Examples
Polynomial regression for the Auto data
The data for this example are drawn from the ISLR2
package for R, associated with James et al.
(2021). The presentation here is close (though not identical) to
that in the original source (James et al., 2021,
secs. 5.1, 5.3), and it demonstrates the use of the
cv() function in the cv package.1
The Auto dataset contains information about 392
cars:
data("Auto", package="ISLR2")
head(Auto)
#> mpg cylinders displacement horsepower weight acceleration year origin
#> 1 18 8 307 130 3504 12.0 70 1
#> 2 15 8 350 165 3693 11.5 70 1
#> 3 18 8 318 150 3436 11.0 70 1
#> 4 16 8 304 150 3433 12.0 70 1
#> 5 17 8 302 140 3449 10.5 70 1
#> 6 15 8 429 198 4341 10.0 70 1
#> name
#> 1 chevrolet chevelle malibu
#> 2 buick skylark 320
#> 3 plymouth satellite
#> 4 amc rebel sst
#> 5 ford torino
#> 6 ford galaxie 500
dim(Auto)
#> [1] 392 9With the exception of origin (which we don’t use here),
these variables are largely self-explanatory, except possibly for units
of measurement: for details see
help("Auto", package="ISLR2").
We’ll focus here on the relationship of mpg (miles per
gallon) to horsepower, as displayed in the following
scatterplot:
mpg vs horsepower for the Auto
data
The relationship between the two variables is monotone, decreasing, and nonlinear. Following James et al. (2021), we’ll consider approximating the relationship by a polynomial regression, with the degree of the polynomial \(p\) ranging from 1 (a linear regression) to 10.2 Polynomial fits for \(p = 1\) to \(5\) are shown in the following figure:
plot(mpg ~ horsepower, data = Auto)
horsepower <- with(Auto,
seq(min(horsepower), max(horsepower),
length = 1000))
for (p in 1:5) {
m <- lm(mpg ~ poly(horsepower, p), data = Auto)
mpg <- predict(m, newdata = data.frame(horsepower = horsepower))
lines(horsepower,
mpg,
col = p + 1,
lty = p,
lwd = 2)
}
legend(
"topright",
legend = 1:5,
col = 2:6,
lty = 1:5,
lwd = 2,
title = "Degree",
inset = 0.02
)
mpg vs horsepower for the Auto
data
The linear fit is clearly inappropriate; the fits for \(p = 2\) (quadratic) through \(4\) are very similar; and the fit for \(p = 5\) may over-fit the data by chasing
one or two relatively high mpg values at the right (but see
the CV results reported below).
The following graph shows two measures of estimated (squared) error as a function of polynomial-regression degree: The mean-squared error (“MSE”), defined as \(\mathsf{MSE} = \frac{1}{n}\sum_{i=1}^n (y_i - \widehat{y}_i)^2\), and the usual residual variance, defined as \(\widehat{\sigma}^2 = \frac{1}{n - p - 1} \sum_{i=1}^n (y_i - \widehat{y}_i)^2\). The former necessarily declines with \(p\) (or, more strictly, can’t increase with \(p\)), while the latter gets slightly larger for the largest values of \(p\), with the “best” value, by a small margin, for \(p = 7\).
library("cv") # for mse() and other functions
var <- mse <- numeric(10)
for (p in 1:10) {
m <- lm(mpg ~ poly(horsepower, p), data = Auto)
mse[p] <- mse(Auto$mpg, fitted(m))
var[p] <- summary(m)$sigma ^ 2
}
plot(
c(1, 10),
range(mse, var),
type = "n",
xlab = "Degree of polynomial, p",
ylab = "Estimated Squared Error"
)
lines(
1:10,
mse,
lwd = 2,
lty = 1,
col = 2,
pch = 16,
type = "b"
)
lines(
1:10,
var,
lwd = 2,
lty = 2,
col = 3,
pch = 17,
type = "b"
)
legend(
"topright",
inset = 0.02,
legend = c(expression(hat(sigma) ^ 2), "MSE"),
lwd = 2,
lty = 2:1,
col = 3:2,
pch = 17:16
)
Estimated squared error as a function of polynomial degree, \(p\)
The code for this graph uses the mse() function from the
cv package to compute the MSE for each fit.
Using cv()
The generic cv() function has an "lm"
method, which by default performs \(k =
10\)-fold CV:
m.auto <- lm(mpg ~ poly(horsepower, 2), data = Auto)
summary(m.auto)
#>
#> Call:
#> lm(formula = mpg ~ poly(horsepower, 2), data = Auto)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -14.714 -2.594 -0.086 2.287 15.896
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 23.446 0.221 106.1 <2e-16 ***
#> poly(horsepower, 2)1 -120.138 4.374 -27.5 <2e-16 ***
#> poly(horsepower, 2)2 44.090 4.374 10.1 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 4.37 on 389 degrees of freedom
#> Multiple R-squared: 0.688, Adjusted R-squared: 0.686
#> F-statistic: 428 on 2 and 389 DF, p-value: <2e-16
cv(m.auto)
#> R RNG seed set to 860198
#> 10-Fold Cross Validation
#> method: Woodbury
#> criterion: mse
#> cross-validation criterion = 19.605
#> bias-adjusted cross-validation criterion = 19.571
#> full-sample criterion = 18.985The "lm" method by default uses mse() as
the CV criterion and the Woodbury matrix identity (Hager, 1989) to update the regression with each
fold deleted without having literally to refit the model. (Computational
details are discussed in a separate vignette.) The function reports the
CV estimate of MSE, a biased-adjusted estimate of the MSE (the bias
adjustment is explained in the final section), and the MSE is also
computed for the original, full-sample regression. Because the division
of the data into 10 folds is random, cv() explicitly
(randomly) generates and saves a seed for R’s pseudo-random number
generator, to make the results replicable. The user can also specify the
seed directly via the seed argument to
cv().
To perform LOO CV, we can set the k argument to
cv() to the number of cases in the data, here
k=392, or, more conveniently, to k="loo" or
k="n":
cv(m.auto, k = "loo")
#> n-Fold Cross Validation
#> method: hatvalues
#> criterion: mse
#> cross-validation criterion = 19.248For LOO CV of a linear model, cv() by default uses the
hatvalues from the model fit to the full data for the LOO updates, and
reports only the CV estimate of MSE. Alternative methods are to use the
Woodbury matrix identity or the “naive” approach of literally refitting
the model with each case omitted. All three methods produce exact
results for a linear model (within the precision of floating-point
computations):
cv(m.auto, k = "loo", method = "naive")
#> n-Fold Cross Validation
#> method: naive
#> criterion: mse
#> cross-validation criterion = 19.248
#> bias-adjusted cross-validation criterion = 19.248
#> full-sample criterion = 18.985
cv(m.auto, k = "loo", method = "Woodbury")
#> n-Fold Cross Validation
#> method: Woodbury
#> criterion: mse
#> cross-validation criterion = 19.248
#> bias-adjusted cross-validation criterion = 19.248
#> full-sample criterion = 18.985The "naive" and "Woodbury" methods also
return the bias-adjusted estimate of MSE and the full-sample MSE, but
bias isn’t an issue for LOO CV.
Comparing competing models
The cv() function also has a method that can be applied
to a list of regression models for the same data, composed using the
models() function. For \(k\)-fold CV, the same folds are used for
the competing models, which reduces random error in their comparison.
This result can also be obtained by specifying a common seed for R’s
random-number generator while applying cv() separately to
each model, but employing a list of models is more convenient for both
\(k\)-fold and LOO CV (where there is
no random component to the composition of the \(n\) folds).
We illustrate with the polynomial regression models of varying degree
for the Auto data (discussed previously), beginning by
fitting and saving the 10 models:
for (p in 1:10) {
command <- paste0("m.", p, "<- lm(mpg ~ poly(horsepower, ", p,
"), data=Auto)")
eval(parse(text = command))
}
objects(pattern = "m\\.[0-9]")
#> [1] "m.1" "m.10" "m.2" "m.3" "m.4" "m.5" "m.6" "m.7" "m.8" "m.9"
summary(m.2) # for example, the quadratic fit
#>
#> Call:
#> lm(formula = mpg ~ poly(horsepower, 2), data = Auto)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -14.714 -2.594 -0.086 2.287 15.896
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 23.446 0.221 106.1 <2e-16 ***
#> poly(horsepower, 2)1 -120.138 4.374 -27.5 <2e-16 ***
#> poly(horsepower, 2)2 44.090 4.374 10.1 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 4.37 on 389 degrees of freedom
#> Multiple R-squared: 0.688, Adjusted R-squared: 0.686
#> F-statistic: 428 on 2 and 389 DF, p-value: <2e-16The convoluted code within the loop to produce the 10 models insures
that the model formulas are of the form, e.g.,
mpg ~ poly(horsepower, 2) rather than
mpg ~ poly(horsepower, p), which will be useful to us in a
separate vignette, where we consider cross-validating the
model-selection process.
We then apply cv() to the list of 10 models (the
data argument is required):
# 10-fold CV
cv.auto.10 <- cv(
models(m.1, m.2, m.3, m.4, m.5,
m.6, m.7, m.8, m.9, m.10),
data = Auto,
seed = 2120
)
cv.auto.10[1:2] # for the linear and quadratic models
#>
#> Model model.1:
#> 10-Fold Cross Validation
#> method: Woodbury
#> cross-validation criterion = 24.246
#> bias-adjusted cross-validation criterion = 24.23
#> full-sample criterion = 23.944
#>
#> Model model.2:
#> 10-Fold Cross Validation
#> method: Woodbury
#> cross-validation criterion = 19.346
#> bias-adjusted cross-validation criterion = 19.327
#> full-sample criterion = 18.985
# LOO CV
cv.auto.loo <- cv(models(m.1, m.2, m.3, m.4, m.5,
m.6, m.7, m.8, m.9, m.10),
data = Auto,
k = "loo")
cv.auto.loo[1:2] # linear and quadratic models
#>
#> Model model.1:
#> n-Fold Cross Validation
#> method: hatvalues
#> cross-validation criterion = 24.232
#> Model model.2:
#> n-Fold Cross Validation
#> method: hatvalues
#> cross-validation criterion = 19.248Because we didn’t supply names for the models in the calls to the
models() function, the names model.1,
model.2, etc., are generated by the function.
Finally, we extract and graph the adjusted MSEs for \(10\)-fold CV and the MSEs for LOO CV (see
the section below on manipulating "cv" objects:
cv.mse.10 <- as.data.frame(cv.auto.10,
rows="cv",
columns="criteria"
)$adjusted.criterion
cv.mse.loo <- as.data.frame(cv.auto.loo,
rows="cv",
columns="criteria"
)$criterion
plot(
c(1, 10),
range(cv.mse.10, cv.mse.loo),
type = "n",
xlab = "Degree of polynomial, p",
ylab = "Cross-Validated MSE"
)
lines(
1:10,
cv.mse.10,
lwd = 2,
lty = 1,
col = 2,
pch = 16,
type = "b"
)
lines(
1:10,
cv.mse.loo,
lwd = 2,
lty = 2,
col = 3,
pch = 17,
type = "b"
)
legend(
"topright",
inset = 0.02,
legend = c("10-Fold CV", "LOO CV"),
lwd = 2,
lty = 2:1,
col = 3:2,
pch = 17:16
)
Cross-validated 10-fold and LOO MSE as a function of polynomial degree, \(p\)
Alternatively, we can use the plot() method for
"cvModList" objects to compare the models, though with
separate graphs for 10-fold and LOO CV:
plot(cv.auto.10, main="Polynomial Regressions, 10-Fold CV",
axis.args=list(labels=1:10), xlab="Degree of Polynomial, p")
plot(cv.auto.loo, main="Polynomial Regressions, LOO CV",
axis.args=list(labels=1:10), xlab="Degree of Polynomial, p")
Cross-validated 10-fold and LOO MSE as a function of polynomial degree, \(p\)
In this example, 10-fold and LOO CV produce generally similar results, and also results that are similar to those produced by the estimated error variance \(\widehat{\sigma}^2\) for each model, reported above (except for the highest-degree polynomials, where the CV results more clearly suggest over-fitting).
Logistic regression for the Mroz data
The Mroz data set from the carData
package (associated with Fox & Weisberg,
2019) has been used by several authors to illustrate binary
logistic regression; see, in particular Fox &
Weisberg (2019). The data were originally drawn from the U.S.
Panel Study of Income Dynamics and pertain to married women. Here are a
few cases in the data set:
data("Mroz", package = "carData")
head(Mroz, 3)
#> lfp k5 k618 age wc hc lwg inc
#> 1 yes 1 0 32 no no 1.2102 10.91
#> 2 yes 0 2 30 no no 0.3285 19.50
#> 3 yes 1 3 35 no no 1.5141 12.04
tail(Mroz, 3)
#> lfp k5 k618 age wc hc lwg inc
#> 751 no 0 0 43 no no 0.88814 9.952
#> 752 no 0 0 60 no no 1.22497 24.984
#> 753 no 0 3 39 no no 0.85321 28.363The response variable in the logistic regression is lfp,
labor-force participation, a factor coded "yes" or
"no". The remaining variables are predictors:
k5, number of children 5 years old of younger in the woman’s household;k618, number of children between 6 and 18 years old;age, in years;wc, wife’s college attendance,"yes"or"no";hc, husband’s college attendance;lwg, the woman’s log wage rate if she is employed, or her imputed wage rate, if she is not (a variable that Fox & Weisberg, 2019 show is problematically defined); andinc, family income, in $1000s, exclusive of wife’s income.
We use the glm() function to fit a binary logistic
regression to the Mroz data:
m.mroz <- glm(lfp ~ ., data = Mroz, family = binomial)
summary(m.mroz)
#>
#> Call:
#> glm(formula = lfp ~ ., family = binomial, data = Mroz)
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 3.18214 0.64438 4.94 7.9e-07 ***
#> k5 -1.46291 0.19700 -7.43 1.1e-13 ***
#> k618 -0.06457 0.06800 -0.95 0.34234
#> age -0.06287 0.01278 -4.92 8.7e-07 ***
#> wcyes 0.80727 0.22998 3.51 0.00045 ***
#> hcyes 0.11173 0.20604 0.54 0.58762
#> lwg 0.60469 0.15082 4.01 6.1e-05 ***
#> inc -0.03445 0.00821 -4.20 2.7e-05 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for binomial family taken to be 1)
#>
#> Null deviance: 1029.75 on 752 degrees of freedom
#> Residual deviance: 905.27 on 745 degrees of freedom
#> AIC: 921.3
#>
#> Number of Fisher Scoring iterations: 4
BayesRule(ifelse(Mroz$lfp == "yes", 1, 0),
fitted(m.mroz, type = "response"))
#> [1] 0.30677
#> attr(,"casewise loss")
#> [1] "y != round(yhat)"In addition to the usually summary output for a GLM, we show the
result of applying the BayesRule() function from the
cv package to predictions derived from the fitted
model. Bayes rule, which predicts a “success” in a binary regression
model when the fitted probability of success [i.e., \(\phi = \Pr(y = 1)\)] is \(\widehat{\phi} \ge .5\) and a “failure” if
\(\widehat{\phi} \lt .5\).3 The first
argument to BayesRule() is the binary {0, 1} response, and
the second argument is the predicted probability of success.
BayesRule() returns the proportion of predictions that are
in error, as appropriate for a “cost” function.
The value returned by BayesRule() is associated with an
“attribute” named "casewise loss" and set to
"y != round(yhat)", signifying that the Bayes rule CV
criterion is computed as the mean of casewise values, here 0 if the
prediction for a case matches the observed value and 1 if it does not
(signifying a prediction error). The mse() function for
numeric responses is also calculated as a casewise average. Some other
criteria, such as the median absolute error, computed by the
medAbsErr() function in the cv package,
aren’t averages of casewise components. The distinction is important
because, to our knowledge, the statistical theory of cross-validation,
for example, in Davison & Hinkley
(1997), Bates, Hastie, & Tibshirani
(2023), and Arlot & Celisse
(2010), is developed for CV criteria like MSE that are means of
casewise components. As a consequence, we limit computation of bias
adjustment and confidence intervals (see below) to criteria that are
casewise averages.
In this example, the fitted logistic regression incorrectly predicts 31% of the responses; we expect this estimate to be optimistic given that the model is used to “predict” the data to which it is fit.
The "glm" method for cv() is largely
similar to the "lm" method, although the default algorithm,
selected explicitly by method="exact", refits the model
with each fold removed (and is thus equivalent to
method="naive" for "lm" models). For
generalized linear models, method="Woodbury" or (for LOO
CV) method="hatvalues" provide approximate results (see the
computational and technical vignette for details):
cv(m.mroz, criterion = BayesRule, seed = 248)
#> R RNG seed set to 248
#> 10-Fold Cross Validation
#> method: exact
#> criterion: BayesRule
#> cross-validation criterion = 0.32404
#> bias-adjusted cross-validation criterion = 0.31952
#> 95% CI for bias-adjusted CV criterion = (0.28607, 0.35297)
#> full-sample criterion = 0.30677
cv(m.mroz,
criterion = BayesRule,
seed = 248,
method = "Woodbury")
#> R RNG seed set to 248
#> 10-Fold Cross Validation
#> method: Woodbury
#> criterion: BayesRule
#> cross-validation criterion = 0.32404
#> bias-adjusted cross-validation criterion = 0.31926
#> 95% CI for bias-adjusted CV criterion = (0.28581, 0.35271)
#> full-sample criterion = 0.30677To ensure that the two methods use the same 10 folds, we specify the
seed for R’s random-number generator explicitly; here, and as is common
in our experience, the "exact" and "Woodbury"
algorithms produce nearly identical results. The CV estimates of
prediction error are slightly higher than the estimate based on all of
the cases.
The printed output includes a 95% confidence interval for the
bias-adjusted Bayes rule CV criterion. Bates et
al. (2023) show that these confidence intervals are unreliable
for models fit to small samples, and by default cv()
computes them only when the sample size is 400 or larger and when the CV
criterion employed is an average of casewise components, as is the case
for Bayes rule. See the final section of the vignette for details of the
computation of confidence intervals for bias-adjusted CV criteria.
Here are results of applying LOO CV to the Mroz model, using both the exact and the approximate methods:
cv(m.mroz, k = "loo", criterion = BayesRule)
#> n-Fold Cross Validation
#> method: exact
#> criterion: BayesRule
#> cross-validation criterion = 0.32005
#> bias-adjusted cross-validation criterion = 0.3183
#> 95% CI for bias-adjusted CV criterion = (0.28496, 0.35164)
#> full-sample criterion = 0.30677
cv(m.mroz,
k = "loo",
criterion = BayesRule,
method = "Woodbury")
#> n-Fold Cross Validation
#> method: Woodbury
#> criterion: BayesRule
#> cross-validation criterion = 0.32005
#> bias-adjusted cross-validation criterion = 0.3183
#> 95% CI for bias-adjusted CV criterion = (0.28496, 0.35164)
#> full-sample criterion = 0.30677
cv(m.mroz,
k = "loo",
criterion = BayesRule,
method = "hatvalues")
#> n-Fold Cross Validation
#> method: hatvalues
#> criterion: BayesRule
#> cross-validation criterion = 0.32005To the number of decimal digits shown, the three methods produce identical results for this example.
