the product of the time-specific weights as the weights that are used in the balance moment conditions. This yields weights that balance the covariate at each time point. This is not the same implementation as is implemented in CBPS::CBMSM(), and results should not be expected to align between the two methods. Any combination of treatment types is supported.

For the over-identified version (i.e., setting over = TRUE), the empirical variance is used in the objective function, whereas the expected variance averaging over the treatment is used with binary and multi-category point treatments.

Sampling Weights

Sampling weights are supported through s.weights in all scenarios.

Missing Data

In the presence of missing data, the following value(s) for missing are allowed:

"ind" (default)

First, for each variable with missingness, a new missingness indicator variable is created which takes the value 1 if the original covariate is NA and 0 otherwise. The missingness indicators are added to the model formula as main effects. The missing values in the covariates are then replaced with the covariate medians (this value is arbitrary and does not affect estimation). The weight estimation then proceeds with this new formula and set of covariates. The covariates output in the resulting weightit object will be the original covariates with the NAs.

M-estimation

M-estimation is supported for the just-identified CBPS (the default, setting over = FALSE) for binary and multi-category treatments. See glm_weightit() and vignette("estimating-effects") for details.

Details

CBPS estimates the coefficients of a generalized linear model (for binary treatments), multinomial logistic regression model (for multi-category treatments), or linear regression model (for continuous treatments) that is used to compute (generalized) propensity scores, from which the weights are computed. It involves replacing (or augmenting, in the case of the over-identified version) the standard regression score equations with the balance constraints in a generalized method of moments estimation. The idea is to nudge the estimation of the coefficients toward those that produce balance in the weighted sample. The just-identified version (with exact = FALSE) does away with the score equations for the coefficients so that only the balance constraints are used. The just-identified version will therefore produce superior balance on the means (i.e., corresponding to the balance constraints) for binary and multi-category treatments and linear terms for continuous treatments than will the over-identified version.

Just-identified CBPS is very similar to entropy balancing and inverse probability tilting. For the ATT, all three methods will yield identical estimates. For other estimands, the results will differ.

Note that WeightIt provides different functionality from the CBPS package in terms of the versions of CBPS available; for extensions to CBPS (e.g., optimal CBPS and CBPS for instrumental variables), the CBPS package may be preferred. Note that for longitudinal treatments, CBPS::CBMSM() uses different methods and produces different results from weightitMSM() called with method = "cbps".

Additional Arguments

The following additional arguments can be specified:

over

logical; whether to request the over-identified CBPS, which combines the generalized linear model regression score equations (for binary treatments), multinomial logistic regression score equations (for multi-category treatments), or linear regression score equations (for continuous treatments) to the balance moment conditions. Default is FALSE to use the just-identified CBPS.

twostep

logical; when over = TRUE, whether to use the two-step approximation to the generalized method of moments variance. Default is TRUE. Ignored when over = FALSE.

link

"string"; the link used in the generalized linear model for the propensity scores when treatment is binary. Default is "logit" for logistic regression, which is used in the original description of the method by Imai and Ratkovic (2014), but others are allowed: "logit", "probit", "cauchit", and "cloglog" all use the binomial likelihood, "log" uses the Poisson likelihood, and "identity" uses the Gaussian likelihood (i.e., the linear probability model). Note that negative weights are possible with these last two and they should be used with caution. Ignored for multi-category, continuous, and longitudinal treatments.

reltol

the relative tolerance for convergence of the optimization. Passed to the control argument of optim(). Default is sqrt(.Machine$double.eps).

maxit

the maximum number of iterations for convergence of the optimization. Passed to the control argument of optim(). Default is 1000.

Additional Outputs

obj

When include.obj = TRUE, the output of the final call to optim() used to produce the model parameters. Note that because of variable transformations, the resulting parameter estimates may not be interpretable.

Note

This method used to rely on functionality in the CBPS package, but no longer does. Slight differences may be found between the two packages in some cases due to numerical imprecision (or, for continuous and longitudinal treatments, due to a difference in the estimator). WeightIt supports arbitrary numbers of groups for the multi-category CBPS and any estimand, whereas CBPS only supports up to four groups and only the ATE. The implementation of the just-identified CBPS for continuous treatments also differs from that of CBPS, and departs slightly from that described by Fong et al. (2018). The treatment mean and treatment variance are treated as random parameters to be estimated and are included in the balance moment conditions. In Fong et al. (2018), the treatment mean and variance are fixed to their empirical counterparts. For continuous treatments with the over-identified CBPS, WeightIt and CBPS use different methods of specifying the GMM variance matrix, which may lead to differing results.

Note that the default method differs between the two implementations; by default WeightIt uses the just-identified CBPS, which is faster to fit, yields better balance, and is compatible with M-estimation for estimating the standard error of the treatment effect, whereas CBPS uses the over-identified CBPS by default. However, both the just-identified and over-identified versions are available in both packages.

References

Binary treatments

Imai, K., & Ratkovic, M. (2014). Covariate balancing propensity score. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 76(1), 243–263.

Multi-Category treatments

Imai, K., & Ratkovic, M. (2014). Covariate balancing propensity score. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 76(1), 243–263.

Continuous treatments

Fong, C., Hazlett, C., & Imai, K. (2018). Covariate balancing propensity score for a continuous treatment: Application to the efficacy of political advertisements. The Annals of Applied Statistics, 12(1), 156–177. doi:10.1214/17-AOAS1101

Longitudinal treatments

Huffman, C., & van Gameren, E. (2018). Covariate Balancing Inverse Probability Weights for Time-Varying Continuous Interventions. Journal of Causal Inference, 6(2). doi:10.1515/jci-2017-0002

Note: one should not cite Imai & Ratkovic (2015) when using CBPS for longitudinal treatments.

Some of the code was inspired by the source code of the CBPS package.

See Also

weightit(), weightitMSM()

method_ebal and method_ipt for entropy balancing and inverse probability tilting, which work similarly.

Examples

data("lalonde", package = "cobalt")

#Balancing covariates between treatment groups (binary)
(W1a <- weightit(treat ~ age + educ + married +
                  nodegree + re74, data = lalonde,
                method = "cbps", estimand = "ATT"))
summary(W1a)
cobalt::bal.tab(W1a)

#Balancing covariates between treatment groups (binary)
#using over-identified CBPS with probit link
(W1b <- weightit(treat ~ age + educ + married +
                  nodegree + re74, data = lalonde,
                method = "cbps", estimand = "ATT",
                over = TRUE, link = "probit"))
summary(W1b)
cobalt::bal.tab(W1b)

#Balancing covariates with respect to race (multi-category)
(W2 <- weightit(race ~ age + educ + married +
                  nodegree + re74, data = lalonde,
                method = "cbps", estimand = "ATE"))
summary(W2)
cobalt::bal.tab(W2)

#Balancing covariates with respect to re75 (continuous)
(W3 <- weightit(re75 ~ age + educ + married +
                  nodegree + re74, data = lalonde,
                method = "cbps"))
summary(W3)
cobalt::bal.tab(W3)

#Longitudinal treatments
data("msmdata")
(W4 <- weightitMSM(list(A_1 ~ X1_0 + X2_0,
                        A_2 ~ X1_1 + X2_1 +
                          A_1 + X1_0 + X2_0),
                   data = msmdata,
                   method = "cbps"))
summary(W4)
cobalt::bal.tab(W4)