Summary: “Introduction to stratamatch” covers the basic functionality of stratamatch: stratifying a data set, assessing the quality of the match, and matching the data within strata. This vignette features more advanced functionality of stratamatch.

This vignette contains:

1. Set-up
2. Splitting the pilot set
3. Fitting the prognostic model
4. Alternative matching schemes

# Set up

library(stratamatch)
library(dplyr)
#>
#> Attaching package: 'dplyr'
#> The following objects are masked from 'package:stats':
#>
#>     filter, lag
#> The following objects are masked from 'package:base':
#>
#>     intersect, setdiff, setequal, union
set.seed(123)
mydata <- make_sample_data(n = 5000)

# Splitting the pilot set

An important consideration for pilot designs like the stratamatch approach is the selection of the pilot set. Ideally, the individuals in the pilot set should be similarto the individuals in the treatment group, so a prognostic model built on this pilot set will not beextrapolating heavily when estimating prognostic scores on the analysis set. To more closely ensure that the selected pilot set is a representative sample of controls, one easy step is to specify a list of categorical or binary covariates and sample the pilot set proportionally based on these covariates.

This can be done in one step using auto_stratify, for example:

a.strat1 <- auto_stratify(mydata, "treat",
prognosis = outcome ~ X1 + X2,
group_by_covariates = c("C1", "B2"), size = 500)
#> Constructing a pilot set by subsampling 10% of controls.
#> Subsampling while balancing on:
#> C1, B2
#> Fitting prognostic model via logistic regression: outcome ~ X1 + X2

Another method is to use the split_pilot_set function, which allows the user to split the pilot set (and examine the balance) before passing the result to auto_stratify to fit the prognostic score.

mysplit <- split_pilot_set(mydata, "treat", group_by_covariates = c("C1", "B2"))
#> Constructing a pilot set by subsampling 10% of controls.
#> Subsampling while balancing on:
#> C1, B2

The result, mysplit, is a list containing a pilot_set and an analysis_set, in this case partitioned while balancing on B1 and C2. At this point, we might pass the result to auto_stratify as follows:

a.strat2 <- auto_stratify(mysplit$analysis_set, "treat", prognosis = outcome ~ X1 + X2, pilot_sample = mysplit$pilot_set, size = 500)
#> Using user-specified set for prognostic score modeling.
#> Fitting prognostic model via logistic regression: outcome ~ X1 + X2

In this case, the pilot set splitting method is the same for a.strat1 and a.strat2, so each should be qualitatively similar.

# Fitting the prognostic model

By default, auto_stratify uses a logistic regression (for binary outcomes) or a linear regression (for continuous outcomes) to fit the prognostic model. auto_stratify is built to accomodate other prognostic score estimation approaches: rather than letting auto_stratify do the model fitting, the user can specify a pre-fit model or a vector of prognostic scores.

A word of caution is advisable: Logistic regression is generally the norm when fitting propensity scores (Stuart (2010)), and most studies discussing the prognostic score have likewise focused on linear or logistic regression (Hansen (2008), Leacy and Stuart (2014), Aikens et al. (2020)), or less commonly the lasso (Antonelli et al. (2018)). The nuances of modeling and checking the fit of the prognostic score are still understudied. In particular, prognostic models generally are fit on only control observations, meaning that they must necessarily extrapolate to the treatment group. Users should, as always, consider diagnostics for their specific model, for their stratified data set (see “Intro to statamatch”), and for their matched dataset (see, as an introduction Stuart (2010)). Additionally, in order to maintain the integrity of the pilot set and prevent over-fitting, users interested in trying many modeling or matching schemes are encouraged to define a single pilot/analysis set split (e.g. with split_pilot_set, above) and use the same pilot set throughout their design process.

To an extent, any prognostic score stratification – even on a poor quality model – will increase the speed of matching for large datasets. In addition, if the strata are sufficiently large, the subsequent within-strata matching step may compensate for a poor-quality prognostic model. To one view, the diagnostic check that matters most is the quality of the final matching result, whereas specific prognostic modeling concerns are perhaps secondary. Nonetheless, simulation and theory results suggest that incorporating a prognostic score into a study design (in combination, generally, with a propensity score) can have favorable statistical properties, such as decreasing variance, increasing power in gamma sensitivity analyses, and decreasing dependence on the propensity score (Stuart (2010), Aikens, Greaves, and Baiocchi (2020), Antonelli et al. (2018), Hansen (2008)). To that end, prognostic models which produce high-quality prognostic score estimates are expected to ultimately produce higher quality designs by improving the prognostic balance of the matched set.

This section contains a few examples to introduce users who may be new to the modeling space. By no means does it begin to cover all of the modeling possibilities, or even all of the nuances of any one model. This is not a tutorial on predictive modeling. Users looking to become more familiar with modeling in R more broadly may be interested in the caret package (Kuhn (2021)), which implements support for a wide variety of predictive models.

## Outcomes: Binary or Continuous

It’s important to select a model which is appropriate to the nature of the outcome of interest. In this tutorial, our sample data has a binary outcome, so we use models appropriate to that outcome. Users with continuous outcomes should use regression models appropriate to continuous outcomes. Other types of outcomes – such as categorical – have not yet been characterized in the prognostic score literature.

## A lasso

The lasso is a sparsifying linear model – it is a mathematical cousin to linear regression, but it functions well when a substaintial number of the measured covariates are actually uninformative to the outcome. This may be a particularly useful and intuitive approach when there are many measured covariates which may be redundant or uninformative.

The code below uses the glmnet package (Friedman, Hastie, and Tibshirani (2010)) to fit a cross-validated lasso on the pilot set based on all the measured covariates. In this example, since the outcome is binary, we will run a logistic lasso. This is done by specifying the family = "binomial" argument to cv.glmnet (although other modeling steps are simular for continuous outcomes.)

The code below does some preprocessing to convert the pilot set data to the right format before passing it to cv.glmnet. glmnet expects the input data to be a model matrix rather than a data frame, and it expects outcomes (y_pilot) to be separated from the covariate data (x_pilot).

library(glmnet)

# fit model on pilot set
x_pilot <- model.matrix(outcome ~ X1 + X2 + B1 + B2 + C1,
data = mysplit$pilot_set) y_pilot <- mysplit$pilot_set %>%
dplyr::select(outcome) %>%
as.matrix()

cvlasso <- cv.glmnet(x_pilot, y_pilot, family = "binomial")

At this point, we can run diagnostics on cvlasso. The Introduction to glmnet vignette contains an accessible starting point. In this simulated data, we happen to know that the only variable that actually affects the outcome is X1. The sparsifying model often does a great job of picking out X1 as the most important variable. We can see this by printing the coefficients:

coef(cvlasso)
#> 8 x 1 sparse Matrix of class "dgCMatrix"
#>                       1
#> (Intercept)  0.02870109
#> (Intercept)  .
#> X1          -0.64374310
#> X2           .
#> B1           .
#> B2           .
#> C1b          .
#> C1c          .

When we are satisfied with our prognostic model, we can estimate the scores on the analysis set with predict and pass the result to auto_stratify.

# estimate scores on analysis set
x_analysis <- model.matrix(outcome ~ X1 + X2 + B1 + B2 + C1,
data = mysplit$analysis_set) lasso_scores <- predict(cvlasso, newx = x_analysis, s = "lambda.min", type = "response") # pass the scores to auto_stratify a.strat_lasso <- auto_stratify(data = mysplit$analysis_set,
treat = "treat",
outcome = "outcome",
prognosis = lasso_scores,
pilot_sample = mysplit$pilot_set, size = 500) ## An elastic net An elastic net is a fairly straightforward extension of the code above – we can use the same form of the pilot set data that we used above. An additional task is to select the alpha “mixing” parameter determining the amount of L2-regularization (1 for lasso, 0 for ridge regression). Here, I set alpha = 0.2. The tutorial for glmnet also contains some advice for selecting the alpha parameter via cross-validation. As above, we specify a logistic elastic net with family = "binomial", since in this case our outcome is binary. cvenet <- cv.glmnet(x_pilot, y_pilot, family = "binomial", alpha = 0.2) enet_scores <- predict(cvenet, newx = x_analysis, s = "lambda.min", type = "response") # pass the scores to auto_stratify a.strat_enet <- auto_stratify(data = mysplit$analysis_set,
treat = "treat",
outcome = "outcome",
prognosis = enet_scores,
pilot_sample = mysplit$pilot_set, size = 500) ## A random forest Random forests are a popular option for both classification and regression modeling, particularly because of their strengths in modeling nonlinear relationships in the data. Below is an example which fits a random forest for our binary outcome using randomForest. A note for users with binary outcomes: randomForest will run regression by default if the outcome column is numeric. To circumvent this (e.g. for 0/1 coded data), the outcome can be cast as a factor. library(randomForest) #> randomForest 4.6-14 #> Type rfNews() to see new features/changes/bug fixes. #> #> Attaching package: 'randomForest' #> The following object is masked from 'package:dplyr': #> #> combine forest <- randomForest(as.factor(outcome) ~ X1 + X2 + B1 + B2, data = mysplit$pilot_set)

Random forests can be somewhat more opaque than linear models in terms of understanding how predictions are made. A good starting point is running importance(forest) to check on which features are weighted heavily in the model.

Below, we extract a “prognostic score” from the random forest. Another note for users with binary outcomes: The predict method for random forest classifiers outputs 0/1 predictions by default. These will be useless for stratification. Instead, we need to specify type = "prob" in the call to predict and extract the probabilities for the “positive” outcome class.

forest_scores <- predict(forest,
newdata = mysplit$analysis_set, type = "prob")[,1] a.strat_forest <- auto_stratify(data = mysplit$analysis_set,
treat = "treat",
outcome = "outcome",
prognosis = forest_scores,
pilot_sample = mysplit$pilot_set, size = 500) # Alternative matching schemes “Intro to Stratamatch” covers only the default functionality of stratamatch, which is a fixed 1:k propensity score match within strata. This tutorial covers some alternative options. ## Distance measure: Mahalanobis Distance Users can opt to use Mahalanobis distance rather then propensity score for the within-strata matching step by specifying the “method” to strata_match. We set k = 2, so that precisely 2 control observations are matched to each treated observation. mahalmatch <- strata_match(a.strat2, model = treat ~ X1 + X2 + B1 + B2, method = "mahal", k = 2) #> This function makes essential use of the optmatch package, which has an academic license. #> For more information, run optmatch::relaxinfo() #> Using Mahalanobis distance: treat ~ X1 + X2 + B1 + B2 summary(mahalmatch) #> Structure of matched sets: #> 1:2 0:1 #> 1146 1176 #> Effective Sample Size: 1528 #> (equivalent number of matched pairs). ## Matching procedure: Full Matching Full matching may be a particularly useful approach when the ratio of treated to control individuals varies within strata, but the researcher still would prefer to use as much of the data as possible. To do full matching, set k = "full". This can be used in combination with mahalanobis distance matching, as shown below: fullmahalmatch <- strata_match(a.strat2, model = treat ~ X1 + X2 + B1 + B2, method = "mahal", k = "full") #> This function makes essential use of the optmatch package, which has an academic license. #> For more information, run optmatch::relaxinfo() #> Using Mahalanobis distance: treat ~ X1 + X2 + B1 + B2 summary(fullmahalmatch) #> Structure of matched sets: #> 5+:1 4:1 3:1 2:1 1:1 1:2 1:3 1:4 1:5+ #> 8 6 14 43 448 135 87 69 204 #> Effective Sample Size: 1331.7 #> (equivalent number of matched pairs). ## Matching with other software stratamatch doesn’t natively support all possible matching schemes. Luckily, it can be fairly straightforward to stratify a data set with auto_stratify and match the results with other software. As an example, the code below uses the optmatch package (Hansen and Klopfer (2006)) to match within-strata using Mahalanobis distance with a propensity score caliper. library(optmatch) #> Loading required package: survival #> The optmatch package has an academic license. Enter relaxinfo() for more information. # mahalanobis distance matrix for within-strata matching mahaldist <- match_on(treat ~ X1 + X2 + B1 + B2, within = exactMatch(treat ~ stratum, data = a.strat2$analysis_set),
data = a.strat2$analysis_set) # add propensity score caliper propdist <- match_on(glm(treat ~ X1 + X2 + B1 + B2, family = "binomial", data = a.strat2$analysis_set))

mahalcaliper <- mahaldist + caliper(propdist, width = 1)

mahalcaliper_match <- pairmatch(mahalcaliper, data = a.strat2\$analysis_set)
#> Warning in fullmatch.BlockedInfinitySparseMatrix(x = x, min.controls = controls, : At least one subproblem matching failed.
#>  (Restrictions impossible to meet?)

summary(mahalcaliper_match)
#> Matches made in 8 of 10 subgroups, accounting for 3691 of 4614 total observations.
#>
#>
#> Structure of matched sets:
#>  1:0  1:1  0:1
#>  192  954 2514
#> Effective Sample Size:  954
#> (equivalent number of matched pairs).

# References

Aikens, Rachael C, Dylan Greaves, and Michael Baiocchi. 2020. “A Pilot Design for Observational Studies: Using Abundant Data Thoughtfully.” Statistics in Medicine. Wiley Online Library.

Aikens, Rachael C, Joseph Rigdon, Justin Lee, Michael Baiocchi, Andrew B Goldstone, Peter Chiu, Y Joseph Woo, and Jonathan H Chen. 2020. “Stratified Pilot Matching in R: The Stratamatch Package.” Statistics arXiv, January. https://arxiv.org/abs/2001.02775.

Antonelli, Joseph, Matthew Cefalu, Nathan Palmer, and Denis Agniel. 2018. “Doubly Robust Matching Estimators for High Dimensional Confounding Adjustment.” Biometrics 74 (4). Wiley Online Library: 1171–9.

Friedman, Jerome, Trevor Hastie, and Rob Tibshirani. 2010. “Regularization Paths for Generalized Linear Models via Coordinate Descent.” Journal of Statistical Software 33 (1). NIH Public Access: 1.

Hansen, Ben B. 2008. “The Prognostic Analogue of the Propensity Score.” Biometrika 95 (2). Oxford University Press: 481–88.

Hansen, Ben B., and Stephanie Olsen Klopfer. 2006. “Optimal Full Matching and Related Designs via Network Flows.” Journal of Computational and Graphical Statistics 15 (3): 609–27.

Kuhn, Max. 2021. Caret: Classification and Regression Training. https://CRAN.R-project.org/package=caret.

Leacy, Finbarr P, and Elizabeth A Stuart. 2014. “On the Joint Use of Propensity and Prognostic Scores in Estimation of the Average Treatment Effect on the Treated: A Simulation Study.” Statistics in Medicine 33 (20). Wiley Online Library: 3488–3508.

Stuart, Elizabeth A. 2010. “Matching Methods for Causal Inference: A Review and a Look Forward.” Statistical Science: A Review Journal of the Institute of Mathematical Statistics 25 (1). NIH Public Access: 1.