Two way ANOVA - overview
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Two way ANOVA | Sign test | Two sample $z$ test | Chi-squared test for the relationship between two categorical variables | Two sample $z$ test | Cochran's Q test |
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Independent/grouping variables | Independent variable | Independent/grouping variable | Independent /column variable | Independent/grouping variable | Independent/grouping variable | |
Two categorical, the first with $I$ independent groups and the second with $J$ independent groups ($I \geqslant 2$, $J \geqslant 2$) | 2 paired groups | One categorical with 2 independent groups | One categorical with $I$ independent groups ($I \geqslant 2$) | One categorical with 2 independent groups | One within subject factor ($\geq 2$ related groups) | |
Dependent variable | Dependent variable | Dependent variable | Dependent /row variable | Dependent variable | Dependent variable | |
One quantitative of interval or ratio level | One of ordinal level | One quantitative of interval or ratio level | One categorical with $J$ independent groups ($J \geqslant 2$) | One quantitative of interval or ratio level | One categorical with 2 independent groups | |
Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | |
ANOVA $F$ tests:
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| 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: 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: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2. | 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 $I.$ | |
Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
ANOVA $F$ tests:
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| H1 two sided: $\mu_1 \neq \mu_2$ H1 right sided: $\mu_1 > \mu_2$ H1 left sided: $\mu_1 < \mu_2$ | 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: $\mu_1 \neq \mu_2$ H1 right sided: $\mu_1 > \mu_2$ H1 left sided: $\mu_1 < \mu_2$ | H1: not all population proportions are equal | |
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):
| $W = $ number of difference scores that is larger than 0 | $z = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $\sigma^2_1$ is the population variance in population 1, $\sigma^2_2$ is the population variance in population 2, $n_1$ 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 $\sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$ is the standard deviation of the sampling distribution of $\bar{y}_1 - \bar{y}_2$. The $z$ value indicates how many of these standard deviations $\bar{y}_1 - \bar{y}_2$ 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 $\mu_2 > \mu_1$. | $X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
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. | $z = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $\sigma^2_1$ is the population variance in population 1, $\sigma^2_2$ is the population variance in population 2, $n_1$ 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 $\sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$ is the standard deviation of the sampling distribution of $\bar{y}_1 - \bar{y}_2$. The $z$ value indicates how many of these standard deviations $\bar{y}_1 - \bar{y}_2$ 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 $\mu_2 > \mu_1$. | 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. | |
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 $W$ if H0 were true | Sampling distribution of $z$ 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 | |
For main and interaction effects together (model):
| The exact distribution of $W$ under the null hypothesis is the Binomial($n$, $P$) distribution, with $n =$ number of positive differences $+$ number of negative differences, and $P = 0.5$.
If $n$ is large, $W$ is approximately normally distributed under the null hypothesis, with mean $nP = n \times 0.5$ and standard deviation $\sqrt{nP(1-P)} = \sqrt{n \times 0.5(1 - 0.5)}$. Hence, if $n$ is large, the standardized test statistic $$z = \frac{W - n \times 0.5}{\sqrt{n \times 0.5(1 - 0.5)}}$$ follows approximately the standard normal distribution if the null hypothesis were true. | Standard normal distribution | Approximately the chi-squared distribution with $(I - 1) \times (J - 1)$ degrees of freedom | Standard normal distribution | If the number of blocks (usually the number of subjects) is large, approximately the chi-squared distribution with $k - 1$ degrees of freedom | |
Significant? | Significant? | Significant? | Significant? | Significant? | Significant? | |
| If $n$ is small, the table for the binomial distribution should be used: Two sided:
If $n$ is large, the table for standard normal probabilities can be used: Two sided:
| Two sided:
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| Two sided:
| If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
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n.a. | n.a. | $C\%$ confidence interval for $\mu_1 - \mu_2$ | n.a. | $C\%$ confidence interval for $\mu_1 - \mu_2$ | n.a. | |
- | - | $(\bar{y}_1 - \bar{y}_2) \pm z^* \times \sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}$
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 $\mu_1 - \mu_2$ can also be used as significance test. | - | $(\bar{y}_1 - \bar{y}_2) \pm z^* \times \sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}$
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 $\mu_1 - \mu_2$ can also be used as significance test. | - | |
Effect size | n.a. | n.a. | n.a. | n.a. | n.a. | |
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n.a. | n.a. | Visual representation | 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 | Equivalent to | n.a. | n.a. | 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. |
Two sided sign test is equivalent to
| - | - | - | 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? | Do people tend to score higher on mental health after a mindfulness course? | Is the average mental health score different between men and women? Assume that in the population, the standard devation of the mental health scores is $\sigma_1 = 2$ amongst men and $\sigma_2 = 2.5$ amongst women. | 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 different between men and women? Assume that in the population, the standard devation of the mental health scores is $\sigma_1 = 2$ amongst men and $\sigma_2 = 2.5$ amongst women. | 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? | |
SPSS | SPSS | n.a. | SPSS | n.a. | SPSS | |
Analyze > General Linear Model > Univariate...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
| - | Analyze > Descriptive Statistics > Crosstabs...
| - | Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
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Jamovi | Jamovi | n.a. | Jamovi | n.a. | Jamovi | |
ANOVA > ANOVA
| Jamovi does not have a specific option for the sign test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the two sided $p$ value that would have resulted from the sign test. Go to:
ANOVA > Repeated Measures ANOVA - Friedman
| - | Frequencies > Independent Samples - $\chi^2$ test of association
| - | 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
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Practice questions | Practice questions | Practice questions | Practice questions | Practice questions | Practice questions | |