Two way ANOVA - overview
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Two way ANOVA | Friedman test | Two sample $t$ test - equal variances not assumed | $z$ test for the difference between two proportions | McNemar's test |
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Independent/grouping variables | Independent/grouping variable | Independent/grouping variable | Independent/grouping 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) | One categorical with 2 independent groups | One categorical with 2 independent groups | 2 paired groups | |
Dependent variable | Dependent variable | Dependent 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 2 independent groups | One categorical with 2 independent groups | |
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: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2. | H0: $\pi_1 = \pi_2$
Here $\pi_1$ is the population proportion of 'successes' for group 1, and $\pi_2$ is the population proportion of 'successes' for group 2. | Let's say that the scores on the dependent variable are scored 0 and 1. Then for each pair of scores, the data allow four options:
Other formulations of the null hypothesis are:
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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 two sided: $\mu_1 \neq \mu_2$ H1 right sided: $\mu_1 > \mu_2$ H1 left sided: $\mu_1 < \mu_2$ | H1 two sided: $\pi_1 \neq \pi_2$ H1 right sided: $\pi_1 > \pi_2$ H1 left sided: $\pi_1 < \pi_2$ | The alternative hypothesis H1 is that for each pair of scores, P(first score of pair is 0 while second score of pair is 1) $\neq$ P(first score of pair is 1 while second score of pair is 0). That is, the probability that a pair of scores switches from 0 to 1 is not the same as the probability that a pair of scores switches from 1 to 0. Other formulations of the alternative hypothesis are:
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Assumptions | Assumptions | Assumptions | Assumptions | Assumptions | |
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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. | $t = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s^2_1$ is the sample variance in group 1, $s^2_2$ is the sample variance in group 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 $\sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}}$ 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 $\mu_2 > \mu_1$. | $z = \dfrac{p_1 - p_2}{\sqrt{p(1 - p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
Here $p_1$ is the sample proportion of successes in group 1: $\dfrac{X_1}{n_1}$, $p_2$ is the sample proportion of successes in group 2: $\dfrac{X_2}{n_2}$, $p$ is the total proportion of successes in the sample: $\dfrac{X_1 + X_2}{n_1 + n_2}$, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. Note: we could just as well compute $p_2 - p_1$ in the numerator, but then the left sided alternative becomes $\pi_2 < \pi_1$, and the right sided alternative becomes $\pi_2 > \pi_1.$ | $X^2 = \dfrac{(b - c)^2}{b + c}$
Here $b$ is the number of pairs in the sample for which the first score is 0 while the second score is 1, and $c$ is the number of pairs in the sample for which the first score is 1 while the second score is 0. | |
Pooled standard deviation | 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 $t$ 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 $t$ distribution with $k$ degrees of freedom, with $k$ equal to $k = \dfrac{\Bigg(\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}\Bigg)^2}{\dfrac{1}{n_1 - 1} \Bigg(\dfrac{s^2_1}{n_1}\Bigg)^2 + \dfrac{1}{n_2 - 1} \Bigg(\dfrac{s^2_2}{n_2}\Bigg)^2}$ or $k$ = the smaller of $n_1$ - 1 and $n_2$ - 1 First definition of $k$ is used by computer programs, second definition is often used for hand calculations. | Approximately the standard normal distribution | If $b + c$ is large enough (say, > 20), approximately the chi-squared distribution with 1 degree of freedom. If $b + c$ is small, the Binomial($n$, $P$) distribution should be used, with $n = b + c$ and $P = 0.5$. In that case the test statistic becomes equal to $b$. | |
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 $X^2 = Q$:
| Two sided:
| Two sided:
| For test statistic $X^2$:
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n.a. | n.a. | Approximate $C\%$ confidence interval for $\mu_1 - \mu_2$ | Approximate $C\%$ confidence interval for $\pi_1 - \pi_2$ | n.a. | |
- | - | $(\bar{y}_1 - \bar{y}_2) \pm t^* \times \sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}$
where the critical value $t^*$ is the value under the $t_{k}$ 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. | Regular (large sample):
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Effect size | n.a. | n.a. | n.a. | n.a. | |
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n.a. | n.a. | Visual representation | n.a. | n.a. | |
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ANOVA table | n.a. | n.a. | n.a. | n.a. | |
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Equivalent to | n.a. | n.a. | Equivalent to | 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. | - | - | When testing two sided: chi-squared test for the relationship between two categorical variables, where both categorical variables have 2 levels. |
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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 average mental health score different between men and women? | Is the proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic. | Does a tv documentary about spiders change whether people are afraid (yes/no) of spiders? | |
SPSS | SPSS | SPSS | SPSS | SPSS | |
Analyze > General Linear Model > Univariate...
| Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
| Analyze > Compare Means > Independent-Samples T Test...
| SPSS does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Analyze > Descriptive Statistics > Crosstabs...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
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Jamovi | Jamovi | Jamovi | Jamovi | Jamovi | |
ANOVA > ANOVA
| ANOVA > Repeated Measures ANOVA - Friedman
| T-Tests > Independent Samples T-Test
| Jamovi does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Frequencies > Independent Samples - $\chi^2$ test of association
| Frequencies > Paired Samples - McNemar test
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Practice questions | Practice questions | Practice questions | Practice questions | Practice questions | |