This is an introduction to the use of cobalt with longitudinal treatments. These occur when there are multiple treatment periods spaced over time, with the potential for time-dependent confounding to occur. A common way to estimate treatment effects in these scenarios is to use marginal structural models (MSM), weighted by balancing weights. The goal of applying weights is to simulate a sequential randomization design, where the probability of being assigned to treatment at each time point is independent of each unit’s prior covariate and treatment history. For introduction to MSMs in general, see Thoemmes & Ong (2016), VanderWeele, Jackson, & Li (2016), Cole & Hernán (2008), or Robins, Hernán, & Brumback (2000). The key issue addressed by this guide and cobalt in general is assessing balance before each treatment period to ensure the removal of confounding.
In preprocessing for MSMs, three types of variables are relevant: baseline covariates, treatments, and intermediate outcomes/time-varying covariates. The goal of balance assessment is to assess whether after preprocessing, the resulting sample is one in which each treatment is independent of baseline covariates, treatment history, and time-varying covariates. The tools in cobalt have been developed to satisfy these goals.
The next section describe how to use cobalt’s tools to assess balance with longitudinal treatments. First, we’ll examine an example data set and identify some tools that can be used to generate weights for MSMs. Next we’ll use bal.tab(), bal.plot(), and love.plot() to assess and present balance.
We’re going to use the iptwExWide data set in the twang package.
| outcome | gender | age | use0 | use1 | use2 | tx1 | tx2 | tx3 |
|---|---|---|---|---|---|---|---|---|
| -0.2782802 | 0 | 43 | 1.1349651 | 0.4674825 | 0.3174825 | 1 | 1 | 1 |
| 0.5319329 | 0 | 50 | 1.1119318 | 0.4559659 | 0.4059659 | 1 | 0 | 1 |
| -0.8173614 | 1 | 36 | -0.8707776 | -0.5353888 | -0.5853888 | 1 | 0 | 0 |
| -0.1530853 | 1 | 63 | 0.2107316 | 0.0053658 | -0.1446342 | 1 | 1 | 1 |
| -0.7344267 | 0 | 24 | 0.0693956 | -0.0653022 | -0.1153022 | 1 | 0 | 1 |
| -0.8519376 | 1 | 20 | -1.6626489 | -0.9313244 | -1.0813244 | 1 | 1 | 1 |
We have the variables outcome, which is the outcome, gender, age, and use0, which are the baseline covariates, use1 and use2, which are time-varying covariates measured after treatment periods 1 and 2, and tx1, tx2, and tx3, which are the treatments at each of the three treatment periods.
The goal of balance assessment in this scenario is to ensure the following:
tx1 is independent from gender, age, and use0tx2 is independent from gender, age, use0, tx1, and use1tx3 is independent from gender, age, use0, tx1, use1, tx2, and use2To estimate the weights, we’ll use WeightIt to fit a series of logistic regressions that generate the weights. See the WeightIt documentation for more information on how to use WeightIt with longitudinal treatments.
library("WeightIt")
Wmsm <- weightitMSM(list(tx1 ~ use0 + gender + age,
tx2 ~ use0 + gender + age + use1 + tx1,
tx3 ~ use0 + gender + age + use1 + tx1 + use2 + tx2),
data = iptwExWide,
method = "ps")Next we’ll use bal.tab() to examine balance before and after applying the weights.
bal.tab()To examine balance on the original data, we can specify the treatment-covariate relationship we want to assess by using either the formula or data frame interfaces to bal.tab(). The formula interface requires a list of formulas, one for each treatment, and a data set containing the relevant variables. The data set must be in the “wide” setup, where each time point receives its own columns and each unit has exactly one row of data. The formula interface is similar to the WeightIt input seen above. The data frame interface requires a list of treatment values for each time period and a data frame or list of covariates for each time period. We’ll use the data frame interface here.
bal.tab(list(iptwExWide[c("use0", "gender", "age")],
iptwExWide[c("use0", "gender", "age", "use1", "tx1")],
iptwExWide[c("use0", "gender", "age", "use1", "tx1", "use2", "tx2")]),
treat.list = iptwExWide[c("tx1", "tx2", "tx3")])## Balance by Time Point
##
## - - - Time: 1 - - -
## Balance Measures
## Type Diff.Un
## use0 Contin. 0.2668
## gender_1 Binary 0.2945
## age Contin. 0.3799
##
## Sample sizes
## Control Treated
## All 294 706
##
## - - - Time: 2 - - -
## Balance Measures
## Type Diff.Un
## use0 Contin. 0.1169
## gender_1 Binary 0.1927
## age Contin. 0.2240
## use1 Contin. 0.0848
## tx1_1 Binary 0.1695
##
## Sample sizes
## Control Treated
## All 492 508
##
## - - - Time: 3 - - -
## Balance Measures
## Type Diff.Un
## use0 Contin. 0.1859
## gender_1 Binary 0.1532
## age Contin. 0.3431
## use1 Contin. 0.1662
## tx1_1 Binary 0.1071
## use2 Contin. 0.1087
## tx2_1 Binary 0.2423
##
## Sample sizes
## Control Treated
## All 415 585
## - - - - - - - - - - -
##
## Balance summary across all time points
## Times Type Max.Diff.Un
## use0 1, 2, 3 Contin. 0.2668
## gender_1 1, 2, 3 Binary 0.2945
## age 1, 2, 3 Contin. 0.3799
## use1 2, 3 Contin. 0.1662
## tx1_1 2, 3 Binary 0.1695
## use2 3 Contin. 0.1087
## tx2_1 3 Binary 0.2423
##
## Sample sizes
## - Time 1
## Control Treated
## All 294 706
## - Time 2
## Control Treated
## All 492 508
## - Time 3
## Control Treated
## All 415 585First we see balance by time point. At each time point, a bal.tab object is produced for that time point. These function just like regular bal.tab objects. Next, we see a balance summary across all time points. This displays each variable, how many times it appears in balance tables, its type, and the greatest imbalance for that variable across all time points. Below this is a summary of sample sizes across time points.
This output will appear no matter what the treatment types are (i.e., binary, continuous, multinomial), but for multinomial treatments or when the treatment types vary, no balance summary will be computed or displayed.
We can use bal.tab() with the weightitMSM object generated above. Setting un = TRUE would produce balance statistics before adjustment, like we did before.
## Balance by Time Point
##
## - - - Time: 1 - - -
## Balance Measures
## Type Diff.Un Diff.Adj
## prop.score Distance 0.7862 0.0251
## use0 Contin. 0.2668 0.0558
## gender_1 Binary 0.2945 0.0224
## age Contin. 0.3799 -0.0019
##
## Effective sample sizes
## Control Treated
## Unadjusted 294.00 706.000
## Adjusted 185.18 573.598
##
## - - - Time: 2 - - -
## Balance Measures
## Type Diff.Un Diff.Adj
## prop.score Distance 0.5288 -0.0065
## use0 Contin. 0.1169 -0.0327
## gender_1 Binary 0.1927 -0.0117
## age Contin. 0.2240 0.0703
## use1 Contin. 0.0848 -0.0311
## tx1_1 Binary 0.1695 -0.0088
##
## Effective sample sizes
## Control Treated
## Unadjusted 492.000 508.000
## Adjusted 318.903 264.487
##
## - - - Time: 3 - - -
## Balance Measures
## Type Diff.Un Diff.Adj
## prop.score Distance 0.6565 0.0229
## use0 Contin. 0.1859 -0.0347
## gender_1 Binary 0.1532 0.0263
## age Contin. 0.3431 0.0182
## use1 Contin. 0.1662 -0.0316
## tx1_1 Binary 0.1071 -0.0171
## use2 Contin. 0.1087 -0.0315
## tx2_1 Binary 0.2423 0.0085
##
## Effective sample sizes
## Control Treated
## Unadjusted 415.000 585.000
## Adjusted 235.672 366.399
## - - - - - - - - - - -
##
## Balance summary across all time points
## Times Type Max.Diff.Un Max.Diff.Adj
## prop.score 1, 2, 3 Distance 0.7862 0.0251
## use0 1, 2, 3 Contin. 0.2668 0.0558
## gender_1 1, 2, 3 Binary 0.2945 0.0263
## age 1, 2, 3 Contin. 0.3799 0.0703
## use1 2, 3 Contin. 0.1662 0.0316
## tx1_1 2, 3 Binary 0.1695 0.0171
## use2 3 Contin. 0.1087 0.0315
## tx2_1 3 Binary 0.2423 0.0085
##
## Effective sample sizes
## - Time 1
## Control Treated
## Unadjusted 294.00 706.000
## Adjusted 185.18 573.598
## - Time 2
## Control Treated
## Unadjusted 492.000 508.000
## Adjusted 318.903 264.487
## - Time 3
## Control Treated
## Unadjusted 415.000 585.000
## Adjusted 235.672 366.399We can use which.time and msm.summary to focus in on specific time points. which.time controls which time periods are displayed; one or more time periods can be entered. NA will not display any time periods (so only the summary will be displayed), and NULL, the default, will display all time periods. Setting msm.summary to FALSE will hide the summary across time points.
## Balance by Time Point
##
## - - - Time: 1 - - -
## Balance Measures
## Type Diff.Adj
## prop.score Distance 0.0251
## use0 Contin. 0.0558
## gender_1 Binary 0.0224
## age Contin. -0.0019
##
## Effective sample sizes
## Control Treated
## Unadjusted 294.00 706.000
## Adjusted 185.18 573.598
##
## - - - Time: 2 - - -
## Balance Measures
## Type Diff.Adj
## prop.score Distance -0.0065
## use0 Contin. -0.0327
## gender_1 Binary -0.0117
## age Contin. 0.0703
## use1 Contin. -0.0311
## tx1_1 Binary -0.0088
##
## Effective sample sizes
## Control Treated
## Unadjusted 492.000 508.000
## Adjusted 318.903 264.487
## - - - - - - - - - - -Note that to add covariates, we must use addl.list (which can be abbreviated as addl), which functions like addl in point treatments. The input to addl.list must be a list of covariates for each time point, or a single data data frame of variables to be assessed at all time points. The same goes for adding distance variables, which must be done with distance.list (which can be abbreviated as distance).
Next we’ll use bal.plot() to more finely examine covariate balance.
bal.plot()We can compare distributions of covariates across treatment groups for each time point using bal.plot(), just as we could with point treatments.
balance for variables that only appear in certain time points will only be displayed at those time points:
As with bal.tab(), which.time can be specified to limit output to chosen time periods.
Finally, we’ll examine using love.plot() with longitudinal treatments to display balance for presentation.
love.plot()love.plot() works with longitudinal treatments just as it does with point treatments, except that the user can choose whether to display separate plots for each time period or one plot with the summary across time periods. As with bal.tab(), the user can set which.time to display only certain time periods. When set to NA, the summary across time points is displayed. The agg.fun argument is automatically set to "max" and cannot be changed.
Here we used WeightIt to generate our MSM weights, but cobalt is compatible with other packages for longitudinal treatments as well. CBMSM objects from the CBPS package and iptw objects from the twang package can be used in place of the weightitMSM object in the above examples. In addition, users who have generated balancing weights outside any of these package can specify an argument to weights in bal.tab() with the formula or data frame methods to assess balance using those weights, or they can use the default method of bal.tab() to supply an object containing any of the objects required for balance assessment.
Note that CBPS estimates and assess balance on MSM weights differently than twang and cobalt. Its focus is on ensuring balance across all treatment history permutations, whereas cobalt focuses on evaluating the possibility of sequential randomization. For this reason, it may appear that CBMSM objects have different balance qualities as measured by the two packages.
Cole, S. R., & Hernán, M. Á. (2008). Constructing Inverse Probability Weights for Marginal Structural Models. American Journal of Epidemiology, 168(6), 656–664. https://doi.org/10.1093/aje/kwn164
Robins, J. M., Hernán, M. Á., & Brumback, B. (2000). Marginal Structural Models and Causal Inference in Epidemiology. Epidemiology, 11(5), 550–560.
Thoemmes, F., & Ong, A. D. (2016). A Primer on Inverse Probability of Treatment Weighting and Marginal Structural Models. Emerging Adulthood, 4(1), 40–59. https://doi.org/10.1177/2167696815621645
VanderWeele, T. J., Jackson, J. W., & Li, S. (2016). Causal inference and longitudinal data: a case study of religion and mental health. Social Psychiatry and Psychiatric Epidemiology, 51(11), 1457–1466. https://doi.org/10.1007/s00127-016-1281-9