# Spearman's rho - overview

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Spearman's rho
Independent variable
One of ordinal level
Dependent variable
One of ordinal level
Null hypothesis
$\rho_s = 0$
$\rho_s$ is the unknown Spearman correlation in the population.

In words:
there is no monotonic relationship between the two variables in the population
Alternative hypothesis
Two sided: $\rho_s \neq 0$
Right sided: $\rho_s > 0$
Left sided: $\rho_s < 0$
Assumptions
Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another

Note: this assumption is only important for the significance test, not for the correlation coefficient itself. The correlation coefficient itself just measures the strength of the monotonic relationship between two variables.
Test statistic
$t = \dfrac{r_s \times \sqrt{N - 2}}{\sqrt{1 - r_s^2}}$
where $r_s$ 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.
Sampling distribution of $t$ if H0 were true
Approximately a $t$ distribution with $N - 2$ degrees of freedom
Significant?
Two sided:
Right sided:
Left sided:
Example context
Is there a monotonic relationship between physical health and mental health?
SPSS
Analyze > Correlate > Bivariate...
• Put your two variables in the box below Variables
• Under Correlation Coefficients, select Spearman
Jamovi
Regression > Correlation Matrix
• Put your two variables in the white box at the right
• Under Correlation Coefficients, select Spearman
• Under Hypothesis, select your alternative hypothesis
Practice questions