Fits subgroup identification model class of Chen, et al (2017)

fit.subgroup(x, y, trt, propensity.func = NULL,
loss = c("sq_loss_lasso", "logistic_loss_lasso", "poisson_loss_lasso",
"cox_loss_lasso", "owl_logistic_loss_lasso",
"owl_logistic_flip_loss_lasso", "owl_hinge_loss", "owl_hinge_flip_loss",
"sq_loss_lasso_gam", "poisson_loss_lasso_gam", "logistic_loss_lasso_gam",
"sq_loss_gam", "poisson_loss_gam", "logistic_loss_gam",
"owl_logistic_loss_gam", "owl_logistic_flip_loss_gam",
"owl_logistic_loss_lasso_gam", "owl_logistic_flip_loss_lasso_gam",
"sq_loss_gbm", "poisson_loss_gbm", "logistic_loss_gbm", "cox_loss_gbm",
"custom"), method = c("weighting", "a_learning"), match.id = NULL,
augment.func = NULL, fit.custom.loss = NULL, cutpoint = 0,
larger.outcome.better = TRUE, reference.trt = NULL, retcall = TRUE,
...)

x The design matrix (not including intercept term) The response vector treatment vector with each element equal to a 0 or a 1, with 1 indicating treatment status is active. function that inputs the design matrix x and the treatment vector trt and outputs the propensity score, ie Pr(trt = 1 | X = x). Function should take two arguments 1) x and 2) trt. See example below. For a randomized controlled trial this can simply be a function that returns a constant equal to the proportion of patients assigned to the treatment group, i.e.: propensity.func = function(x, trt) 0.5. choice of both the M function from Chen, et al (2017) and potentially the penalty used for variable selection. All loss options starting with sq_loss use M(y, v) = (v - y) ^ 2, all options starting with logistic_loss use the logistic loss: M(y, v) = y * log(1 + exp{-v}), and all options starting with cox_loss use the negative partial likelihood loss for the Cox PH model. All options ending with lasso have a lasso penalty added to the loss for variable selection. sq_loss_lasso_gam and logistic_loss_lasso_gam first use the lasso to select variables and then fit a generalized additive model with nonparametric additive terms for each selected variable. sq_loss_gam involves a squared error loss with a generalized additive model and no variable selection. sq_loss_gbm involves a squared error loss with a gradient-boosted decision trees model for the benefit score; this allows for flexible estimation using machine learning and can be useful when the underlying treatment-covariate interaction is complex. Continuous Outcomes "sq_loss_lasso" - M(y, v) = (v - y) ^ 2 with linear model and lasso penalty "owl_logistic_loss_lasso" - M(y, v) = ylog(1 + exp{-v}) (method of Regularized Outcome Weighted Subgroup Identification) "owl_logistic_flip_loss_lasso" - M(y, v) = |y|log(1 + exp{-sign(y)v}) "owl_hinge_loss" - M(y, v) = ymax(0, 1 - v) (method of Estimating individualized treatment rules using outcome weighted learning) "owl_hinge_flip_loss" - M(y, v) = |y|max(0, 1 - sign(y)v) "sq_loss_lasso_gam" - M(y, v) = (v - y) ^ 2 with variables selected by lasso penalty and generalized additive model fit on the selected variables "sq_loss_gam" - M(y, v) = (v - y) ^ 2 with generalized additive model fit on all variables "owl_logistic_loss_gam" - M(y, v) = ylog(1 + exp{-v}) with generalized additive model fit on all variables "owl_logistic_flip_loss_gam" - M(y, v) = |y|log(1 + exp{-sign(y)v}) with generalized additive model fit on all variables "owl_logistic_loss_lasso_gam" - M(y, v) = ylog(1 + exp{-v}) with variables selected by lasso penalty and generalized additive model fit on the selected variables "owl_logistic_flip_loss_lasso_gam" - M(y, v) = |y|log(1 + exp{-sign(y)v}) with variables selected by lasso penalty and generalized additive model fit on the selected variables "sq_loss_gbm" - M(y, v) = (v - y) ^ 2 with gradient-boosted decision trees model Binary Outcomes All losses for continuous outcomes can be used plus the following: "logistic_loss_lasso" - M(y, v) = -[yv - log(1 + exp{-v})] with with linear model and lasso penalty "logistic_loss_lasso_gam" - M(y, v) = -[yv - log(1 + exp{-v})] with variables selected by lasso penalty and generalized additive model fit on the selected variables "logistic_loss_gam" - M(y, v) = -[yv - log(1 + exp{-v})] with generalized additive model fit on all variables "logistic_loss_gbm" - M(y, v) = -[yv - log(1 + exp{-v})] with gradient-boosted decision trees model Count Outcomes All losses for continuous outcomes can be used plus the following: "poisson_loss_lasso" - M(y, v) = -[yv - exp(v)] with with linear model and lasso penalty "poisson_loss_lasso_gam" - M(y, v) = -[yv - exp(v)] with variables selected by lasso penalty and generalized additive model fit on the selected variables "poisson_loss_gam" - M(y, v) = -[yv - exp(v)] with generalized additive model fit on all variables "poisson_loss_gbm" - M(y, v) = -[yv - exp(v)] with gradient-boosted decision trees model Time-to-Event Outcomes "cox_loss_lasso" - M corresponds to the negative partial likelihood of the cox model with linear model and additionally a lasso penalty "cox_loss_gbm" - M corresponds to the negative partial likelihood of the cox model with gradient-boosted decision trees model subgroup ID model type. Either the weighting or A-learning method of Chen et al, (2017) a (character, factor, or integer) vector with length equal to the number of observations in x indicating using integers or levels of a factor vector which patients are in which matched groups. Defaults to NULL and assumes the samples are not from a matched cohort. Matched case-control groups can be created using any method (propensity score matching, optimal matching, etc). If each case is matched with a control or multiple controls, this would indicate which case-control pairs or groups go together. If match.id is supplied, then it is unecessary to specify a function via the propensity.func argument. A quick usage example: if the first patient is a case and the second and third are controls matched to it, and the fouth patient is a case and the fifth through seventh patients are matched with it, then the user should specify match.id = c(1,1,1,2,2,2,2) or match.id = c(rep("Grp1", 3),rep("Grp2", 4))  function which inputs the response y, the covariates x, and trt and outputs predicted values (on the link scale) for the response using a model constructed with x. augment.func() can also be simply a function of x and y. This function is used for efficiency augmentation. When the form of the augmentation function is correct, it can provide efficient estimation of the subgroups. Some examples of possible augmentation functions are: Example 1: augment.func <- function(x, y) {lmod <- lm(y ~ x); return(fitted(lmod))} Example 2:augment.func <- function(x, y, trt) { data <- data.frame(x, y, trt) lmod <- lm(y ~ x * trt) ## get predictions when trt = 1 data$trt <- 1 preds_1 <- predict(lmod, data) ## get predictions when trt = -1 data$trt <- -1 preds_n1 <- predict(lmod, data) ## return predictions averaged over trt return(0.5 * (preds_1 + preds_n1)) }  For binary and time-to-event outcomes, make sure that predictions are returned on the scale of the predictors Example 3:augment.func <- function(x, y) { bmod <- glm(y ~ x, family = binomial()) return(predict(bmod, type = "link")) }  A function which minimizes a user-specified custom loss function M(y,v) to be used in model fitting. If provided, fit.custom.loss should take the modified design matrix (which includes an intercept term) as an argument and the responses and optimize a custom weighted loss function. The loss function $$M(y, v)$$ to be minimized MUST meet the following two criteria: $$D_M(y, v) = \partial M(y, v)/\partial v$$ must be increasing in v for each fixed y. $$D_M(y, v)$$ is the partial derivative of the loss function M(y, v) with respect to v $$D_M(y, 0)$$ is monotone in y An example of a valid loss function is $$M(y, v) = (y - v)^2$$. In this case $$D_M(y, v) = -2(y - v)$$. See Chen et al. (2017) for more details on the restrictions on the loss function $$M(y, v)$$. The provided function MUST return a list with the following elements: predict a function that inputs a design matrix and a 'type' argument for the type of predictions and outputs a vector of predictions on the scale of the linear predictor. Note that the matrix provided to 'fit.custom.loss' has a column appended to the first column of x corresponding to the treatment main effect. Thus, the prediction function should deal with this, e.g. predict(model, cbind(1, x)) model a fitted model object returned by the underlying fitting function coefficients if the underlying fitting function yields a vector of coefficient estimates, they should be provided here The provided function MUST be a function with the following arguments: x design matrix y vector of responses weights vector for observations weights. The underlying loss function MUST have samples weighted according to this vector. See below example ... additional arguments passed via '...'. This can be used so that users can specify more arguments to the underlying fitting function if so desired. The provided function can also optionally take the following arguments: match.id vector of case/control cluster IDs. This is useful if cross validation is used in the underlying fitting function in which case it is advisable to sample whole clusters randomly instead of individual observations. offset if efficiency augmentation is used, the predictions from the outcome model from augment.func will be provided via the offset argument, which can be used as an offset in the underlying fitting function as a means of incorporating the efficiency augmentation model's predictions trt vector of treatment statuses family family of outcome n.trts numer of treatment levels. Can be useful if there are more than 2 treatment levels Example 1: Here we minimize $$M(y, v) = (y - v)^2$$ fit.custom.loss <- function(x, y, weights, ...) { df <- data.frame(y = y, x) # minimize squared error loss with NO lasso penalty lmf <- lm(y ~ x - 1, weights = weights, data = df, ...) # save coefficients cfs = coef(lmf) # create prediction function. Notice # how a column of 1's is appended # to ensure treatment main effects are included # in predictions prd = function(x, type = "response") { dfte <- cbind(1, x) colnames(dfte) <- names(cfs) predict(lmf, data.frame(dfte)) } # return lost of required components list(predict = prd, model = lmf, coefficients = cfs) }  Example 2: $$M(y, v) = y\exp(-v)$$ fit.expo.loss <- function(x, y, weights, ...) { ## define loss function to be minimized expo.loss <- function(beta, x, y, weights) { sum(weights * y * exp(-drop(tcrossprod(x, t(beta) ))) } # use optim() to minimize loss function opt <- optim(rep(0, NCOL(x)), fn = expo.loss, x = x, y = y, weights = weights) coefs <- opt$par pred <- function(x, type = "response") { tcrossprod(cbind(1, x), t(coefs)) } # return list of required components list(predict = pred, model = opt, coefficients = coefs) }  numeric value for patients with benefit scores above which (or below which if larger.outcome.better = FALSE) will be recommended to be in the treatment group. Can also set cutpoint = "median", which will use the median value of the benefit scores as the cutpoint or can set specific quantile values via "quantx" where "x" is a number between 0 and 100 representing the quantile value; e.g. cutpoint = "quant75" will use the 75th perent upper quantile of the benefit scores as the quantile. boolean value of whether a larger outcome is better/preferable. Set to TRUE if a larger outcome is better/preferable and set to FALSE if a smaller outcome is better/preferable. Defaults to TRUE. which treatment should be treated as the reference treatment. Defaults to the first level of trt if trt is a factor or the first alphabetical or numerically first treatment level. Not used for multiple treatment fitting with OWL-type losses. boolean value. if TRUE then the passed arguments will be saved. Do not set to FALSE if the validate.subgroup() function will later be used for your fitted subgroup model. Only set to FALSE if memory is limited as setting to TRUE saves the design matrix to the fitted object options to be passed to underlying fitting function. For all loss options with 'lasso', this will be passed to cv.glmnet. For all loss options with 'gam', this will be passed to gam from the mgcv package Note that for all loss options that use gam() from the mgcv package, the user cannot supply the gam argument method because it is also an argument of fit.subgroup, so instead, to change the gam method argument, supply method.gam, ie method.gam = "REML". For all loss options with 'hinge', this will be passed to both weighted.ksvm and ipop from the kernlab package ## Value An object of class "subgroup_fitted". predict A function that returns predictions of the covariate-conditional treatment effects model An object returned by the underlying fitting function used. For example, if the lasso use used to fit the underlying subgroup identification model, this will be an object returned by cv.glmnet. coefficients If the underlying subgroup identification model is parametric, coefficients will contain the estimated coefficients of the model. call The call that produced the returned object. If retcall = TRUE, this will contain all objects supplied to fit.subgroup() family The family corresponding to the outcome provided loss The loss function used method The method used (either weighting or A-learning) propensity.func The propensity score function used larger.outcome.better If larger outcomes are preferred for this model cutpoint Benefit score cutoff value used for determining subgroups var.names The names of all variables used n.trts The number of treatment levels comparison.trts All treatment levels other than the reference level reference.trt The reference level for the treatment. This should usually be the control group/level trts All treatment levels trt.received The vector of treatment assignments pi.x A vector of propensity scores y A vector of outcomes benefit.scores A vector of conditional treatment effects, i.e. benefit scores recommended.trts A vector of treatment recommendations (i.e. for each patient, which treatment results in the best expected potential outcomes) subgroup.trt.effects (Biased) estimates of the conditional treatment effects and conditional outcomes. These are essentially just empirical averages within different combinations of treatment assignments and treatment recommendations individual.trt.effects estimates of the individual treatment effects as returned by treat.effects ## References Chen, S., Tian, L., Cai, T. and Yu, M. (2017), A general statistical framework for subgroup identification and comparative treatment scoring. Biometrics. doi:10.1111/biom.12676 http://onlinelibrary.wiley.com/doi/10.1111/biom.12676/abstract Xu, Y., Yu, M., Zhao, Y. Q., Li, Q., Wang, S., & Shao, J. (2015), Regularized outcome weighted subgroup identification for differential treatment effects. Biometrics, 71(3), 645-653. doi: 10.1111/biom.12322 http://onlinelibrary.wiley.com/doi/10.1111/biom.12322/full Zhao, Y., Zeng, D., Rush, A. J., & Kosorok, M. R. (2012), Estimating individualized treatment rules using outcome weighted learning. Journal of the American Statistical Association, 107(499), 1106-1118. doi: 10.1080/01621459.2012.695674 http://dx.doi.org/10.1080/01621459.2012.695674 ## See also validate.subgroup for function which creates validation results for subgroup identification models, predict.subgroup_fitted for a prediction function for fitted models from fit.subgroup, plot.subgroup_fitted for a function which plots results from fitted models, and print.subgroup_fitted for arguments for printing options for fit.subgroup(). from fit.subgroup. ## Examples library(personalized) set.seed(123) n.obs <- 500 n.vars <- 15 x <- matrix(rnorm(n.obs * n.vars, sd = 3), n.obs, n.vars) # simulate non-randomized treatment xbetat <- 0.5 + 0.5 * x[,7] - 0.5 * x[,9] trt.prob <- exp(xbetat) / (1 + exp(xbetat)) trt01 <- rbinom(n.obs, 1, prob = trt.prob) trt <- 2 * trt01 - 1 # simulate response # delta below drives treatment effect heterogeneity delta <- 2 * (0.5 + x[,2] - x[,3] - x[,11] + x[,1] * x[,12] ) xbeta <- x[,1] + x[,11] - 2 * x[,12]^2 + x[,13] + 0.5 * x[,15] ^ 2 xbeta <- xbeta + delta * trt # continuous outcomes y <- drop(xbeta) + rnorm(n.obs, sd = 2) # binary outcomes y.binary <- 1 * (xbeta + rnorm(n.obs, sd = 2) > 0 ) # count outcomes y.count <- round(abs(xbeta + rnorm(n.obs, sd = 2))) # time-to-event outcomes surv.time <- exp(-20 - xbeta + rnorm(n.obs, sd = 1)) cens.time <- exp(rnorm(n.obs, sd = 3)) y.time.to.event <- pmin(surv.time, cens.time) status <- 1 * (surv.time <= cens.time) # create function for fitting propensity score model prop.func <- function(x, trt) { # fit propensity score model propens.model <- cv.glmnet(y = trt, x = x, family = "binomial") pi.x <- predict(propens.model, s = "lambda.min", newx = x, type = "response")[,1] pi.x } #################### Continuous outcomes ################################ subgrp.model <- fit.subgroup(x = x, y = y, trt = trt01, propensity.func = prop.func, loss = "sq_loss_lasso", nfolds = 10) # option for cv.glmnet summary(subgrp.model)#> family: gaussian #> loss: sq_loss_lasso #> method: weighting #> cutpoint: 0 #> propensity #> function: propensity.func #> #> benefit score: f(x), #> Trt recom = 1*I(f(x)>c)+0*I(f(x)<=c) where c is 'cutpoint' #> #> Average Outcomes: #> Recommended 0 Recommended 1 #> Received 0 -6.7529 (n = 82) -24.5033 (n = 119) #> Received 1 -26.9737 (n = 121) -1.834 (n = 178) #> #> Treatment effects conditional on subgroups: #> Est of E[Y|T=0,T=Recom]-E[Y|T=/=0,T=Recom] #> 20.2208 (n = 203) #> Est of E[Y|T=1,T=Recom]-E[Y|T=/=1,T=Recom] #> 22.6694 (n = 297) #> #> NOTE: The above average outcomes are biased estimates of #> the expected outcomes conditional on subgroups. #> Use 'validate.subgroup()' to obtain unbiased estimates. #> #> --------------------------------------------------- #> #> Benefit score quantiles (f(X) for 1 vs 0): #> 0% 25% 50% 75% 100% #> -27.548 -3.989 2.619 8.564 24.586 #> #> --------------------------------------------------- #> #> Summary of individual treatment effects: #> E[Y|T=1, X] - E[Y|T=0, X] #> #> Min. 1st Qu. Median Mean 3rd Qu. Max. #> -55.095 -7.977 5.237 4.856 17.128 49.171 #> #> --------------------------------------------------- #> #> 8 out of 15 interactions selected in total by the lasso (cross validation criterion). #> #> The first estimate is the treatment main effect, which is always selected. #> Any other variables selected represent treatment-covariate interactions. #> #> Trt1 V2 V3 V6 V8 V10 V11 V13 V15 #> Estimate 2.6841 2.0546 -1.8871 0.2005 0.349 0.331 -0.6036 1.2472 0.3743 # estimates of the individual-specific # treatment effect estimates: subgrp.model$individual.trt.effects#> Summary of individual treatment effects:
#> E[Y|T=1, X] - E[Y|T=0, X]
#>
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
#> -55.095  -7.977   5.237   4.856  17.128  49.171
# fit lasso + gam model with REML option for gam

subgrp.modelg <- fit.subgroup(x = x, y = y,
trt = trt01,
propensity.func = prop.func,
loss   = "sq_loss_lasso_gam",
method.gam = "REML",     # option for gam
nfolds = 5)              # option for cv.glmnet

subgrp.modelg#> family:    gaussian
#> loss:      sq_loss_lasso_gam
#> method:    weighting
#> cutpoint:  0
#> propensity
#> function:  propensity.func
#>
#> benefit score: f(x),
#> Trt recom = 1*I(f(x)>c)+0*I(f(x)<=c) where c is 'cutpoint'
#>
#> Average Outcomes:
#>                 Recommended 0     Recommended 1
#> Received 0 -12.6754 (n = 132) -25.3863 (n = 69)
#> Received 1 -21.0867 (n = 141) -2.7467 (n = 158)
#>
#> Treatment effects conditional on subgroups:
#> Est of E[Y|T=0,T=Recom]-E[Y|T=/=0,T=Recom]
#>                           8.4113 (n = 273)
#> Est of E[Y|T=1,T=Recom]-E[Y|T=/=1,T=Recom]
#>                          22.6397 (n = 227)
#>
#> NOTE: The above average outcomes are biased estimates of
#>       the expected outcomes conditional on subgroups.
#>       Use 'validate.subgroup()' to obtain unbiased estimates.
#>
#> ---------------------------------------------------
#>
#> Benefit score quantiles (f(X) for 1 vs 0):
#>      0%     25%     50%     75%    100%
#> -58.022 -11.497  -2.069   9.130  45.392
#>
#> ---------------------------------------------------
#>
#> Summary of individual treatment effects:
#> E[Y|T=1, X] - E[Y|T=0, X]
#>
#>     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
#> -116.043  -22.994   -4.138   -2.356   18.260   90.785
####################  Using an augmentation function #####################
## augmentation funcions involve modeling the conditional mean E[Y|T, X]
## and returning predictions that are averaged over the treatment values
## return <- 1/2 * (hat{E}[Y|T=1, X] + hat{E}[Y|T=-1, X])
##########################################################################

augment.func <- function(x, y, trt) {
data <- data.frame(x, y, trt)
xm <- model.matrix(y~trt*x-1, data = data)

lmod <- cv.glmnet(y = y, x = xm)
## get predictions when trt = 1
data$trt <- 1 xm <- model.matrix(y~trt*x-1, data = data) preds_1 <- predict(lmod, xm, s = "lambda.min") ## get predictions when trt = -1 data$trt <- -1
xm <- model.matrix(y~trt*x-1, data = data)
preds_n1  <- predict(lmod, xm, s = "lambda.min")

## return predictions averaged over trt
return(0.5 * (preds_1 + preds_n1))
}subgrp.model.aug <- fit.subgroup(x = x, y = y,
trt = trt01,
propensity.func = prop.func,
augment.func    = augment.func,
loss   = "sq_loss_lasso",
nfolds = 10)              # option for cv.glmnet

summary(subgrp.model.aug)#> family:    gaussian
#> loss:      sq_loss_lasso
#> method:    weighting
#> cutpoint:  0
#> augmentation
#> function: augment.func
#> propensity
#> function:  propensity.func
#>
#> benefit score: f(x),
#> Trt recom = 1*I(f(x)>c)+0*I(f(x)<=c) where c is 'cutpoint'
#>
#> Average Outcomes:
#>                 Recommended 0      Recommended 1
#> Received 0   -8.3084 (n = 82) -23.4351 (n = 119)
#> Received 1 -26.5609 (n = 116)  -2.4744 (n = 183)
#>
#> Treatment effects conditional on subgroups:
#> Est of E[Y|T=0,T=Recom]-E[Y|T=/=0,T=Recom]
#>                          18.2525 (n = 198)
#> Est of E[Y|T=1,T=Recom]-E[Y|T=/=1,T=Recom]
#>                          20.9606 (n = 302)
#>
#> NOTE: The above average outcomes are biased estimates of
#>       the expected outcomes conditional on subgroups.
#>       Use 'validate.subgroup()' to obtain unbiased estimates.
#>
#> ---------------------------------------------------
#>
#> Benefit score quantiles (f(X) for 1 vs 0):
#>      0%     25%     50%     75%    100%
#> -29.635  -4.645   2.712   9.598  33.270
#>
#> ---------------------------------------------------
#>
#> Summary of individual treatment effects:
#> E[Y|T=1, X] - E[Y|T=0, X]
#>
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
#> -59.270  -9.289   5.423   5.839  19.195  66.540
#>
#> ---------------------------------------------------
#>
#> 9 out of 15 interactions selected in total by the lasso (cross validation criterion).
#>
#> The first estimate is the treatment main effect, which is always selected.
#> Any other variables selected represent treatment-covariate interactions.
#>
#>            Trt1    V2      V3    V6     V8      V9    V10     V11    V13    V15
#> Estimate 3.0734 1.977 -2.2989 0.502 0.7772 -0.1928 0.4402 -0.9236 1.1415 0.9552
####################  Binary outcomes ####################################

# use logistic loss for binary outcomes
subgrp.model.bin <- fit.subgroup(x = x, y = y.binary,
trt = trt01,
propensity.func = prop.func,
loss   = "logistic_loss_lasso",
type.measure = "auc",    # option for cv.glmnet
nfolds = 5)              # option for cv.glmnet

subgrp.model.bin#> family:    binomial
#> loss:      logistic_loss_lasso
#> method:    weighting
#> cutpoint:  0
#> propensity
#> function:  propensity.func
#>
#> benefit score: f(x),
#> Trt recom = 1*I(f(x)>c)+0*I(f(x)<=c) where c is 'cutpoint'
#>
#> Average Outcomes:
#>               Recommended 0    Recommended 1
#> Received 0  0.5271 (n = 83) 0.1744 (n = 118)
#> Received 1 0.1856 (n = 127)  0.546 (n = 172)
#>
#> Treatment effects conditional on subgroups:
#> Est of E[Y|T=0,T=Recom]-E[Y|T=/=0,T=Recom]
#>                           0.3414 (n = 210)
#> Est of E[Y|T=1,T=Recom]-E[Y|T=/=1,T=Recom]
#>                           0.3716 (n = 290)
#>
#> NOTE: The above average outcomes are biased estimates of
#>       the expected outcomes conditional on subgroups.
#>       Use 'validate.subgroup()' to obtain unbiased estimates.
#>
#> ---------------------------------------------------
#>
#> Benefit score quantiles (f(X) for 1 vs 0):
#>      0%     25%     50%     75%    100%
#> -1.9448 -0.3378  0.1509  0.6138  2.1915
#>
#> ---------------------------------------------------
#>
#> Summary of individual treatment effects:
#> E[Y|T=1, X] - E[Y|T=0, X]
#>
#>     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
#> -0.74976 -0.16730  0.07530  0.06992  0.29760  0.79896

####################  Count outcomes #####################################

# use poisson loss for count/poisson outcomes
subgrp.model.poisson <- fit.subgroup(x = x, y = y.count,
trt = trt01,
propensity.func = prop.func,
loss   = "poisson_loss_lasso",
type.measure = "mse",    # option for cv.glmnet
nfolds = 5)              # option for cv.glmnet

subgrp.model.poisson#> family:    poisson
#> loss:      poisson_loss_lasso
#> method:    weighting
#> cutpoint:  0
#> propensity
#> function:  propensity.func
#>
#> benefit score: f(x),
#> Trt recom = 1*I(f(x)>c)+0*I(f(x)<=c) where c is 'cutpoint'
#>
#> Average Outcomes:
#>                Recommended 0     Recommended 1
#> Received 0  26.996 (n = 130)  19.5423 (n = 71)
#> Received 1 18.5265 (n = 186) 27.7822 (n = 113)
#>
#> Treatment effects conditional on subgroups:
#> Est of E[Y|T=0,T=Recom]-E[Y|T=/=0,T=Recom]
#>                           8.4695 (n = 316)
#> Est of E[Y|T=1,T=Recom]-E[Y|T=/=1,T=Recom]
#>                           8.2399 (n = 184)
#>
#> NOTE: The above average outcomes are biased estimates of
#>       the expected outcomes conditional on subgroups.
#>       Use 'validate.subgroup()' to obtain unbiased estimates.
#>
#> ---------------------------------------------------
#>
#> Benefit score quantiles (f(X) for 1 vs 0):
#>      0%     25%     50%     75%    100%
#> -2.8807 -1.0340 -0.3018  0.2388  2.8825
#>
#> ---------------------------------------------------
#>
#> Summary of individual treatment effects:
#> E[Y|T=1, X] - E[Y|T=0, X]
#>
#>     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
#> -17.7706  -2.4567  -0.6127  -1.1499   0.4822  17.8026

####################  Time-to-event outcomes #############################

library(survival)subgrp.model.cox <- fit.subgroup(x = x, y = Surv(y.time.to.event, status),
trt = trt01,
propensity.func = prop.func,
loss   = "cox_loss_lasso",
nfolds = 5)              # option for cv.glmnet

subgrp.model.cox#> family:    cox
#> loss:      cox_loss_lasso
#> method:    weighting
#> cutpoint:  0
#> propensity
#> function:  propensity.func
#>
#> benefit score: f(x),
#> Trt recom = 1*I(f(x)>c)+0*I(f(x)<=c) where c is 'cutpoint'
#>
#> Average Outcomes:
#>                 Recommended 0     Recommended 1
#> Received 0 175.2226 (n = 140)   8.4595 (n = 61)
#> Received 1 180.6631 (n = 203) 134.9956 (n = 96)
#>
#> Treatment effects conditional on subgroups:
#> Est of E[Y|T=0,T=Recom]-E[Y|T=/=0,T=Recom]
#>                          -5.4405 (n = 343)
#> Est of E[Y|T=1,T=Recom]-E[Y|T=/=1,T=Recom]
#>                         126.5361 (n = 157)
#>
#> NOTE: The above average outcomes are biased estimates of
#>       the expected outcomes conditional on subgroups.
#>       Use 'validate.subgroup()' to obtain unbiased estimates.
#>
#> ---------------------------------------------------
#>
#> Benefit score quantiles (f(X) for 1 vs 0):
#>       0%      25%      50%      75%     100%
#> -0.87626 -0.34859 -0.14888  0.06438  0.87688
#>
#> ---------------------------------------------------
#>
#> Summary of individual treatment effects:
#> E[Y|T=1, X] / E[Y|T=0, X]
#>
#> Note: for survival outcomes, the above ratio is
#> E[g(Y)|T=1, X] / E[g(Y)|T=0, X],
#> where g() is a monotone increasing function of Y,
#> the survival time
#>
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
#>  0.4161  0.9376  1.1605  1.2129  1.4171  2.4019

####################  Using custom loss functions ########################

## Use custom loss function for binary outcomes

fit.custom.loss.bin <- function(x, y, weights, offset, ...) {
df <- data.frame(y = y, x)

# minimize logistic loss with NO lasso penalty
# with allowance for efficiency augmentation
glmf <- glm(y ~ x - 1, weights = weights,
offset = offset, # offset term allows for efficiency augmentation
family = binomial(), ...)

# save coefficients
cfs = coef(glmf)

# create prediction function.
prd = function(x, type = "response") {
dfte <- cbind(1, x)
colnames(dfte) <- names(cfs)
## predictions must be returned on the scale
## of the linear predictor
}
# return lost of required components
list(predict = prd, model = glmf, coefficients = cfs)
}subgrp.model.bin.cust <- fit.subgroup(x = x, y = y.binary,
trt = trt01,
propensity.func = prop.func,
fit.custom.loss = fit.custom.loss.bin)#> Warning: non-integer #successes in a binomial glm!
subgrp.model.bin.cust#> family:    gaussian
#> loss:      custom
#> method:    weighting
#> cutpoint:  0
#> propensity
#> function:  propensity.func
#>
#> benefit score: f(x),
#> Trt recom = 1*I(f(x)>c)+0*I(f(x)<=c) where c is 'cutpoint'
#>
#> Average Outcomes:
#>               Recommended 0    Recommended 1
#> Received 0 0.1919 (n = 121)   0.471 (n = 80)
#> Received 1 0.1819 (n = 164) 0.6108 (n = 135)
#>
#> Treatment effects conditional on subgroups:
#> Est of E[Y|T=0,T=Recom]-E[Y|T=/=0,T=Recom]
#>                             0.01 (n = 285)
#> Est of E[Y|T=1,T=Recom]-E[Y|T=/=1,T=Recom]
#>                           0.1397 (n = 215)
#>
#> NOTE: The above average outcomes are biased estimates of
#>       the expected outcomes conditional on subgroups.
#>       Use 'validate.subgroup()' to obtain unbiased estimates.
#>
#> ---------------------------------------------------
#>
#> Benefit score quantiles (f(X) for 1 vs 0):
#>      0%     25%     50%     75%    100%
#> -5.1588 -1.2724 -0.3145  0.7797  4.0227
#>
#> ---------------------------------------------------
#>

## try exponential loss for
## positive outcomes

fit.expo.loss <- function(x, y, weights, ...)
{
expo.loss <- function(beta, x, y, weights) {
sum(weights * y * exp(-drop(x %*% beta)))
}

# use optim() to minimize loss function
opt <- optim(rep(0, NCOL(x)), fn = expo.loss, x = x, y = y, weights = weights)

coefs <- opt\$par

pred <- function(x, type = "response") {
tcrossprod(cbind(1, x), t(coefs))
}

# return list of required components
list(predict = pred, model = opt, coefficients = coefs)
}# use exponential loss for positive outcomes
subgrp.model.expo <- fit.subgroup(x = x, y = y.count,
trt = trt01,
propensity.func = prop.func,
fit.custom.loss = fit.expo.loss)

subgrp.model.expo#> family:    gaussian
#> loss:      custom
#> method:    weighting
#> cutpoint:  0
#> propensity
#> function:  propensity.func
#>
#> benefit score: f(x),
#> Trt recom = 1*I(f(x)>c)+0*I(f(x)<=c) where c is 'cutpoint'
#>
#> Average Outcomes:
#>                Recommended 0     Recommended 1
#> Received 0 30.4423 (n = 142)  15.3946 (n = 59)
#> Received 1 15.0464 (n = 112) 27.9179 (n = 187)
#>
#> Treatment effects conditional on subgroups:
#> Est of E[Y|T=0,T=Recom]-E[Y|T=/=0,T=Recom]
#>                          15.3959 (n = 254)
#> Est of E[Y|T=1,T=Recom]-E[Y|T=/=1,T=Recom]
#>                          12.5233 (n = 246)
#>
#> NOTE: The above average outcomes are biased estimates of
#>       the expected outcomes conditional on subgroups.
#>       Use 'validate.subgroup()' to obtain unbiased estimates.
#>
#> ---------------------------------------------------
#>
#> Benefit score quantiles (f(X) for 1 vs 0):
#>        0%       25%       50%       75%      100%
#> -1.414203 -0.326411 -0.003262  0.379219  1.989184
#>
#> ---------------------------------------------------
#>