# ices: Interrupted Coefficient Estimation Selection In ADVICE: Automatic Direct Variable Selection via Interrupted Coefficient Estimation

## Description

This function provides an alternative multiple regression fitting procedure which simultaneously estimates and selects variables. The resulting coefficient estimates will tend to be slightly biased, but in a sparse setting, they can be quite accurate. A full regression model is specified by the user, and the function usually returns coefficient estimates for a reduced model, i.e., a model for which some of the coefficient estimates are exactly 0.

## Usage

 `1` ``` ices(formula, data, model = TRUE, x = FALSE, y = FALSE, qr = TRUE) ```

## Arguments

 `formula` a formula object specifying the full regression model. `data` a data frame containing observations on the response variable and the predictor variables. `model, x, y, qr` logicals. If `TRUE` the corresponding components of the fit (the model frame, the model matrix, the response, the QR decomposition) are returned.

## Value

a QRS class object

 `coefficients` a named numeric vector of coefficients `residuals` a numeric vector containing the response minus the fitted values. `effects` a numeric vector of containing the projections of the response variable under the orthogonal Q matrix coming from the QR decomposition of the model matrix. `rank` the numeric rank of the fitted linear model. `fitted.values` the estimated response values according to the fitted interrupted coefficient estimation selection regression model. `sigma2` the estimated noise variance based on the n-p residual effects, where p is the size of the full model. `std_error` a numeric vector of standard errors. `df.residual` residual degrees of freedom. `x` a numeric matrix containing the model matrix. `y` a numeric vector containing the response variable values. `qr` the QR decomposition object coming from the model matrix (after re-ordering columns). `coefOrder` permutation of the sequence 1:p which gives the ascending order of the coefficients of the linear model object, as a result of the pre-screening. `call` the matched call. `terms` the terms object used. `names` a character vector containing the column names of the model matrix. `model` if requested (the default), the model frame used in the case of the full regression model.

## Author(s)

`lm.R`, `QRS.R`
 ```1 2 3 4 5``` ``` myRegressionData <- rmultreg(50, k=10, p=.25, sdnoise = .5) pairs(myRegressionData\$data) out <- ices(y ~ ., data = myRegressionData\$data) # fit model to simulated data confint(out) # calculate 95 % confidence intervals for all coefficients myRegressionData\$coefficients # compare with true coefficients ```