Sign test  overview
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Sign test  Chisquared test for the relationship between two categorical variables 


Independent variable  Independent /column variable  
2 paired groups  One categorical with $I$ independent groups ($I \geqslant 2$)  
Dependent variable  Dependent /row variable  
One of ordinal level  One categorical with $J$ independent groups ($J \geqslant 2$)  
Null hypothesis  Null hypothesis  
 H_{0}: 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:
 
Alternative hypothesis  Alternative hypothesis  
 H_{1}: 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:
 
Assumptions  Assumptions  

 
Test statistic  Test statistic  
$W = $ number of difference scores that is larger than 0  $X^2 = \sum{\frac{(\mbox{observed cell count}  \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
where 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  
Sampling distribution of $W$ if H_{0} were true  Sampling distribution of $X^2$ if H_{0} were true  
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 $\sqrt{np(1p)} = \sqrt{n \times 0.5(1  0.5)}$. 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.  Approximately the chisquared distribution with $(I  1) \times (J  1)$ degrees of freedom  
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:

 
Equivalent to  n.a.  
Two sided sign test is equivalent to
   
Example context  Example context  
Do people tend to score higher on mental health after a mindfulness course?  Is there an association between economic class and gender? Is the distribution of economic class different between men and women?  
SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
 Analyze > Descriptive Statistics > Crosstabs...
 
Jamovi  Jamovi  
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
 
Practice questions  Practice questions  