Spearman's rho  overview
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Spearman's rho  KruskalWallis test  $z$ test for the difference between two proportions 
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Variable 1  Independent/grouping variable  Independent/grouping variable  
One of ordinal level  One categorical with $I$ independent groups ($I \geqslant 2$)  One categorical with 2 independent groups  
Variable 2  Dependent variable  Dependent variable  
One of ordinal level  One of ordinal level  One categorical with 2 independent groups  
Null hypothesis  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.  If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations:
Formulation 1:
 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  Alternative hypothesis  
H_{1} two sided: $\rho_s \neq 0$ H_{1} right sided: $\rho_s > 0$ H_{1} left sided: $\rho_s < 0$  If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations:
Formulation 1:
 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  Assumptions  


 
Test statistic  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.  $H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i}  3(N + 1)$  $z = \dfrac{p_1  p_2}{\sqrt{p(1  p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
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 $H$ if H_{0} were true  Sampling distribution of $z$ if H_{0} were true  
Approximately the $t$ distribution with $N  2$ degrees of freedom  For large samples, approximately the chisquared distribution with $I  1$ degrees of freedom. For small samples, the exact distribution of $H$ should be used.  Approximately the standard normal distribution  
Significant?  Significant?  Significant?  
Two sided:
 For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
 Two sided:
 
n.a.  n.a.  Approximate $C\%$ confidence interval for $\pi_1  \pi_2$  
    Regular (large sample):
 
n.a.  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  Example context  
Is there a monotonic relationship between physical health and mental health?  Do people from different religions tend to score differently on social economic status?  Is the proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic.  
SPSS  SPSS  SPSS  
Analyze > Correlate > Bivariate...
 Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
 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  Jamovi  
Regression > Correlation Matrix
 ANOVA > One Way ANOVA  KruskalWallis
 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  Practice questions  