correctR is a lightweight package that implements a
small number of corrected test statistics for cases when samples of two
machine learning model metrics (e.g., classification accuracy) are not
independent (and therefore are correlated), such as in the case of
resampling and \(k\)-fold
cross-validation. We demonstrate the basic functionality here using some
trivial examples for the following corrected tests that are currently
implemented in correctR:
These corrections were all originally proposed by Nadeau and Bengio (2003)1 with additional representations in Bouckaert and Frank (2004)2.
In random subsampling, the standard \(t\)-test inflates Type I error when used in
conjunction with random subsampling due to an underestimation of the
variance, as found by Dietterich (1998)3. Nadeau and Bengio
(2003) proposed a solution (which we implement as
resampled_ttest in correctR) in the form
of:
\[ t = \frac{\frac{1}{n} \sum_{j=1}^{n}x_{j}}{\sqrt{(\frac{1}{n} + \frac{n_{2}}{n_{1}})\sigma^{2}}} \]
where \(n\) is the number of resamples (NOTE: \(n\) is not sample size), \(n_{1}\) is the number of samples in the training data, and \(n_{2}\) is the number of samples in the test data. \(\sigma^{2}\) is the variance estimate used in the standard paired \(t\)-test (which simply has \(\frac{\sigma}{\sqrt{n}}\) in the denominator where \(n\) is the sample size in this case).
There is an alternate formulation of the random subsampling
correction, devised in terms of the unbiased estimator \(\rho\), discussed in Corani et al. (2016)4 which we
implement as kfold_tttest in correctR:
\[ t = \frac{\frac{1}{n} \sum_{j=1}^{n}x_{j}}{\sqrt{(\frac{1}{n} + \frac{\rho}{1-\rho})\sigma^{2}}} \]
where \(n\) is the number of resamples and \(\rho = \frac{1}{k}\) where \(k\) is the number of folds in the \(k\)-fold cross-validation procedure. This formulation stems from the fact that Nadeau and Bengio (2003) proved there is no unbiased estimator, but it can be approximated with \(\rho = \frac{1}{k}\).
Repeated \(k\)-fold cross-validation
is more complex than the previous case(s) as we now have \(r\) repeats for every fold \(k\). Bouckaert and Frank (2004) present a
nice representation of the corrected test for this case which we
implement as repkfold_ttest in correctR:
\[ t = \frac{\frac{1}{k \cdot r} \sum_{i=1}^{k} \sum_{j=1}^{r} x_{ij}}{\sqrt{(\frac{1}{k \cdot r} + \frac{n_{2}}{n_{1}})\sigma^{2}}} \]
In the real world, we would have proper results obtained through
fitting two models according to one or more of the procedures outlined
above. For simplicity here, we are just going to simulate three datasets
so we can get to the package functionality cleaner and easier. We are
going to assume we are in a classification context and generate
classification accuracy values. These values are purposefully
egregious—we are going to (in the case of the random subsampling) just
fix the train set sample size (n1) to 80 and the test set
sample size (n2) to 20, and assume (using the same data)
for the \(k\)-fold cross-validation
correction that the same numbers were obtained on such a method. Again,
the values are not important here, it is the interface for performing
corrections that is the focus.
In the case of repeated \(k\)-fold
cross-validation, take note of the column names. While your
data.frame you pass in to repkfold_ttest can
have more than the four columns specified here, it must
contain at least these four with the exact corresponding names. The
function explicitly searches for them. They are:
"model" — contains a label for each of the two models
to compare"values" — the numerical values of the performance
metric (i.e., classification accuracy)"k" — which fold the values correspond to"r" — which repeat of the fold the values correspond
toset.seed(123) # For reproducibility
# Data for random subsampling and k-fold cross-validation corrections
x <- stats::rnorm(30, mean = 0.6, sd = 0.1)
y <- stats::rnorm(30, mean = 0.4, sd = 0.1)
# Data for repeated k-fold cross-validation correction
tmp <- data.frame(model = rep(c(1, 2), each = 60),
values = c(stats::rnorm(60, mean = 0.6, sd = 0.1),
stats::rnorm(60, mean = 0.4, sd = 0.1)),
k = rep(c(1, 1, 2, 2), times = 15),
r = rep(c(1, 2), times = 30))We can fit all the corrections in one-line functions:
rss <- resampled_ttest(x = x, y = y, n = 30, n1 = 80, n2 = 20)
kcv <- kfold_ttest(x = x, y = y, n = 100, k = 30)
rkcv <- repkfold_ttest(data = tmp, n1 = 80, n2 = 20, k = 2, r = 2)All the functions return a data.frame with two named
columns: "statistic" (the \(t\)-statistic) and "p.value"
(the associated \(p\)-value), meaning
they can be easily integrated into complex machine pipelines. Here is an
example for resampled_ttest:
Note that all three functions express the hypothesis test as a
two-tailed test by default. If we wanted to specify a one-tailed
(directional) hypothesis, we can set tailed = "one" in any
of the functions. Note that if we do so, we must specify the direction.
In the case of resampled_ttest and
kfold_ttest, this is as simple as setting
greater = "x" if we expect \(x
> y\), or greater = "y" if we expect \(y > x\), such as:
resampled_ttest(x = x, y = y, n = 30, n1 = 80, n2 = 20,
tailed = "one", greater = "x")
## statistic p.value
## 1 2.407318 0.01132991kfold_ttest(x = x, y = y, n = 100, k = 30,
tailed = "one", greater = "x")
## statistic p.value
## 1 6.052149 1.281991e-08In the case of repkfold_ttest, since we have a data
frame, we need to pass in the value in the model column
which corresponds to the model we expect to demonstrate greater values,
such as:
Nadeau, C., and Bengio, Y. Inference for the Generalization Error. Machine Learning, 52, 239-281, (2003).↩︎
Bouckaert, R. R., and Frank, E. Evaluating the Replicability of Significance Tests for Comparing Learning Algorithms. Advances in Knowledge Discovery and Data Mining. PAKDD 2004. Lecture Notes in Computer Science, 3056, (2004).↩︎
Dietterich, T. G. (1998). Approximate Statistical Tests for Comparing Supervised Classification Learning Algorithms. Neural Computation, 10(7)↩︎
Corani, G., Benavoli, A., Demsar, J., Mangili, F., and Zaffalon, M. Statistical comparison of classifiers through Bayesian hierarchical modelling. Machine Learning, 106, (2017).↩︎