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Approximate Statistical Tests for Comparing Supervised Classification Learning Algorithms
, 1998
"... This article reviews five approximate statistical tests for determining whether one learning algorithm outperforms another on a particular learning task. These tests are compared experimentally to determine their probability of incorrectly detecting a difference when no difference exists (type I err ..."
Abstract

Cited by 666 (8 self)
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This article reviews five approximate statistical tests for determining whether one learning algorithm outperforms another on a particular learning task. These tests are compared experimentally to determine their probability of incorrectly detecting a difference when no difference exists (type I error). Two widely used statistical tests are shown to have high probability of type I error in certain situations and should never be used: a test for the difference of two proportions and a paireddifferences t test based on taking several random traintest splits. A third test, a paireddifferences t test based on 10fold crossvalidation, exhibits somewhat elevated probability of type I error. A fourth test, McNemar’s test, is shown to have low type I error. The fifth test is a new test, 5 × 2 cv, based on five iterations of twofold crossvalidation. Experiments show that this test also has acceptable type I error. The article also measures the power (ability to detect algorithm differences when they do exist) of these tests. The crossvalidated t test is the most powerful. The 5×2 cv test is shown to be slightly more powerful than McNemar’s test. The choice of the best test is determined by the computational cost of running the learning algorithm. For algorithms that can be executed only once, McNemar’s test is the only test with acceptable type I error. For algorithms that can be executed 10 times, the 5×2 cv test is recommended, because it is slightly more powerful and because it directly measures variation due to the choice of training set.
Inference for the Generalization Error
, 2001
"... In order to compare learning algorithms, experimental results reported in the machine learning literature often use statistical tests of signicance to support the claim that a new learning algorithm generalizes better. Such tests should take into account the variability due to the choice of training ..."
Abstract

Cited by 167 (4 self)
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In order to compare learning algorithms, experimental results reported in the machine learning literature often use statistical tests of signicance to support the claim that a new learning algorithm generalizes better. Such tests should take into account the variability due to the choice of training set and not only that due to the test examples, as is often the case. This could lead to gross underestimation of the variance of the crossvalidation estimator, and to the wrong conclusion that the new algorithm is signicantly better when it is not. We perform a theoretical investigation of the variance of a crossvalidation estimator of the generalization error that takes into account the variability due to the randomness of the training set as well as test examples. Our analysis shows that all the variance estimators that are based only on the results of the crossvalidation experiment must be biased. This analysis allows us to propose new estimators of this variance. We show, via simulations, that tests of hypothesis about the generalization error using those new variance estimators have better properties than tests involving variance estimators currently in use and listed in (Dietterich, 1998). In particular, the new tests have correct size and good power. That is, the new tests do not reject the null hypothesis too often when the hypothesis is true, but they tend to frequently reject the null hypothesis when the latter is false.
Abstract
"... In order to to compare learning algorithms, experimental results reported in the machine learning litterature often use statistical tests of significance. Unfortunately, most of these tests do not take into account the variability due to the choice of training set. We perform a theoretical investiga ..."
Abstract
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In order to to compare learning algorithms, experimental results reported in the machine learning litterature often use statistical tests of significance. Unfortunately, most of these tests do not take into account the variability due to the choice of training set. We perform a theoretical investigation of the variance of the crossvalidation estimate of the generalization error that takes into account the variability due to the choice of training sets. This allows us to propose two new ways to estimate this variance. We show, via simulations, that these new statistics perform well relative to the statistics considered by Dietterich (Dietterich, 1998). 1