catefitsurv {precmed}

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

Description

Provides singly robust and doubly robust estimation of CATE score for survival data with up to 5 scoring methods among the following: Random forest, boosting, poisson regression, two regressions, and contrast regression.

Usage

catefitsurv(
  data,
  score.method,
  cate.model,
  ps.model,
  ps.method = "glm",
  initial.predictor.method = "randomForest",
  ipcw.model = NULL,
  ipcw.method = "breslow",
  minPS = 0.01,
  maxPS = 0.99,
  followup.time = NULL,
  tau0 = NULL,
  higher.y = TRUE,
  prop.cutoff = seq(0.5, 1, length = 6),
  surv.min = 0.025,
  tree.depth = 2,
  n.trees.rf = 1000,
  n.trees.boosting = 200,
  B = 3,
  Kfold = 5,
  plot.gbmperf = TRUE,
  error.maxNR = 0.001,
  max.iterNR = 100,
  tune = c(0.5, 2),
  seed = NULL,
  verbose = 0,
  ...
)

Arguments

data

A data frame containing the variables in the outcome, propensity score, and inverse probability of censoring models (if specified); 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: 'randomForest', 'boosting', 'poisson', 'twoReg', and 'contrastReg'.

cate.model

A standard Surv 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 (PS) 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 randomized controlled trial, specify ps.model = ~1 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 interactions are not specified in ps.model). Relevant only when ps.model has more than one variable.

initial.predictor.method

A character vector for the method used to get initial outcome predictions conditional on the covariates specified in cate.model. Only applies when score.method includes 'twoReg' or 'contrastReg'. Allowed values include one of 'randomForest', 'boosting' and 'logistic' (fastest). Default is 'randomForest'.

ipcw.model

A formula describing the inverse probability of censoring weighting (IPCW) model to be fitted. The left-hand side must be empty. Default is ipcw.model = NULL, which corresponds to specifying the IPCW model with the same covariates as the outcome model cate.model plus the treatment.

ipcw.method

A character value for the censoring model. Allowed values are: 'breslow' (Cox regression with Breslow estimator of the baseline survivor function), 'aft (exponential)', 'aft (weibull)', 'aft (lognormal)' or 'aft (loglogistic)' (accelerated failure time model with different distributions for y variable). Default is 'breslow'.

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.

followup.time

A column name in data specifying the maximum follow-up time, interpreted as the potential censoring time. Default is followup.time = NULL, which corresponds to unknown potential censoring time.

tau0

The truncation time for defining restricted mean time lost. Default is NULL, which corresponds to setting the truncation time as the maximum survival time in the data.

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).

surv.min

Lower truncation limit for the probability of being censored. It must be a positive value and should be chosen close to 0. Default is 0.025.

tree.depth

A positive integer specifying the depth of individual trees in boosting (usually 2-3). Used only if score.method = 'boosting' or if initial.predictor.method = 'boosting' with score.method = 'twoReg' or 'contrastReg'. Default is 2.

n.trees.rf

A positive integer specifying the maximum number of trees in random forest. Used if score.method = 'ranfomForest' or if initial.predictor.method = 'randomForest' with score.method = 'twoReg' or 'contrastReg'. Only applies for survival outcomes. Default is 1000.

n.trees.boosting

A positive integer specifying the maximum number of trees in boosting (usually 100-1000). Used if score.method = 'boosting' or if initial.predictor.method = 'boosting' with score.method = 'twoReg' or 'contrastReg'. 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 used in cross-fitting to partition the data in score.method = 'twoReg' and 'contrastReg'. Default is 5.

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.

max.iterNR

A positive integer indicating the maximum number of iterations in the Newton Raphson algorithm. Used only if score.method = 'contrastReg'. Default is 150.

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()

Details

The CATE score represents an individual-level treatment effect for survival data, estimated with random forest, boosting, Poisson regression, and the doubly robust estimator (two regressions, Yadlowsky, 2020) applied separately by treatment group or with the other doubly robust estimators (contrast regression, Yadlowsky, 2020) applied to the entire data set.

catefitsurv() 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.

catefitsurv() 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 catefitsurv() are derived based on the CATE score which is estimated using the same data sample. Therefore, overfitting may be an issue. catecvsurv() is more suitable to inspect the estimated ATEs across scoring methods as it implements internal cross validation to reduce optimism.

Value

Returns an object of the class catefit containing the following components:

• ate.randomForest: A vector of numerical values of length prop.cutoff containing the estimated ATE by the RMTL ratio in nested subgroups (defined by prop.cutoff) constructed based on the estimated CATE scores with random forest method. Only provided if score.method includes 'randomForest'.

• ate.boosting: Same as ate.randomForest, but with the nested subgroups based the estimated CATE scores with boosting. Only provided if score.method includes 'boosting'.

• ate.poisson: Same as ate.randomForest, but with the nested subgroups based the estimated CATE scores with poisson regression. Only provided if score.method includes 'poisson'.

• ate.twoReg: Same as ate.randomForest, 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.randomForest, but with the nested subgroups based the estimated CATE scores with contrast regression. Only provided if score.method includes 'contrastReg'.

• hr.randomForest: A vector of numerical values of length prop.cutoff containing the adjusted hazard ratio in nested subgroups (defined by prop.cutoff) constructed based on the estimated CATE scores with random forest method. Only provided if score.method includes 'randomForest'.

• hr.boosting: Same as hr.randomForest, but with the nested subgroups based the estimated CATE scores with boosting. Only provided if score.method includes 'boosting'.

• hr.poisson: Same as hr.randomForest, but with the nested subgroups based the estimated CATE scores with poisson regression. Only provided if score.method includes 'poisson'.

• hr.twoReg: Same as hr.randomForest, but with the nested subgroups based the estimated CATE scores with two regressions. Only provided if score.method includes 'twoReg'.

• hr.contrastReg: Same as hr.randomForest, but with the nested subgroups based the estimated CATE scores with contrast regression. Only provided if score.method includes 'contrastReg'.

• score.randomForest: A vector of numerical values of length n (number of observations in data) containing the estimated log-CATE scores according to random forest. Only provided if score.method includes 'randomForest'.

• score.boosting: Same as score.randomForest, but with estimated log-CATE score according to boosting. Only provided if score.method includes 'boosting'.

• score.poisson: Same as score.randomForest, but with estimated log-CATE score according to the Poisson regression. Only provided if score.method includes 'poisson'.

• score.twoReg: Same as score.randomForest, but with estimated log-CATE score according to two regressions. Only provided if score.method includes 'twoReg'.

• score.contrastReg: Same as score.randomForest, but with estimated log-CATE score according to contrast regression. Only provided if score.method includes 'contrastReg'.

• fit: Additional details on model fitting if score.method includes 'randomForest', 'boosting' or 'contrastReg':

  • result.randomForest: Details on the random forest model fitted to observations with treatment = 0 ($fit0.rf) and to observations with treatment = 1 ($fit1.rf). Only provided if score.method includes 'randomForest'.

  • result.boosting: Details on the boosting model fitted to observations with treatment = 0, ($fit0.boosting) and ($fit0.gam) and to observations with treatment = 1, ($fit1.boosting) and ($fit1.gam). Only provided if score.method includes 'boosting'.

  • result.contrastReg$converge.contrastReg: Whether the contrast regression algorithm converged or not. Only provided if score.method includes 'contrastReg'.

• coefficients: A data frame with the coefficients of the estimated log-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. 'randomForest' and 'boosting' do not have coefficient results because tree-based methods typically do not express the log-CATE as a linear combination of coefficients and covariates.

• errors/warnings: A nested list of errors and warnings that were wrapped during the calculation of ATE. Errors and warnings are organized by score.method.

References

Yadlowsky, S., Pellegrini, F., Lionetto, F., Braune, S., & Tian, L. (2020). Estimation and validation of ratio-based conditional average treatment effects using observational data. Journal of the American Statistical Association, 1-18. https://www.tandfonline.com/doi/full/10.1080/01621459.2020.1772080

See Also

catecvsurv()

Examples

library(survival)

tau0 <- with(survivalExample, min(quantile(y[trt == "drug1"], 0.95),
                               quantile(y[trt == "drug0"], 0.95)))

fit <- catefitsurv(data = survivalExample,
                   score.method = "randomForest",
                   cate.model = Surv(y, d) ~ age + female + previous_cost +
                                             previous_number_relapses,
                   ps.model = trt ~ age + previous_treatment,
                   ipcw.model = ~ age + previous_cost + previous_treatment,
                   tau0 = tau0,
                   seed = 999)

coef(fit)