catecvmean {precmed}

Cross-validation of the conditional average treatment effect (CATE) score for continuous 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 6 methods among the following: Linear regression, boosting, two regressions, contrast regression, random forest and generalized additive model (see score.method).

Usage

catecvmean(
  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),
  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.score = NULL,
  xvar.smooth.init = NULL,
  tree.depth = 2,
  n.trees.rf = 1000,
  n.trees.boosting = 200,
  B = 3,
  Kfold = 6,
  plot.gbmperf = TRUE,
  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 interactions are not specified in ps.model). Relevant only when ps.model has more than one variable.

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 cate.model in score.method = 'twoReg' and 'contrastReg'. Allowed values include one of 'poisson' (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 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 (in [0, 1]) specifying percentiles of the estimated 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 (in (0, 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.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 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 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 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 for continuous data, estimated with boosting, linear regression, random forest, generalized additive model 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.

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. 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.

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

Value

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

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

  • ate.est.train.high.cv: A matrix of numerical values with length(prop.cutoff) rows and cv.n columns. The ith column/jth row 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[j]x100% percentile of the estimated CATE score in the training set in the ith 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 column/jth row 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[j]x100% percentile of the estimated CATE score in the training set in the ith 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 ith column contains the estimated ATE in length(prop.multi) - 1 mutually exclusive subgroups defined by prop.multi in the training set in ith 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.gaussian output if score.method includes 'boosting'.

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

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

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

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

• 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.

• 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.

• 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.

• 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 'continuous' 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 catefitmean() function.

Examples

# Not implemented yet!