csim: A Constrained single index model (main function)
In syhyunpark/csim: Constrained single index models (CSIM)

Description Usage Arguments Details Value Author(s) See Also Examples

csim is the main function for fitting the constrained single index model (CSIM). csim uses pre-treatment covariates X for modeling a scalar-valued treatment outcome y. The single index variable is defined to be a linear combinations of X. The differential treatment effect is estimated by treatment-specific nonparametrically-defined link functions on the single index variable. For simultaneous variable selection for the treatment effect modifiers, csim estimates a sparse single index coefficient vector via a L1 regularization. The estimated single index variable is useful for comparing differential treatment efficacy; the resulting csim object can be used to estimate an optimal treatment selection rule for a new patient with pretreatment clinical information.

csim(y, Tr, X, ortho.constr = TRUE, sparse = TRUE, type = "AIC",
  seed = 1234, lam.by = 0.03, n.lam = 100, n.max = 10,
  eps = 10^-4, it.max = 70, trace = F, nbasis.t = NULL,
  rho.grid = c(0, 0.25, 0.5), eff.aug = FALSE, X.aug = NULL,
  linear.link = FALSE, coef.ini = NULL, i.fx = NULL,
  mc.ini = FALSE, unit.norm = FALSE, plots = TRUE, boot.CI = FALSE,
  n.boot = 500)

`y`	treatment outcomes, n-by-1 vector
`Tr`	treatment indicators, n-by-1 vector; each element represents one of the K available treatment options
`X`	a pretreatment covarate matrix, n-by-p matrix
`ortho.constr`	the constraint that separates the interaction effects from the main effect (without this, the interaction effect can be confounded by the main effect); the default is `TRUE`.
`sparse`	if `TRUE`, apply L1 regularization when estimating the single index coefficients; the default is `TRUE`.
`type`	when `sparse=TRUE`, can choose bewteen `"CV"` and `"BIC"`, for the sparsity tuninig parameter selection.
`seed`	when `type="CV"`, randomization seed for cross validation.
`lam.by`	a value specifying the grid of the sparsity tuning parameters [1, 1+lam.by, ... 1 + n.lam*lam.by].
`n.lam`	a value specifying the grid of the sparsity tuning parameters [1, 1+lam.by, ... 1 + n.lam*lam.by].
`n.max`	a maximum number of nonzero (active) coeffients in CSIM.
`eps`	a value specifying the converge criterion of algorithm.
`it.max`	an integer value specifying the maximum number of iterations for each coordinate.
`trace`	if `TRUE`, show the trace of the fitting procedure; the default is `FALSE`.
`nbasis.t`	a length K+1 vector; each element specifies the number of B-spline basis funtions for approximating the treatment-specific link function; the last element is for the "main effect" link function; the default is `nbasis.t=NULL`, and will be determined depending on the sample size.
`rho.grid`	a grid vector of (ridge-type) smoothing parameters for approximating the link functions.
`eff.aug`	if `TRUE`, perform efficiency augmentation (using a L1 regularized linear regression for the main effects of X); the default is `FALSE`.
`X.aug`	a design matrix to be used for efficinecy augmentation; the default is `NULL`.
`linear.link`	if `TRUE`, restrict the link functions to be linear functions; the default is `FALSE`.
`coef.ini`	an initial solution for alpha.coef; the default is `NULL`.
`i.fx`	an index to be fixed throughout estimation for model identifiability; the default is `NULL`.
`mc.ini`	if `TRUE`, use an estimate obtained from the modified covariate method as an initial solution; this is only applicable for K=2 case; the default is `FALSE`.
`unit.norm`	if `TRUE`, set the estimated coefficient vector to have a unit norm for model identifiability; the default is to set the component with the largest magnitude to be 1.
`plots`	if `TRUE`, produce a plot of the estimated link functions and the coefficient estimate.
`boot.CI`	if `TRUE`, return a `boot` object that can be used to construct bootstrap confidence intervals for the single index coefficients; the default is `FALSE`.
`n.boot`	when `boot.CI=TRUE`, a value specifying the number of bootstrap replications.

The sequence of the model coefficients implied by the tuning paramters lam is fit by block coordinate descent algorithm.

a list of information of the final fitted model including

`coef.obj`	an object containing information about the estimated coefficients.
`link.fn.obj`	an object containing information about the estimated link functions.
`alpha.coef`	the estimated single index coefficients.
`eta.coef`	the estimated main effect coefficients if `eff.aug=TRUE`.
`beta.t.coef`	the list of the estimated treatment-specific B-spline coefficient vectors.
`beta.0.coef`	the estimated B-spline coefficient vector for the main effect of the single index variable.
`smoother`	an object containing information about the estimated link functions, including the knot sequences used in the B-spline approximation `knots.t.`
`link.fn.plot`	a plot object for the link functions.
`coef.plot`	a plot object for the single index coefficients.
`boot.results`	a `boot` object that can be used to construct bootstrap confidence intervals for the single index coefficients.

Park, Petkova, Tarpey, Ogden

pred.csim, fit.csim, fit.csim.cv

## generate a training set
dat.train <- dataGenerationFn(n=500, p=500, w=3, delta = 1);
y.train <- dat.train$y;
Tr.train <- dat.train$Tr;
X.train <- dat.train$X

## generate a  testing set, under the same setting as in the trainint set.
dat.test <- dataGenerationFn(n=10000, p= dat.train$p, w= dat.train$w, delta = dat.train$delta,
                             true.alpha = dat.train$true.alpha, true.eta = dat.train$true.eta);
y.test <- dat.test$y;
Tr.test <- dat.test$Tr;
X.test <- dat.test$X;

# the "signal" to "noise"  ratio (SNR)
dat.test$SNR
# the optimal treatment selection rule
optTr <- dat.test$optTr
value.opt <- dat.test$value.opt
# the true single index coefficients
true.alpha <- dat.test$true.alpha


# 1. fit a constrained single index model, based on the training set
csim.obj   <- csim(y.train, Tr.train, X.train, type = "AIC", rho.grid =0, eff.aug = FALSE)
csim.obj$alpha.coef
# compare the estimate to the true single index coefficients
true.alpha

## can also perform a 5 fold cross validation
#csim.obj2  <- csim(y.train, Tr.train, X.train, type = "CV", rho.grid =0, eff.aug = FALSE)
#csim.obj2$alpha.coef
## can also obtain an un-regularized estimator.
#csim.obj3   <- csim(y.train, Tr.train, X.train, sparse = FALSE, rho.grid =0, eff.aug = FALSE)
#csim.obj3$alpha.coef

## performance assessment:
# y.test, Tr.test, X.test are testing data
pred.test <- pred.csim(csim.obj, X.test)$pred.new
performance.obj <- performance_measure(pred.test, y.test, Tr.test, value.opt, optTr)

# proprortion of correct decisions
performance.obj$pcd;
# the ratio of the Value of ITR (i.e., Value/ Value.opt)
performance.obj$value.s;
# proportion of correctly (C.) selected treatment effect modifiers:
sum(csim.obj$alpha.coef[true.alpha!=0] != 0) /sum(true.alpha!=0);
# proportion of covariates incorrectly (I.C.) selected as treatment effect modifiers:
sum(csim.obj$alpha.coef[!(true.alpha!=0)] != 0)/ sum(true.alpha==0);


# 2. fit a modified covariate model (with efficiency augmentation)
mc.obj <- mc(y.train, Tr.train, X.train, eff.aug  = TRUE, use.lasso=TRUE)
mc.obj$alpha.coef

pred.test <- pred.mc(mc.obj, X.test)$pred.new
performance.obj <- performance_measure(pred.test, y.test, Tr.test, value.opt, optTr)
performance.obj$pcd
performance.obj$value.s

# proportion of correctly (C.) selected treatment effect modifiers:
sum(mc.obj$alpha.coef[true.alpha!=0]!= 0) /sum(true.alpha!=0);
# proportion of covariates incorrectly (I.C.) selected as treatment effect modifiers:
sum(mc.obj$alpha.coef[!(true.alpha!=0)]!= 0)/ sum(true.alpha==0);

## 3. fit a system of K separate sparse addtive models, one for each treatment group
#K.SAM.obj <- K.SAM(y.train, Tr.train, X.train)
#pred.test <- pred.K.SAM(K.SAM.obj, X.test)$pred.new
#performance.obj <- performance_measure(pred.test, y.test, Tr.test, value.opt, optTr)
#performance.obj$pcd
#performance.obj$value.s