# Using `cobalt` with Longitudinal Treatments

#### Noah Greifer

#### 2025-08-20

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 and Ong (2016), VanderWeele, Jackson, and Li (2016), Cole and Hernán (2008), or Robins, Hernán, and 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.

## Setup

We’re going to use the `msmdata` data set in the `WeightIt` package.

    library("cobalt")
    data("msmdata", package = "WeightIt")
    head(msmdata)

X1_0 | X2_0 | A_1 | X1_1 | X2_1 | A_2 | X1_2 | X2_2 | A_3 | Y_B
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
2 | 0 | 1 | 5 | 1 | 0 | 4 | 1 | 0 | 0
4 | 0 | 1 | 9 | 0 | 1 | 10 | 0 | 1 | 1
4 | 1 | 0 | 5 | 0 | 1 | 4 | 0 | 0 | 1
4 | 1 | 0 | 4 | 0 | 0 | 6 | 1 | 0 | 1
6 | 1 | 1 | 5 | 0 | 1 | 6 | 0 | 0 | 1
5 | 1 | 0 | 4 | 0 | 1 | 4 | 0 | 1 | 0

We have the variables `Y_B`, the outcome; `X1_0` and `X2_0`, the baseline covariates; `X1_1` and `X2_1`, time-varying covariates measured after treatment period 1; `X1_1` and `X2_1`, covariates measured after treatment period 2; and `A_1`, `A_2`, and `A_3`, the treatments at each of the three treatment periods.

The goal of balance assessment in this scenario is to ensure the following:

1. `A_1` is independent from `X1_0` and `X2_0`
2. `A_2` is independent from `X1_0`, `X2_0`, `A_1,`X1_1`, and`X2_1`
3. `tx3` is independent from `X1_0`, `X2_0`, `A_1,`X1_1`,`X2_1`,`A_2, `X1_2`, and `X2_2`

Note these conditions are different from and weaker than those described by Jackson (2016). See his `confoundr` package for implementing the diagnostics he describes.

## `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 point and a data frame or list of covariates for each time point. We’ll use the formula interface here.

    bal.tab(list(A_1 ~ X1_0 + X2_0,
                 A_2 ~ X1_1 + X2_1 +
                     A_1 + X1_0 + X2_0,
                 A_3 ~ X1_2 + X2_2 +
                     A_2 + X1_1 + X2_1 +
                     A_1 + X1_0 + X2_0),
            data = msmdata)

    ## Balance summary across all time points
    ##        Times    Type Max.Diff.Un
    ## X1_0 1, 2, 3 Contin.      0.6897
    ## X2_0 1, 2, 3  Binary      0.3253
    ## X1_1    2, 3 Contin.      0.8736
    ## X2_1    2, 3  Binary      0.2994
    ## A_1     2, 3  Binary      0.1267
    ## X1_2       3 Contin.      0.4749
    ## X2_2       3  Binary      0.5945
    ## A_2        3  Binary      0.1620
    ##
    ## Sample sizes
    ##  - Time 1
    ##     Control Treated
    ## All    3306    4194
    ##  - Time 2
    ##     Control Treated
    ## All    3701    3799
    ##  - Time 3
    ##     Control Treated
    ## All    4886    2614

Here we see a summary of balance 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. To request balance on individual time points, we can use the `which.time` argument, which can be set to one or more numbers or `.all` or `.none` (the default). Below we’ll request balance on all time points by setting `which.time = .all`. Doing so hides the balance summary across time points, but this can be requested again by setting `msm.summary = TRUE`.

    bal.tab(list(A_1 ~ X1_0 + X2_0,
                 A_2 ~ X1_1 + X2_1 +
                     A_1 + X1_0 + X2_0,
                 A_3 ~ X1_2 + X2_2 +
                     A_2 + X1_1 + X2_1 +
                     A_1 + X1_0 + X2_0),
            data = msmdata,
            which.time = .all)

    ## Balance by Time Point
    ##
    ##  - - - Time: 1 - - -
    ## Balance Measures
    ##         Type Diff.Un
    ## X1_0 Contin.  0.6897
    ## X2_0  Binary -0.3253
    ##
    ## Sample sizes
    ##     Control Treated
    ## All    3306    4194
    ##
    ##  - - - Time: 2 - - -
    ## Balance Measures
    ##         Type Diff.Un
    ## X1_1 Contin.  0.8736
    ## X2_1  Binary -0.2994
    ## A_1   Binary  0.1267
    ## X1_0 Contin.  0.5276
    ## X2_0  Binary -0.0599
    ##
    ## Sample sizes
    ##     Control Treated
    ## All    3701    3799
    ##
    ##  - - - Time: 3 - - -
    ## Balance Measures
    ##         Type Diff.Un
    ## X1_2 Contin.  0.4749
    ## X2_2  Binary -0.5945
    ## A_2   Binary  0.1620
    ## X1_1 Contin.  0.5727
    ## X2_1  Binary -0.0405
    ## A_1   Binary  0.1000
    ## X1_0 Contin.  0.3614
    ## X2_0  Binary -0.0402
    ##
    ## Sample sizes
    ##     Control Treated
    ## All    4886    2614
    ##  - - - - - - - - - - -

Here 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.

This output will appear no matter what the treatment types are (i.e., binary, continuous, multi-category), but for multi-category treatments or when the treatment types vary or for multiply imputed data, no balance summary will be computed or displayed.

To estimate the weights, we’ll use `WeightIt::weightitMSM()` 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.

    Wmsm <- WeightIt::weightitMSM(
        list(A_1 ~ X1_0 + X2_0,
             A_2 ~ X1_1 + X2_1 +
                 A_1 + X1_0 + X2_0,
             A_3 ~ X1_2 + X2_2 +
                 A_2 + X1_1 + X2_1 +
                 A_1 + X1_0 + X2_0),
        data = msmdata,
        method = "glm")

We can use `bal.tab()` with the `weightitMSM` object generated above. Setting `un = TRUE` would produce balance statistics before adjustment, like we did before. We’ll set `which.time = .all` and `msm.summary = TRUE` to see balance for each time point and across time points.

    bal.tab(Wmsm, un = TRUE, which.time = .all, msm.summary = TRUE)

    ## Balance by Time Point
    ##
    ##  - - - Time: 1 - - -
    ## Balance Measures
    ##         Type Diff.Un Diff.Adj
    ## X1_0 Contin.  0.6897   0.0026
    ## X2_0  Binary -0.3253  -0.0239
    ##
    ## Effective sample sizes
    ##            Control Treated
    ## Unadjusted 3306.    4194.
    ## Adjusted    845.79   899.4
    ##
    ##  - - - Time: 2 - - -
    ## Balance Measures
    ##         Type Diff.Un Diff.Adj
    ## X1_1 Contin.  0.8736   0.0531
    ## X2_1  Binary -0.2994  -0.0299
    ## A_1   Binary  0.1267   0.0065
    ## X1_0 Contin.  0.5276   0.0183
    ## X2_0  Binary -0.0599  -0.0299
    ##
    ## Effective sample sizes
    ##            Control Treated
    ## Unadjusted 3701.   3799.
    ## Adjusted    912.87  829.87
    ##
    ##  - - - Time: 3 - - -
    ## Balance Measures
    ##         Type Diff.Un Diff.Adj
    ## X1_2 Contin.  0.4749   0.0643
    ## X2_2  Binary -0.5945   0.0096
    ## A_2   Binary  0.1620  -0.0054
    ## X1_1 Contin.  0.5727   0.0657
    ## X2_1  Binary -0.0405  -0.0248
    ## A_1   Binary  0.1000  -0.0262
    ## X1_0 Contin.  0.3614   0.0342
    ## X2_0  Binary -0.0402   0.0147
    ##
    ## Effective sample sizes
    ##            Control Treated
    ## Unadjusted 4886.   2614.
    ## Adjusted   1900.26  600.12
    ##  - - - - - - - - - - -
    ##
    ## Balance summary across all time points
    ##        Times    Type Max.Diff.Un Max.Diff.Adj
    ## X1_0 1, 2, 3 Contin.      0.6897       0.0342
    ## X2_0 1, 2, 3  Binary      0.3253       0.0299
    ## X1_1    2, 3 Contin.      0.8736       0.0657
    ## X2_1    2, 3  Binary      0.2994       0.0299
    ## A_1     2, 3  Binary      0.1267       0.0262
    ## X1_2       3 Contin.      0.4749       0.0643
    ## X2_2       3  Binary      0.5945       0.0096
    ## A_2        3  Binary      0.1620       0.0054
    ##
    ## Effective sample sizes
    ##  - Time 1
    ##            Control Treated
    ## Unadjusted 3306.    4194.
    ## Adjusted    845.79   899.4
    ##  - Time 2
    ##            Control Treated
    ## Unadjusted 3701.   3799.
    ## Adjusted    912.87  829.87
    ##  - Time 3
    ##            Control Treated
    ## Unadjusted 4886.   2614.
    ## Adjusted   1900.26  600.12

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.

    bal.plot(Wmsm, var.name = "X1_0", which = "both",
             type = "histogram")

Balance for variables that only appear in certain time points will only be displayed at those time points:

    bal.plot(Wmsm, var.name = "X2_1", which = "both")

As with `bal.tab()`, `which.time` can be specified to limit output to chosen time points.

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 point or one plot with the summary across time points. As with `bal.tab()`, the user can set `which.time` to display only certain time points, including setting it to `.all` to display all time points (note that not all variables will be present in all time points). When set to `.none` (the default), the summary across time points is displayed. The `agg.fun` argument is set to `"max"` by default.

    love.plot(Wmsm, binary = "std")

    love.plot(Wmsm, binary = "std", which.time = .all)

## Other Packages

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 assesses balance on MSM weights differently from `twang` and `cobalt`. Its focus is on ensuring balance across all treatment history permutations, whereas `cobalt` focuses on evaluating the similarity to sequential randomization. For this reason, it may appear that `CBMSM` objects have different balance qualities as measured by the two packages.

## References

Cole, Stephen R., and Miguel A Hernán. 2008. “Constructing Inverse Probability Weights for Marginal Structural Models.” American Journal of Epidemiology 168 (6): 656–64. https://doi.org/10.1093/aje/kwn164.
Jackson, John W. 2016. “Diagnostics for Confounding of Time-Varying and Other Joint Exposures:” Epidemiology 27 (6): 859–69. https://doi.org/10.1097/EDE.0000000000000547.
Robins, James M., Miguel Ángel Hernán, and Babette Brumback. 2000. “Marginal Structural Models and Causal Inference in Epidemiology.” Epidemiology 11 (5): 550–60. https://doi.org/10.1097/00001648-200009000-00011.
Thoemmes, Felix J., and Anthony D. Ong. 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, Tyler J., John W. Jackson, and Shanshan Li. 2016. “Causal Inference and Longitudinal Data: A Case Study of Religion and Mental Health.” Social Psychiatry and Psychiatric Epidemiology 51 (11): 1457–66. https://doi.org/10.1007/s00127-016-1281-9.