Two way ANOVA overview

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Two way ANOVA	One sample $z$ test for the mean	Binomial test for a single proportion $z$ test for a single proportion $z$ test for the difference between two proportions Goodness of fit test Chi-squared test for the relationship between two categorical variables One sample $z$ test for the mean One sample $t$ test for the mean Paired sample $t$ test Two sample $z$ test Two sample $t$ test - equal variances not assumed Two sample $t$ test - equal variances assumed One way ANOVA Two way ANOVA Pearson correlation Regression (OLS)Logistic regression Mann-Whitney-Wilcoxon test Kruskal-Wallis test Sign test McNemar's test Cochran's Q test Marginal Homogeneity test / Stuart-Maxwell test Friedman test Wilcoxon signed-rank test One sample Wilcoxon signed-rank test Spearman's rho
Independent/grouping variables	Independent variable
Two categorical, the first with $I$ independent groups and the second with $J$ independent groups ($I \geqslant 2$, $J \geqslant 2$)	None
Dependent variable	Dependent variable
One quantitative of interval or ratio level	One quantitative of interval or ratio level
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: $\mu = \mu_0$ Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null 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 two sided: $\mu \neq \mu_0$ H₁ right sided: $\mu > \mu_0$ H₁ left sided: $\mu < \mu_0$
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)	Scores are normally distributed in the population Population standard deviation $\sigma$ is known Sample is a simple random sample from the population. That is, observations are independent of one another
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.	$z = \dfrac{\bar{y} - \mu_0}{\sigma / \sqrt{N}}$ Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $\sigma$ is the population standard deviation, and $N$ is the sample size. The denominator $\sigma / \sqrt{N}$ is the standard deviation of the sampling distribution of $\bar{y}$. The $z$ value indicates how many of these standard deviations $\bar{y}$ is removed from $\mu_0$.
Pooled standard deviation	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 $z$ 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.	Standard normal distribution
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 $\alpha$	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.	$C\%$ confidence interval for $\mu$
-	$\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$ 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). The confidence interval for $\mu$ can also be used as significance test.
Effect size	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 $\eta^2_{partial}$: $$ \begin{align} \eta^2_{partial\,A} &= \frac{\mbox{sum of squares A}}{\mbox{sum of squares A} + \mbox{sum of squares error}}\\ \\ \eta^2_{partial\,B} &= \frac{\mbox{sum of squares B}}{\mbox{sum of squares B} + \mbox{sum of squares error}}\\ \\ \eta^2_{partial\,int} &= \frac{\mbox{sum of squares int}}{\mbox{sum of squares int} + \mbox{sum of squares error}} \end{align} $$	Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{\sigma}$$ Cohen's $d$ indicates how many standard deviations $\sigma$ the sample mean $\bar{y}$ is removed from $\mu_0.$
n.a.	Visual representation
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ANOVA table	n.a.
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Equivalent to	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
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 the average mental health score of office workers different from $\mu_0 = 50$? Assume that the standard deviation of the mental health scores in the population is $\sigma = 3.$
SPSS	n.a.
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	n.a.
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	-
Practice questions	Practice questions

Two way ANOVA - overview