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 | Chi-squared test for the relationship between two categorical variables | One sample $z$ test for the mean | Friedman test | Two sample $t$ test - equal variances assumed | Friedman test | Mann-Whitney-Wilcoxon test |
|
---|---|---|---|---|---|---|---|
Independent/grouping variables | Independent /column variable | Independent variable | Independent/grouping variable | Independent/grouping variable | Independent/grouping variable | Independent/grouping variable | |
Two categorical, the first with $I$ independent groups and the second with $J$ independent groups ($I \geqslant 2$, $J \geqslant 2$) | One categorical with $I$ independent groups ($I \geqslant 2$) | None | One within subject factor ($\geq 2$ related groups) | One categorical with 2 independent groups | One within subject factor ($\geq 2$ related groups) | One categorical with 2 independent groups | |
Dependent variable | Dependent /row variable | Dependent variable | Dependent variable | Dependent variable | Dependent variable | Dependent variable | |
One quantitative of interval or ratio level | One categorical with $J$ independent groups ($J \geqslant 2$) | One quantitative of interval or ratio level | One of ordinal level | One quantitative of interval or ratio level | One of ordinal level | One of ordinal level | |
Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | |
ANOVA $F$ tests:
| 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: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis. | 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: 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. | If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
Formulation 1:
| |
Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
ANOVA $F$ tests:
| 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 two sided: $\mu \neq \mu_0$ H1 right sided: $\mu > \mu_0$ H1 left sided: $\mu < \mu_0$ | 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: the population scores in some of the related groups are systematically higher or lower than the population scores in other related groups | If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
Formulation 1:
| |
Assumptions | Assumptions | Assumptions | Assumptions | Assumptions | Assumptions | Assumptions | |
|
|
|
|
|
|
| |
Test statistic | Test statistic | Test statistic | Test statistic | Test statistic | Test statistic | Test statistic | |
For main and interaction effects together (model):
| $X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
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. | $z = \dfrac{\bar{y} - \mu_0}{\sigma / \sqrt{N}}$
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 $\mu_0$. | $Q = \dfrac{12}{N \times k(k + 1)} \sum R^2_i - 3 \times N(k + 1)$
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. | $t = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s_p$ is the pooled standard deviation, $n_1$ 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 $s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$ 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 $\mu_2 > \mu_1$. | $Q = \dfrac{12}{N \times k(k + 1)} \sum R^2_i - 3 \times N(k + 1)$
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. | Two different types of test statistics can be used; both will result in the same test outcome. The first is the Wilcoxon rank sum statistic $W$:
Note: we could just as well base W and U on group 2. This would only 'flip' the right and left sided alternative hypotheses. Also, tables with critical values for $U$ are often based on the smaller of $U$ for group 1 and for group 2. | |
Pooled standard deviation | n.a. | n.a. | n.a. | 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} $ | - | - | - | $s_p = \sqrt{\dfrac{(n_1 - 1) \times s^2_1 + (n_2 - 1) \times s^2_2}{n_1 + n_2 - 2}}$ | - | - | |
Sampling distribution of $F$ if H0 were true | Sampling distribution of $X^2$ if H0 were true | Sampling distribution of $z$ if H0 were true | Sampling distribution of $Q$ if H0 were true | Sampling distribution of $t$ if H0 were true | Sampling distribution of $Q$ if H0 were true | Sampling distribution of $W$ and of $U$ if H0 were true | |
For main and interaction effects together (model):
| Approximately the chi-squared distribution with $(I - 1) \times (J - 1)$ degrees of freedom | Standard normal distribution | 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. | $t$ distribution with $n_1 + n_2 - 2$ degrees of freedom | 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. | Sampling distribution of $W$:
Sampling distribution of $U$: For small samples, the exact distribution of $W$ or $U$ should be used. Note: if ties are present in the data, the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated. | |
Significant? | Significant? | Significant? | Significant? | Significant? | Significant? | Significant? | |
|
| Two sided:
| If the number of blocks $N$ is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
| Two sided:
| If the number of blocks $N$ is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
| For large samples, the table for standard normal probabilities can be used: Two sided:
| |
n.a. | n.a. | $C\%$ confidence interval for $\mu$ | n.a. | $C\%$ confidence interval for $\mu_1 - \mu_2$ | n.a. | n.a. | |
- | - | $\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
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. | - | $(\bar{y}_1 - \bar{y}_2) \pm t^* \times s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$
where the critical value $t^*$ is the value under the $t_{n_1 + n_2 - 2}$ 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. | Effect size | n.a. | Effect size | n.a. | n.a. | |
| - | 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 $\mu_0.$ | - | Cohen's $d$: Standardized difference between the mean in group $1$ and in group $2$: $$d = \frac{\bar{y}_1 - \bar{y}_2}{s_p}$$ Cohen's $d$ indicates how many standard deviations $s_p$ the two sample means are removed from each other. | - | - | |
n.a. | n.a. | Visual representation | n.a. | Visual representation | n.a. | n.a. | |
- | - | - | - | - | |||
ANOVA table | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | |
- | - | - | - | - | - | ||
Equivalent to | n.a. | n.a. | n.a. | Equivalent to | n.a. | Equivalent to | |
OLS regression with two categorical independent variables and the interaction term, transformed into $(I - 1)$ + $(J - 1)$ + $(I - 1) \times (J - 1)$ code variables. | - | - | - | One way ANOVA with an independent variable with 2 levels ($I$ = 2):
| - | If there are no ties in the data, the two sided Mann-Whitney-Wilcoxon test is equivalent to the Kruskal-Wallis test with an independent variable with 2 levels ($I$ = 2). | |
Example context | Example context | 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 an association between economic class and gender? Is the distribution of economic class different between men and women? | 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 $\sigma = 3.$ | 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? Assume that in the population, the standard deviation of mental health scores is equal amongst men and women. | 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)? | Do men tend to score higher on social economic status than women? | |
SPSS | SPSS | n.a. | SPSS | SPSS | SPSS | SPSS | |
Analyze > General Linear Model > Univariate...
| Analyze > Descriptive Statistics > Crosstabs...
| - | Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
| Analyze > Compare Means > Independent-Samples T Test...
| Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples...
| |
Jamovi | Jamovi | n.a. | Jamovi | Jamovi | Jamovi | Jamovi | |
ANOVA > ANOVA
| Frequencies > Independent Samples - $\chi^2$ test of association
| - | ANOVA > Repeated Measures ANOVA - Friedman
| T-Tests > Independent Samples T-Test
| ANOVA > Repeated Measures ANOVA - Friedman
| T-Tests > Independent Samples T-Test
| |
Practice questions | Practice questions | Practice questions | Practice questions | Practice questions | Practice questions | Practice questions | |