PTE_bootstrap_inference {PTE}

Bootstrap inference for a prespecified personalization / recommendation model

Description

Runs B bootstrap samples using a prespecified model then computes the two I estimates based on cross validation. p values of the two I estimates are computed for a given H_0: \mu_{I_0} = \mu_0 and confidence intervals are provided using the basic, percentile methods by default and the BCa method as well if desired.

Usage

PTE_bootstrap_inference(X, y, regression_type = "continuous",
  incidence_metric = "odds_ratio", personalized_model_build_function = NULL,
  censored = NULL, predict_function = function(mod, Xyleftout) {    
  predict(mod, Xyleftout) }, difference_function = NULL,
  cleanup_mod_function = NULL, y_higher_is_better = TRUE, verbose = FALSE,
  full_verbose = FALSE, H_0_mu_equals = NULL, pct_leave_out = 0.1,
  m_prop = 1, B = 3000, alpha = 0.05, run_bca_bootstrap = FALSE,
  display_adversarial_score = FALSE, num_cores = NULL)

Arguments

X

A n \times p dataframe of covariates where one column is labeled "treatment" and it is a binary vector of treatment allocations in the study.

y

An n-length numeric vector which is the response

regression_type

A string indicating the regression problem. Legal values are "continous" (the response y is a real number with no missing data, the default), "incidence" (the reponse y is either 0 or 1) and "survival". If the type is "survival", the user must also supply additional data via the parameter censored.

incidence_metric

Ignored unless the regression_type is "incidence" and difference_function is set to NULL (in the latter case, you have specified a more custom metric). Then, this parameter allows the user to select which of the three standard metrics to use for comparison: "probability_difference", "risk_ratio", "odds_ratio" where the default is "odds_ratio".

personalized_model_build_function

An R function that will be evaluated to construct the personalized medicine / recommendation model. In the formula for the model, the response is "y", the treatment vector is "treatment" and the data is "Xytrain". This function must return some type of object that can be used for prediction later via predict_function. Here are the defaults for each regression_type. They are linear models with first order interactions:

personalized_model_build_function = switch(regression_type, continuous = function(Xytrain) #defalt is OLS regression lm(y ~ . * treatment, data = Xytrain) , incidence = function(Xytrain) #default is logistic regression glm(y ~ . * treatment, data = Xytrain, family = "binomial") , survival = function(Xytrain) #default is Weibull regression survreg(Surv(Xytrain$y, Xytrain$censored) ~ (. - censored) * treatment, data = Xytrain, dist = "weibull") )

censored

Only required if the regression_type is "survival". In this case, this vector is of length n and is binary where 0 indicates censored and 1 indicates uncensored. In a clinical trial, someone who is still alive at the end of the study or was lost to follow up will receive a censor value of 0, while someone who died during the study will receive a censor value of 1. n and is binary where 0 indicates censorship (e.g. the patient died).

predict_function

An R function that will be evaluated on left out data after the model is built with the training data. This function uses the object "mod" that is the result of the personalized_model_build_function and it must make use of "Xyleftout", a subset of observations from X. This function must return a scalar numeric quantity for comparison. The default function is predict(mod, obs_left_out) e.g. the default looks like:

function(mod, Xyleftout) predict(mod, Xyleftout)

difference_function

A function which takes the result of one out of sample experiment (boostrap or not) of all n samples and converts it into a difference that will be used as a score in a score distribution to determine if the personalization model is statistically significantly able to distinguish subjects. The function looks as follows:

function(results, indices_1_1, indices_0_0, indices_0_1, indices_1_0) ... c(rec_vs_non_rec_diff_score, rec_vs_all_score, rec_vs_best_score)

where results is a matrix consisting of columns of the estimated response of the treatment administered, the estimated response of the counterfactual treatment, the administered treatment, the recommended treatment based on the personalization model, the real response, and if this subject was censored (0 if so). Here are a couple of example entries:

est_true est_counterfactual given_tx rec_tx real_y censored 166.8 152.2 1 1 324 1 1679.1 2072.0 1 0 160 0

The arguments indices_1_1, indices_0_0, indices_0_1, indices_1_0 give the indices of the subjects whose treatment was administered as 1 and whose optimal was 1, whose treatment was administered 0 and whose optimal was 0, etc.

This function should return three numeric scores: the recommend vs. the non-recommended (adversarial), the recommended vs. all (all) and the recommended vs. the best average treatment (best) as a 3-dimensional vector as illustrated above.

By default, this parameter is NULL which means for continuous and incidence the average difference is used and for survival, the median Kaplan-Meier survival is used.

cleanup_mod_function

A function that is called at the end of a cross validation iteration to cleanup the model in some way. This is used for instance if you would like to release the memory your model is using but generally does not apply. The default is NA for "no function."

y_higher_is_better

True if a response value being higher is clinically "better" than one that is lower (e.g. cognitive ability in a drug trial for the mentally ill). False if the response value being lower is clinically "better" than one that is higher (e.g. amount of weight lost in a weight-loss trial). Default is TRUE.

verbose

Prints out a dot for each bootstrap sample. This only works on some platforms.

full_verbose

Prints out full information for each cross validation model for each bootstrap sample. This only works on some platforms.

H_0_mu_equals

The \mu_{I_0} value in H_0. Default is NULL which specifies 0 for regression types continuous, survival and incidence (with incidence metric "probability_difference") or 1 if the regression type is incidence and the incidence metric is "risk_ratio" or "odds_ratio". These defaults essentially answer the question: does my allocation procedure do better than the business-as-usual / naive allocation procedure?

pct_leave_out

In the cross-validation, the proportion of the original dataset left out to estimate out-of-sample metrics. The default is 0.1 which corresponds to 10-fold cross validation.

m_prop

Within each bootstrap sample, the proportion of the total number of rows of X to sample without replacement. m_prop < 1 ensures the number of rows sampled is less than n which fixes the consistency of the bootstrap estimator of a non-smooth functional. The default is 1 since non-smoothness may not be a common issue.

B

The number of bootstrap samples to take. We recommend making this as high as you can tolerate given speed considerations. The default is 3000.

alpha

Defines the confidence interval size (1 - alpha). Defaults to 0.05.

run_bca_bootstrap

Do the BCA bootstrap as well. This takes double the time. It defaults to FALSE.

display_adversarial_score

The adversarial score records the personalization metric versus the deliberate opposite of the personalization. This does not correspond to any practical situation but it is useful for debugging. Default is FALSE.

num_cores

The number of cores to use in parallel to run the bootstrap samples more rapidly. Defaults to NULL which automatically sets it to one if there is one available processor or if there are multiple available processors, the number of available processors save one.

Value

A results object of type "PTE_bootstrap_results" that contains much information about the observed results and the bootstrap runs, including hypothesis testing and confidence intervals.

Author(s)

Adam Kapelner

Examples

## Not run: 
	library(PTE)
	B = 1000 #lower this for quicker demos

	##response: continuous
	data(continuous_example)
	X = continuous_example$X
 y = continuous_example$y
	pte_results = PTE_bootstrap_inference(X, y, regression_type = "continuous", B = B)
	pte_results

	##response: incidence
	data(continuous_example)
	X = continuous_example$X
 y = continuous_example$y
	y = ifelse(y > quantile(y, 0.75), 1, 0) #force incidence and pretend y came to you this way
#there are three ways to assess incidence effects below: 
	#	odds ratio, risk ratio and probability difference 
	pte_results = PTE_bootstrap_inference(X, y, regression_type = "incidence", B = B)
	pte_results
	pte_results = PTE_bootstrap_inference(X, y, regression_type = "incidence", B = B, 
                                      incidence_metric = "risk_ratio")
	pte_results
	pte_results = PTE_bootstrap_inference(X, y, regression_type = "incidence", B = B, 
	                                      incidence_metric = "probability_difference")
	pte_results

	##response: survival
	data(survival_example)
	X = survival_example$X
	y = survival_example$y
 censored = survival_example$censored
	pte_results = PTE_bootstrap_inference(X, y, censored = censored, 
    	regression_type = "survival", 
        B = 1000)
	pte_results

## End(Not run)