Spearman's rho  overview
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Spearman's rho  McNemar's test 


Variable 1  Independent variable  
One of ordinal level  2 paired groups  
Variable 2  Dependent variable  
One of ordinal level  One categorical with 2 independent groups  
Null hypothesis  Null hypothesis  
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.  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:
 
Alternative hypothesis  Alternative hypothesis  
H_{1} two sided: $\rho_s \neq 0$ H_{1} right sided: $\rho_s > 0$ H_{1} left sided: $\rho_s < 0$  The alternative hypothesis H_{1} 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:
 
Assumptions  Assumptions  

 
Test statistic  Test statistic  
$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.  $X^2 = \dfrac{(b  c)^2}{b + c}$
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.  
Sampling distribution of $t$ if H_{0} were true  Sampling distribution of $X^2$ if H_{0} were true  
Approximately the $t$ distribution with $N  2$ degrees of freedom  If $b + c$ is large enough (say, > 20), approximately the chisquared 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?  
Two sided:
 For test statistic $X^2$:
 
n.a.  Equivalent to  
 
 
Example context  Example context  
Is there a monotonic relationship between physical health and mental health?  Does a tv documentary about spiders change whether people are afraid (yes/no) of spiders?  
SPSS  SPSS  
Analyze > Correlate > Bivariate...
 Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
 
Jamovi  Jamovi  
Regression > Correlation Matrix
 Frequencies > Paired Samples  McNemar test
 
Practice questions  Practice questions  