Estimation of the conditional average treatment effect (CATE) score for continuous data

Provides singly robust and doubly robust estimation of CATE score with up to 6 scoring methods among the following: Linear regression, boosting, two regressions, contrast regression, random forest and generalized additive model.

Usage

catefitmean(
  data,
  score.method,
  cate.model,
  ps.model,
  ps.method = "glm",
  init.model = NULL,
  initial.predictor.method = "boosting",
  minPS = 0.01,
  maxPS = 0.99,
  higher.y = TRUE,
  prop.cutoff = seq(0.5, 1, length = 6),
  xvar.smooth.score = NULL,
  xvar.smooth.init = NULL,
  tree.depth = 2,
  n.trees.rf = 1000,
  n.trees.boosting = 200,
  B = 3,
  Kfold = 6,
  plot.gbmperf = FALSE,
  error.maxNR = 0.001,
  tune = c(0.5, 2),
  seed = NULL,
  verbose = 0,
  ...
)

Arguments

data: A data frame containing the variables in the outcome and propensity score models; a data frame with n rows (1 row per observation).
score.method: A vector of one or multiple methods to estimate the CATE score. Allowed values are: 'boosting', 'gaussian', 'twoReg', 'contrastReg', 'randomForest', 'gam'.
cate.model: A formula describing the outcome model to be fitted. The outcome must appear on the left-hand side.
ps.model: A formula describing the propensity score model to be fitted. The treatment must appear on the left-hand side. The treatment must be a numeric vector coded as 0/1. If data are from a RCT, specify ps.model as an intercept-only model.
ps.method: A character value for the method to estimate the propensity score. Allowed values include one of: 'glm' for logistic regression with main effects only (default), or 'lasso' for a logistic regression with main effects and LASSO penalization on two-way interactions (added to the model if not specified in ps.model).
init.model: A formula describing the initial predictor model. The outcome must appear on the left-hand side. It must be specified when score.method = contrastReg or twoReg.
initial.predictor.method: A character vector for the method used to get initial outcome predictions conditional on the covariates in init.model in score.method = 'twoReg' and 'contrastReg'. Allowed values include one of 'gaussian' (fastest), 'boosting' and 'gam'. Default is 'boosting'.
minPS: A numerical value (in `[0, 1]`) below which estimated propensity scores should be truncated. Default is 0.01.
maxPS: A numerical value (in `(0, 1]`) above which estimated propensity scores should be truncated. Must be strictly greater than minPS. Default is 0.99.
higher.y: A logical value indicating whether higher (TRUE) or lower (FALSE) values of the outcome are more desirable. Default is TRUE.
prop.cutoff: A vector of numerical values (in `(0, 1]`) specifying percentiles of the estimated log CATE scores to define nested subgroups. Each element represents the cutoff to separate observations in nested subgroups (below vs above cutoff). The length of prop.cutoff is the number of nested subgroups. An equally-spaced sequence of proportions ending with 1 is recommended. Default is seq(0.5, 1, length = 6).
xvar.smooth.score: A vector of characters indicating the name of the variables used as the smooth terms if score.method = 'gam'. The variables must be selected from the variables listed in cate.model. Default is NULL, which uses all variables in cate.model.
xvar.smooth.init: A vector of characters indicating the name of the variables used as the smooth terms if initial.predictor.method = 'gam'. The variables must be selected from the variables listed in init.model. Default is NULL, which uses all variables in init.model.
tree.depth: A positive integer specifying the depth of individual trees in boosting (usually 2-3). Used only if score.method = 'boosting' or if score.method = 'twoReg' or 'contrastReg' and initial.predictor.method = 'boosting'. Default is 2.
n.trees.rf: A positive integer specifying the number of trees. Used only if score.method = 'randomForest'. Default is 1000.
n.trees.boosting: A positive integer specifying the maximum number of trees in boosting (usually 100-1000). Used only if score.method = 'boosting' or if score.method = 'twoReg' or 'contrastReg' and initial.predictor.method = 'boosting'. Default is 200.
B: A positive integer specifying the number of time cross-fitting is repeated in score.method = 'twoReg' and 'contrastReg'. Default is 3.
Kfold: A positive integer specifying the number of folds (parts) used in cross-fitting to partition the data in score.method = 'twoReg' and 'contrastReg'. Default is 6.
plot.gbmperf: A logical value indicating whether to plot the performance measures in boosting. Used only if score.method = 'boosting' or if score.method = 'twoReg' or 'contrastReg' and initial.predictor.method = 'boosting'. Default is TRUE.
error.maxNR: A numerical value > 0 indicating the minimum value of the mean absolute error in Newton Raphson algorithm. Used only if score.method = 'contrastReg'. Default is 0.001.
tune: A vector of 2 numerical values > 0 specifying tuning parameters for the Newton Raphson algorithm. tune[1] is the step size, tune[2] specifies a quantity to be added to diagonal of the slope matrix to prevent singularity. Used only if score.method = 'contrastReg'. Default is c(0.5, 2).
seed: An optional integer specifying an initial randomization seed for reproducibility. Default is NULL, corresponding to no seed.
verbose: An integer value indicating what kind of intermediate progress messages should be printed. 0 means no outputs. 1 means only progress and run time. 2 means progress, run time, and all errors and warnings. Default is 0.
...: Additional arguments for gbm()

Value

Returns a list containing the following components:

ate.gaussian: A vector of numerical values of length prop.cutoff containing the estimated ATE in nested subgroups (defined by prop.cutoff) constructed based on the estimated CATE scores with Poisson regression. Only provided if score.method includes 'gaussian'.
ate.boosting: Same as ate.gaussian, but with the nested subgroups based the estimated CATE scores with boosting. Only provided if score.method includes 'boosting'.
ate.twoReg: Same as ate.gaussian, but with the nested subgroups based the estimated CATE scores with two regressions. Only provided if score.method includes 'twoReg'.
ate.contrastReg: Same as ate.gaussian, but with the nested subgroups based the estimated CATE scores with contrast regression. Only provided if score.method includes 'contrastReg'.
ate.randomForest: Same as ate.gaussian, but with the nested subgroups based the estimated CATE scores with random forest. Only provided if score.method includes 'gam'.
ate.gam: Same as ate.gaussian, but with the nested subgroups based the estimated CATE scores with generalized additive model. Only provided if score.method includes 'gam'.
score.gaussian: A vector of numerical values of length n (number of observations in data) containing the estimated CATE scores according to the linear regression. Only provided if score.method includes 'gaussian'.
score.boosting: Same as score.gaussian, but with estimated CATE score according to boosting. Only provided if score.method includes 'boosting'.
score.twoReg: Same as score.gaussian, but with estimated CATE score according to two regressions. Only provided if score.method includes 'twoReg'.
score.contrastReg: Same as score.gaussian, but with estimated CATE score according to contrast regression. Only provided if score.method includes 'contrastReg'.
score.randomForest: Same as score.gaussian, but with estimated CATE score according to random forest. Only provided if score.method includes 'randomForest'.
score.gam: Same as score.gaussian, but with estimated CATE score according to generalized additive model. Only provided if score.method includes 'gam'.
fit: Additional details on model fitting if score.method includes 'boosting' or 'contrastReg':
- result.boosting: Details on the boosting model fitted to observations with treatment = 0 ($fit0.boosting) and to observations with treatment = 1 ($fit1.boosting). Only provided if score.method includes 'boosting'.
- result.randomForest: Details on the boosting model fitted to observations with treatment = 0 ($fit0.randomForest) and to observations with treatment = 1 ($fit1.randomForest). Only provided if score.method includes 'randomForest'.
- result.gam: Details on the boosting model fitted to observations with treatment = 0 ($fit0.gam) and to observations with treatment = 1 ($fit1.gam). Only provided if score.method includes 'gam'.
- result.contrastReg$sigma.contrastReg: Variance-covariance matrix of the estimated CATE coefficients in contrast regression. Only provided if score.method includes 'contrastReg'.
coefficients: A data frame with the coefficients of the estimated CATE score by score.method. The data frame has number of rows equal to the number of covariates in cate.model and number of columns equal to length(score.method). If score.method includes 'contrastReg', the data frame has an additional column containing the standard errors of the coefficients estimated with contrast regression. 'boosting', 'randomForest', 'gam' do not have coefficient results because these methods do not express the CATE as a linear combination of coefficients and covariates.

Details

The CATE score represents an individual-level treatment effect, estimated with either linear regression, boosting, random forest and generalized additive model applied separately by treatment group or with two doubly robust estimators, two regressions and contrast regression (Yadlowsky, 2020) applied to the entire dataset.

catefitmean() provides the coefficients of the CATE score for each scoring method requested through score.method. Currently, contrast regression is the only method which allows for inference of the CATE coefficients by providing standard errors of the coefficients. The coefficients can be used to learn the effect size of each variable and predict the CATE score for a new observation.

catefitmean() also provides the predicted CATE score of each observation in the data set, for each scoring method. The predictions allow ranking the observations from potentially high responders to the treatment to potentially low or standard responders.

The estimated ATE among nested subgroups of high responders are also provided by scoring method. Note that the ATEs in catefitmean() are derived based on the CATE score which is estimated using the same data sample. Therefore, overfitting may be an issue. catefitmean() is more suitable to inspect the estimated ATEs across scoring methods as it implements internal cross validation to reduce optimism.

References