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, ATC, and ATO.
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.
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
NAand 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 resultingweightitobject will be the original covariates with theNAs.
M-estimation
M-estimation is supported for the just-identified CBPS (the default, setting
over = FALSE) for binary and multi-category treatments. Otherwise (i.e.,
for continuous or longitudinal treatments or when over = TRUE),
M-estimation is not supported. 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 maximum likelihood 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 over = FALSE) does away with the maximum likelihood score equations for the coefficients so that only
the balance constraints are used, which 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".
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.
When the rootSolve package is installed, the optimization process will
be slightly faster and more accurate because starting values are provided by
an initial call to rootSolve::multiroot(). However, the package is not
required.
Additional Arguments
The following additional arguments can be specified:
overlogical; 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 isFALSEto use the just-identified CBPS.twosteplogical; whenover = TRUE, whether to use the two-step approximation to the generalized method of moments variance. Default isTRUE. Setting toFALSEincreases computation time but may improve estimation. Ignored with a warning whenover = FALSE.linkthe 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, including"probit","cauchit","cloglog","loglog","log","clog", and"identity". Note that negative weights are possible with these last three and they should be used with caution. An object of class"link-glm"can also be supplied. The argument is passed toquasibinomial(). Ignored for multi-category, continuous, and longitudinal treatments.reltolthe relative tolerance for convergence of the optimization. Passed to the
controlargument ofoptim(). Default is1e-10.maxitthe maximum number of iterations for convergence of the optimization. Passed to the
controlargument ofoptim(). Default is 1000 for binary and multi-category treatments and 10000 for continuous and longitudinal treatments.solverthe solver to use to estimate the parameters of the just-identified CBPS. Allowable options include
"multiroot"to userootSolve::multiroot()and"optim"to usestats::optim()."multiroot"is the default when rootSolve is installed, as it tends to be much faster and more accurate; otherwise,"optim"is the default and requires no dependencies. Regardless ofsolver, the output ofoptim()is returned wheninclude.obj = TRUE(see below). Whenover = TRUE, the parameter estimates of the just-identified CBPS are used as starting values for the over-identified CBPS.momentsinteger; the highest power of each covariate to be balanced. For example, ifmoments = 3, each covariate, its square, and its cube will be balanced. Can also be a named vector with a value for each covariate (e.g.,moments = c(x1 = 2, x2 = 4)). Values greater than 1 for categorical covariates are ignored. Default is 1 to balance covariate means.intlogical; whether first-order interactions of the covariates are to be balanced. Default isFALSE.quantilea named list of quantiles (values between 0 and 1) for each continuous covariate, which are used to create additional variables that when balanced ensure balance on the corresponding quantile of the variable. For example, setting
quantile = list(x1 = c(.25, .5. , .75))ensures the 25th, 50th, and 75th percentiles ofx1in each treatment group will be balanced in the weighted sample. Can also be a single number (e.g.,.5) or a vector (e.g.,c(.25, .5, .75)) to request the same quantile(s) for all continuous covariates. Only allowed with binary and multi-category treatments.
Additional Outputs
objWhen
include.obj = TRUE, the output of the final call tooptim()used to produce the model parameters. Note that because of variable transformations, the resulting parameter estimates may not be interpretable.
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
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"))
#> A weightit object
#> - method: "cbps" (covariate balancing propensity score weighting)
#> - number of obs.: 614
#> - sampling weights: none
#> - treatment: 2-category
#> - estimand: ATT (focal: 1)
#> - covariates: age, educ, married, nodegree, re74
summary(W1a)
#> Summary of weights
#>
#> - Weight ranges:
#>
#> Min Max
#> treated 1. || 1.
#> control 0.017 |---------------------------| 2.263
#>
#> - Units with the 5 most extreme weights by group:
#>
#> 1 2 3 4 5
#> treated 1 1 1 1 1
#> 410 404 224 111 84
#> control 1.464 1.485 1.576 1.743 2.263
#>
#> - Weight statistics:
#>
#> Coef of Var MAD Entropy # Zeros
#> treated 0.000 0.000 0.000 0
#> control 0.839 0.707 0.341 0
#>
#> - Effective Sample Sizes:
#>
#> Control Treated
#> Unweighted 429. 185
#> Weighted 252.12 185
cobalt::bal.tab(W1a)
#> Balance Measures
#> Type Diff.Adj
#> prop.score Distance 0.0164
#> age Contin. -0.0000
#> educ Contin. 0.0000
#> married Binary -0.0000
#> nodegree Binary 0.0000
#> re74 Contin. -0.0000
#>
#> Effective sample sizes
#> Control Treated
#> Unadjusted 429. 185
#> Adjusted 252.12 185
#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"))
#> A weightit object
#> - method: "cbps" (covariate balancing propensity score weighting)
#> - number of obs.: 614
#> - sampling weights: none
#> - treatment: 2-category
#> - estimand: ATT (focal: 1)
#> - covariates: age, educ, married, nodegree, re74
summary(W1b)
#> Summary of weights
#>
#> - Weight ranges:
#>
#> Min Max
#> treated 1. || 1.
#> control 0.012 |---------------------------| 2.053
#>
#> - Units with the 5 most extreme weights by group:
#>
#> 1 2 3 4 5
#> treated 1 1 1 1 1
#> 410 404 224 111 84
#> control 1.368 1.378 1.472 1.607 2.053
#>
#> - Weight statistics:
#>
#> Coef of Var MAD Entropy # Zeros
#> treated 0.00 0.000 0.000 0
#> control 0.81 0.693 0.326 0
#>
#> - Effective Sample Sizes:
#>
#> Control Treated
#> Unweighted 429. 185
#> Weighted 259.24 185
cobalt::bal.tab(W1b)
#> Balance Measures
#> Type Diff.Adj
#> prop.score Distance 0.0334
#> age Contin. -0.0021
#> educ Contin. 0.0028
#> married Binary 0.0011
#> nodegree Binary 0.0041
#> re74 Contin. -0.0284
#>
#> Effective sample sizes
#> Control Treated
#> Unadjusted 429. 185
#> Adjusted 259.24 185
#Balancing covariates with respect to race (multi-category)
(W2 <- weightit(race ~ age + educ + married +
nodegree + re74, data = lalonde,
method = "cbps", estimand = "ATE"))
#> A weightit object
#> - method: "cbps" (covariate balancing propensity score weighting)
#> - number of obs.: 614
#> - sampling weights: none
#> - treatment: 3-category (black, hispan, white)
#> - estimand: ATE
#> - covariates: age, educ, married, nodegree, re74
summary(W2)
#> Summary of weights
#>
#> - Weight ranges:
#>
#> Min Max
#> black 1.501 |------------------| 17.966
#> hispan 1.631 |--------------------------| 24.561
#> white 1.131 |--| 4.134
#>
#> - Units with the 5 most extreme weights by group:
#>
#> 203 164 163 153 152
#> black 6.799 6.838 7.267 9.897 17.966
#> 67 43 39 36 28
#> hispan 16.762 19.853 22.019 23.878 24.561
#> 270 195 190 172 169
#> white 3.688 3.781 3.848 3.895 4.134
#>
#> - Weight statistics:
#>
#> Coef of Var MAD Entropy # Zeros
#> black 0.635 0.387 0.133 0
#> hispan 0.582 0.447 0.155 0
#> white 0.389 0.327 0.071 0
#>
#> - Effective Sample Sizes:
#>
#> black hispan white
#> Unweighted 243. 72. 299.
#> Weighted 173.37 53.95 259.76
cobalt::bal.tab(W2)
#> Balance summary across all treatment pairs
#> Type Max.Diff.Adj
#> age Contin. 0
#> educ Contin. 0
#> married Binary 0
#> nodegree Binary 0
#> re74 Contin. 0
#>
#> Effective sample sizes
#> black hispan white
#> Unadjusted 243. 72. 299.
#> Adjusted 173.37 53.95 259.76
#Balancing covariates with respect to re75 (continuous)
(W3 <- weightit(re75 ~ age + educ + married +
nodegree + re74, data = lalonde,
method = "cbps"))
#> A weightit object
#> - method: "cbps" (covariate balancing propensity score weighting)
#> - number of obs.: 614
#> - sampling weights: none
#> - treatment: continuous
#> - covariates: age, educ, married, nodegree, re74
summary(W3)
#> Summary of weights
#>
#> - Weight ranges:
#>
#> Min Max
#> all 0.01 |---------------------------| 20.946
#>
#> - Units with the 5 most extreme weights:
#>
#> 485 484 483 482 481
#> all 10.209 13.112 13.974 17.816 20.946
#>
#> - Weight statistics:
#>
#> Coef of Var MAD Entropy # Zeros
#> all 1.454 0.535 0.396 0
#>
#> - Effective Sample Sizes:
#>
#> Total
#> Unweighted 614.
#> Weighted 197.36
cobalt::bal.tab(W3)
#> Balance Measures
#> Type Corr.Adj
#> age Contin. -0
#> educ Contin. 0
#> married Binary -0
#> nodegree Binary -0
#> re74 Contin. 0
#>
#> Effective sample sizes
#> Total
#> Unadjusted 614.
#> Adjusted 197.36
# \donttest{
#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"))
#> A weightitMSM object
#> - method: "cbps" (covariate balancing propensity score weighting)
#> - number of obs.: 7500
#> - sampling weights: none
#> - number of time points: 2 (A_1, A_2)
#> - treatment:
#> + time 1: 2-category
#> + time 2: 2-category
#> - covariates:
#> + baseline: X1_0, X2_0
#> + after time 1: X1_1, X2_1, A_1, X1_0, X2_0
summary(W4)
#> Time 1
#> Summary of weights
#>
#> - Weight ranges:
#>
#> Min Max
#> treated 1.05 |---------------------------| 110.88
#> control 1.239 |---------------------| 86.951
#>
#> - Units with the 5 most extreme weights by group:
#>
#> 2115 2177 2058 1611 90
#> treated 50.114 50.114 56.079 96.45 110.88
#> 2778 1532 2516 832 586
#> control 52.431 55.212 55.212 58.593 86.951
#>
#> - Weight statistics:
#>
#> Coef of Var MAD Entropy # Zeros
#> treated 1.107 0.552 0.291 0
#> control 1.033 0.600 0.315 0
#>
#> - Effective Sample Sizes:
#>
#> Control Treated
#> Unweighted 3306. 4194.
#> Weighted 1598.88 1884.92
#>
#> Time 2
#> Summary of weights
#>
#> - Weight ranges:
#>
#> Min Max
#> treated 1.05 |-----------------------| 96.45
#> control 1.239 |---------------------------| 110.88
#>
#> - Units with the 5 most extreme weights by group:
#>
#> 2004 1954 460 1933 82
#> treated 42.387 42.822 50.114 50.114 96.45
#> 1753 1689 1386 911 664
#> control 55.212 56.079 58.593 86.951 110.88
#>
#> - Weight statistics:
#>
#> Coef of Var MAD Entropy # Zeros
#> treated 0.986 0.588 0.292 0
#> control 1.165 0.583 0.329 0
#>
#> - Effective Sample Sizes:
#>
#> Control Treated
#> Unweighted 3701. 3799
#> Weighted 1570.46 1926
#>
cobalt::bal.tab(W4)
#> Balance summary across all time points
#> Times Type Max.Diff.Adj
#> X1_0 1, 2 Contin. 0
#> X2_0 1, 2 Binary 0
#> X1_1 2 Contin. 0
#> X2_1 2 Binary 0
#> A_1 2 Binary 0
#>
#> Effective sample sizes
#> - Time 1
#> Control Treated
#> Unadjusted 3306. 4194.
#> Adjusted 1598.88 1884.92
#> - Time 2
#> Control Treated
#> Unadjusted 3701. 3799
#> Adjusted 1570.46 1926
# }