Spearman's rho  overview
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Spearman's rho  $z$ test for the difference between two proportions 


Variable 1  Independent/grouping variable  
One of ordinal level  One categorical with 2 independent 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.  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.  
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} two sided: $\pi_1 \neq \pi_2$ H_{1} right sided: $\pi_1 > \pi_2$ H_{1} left sided: $\pi_1 < \pi_2$  
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.  $z = \dfrac{p_1  p_2}{\sqrt{p(1  p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
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 $\pi_2 > \pi_1.$  
Sampling distribution of $t$ if H_{0} were true  Sampling distribution of $z$ if H_{0} were true  
Approximately the $t$ distribution with $N  2$ degrees of freedom  Approximately the standard normal distribution  
Significant?  Significant?  
Two sided:
 Two sided:
 
n.a.  Approximate $C\%$ confidence interval for $\pi_1  \pi_2$  
  Regular (large sample):
 
n.a.  Equivalent to  
  When testing two sided: chisquared test for the relationship between two categorical variables, where both categorical variables have 2 levels.  
Example context  Example context  
Is there a monotonic relationship between physical health and mental health?  Is the proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic.  
SPSS  SPSS  
Analyze > Correlate > Bivariate...
 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...
 
Jamovi  Jamovi  
Regression > Correlation Matrix
 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  