Spearman's rho  overview
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Spearman's rho  One sample $t$ test for the mean 


Variable 1  Independent variable  
One of ordinal level  None  
Variable 2  Dependent variable  
One of ordinal level  One quantitative of interval or ratio level  
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}: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis.  
Alternative hypothesis  Alternative hypothesis  
H_{1} two sided: $\rho_s \neq 0$ H_{1} right sided: $\rho_s > 0$ H_{1} left sided: $\rho_s < 0$  H_{1} two sided: $\mu \neq \mu_0$ H_{1} right sided: $\mu > \mu_0$ H_{1} left sided: $\mu < \mu_0$  
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.  $t = \dfrac{\bar{y}  \mu_0}{s / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $s$ is the sample standard deviation, and $N$ is the sample size. The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ is removed from $\mu_0$.  
Sampling distribution of $t$ if H_{0} were true  Sampling distribution of $t$ if H_{0} were true  
Approximately the $t$ distribution with $N  2$ degrees of freedom  $t$ distribution with $N  1$ degrees of freedom  
Significant?  Significant?  
Two sided:
 Two sided:
 
n.a.  $C\%$ confidence interval for $\mu$  
  $\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
where the critical value $t^*$ is the value under the $t_{N1}$ distribution with the area $C / 100$ between $t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20). The confidence interval for $\mu$ can also be used as significance test.  
n.a.  Effect size  
  Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean $\bar{y}$ is removed from $\mu_0.$  
n.a.  Visual representation  
  
Example context  Example context  
Is there a monotonic relationship between physical health and mental health?  Is the average mental health score of office workers different from $\mu_0 = 50$?  
SPSS  SPSS  
Analyze > Correlate > Bivariate...
 Analyze > Compare Means > OneSample T Test...
 
Jamovi  Jamovi  
Regression > Correlation Matrix
 TTests > One Sample TTest
 
Practice questions  Practice questions  