# Two way ANOVA - overview

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Two way ANOVA | Two sample $t$ test - equal variances assumed | Spearman's rho |
You cannot compare more than 3 methods |
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Independent/grouping variables | Independent/grouping variable | Variable 1 | |

Two categorical, the first with $I$ independent groups and the second with $J$ independent groups ($I \geqslant 2$, $J \geqslant 2$) | One categorical with 2 independent groups | One of ordinal level | |

Dependent variable | Dependent variable | Variable 2 | |

One quantitative of interval or ratio level | One quantitative of interval or ratio level | One of ordinal level | |

Null hypothesis | Null hypothesis | Null hypothesis | |

ANOVA $F$ tests:
- H
_{0}for main and interaction effects together (model): no main effects and interaction effect - H
_{0}for independent variable A: no main effect for A - H
_{0}for independent variable B: no main effect for B - H
_{0}for the interaction term: no interaction effect between A and B
| H_{0}: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2. | H_{0}: $\rho_s = 0$
Here $\rho_s$ is the Spearman correlation in the population. The Spearman correlation is a measure for the strength and direction of the monotonic relationship between two variables of at least ordinal measurement level. In words, the null hypothesis would be: H _{0}: there is no monotonic relationship between the two variables in the population.
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Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |

ANOVA $F$ tests:
- H
_{1}for main and interaction effects together (model): there is a main effect for A, and/or for B, and/or an interaction effect - H
_{1}for independent variable A: there is a main effect for A - H
_{1}for independent variable B: there is a main effect for B - H
_{1}for the interaction term: there is an interaction effect between A and B
| H_{1} two sided: $\mu_1 \neq \mu_2$H _{1} right sided: $\mu_1 > \mu_2$H _{1} left sided: $\mu_1 < \mu_2$
| H_{1} two sided: $\rho_s \neq 0$H _{1} right sided: $\rho_s > 0$H _{1} left sided: $\rho_s < 0$ | |

Assumptions | Assumptions | Assumptions | |

- Within each of the $I \times J$ populations, the scores on the dependent variable are normally distributed
- The standard deviation of the scores on the dependent variable is the same in each of the $I \times J$ populations
- For each of the $I \times J$ groups, the sample is an independent and simple random sample from the population defined by that group. That is, within and between groups, observations are independent of one another
- Equal sample sizes for each group make the interpretation of the ANOVA output easier (unequal sample sizes result in overlap in the sum of squares; this is advanced stuff)
| - Within each population, the scores on the dependent variable are normally distributed
- The standard deviation of the scores on the dependent variable is the same in both populations: $\sigma_1 = \sigma_2$
- Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations are independent of one another
| - Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
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Test statistic | Test statistic | Test statistic | |

For main and interaction effects together (model):
- $F = \dfrac{\mbox{mean square model}}{\mbox{mean square error}}$
- $F = \dfrac{\mbox{mean square A}}{\mbox{mean square error}}$
- $F = \dfrac{\mbox{mean square B}}{\mbox{mean square error}}$
- $F = \dfrac{\mbox{mean square interaction}}{\mbox{mean square error}}$
| $t = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
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 $\mu_2 > \mu_1$. | $t = \dfrac{r_s \times \sqrt{N - 2}}{\sqrt{1 - r_s^2}} $ Here $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores. | |

Pooled standard deviation | Pooled standard deviation | 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 H_{0} were true | Sampling distribution of $t$ if H_{0} were true | Sampling distribution of $t$ if H_{0} were true | |

For main and interaction effects together (model):
- $F$ distribution with $(I - 1) + (J - 1) + (I - 1) \times (J - 1)$ (df model, numerator) and $N - (I \times J)$ (df error, denominator) degrees of freedom
- $F$ distribution with $I - 1$ (df A, numerator) and $N - (I \times J)$ (df error, denominator) degrees of freedom
- $F$ distribution with $J - 1$ (df B, numerator) and $N - (I \times J)$ (df error, denominator) degrees of freedom
- $F$ distribution with $(I - 1) \times (J - 1)$ (df interaction, numerator) and $N - (I \times J)$ (df error, denominator) degrees of freedom
| $t$ distribution with $n_1 + n_2 - 2$ degrees of freedom | Approximately the $t$ distribution with $N - 2$ degrees of freedom | |

Significant? | Significant? | Significant? | |

- Check if $F$ observed in sample is equal to or larger than critical value $F^*$ or
- Find $p$ value corresponding to observed $F$ and check if it is equal to or smaller than $\alpha$
| Two sided:
- Check if $t$ observed in sample is at least as extreme as critical value $t^*$ or
- Find two sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
- Check if $t$ observed in sample is equal to or larger than critical value $t^*$ or
- Find right sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
- Check if $t$ observed in sample is equal to or smaller than critical value $t^*$ or
- Find left sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
| Two sided:
- Check if $t$ observed in sample is at least as extreme as critical value $t^*$ or
- Find two sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
- Check if $t$ observed in sample is equal to or larger than critical value $t^*$ or
- Find right sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
- Check if $t$ observed in sample is equal to or smaller than critical value $t^*$ or
- Find left sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
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n.a. | $C\%$ confidence interval for $\mu_1 - \mu_2$ | n.a. | |

- | $(\bar{y}_1 - \bar{y}_2) \pm t^* \times s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$
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 | Effect size | n.a. | |

*Proportion variance explained $R^2$:* Proportion variance of the dependent variable $y$ explained by the independent variables and the interaction effect together: $$ \begin{align} R^2 &= \dfrac{\mbox{sum of squares model}}{\mbox{sum of squares total}} \end{align} $$ $R^2$ is the proportion variance explained in the sample. It is a positively biased estimate of the proportion variance explained in the population.
*Proportion variance explained $\eta^2$:* Proportion variance of the dependent variable $y$ explained by an independent variable or interaction effect: $$ \begin{align} \eta^2_A &= \dfrac{\mbox{sum of squares A}}{\mbox{sum of squares total}}\\ \\ \eta^2_B &= \dfrac{\mbox{sum of squares B}}{\mbox{sum of squares total}}\\ \\ \eta^2_{int} &= \dfrac{\mbox{sum of squares int}}{\mbox{sum of squares total}} \end{align} $$ $\eta^2$ is the proportion variance explained in the sample. It is a positively biased estimate of the proportion variance explained in the population.
*Proportion variance explained $\omega^2$:* Corrects for the positive bias in $\eta^2$ and is equal to: $$ \begin{align} \omega^2_A &= \dfrac{\mbox{sum of squares A} - \mbox{degrees of freedom A} \times \mbox{mean square error}}{\mbox{sum of squares total} + \mbox{mean square error}}\\ \\ \omega^2_B &= \dfrac{\mbox{sum of squares B} - \mbox{degrees of freedom B} \times \mbox{mean square error}}{\mbox{sum of squares total} + \mbox{mean square error}}\\ \\ \omega^2_{int} &= \dfrac{\mbox{sum of squares int} - \mbox{degrees of freedom int} \times \mbox{mean square error}}{\mbox{sum of squares total} + \mbox{mean square error}}\\ \end{align} $$ $\omega^2$ is a better estimate of the explained variance in the population than $\eta^2$. Only for balanced designs (equal sample sizes).
*Proportion variance explained $\eta^2_{partial}$:*$$ \begin{align} \eta^2_{partial\,A} &= \frac{\mbox{sum of squares A}}{\mbox{sum of squares A} + \mbox{sum of squares error}}\\ \\ \eta^2_{partial\,B} &= \frac{\mbox{sum of squares B}}{\mbox{sum of squares B} + \mbox{sum of squares error}}\\ \\ \eta^2_{partial\,int} &= \frac{\mbox{sum of squares int}}{\mbox{sum of squares int} + \mbox{sum of squares error}} \end{align} $$
| 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. | Visual representation | n.a. | |

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ANOVA table | n.a. | n.a. | |

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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. | One way ANOVA with an independent variable with 2 levels ($I$ = 2):
- two sided two sample $t$ test is equivalent to ANOVA $F$ test when $I$ = 2
- two sample $t$ test is equivalent to $t$ test for contrast when $I$ = 2
- two sample $t$ test is equivalent to $t$ test multiple comparisons when $I$ = 2
- two sided two sample $t$ test is equivalent to $F$ test regression model
- two sample $t$ test is equivalent to $t$ test for regression coefficient $\beta_1$
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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 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 monotonic relationship between physical health and mental health? | |

SPSS | SPSS | SPSS | |

Analyze > General Linear Model > Univariate...
- Put your dependent (quantitative) variable in the box below Dependent Variable and your two independent (grouping) variables in the box below Fixed Factor(s)
| Analyze > Compare Means > Independent-Samples T Test...
- Put your dependent (quantitative) variable in the box below Test Variable(s) and your independent (grouping) variable in the box below Grouping Variable
- Click on the Define Groups... button. If you can't click on it, first click on the grouping variable so its background turns yellow
- Fill in the value you have used to indicate your first group in the box next to Group 1, and the value you have used to indicate your second group in the box next to Group 2
- Continue and click OK
| Analyze > Correlate > Bivariate...
- Put your two variables in the box below Variables
- Under Correlation Coefficients, select Spearman
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Jamovi | Jamovi | Jamovi | |

ANOVA > ANOVA
- Put your dependent (quantitative) variable in the box below Dependent Variable and your two independent (grouping) variables in the box below Fixed Factors
| T-Tests > Independent Samples T-Test
- Put your dependent (quantitative) variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
- Under Tests, select Student's (selected by default)
- Under Hypothesis, select your alternative hypothesis
| Regression > Correlation Matrix
- Put your two variables in the white box at the right
- Under Correlation Coefficients, select Spearman
- Under Hypothesis, select your alternative hypothesis
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Practice questions | Practice questions | Practice questions | |