McNemar's test overview

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McNemar's test	Logistic regression	$z$ test for the difference between two proportions
Independent variable	Independent variables	Independent/grouping variable
2 paired groups	One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables	One categorical with 2 independent groups
Dependent variable	Dependent variable	Dependent variable
One categorical with 2 independent groups	One categorical with 2 independent groups	One categorical with 2 independent groups
Null hypothesis	Null hypothesis	Null hypothesis
Let's say that the scores on the dependent variable are scored 0 and 1. Then for each pair of scores, the data allow four options: First score of pair is 0, second score of pair is 0 First score of pair is 0, second score of pair is 1 (switched) First score of pair is 1, second score of pair is 0 (switched) First score of pair is 1, second score of pair is 1 The null hypothesis H₀ is that for each pair of scores, P(first score of pair is 0 while second score of pair is 1) = P(first score of pair is 1 while second score of pair is 0). That is, the probability that a pair of scores switches from 0 to 1 is the same as the probability that a pair of scores switches from 1 to 0. Other formulations of the null hypothesis are: H₀: $\pi_1 = \pi_2$, where $\pi_1$ is the population proportion of ones for the first paired group and $\pi_2$ is the population proportion of ones for the second paired group H₀: for each pair of scores, P(first score of pair is 1) = P(second score of pair is 1)	Model chi-squared test for the complete regression model: H₀: $\beta_1 = \beta_2 = \ldots = \beta_K = 0$ Wald test for individual regression coefficient $\beta_k$: H₀: $\beta_k = 0$ or in terms of odds ratio: H₀: $e^{\beta_k} = 1$ Likelihood ratio chi-squared test for individual regression coefficient $\beta_k$: H₀: $\beta_k = 0$ or in terms of odds ratio: H₀: $e^{\beta_k} = 1$ in the regression equation $ \ln \big(\frac{\pi_{y = 1}}{1 - \pi_{y = 1}} \big) = \beta_0 + \beta_1 \times x_1 + \beta_2 \times x_2 + \ldots + \beta_K \times x_K $. Here $ x_i$ represents independent variable $ i$, $\beta_i$ is the regression weight for independent variable $ x_i$, and $\pi_{y = 1}$ represents the true probability that the dependent variable $ y = 1$ (or equivalently, the proportion of $ y = 1$ in the population) given the scores on the independent variables.	H₀: $\pi_1 = \pi_2$ Here $\pi_1$ is the population proportion of 'successes' for group 1, and $\pi_2$ is the population proportion of 'successes' for group 2.
Alternative hypothesis	Alternative hypothesis	Alternative hypothesis
The alternative hypothesis H₁ is that for each pair of scores, P(first score of pair is 0 while second score of pair is 1) $\neq$ P(first score of pair is 1 while second score of pair is 0). That is, the probability that a pair of scores switches from 0 to 1 is not the same as the probability that a pair of scores switches from 1 to 0. Other formulations of the alternative hypothesis are: H₁: $\pi_1 \neq \pi_2$ H₁: for each pair of scores, P(first score of pair is 1) $\neq$ P(second score of pair is 1)	Model chi-squared test for the complete regression model: H₁: not all population regression coefficients are 0 Wald test for individual regression coefficient $\beta_k$: H₁: $\beta_k \neq 0$ or in terms of odds ratio: H₁: $e^{\beta_k} \neq 1$ If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$ (see 'Test statistic'), also one sided alternatives can be tested: H₁ right sided: $\beta_k > 0$ H₁ left sided: $\beta_k < 0$ Likelihood ratio chi-squared test for individual regression coefficient $\beta_k$: H₁: $\beta_k \neq 0$ or in terms of odds ratio: H₁: $e^{\beta_k} \neq 1$	H₁ two sided: $\pi_1 \neq \pi_2$ H₁ right sided: $\pi_1 > \pi_2$ H₁ left sided: $\pi_1 < \pi_2$
Assumptions	Assumptions	Assumptions
Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another	In the population, the relationship between the independent variables and the log odds $\ln (\frac{\pi_{y=1}}{1 - \pi_{y=1}})$ is linear The residuals are independent of one another Often ignored additional assumption: Variables are measured without error Also pay attention to: Multicollinearity Outliers	Sample size is large enough for $z$ to be approximately normally distributed. Rule of thumb: Significance test: number of successes and number of failures are each 5 or more in both sample groups Regular (large sample) 90%, 95%, or 99% confidence interval: number of successes and number of failures are each 10 or more in both sample groups Plus four 90%, 95%, or 99% confidence interval: sample sizes of both groups are 5 or more Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations are independent of one another
Test statistic	Test statistic	Test statistic
$X^2 = \dfrac{(b - c)^2}{b + c}$ Here $b$ is the number of pairs in the sample for which the first score is 0 while the second score is 1, and $c$ is the number of pairs in the sample for which the first score is 1 while the second score is 0.	Model chi-squared test for the complete regression model: $X^2 = D_{null} - D_K = \mbox{null deviance} - \mbox{model deviance} $ $D_{null}$, the null deviance, is conceptually similar to the total variance of the dependent variable in OLS regression analysis. $D_K$, the model deviance, is conceptually similar to the residual variance in OLS regression analysis. Wald test for individual $\beta_k$: The wald statistic can be defined in two ways: Wald $ = \dfrac{b_k^2}{SE^2_{b_k}}$ Wald $ = \dfrac{b_k}{SE_{b_k}}$ SPSS uses the first definition. Likelihood ratio chi-squared test for individual $\beta_k$: $X^2 = D_{K-1} - D_K$ $D_{K-1}$ is the model deviance, where independent variable $k$ is excluded from the model. $D_{K}$ is the model deviance, where independent variable $k$ is included in the model.	$z = \dfrac{p_1 - p_2}{\sqrt{p(1 - p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$ Here $p_1$ is the sample proportion of successes in group 1: $\dfrac{X_1}{n_1}$, $p_2$ is the sample proportion of successes in group 2: $\dfrac{X_2}{n_2}$, $p$ is the total proportion of successes in the sample: $\dfrac{X_1 + X_2}{n_1 + n_2}$, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. Note: we could just as well compute $p_2 - p_1$ in the numerator, but then the left sided alternative becomes $\pi_2 < \pi_1$, and the right sided alternative becomes $\pi_2 > \pi_1.$
Sampling distribution of $X^2$ if H₀ were true	Sampling distribution of $X^2$ and of the Wald statistic if H₀ were true	Sampling distribution of $z$ if H₀ were true
If $b + c$ is large enough (say, > 20), approximately the chi-squared distribution with 1 degree of freedom. If $b + c$ is small, the Binomial($n$, $P$) distribution should be used, with $n = b + c$ and $P = 0.5$. In that case the test statistic becomes equal to $b$.	Sampling distribution of $X^2$, as computed in the model chi-squared test for the complete model: chi-squared distribution with $K$ (number of independent variables) degrees of freedom Sampling distribution of the Wald statistic: If defined as Wald $ = \dfrac{b_k^2}{SE^2_{b_k}}$: approximately the chi-squared distribution with 1 degree of freedom If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$: approximately the standard normal distribution Sampling distribution of $X^2$, as computed in the likelihood ratio chi-squared test for individual $\beta_k$: chi-squared distribution with 1 degree of freedom	Approximately the standard normal distribution
Significant?	Significant?	Significant?
For test statistic $X^2$: Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$ If $b + c$ is small, the table for the binomial distribution should be used, with as test statistic $b$: Check if $b$ observed in sample is in the rejection region or Find two sided $p$ value corresponding to observed $b$ and check if it is equal to or smaller than $\alpha$	For the model chi-squared test for the complete regression model and likelihood ratio chi-squared test for individual $\beta_k$: Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$ For the Wald test: If defined as Wald $ = \dfrac{b_k^2}{SE^2_{b_k}}$: same procedure as for the chi-squared tests. Wald can be interpret as $X^2$ If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$: same procedure as for any $z$ test. Wald can be interpreted as $z$.	Two sided: Check if $z$ observed in sample is at least as extreme as critical value $z^$ or Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $z$ observed in sample is equal to or larger than critical value $z^$ or Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
n.a.	Wald-type approximate $C\%$ confidence interval for $\beta_k$	Approximate $C\%$ confidence interval for $\pi_1 - \pi_2$
-	$b_k \pm z^* \times SE_{b_k}$ where the critical value $z^$ is the value under the normal curve with the area $C / 100$ between $-z^$ and $z^$ (e.g. $z^$ = 1.96 for a 95% confidence interval).	Regular (large sample): $(p_1 - p_2) \pm z^* \times \sqrt{\dfrac{p_1(1 - p_1)}{n_1} + \dfrac{p_2(1 - p_2)}{n_2}}$ where the critical value $z^$ is the value under the normal curve with the area $C / 100$ between $-z^$ and $z^$ (e.g. $z^$ = 1.96 for a 95% confidence interval) With plus four method: $(p_{1.plus} - p_{2.plus}) \pm z^* \times \sqrt{\dfrac{p_{1.plus}(1 - p_{1.plus})}{n_1 + 2} + \dfrac{p_{2.plus}(1 - p_{2.plus})}{n_2 + 2}}$ where $p_{1.plus} = \dfrac{X_1 + 1}{n_1 + 2}$, $p_{2.plus} = \dfrac{X_2 + 1}{n_2 + 2}$, and the critical value $z^$ is the value under the normal curve with the area $C / 100$ between $-z^$ and $z^$ (e.g. $z^$ = 1.96 for a 95% confidence interval)
n.a.	Goodness of fit measure $R^2_L$	n.a.
-	$R^2_L = \dfrac{D_{null} - D_K}{D_{null}}$ There are several other goodness of fit measures in logistic regression. In logistic regression, there is no single agreed upon measure of goodness of fit.	-
Equivalent to	n.a.	Equivalent to
Stuart-Maxwell test, with a categorical dependent variable consisting of two independent groups Cochran's Q test, with two related groups Two sided sign test: $b = W$, and $X^2 = z^2$	-	When testing two sided: chi-squared test for the relationship between two categorical variables, where both categorical variables have 2 levels.
Example context	Example context	Example context
Does a tv documentary about spiders change whether people are afraid (yes/no) of spiders?	Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes?	Is the proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic.
SPSS	SPSS	SPSS
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples... Put the two paired variables in the boxes below Variable 1 and Variable 2 Under Test Type, select the McNemar test	Analyze > Regression > Binary Logistic... Put your dependent variable in the box below Dependent and your independent (predictor) variables in the box below Covariate(s)	SPSS does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to: Analyze > Descriptive Statistics > Crosstabs... Put your independent (grouping) variable in the box below Row(s), and your dependent variable in the box below Column(s) Click the Statistics... button, and click on the square in front of Chi-square Continue and click OK
Jamovi	Jamovi	Jamovi
Frequencies > Paired Samples - McNemar test Put one of the two paired variables in the box below Rows and the other paired variable in the box below Columns	Regression > 2 Outcomes - Binomial Put your dependent variable in the box below Dependent Variable and your independent variables of interval/ratio level in the box below Covariates If you also have code (dummy) variables as independent variables, you can put these in the box below Covariates as well Instead of transforming your categorical independent variable(s) into code variables, you can also put the untransformed categorical independent variables in the box below Factors. Jamovi will then make the code variables for you 'behind the scenes'	Jamovi does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to: Frequencies > Independent Samples - $\chi^2$ test of association Put your independent (grouping) variable in the box below Rows, and your dependent variable in the box below Columns
Practice questions	Practice questions	Practice questions

McNemar's test - overview