Extending the cv package
In cv: Cross-Validating Regression Models

knitr::opts_chunk$set(
  collapse = TRUE,
  message = TRUE,
  warning = TRUE,
  fig.align = "center",
  fig.height = 6,
  fig.width = 7,
  fig.path = "fig/",
  dev = "png",
  comment = "#>" #,
  # eval = nzchar(Sys.getenv("REBUILD_VIGNETTES"))
)

# save some typing
knitr::set_alias(w = "fig.width",
                 h = "fig.height",
                 cap = "fig.cap")

# colorize text: use inline as `r colorize(text, color)`
colorize <- function(x, color) {
  if (knitr::is_latex_output()) {
    sprintf("\\textcolor{%s}{%s}", color, x)
  } else if (knitr::is_html_output()) {
    sprintf("<span style='color: %s;'>%s</span>", color,
      x)
  } else x
}


.opts <- options(digits = 5)

The cv package is designed to be extensible in several directions. In this vignette, we discuss three kinds of extensions, ordered by increasing general complexity: (1) adding a cross-validation cost criterion; (2) adding a model class that's not directly accommodated by the cv() default method or by another directly inherited method, with separate consideration of mixed-effects models; and (3) adding a new model-selection procedure suitable for use with selectModel().

Adding a cost criterion

A cost criterion suitable for use with cv() or cvSelect() should take two arguments, y (the observed response vector) and yhat (a vector of fitted or predicted response values), and return a numeric index of lack of fit. The cv package supplies several such criteria: mse(y, yhat), which returns the mean-squared prediction error for a numeric response; rmse(y, yhat), which returns the (square-)root mean-squared error; medAbsErr(y, yhat), which returns the median absolute error; and BayesRule(y, yhat) (and its non-error-checking version, BayesRule2(y, yhat)), suitable for use with a binary regression model, where y is the binary response coded 0 for a "failure" or 1 for a "success"; where yhat is the predicted probability of success; and where the proportion of incorrectly classified cases is returned.

To illustrate using a different prediction cost criterion, we'll base a cost criterion on the area under the receiver operating characteristic ("ROC") curve for a logistic regression. The ROC curve is a graphical representation of the classification power of a binary regression model, and the area under the ROC curve ("AUC"), which varies from 0 to 1, is a common summary measure based on the ROC [see @Wikipedia-ROC:2023]. The Metrics package [@HamnerFrasco:2018] includes a variety of measures useful for model selection, including an auc() function. We convert the AUC into a cost measure by taking its complement:

AUCcomp <- function(y, yhat) 1 - Metrics::auc(y, yhat)

We then apply AUCcomp() to the the Mroz logistic regression, discussed in the introductory vignette on cross-validating regression models, which we reproduce here. Using the Mroz data frame from the carData package [@FoxWeisberg:2019]:

data("Mroz", package="carData")
m.mroz <- glm(lfp ~ ., data=Mroz, family=binomial)
summary(m.mroz)

AUCcomp(with(Mroz, as.numeric(lfp == "yes")), fitted(m.mroz))

Cross-validating this cost measure is straightforward:

library("cv")
cv(m.mroz, criterion=AUCcomp, seed=3639)

As expected, the cross-validated complement to the AUC is somewhat less optimistic than the criterion computed from the model fit to the whole data set.

As we explain in the vignette "Cross-validating regression models," the cv() function differentiates between CV criteria that are averages of casewise components and criteria that are not. Computation of bias corrections and confidence intervals is limited to the former. We show in the technical and computational vignette that the AUC, and hence its complement, cannot be expressed as averages of casewise components.

cv() looks for a "casewise loss" attribute of the value returned by a CV criterion function. If this attribute exists, then the criterion is treated as the mean of casewise components, and cv() uses the unexported function getLossFn() to construct a function that returns the casewise components of the criterion.

We illustrate with the mse():

mse

cv:::getLossFn(mse(rnorm(100), rnorm(100)))

For this scheme to work, the "casewise loss" attribute must be a character string (or vector of character strings), here "(y - yhat)^2", that evaluates to an expression that is a function of y and yhat, and that computes the vector of casewise components of the CV criterion.

Adding a model class not covered by the default `cv()` method

Independently sampled cases

Suppose that we want to cross-validate a multinomial logistic regression model fit by the multinom() function in the nnet package [@VenablesRipley:2002]. We borrow an example from @Fox:2016 [Sec. 14.2.1], with data from the British Election Panel Study on vote choice in the 2001 British election. Data for the example are in the BEPS data frame in the carData package:

data("BEPS", package="carData")
head(BEPS)

The polytomous (multi-category) response variable is vote, a factor with levels "Conservative", "Labour", and "Liberal Democrat". The predictors of vote are:

age, in years;
econ.cond.national and econ.cond.household, the respondent's ratings of the state of the economy, on 1 to 5 scales.
Blair, Hague, and Kennedy, ratings of the leaders of the Labour, Conservative, and Liberal Democratic parties, on 1 to 5 scales.
Europe, an 11-point scale on attitude towards European integration, with high scores representing "Euro-skepticism."
political.knowledge, knowledge of the parties' positions on European integration, with scores from 0 to 3.
gender, "female" or "male".

The model fit to the data includes an interaction between Europe and political.knowledge; the other predictors enter the model additively:

library("nnet")
m.beps <- multinom(
  vote ~ age + gender + economic.cond.national +
    economic.cond.household + Blair + Hague + Kennedy +
    Europe * political.knowledge,
  data = BEPS
)

car::Anova(m.beps)

Most of the predictors, including the Europe $\times$ political.knowledge interaction, are associated with very small $p$-values; the Anova() function is from the car package [@FoxWeisberg:2019].

Here's an "effect plot", using the the effects package [@FoxWeisberg:2019] to visualize the Europe $\times$ political.knowledge interaction in a "stacked-area" graph:

plot(
  effects::Effect(
    c("Europe", "political.knowledge"),
    m.beps,
    xlevels = list(Europe = 1:11, political.knowledge = 0:3),
    fixed.predictors = list(given.values = c(gendermale = 0.5))
  ),
  lines = list(col = c("blue", "red", "orange")),
  axes = list(x = list(rug = FALSE), y = list(style = "stacked"))
)

To cross-validate this multinomial-logit model we need an appropriate cost criterion. None of the criteria supplied by the cv package---for example, neither mse(), which is appropriate for a numeric response, nor BayesRule(), which is appropriate for a binary response---will do. One possibility is to adapt Bayes rule to a polytomous response:

head(BEPS$vote)
yhat <- predict(m.beps, type = "class")
head(yhat)

BayesRuleMulti <- function(y, yhat) {
  result <- mean(y != yhat)
  attr(result, "casewise loss") <- "y != yhat"
  result
}

BayesRuleMulti(BEPS$vote, yhat)

The predict() method for "multinom" models called with argument type="class" reports the Bayes-rule prediction for each case---that is, the response category with the highest predicted probability. Our BayesRuleMulti() function calculates the proportion of misclassified cases. Because this value is the mean of casewise components, we attach a "casewise loss" attribute to the result (as explained in the preceding section).

The marginal proportions for the response categories are

xtabs(~ vote, data=BEPS)/nrow(BEPS)

and so the marginal Bayes-rule prediction, that everyone will vote Labour, produces an error rate of $1 - 0.47213 = 0.52787$. The multinomial-logit model appears to do substantially better than that, but does its performance hold up to cross-validation?

We check first whether the default cv() method works "out-of-the-box" for the "multinom" model:

cv(m.beps, seed=3465, criterion=BayesRuleMulti)

The default method of GetResponse() (a function supplied by the cv package---see ?GetResponse) fails for a "multinom" object. A straightforward solution is to supply a GetResponse.multinom() method that returns the factor response [using the get_response() function from the insight package, @LudeckeWaggonerMakowski:2019],

GetResponse.multinom <- function(model, ...) {
  insight::get_response(model)
}

head(GetResponse(m.beps))

and to try again:

cv(m.beps, seed=3465, criterion=BayesRuleMulti)

A traceback() (not shown) reveals that the problem is that the default method of cv() calls the "multinom" method for predict() with the argument type="response", when the correct argument should be type="class". We therefore must write a "multinom" method for cv(), but that proves to be very simple:

cv.multinom <-
  function (model, data, criterion = BayesRuleMulti, k, reps,
            seed, ...) {
    model <- update(model, trace = FALSE)
    NextMethod(
      type = "class",
      criterion = criterion,
      criterion.name = deparse(substitute(criterion))
    )
  }

That is, we simply call the default cv() method with the type argument properly set. In addition to supplying the correct type argument, our method sets the default criterion for the cv.multinom() method to BayesRuleMulti. Adding the argument criterion.name=deparse(substitute(criterion)) is inessential, but it insures that printed output will include the name of the criterion function that's employed, whether it's the default BayesRuleMulti or something else. Prior to invoking NextMethod(), we called update() with trace=FALSE to suppress the iteration history reported by default by multinom()---it would be tedious to see the iteration history for each fold.

Then:

summary(cv(m.beps, seed=3465))

The cross-validated polytomous Bayes-rule criterion confirms that the fitted model does substantially better than the marginal Bayes-rule prediction that everyone votes for Labour.

Calling cvCompute()

cv() methods for independently sampled cases, such as cv.default(), cv.lm(), and cv.glm(), work by setting up calls to the cvCompute() function, which is exported from the cv package to support development of cv() methods for additional classes of regression models. In most cases, however, such as the preceding cv.multinom() example, it will suffice and be much simpler to set up a suitable call to cv.default() via NextMethod().

To illustrate how to use cvCompute() directly, we write an alternative, and necessarily more complicated, version of cv.multinom().

cv.multinom <- function(model,
                        data = insight::get_data(model),
                        criterion = BayesRuleMulti,
                        k = 10,
                        reps = 1,
                        seed = NULL,
                        details = k <= 10,
                        confint = n >= 400,
                        level = 0.95,
                        ncores = 1,
                        start = FALSE,
                        ...) {
  f <- function(i) {
    # helper function to compute to compute fitted values,
    #  etc., for each fold i

    indices.i <- fold(folds, i)
    model.i <- if (start) {
      update(model,
             data = data[-indices.i,],
             start = b,
             trace = FALSE)
    } else {
      update(model, data = data[-indices.i,], trace = FALSE)
    }
    fit.all.i <- predict(model.i, newdata = data, type = "class")
    fit.i <- fit.all.i[indices.i]
    # returns:
    #  fit.i: fitted values for the i-th fold
    #  crit.all.i: CV criterion for all cases based on model with
    #              i-th fold omitted
    #  coef.i: coefficients for the model with i-th fold omitted
    list(
      fit.i = fit.i,
      crit.all.i = criterion(y, fit.all.i),
      coef.i = coef(model.i)
    )
  }

  fPara <- function(i, multinom, ...) {
    # helper function for parallel computation
    #   argument multinom makes multinom() locally available
    #   ... is necessary but not used
    indices.i <- fold(folds, i)
    model.i <- if (start) {
      update(model,
             data = data[-indices.i,],
             start = b,
             trace = FALSE)
    } else {
      update(model, data = data[-indices.i,], trace = FALSE)
    }
    fit.all.i <- predict(model.i, newdata = data, type = "class")
    fit.i <- fit.all.i[indices.i]
    list(
      fit.i = fit.i,
      crit.all.i = criterion(y, fit.all.i),
      coef.i = coef(model.i)
    )
  }

  n <- nrow(data)

  # see ?cvCompute for definitions of arguments
  cvCompute(
    model = model,
    data = data,
    criterion = criterion,
    criterion.name = deparse(substitute(criterion)),
    k = k,
    reps = reps,
    seed = seed,
    details = details,
    confint = confint,
    level = level,
    ncores = ncores,
    type = "class",
    start = start,
    f = f,
    fPara = fPara,
    multinom = nnet::multinom
  )
}

Notice that separate "helper" functions are defined for non-parallel and parallel computations.[^multinom-parallel] The new version of cv.multinom() produces the same results as the version that calls cv.default():[^multinom-scoping]

summary(cv(m.beps, seed=3465))

[^multinom-parallel]: Try the following, for example, with both versions of cv.multinom() (possibly replacing ncores=2 with a larger number):

system.time(print(cv1 <- cv(m.beps, k="loo")))
system.time(print(cv2 <- cv(m.beps, k="loo", ncores=2)))
all.equal(cv1, cv2)

[^multinom-scoping]: A subtle point is that we added a multinom argument to the local function fPara(), which is passed to the fPara argument of cvCompute(). There is also a multinom argument to cvCompute(), which is set to the multinom function in the nnet package. The multinom argument isn't directly defined in cvCompute() (examine the definition of this function), but is passed through the ... argument. cvCompute(), in turn, will pass multinom to fPara() via ..., allowing fPara() to find this function when it calls update() to refit the model with each fold i omitted. This scoping issue arises because cvCompute() uses foreach() for parallel computations, even though the nnet package is attached to the search path in the current R session via library("nnet"). cv.default() is able to handle the scoping issue transparently by automatically locating multinom().

Mixed-effects models

Adding a cv() method for a mixed-model class is somewhat more complicated. We provide the cvMixed() function to facilitate this process, and to see how that works, consider the "lme" method from the cv package:

cv:::cv.lme

Notice that cv.lme() sets up a call to cvMixed(), which does the computational work.

Most of the arguments of cvMixed() are familiar:

model is the mixed-model object, here of class "lme".
package is the name of the package in which the mixed-modeling function used to fit the model, here lme(), resides---i.e., "nlme"; cvMixed() uses this argument to retrieve the package namespace.
data is the data set to which the model is fit, by default extracted by the get_data() function in the insight package.
criterion is the CV criterion, defaulting to the mse() function.
k is the number of CV folds, defaulting to "loo" for CV by clusters and 10 for CV by cases.
reps is the number of times the CV process is repeated, defaulting to 1.
seed is the seed for R's random-number generator, defaulting to a randomly selected (and saved) value.
ncores is the number of cores to use for parallel computation; if 1, the default, then the computation isn't parallelized.
clusterVariables is a character vector of the names of variables defining clusters; if missing, then CV is based on cases rather than clusters.

The remaining arguments are unfamiliar:

predict.clusters.args is a named list of arguments to be passed to the predict() function to obtain predictions for the full data set from a model fit to a subset of the data for cluster-based CV. The first two arguments should be object and newdata. It is typically necessary to tell cvMixed() how to base predictions only on fixed effects; in the case of "lme" models, this is done by setting level = 0.
Similarly, predict.cases.args is a named list of arguments to be passed to predict() for case-based CV. Setting level = 1 includes random effects in the predictions.
fixed.effects is used to compute detailed fold-based statistics.

Finally, any additional arguments, absorbed by ..., are passed to update() when the model is refit with each fold omitted. cvMixed() returns an object of class "cv".

Now imagine that we want to support a new class of mixed-effects models. To be concrete, we illustrate with the glmmPQL() function in the MASS package [@VenablesRipley:2002], which fits generalized-linear mixed-effects models by penalized quasi-likelihood.[^glmmPQL] Not coincidentally, the arguments of glmmPQL() are similar to those of lme() (with an additional family argument), because the former iteratively invokes the latter; so cv.glmmPQL() should resemble cv.lme().

[^glmmPQL]: This example is somewhat artificial in that glmmPQL() has largely been superseded by computationally superior functions, such the glmer() function in the lme4 package. There is, however, one situation in which glmmPQL() might prove useful: to specify serial dependency in case-level errors within clusters for longitudinal data, which is not currently supported by glmer().

As it turns out, neither the default method for GetResponse() nor insight::get_data() work for "glmmPQL" objects. These objects include a "data" element, however, and so we can simply extract this element as the default for the data argument of our cv.glmmPQL() method.

To get the response variable is more complicated: We refit the fixed part of the model as a GLM with only the regression constant on the right-hand side, and extract the response from that; because all we need is the response variable, we limit the number of GLM iterations to 1 and suppress warning messages about non-convergence:

GetResponse.glmmPQL <- function(model, ...) {
  f <- formula(model)
  f[[3]] <- 1 # regression constant only on RHS
  model <-
    suppressWarnings(glm(
      f,
      data = model$data,
      family = model$family,
      control = list(maxit = 1)
    ))
  cv::GetResponse(model)
}

Writing the cv() method is then straightforward:

cv.glmmPQL <- function(model,
                       data = model$data,
                       criterion = mse,
                       k,
                       reps = 1,
                       seed,
                       ncores = 1,
                       clusterVariables,
                       ...) {
  cvMixed(
    model,
    package = "MASS",
    data = data,
    criterion = criterion,
    k = k,
    reps = reps,
    seed = seed,
    ncores = ncores,
    clusterVariables = clusterVariables,
    predict.clusters.args = list(
      object = model,
      newdata = data,
      level = 0,
      type = "response"
    ),
    predict.cases.args = list(
      object = model,
      newdata = data,
      level = 1,
      type = "response"
    ),
    fixed.effects = nlme::fixef, 
    verbose = FALSE,
    ...
  )
}

We set the argument verbose=FALSE to suppress glmmPQL()'s iteration counter when cvMixed() calls update().

Let's apply our newly minted method to a logistic regression with a random intercept in an example that appears in ?glmmPQL:

library("MASS")
m.pql <- glmmPQL(
  y ~ trt + I(week > 2),
  random = ~ 1 | ID,
  family = binomial,
  data = bacteria
)
summary(m.pql)

We compare this result to that obtained from glmer() in the lme4 package:

library("lme4")
m.glmer <- glmer(y ~ trt + I(week > 2) + (1 | ID),
                 family = binomial, data = bacteria)
summary(m.glmer)

# comparison of fixed effects:
car::compareCoefs(m.pql, m.glmer)

The two sets of estimates are similar, but not identical

Finally, we try out our cv.glmmPQL() method, cross-validating both by clusters and by cases,

summary(cv(m.pql, clusterVariables="ID", criterion=BayesRule))

summary(cv(m.pql, data=bacteria, criterion=BayesRule, seed=1490))

and again compare to glmer():

summary(cv(m.glmer, clusterVariables="ID", criterion=BayesRule))

summary(cv(m.glmer, data=bacteria, criterion=BayesRule, seed=1490))

Adding a model-selection procedure

The selectStepAIC() function supplied by the cv package, which is based on the stepAIC() function from the nnet package [@VenablesRipley:2002] for stepwise model selection, is suitable for the procedure argument of cvSelect(). The use of selectStepAIC() is illustrated in the vignette on cross-validating model selection.

We'll employ selectStepAIC() as a "template" for writing a CV model-selection procedure. To see the code for this function, type cv::selectStepAIC at the R command prompt, or examine the sources for the cv package at https://github.com/gmonette/cv (the code for selectStepAIC() is in https://github.com/gmonette/cv/blob/main/R/cv-select.R).

Another approach to model selection is all-subsets regression. The regsubsets() function in the leaps package [@LumleyMiller:2020] implements an efficient algorithm for selecting the best-fitting linear least-squares regressions for subsets of predictors of all sizes, from 1 through the maximum number of candidate predictors.[^1] To illustrate the use of regsubsets(), we employ the swiss data frame supplied by the leaps package:

[^1]: The regsubsets() function computes several measures of model predictive performance, including the $R^2$ and $R^2$ adjusted for degrees of freedom, the residual sums of squares, Mallows's $C_p$, and the BIC. Several of these are suitable for comparing models with differing numbers of coefficients---we use the BIC below---but all necessarily agree when comparing models with the same number of coefficients.

library("leaps")
head(swiss)
nrow(swiss)

The data set includes the following variables, for each of 47 French-speaking Swiss provinces circa 1888:

Fertility: A standardized fertility measure.
Agriculture: The percentage of the male population engaged in agriculture.
Examination: The percentage of draftees into the Swiss army receiving the highest grade on an examination.
Education: The percentage of draftees with more than a primary-school education.
Catholic: The percentage of the population who were Catholic.
Infant.Mortality: The infant-mortality rate, expressed as the percentage of live births surviving less than a year.

Following @LumleyMiller:2020, we treat Fertility as the response and the other variables as predictors in a linear least-squares regression:

m.swiss <- lm(Fertility ~ ., data=swiss)
summary(m.swiss)

summary(cv(m.swiss, seed=8433))

Thus, the MSE for the model fit to the complete data is considerably smaller than the CV estimate of the MSE. Can we do better by selecting a subset of the predictors, taking account of the additional uncertainty induced by model selection?

First, let's apply best-subset selection to the complete data set:

#| fig.cap = "Selecting the best model of each size."
swiss.sub <- regsubsets(Fertility ~ ., data=swiss)
summary(swiss.sub)

(bics <- summary(swiss.sub)$bic)
which.min(bics)

car::subsets(swiss.sub, legend="topright")

The graph, produced by the subsets() function in the car package, shows that the model with the smallest BIC is the "best" model with 4 predictors, including Agriculture, Education, Catholic, and Infant.Mortality, but not Examination:

m.best <- update(m.swiss, . ~ . - Examination)
summary(m.best)

summary(cv(m.best, seed=8433)) # use same folds as before

The MSE for the selected model is (of course) slightly higher than for the full model fit previously, but the cross-validated MSE is a bit lower; as we explain in the vignette on cross-validating model selection, however, it isn't kosher to select and cross-validate a model on the same data.

Here's a function named selectSubsets(), meant to be used with cvSelect(), suitable for cross-validating the model-selection process:

selectSubsets <- function(data = insight::get_data(model),
                          model,
                          indices,
                          criterion = mse,
                          details = TRUE,
                          seed,
                          save.model = FALSE,
                          ...) {
  if (inherits(model, "lm", which = TRUE) != 1)
    stop("selectSubsets is appropriate only for 'lm' models")

  y <- GetResponse(model)
  formula <- formula(model)
  X <- model.matrix(model)

  if (missing(indices)) {
    if (missing(seed) || is.null(seed))
      seed <- sample(1e6, 1L)
    # select the best model from the full data by BIC
    sel <- leaps::regsubsets(formula, data = data, ...)
    bics <- summary(sel)$bic
    best <- coef(sel, 1:length(bics))[[which.min(bics)]]
    x.names <- names(best)
    # fit the best model; intercept is already in X, hence - 1:
    m.best <- lm(y ~ X[, x.names] - 1)
    fit.all <- predict(m.best, newdata = data)
    return(list(
      criterion = criterion(y, fit.all),
      model = if (save.model)
        m.best # return best model
      else
        NULL
    ))
  }

  # select the best model omitting the i-th fold (given by indices)
  sel.i <- leaps::regsubsets(formula, data[-indices,], ...)
  bics.i <- summary(sel.i)$bic
  best.i <- coef(sel.i, 1:length(bics.i))[[which.min(bics.i)]]
  x.names.i <- names(best.i)
  m.best.i <- lm(y[-indices] ~ X[-indices, x.names.i] - 1)
  # predict() doesn't work here:
  fit.all.i <- as.vector(X[, x.names.i] %*% coef(m.best.i))
  fit.i <- fit.all.i[indices]
  # return the fitted values for i-th fold, CV criterion for all cases,
  #   and the regression coefficients
  list(
    fit.i = fit.i,
    # fitted values for i-th fold
    crit.all.i = criterion(y, fit.all.i),
    # CV crit for all cases
    coefficients = if (details) {
      # regression coefficients
      coefs <- coef(m.best.i)

      # fix coefficient names
      names(coefs) <- sub("X\\[-indices, x.names.i\\]", "",
                          names(coefs))

      coefs
    }  else {
      NULL
    }
  )
}

A slightly tricky point is that because of scoping issues, predict() doesn't work with the model fit omitting the $i$th fold, and so the fitted values for all cases are computed directly as $\widehat{\mathbf{y}}{-i} = \mathbf{X} \mathbf{b}{-i}$, where $\mathbf{X}$ is the model-matrix for all of the cases, and $\mathbf{b}_{-i}$ is the vector of least-squares coefficients for the selected model with the $i$th fold omitted.

Additionally, the command lm(y[-indices] ~ X[-indices, x.names.i] - 1), which is the selected model with the $i$th fold deleted, produces awkward coefficient names like "X[-indices, x.names.i]Infant.Mortality". Purely for aesthetic reasons, the command sub("X\\[-indices, x.names.i\\]", "", names(coefs)) fixes these awkward names, removing the extraneous text, "X[-indices, x.names.i]".

Applying selectSubsets() to the full data produces the full-data cross-validated MSE (which we obtained previously):

selectSubsets(model=m.swiss)

Similarly, applying the function to an imaginary "fold" of 5 cases returns the MSE for the cases in the fold, based on the model selected and fit to the cases omitting the fold; the MSE for all of the cases, based on the same model; and the coefficients of the selected model, which includes 4 or the 5 predictors (and the intercept):

selectSubsets(model=m.swiss, indices=seq(5, 45, by=10))

Then, using selectSubsets() in cross-validation, invoking the cv.function() method for cv(), we get:

cv.swiss <- cv(
  selectSubsets,
  working.model = m.swiss,
  data = swiss,
  seed = 8433 # use same folds
)
summary(cv.swiss)

Cross-validation shows that model selection exacts a penalty in MSE. Examining the models selected for the 10 folds reveals that there is some uncertainty in identifying the predictors in the "best" model, with Agriculture sometimes appearing and sometimes not: