Two way ANOVA - overview
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Two way ANOVA | Friedman test | Goodness of fit test | Cochran's Q test | One sample $z$ test for the mean | Goodness of fit test |
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Independent/grouping variables | Independent/grouping variable | Independent variable | Independent/grouping variable | Independent variable | Independent variable | |
Two categorical, the first with $I$ independent groups and the second with $J$ independent groups ($I \geqslant 2$, $J \geqslant 2$) | One within subject factor ($\geq 2$ related groups) | None | One within subject factor ($\geq 2$ related groups) | None | None | |
Dependent variable | Dependent variable | Dependent variable | Dependent variable | Dependent variable | Dependent variable | |
One quantitative of interval or ratio level | One of ordinal level | One categorical with $J$ independent groups ($J \geqslant 2$) | One categorical with 2 independent groups | One quantitative of interval or ratio level | One categorical with $J$ independent groups ($J \geqslant 2$) | |
Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | |
ANOVA $F$ tests:
| 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: $\pi_1 = \pi_2 = \ldots = \pi_I$
Here $\pi_1$ is the population proportion of 'successes' for group 1, $\pi_2$ is the population proportion of 'successes' for group 2, and $\pi_I$ is the population proportion of 'successes' for group $I.$ | H0: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis. |
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Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
ANOVA $F$ tests:
| H1: the population scores in some of the related groups are systematically higher or lower than the population scores in other related groups |
| H1: not all population proportions are equal | H1 two sided: $\mu \neq \mu_0$ H1 right sided: $\mu > \mu_0$ H1 left sided: $\mu < \mu_0$ |
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Assumptions | Assumptions | Assumptions | Assumptions | Assumptions | Assumptions | |
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Test statistic | Test statistic | Test statistic | Test statistic | Test statistic | Test statistic | |
For main and interaction effects together (model):
| $Q = \dfrac{12}{N \times k(k + 1)} \sum R^2_i - 3 \times N(k + 1)$
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. | $X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here the expected cell count for one cell = $N \times \pi_j$, the observed cell count is the observed sample count in that same cell, and the sum is over all $J$ cells. | If a failure is scored as 0 and a success is scored as 1:
$Q = k(k - 1) \dfrac{\sum_{groups} \Big (\mbox{group total} - \frac{\mbox{grand total}}{k} \Big)^2}{\sum_{blocks} \mbox{block total} \times (k - \mbox{block total})}$ Here $k$ is the number of related groups (usually the number of repeated measurements), a group total is the sum of the scores in a group, a block total is the sum of the scores in a block (usually a subject), and the grand total is the sum of all the scores. Before computing $Q$, first exclude blocks with equal scores in all $k$ groups. | $z = \dfrac{\bar{y} - \mu_0}{\sigma / \sqrt{N}}$
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 $\mu_0$. | $X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here the expected cell count for one cell = $N \times \pi_j$, the observed cell count is the observed sample count in that same cell, and the sum is over all $J$ cells. | |
Pooled standard deviation | n.a. | 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 $X^2$ if H0 were true | Sampling distribution of $Q$ if H0 were true | Sampling distribution of $z$ if H0 were true | Sampling distribution of $X^2$ if H0 were true | |
For main and interaction effects together (model):
| 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 chi-squared distribution with $J - 1$ degrees of freedom | If the number of blocks (usually the number of subjects) is large, approximately the chi-squared distribution with $k - 1$ degrees of freedom | Standard normal distribution | Approximately the chi-squared distribution with $J - 1$ degrees of freedom | |
Significant? | Significant? | Significant? | Significant? | Significant? | Significant? | |
| If the number of blocks $N$ is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
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| If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
| Two sided:
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n.a. | n.a. | n.a. | n.a. | $C\%$ confidence interval for $\mu$ | 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. | - | |
Effect size | n.a. | n.a. | n.a. | Effect size | 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 $\mu_0.$ | - | |
n.a. | n.a. | n.a. | n.a. | Visual representation | n.a. | |
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ANOVA table | n.a. | n.a. | n.a. | n.a. | n.a. | |
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Equivalent to | 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. | - | - | Friedman test, with a categorical dependent variable consisting of two independent groups. | - | - | |
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 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 proportion of people with a low, moderate, and high social economic status in the population different from $\pi_{low} = 0.2,$ $\pi_{moderate} = 0.6,$ and $\pi_{high} = 0.2$? | Subjects perform three different tasks, which they can either perform correctly or incorrectly. Is there a difference in task performance between the three different tasks? | 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 the proportion of people with a low, moderate, and high social economic status in the population different from $\pi_{low} = 0.2,$ $\pi_{moderate} = 0.6,$ and $\pi_{high} = 0.2$? | |
SPSS | SPSS | SPSS | SPSS | n.a. | SPSS | |
Analyze > General Linear Model > Univariate...
| Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
| Analyze > Nonparametric Tests > Legacy Dialogs > Chi-square...
| Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
| - | Analyze > Nonparametric Tests > Legacy Dialogs > Chi-square...
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Jamovi | Jamovi | Jamovi | Jamovi | n.a. | Jamovi | |
ANOVA > ANOVA
| ANOVA > Repeated Measures ANOVA - Friedman
| Frequencies > N Outcomes - $\chi^2$ Goodness of fit
| Jamovi does not have a specific option for the Cochran's Q test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the $p$ value that would have resulted from the Cochran's Q test. Go to:
ANOVA > Repeated Measures ANOVA - Friedman
| - | Frequencies > N Outcomes - $\chi^2$ Goodness of fit
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Practice questions | Practice questions | Practice questions | Practice questions | Practice questions | Practice questions | |