Using `cobalt` with Longitudinal Treatments

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
## 1    2    0   1    5    1   0    4    1   0   0
## 2    4    0   1    9    0   1   10    0   1   1
## 3    4    1   0    5    0   1    4    0   0   1
## 4    4    1   0    4    0   0    6    1   0   1
## 5    6    1   1    5    0   1    6    0   0   1
## 6    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:

A_1 is independent from X1_0 and X2_0
A_2 is independent from X1_0, X2_0, A_1,X1_1, andX2_1`
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
## prop.score Distance  0.9851   0.0409
## 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
## prop.score Distance  1.1546   0.0773
## 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
## prop.score Distance  1.6762   0.0327
## 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
## prop.score 1, 2, 3 Distance      1.6762       0.0773
## 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.