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.

### 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`NA`

s.

### 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"`

.

## 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.

## Additional Arguments

`moments`

and `int`

are accepted. See `weightit()`

for details.

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.`quantile`

A 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 of`x1`

in each treatment group will be balanced in the weighted sample. Can also be a single number (e.g.,`.5`

) or an unnamed list of length 1 (e.g.,`list(c(.25, .5, .75))`

) to request the same quantile(s) for all continuous covariates, or a named vector (e.g.,`c(x1 = .5, x2 = .75)`

to request one quantile for each covariate. Only allowed with binary and multi-category treatments.

## 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.

## 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 1.0000 || 1.0000
#> control 0.0172 |---------------------------| 2.2625
#>
#> - Units with the 5 most extreme weights by group:
#>
#> 5 4 3 2 1
#> treated 1 1 1 1 1
#> 589 595 269 409 296
#> control 1.4644 1.4848 1.5763 1.7434 2.2625
#>
#> - 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 1.0000 || 1.0000
#> control 0.0181 |---------------------------| 1.8803
#>
#> - Units with the 5 most extreme weights by group:
#>
#> 5 4 3 2 1
#> treated 1 1 1 1 1
#> 411 595 269 409 296
#> control 1.2702 1.3139 1.4024 1.5131 1.8803
#>
#> - Weight statistics:
#>
#> Coef of Var MAD Entropy # Zeros
#> treated 0.000 0.000 0.000 0
#> control 0.793 0.685 0.315 0
#>
#> - Effective Sample Sizes:
#>
#> Control Treated
#> Unweighted 429. 185
#> Weighted 263.58 185
cobalt::bal.tab(W1b)
#> Balance Measures
#> Type Diff.Adj
#> prop.score Distance 0.0404
#> age Contin. 0.0233
#> educ Contin. -0.0201
#> married Binary 0.0048
#> nodegree Binary 0.0119
#> re74 Contin. -0.0609
#>
#> Effective sample sizes
#> Control Treated
#> Unadjusted 429. 185
#> Adjusted 263.58 185
#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 1.5007 |------------------| 17.9659
#> hispan 1.6315 |--------------------------| 24.5609
#> white 1.1311 |--| 4.1340
#>
#> - Units with the 5 most extreme weights by group:
#>
#> 226 231 485 181 182
#> black 6.7985 6.8385 7.2674 9.8974 17.9659
#> 392 564 269 345 371
#> hispan 16.762 19.8529 22.0193 23.8782 24.5609
#> 398 432 437 404 599
#> white 3.6882 3.7815 3.8476 3.8952 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 0.0151 |---------------------------| 43.9954
#>
#> - Units with the 5 most extreme weights:
#>
#> 482 180 481 483 185
#> all 10.8236 11.0877 11.97 13.1312 43.9954
#>
#> - Weight statistics:
#>
#> Coef of Var MAD Entropy # Zeros
#> all 1.942 0.528 0.454 0
#>
#> - Effective Sample Sizes:
#>
#> Total
#> Unweighted 614.
#> Weighted 128.86
cobalt::bal.tab(W3)
#> Balance Measures
#> Type Corr.Adj
#> age Contin. 0.1773
#> educ Contin. 0.1071
#> married Binary 0.4522
#> nodegree Binary 0.3308
#> re74 Contin. 0.5514
#>
#> Effective sample sizes
#> Total
#> Unadjusted 614.
#> Adjusted 128.86
# \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 1.0503 |---------------------------| 110.8801
#> control 1.2386 |---------------------| 86.9513
#>
#> - Units with the 5 most extreme weights by group:
#>
#> 3880 168 2859 3774 3653
#> treated 50.1138 50.1138 56.0785 96.4499 110.8801
#> 5695 6284 3500 1875 1362
#> control 52.431 55.2117 55.2117 58.5933 86.9513
#>
#> - 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 1.0503 |-----------------------| 96.4499
#> control 1.2386 |---------------------------| 110.8801
#>
#> - Units with the 5 most extreme weights by group:
#>
#> 3729 871 3880 168 3774
#> treated 42.3872 42.822 50.1138 50.1138 96.4499
#> 3500 2859 1875 1362 3653
#> control 55.2117 56.0785 58.5933 86.9513 110.8801
#>
#> - 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 1570.47 1926.01
#>
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 1570.47 1926.01
# }
```