Two way ANOVA  overview
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Two way ANOVA  Friedman test  Goodness of fit test  $z$ test for the difference between two proportions  Marginal Homogeneity test / StuartMaxwell test 


Independent/grouping variables  Independent/grouping variable  Independent 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)  None  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 categorical with $J$ independent groups ($J \geqslant 2$)  One categorical with 2 independent groups  One categorical with $J$ independent groups ($J \geqslant 2$)  
Null hypothesis  Null hypothesis  Null hypothesis  Null hypothesis  Null hypothesis  
ANOVA $F$ tests:
 H_{0}: 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. 
 H_{0}: $\pi_1 = \pi_2$
Here $\pi_1$ is the population proportion of 'successes' for group 1, and $\pi_2$ is the population proportion of 'successes' for group 2.  H_{0}: for each category $j$ of the dependent variable, $\pi_j$ for the first paired group = $\pi_j$ for the second paired group.
Here $\pi_j$ is the population proportion in category $j.$  
Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  
ANOVA $F$ tests:
 H_{1}: the population scores in some of the related groups are systematically higher or lower than the population scores in other related groups 
 H_{1} two sided: $\pi_1 \neq \pi_2$ H_{1} right sided: $\pi_1 > \pi_2$ H_{1} left sided: $\pi_1 < \pi_2$  H_{1}: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group.  
Assumptions  Assumptions  Assumptions  Assumptions  Assumptions  




 
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.  $z = \dfrac{p_1  p_2}{\sqrt{p(1  p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
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 $\pi_2 > \pi_1.$  Computing the test statistic is a bit complicated and involves matrix algebra. Unless you are following a technical course, you probably won't need to calculate it by hand.  
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 H_{0} were true  Sampling distribution of $Q$ if H_{0} were true  Sampling distribution of $X^2$ if H_{0} were true  Sampling distribution of $z$ if H_{0} were true  Sampling distribution of the test statistic if H_{0} were true  
For main and interaction effects together (model):
 If the number of blocks $N$ is large, approximately the chisquared distribution with $k  1$ degrees of freedom.
For small samples, the exact distribution of $Q$ should be used.  Approximately the chisquared distribution with $J  1$ degrees of freedom  Approximately the standard normal distribution  Approximately the chisquared distribution with $J  1$ degrees of freedom  
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:
 If we denote the test statistic as $X^2$:
 
n.a.  n.a.  n.a.  Approximate $C\%$ confidence interval for $\pi_1  \pi_2$  n.a.  
      Regular (large sample):
   
Effect size  n.a.  n.a.  n.a.  n.a.  
         
ANOVA table  n.a.  n.a.  n.a.  n.a.  
        
Equivalent to  n.a.  n.a.  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.      When testing two sided: chisquared test for the relationship between two categorical variables, where both categorical variables have 2 levels.    
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 (preintervention), measurement point 2 (1 week postintervention), and measurement point 3 (6 weeks postintervention)?  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 $\pi_{high} = 0.2$?  Is the proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic.  Subjects are asked to taste three different types of mayonnaise, and to indicate which of the three types of mayonnaise they like best. They then have to drink a glass of beer, and taste and rate the three types of mayonnaise again. Does drinking a beer change which type of mayonnaise people like best?  
SPSS  SPSS  SPSS  SPSS  SPSS  
Analyze > General Linear Model > Univariate...
 Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
 Analyze > Nonparametric Tests > Legacy Dialogs > Chisquare...
 SPSS does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chisquared test instead. The $p$ value resulting from this chisquared 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...
 
Jamovi  Jamovi  Jamovi  Jamovi  n.a.  
ANOVA > ANOVA
 ANOVA > Repeated Measures ANOVA  Friedman
 Frequencies > N Outcomes  $\chi^2$ Goodness of fit
 Jamovi does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chisquared test instead. The $p$ value resulting from this chisquared 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
   
Practice questions  Practice questions  Practice questions  Practice questions  Practice questions  