cv.glm {boot} | R Documentation

## Cross-validation for Generalized Linear Models

### Description

This function calculates the estimated K-fold cross-validation prediction error for generalized linear models.

### Usage

    cv.glm(data, glmfit, cost, K)

### Arguments

`data` | A matrix or data frame containing the data. The rows should be cases and the columns correspond to variables, one of which is the response.
`glmfit` | An object of class `"glm"` containing the results of a generalized linear model fitted to `data`.
`cost` | A function of two vector arguments specifying the cost function for the cross-validation. The first argument to `cost` should correspond to the observed responses and the second argument should correspond to the predicted or fitted responses from the generalized linear model. `cost` must return a non-negative scalar value. The defualt is the average squared error function.
`K` | The number of groups into which the data should be split to estimate the cross-validation prediction error. The value of `K` must be such that all groups are of approximately equal size. If the supplied value of `K` does not satisfy this criterion then it will be set to the closest integer which does and a warning is generated specifying the value of `K` used. The default is to set `K` equal to the number of observations in `data` which gives the usual leave-one-out cross-validation.

### Details

The data is divided randomly into `K` groups. For each group the generalized linear model is fit to `data` omitting that group, then the function `cost` is applied to the observed responses in the group that was omitted from the fit and the prediction made by the fitted models for those observations.

When `K` is the number of observations leave-one-out cross-validation is used and all the possible splits of the data are used. When `K` is less than the number of observations the `K` splits to be used are found by randomly partitioning the data into `K` groups of approximately equal size. In this latter case a certain amount of bias is introduced. This can be reduced by using a simple adjustment (see equation 6.48 in Davison and Hinkley, 1997). The second value returned in `delta` is the estimate adjusted by this method.

### Value

The returned value is a list with the following components.

`call` | The original call to `cv.glm`.
`K` | The value of `K` used for the K-fold cross validation.
`delta` | A vector of length two. The first component is the raw cross-validation estimate of prediction error. The second component is the adjusted cross-validation estimate. The adjustment is designed to compensate for the bias introduced by not using leave-one-out cross-validation.
`seed` | The value of `.Random.seed` when `cv.glm` was called.

### Side Effects

The value of `.Random.seed` is updated.

### References

Brieman, L., Friedman, J.H., Olshen, R.A. and Stone, C.J. (1984) Classification and Regression Trees. Wadsworth.

Burman, P. (1989) A comparitive study of ordinary cross-validation, v-fold cross-validation and repeated learning-testing methods. Biometrika, 76, 503–514

Davison, A.C. and Hinkley, D.V. (1997) Bootstrap Methods and Their Application. Cambridge University Press.

Efron, B. (1986) How biased is the apparent error rate of a prediction rule? Journal of the American Statistical Association, 81, 461–470.

Stone, M. (1974) Cross-validation choice and assessment of statistical predictions (with Discussion). Journal of the Royal Statistical Society, B, 36, 111–147.

### See Also

`glm`, `glm.diag`, `predict`

### Examples

    # leave-one-out and 6-fold cross-validation prediction error for
    # the mammals data set.
    data(mammals, package="MASS")
    mammals.glm <- glm(log(brain)~log(body),data=mammals)
    cv.err <- cv.glm(mammals,mammals.glm)
    cv.err.6 <- cv.glm(mammals, mammals.glm, K=6)

    # As this is a linear model we could calculate the leave-one-out
    # cross-validation estimate without any extra model-fitting.
    muhat <- mammals.glm$fitted
    mammals.diag <- glm.diag(mammals.glm)
    cv.err <- mean((mammals.glm$y-muhat)^2/(1-mammals.diag$h)^2)

    # leave-one-out and 11-fold cross-validation prediction error for
    # the nodal data set.  Since the response is a binary variable an
    # appropriate cost function is
    cost <- function(r, pi=0) mean(abs(r-pi)>0.5)

    data(nodal)
    nodal.glm <- glm(r~stage+xray+acid,binomial,data=nodal)
    cv.err <- cv.glm(nodal, nodal.glm, cost, K=nrow(nodal))$delta
    cv.11.err <- cv.glm(nodal, nodal.glm, cost, K=11)$delta

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