Cross-validating model selection
John Fox and Georges Monette
2024-09-22
As Hastie, Tibshirani, & Friedman (2009,
sec. 7.10.2: “The Wrong and Right Way to Do Cross-validation”)
explain, if the whole data are used to select or fine-tune a statistical
model, subsequent cross-validation of the model is intrinsically
misleading, because the model is selected to fit the whole data,
including the part of the data that remains when each fold is
removed.
A preliminary example
The following example is similar in spirit to one employed by Hastie et al. (2009). Suppose that we randomly
generate \(n = 1000\) independent
observations for a response variable variable \(y \sim N(\mu = 10, \sigma^2 = 0)\), and
independently sample \(1000\)
observations for \(p = 100\)
“predictors,” \(x_1, \ldots, x_{100}\),
each from \(x_j \sim N(0, 1)\). The
response has nothing to do with the predictors and so the population
linear-regression model \(y_i = \alpha +
\beta_1 x_{i1} + \cdots + \beta_{100} x_{i,100} + \varepsilon_i\)
has \(\alpha = 10\) and all \(\beta_j = 0\).
set.seed(24361) # for reproducibility
D <- data.frame(y = rnorm(1000, mean = 10),
X = matrix(rnorm(1000 * 100), 1000, 100))
head(D[, 1:6])
#> y X.1 X.2 X.3 X.4 X.5
#> 1 10.0316 -1.23886 -0.26487 -0.03539 -2.576973 0.811048
#> 2 9.6650 0.12287 -0.17744 0.37290 -0.935138 0.628673
#> 3 10.0232 -0.95052 -0.73487 -1.05978 0.882944 0.023918
#> 4 8.9910 1.13571 0.32411 0.11037 1.376303 -0.422114
#> 5 9.0712 1.49474 1.87538 0.10575 0.292140 -0.184568
#> 6 11.3493 -0.18453 -0.78037 -1.23804 -0.010949 0.691034
Least-squares provides accurate estimates of the regression constant
\(\alpha = 10\) and the error variance
\(\sigma^2 = 1\) for the “null model”
including only the regression constant; moreover, the omnibus \(F\)-test of the correct null hypothesis
that all of the \(\beta\)s are 0 for
the “full model” with all 100 \(x\)s is
associated with a large \(p\)-value:
m.full <- lm(y ~ ., data = D)
m.null <- lm(y ~ 1, data = D)
anova(m.null, m.full)
#> Analysis of Variance Table
#>
#> Model 1: y ~ 1
#> Model 2: y ~ X.1 + X.2 + X.3 + X.4 + X.5 + X.6 + X.7 + X.8 + X.9 + X.10 +
#> X.11 + X.12 + X.13 + X.14 + X.15 + X.16 + X.17 + X.18 + X.19 +
#> X.20 + X.21 + X.22 + X.23 + X.24 + X.25 + X.26 + X.27 + X.28 +
#> X.29 + X.30 + X.31 + X.32 + X.33 + X.34 + X.35 + X.36 + X.37 +
#> X.38 + X.39 + X.40 + X.41 + X.42 + X.43 + X.44 + X.45 + X.46 +
#> X.47 + X.48 + X.49 + X.50 + X.51 + X.52 + X.53 + X.54 + X.55 +
#> X.56 + X.57 + X.58 + X.59 + X.60 + X.61 + X.62 + X.63 + X.64 +
#> X.65 + X.66 + X.67 + X.68 + X.69 + X.70 + X.71 + X.72 + X.73 +
#> X.74 + X.75 + X.76 + X.77 + X.78 + X.79 + X.80 + X.81 + X.82 +
#> X.83 + X.84 + X.85 + X.86 + X.87 + X.88 + X.89 + X.90 + X.91 +
#> X.92 + X.93 + X.94 + X.95 + X.96 + X.97 + X.98 + X.99 + X.100
#> Res.Df RSS Df Sum of Sq F Pr(>F)
#> 1 999 974
#> 2 899 888 100 85.2 0.86 0.82
summary(m.null)
#>
#> Call:
#> lm(formula = y ~ 1, data = D)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -3.458 -0.681 0.019 0.636 2.935
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 9.9370 0.0312 318 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.987 on 999 degrees of freedom
Next, using the stepAIC() function in the
MASS package (Venables &
Ripley, 2002), let us perform a forward stepwise regression to
select a “best” model, starting with the null model, and using AIC as
the model-selection criterion (see the help page for
stepAIC() for details):
library("MASS") # for stepAIC()
m.select <- stepAIC(
m.null,
direction = "forward",
trace = FALSE,
scope = list(lower = ~ 1, upper = formula(m.full))
)
summary(m.select)
#>
#> Call:
#> lm(formula = y ~ X.99 + X.90 + X.87 + X.40 + X.65 + X.91 + X.53 +
#> X.45 + X.31 + X.56 + X.61 + X.60 + X.46 + X.35 + X.92, data = D)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -3.262 -0.645 0.024 0.641 3.118
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 9.9372 0.0310 320.80 <2e-16 ***
#> X.99 -0.0910 0.0308 -2.95 0.0032 **
#> X.90 -0.0820 0.0314 -2.62 0.0090 **
#> X.87 -0.0694 0.0311 -2.24 0.0256 *
#> X.40 -0.0476 0.0308 -1.55 0.1221
#> X.65 -0.0552 0.0315 -1.76 0.0795 .
#> X.91 0.0524 0.0308 1.70 0.0894 .
#> X.53 -0.0492 0.0305 -1.61 0.1067
#> X.45 0.0554 0.0318 1.74 0.0818 .
#> X.31 0.0452 0.0311 1.46 0.1457
#> X.56 0.0543 0.0327 1.66 0.0972 .
#> X.61 -0.0508 0.0317 -1.60 0.1091
#> X.60 -0.0513 0.0319 -1.61 0.1083
#> X.46 0.0516 0.0327 1.58 0.1153
#> X.35 0.0470 0.0315 1.49 0.1358
#> X.92 0.0443 0.0310 1.43 0.1533
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.973 on 984 degrees of freedom
#> Multiple R-squared: 0.0442, Adjusted R-squared: 0.0296
#> F-statistic: 3.03 on 15 and 984 DF, p-value: 8.34e-05
library("cv")
mse(D$y, fitted(m.select))
#> [1] 0.93063
#> attr(,"casewise loss")
#> [1] "(y - yhat)^2"
The resulting model has 15 predictors, a very modest \(R^2 = .044\), but a small \(p\)-value for its omnibus \(F\)-test (which, of course, is entirely
spurious because the same data were used to select and test the model).
The MSE for the selected model is smaller than the true error variance
\(\sigma^2 = 1\), as is the estimated
error variance for the selected model, \(\widehat{\sigma}^2 = 0.973^2 = 0.947\).
If we cross-validate the selected model, we also obtain an optimistic
estimate of its predictive power (although the confidence interval for
the bias-adjusted MSE includes 1):
library("cv")
summary(cv(m.select, seed = 2529))
#> R RNG seed set to 2529
#> 10-Fold Cross Validation
#> method: Woodbury
#> criterion: mse
#> cross-validation criterion = 0.95937
#> bias-adjusted cross-validation criterion = 0.95785
#> 95% CI for bias-adjusted CV criterion = (0.87661, 1.0391)
#> full-sample criterion = 0.93063
The "function" method of cv() allows us to
cross-validate the whole model-selection procedure, where first argument
to cv() is a model-selection function capable of refitting
the model with a fold omitted and returning a CV criterion. The
selectStepAIC() function, in the cv
package and based on stepAIC(), is suitable for use with
cv():
cv.select <- cv(
selectStepAIC,
data = D,
seed = 3791,
working.model = m.null,
direction = "forward",
scope = list(lower = ~ 1, upper = formula(m.full))
)
#> R RNG seed set to 3791
summary(cv.select)
#> 10-Fold Cross Validation
#> criterion: mse
#> cross-validation criterion = 1.0687
#> bias-adjusted cross-validation criterion = 1.0612
#> 95% CI for bias-adjusted CV criterion = (0.97172, 1.1506)
#> full-sample criterion = 0.93063
The other arguments to cv() are:
data, the data set to which the model is fit;seed, an optional seed for R’s pseudo-random-number
generator; as for cv(), if the seed isn’t supplied by the
user, a seed is randomly selected and saved;- additional arguments required by the model-selection function, here
the starting
working.model argument, the
direction of model selection, and the scope of
models considered (from the model with only a regression constant to the
model with all 100 predictors).
By default, cv() performs 10-fold CV, and produces an
estimate of MSE for the model-selection procedure even larger
than the true error variance, \(\sigma^2 =
1\).
Also by default, when the number of folds is 10 or fewer,
cv() saves details data about the folds. In this example,
the compareFolds() function reveals that the variables
retained by the model-selection process in the several folds are quite
different:
compareFolds(cv.select)
#> CV criterion by folds:
#> fold.1 fold.2 fold.3 fold.4 fold.5 fold.6 fold.7 fold.8 fold.9 fold.10
#> 1.26782 1.12837 1.04682 1.31007 1.06899 0.87916 0.88380 0.95026 1.21070 0.94130
#>
#> Coefficients by folds:
#> (Intercept) X.87 X.90 X.99 X.91 X.54 X.53 X.56
#> Fold 1 9.9187 -0.0615 -0.0994 -0.0942 0.0512 0.0516
#> Fold 2 9.9451 -0.0745 -0.0899 -0.0614 0.0587 0.0673
#> Fold 3 9.9423 -0.0783 -0.0718 -0.0987 0.0601 0.0512
#> Fold 4 9.9410 -0.0860 -0.0831 -0.0867 0.0570 -0.0508
#> Fold 5 9.9421 -0.0659 -0.0849 -0.1004 0.0701 0.0511 -0.0487 0.0537
#> Fold 6 9.9633 -0.0733 -0.0874 -0.0960 0.0555 0.0629 -0.0478
#> Fold 7 9.9279 -0.0618 -0.0960 -0.0838 0.0533 -0.0464
#> Fold 8 9.9453 -0.0610 -0.0811 -0.0818 0.0497 -0.0612 0.0560
#> Fold 9 9.9173 -0.0663 -0.0894 -0.1100 0.0504 0.0524 0.0747
#> Fold 10 9.9449 -0.0745 -0.0906 -0.0891 0.0535 0.0482 -0.0583 0.0642
#> X.40 X.45 X.65 X.68 X.92 X.15 X.26 X.46 X.60
#> Fold 1 -0.0590 -0.0456 0.0658 0.0608
#> Fold 2 0.0607 0.0487
#> Fold 3 -0.0496 -0.0664 0.0494
#> Fold 4 -0.0597 0.0579 -0.0531 0.0519 -0.0566 -0.0519
#> Fold 5 0.0587 0.0527 -0.0603
#> Fold 6 -0.0596 0.0552 0.0474
#> Fold 7 0.0572 0.0595
#> Fold 8 0.0547 -0.0617 0.0453 0.0493 -0.0613 0.0591 0.0703 -0.0588
#> Fold 9 -0.0552 0.0573 -0.0635 0.0492 -0.0513 0.0484 -0.0507
#> Fold 10 -0.0558 0.0529 0.0710
#> X.61 X.8 X.28 X.29 X.31 X.35 X.70 X.89 X.17
#> Fold 1 -0.0490 0.0616 -0.0537 0.0638
#> Fold 2 0.0671 0.0568 0.0523
#> Fold 3 -0.0631 0.0616
#> Fold 4 0.0659 -0.0549 0.0527 0.0527
#> Fold 5 0.0425 0.0672 0.0613 0.0493
#> Fold 6 0.0559 -0.0629 0.0498 0.0487
#> Fold 7 0.0611 0.0472
#> Fold 8 -0.0719 0.0586
#> Fold 9 0.0525
#> Fold 10 -0.0580 0.0603
#> X.25 X.4 X.64 X.81 X.97 X.11 X.2 X.33 X.47
#> Fold 1 0.0604 0.0575
#> Fold 2 0.0478 0.0532 0.0518
#> Fold 3 0.0574 0.0473
#> Fold 4 0.0628
#> Fold 5 0.0518
#> Fold 6 0.0521
#> Fold 7 0.0550
#> Fold 8
#> Fold 9 0.0556 0.0447
#> Fold 10 0.0516
#> X.6 X.72 X.73 X.77 X.79 X.88
#> Fold 1 0.0476
#> Fold 2 0.0514
#> Fold 3
#> Fold 4 -0.0473
#> Fold 5 0.0586 0.07
#> Fold 6 -0.0489
#> Fold 7
#> Fold 8
#> Fold 9
#> Fold 10
Polynomial regression for the Auto data revisited: meta
cross-validation
In the introductory vignette on cross-validating regression models,
following
James, Witten, Hastie, & Tibshirani
(2021, secs. 5.1, 5.3), we fit polynomial regressions up to
degree 10 to the relationship of
mpg to
horsepower for the
Auto data, saving the
results in
m.1 through
m.10. We then used
cv() to compare the cross-validated MSE for the 10 models,
discovering that the 7th degree polynomial had the smallest MSE (by a
small margin); repeating the relevant graph:
If we then select the 7th degree polynomial model, intending to use
it for prediction, the CV estimate of the MSE for this model will be
optimistic. One solution is to cross-validate the process of using CV to
select the “best” model—that is, to apply CV to CV recursively, a
process that we term “meta cross-validation.” The function
selectModelList(), which is suitable for use with
cv(), implements this idea.
Applying selectModelList() to the Auto
polynomial-regression models, and using 10-fold CV, we obtain:
metaCV.auto <- cv(
selectModelList,
Auto,
working.model = models(m.1, m.2, m.3, m.4, m.5,
m.6, m.7, m.8, m.9, m.10),
save.model = TRUE,
seed = 2120
)
#> R RNG seed set to 2120
summary(metaCV.auto)
#> 10-Fold Cross Validation
#> criterion: mse
#> cross-validation criterion = 19.105
#> bias-adjusted cross-validation criterion = 19.68
#> full-sample criterion = 18.746
(m.sel <- cvInfo(metaCV.auto, "selected model"))
#>
#> Call:
#> lm(formula = mpg ~ poly(horsepower, 7), data = Auto)
#>
#> Coefficients:
#> (Intercept) poly(horsepower, 7)1 poly(horsepower, 7)2
#> 23.45 -120.14 44.09
#> poly(horsepower, 7)3 poly(horsepower, 7)4 poly(horsepower, 7)5
#> -3.95 -5.19 13.27
#> poly(horsepower, 7)6 poly(horsepower, 7)7
#> -8.55 7.98
cv(m.sel, seed = 2120) # same seed for same folds
#> R RNG seed set to 2120
#> cross-validation criterion (mse) = 18.898
As expected, meta CV produces a larger estimate of MSE for the
selected 7th degree polynomial model than CV applied directly to this
model.
We can equivalently call cv() with the list of models as
the first argument and set meta=TRUE:
metaCV.auto.alt <- 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,
meta = TRUE,
save.model = TRUE
)
#> R RNG seed set to 2120
all.equal(metaCV.auto, metaCV.auto.alt)
#> [1] "Component \"criterion\": 1 string mismatch"
Mroz’s logistic regression revisited
Next, let’s apply model selection to Mroz’s logistic regression for
married women’s labor-force participation, also discussed in the
introductory vignette on cross-validating regression models. First,
recall the logistic regression model that we fit to the
Mroz data:
data("Mroz", package = "carData")
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
Applying stepwise model selection Mroz’s logistic regression, using
BIC as the model-selection criterion (via the argument
k=log(nrow(Mroz)) to stepAIC()) selects 5 of
the 7 original predictors:
m.mroz.sel <- stepAIC(m.mroz, k = log(nrow(Mroz)),
trace = FALSE)
summary(m.mroz.sel)
#>
#> Call:
#> glm(formula = lfp ~ k5 + age + wc + lwg + inc, family = binomial,
#> data = Mroz)
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 2.9019 0.5429 5.35 9.0e-08 ***
#> k5 -1.4318 0.1932 -7.41 1.3e-13 ***
#> age -0.0585 0.0114 -5.13 2.9e-07 ***
#> wcyes 0.8724 0.2064 4.23 2.4e-05 ***
#> lwg 0.6157 0.1501 4.10 4.1e-05 ***
#> inc -0.0337 0.0078 -4.32 1.6e-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: 906.46 on 747 degrees of freedom
#> AIC: 918.5
#>
#> Number of Fisher Scoring iterations: 3
BayesRule(Mroz$lfp == "yes",
predict(m.mroz.sel, type = "response"))
#> [1] 0.31873
#> attr(,"casewise loss")
#> [1] "y != round(yhat)"
Bayes rule applied to the selected model misclassifies 32% of the
cases in the Mroz data.
Cross-validating the selected model produces a similar, slightly
larger, estimate of misclassification, about 33%:
summary(cv(m.mroz.sel, criterion = BayesRule, seed = 345266))
#> R RNG seed set to 345266
#> 10-Fold Cross Validation
#> method: exact
#> criterion: BayesRule
#> cross-validation criterion = 0.33068
#> bias-adjusted cross-validation criterion = 0.33332
#> 95% CI for bias-adjusted CV criterion = (0.2997, 0.36695)
#> full-sample criterion = 0.31873
Is this estimate of predictive performance optimistic?
We proceed to apply the model-selection procedure by
cross-validation, producing more or less the same result:
m.mroz.sel.cv <- cv(
selectStepAIC,
Mroz,
seed = 6681,
criterion = BayesRule,
working.model = m.mroz,
AIC = FALSE
)
#> R RNG seed set to 6681
summary(m.mroz.sel.cv)
#> 10-Fold Cross Validation
#> criterion: BayesRule
#> cross-validation criterion = 0.33068
#> bias-adjusted cross-validation criterion = 0.33452
#> 95% CI for bias-adjusted CV criterion = (0.3009, 0.36815)
#> full-sample criterion = 0.31873
Setting AIC=FALSE in the call to cv() uses
the BIC rather than the AIC as the model-selection criterion. As it
turns out, exactly the same predictors are selected when each of the 10
folds are omitted, and the several coefficient estimates are very
similar, as we show using compareFolds():
compareFolds(m.mroz.sel.cv)
#> CV criterion by folds:
#> fold.1 fold.2 fold.3 fold.4 fold.5 fold.6 fold.7 fold.8 fold.9 fold.10
#> 0.27632 0.40789 0.34211 0.36000 0.28000 0.37333 0.32000 0.29333 0.33333 0.32000
#>
#> Coefficients by folds:
#> (Intercept) age inc k5 lwg wcyes
#> Fold 1 2.5014 -0.0454 -0.0388 -1.3613 0.5653 0.85
#> Fold 2 3.0789 -0.0659 -0.0306 -1.5335 0.6923 0.79
#> Fold 3 3.0141 -0.0595 -0.0305 -1.3994 0.5428 0.86
#> Fold 4 2.7251 -0.0543 -0.0354 -1.4474 0.6298 1.09
#> Fold 5 2.7617 -0.0566 -0.0320 -1.4752 0.6324 0.74
#> Fold 6 3.0234 -0.0621 -0.0348 -1.4537 0.6618 0.94
#> Fold 7 2.9615 -0.0600 -0.0351 -1.4127 0.5835 0.97
#> Fold 8 2.9598 -0.0603 -0.0329 -1.3865 0.6210 0.69
#> Fold 9 3.2481 -0.0650 -0.0381 -1.4138 0.6093 0.94
#> Fold 10 2.7724 -0.0569 -0.0295 -1.4503 0.6347 0.85
In this example, therefore, we appear to obtain a realistic estimate
of model performance directly from the selected model, because there is
little added uncertainty induced by model selection.
Cross-validating choice of transformations in regression
The cv package also provides a cv()
procedure, selectTrans(), for choosing transformations of
the predictors and the response in regression.
Some background: As Weisberg (2014, sec.
8.2) explains, there are technical advantages to having (numeric)
predictors in linear regression analysis that are themselves linearly
related. If the predictors aren’t linearly related, then the
relationships between them can often be straightened by power
transformations. Transformations can be selected after graphical
examination of the data, or by analytic methods. Once the relationships
between the predictors are linearized, it can be advantageous similarly
to transform the response variable towards normality.
Selecting transformations analytically raises the possibility of
automating the process, as would be required for cross-validation. One
could, in principle, apply graphical methods to select transformations
for each fold, but because a data analyst couldn’t forget the choices
made for previous folds, the process wouldn’t really be applied
independently to the folds.
To illustrate, we adapt an example appearing in several places in
Fox & Weisberg (2019) (for example in
Chapter 3 on transforming data), using data on the prestige and other
characteristics of 102 Canadian occupations circa 1970. The data are in
the Prestige data frame in the carData
package:
data("Prestige", package = "carData")
head(Prestige)
#> education income women prestige census type
#> gov.administrators 13.11 12351 11.16 68.8 1113 prof
#> general.managers 12.26 25879 4.02 69.1 1130 prof
#> accountants 12.77 9271 15.70 63.4 1171 prof
#> purchasing.officers 11.42 8865 9.11 56.8 1175 prof
#> chemists 14.62 8403 11.68 73.5 2111 prof
#> physicists 15.64 11030 5.13 77.6 2113 prof
summary(Prestige)
#> education income women prestige census
#> Min. : 6.38 Min. : 611 Min. : 0.00 Min. :14.8 Min. :1113
#> 1st Qu.: 8.45 1st Qu.: 4106 1st Qu.: 3.59 1st Qu.:35.2 1st Qu.:3120
#> Median :10.54 Median : 5930 Median :13.60 Median :43.6 Median :5135
#> Mean :10.74 Mean : 6798 Mean :28.98 Mean :46.8 Mean :5402
#> 3rd Qu.:12.65 3rd Qu.: 8187 3rd Qu.:52.20 3rd Qu.:59.3 3rd Qu.:8312
#> Max. :15.97 Max. :25879 Max. :97.51 Max. :87.2 Max. :9517
#> type
#> bc :44
#> prof:31
#> wc :23
#> NA's: 4
#>
#>
The variables in the Prestige data set are:
education: average years of education for incumbents in
the occupation, from the 1971 Canadian Census.income: average dollars of annual income for the
occupation, from the Census.women: percentage of occupational incumbents who were
women, also from the Census.prestige: the average prestige rating of the occupation
on a 0–100 “thermometer” scale, in a Canadian social survey conducted
around the same time.type, type of occupation, and census, the
Census occupational code, which are not used in our example.
The object of a regression analysis for the Prestige
data (and their original purpose) is to predict occupational prestige
from the other variables in the data set.
A scatterplot matrix (using the scatterplotMatrix()
function in the car package) of the numeric variables
in the data reveals that the distributions of income and
women are positively skewed, and that some of the
relationships among the three predictors, and between the predictors and
the response (i.e., prestige), are nonlinear:
library("car")
#> Loading required package: carData
#>
#> Attaching package: 'car'
#> The following object is masked from 'package:dplyr':
#>
#> recode
scatterplotMatrix(
~ prestige + income + education + women,
data = Prestige,
smooth = list(spread = FALSE)
)
The powerTransform() function in the
car package transforms variables towards multivariate
normality by a generalization of Box and Cox’s maximum-likelihood-like
approach (Box & Cox, 1964). Several
“families” of power transformations can be used, including the original
Box-Cox family, simple powers (and roots), and two adaptations of the
Box-Cox family to data that may include negative values and zeros: the
Box-Cox-with-negatives family and the Yeo-Johnson family; see Weisberg (2014, Chapter 8), and Fox & Weisberg (2019, Chapter 3) for
details. Because women has some zero values, we use the
Yeo-Johnson family:
trans <- powerTransform(cbind(income, education, women) ~ 1,
data = Prestige,
family = "yjPower")
summary(trans)
#> yjPower Transformations to Multinormality
#> Est Power Rounded Pwr Wald Lwr Bnd Wald Upr Bnd
#> income 0.2678 0.33 0.1051 0.4304
#> education 0.5162 1.00 -0.2822 1.3145
#> women 0.1630 0.16 0.0112 0.3149
#>
#> Likelihood ratio test that all transformation parameters are equal to 0
#> LRT df pval
#> LR test, lambda = (0 0 0) 15.739 3 0.00128
We thus have evidence of the desirability of transforming
income (by the \(1/3\)
power) and women (by the \(0.16\) power—which is close to the “0”
power, i.e., the log transformation), but not education.
Applying the “rounded” power transformations makes the predictors
better-behaved:
P <- Prestige[, c("prestige", "income", "education", "women")]
(lambdas <- trans$roundlam)
#> income education women
#> 0.33000 1.00000 0.16302
names(lambdas) <- c("income", "education", "women")
for (var in c("income", "education", "women")) {
P[, var] <- yjPower(P[, var], lambda = lambdas[var])
}
summary(P)
#> prestige income education women
#> Min. :14.8 Min. :22.2 Min. : 6.38 Min. :0.00
#> 1st Qu.:35.2 1st Qu.:44.2 1st Qu.: 8.45 1st Qu.:1.73
#> Median :43.6 Median :50.3 Median :10.54 Median :3.36
#> Mean :46.8 Mean :50.8 Mean :10.74 Mean :3.50
#> 3rd Qu.:59.3 3rd Qu.:56.2 3rd Qu.:12.65 3rd Qu.:5.59
#> Max. :87.2 Max. :83.6 Max. :15.97 Max. :6.83
scatterplotMatrix(
~ prestige + income + education + women,
data = P,
smooth = list(spread = FALSE)
)
Comparing the MSE for the regressions with the original and
transformed predictors shows a advantage to the latter:
m.pres <- lm(prestige ~ income + education + women, data = Prestige)
m.pres.trans <- lm(prestige ~ income + education + women, data = P)
mse(Prestige$prestige, fitted(m.pres))
#> [1] 59.153
#> attr(,"casewise loss")
#> [1] "(y - yhat)^2"
mse(P$prestige, fitted(m.pres.trans))
#> [1] 50.6
#> attr(,"casewise loss")
#> [1] "(y - yhat)^2"
Similarly, component+residual plots for the two regressions, produced
by the crPlots() function in the car
package, suggest that the partial relationship of prestige
to income is more nearly linear in the transformed data,
but the transformation of women fails to capture what
appears to be a slight quadratic partial relationship; the partial
relationship of prestige to education is close
to linear in both regressions:
Having transformed the predictors towards multinormality, we now
consider whether there’s evidence for transforming the response (using
powerTransform() for Box and Cox’s original method), and we
discover that there’s not:
summary(powerTransform(m.pres.trans))
#> bcPower Transformation to Normality
#> Est Power Rounded Pwr Wald Lwr Bnd Wald Upr Bnd
#> Y1 1.0194 1 0.6773 1.3615
#>
#> Likelihood ratio test that transformation parameter is equal to 0
#> (log transformation)
#> LRT df pval
#> LR test, lambda = (0) 32.217 1 1.38e-08
#>
#> Likelihood ratio test that no transformation is needed
#> LRT df pval
#> LR test, lambda = (1) 0.012384 1 0.911
The selectTrans() function in the cv
package automates the process of selecting predictor and response
transformations. The function takes a data set and
“working” model as arguments, along with the candidate
predictors and response for transformation,
and the transformation family to employ. If the
predictors argument is missing then only the response is
transformed, and if the response argument is missing, only
the supplied predictors are transformed. The default family
for transforming the predictors is "bcPower"—the original
Box-Cox family—as is the default family.y for transforming
the response; here we specify family="yjPower because of
the zeros in women. selectTrans() returns the
result of applying a lack-of-fit criterion to the model after the
selected transformation is applied, with the default
criterion=mse:
selectTrans(
data = Prestige,
model = m.pres,
predictors = c("income", "education", "women"),
response = "prestige",
family = "yjPower"
)
#> $criterion
#> [1] 50.6
#> attr(,"casewise loss")
#> [1] "(y - yhat)^2"
#>
#> $model
#> NULL
selectTrans() also takes an optional
indices argument, making it suitable for doing computations
on a subset of the data (i.e., a CV fold), and hence for use with
cv() (see ?selectTrans for details):
cvs <- cv(
selectTrans,
data = Prestige,
working.model = m.pres,
seed = 1463,
predictors = c("income", "education", "women"),
response = "prestige",
family = "yjPower"
)
#> R RNG seed set to 1463
summary(cvs)
#> 10-Fold Cross Validation
#> criterion: mse
#> cross-validation criterion = 54.487
#> bias-adjusted cross-validation criterion = 54.308
#> full-sample criterion = 50.6
cv(m.pres, seed = 1463) # untransformed model with same folds
#> R RNG seed set to 1463
#> cross-validation criterion (mse) = 63.293
compareFolds(cvs)
#> CV criterion by folds:
#> fold.1 fold.2 fold.3 fold.4 fold.5 fold.6 fold.7 fold.8 fold.9 fold.10
#> 63.453 79.257 20.634 94.569 19.902 55.821 26.555 75.389 55.702 50.215
#>
#> Coefficients by folds:
#> lam.education lam.income lam.women lambda
#> Fold 1 1.000 0.330 0.330 1
#> Fold 2 1.000 0.330 0.169 1
#> Fold 3 1.000 0.330 0.330 1
#> Fold 4 1.000 0.330 0.330 1
#> Fold 5 1.000 0.330 0.000 1
#> Fold 6 1.000 0.330 0.330 1
#> Fold 7 1.000 0.330 0.330 1
#> Fold 8 1.000 0.330 0.000 1
#> Fold 9 1.000 0.330 0.000 1
#> Fold 10 1.000 0.330 0.000 1
The results suggest that the predictive power of the transformed
regression is reliably greater than that of the untransformed regression
(though in both case, the cross-validated MSE is considerably higher
than the MSE computed for the whole data). Examining the selected
transformations for each fold reveals that the predictor
education and the response prestige are never
transformed; that the \(1/3\) power is
selected for income in all of the folds; and that the
transformation selected for women varies narrowly across
the folds between the \(0\)th power
(i.e., log) and the \(1/3\) power.
Selecting both transformations and predictors
As we mentioned, Hastie et al. (2009, sec.
7.10.2: “The Wrong and Right Way to Do Cross-validation”) explain
that honest cross-validation has to take account of model specification
and selection. Statistical modeling is at least partly a craft, and one
could imagine applying that craft to successive partial data sets, each
with a fold removed. The resulting procedure would be tedious, though
possibly worth the effort, but it would also be difficult to realize in
practice: After all, we can hardly erase our memory of statistical
modeling choices between analyzing partial data sets.
Alternatively, if we’re able to automate the process of model
selection, then we can more realistically apply CV mechanically. That’s
what we did in the preceding two sections, first for predictor selection
and then for selection of transformations in regression. In this
section, we consider the case where we both select variable
transformations and then proceed to select predictors. It’s insufficient
to apply these steps sequentially, first, for example, using
cv() with selectTrans() and then with
selectStepAIC(); rather we should apply the whole
model-selection procedure with each fold omitted. The
selectTransAndStepAIC() function, also supplied by the
cv package, does exactly that.
To illustrate this process, we return to the Auto data
set:
summary(Auto)
#> mpg cylinders displacement horsepower weight
#> Min. : 9.0 Min. :3.00 Min. : 68 Min. : 46.0 Min. :1613
#> 1st Qu.:17.0 1st Qu.:4.00 1st Qu.:105 1st Qu.: 75.0 1st Qu.:2225
#> Median :22.8 Median :4.00 Median :151 Median : 93.5 Median :2804
#> Mean :23.4 Mean :5.47 Mean :194 Mean :104.5 Mean :2978
#> 3rd Qu.:29.0 3rd Qu.:8.00 3rd Qu.:276 3rd Qu.:126.0 3rd Qu.:3615
#> Max. :46.6 Max. :8.00 Max. :455 Max. :230.0 Max. :5140
#>
#> acceleration year origin name
#> Min. : 8.0 Min. :70 Min. :1.00 amc matador : 5
#> 1st Qu.:13.8 1st Qu.:73 1st Qu.:1.00 ford pinto : 5
#> Median :15.5 Median :76 Median :1.00 toyota corolla : 5
#> Mean :15.5 Mean :76 Mean :1.58 amc gremlin : 4
#> 3rd Qu.:17.0 3rd Qu.:79 3rd Qu.:2.00 amc hornet : 4
#> Max. :24.8 Max. :82 Max. :3.00 chevrolet chevette: 4
#> (Other) :365
xtabs( ~ year, data = Auto)
#> year
#> 70 71 72 73 74 75 76 77 78 79 80 81 82
#> 29 27 28 40 26 30 34 28 36 29 27 28 30
xtabs( ~ origin, data = Auto)
#> origin
#> 1 2 3
#> 245 68 79
xtabs( ~ cylinders, data = Auto)
#> cylinders
#> 3 4 5 6 8
#> 4 199 3 83 103
We previously used the Auto here in a preliminary
example where we employed CV to inform the selection of the order of a
polynomial regression of mpg on horsepower.
Here, we consider more generally the problem of predicting
mpg from the other variables in the Auto data.
We begin with a bit of data management, and then examine the pairwise
relationships among the numeric variables in the data set:
Auto$cylinders <- factor(Auto$cylinders,
labels = c("3.4", "3.4", "5.6", "5.6", "8"))
Auto$year <- as.factor(Auto$year)
Auto$origin <- factor(Auto$origin,
labels = c("America", "Europe", "Japan"))
rownames(Auto) <- make.names(Auto$name, unique = TRUE)
Auto$name <- NULL
scatterplotMatrix(
~ mpg + displacement + horsepower + weight + acceleration,
smooth = list(spread = FALSE),
data = Auto
)
A comment before we proceed: origin is clearly
categorical and so converting it to a factor is natural, but we could
imagine treating cylinders and year as numeric
predictors. There are, however, only 5 distinct values of
cylinders (ranging from 3 to 8), but cars with 3 or 5
cylinders are rare. and none of the cars has 7 cylinders. There are
similarly only 13 distinct years between 1970 and 1982 in the data, and
the relationship between mpg and year is
difficult to characterize. It’s apparent that most these variables are
positively skewed and that many of the pairwise relationships among them
are nonlinear.
We begin with a “working model” that specifies linear partial
relationships of the response to the numeric predictors:
m.auto <- lm(mpg ~ ., data = Auto)
summary(m.auto)
#>
#> Call:
#> lm(formula = mpg ~ ., data = Auto)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -9.006 -1.745 -0.092 1.525 10.950
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 37.034132 1.969393 18.80 < 2e-16 ***
#> cylinders5.6 -2.602941 0.655200 -3.97 8.5e-05 ***
#> cylinders8 -0.582458 1.171452 -0.50 0.61934
#> displacement 0.017425 0.006734 2.59 0.01004 *
#> horsepower -0.041353 0.013379 -3.09 0.00215 **
#> weight -0.005548 0.000632 -8.77 < 2e-16 ***
#> acceleration 0.061527 0.088313 0.70 0.48643
#> year71 0.968058 0.837390 1.16 0.24841
#> year72 -0.601435 0.825115 -0.73 0.46652
#> year73 -0.687689 0.740272 -0.93 0.35351
#> year74 1.375576 0.876500 1.57 0.11741
#> year75 0.929929 0.859072 1.08 0.27974
#> year76 1.559893 0.822505 1.90 0.05867 .
#> year77 2.909416 0.841729 3.46 0.00061 ***
#> year78 3.175198 0.798940 3.97 8.5e-05 ***
#> year79 5.019299 0.845759 5.93 6.8e-09 ***
#> year80 9.099763 0.897293 10.14 < 2e-16 ***
#> year81 6.688660 0.885218 7.56 3.3e-13 ***
#> year82 8.071125 0.870668 9.27 < 2e-16 ***
#> originEurope 2.046664 0.517124 3.96 9.1e-05 ***
#> originJapan 2.144887 0.507717 4.22 3.0e-05 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 2.92 on 371 degrees of freedom
#> Multiple R-squared: 0.867, Adjusted R-squared: 0.86
#> F-statistic: 121 on 20 and 371 DF, p-value: <2e-16
Anova(m.auto)
#> Anova Table (Type II tests)
#>
#> Response: mpg
#> Sum Sq Df F value Pr(>F)
#> cylinders 292 2 17.09 7.9e-08 ***
#> displacement 57 1 6.70 0.0100 *
#> horsepower 82 1 9.55 0.0021 **
#> weight 658 1 76.98 < 2e-16 ***
#> acceleration 4 1 0.49 0.4864
#> year 3017 12 29.40 < 2e-16 ***
#> origin 190 2 11.13 2.0e-05 ***
#> Residuals 3173 371
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
crPlots(m.auto)
