Binomial test for a single proportion  overview
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Binomial test for a single proportion  Friedman test 


Independent variable  Independent/grouping variable  
None  One within subject factor ($\geq 2$ related groups)  
Dependent variable  Dependent variable  
One categorical with 2 independent groups  One of ordinal level  
Null hypothesis  Null hypothesis  
H_{0}: $\pi = \pi_0$
Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis.  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.  
Alternative hypothesis  Alternative hypothesis  
H_{1} two sided: $\pi \neq \pi_0$ H_{1} right sided: $\pi > \pi_0$ H_{1} left sided: $\pi < \pi_0$  H_{1}: the population scores in some of the related groups are systematically higher or lower than the population scores in other related groups  
Assumptions  Assumptions  

 
Test statistic  Test statistic  
$X$ = number of successes in the sample  $Q = \dfrac{12}{N \times k(k + 1)} \sum R^2_i  3 \times N(k + 1)$
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.  
Sampling distribution of $X$ if H0 were true  Sampling distribution of $Q$ if H_{0} were true  
Binomial($n$, $P$) distribution.
Here $n = N$ (total sample size), and $P = \pi_0$ (population proportion according to the null hypothesis).  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.  
Significant?  Significant?  
Two sided:
 If the number of blocks $N$ is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
 
Example context  Example context  
Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$?  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)?  
SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
 Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
 
Jamovi  Jamovi  
Frequencies > 2 Outcomes  Binomial test
 ANOVA > Repeated Measures ANOVA  Friedman
 
Practice questions  Practice questions  