Spearman's rho  overview
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Spearman's rho  Marginal Homogeneity test / StuartMaxwell test 


Variable 1  Independent variable  
One of ordinal level  2 paired groups  
Variable 2  Dependent variable  
One of ordinal level  One categorical with $J$ independent groups ($J \geqslant 2$)  
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.  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  
H_{1} two sided: $\rho_s \neq 0$ H_{1} right sided: $\rho_s > 0$ H_{1} left sided: $\rho_s < 0$  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  

 
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.  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.  
Sampling distribution of $t$ if H_{0} were true  Sampling distribution of the test statistic if H_{0} were true  
Approximately the $t$ distribution with $N  2$ degrees of freedom  Approximately the chisquared distribution with $J  1$ degrees of freedom  
Significant?  Significant?  
Two sided:
 If we denote the test statistic as $X^2$:
 
Example context  Example context  
Is there a monotonic relationship between physical health and mental health?  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  
Analyze > Correlate > Bivariate...
 Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
 
Jamovi  n.a.  
Regression > Correlation Matrix
   
Practice questions  Practice questions  