Two way ANOVA - overview
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Two way ANOVA | Cochran's Q test | $z$ test for a single proportion | One sample $t$ test for the mean |
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Independent/grouping variables | 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 | None | |
Dependent variable | Dependent variable | Dependent variable | Dependent variable | |
One quantitative of interval or ratio level | One categorical with 2 independent groups | One categorical with 2 independent groups | One quantitative of interval or ratio level | |
Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | |
ANOVA $F$ tests:
| 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: $\pi = \pi_0$
Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis. | 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 | Alternative hypothesis | Alternative hypothesis | |
ANOVA $F$ tests:
| H1: not all population proportions are equal | H1 two sided: $\pi \neq \pi_0$ H1 right sided: $\pi > \pi_0$ H1 left sided: $\pi < \pi_0$ | H1 two sided: $\mu \neq \mu_0$ H1 right sided: $\mu > \mu_0$ H1 left sided: $\mu < \mu_0$ | |
Assumptions | Assumptions | Assumptions | Assumptions | |
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Test statistic | Test statistic | Test statistic | Test statistic | |
For main and interaction effects together (model):
| 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. | $z = \dfrac{p - \pi_0}{\sqrt{\dfrac{\pi_0(1 - \pi_0)}{N}}}$
Here $p$ is the sample proportion of successes: $\dfrac{X}{N}$, $N$ is the sample size, and $\pi_0$ is the population proportion of successes according to the null hypothesis. | $t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$
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 $\mu_0$. | |
Pooled standard deviation | 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 $z$ if H0 were true | Sampling distribution of $t$ if H0 were true | |
For main and interaction effects together (model):
| If the number of blocks (usually the number of subjects) is large, approximately the chi-squared distribution with $k - 1$ degrees of freedom | Approximately the standard normal distribution | $t$ distribution with $N - 1$ degrees of freedom | |
Significant? | Significant? | Significant? | Significant? | |
| 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:
| Two sided:
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n.a. | n.a. | Approximate $C\%$ confidence interval for $\pi$ | $C\%$ confidence interval for $\mu$ | |
- | - | Regular (large sample):
| $\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
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 | n.a. | n.a. | Effect size | |
| - | - | 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. | n.a. | n.a. | Visual representation | |
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ANOVA table | n.a. | n.a. | n.a. | |
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Equivalent to | Equivalent to | 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. | Friedman test, with a categorical dependent variable consisting of two independent groups. |
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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? | 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 proportion of smokers amongst office workers different from $\pi_0 = 0.2$? Use the normal approximation for the sampling distribution of the test statistic. | Is the average mental health score of office workers different from $\mu_0 = 50$? | |
SPSS | SPSS | SPSS | SPSS | |
Analyze > General Linear Model > Univariate...
| Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
| Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
| Analyze > Compare Means > One-Sample T Test...
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Jamovi | Jamovi | Jamovi | Jamovi | |
ANOVA > ANOVA
| 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 > 2 Outcomes - Binomial test
| T-Tests > One Sample T-Test
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Practice questions | Practice questions | Practice questions | Practice questions | |