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
Two categorical, the first with $I$ independent groups and the second with $J$ independent groups ($I \geqslant 2$, $J \geqslant 2$)
Dependent variable
Dependent variable
One of ordinal level
One quantitative of interval or ratio level
Null hypothesis
Null hypothesis
H_{0}: P(first score of a pair exceeds second score of a pair) = P(second score of a pair exceeds first score of a pair)
If the dependent variable is measured on a continuous scale, this can also be formulated as:
H_{0}: the population median of the difference scores is equal to zero
A difference score is the difference between the first score of a pair and the second score of a pair.
ANOVA $F$ tests:
H_{0} for main and interaction effects together (model): no main effects and interaction effect
H_{0} for independent variable A: no main effect for A
H_{0} for independent variable B: no main effect for B
H_{0} for the interaction term: no interaction effect between A and B
Like in one way ANOVA, we can also perform $t$ tests for specific contrasts and multiple comparisons. This is more advanced stuff.
Alternative hypothesis
Alternative hypothesis
H_{1} two sided: P(first score of a pair exceeds second score of a pair) $\neq$ P(second score of a pair exceeds first score of a pair)
H_{1} right sided: P(first score of a pair exceeds second score of a pair) > P(second score of a pair exceeds first score of a pair)
H_{1} left sided: P(first score of a pair exceeds second score of a pair) < P(second score of a pair exceeds first score of a pair)
If the dependent variable is measured on a continuous scale, this can also be formulated as:
H_{1} two sided: the population median of the difference scores is different from zero
H_{1} right sided: the population median of the difference scores is larger than zero
H_{1} left sided: the population median of the difference scores is smaller than zero
ANOVA $F$ tests:
H_{1} for main and interaction effects together (model): there is a main effect for A, and/or for B, and/or an interaction effect
H_{1} for independent variable A: there is a main effect for A
H_{1} for independent variable B: there is a main effect for B
H_{1} for the interaction term: there is an interaction effect between A and B
Assumptions
Assumptions
Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Within each of the $I \times J$ populations, the scores on the dependent variable are normally distributed
The standard deviation of the scores on the dependent variable is the same in each of the $I \times J$ populations
For each of the $I \times J$ groups, the sample is an independent and simple random sample from the population defined by that group. That is, within and between groups, observations are independent of one another
Equal sample sizes for each group make the interpretation of the ANOVA output easier (unequal sample sizes result in overlap in the sum of squares; this is advanced stuff)
Test statistic
Test statistic
$W = $ number of difference scores that is larger than 0
For main and interaction effects together (model):
Note: mean square error is also known as mean square residual or mean square within.
n.a.
Pooled standard deviation

$
\begin{aligned}
s_p &= \sqrt{\dfrac{\sum\nolimits_{subjects} (\mbox{subject's score}  \mbox{its group mean})^2}{N  (I \times J)}}\\
&= \sqrt{\dfrac{\mbox{sum of squares error}}{\mbox{degrees of freedom error}}}\\
&= \sqrt{\mbox{mean square error}}
\end{aligned}
$
Sampling distribution of $W$ if H_{0} were true
Sampling distribution of $F$ if H_{0} were true
The exact distribution of $W$ under the null hypothesis is the Binomial($n$, $P$) distribution, with $n =$ number of positive differences $+$ number of negative differences, and $P = 0.5$.
If $n$ is large, $W$ is approximately normally distributed under the null hypothesis, with mean $nP = n \times 0.5$ and standard deviation $\sqrt{nP(1P)} = \sqrt{n \times 0.5(1  0.5)}$. Hence, if $n$ is large, the standardized test statistic
$$z = \frac{W  n \times 0.5}{\sqrt{n \times 0.5(1  0.5)}}$$
follows approximately the standard normal distribution if the null hypothesis were true.
For main and interaction effects together (model):
$F$ distribution with $(I  1) + (J  1) + (I  1) \times (J  1)$ (df model, numerator) and $N  (I \times J)$ (df error, denominator) degrees of freedom
For independent variable A:
$F$ distribution with $I  1$ (df A, numerator) and $N  (I \times J)$ (df error, denominator) degrees of freedom
For independent variable B:
$F$ distribution with $J  1$ (df B, numerator) and $N  (I \times J)$ (df error, denominator) degrees of freedom
For the interaction term:
$F$ distribution with $(I  1) \times (J  1)$ (df interaction, numerator) and $N  (I \times J)$ (df error, denominator) degrees of freedom
Here $N$ is the total sample size.
Significant?
Significant?
If $n$ is small, the table for the binomial distribution should be used:
Two sided:
Check if $W$ observed in sample is in the rejection region or
Find two sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$
Right sided:
Check if $W$ observed in sample is in the rejection region or
Find right sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$
Left sided:
Check if $W$ observed in sample is in the rejection region or
Find left sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$
If $n$ is large, the table for standard normal probabilities can be used:
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 $\alpha$
Check if $F$ observed in sample is equal to or larger than critical value $F^*$ or
Find $p$ value corresponding to observed $F$ and check if it is equal to or smaller than $\alpha$
n.a.
Effect size

Proportion variance explained $R^2$:
Proportion variance of the dependent variable $y$ explained by the independent variables and the interaction effect together:
$$
\begin{align}
R^2 &= \dfrac{\mbox{sum of squares model}}{\mbox{sum of squares total}}
\end{align}
$$
$R^2$ is the proportion variance explained in the sample. It is a positively biased estimate of the proportion variance explained in the population.
Proportion variance explained $\eta^2$:
Proportion variance of the dependent variable $y$ explained by an independent variable or interaction effect:
$$
\begin{align}
\eta^2_A &= \dfrac{\mbox{sum of squares A}}{\mbox{sum of squares total}}\\
\\
\eta^2_B &= \dfrac{\mbox{sum of squares B}}{\mbox{sum of squares total}}\\
\\
\eta^2_{int} &= \dfrac{\mbox{sum of squares int}}{\mbox{sum of squares total}}
\end{align}
$$
$\eta^2$ is the proportion variance explained in the sample. It is a positively biased estimate of the proportion variance explained in the population.
Proportion variance explained $\omega^2$:
Corrects for the positive bias in $\eta^2$ and is equal to:
$$
\begin{align}
\omega^2_A &= \dfrac{\mbox{sum of squares A}  \mbox{degrees of freedom A} \times \mbox{mean square error}}{\mbox{sum of squares total} + \mbox{mean square error}}\\
\\
\omega^2_B &= \dfrac{\mbox{sum of squares B}  \mbox{degrees of freedom B} \times \mbox{mean square error}}{\mbox{sum of squares total} + \mbox{mean square error}}\\
\\
\omega^2_{int} &= \dfrac{\mbox{sum of squares int}  \mbox{degrees of freedom int} \times \mbox{mean square error}}{\mbox{sum of squares total} + \mbox{mean square error}}\\
\end{align}
$$
$\omega^2$ is a better estimate of the explained variance in the population than
$\eta^2$. Only for balanced designs (equal sample sizes).
Proportion variance explained $\eta^2_{partial}$:
$$
\begin{align}
\eta^2_{partial\,A} &= \frac{\mbox{sum of squares A}}{\mbox{sum of squares A} + \mbox{sum of squares error}}\\
\\
\eta^2_{partial\,B} &= \frac{\mbox{sum of squares B}}{\mbox{sum of squares B} + \mbox{sum of squares error}}\\
\\
\eta^2_{partial\,int} &= \frac{\mbox{sum of squares int}}{\mbox{sum of squares int} + \mbox{sum of squares error}}
\end{align}
$$
OLS regression with two categorical independent variables and the interaction term, transformed into $(I  1)$ + $(J  1)$ + $(I  1) \times (J  1)$ code variables.
Example context
Example context
Do people tend to score higher on mental health after a mindfulness course?
Is the average mental health score different between people from a low, moderate, and high economic class? And is the average mental health score different between men and women? And is there an interaction effect between economic class and gender?
Put the two paired variables in the boxes below Variable 1 and Variable 2
Under Test Type, select the Sign test
Analyze > General Linear Model > Univariate...
Put your dependent (quantitative) variable in the box below Dependent Variable and your two independent (grouping) variables in the box below Fixed Factor(s)
Jamovi
Jamovi
Jamovi does not have a specific option for the sign test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the two sided $p$ value that would have resulted from the sign test. Go to:
ANOVA > Repeated Measures ANOVA  Friedman
Put the two paired variables in the box below Measures
ANOVA > ANOVA
Put your dependent (quantitative) variable in the box below Dependent Variable and your two independent (grouping) variables in the box below Fixed Factors