Two way ANOVA - overview
This page offers structured overviews of one or more selected methods. Add additional methods for comparisons (max. of 3) 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 | One sample $t$ test for the mean | Marginal Homogeneity test / Stuart-Maxwell test |
You cannot compare more than 3 methods |
---|---|---|---|
Independent/grouping variables | Independent variable | Independent variable | |
Two categorical, the first with $I$ independent groups and the second with $J$ independent groups ($I \geqslant 2$, $J \geqslant 2$) | None | 2 paired groups | |
Dependent variable | Dependent variable | Dependent variable | |
One quantitative of interval or ratio level | One quantitative of interval or ratio level | One categorical with $J$ independent groups ($J \geqslant 2$) | |
Null hypothesis | Null hypothesis | Null hypothesis | |
ANOVA $F$ tests:
| H0: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis. | H0: for each category $j$ of the dependent variable, $\pi_j$ for the first paired group = $\pi_j$ for the second paired group.
Here $\pi_j$ is the population proportion in category $j.$ | |
Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
ANOVA $F$ tests:
| H1 two sided: $\mu \neq \mu_0$ H1 right sided: $\mu > \mu_0$ H1 left sided: $\mu < \mu_0$ | H1: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group. | |
Assumptions | Assumptions | Assumptions | |
|
|
| |
Test statistic | Test statistic | Test statistic | |
For main and interaction effects together (model):
| $t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $s$ is the sample standard deviation, and $N$ is the sample size. The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ is removed from $\mu_0$. | Computing the test statistic is a bit complicated and involves matrix algebra. Unless you are following a technical course, you probably won't need to calculate it by hand. | |
Pooled standard deviation | 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 $t$ if H0 were true | Sampling distribution of the test statistic if H0 were true | |
For main and interaction effects together (model):
| $t$ distribution with $N - 1$ degrees of freedom | Approximately the chi-squared distribution with $J - 1$ degrees of freedom | |
Significant? | Significant? | Significant? | |
| Two sided:
| If we denote the test statistic as $X^2$:
| |
n.a. | $C\%$ confidence interval for $\mu$ | n.a. | |
- | $\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
where the critical value $t^*$ is the value under the $t_{N-1}$ 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 $\mu$ can also be used as significance test. | - | |
Effect size | Effect size | n.a. | |
| Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean $\bar{y}$ is removed from $\mu_0.$ | - | |
n.a. | Visual representation | n.a. | |
- | - | ||
ANOVA table | n.a. | n.a. | |
- | - | ||
Equivalent to | 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 | |
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 $\mu_0 = 50$? | Subjects are asked to taste three different types of mayonnaise, and to indicate which of the three types of mayonnaise they like best. They then have to drink a glass of beer, and taste and rate the three types of mayonnaise again. Does drinking a beer change which type of mayonnaise people like best? | |
SPSS | SPSS | SPSS | |
Analyze > General Linear Model > Univariate...
| Analyze > Compare Means > One-Sample T Test...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
| |
Jamovi | Jamovi | n.a. | |
ANOVA > ANOVA
| T-Tests > One Sample T-Test
| - | |
Practice questions | Practice questions | Practice questions | |