This page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the righthand column. To practice with a specific method click the button at the bottom row of the table
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
Null hypothesis 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 :
$\pi_1 = \pi_2$, where $\pi_1$ is the population proportion of ones in the first paired group and $\pi_2$ is the population proportion of ones in the second paired group
For each pair of scores, P(first score of pair is 1) = P(second score of pair is 1)
Alternative hypothesis
Alternative hypothesis
Model chisquared test for the complete regression model:
not all population regression coefficients are 0
Wald test for individual $\beta_k$:
$\beta_k \neq 0$
or in terms of odds ratio:
$e^{\beta_k} \neq 1$
If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$ (see 'Test statistic'), also one sided alternatives can be tested:
right sided: $\beta_k > 0$
left sided: $\beta_k < 0$
Likelihood ratio chisquared test for individual $\beta_k$:
$\beta_k \neq 0$
or in terms of odds ratio:
$e^{\beta_k} \neq 1$
Alternative hypothesis 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 that, for each pair of scores:
$\pi_1 \neq \pi_2$
For each pair of scores, P(first score of pair is 1) $\neq$ P(second score of pair is 1)
Assumptions
Assumptions
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 of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Test statistic
Test statistic
Model chisquared 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 chisquared test for individual $\beta_k$:
$X^2 = D_{K1}  D_K$
$D_{K1}$ 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.
$X^2 = \dfrac{(b  c)^2}{b + c}$
$b$ 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
Sampling distribution of $X^2$ and of the Wald statistic if H0 were true
Sampling distribution of $X^2$ if H0 were true
Sampling distribution of $X^2$, as computed in the model chisquared test for the complete model:
chisquared 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 a chisquared distribution with 1 degree of freedom
If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$: approximately a standard normal distribution
Sampling distribution of $X^2$, as computed in the likelihood ratio chisquared test for individual $\beta_k$:
chisquared distribution with 1 degree of freedom
If $b + c$ is large enough (say, > 20), approximately a chisquared 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 $b$.
Significant?
Significant?
For the model chisquared test for the complete regression model and likelihood ratio chisquared test for individual $\beta_k$:
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 $\alpha$
Waldtype approximate $C\%$ confidence interval for $\beta_k$
n.a.
$b_k \pm z^* \times SE_{b_k}$
where $z^*$ 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)

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.

n.a.
Equivalent to

Marginal homogeneity test, with a categorical dependent variable consisting of two independent groups
Put the two paired variables in the boxes below Variable 1 and Variable 2
Under Test Type, select the McNemar test
Jamovi
Jamovi
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'
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