This page explains the details of estimating optimization-based weights (also known as stable balancing weights) by setting `method = "optweight"`

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 weights by solving a quadratic programming problem subject to approximate or exact balance constraints. This method relies on optweightoptweight from the optweight package.

Because `optweight()`

offers finer control and uses the same syntax as `weightit()`

, it is recommended that optweightoptweight be used instead of `weightit()`

with `method = "optweight"`

.

### Binary Treatments

For binary treatments, this method estimates the weights using optweightoptweight. The following estimands are allowed: ATE, ATT, and ATC. The weights are taken from the output of the `optweight`

fit object.

### Multi-Category Treatments

For multi-category treatments, this method estimates the weights using optweightoptweight. The following estimands are allowed: ATE and ATT. The weights are taken from the output of the `optweight`

fit object.

### Continuous Treatments

For binary treatments, this method estimates the weights using optweightoptweight. The weights are taken from the output of the `optweight`

fit object.

### Longitudinal Treatments

For longitudinal treatments, `optweight()`

estimates weights that simultaneously satisfy balance constraints at all time points, so only one model is fit to obtain the weights. Using `method = "optweight"`

in `weightitMSM()`

causes `is.MSM.method`

to be set to `TRUE`

by default. Setting it to `FALSE`

will run one model for each time point and multiply the weights together, a method that is not recommended. NOTE: neither use of optimization-based weights with longitudinal treatments has been validated!

### 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.

## Details

Stable balancing weights are weights that solve a constrained optimization problem, where the constraints correspond to covariate balance and the loss function is the variance (or other norm) of the weights. These weights maximize the effective sample size of the weighted sample subject to user-supplied balance constraints. An advantage of this method over entropy balancing is the ability to allow approximate, rather than exact, balance through the `tols`

argument, which can increase precision even for slight relaxations of the constraints.

`plot()`

can be used on the output of `weightit()`

with `method = "optweight"`

to display the dual variables; see Examples and `plot.weightit()`

for more details.

## Note

The specification of `tols`

differs between `weightit()`

and `optweight()`

. In `weightit()`

, one tolerance value should be included per level of each factor variable, whereas in `optweight()`

, all levels of a factor are given the same tolerance, and only one value needs to be supplied for a factor variable. Because of the potential for confusion and ambiguity, it is recommended to only supply one value for `tols`

in `weightit()`

that applies to all variables. For finer control, use `optweight()`

directly.

Seriously, just use optweightoptweight. The syntax is almost identical and it's compatible with cobalt, too.

## Additional Arguments

`moments`

and `int`

are accepted. See `weightit()`

for details.

`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.

All arguments to `optweight()`

can be passed through `weightit()`

or `weightitMSM()`

, with the following exception:

`targets`

cannot be used and is ignored.

All arguments take on the defaults of those in `optweight()`

.

## Additional Outputs

`info`

A list with one entry:

`duals`

A data frame of dual variables for each balance constraint.

`obj`

When

`include.obj = TRUE`

, the output of the call to optweightoptweight.

## References

### Binary treatments

Wang, Y., & Zubizarreta, J. R. (2020). Minimal dispersion approximately balancing weights: Asymptotic properties and practical considerations. *Biometrika*, 107(1), 93–105. doi:10.1093/biomet/asz050

Zubizarreta, J. R. (2015). Stable Weights that Balance Covariates for Estimation With Incomplete Outcome Data. *Journal of the American Statistical Association*, 110(511), 910–922. doi:10.1080/01621459.2015.1023805

### Multi-Category Treatments

de los Angeles Resa, M., & Zubizarreta, J. R. (2020). Direct and stable weight adjustment in non-experimental studies with multivalued treatments: Analysis of the effect of an earthquake on post-traumatic stress. *Journal of the Royal Statistical Society: Series A (Statistics in Society)*, n/a(n/a). doi:10.1111/rssa.12561

### Continuous treatments

Greifer, N. (2020). *Estimating Balancing Weights for Continuous Treatments Using Constrained Optimization*. doi:10.17615/DYSS-B342

## Examples

```
data("lalonde", package = "cobalt")
#Balancing covariates between treatment groups (binary)
(W1 <- weightit(treat ~ age + educ + married +
nodegree + re74, data = lalonde,
method = "optweight", estimand = "ATT",
tols = 0))
#> A weightit object
#> - method: "optweight" (stable balancing weights)
#> - number of obs.: 614
#> - sampling weights: none
#> - treatment: 2-category
#> - estimand: ATT (focal: 1)
#> - covariates: age, educ, married, nodegree, re74
summary(W1)
#> Summary of weights
#>
#> - Weight ranges:
#>
#> Min Max
#> treated 1 || 1.0000
#> control 0 |---------------------------| 3.0426
#>
#> - Units with the 5 most extreme weights by group:
#>
#> 5 4 3 2 1
#> treated 1 1 1 1 1
#> 411 589 269 409 296
#> control 2.5261 2.5415 2.6434 2.7396 3.0426
#>
#> - Weight statistics:
#>
#> Coef of Var MAD Entropy # Zeros
#> treated 0.000 0.000 0.000 0
#> control 0.788 0.697 0.393 83
#>
#> - Effective Sample Sizes:
#>
#> Control Treated
#> Unweighted 429. 185
#> Weighted 264.88 185
cobalt::bal.tab(W1)
#> Balance Measures
#> Type Diff.Adj
#> age Contin. -0
#> educ Contin. -0
#> married Binary -0
#> nodegree Binary 0
#> re74 Contin. -0
#>
#> Effective sample sizes
#> Control Treated
#> Unadjusted 429. 185
#> Adjusted 264.88 185
plot(W1)
#Balancing covariates with respect to race (multi-category)
(W2 <- weightit(race ~ age + educ + married +
nodegree + re74, data = lalonde,
method = "optweight", estimand = "ATE",
tols = .01))
#> A weightit object
#> - method: "optweight" (stable balancing weights)
#> - 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 0.4429 |-----------------------| 3.5741
#> hispan 0.0000 |-------------------| 2.5848
#> white 0.2574 |---------| 1.6593
#>
#> - Units with the 5 most extreme weights by group:
#>
#> 184 190 485 181 182
#> black 2.3351 2.3723 2.5586 2.8367 3.5741
#> 392 345 269 564 371
#> hispan 2.0459 2.0984 2.1887 2.1982 2.5848
#> 68 589 324 599 531
#> white 1.5706 1.5706 1.5725 1.5968 1.6593
#>
#> - Weight statistics:
#>
#> Coef of Var MAD Entropy # Zeros
#> black 0.550 0.443 0.130 0
#> hispan 0.566 0.449 0.176 2
#> white 0.353 0.295 0.065 0
#>
#> - Effective Sample Sizes:
#>
#> black hispan white
#> Unweighted 243. 72. 299.
#> Weighted 186.76 54.7 266.01
cobalt::bal.tab(W2)
#> Balance summary across all treatment pairs
#> Type Max.Diff.Adj
#> age Contin. 0.01
#> educ Contin. 0.01
#> married Binary 0.01
#> nodegree Binary 0.01
#> re74 Contin. 0.01
#>
#> Effective sample sizes
#> black hispan white
#> Unadjusted 243. 72. 299.
#> Adjusted 186.76 54.7 266.01
plot(W2)
#Balancing covariates with respect to re75 (continuous)
```