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Compare predictive accuracies of two classification models

`testcholdout`

statistically assesses
the accuracies of two classification models. The function first compares
their predicted labels against the true labels, and then it detects
whether the difference between the misclassification rates is statistically
significant.

You can assess whether the accuracies of the classification
models are different, or whether one classification model performs
better than another. `testcholdout`

can conduct several McNemar test variations,
including the asymptotic test, the exact-conditional test, and the
mid-*p*-value test. For cost-sensitive assessment, available
tests include a chi-square test (requires an Optimization Toolbox™ license)
and a likelihood ratio test.

returns
the test decision, by conducting the mid-`h`

= testcholdout(`YHat1`

,`YHat2`

,`Y`

)*p*-value McNemar test, from
testing the null hypothesis that the predicted class labels `YHat1`

and `YHat2`

have
equal accuracy for predicting the true class labels `Y`

.
The alternative hypothesis is that the labels have unequal accuracy.

`h`

= `1`

indicates to reject
the null hypothesis at the 5% significance level. `h`

= `0`

indicates
to not reject the null hypothesis at 5% level.

returns
the result of the hypothesis test with additional options specified
by one or more `h`

= testcholdout(`YHat1`

,`YHat2`

,`Y`

,`Name,Value`

)`Name,Value`

pair arguments. For example,
you can specify the type of alternative hypothesis, specify the type
of test, or supply a cost matrix.

Train two classification models using different algorithms. Conduct a statistical test comparing the misclassification rates of the two models on a held-out set.

Load the `ionosphere`

data set.

`load ionosphere`

Create a partition that evenly splits the data into training and testing sets.

rng(1); % For reproducibility CVP = cvpartition(Y,'holdout',0.5); idxTrain = training(CVP); % Training-set indices idxTest = test(CVP); % Test-set indices

`CVP`

is a cross-validation partition object that specifies the training and test sets.

Train an SVM model and an ensemble of 100 bagged classification trees. For the SVM model, specify to use the radial basis function kernel and a heuristic procedure to determine the kernel scale.

MdlSVM = fitcsvm(X(idxTrain,:),Y(idxTrain),'Standardize',true,... 'KernelFunction','RBF','KernelScale','auto'); t = templateTree('Reproducible',true); % For reproducibility of random predictor selections MdlBag = fitcensemble(X(idxTrain,:),Y(idxTrain),'Method','Bag','Learners',t);

`MdlSVM`

is a trained `ClassificationSVM`

model. `MdlBag`

is a trained `ClassificationBaggedEnsemble`

model.

Label the test-set observations using the trained models.

YhatSVM = predict(MdlSVM,X(idxTest,:)); YhatBag = predict(MdlBag,X(idxTest,:));

`YhatSVM`

and `YhatBag`

are vectors continuing the predicted class labels of the respective models.

Test whether the two models have equal predictive accuracies.

h = testcholdout(YhatSVM,YhatBag,Y(idxTest))

`h = `*logical*
0

`h = 0`

indicates to not reject the null hypothesis that the two models have equal predictive accuracies.

Train two classification models using the same algorithm, but adjust a hyperparameter to make the algorithm more complex. Conduct a statistical test to assess whether the simpler model has better accuracy in held-out data than the more complex model.

Load the `ionosphere`

data set.

`load ionosphere;`

Create a partition that evenly splits the data into training and testing sets.

rng(1); % For reproducibility CVP = cvpartition(Y,'holdout',0.5); idxTrain = training(CVP); % Training-set indices idxTest = test(CVP); % Test-set indices

`CVP`

is a cross-validation partition object that specifies the training and test sets.

Train two SVM models: one that uses a linear kernel (the default for binary classification) and one that uses the radial basis function kernel. Use the default kernel scale of 1.

MdlLinear = fitcsvm(X(idxTrain,:),Y(idxTrain),'Standardize',true); MdlRBF = fitcsvm(X(idxTrain,:),Y(idxTrain),'Standardize',true,... 'KernelFunction','RBF');

`MdlLinear`

and `MdlRBF`

are trained `ClassificationSVM`

models.

Label the test-set observations using the trained models.

YhatLinear = predict(MdlLinear,X(idxTest,:)); YhatRBF = predict(MdlRBF,X(idxTest,:));

`YhatLinear`

and `YhatRBF`

are vectors continuing the predicted class labels of the respective models.

Test the null hypothesis that the simpler model (`MdlLinear`

) is at most as accurate as the more complex model (`MdlRBF`

). Because the test-set size is large, conduct the asymptotic McNemar test, and compare the results with the mid- *p*-value test (the cost-insensitive testing default). Request to return *p*-values and misclassification rates.

Asymp = zeros(4,1); % Preallocation MidP = zeros(4,1); [Asymp(1),Asymp(2),Asymp(3),Asymp(4)] = testcholdout(YhatLinear,YhatRBF,Y(idxTest),... 'Alternative','greater','Test','asymptotic'); [MidP(1),MidP(2),MidP(3),MidP(4)] = testcholdout(YhatLinear,YhatRBF,Y(idxTest),... 'Alternative','greater'); table(Asymp,MidP,'RowNames',{'h' 'p' 'e1' 'e2'})

`ans=`*4×2 table*
Asymp MidP
__________ __________
h 1 1
p 7.2801e-09 2.7649e-10
e1 0.13714 0.13714
e2 0.33143 0.33143

The *p*-value is close to zero for both tests, which indicates strong evidence to reject the null hypothesis that the simpler model is less accurate than the more complex model. No matter what test you specify, `testcholdout`

returns the same type of misclassification measure for both models.

For data sets with imbalanced class representations, or if the false-positive and false-negative costs are imbalanced, you can statistically compare the predictive performance of two classification models by including a cost matrix in the analysis.

Load the `arrhythmia`

data set. Determine the class representations in the data.

```
load arrhythmia;
Y = categorical(Y);
tabulate(Y);
```

Value Count Percent 1 245 54.20% 2 44 9.73% 3 15 3.32% 4 15 3.32% 5 13 2.88% 6 25 5.53% 7 3 0.66% 8 2 0.44% 9 9 1.99% 10 50 11.06% 14 4 0.88% 15 5 1.11% 16 22 4.87%

There are 16 classes, however some are not represented in the data set (for example, class 13). Most observations are classified as not having arrhythmia (class 1). The data set is highly discrete with imbalanced classes.

Combine all observations with arrhythmia (classes 2 through 15) into one class. Remove those observations with unknown arrhythmia status (class 16) from the data set.

idx = (Y ~= '16'); Y = Y(idx); X = X(idx,:); Y(Y ~= '1') = 'WithArrhythmia'; Y(Y == '1') = 'NoArrhythmia'; Y = removecats(Y);

Create a partition that evenly splits the data into training and test sets.

rng(1); % For reproducibility CVP = cvpartition(Y,'holdout',0.5); idxTrain = training(CVP); % Training-set indices idxTest = test(CVP); % Test-set indices

`CVP`

is a cross-validation partition object that specifies the training and test sets.

Create a cost matrix such that misclassifying a patient with arrhythmia into the "no arrhythmia" class is five times worse than misclassifying a patient without arrhythmia into the arrhythmia class. Classifying correctly incurs no cost. The rows indicate the true class and the columns indicate predicted class. When you conduct a cost-sensitive analysis, a good practice is to specify the order of the classes.

Cost = [0 1;5 0]; ClassNames = {'NoArrhythmia','WithArrhythmia'};

Train two boosting ensembles of 50 classification trees, one that uses AdaBoostM1 and another that uses LogitBoost. Because there are missing values in the data set, specify to use surrogate splits. Train the models using the cost matrix.

t = templateTree('Surrogate','on'); numTrees = 50; MdlAda = fitcensemble(X(idxTrain,:),Y(idxTrain),'Method','AdaBoostM1',... 'NumLearningCycles',numTrees,'Learners',t,... 'Cost',Cost,'ClassNames',ClassNames); MdlLogit = fitcensemble(X(idxTrain,:),Y(idxTrain),'Method','LogitBoost',... 'NumLearningCycles',numTrees,'Learners',t,... 'Cost',Cost,'ClassNames',ClassNames);

`MdlAda`

and `MdlLogit`

are trained `ClassificationEnsemble`

models.

Label the test-set observations using the trained models.

YhatAda = predict(MdlAda,X(idxTest,:)); YhatLogit = predict(MdlLogit,X(idxTest,:));

`YhatLinear`

and `YhatRBF`

are vectors containing the predicted class labels of the respective models.

Test whether the AdaBoostM1 ensemble (`MdlAda`

) and the LogitBoost ensemble (`MdlLogit`

) have equal predictive accuracy. Supply the cost matrix. Conduct the asymptotic, likelihood ratio, cost-sensitive test (the default when you pass in a cost matrix). Request to return *p*-values and misclassification costs.

`[h,p,e1,e2] = testcholdout(YhatAda,YhatLogit,Y(idxTest),'Cost',Cost)`

`h = `*logical*
0

p = 0.2795

e1 = 0.5581

e2 = 0.4744

`h = 0`

indicates to not reject the null hypothesis that the two models have equal predictive accuracies.

`YHat1`

— Predicted class labelscategorical array | character array | string array | logical vector | numeric vector | cell array of character vectors

Predicted class labels of the first classification model, specified as a categorical, character, or string array, logical or numeric vector, or cell array of character vectors.

If `YHat1`

is a character array, then each
element must correspond to one row of the array.

`YHat1`

, `YHat2`

, and `Y`

must
have equal lengths.

It is a best practice for `YHat1`

, `YHat2`

,
and `Y`

to share the same data type.

**Data Types: **`categorical`

| `char`

| `string`

| `logical`

| `single`

| `double`

| `cell`

`YHat2`

— Predicted class labelscategorical array | character array | string array | logical vector | numeric vector | cell array of character vectors

Predicted class labels of the second classification model, specified as a categorical, character, or string array, logical or numeric vector, or cell array of character vectors.

If `YHat2`

is a character array, then each
element must correspond to one row of the array.

`YHat1`

, `YHat2`

, and `Y`

must
have equal lengths.

It is a best practice for `YHat1`

, `YHat2`

,
and `Y`

to share the same data type.

**Data Types: **`categorical`

| `char`

| `string`

| `logical`

| `single`

| `double`

| `cell`

`Y`

— True class labelscategorical array | character array | string array | logical vector | numeric vector | cell array of character vectors

True class labels, specified as a categorical, character, or string array, logical or numeric vector, or cell array of character vectors.

If `Y`

is a character array, then each element
must correspond to one row of the array.

`YHat1`

, `YHat2`

, and `Y`

must
have equal lengths.

It is a best practice for `YHat1`

, `YHat2`

,
and `Y`

to share the same data type.

**Data Types: **`categorical`

| `char`

| `string`

| `logical`

| `single`

| `double`

| `cell`

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

```
'Alternative','greater','Test','asymptotic','Cost',[0
2;1 0]
```

specifies to test whether the first set of first
predicted class labels is more accurate than the second set, to conduct
the asymptotic McNemar test, and to penalize misclassifying observations
with the true label `ClassNames{1}`

twice as much
as for misclassifying observations with the true label `ClassNames{2}`

.`Alpha`

— Hypothesis test significance level`0.05`

(default) | scalar value in the interval (0,1)Hypothesis test significance level, specified as the comma-separated
pair consisting of `'Alpha'`

and a scalar value in
the interval (0,1).

**Example: **`'Alpha',0.1`

**Data Types: **`single`

| `double`

`Alternative`

— Alternative hypothesis to assess`'unequal'`

(default) | `'greater'`

| `'less'`

Alternative hypothesis to assess, specified as the comma-separated
pair consisting of `'Alternative'`

and one of the
values listed in the table.

Value | Alternative hypothesis |
---|---|

`'unequal'` (default) | For predicting `Y` , `YHat1` and `YHat2` have
unequal accuracies. |

`'greater'` | For predicting `Y` , `YHat1` is
more accurate than `YHat2` . |

`'less'` | For predicting `Y` , `YHat1` is
less accurate than `YHat2` . |

**Example: **`'Alternative','greater'`

`ClassNames`

— Class namescategorical array | character array | string array | logical vector | numeric vector | cell array of character vectors

Class names, specified as the comma-separated pair consisting of
`'ClassNames'`

and a categorical, character, or
string array, logical or numeric vector, or cell array of character
vectors. You must set `ClassNames`

using the data
type of `Y`

.

If `ClassNames`

is a character array, then each
element must correspond to one row of the array.

Use `ClassNames`

to:

Specify the order of any input argument dimension that corresponds to class order. For example, use

`ClassNames`

to specify the order of the dimensions of`Cost`

.Select a subset of classes for testing. For example, suppose that the set of all distinct class names in

`Y`

is`{'a','b','c'}`

. To train and test models using observations from classes`'a'`

and`'c'`

only, specify`'ClassNames',{'a','c'}`

.

The default is the set of all distinct class names in
`Y`

.

**Example: **`'ClassNames',{'b','g'}`

**Data Types: **`single`

| `double`

| `logical`

| `char`

| `string`

| `cell`

| `categorical`

`Cost`

— Misclassification costsquare matrix | structure array

Misclassification cost, specified as the comma-separated pair consisting of
`'Cost'`

and a square matrix or structure array.

If you specify the square matrix

`Cost`

, then`Cost(i,j)`

is the cost of classifying a point into class`j`

if its true class is`i`

. That is, the rows correspond to the true class and the columns correspond to the predicted class. To specify the class order for the corresponding rows and columns of`Cost`

, additionally specify the`ClassNames`

name-value pair argument.If you specify the structure

`S`

, then`S`

must have two fields:`S.ClassNames`

, which contains the class names as a variable of the same data type as`Y`

. You can use this field to specify the order of the classes.`S.ClassificationCosts`

, which contains the cost matrix, with rows and columns ordered as in`S.ClassNames`

.

If you specify `Cost`

, then `testcholdout`

cannot conduct
one-sided, exact, or mid-*p* tests. You must also specify
`'Alternative','unequal','Test','asymptotic'`

. For cost-sensitive
testing options, see the `CostTest`

name-value pair argument.

A best practice is to supply the same cost matrix used to train the classification models.

The default is `Cost(i,j) = 1`

if ```
i
~= j
```

, and `Cost(i,j) = 0`

if ```
i
= j
```

.

**Example: **`'Cost',[0 1 2 ; 1 0 2; 2 2 0]`

**Data Types: **`single`

| `double`

| `struct`

`CostTest`

— Cost-sensitive test type`'likelihood'`

(default) | `'chisquare'`

Cost-sensitive test type, specified as the comma-separated pair
consisting of `'CostTest'`

and `'chisquare'`

or `'likelihood'`

.
Unless you specify a cost matrix using the `Cost`

name-value
pair argument, `testcholdout`

ignores `CostTest`

.

This table summarizes the available options for cost-sensitive testing.

Value | Asymptotic test type | Requirements |
---|---|---|

`'chisquare'` | Chi-square test | Optimization Toolbox license to implement `quadprog` (Optimization Toolbox) |

`'likelihood'` | Likelihood ratio test | None |

For more details, see Cost-Sensitive Testing.

**Example: **`'CostTest','chisquare'`

`Test`

— Test to conduct`'asymptotic'`

| `'exact'`

| `'midp'`

Test to conduct, specified as the comma-separated pair consisting
of `'Test'`

and `'asymptotic'`

, `'exact'`

,
and `'midp'`

. This table summarizes the available
options for cost-insensitive testing.

Value | Description |
---|---|

`'asymptotic'` | Asymptotic McNemar test |

`'exact'` | Exact-conditional McNemar test |

`'midp'` (default) | Mid-p-value McNemar test |

For more details, see McNemar Tests.

For cost-sensitive testing, `Test`

must be `'asymptotic'`

.
When you specify the `Cost`

name-value pair argument,
and choose a cost-sensitive test using the `CostTest`

name-value
pair argument, `'asymptotic'`

is the default.

**Example: **`'Test','asymptotic'`

**Note**

`NaN`

s, `<undefined>`

values, empty character vectors
(`''`

), empty strings (`""`

), and
`<missing>`

values indicate missing data values.
`testcholdout`

:

Treats missing values in

`YHat1`

and`YHat2`

as misclassified observations.Removes missing values in

`Y`

and the corresponding values of`YHat1`

and`YHat2`

`h`

— Hypothesis test result`1`

| `0`

Hypothesis test result, returned as a logical value.

`h = 1`

indicates the rejection of the null
hypothesis at the `Alpha`

significance level.

`h = 0`

indicates failure to reject the null hypothesis at the
`Alpha`

significance level.

**Data Types: **`logical`

`p`

— scalar in the interval [0,1]

*p*-value of the test, returned as a scalar
in the interval [0,1]. `p`

is the probability that
a random test statistic is at least as extreme as the observed test
statistic, given that the null hypothesis is true.

`testcholdout`

estimates `p`

using
the distribution of the test statistic, which varies with the type
of test. For details on test statistics derived from the available
variants of the McNemar test, see McNemar Tests. For details on test statistics derived from
cost-sensitive tests, see Cost-Sensitive Testing.

`e1`

— Classification lossscalar

Classification
loss that summarizes the accuracy of the first set of class
labels (`YHat1`

) predicting the true class labels
(`Y`

), returned as a scalar.

For cost-insensitive testing, `e1`

is the misclassification
rate. That is, `e1`

is the proportion of misclassified
observations, which is a scalar in the interval [0,1].

For cost-sensitive testing, `e1`

is the misclassification
cost. That is, `e1`

is the weighted average of the
misclassification costs, in which the weights are the respective estimated
proportions of misclassified observations.

`e2`

— Classification lossscalar

Classification
loss that summarizes the accuracy of the second set of class
labels (`YHat2`

) predicting the true class labels
(`Y`

), returned as a scalar.

For cost-insensitive testing, `e2`

is the misclassification
rate. That is, `e2`

is the proportion of misclassified
observations, which is a scalar in the interval [0,1].

For cost-sensitive testing, `e2`

is the misclassification
cost. That is, `e2`

is the weighted average of the
costs of misclassification, in which the weights are the respective
estimated proportions of misclassified observations.

Conduct *cost-sensitive testing* when the cost of
misclassification is imbalanced. By conducting a cost-sensitive analysis, you can
account for the cost imbalance when you train the classification models and when you
statistically compare them.

If the cost of misclassification is imbalanced, then the misclassification rate tends to be a poorly performing classification loss. Use misclassification cost instead to compare classification models.

Misclassification costs are often imbalanced in applications. For example, consider classifying subjects based on a set of predictors into two categories: healthy and sick. Misclassifying a sick subject as healthy poses a danger to the subject's life. However, misclassifying a healthy subject as sick typically causes some inconvenience, but does not pose significant danger. In this situation, you assign misclassification costs such that misclassifying a sick subject as healthy is more costly than misclassifying a healthy subject as sick.

The definitions that follow summarize the cost-sensitive tests. In the definitions:

*n*and $${\widehat{\pi}}_{ijk}$$ are the number and estimated proportion of test-sample observations with the following characteristics._{ijk}*k*is the true class,*i*is the label assigned by the first classification model, and*j*is the label assigned by the second classification model. The unknown true value of $${\widehat{\pi}}_{ijk}$$ is*π*. The test-set sample size is $$\sum _{i,j,k}{n}_{ijk}={n}_{test}.$$ Additionally, $$\sum _{i,j,k}{\pi}_{ijk}=\sum _{i,j,k}{\widehat{\pi}}_{ijk}=1.$$_{ijk}*c*is the relative cost of assigning label_{ij}*j*to an observation with true class*i*.*c*= 0,_{ii}*c*≥ 0, and, for at least one (_{ij}*i*,*j*) pair,*c*> 0._{ij}All subscripts take on integer values from 1 through

*K*, which is the number of classes.The expected difference in the misclassification costs of the two classification models is

$$\delta ={\displaystyle \sum _{i=1}^{K}{\displaystyle \sum _{j=1}^{K}{\displaystyle \sum}_{k=1}^{K}\left({c}_{ki}-{c}_{kj}\right){\pi}_{ijk}}}.$$

The hypothesis test is

$$\begin{array}{c}{H}_{0}:\delta =0\\ {H}_{1}:\delta \ne 0\end{array}.$$

The available cost-sensitive tests are appropriate for two-tailed testing.

Available asymptotic tests that address imbalanced costs are a
*chi-square test* and a *likelihood ratio
test*.

Chi-square test — The chi-square test statistic is based on the Pearson and Neyman chi-square test statistics, but with a Laplace correction factor to account for any

*n*= 0. The test statistic is_{ijk}$${t}_{{\chi}^{2}}^{\ast}=\sum _{i\ne j}\sum _{k}\frac{{\left({n}_{ijk}+1-\left({n}_{test}+{K}^{3}\right){\widehat{\pi}}_{ijk}^{(1)}\right)}^{2}}{{n}_{ijk}+1}.$$

If $$1-{F}_{{\chi}^{2}}\left({t}_{{\chi}^{2}}^{\ast};1\right)<\alpha $$, then reject

*H*_{0}.$${\widehat{\pi}}_{ijk}^{(1)}$$ are estimated by minimizing $${t}_{{\chi}^{2}}^{\ast}$$ under the constraint that

*δ*= 0.$${F}_{{\chi}^{2}}(x;1)$$ is the

*χ*^{2}cdf with one degree of freedom evaluated at*x*.

Likelihood ratio test — The likelihood ratio test is based on

*N*, which are binomial random variables with sample size_{ijk}*n*and success probability_{test}*π*. The random variables represent the random number of observations with: true class_{ijk}*k*, label*i*assigned by the first classification model, and label*j*assigned by the second classification model. Jointly, the distribution of the random variables is multinomial.The test statistic is

$${t}_{LRT}^{\ast}=2\mathrm{log}\left[\frac{P\left(\underset{i,j,k}{\cap}{N}_{ijk}={n}_{ijk};{n}_{test},{\widehat{\pi}}_{ijk}={\widehat{\pi}}_{ijk}^{(2)}\right)}{P\left(\underset{i,j,k}{\cap}{N}_{ijk}={n}_{ijk};{n}_{test},{\widehat{\pi}}_{ijk}={\widehat{\pi}}_{ijk}^{(3)}\right)}\right].$$

If $$1-{F}_{{\chi}^{2}}\left({t}_{LRT}^{\ast};1\right)<\alpha ,$$ then reject

*H*_{0}.$${\widehat{\pi}}_{ijk}^{(2)}=\frac{{n}_{ijk}}{{n}_{test}}$$ is the unrestricted MLE of

*π*._{ijk}$${\widehat{\pi}}_{ijk}^{(3)}=\frac{{n}_{ijk}}{{n}_{test}+\lambda ({c}_{ki}-{c}_{kj})}$$ is the MLE under the null hypothesis that

*δ*= 0.*λ*is the solution to$$\sum _{i,j,k}\frac{{n}_{ijk}({c}_{ki}-{c}_{kj})}{{n}_{test}+\lambda ({c}_{ki}-{c}_{kj})}=0.$$

$${F}_{{\chi}^{2}}(x;1)$$ is the

*χ*^{2}cdf with one degree of freedom evaluated at*x*.

*McNemar Tests* are hypothesis
tests that compare two population proportions while addressing the
issues resulting from two dependent, matched-pair samples.

One way to compare the predictive accuracies of two classification models is:

Partition the data into training and test sets.

Train both classification models using the training set.

Predict class labels using the test set.

Summarize the results in a two-by-two table similar to this figure.

*n*are the number of concordant pairs, that is, the number of observations that both models classify the same way (correctly or incorrectly)._{ii}*n*,_{ij}*i*≠*j*, are the number of discordant pairs, that is, the number of observations that models classify differently (correctly or incorrectly).

The misclassification rates for Models 1 and 2 are $${\widehat{\pi}}_{2\u2022}={n}_{2\u2022}/n$$ and $${\widehat{\pi}}_{\u20222}={n}_{\u20222}/n$$, respectively. A two-sided test for comparing the accuracy of the two models is

$$\begin{array}{c}{H}_{0}:{\pi}_{\u20222}={\pi}_{2\u2022}\\ {H}_{1}:{\pi}_{\u20222}\ne {\pi}_{2\u2022}\end{array}.$$

The null hypothesis suggests that the population exhibits marginal
homogeneity, which reduces the null hypothesis to $${H}_{0}:{\pi}_{12}={\pi}_{21}.$$ Also, under the null hypothesis,
*N*_{12} ~
Binomial(*n*_{12} +
*n*_{21},0.5) [1].

These facts are the basis for the available McNemar test variants: the
*asymptotic*, *exact-conditional*, and
*mid-p-value* McNemar tests. The definitions that follow summarize
the available variants.

Asymptotic — The asymptotic McNemar test statistics and rejection regions (for significance level

*α*) are:For one-sided tests, the test statistic is

$${t}_{a1}^{\ast}=\frac{{n}_{12}-{n}_{21}}{\sqrt{{n}_{12}+{n}_{21}}}.$$

If $$1-\Phi \left(\left|{t}_{1}^{\ast}\right|\right)<\alpha ,$$ where

*Φ*is the standard Gaussian cdf, then reject*H*_{0}.For two-sided tests, the test statistic is

$${t}_{a2}^{\ast}=\frac{{\left({n}_{12}-{n}_{21}\right)}^{2}}{{n}_{12}+{n}_{21}}.$$

If $$1-{F}_{{\chi}^{2}}\left({t}_{2}^{\ast};m\right)<\alpha $$, where $${F}_{{\chi}^{2}}(x;m)$$ is the

*χ*_{m}^{2}cdf evaluated at*x*, then reject*H*_{0}.

The asymptotic test requires large-sample theory, specifically, the Gaussian approximation to the binomial distribution.

The total number of discordant pairs, $${n}_{d}={n}_{12}+{n}_{21}$$, must be greater than 10 ([1], Ch. 10.1.4).

In general, asymptotic tests do not guarantee nominal coverage. The observed probability of falsely rejecting the null hypothesis can exceed

*α*, as suggested in simulation studies in [18]. However, the asymptotic McNemar test performs well in terms of statistical power.

Exact-Conditional — The exact-conditional McNemar test statistics and rejection regions (for significance level

*α*) are ([36], [38]):For one-sided tests, the test statistic is

$${t}_{1}^{\ast}={n}_{12}.$$

If $${F}_{\text{Bin}}\left({t}_{1}^{\ast};{n}_{d},0.5\right)<\alpha $$, where $${F}_{\text{Bin}}\left(x;n,p\right)$$ is the binomial cdf with sample size

*n*and success probability*p*evaluated at*x*, then reject*H*_{0}.For two-sided tests, the test statistic is

$${t}_{2}^{\ast}=\mathrm{min}({n}_{12},{n}_{21}).$$

If $${F}_{\text{Bin}}\left({t}_{2}^{\ast};{n}_{d},0.5\right)<\alpha /2$$, then reject

*H*_{0}.

The exact-conditional test always attains nominal coverage. Simulation studies in [18] suggest that the test is conservative, and then show that the test lacks statistical power compared to other variants. For small or highly discrete test samples, consider using the mid-

*p*-value test ([1], Ch. 3.6.3).Mid-

*p*-value test — The mid-*p*-value McNemar test statistics and rejection regions (for significance level*α*) are ([32]):For one-sided tests, the test statistic is

$${t}_{1}^{\ast}={n}_{12}.$$

If $${F}_{\text{Bin}}\left({t}_{1}^{\ast}-1;{n}_{12}+{n}_{21},0.5\right)+0.5{f}_{\text{Bin}}\left({t}_{1}^{\ast};{n}_{12}+{n}_{21},0.5\right)<\alpha $$, where $${F}_{\text{Bin}}\left(x;n,p\right)$$ and $${f}_{\text{Bin}}\left(x;n,p\right)$$ are the binomial cdf and pdf, respectively, with sample size

*n*and success probability*p*evaluated at*x*, then reject*H*_{0}.For two-sided tests, the test statistic is

$${t}_{2}^{\ast}=\mathrm{min}({n}_{12},{n}_{21}).$$

If $${F}_{\text{Bin}}\left({t}_{2}^{\ast}-1;{n}_{12}+{n}_{21}-1,0.5\right)+0.5{f}_{\text{Bin}}\left({t}_{2}^{\ast};{n}_{12}+{n}_{21},0.5\right)<\alpha /2$$, then reject

*H*_{0}.

The mid-

*p*-value test addresses the over-conservative behavior of the exact-conditional test. The simulation studies in [18] demonstrate that this test attains nominal coverage, and has good statistical power.

*Classification losses* indicate the accuracy of a
classification model or set of predicted labels. Two classification losses are the
misclassification rate and cost.

`testcholdout`

returns the classification losses (see
`e1`

and `e2`

) under the alternative hypothesis
(that is, the unrestricted classification losses).
*n _{ijk}* is the number of test-sample observations
with: true class

The

*misclassification rate*, or classification error, is a scalar in the interval [0,1] representing the proportion of misclassified observations. That is, the misclassification rate for the first classification model is$${e}_{1}={\displaystyle \sum _{j=1}^{K}{\displaystyle \sum _{k=1}^{K}{\displaystyle \sum _{i\ne k}^{}{\widehat{\pi}}_{ijk}}}}.$$

For the misclassification rate of the second classification model (

*e*_{2}), switch the indices*i*and*j*in the formula.Classification accuracy decreases as the misclassification rate increases to 1.

The

*misclassification cost*is a nonnegative scalar that is a measure of classification quality relative to the values of the specified cost matrix. Its interpretation depends on the specified costs of misclassification. The misclassification cost is the weighted average of the costs of misclassification (specified in a cost matrix,*C*) in which the weights are the respective estimated proportions of misclassified observations. The misclassification cost for the first classification model is$${e}_{1}={\displaystyle \sum _{j=1}^{K}{\displaystyle \sum _{k=1}^{K}{\displaystyle \sum _{i\ne k}^{}{\widehat{\pi}}_{ijk}{c}_{ki},}}}$$

where

*c*is the cost of classifying an observation into class_{kj}*j*if its true class is*k*. For the misclassification cost of the second classification model (*e*_{2}), switch the indices*i*and*j*in the formula.In general, for a fixed cost matrix, classification accuracy decreases as the misclassification cost increases.

It is a good practice to obtain predicted class labels by passing any trained classification model and new predictor data to the

`predict`

method. For example, for predicted labels from an SVM model, see`predict`

.Cost-sensitive tests perform numerical optimization, which requires additional computational resources. The likelihood ratio test conducts numerical optimization indirectly by finding the root of a Lagrange multiplier in an interval. For some data sets, if the root lies close to the boundaries of the interval, then the method can fail. Therefore, if you have an Optimization Toolbox license, consider conducting the cost-sensitive chi-square test instead. For more details, see

`CostTest`

and Cost-Sensitive Testing.

[1] Agresti, A. *Categorical Data Analysis*,
2nd Ed. John Wiley & Sons, Inc.: Hoboken, NJ, 2002.

[2] Fagerlan, M.W., S. Lydersen, and P. Laake. “The McNemar Test for Binary
Matched-Pairs Data: Mid-p and Asymptotic Are Better Than Exact Conditional.”
*BMC Medical Research Methodology*. Vol. 13, 2013, pp.
1–8.

[3] Lancaster, H.O. “Significance Tests in Discrete
Distributions.” *JASA*, Vol. 56, Number
294, 1961, pp. 223–234.

[4] McNemar, Q. “Note on the Sampling Error of the
Difference Between Correlated Proportions or Percentages.” *Psychometrika*,
Vol. 12, Number 2, 1947, pp. 153–157.

[5] Mosteller, F. “Some Statistical Problems in Measuring
the Subjective Response to Drugs.” *Biometrics*,
Vol. 8, Number 3, 1952, pp. 220–226.

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