catecvcount {precmed}

Cross-validation of the conditional average treatment effect (CATE) score for count outcomes

Description

Provides doubly robust estimation of the average treatment effect (ATE) in nested and mutually exclusive subgroups of patients defined by an estimated conditional average treatment effect (CATE) score via cross-validation (CV). The CATE score can be estimated with up to 5 methods among the following: Poisson regression, boosting, two regressions, contrast regression, and negative binomial (see score.method).

Usage

catecvcount(
  data,
  score.method,
  cate.model,
  ps.model,
  ps.method = "glm",
  initial.predictor.method = "boosting",
  minPS = 0.01,
  maxPS = 0.99,
  higher.y = TRUE,
  prop.cutoff = seq(0.5, 1, length = 6),
  prop.multi = c(0, 1/3, 2/3, 1),
  abc = TRUE,
  train.prop = 3/4,
  cv.n = 10,
  error.max = 0.1,
  max.iter = 5000,
  xvar.smooth = NULL,
  tree.depth = 2,
  n.trees.boosting = 200,
  B = 3,
  Kfold = 5,
  error.maxNR = 0.001,
  max.iterNR = 150,
  tune = c(0.5, 2),
  seed = NULL,
  plot.gbmperf = TRUE,
  verbose = 0,
  ...
)

Arguments

data

A data frame containing the variables in the outcome and propensity score model; 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', 'poisson', 'twoReg', 'contrastReg', and 'negBin'.

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 or 1. If data are from a randomized trial, 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 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 in cate.model. Only applies when score.method includes 'twoReg' or 'contrastReg'. Allowed values include one of 'poisson' (fastest), 'boosting' and 'gam'. Default is 'boosting'.

minPS

A numerical value between 0 and 1 below which estimated propensity scores should be truncated. Default is 0.01.

maxPS

A numerical value between 0 and 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 between 0 and 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).

prop.multi

A vector of numerical values between 0 and 1 specifying percentiles of the estimated log CATE scores to define mutually exclusive subgroups. It should start with 0, end with 1, and be of length(prop.multi) > 2. Each element represents the cutoff to separate the observations into length(prop.multi) - 1 mutually exclusive subgroups. Default is c(0, 1/3, 2/3, 1).

abc

A logical value indicating whether the area between curves (ABC) should be calculated at each cross-validation iterations, for each score.method. Default is TRUE.

train.prop

A numerical value between 0 and 1 indicating the proportion of total data used for training. Default is 3/4.

cv.n

A positive integer value indicating the number of cross-validation iterations. Default is 10.

error.max

A numerical value > 0 indicating the tolerance (maximum value of error) for the largest standardized absolute difference in the covariate distributions or in the doubly robust estimated rate ratios between the training and validation sets. This is used to define a balanced training-validation splitting. Default is 0.1.

max.iter

A positive integer value indicating the maximum number of iterations when searching for a balanced training-validation split. Default is 5,000.

xvar.smooth

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 cate.model. Default is NULL, which uses all variables in cate.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 initial.predictor.method = 'boosting' with score.method = 'twoReg' or 'contrastReg'. Default is 2.

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.

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.

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.

verbose

An integer value indicating what kind of intermediate progress messages should be printed. 0 means no outputs. 1 means only progress bar and run time. 2 means progress bar, run time, and all errors and warnings. Default is 0.

...

Additional arguments for gbm()

Details

The CATE score represents an individual-level treatment effect expressed as a rate ratio for count outcomes. It can be estimated with boosting, Poisson regression, negative binomial regression, and the doubly robust estimator two regressions (Yadlowsky, 2020) applied separately by treatment group or with the other doubly robust estimator contrast regression (Yadlowsky, 2020) applied to the entire data set.

Internal CV is applied to reduce optimism in choosing the CATE estimation method that captures the most treatment effect heterogeneity. The CV is applied by repeating the following steps cv.n times:

1. Split the data into a training and validation set according to train.prop. The training and validation sets must be balanced with respect to covariate distributions and doubly robust rate ratio estimates (see error.max).

2. Estimate the CATE score in the training set with the specified scoring method.

3. Predict the CATE score in the validation set using the scoring model fitted from the training set.

4. Build nested subgroups of treatment responders in the training and validation sets, separately, and estimate the ATE within each nested subgroup. For each element i of prop.cutoff (e.g., prop.cutoff[i] = 0.6), take the following steps:

   1. Identify high responders as observations with the 60% (i.e., prop.cutoff[i]x100%) highest (if higher.y = TRUE) or lowest (if higher.y = FALSE) estimated CATE scores.

   2. Estimate the ATE in the subgroup of high responders using a doubly robust estimator.

   3. Conversely, identify low responders as observations with the 40% (i.e., 1 - prop.cutoff[i]x100%) lowest (if higher.y = TRUE) or highest (if higher.y = FALSE) estimated CATE scores.

   4. Estimate the ATE in the subgroup of low responders using a doubly robust estimator.

5. If abc = TRUE, calculate the area between the ATE and the series of ATEs in nested subgroups of high responders in the validation set.

6. Build mutually exclusive subgroups of treatment responders in the training and validation sets, separately, and estimate the ATE within each subgroup. Mutually exclusive subgroups are built by splitting the estimated CATE scores according to prop.multi.

Value

Returns a list containing the following components saved as a "precmed" object:

• ate.poisson: A list of results output if score.method includes 'poisson':

  • ate.est.train.high.cv: A matrix of numerical values with length(prop.cutoff) rows and cv.n columns. The ith row/jth column cell contains the estimated ATE in the nested subgroup of high responders defined by CATE score above (if higher.y = TRUE) or below (if higher.y = FALSE) the prop.cutoff[i]x100% percentile of the estimated CATE score in the training set in the jth cross-validation iteration.

  • ate.est.train.low.cv: A matrix of numerical values with length(prop.cutoff) - 1 rows and cv.n columns. The ith row/jth column cell contains the estimated ATE in the nested subgroup of low responders defined by CATE score below (if higher.y = TRUE) or above (if higher.y = FALSE) the prop.cutoff[i]x100% percentile of the estimated CATE score in the training set in the jth cross-validation iteration.

  • ate.est.valid.high.cv: Same as ate.est.train.high.cv, but in the validation set.

  • ate.est.valid.low.cv: Same as ate.est.train.low.cv, but in the validation set.

  • ate.est.train.group.cv: A matrix of numerical values with length(prop.multi) - 1 rows and cv.n columns. The jth column contains the estimated ATE in length(prop.multi) - 1 mutually exclusive subgroups defined by prop.multi in the training set in jth cross-validation iteration.

  • ate.est.valid.group.cv: Same as ate.est.train.group.cv, but in the validation set.

  • abc.valid: A vector of numerical values of length cv.n. The ith element returns the ABC of the validation curve in the ith cross-validation iteration. Only returned if abc = TRUE.

• ate.boosting: A list of results similar to ate.poisson output if score.method includes 'boosting'.

• ate.twoReg: A list of results similar to ate.poisson output if score.method includes 'twoReg'.

• ate.contrastReg: A list of results similar to ate.poisson output if score.method includes 'contrastReg'. This method has an additional element in the list of results:

  • converge.contrastReg.cv: A vector of logical value of length cv.n. The ith element indicates whether the algorithm converged in the ith cross-validation iteration.

• ate.negBin: A list of results similar to ate.poisson output if score.method includes 'negBin'.

• props: A list of 3 elements:

  • prop.onlyhigh: The original argument prop.cutoff, reformatted as necessary.

  • prop.bi: The original argument prop.cutoff, similar to prop.onlyhigh but reformatted to exclude 1.

  • prop.multi: The original argument prop.multi, reformatted as necessary to include 0 and 1.

• overall.ate.valid: A vector of numerical values of length cv.n. The ith element contains the ATE in the validation set of the ith cross-validation iteration, estimated with the doubly robust estimator.

• overall.ate.train: A vector of numerical values of length cv.n. The ith element contains the ATE in the training set of the ith cross-validation iteration, estimated with the doubly robust estimator.

• fgam: The formula used in GAM if initial.predictor.method = 'gam'.

• higher.y: The original higher.y argument.

• abc: The original abc argument.

• cv.n: The original cv.n argument.

• response: The type of response. Always 'count' for this function.

• formulas:A list of 3 elements: (1) cate.model argument, (2) ps.model argument and (3) original labels of the left-hand side variable in ps.model (treatment) if it was not 0/1.

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

plot.precmed(), boxplot.precmed(), abc() methods for "precmed" objects, and catefitcount() function.

Examples

catecv <- catecvcount(data = countExample,
                      score.method = "poisson",
                      cate.model = y ~ age + female + previous_treatment +
                                   previous_cost + previous_number_relapses +
                                   offset(log(years)),
                      ps.model = trt ~ age + previous_treatment,
                      higher.y = FALSE,
                      cv.n = 5,
                      seed = 999,
                      plot.gbmperf = FALSE,
                      verbose = 1)

plot(catecv, ylab = "Rate ratio of drug1 vs drug0 in each subgroup")
boxplot(catecv, ylab = "Rate ratio of drug1 vs drug0 in each subgroup")
abc(catecv)