# R/shrink.heur.R In apricom: Tools for the a Priori Comparison of Regression Modelling Strategies

#' Shrinkage After Estimation Using Heuristic Formulae
#'
#' Shrink regression coefficients using heuristic formulae, first described
#' by Van Houwelingen and Le Cessie (Stat. Med., 1990)
#'
#' This function can be used to estimate shrunken regression coefficients based on
#' heuristic formulae (see References). A linear or logistic regression model with
#' an intercept is fitted to the data, and a shrinkage factor is estimated. The
#' shrinkage factor is then applied to the regression coefficients. If
#' \code{int.adj == FALSE} the intercept value is estimated as described in
#' Harrell 2001.If \code{int.adj == TRUE} the intercept value will be re-estimated
#' by refitting the model with the shrunken coefficients.
#'
#' The heuristic formula work by applying the
#' number of model degrees of freedom (or the number of predictors) as a penalty,
#' and so as the model becomes more complex, the necessary shrinkage increases, and
#' the shrinkage factor becomes closer to zero.
#'
#' @importFrom stats glm.fit binomial
#'
#' @param dataset   a dataset for regression analysis. Data should be in the form
#' of a matrix, with the outcome variable as the final column. Application of the
#' \code{\link{datashape}} function beforehand is recommended, especially if
#'  categorical predictors are present. For regression with an intercept included
#'  a column vector of 1s should be included before the dataset (see examples)
#' @param model     type of regression model. Either "linear" or "logistic".
#' @param DF        the number of degrees of freedom or number of predictors in the model.
#'                  If DF is missing the value will be automatically estimated. This
#'                  may be inaccurate for complex models with non-linear terms.
#' @param int       logical. If TRUE the model will include a regression intercept.
#' @param int.adj   logical. If TRUE the regression intercept will be re-estimated
#'          after shrinkage of the regression coefficients. If FALSE the regression
#'          intercept will be re-estimated as described by Harrell 2001.
#'
#' @return \code{shrink.heur} returns a list containing the following:
#' @return \item{raw.coeff}{the raw regression model coefficients, pre-shrinkage.}
#' @return \item{shrunk.coeff}{the shrunken regression model coefficients}
#' @return \item{lambda}{the heuristic shrinkage factor}
#' @return \item{DF}{the number of degrees of freedom or number of predictors in the model}
#'
#' @note Warning: In poorly fitting models that includea large number of predictors
#'       the number of degrees of freedom may approch or exceed the model chi square.
#'       In such cases the shrinkage factor will be very small or even negative,
#'       and a different model building strategy is recommended.
#'
#' @examples
#'## Example 1: Linear regression using the iris dataset
#'## shrinkage using a heuristic formula
#' data(iris)
#' iris.data <- as.matrix(iris[, 1:4])
#' iris.data <- cbind(1, iris.data)
#' set.seed(123)
#' shrink.heur(dataset = iris.data, model = "linear")
#'
#'## Example 2: logistic regression using a subset of the mtcars data
#'## shrinkage using a heuristic formula
#' data(mtcars)
#' mtc.data <- cbind(1,datashape(mtcars, y = 8, x = c(1,6,9)))
#' set.seed(321)
#' shrink.heur(dataset = mtc.data, model = "logistic", DF = 3,
#' int = TRUE, int.adj = TRUE)
#'
#' @references Harrell, F. E. \emph{"Regression modeling strategies: with applications
#'              to linear models, logistic regression, and survival analysis."} \emph{Springer}, (2001).
#' @references Harrell, F. E., Kerry L. Lee, and Daniel B. Mark. \emph{"Tutorial in
#'            biostatistics multivariable prognostic models: issues in developing models,
#'            evaluating assumptions and adequacy, and measuring and reducing errors."}
#'            \emph{Statistics in medicine} (1996) 15:361-387.
#' @references Steyerberg, E. \emph{"Clinical Prediction Models"} \emph{Springer} (2009)
#' @references Van Houwelingen, J. C. and Le Cessie, S., \emph{"Predictive value of statistical models."}
#'             \emph{Statistics in medicine} (1990) 9:1303:1325.

shrink.heur <- function(dataset, model, DF, int = TRUE, int.adj = FALSE){

if(missing(model)) stop("Type of model must be specified in the function call")
if(int == FALSE) stop("For heuristic methods, an intercept must be included in
the regression")
if(missing(DF)) warning("The number of degrees of freedom used by predictors was
estimated automatically")
if (missing(DF)) DF <- dim(dataset)[2] - 2

dataset <- as.matrix(dataset)
nc <- dim(dataset)[2]

if (model == "linear") {

### HEURISTIC FORMULA FOR LINEAR REGRESSION WITH OLS ###

Rsq <- function(dataset){
nc <- dim(dataset)[2]
b <- ols.rgr(dataset)
y <- dataset[, nc]
X <- dataset[, 1:(nc - 1)]
yhat <- (X %*% b)
res <- y - yhat
sse0 <- t(res) %*% res
scale(y, center = TRUE, scale = FALSE)
Syy <- t(y) %*% y
r <- 1 - (sse0 / Syy)
}
b <- ols.rgr(dataset)
R2<- Rsq(dataset)
n <- dim(dataset)[1]
AdjR2 <- 1 - (1 - R2) * (n - 1) / (n - DF - 1)
s <- (n - DF - 1) / (n - 1) * AdjR2 / R2

} else if (model == "logistic") {

### HEURISTIC FORMULA FOR GLMS WITH MAXIMUM LIKELIHOOD ESTIMATION ###

# calculate the model coefficients using MLE
b <- ml.rgr(dataset)
# do the same but with an empty model (intercept only)
y <- dataset[, nc]
d0 <- cbind(1, y)
b0 <- ml.rgr(d0)
b0 <- matrix(c(b0, rep(0, nc - 2)), ncol = 1)
model.chisq <- loglikelihood(b0, dataset) - loglikelihood(b, dataset)
s <- ((model.chisq - DF) / model.chisq)
}

### APPLY SHRINKAGE FACTOR ### As described in Harrell 2001 (book p 64)
b.shrunk <- b
y <- dataset[, nc]
bl <- dim(b.shrunk)[1]
b.shrunk[1] <- (1 - s) * mean(y) + s * b[1]
b.shrunk[2:bl] <- b[2:bl] * s

} else {
b.shrunk <- matrix(s * b[2:length(b)], ncol = 1)
if (model == "linear") {
new.int <- mean((dataset[, nc]) - ((dataset[, 2:(nc - 1)]) %*% b.shrunk))
b.shrunk <- c(new.int, b.shrunk)
} else {
X.i <-  matrix(dataset[, 1], ncol = 1)
Y <- dataset[, nc]
offs <- as.vector((dataset[, 2:(nc - 1)]) %*% b.shrunk)
new.int <- glm.fit(X.i, Y, family = binomial(link = "logit"),
offset = offs)\$coefficients
b.shrunk <- c(new.int, b.shrunk)
}
}

return(list(raw.coeff = b, shrunk.coeff = matrix(b.shrunk, ncol = 1),
lambda = s, DF = DF))
}


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apricom documentation built on May 2, 2019, 6:21 a.m.