Two way ANOVA overview

This page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table

Two way ANOVA	Friedman test	Two sample $t$ test - equal variances not assumed	Chi-squared test for the relationship between two categorical variables	Spearman's rho
Independent/grouping variables	Independent/grouping variable	Independent/grouping variable	Independent /column variable	Variable 1
Two categorical, the first with $I$ independent groups and the second with $J$ independent groups ($I \geqslant 2$, $J \geqslant 2$)	One within subject factor ($\geq 2$ related groups)	One categorical with 2 independent groups	One categorical with $I$ independent groups ($I \geqslant 2$)	One of ordinal level
Dependent variable	Dependent variable	Dependent variable	Dependent /row variable	Variable 2
One quantitative of interval or ratio level	One of ordinal level	One quantitative of interval or ratio level	One categorical with $J$ independent groups ($J \geqslant 2$)	One of ordinal level
Null hypothesis	Null hypothesis	Null hypothesis	Null hypothesis	Null hypothesis
ANOVA $F$ tests: H₀ for main and interaction effects together (model): no main effects and interaction effect H₀ for independent variable A: no main effect for A H₀ for independent variable B: no main effect for B H₀ 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.	H₀: the population scores in any of the related groups are not systematically higher or lower than the population scores in any of the other related groups Usually the related groups are the different measurement points. Several different formulations of the null hypothesis can be found in the literature, and we do not agree with all of them. Make sure you (also) learn the one that is given in your text book or by your teacher.	H₀: $\mu_1 = \mu_2$ Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2.	H₀: there is no association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable: H₀: the distribution of the dependent variable is the same in each of the $I$ populations If there is one random sample of size $N$ from the total population: H₀: the row and column variables are independent	H₀: $\rho_s = 0$ Here $\rho_s$ is the Spearman correlation in the population. The Spearman correlation is a measure for the strength and direction of the monotonic relationship between two variables of at least ordinal measurement level. In words, the null hypothesis would be: H₀: there is no monotonic relationship between the two variables in the population.
Alternative hypothesis	Alternative hypothesis	Alternative hypothesis	Alternative hypothesis	Alternative hypothesis
ANOVA $F$ tests: H₁ for main and interaction effects together (model): there is a main effect for A, and/or for B, and/or an interaction effect H₁ for independent variable A: there is a main effect for A H₁ for independent variable B: there is a main effect for B H₁ for the interaction term: there is an interaction effect between A and B	H₁: the population scores in some of the related groups are systematically higher or lower than the population scores in other related groups	H₁ two sided: $\mu_1 \neq \mu_2$ H₁ right sided: $\mu_1 > \mu_2$ H₁ left sided: $\mu_1 < \mu_2$	H₁: there is an association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable: H₁: the distribution of the dependent variable is not the same in all of the $I$ populations If there is one random sample of size $N$ from the total population: H₁: the row and column variables are dependent	H₁ two sided: $\rho_s \neq 0$ H₁ right sided: $\rho_s > 0$ H₁ left sided: $\rho_s < 0$
Assumptions	Assumptions	Assumptions	Assumptions	Assumptions
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)	Sample of 'blocks' (usually the subjects) is a simple random sample from the population. That is, blocks are independent of one another	Within each population, the scores on the dependent variable are normally distributed 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	Sample size is large enough for $X^2$ to be approximately chi-squared distributed under the null hypothesis. Rule of thumb: 2 $\times$ 2 table: all four expected cell counts are 5 or more Larger than 2 $\times$ 2 tables: average of the expected cell counts is 5 or more, smallest expected cell count is 1 or more There are $I$ independent simple random samples from each of $I$ populations defined by the independent variable, or there is one simple random sample from the total population	Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another Note: this assumption is only important for the significance test, not for the correlation coefficient itself. The correlation coefficient itself just measures the strength of the monotonic relationship between two variables.
Test statistic	Test statistic	Test statistic	Test statistic	Test statistic
For main and interaction effects together (model): $F = \dfrac{\mbox{mean square model}}{\mbox{mean square error}}$ For independent variable A: $F = \dfrac{\mbox{mean square A}}{\mbox{mean square error}}$ For independent variable B: $F = \dfrac{\mbox{mean square B}}{\mbox{mean square error}}$ For the interaction term: $F = \dfrac{\mbox{mean square interaction}}{\mbox{mean square error}}$ Note: mean square error is also known as mean square residual or mean square within.	$Q = \dfrac{12}{N \times k(k + 1)} \sum R^2_i - 3 \times N(k + 1)$ Here $N$ is the number of 'blocks' (usually the subjects - so if you have 4 repeated measurements for 60 subjects, $N$ equals 60), $k$ is the number of related groups (usually the number of repeated measurements), and $R_i$ is the sum of ranks in group $i$. Remember that multiplication precedes addition, so first compute $\frac{12}{N \times k(k + 1)} \times \sum R^2_i$ and then subtract $3 \times N(k + 1)$. Note: if ties are present in the data, the formula for $Q$ is more complicated.	$t = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}}$ Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s^2_1$ is the sample variance in group 1, $s^2_2$ is the sample variance in group 2, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis. The denominator $\sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}}$ is the standard error of the sampling distribution of $\bar{y}_1 - \bar{y}_2$. The $t$ value indicates how many standard errors $\bar{y}_1 - \bar{y}_2$ is removed from 0. Note: we could just as well compute $\bar{y}_2 - \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$.	$X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$ Here for each cell, the expected cell count = $\dfrac{\mbox{row total} \times \mbox{column total}}{\mbox{total sample size}}$, the observed cell count is the observed sample count in that same cell, and the sum is over all $I \times J$ cells.	$t = \dfrac{r_s \times \sqrt{N - 2}}{\sqrt{1 - r_s^2}} $ Here $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores.
Pooled standard deviation	n.a.	n.a.	n.a.	n.a.
$ \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 $F$ if H₀ were true	Sampling distribution of $Q$ if H₀ were true	Sampling distribution of $t$ if H₀ were true	Sampling distribution of $X^2$ if H₀ were true	Sampling distribution of $t$ if H₀ 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.	If the number of blocks $N$ is large, approximately the chi-squared distribution with $k - 1$ degrees of freedom. For small samples, the exact distribution of $Q$ should be used.	Approximately the $t$ distribution with $k$ degrees of freedom, with $k$ equal to $k = \dfrac{\Bigg(\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}\Bigg)^2}{\dfrac{1}{n_1 - 1} \Bigg(\dfrac{s^2_1}{n_1}\Bigg)^2 + \dfrac{1}{n_2 - 1} \Bigg(\dfrac{s^2_2}{n_2}\Bigg)^2}$ or $k$ = the smaller of $n_1$ - 1 and $n_2$ - 1 First definition of $k$ is used by computer programs, second definition is often used for hand calculations.	Approximately the chi-squared distribution with $(I - 1) \times (J - 1)$ degrees of freedom	Approximately the $t$ distribution with $N - 2$ degrees of freedom
Significant?	Significant?	Significant?	Significant?	Significant?
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 $\alpha$	If the number of blocks $N$ is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$: 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 $\alpha$	Two sided: Check if $t$ observed in sample is at least as extreme as critical value $t^$ or Find two sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $t$ observed in sample is equal to or larger than critical value $t^$ or Find right sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $t$ observed in sample is equal to or smaller than critical value $t^*$ or Find left sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$	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 $\alpha$	Two sided: Check if $t$ observed in sample is at least as extreme as critical value $t^$ or Find two sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $t$ observed in sample is equal to or larger than critical value $t^$ or Find right sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $t$ observed in sample is equal to or smaller than critical value $t^*$ or Find left sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
n.a.	n.a.	Approximate $C\%$ confidence interval for $\mu_1 - \mu_2$	n.a.	n.a.
-	-	$(\bar{y}_1 - \bar{y}_2) \pm t^* \times \sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}$ where the critical value $t^$ is the value under the $t_{k}$ distribution with the area $C / 100$ between $-t^$ and $t^$ (e.g. $t^$ = 2.086 for a 95% confidence interval when df = 20). The confidence interval for $\mu_1 - \mu_2$ can also be used as significance test.	-	-
Effect size	n.a.	n.a.	n.a.	n.a.
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 $\eta^2_{partial}$: $$ \begin{align} \eta^2_{partial\,A} &= \frac{\mbox{sum of squares A}}{\mbox{sum of squares A} + \mbox{sum of squares error}}\\ \\ \eta^2_{partial\,B} &= \frac{\mbox{sum of squares B}}{\mbox{sum of squares B} + \mbox{sum of squares error}}\\ \\ \eta^2_{partial\,int} &= \frac{\mbox{sum of squares int}}{\mbox{sum of squares int} + \mbox{sum of squares error}} \end{align} $$	-	-	-	-
n.a.	n.a.	Visual representation	n.a.	n.a.
-	-		-	-
ANOVA table	n.a.	n.a.	n.a.	n.a.
	-	-	-	-
Equivalent to	n.a.	n.a.	n.a.	n.a.
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	Example context	Example context	Example context
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?	Is there a difference in depression level between measurement point 1 (pre-intervention), measurement point 2 (1 week post-intervention), and measurement point 3 (6 weeks post-intervention)?	Is the average mental health score different between men and women?	Is there an association between economic class and gender? Is the distribution of economic class different between men and women?	Is there a monotonic relationship between physical health and mental health?
SPSS	SPSS	SPSS	SPSS	SPSS
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)	Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples... Put the $k$ variables containing the scores for the $k$ related groups in the white box below Test Variables Under Test Type, select the Friedman test	Analyze > Compare Means > Independent-Samples T Test... Put your dependent (quantitative) variable in the box below Test Variable(s) and your independent (grouping) variable in the box below Grouping Variable Click on the Define Groups... button. If you can't click on it, first click on the grouping variable so its background turns yellow Fill in the value you have used to indicate your first group in the box next to Group 1, and the value you have used to indicate your second group in the box next to Group 2 Continue and click OK	Analyze > Descriptive Statistics > Crosstabs... Put one of your two categorical variables in the box below Row(s), and the other categorical 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	Analyze > Correlate > Bivariate... Put your two variables in the box below Variables Under Correlation Coefficients, select Spearman
Jamovi	Jamovi	Jamovi	Jamovi	Jamovi
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	ANOVA > Repeated Measures ANOVA - Friedman Put the $k$ variables containing the scores for the $k$ related groups in the box below Measures	T-Tests > Independent Samples T-Test Put your dependent (quantitative) variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable Under Tests, select Welch's Under Hypothesis, select your alternative hypothesis	Frequencies > Independent Samples - $\chi^2$ test of association Put one of your two categorical variables in the box below Rows, and the other categorical variable in the box below Columns	Regression > Correlation Matrix Put your two variables in the white box at the right Under Correlation Coefficients, select Spearman Under Hypothesis, select your alternative hypothesis
Practice questions	Practice questions	Practice questions	Practice questions	Practice questions

Two way ANOVA - overview