Spearman's rho

This page offers all the basic information you need about Spearman's rho and its significance test. It is part of Statkat’s wiki module, containing similarly structured info pages for many different statistical methods. The info pages give information about null and alternative hypotheses, assumptions, test statistics and confidence intervals, how to find p values, SPSS how-to’s and more.

To compare Spearman's rho with other statistical methods, go to Statkat's or practice with Spearman's rho at Statkat's

1. When to use
2. Null hypothesis
3. Alternative hypothesis
4. Assumptions
5. Test statistic
6. Sampling distribution
7. Significant?
8. Example context
9. SPSS
10. Jamovi

When to use?

Deciding which statistical method to use to analyze your data can be a challenging task. Whether a statistical method is appropriate for your data is partly determined by the measurement level of your variables. Spearman's rho requires the following variable types:

Variable types required for Spearman's rho :

Variable 1:
One of ordinal level

Variable 2:
One of ordinal level

Note that theoretically, it is always possible to 'downgrade' the measurement level of a variable. For instance, a test that can be performed on a variable of ordinal measurement level can also be performed on a variable of interval measurement level, in which case the interval variable is downgraded to an ordinal variable. However, downgrading the measurement level of variables is generally a bad idea since it means you are throwing away important information in your data (an exception is the downgrade from ratio to interval level, which is generally irrelevant in data analysis).

If you are not sure which method you should use, you might like the assistance of our method selection tool or our method selection table.

Null hypothesis

The test for Spearman's rho tests the following null hypothesis (H₀):

H₀: $\rho_s = 0$

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:

H₀: there is no monotonic relationship between the two variables in the population.

Alternative hypothesis

The test for Spearman's rho tests the above null hypothesis against the following alternative hypothesis (H₁ or H_a):

H₁ two sided: $\rho_s \neq 0$
H₁ right sided: $\rho_s > 0$
H₁ left sided: $\rho_s < 0$

Assumptions

Statistical tests always make assumptions about the sampling procedure that was used to obtain the sample data. So called parametric tests also make assumptions about how data are distributed in the population. Non-parametric tests are more 'robust' and make no or less strict assumptions about population distributions, but are generally less powerful. Violation of assumptions may render the outcome of statistical tests useless, although violation of some assumptions (e.g. independence assumptions) are generally more problematic than violation of other assumptions (e.g. normality assumptions in combination with large samples).

The test for Spearman's rho makes the following 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

The test for Spearman's rho is based on the following 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.

Sampling distribution

Sampling distribution of $t$ if H₀ were true:

Approximately the $t$ distribution with $N - 2$ degrees of freedom

Significant?

This is how you find out if your test result is significant:

Two sided:

Check if $t$ observed in sample is at least as extreme as critical value $t^*$ or
Find two sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$

Right sided:

Check if $t$ observed in sample is equal to or larger than critical value $t^*$ or
Find right sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$

Left sided:

Check if $t$ observed in sample is equal to or smaller than critical value $t^*$ or
Find left sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$

Example context

The test for Spearman's rho could for instance be used to answer the question:

Is there a monotonic relationship between physical health and mental health?

SPSS

How to compute Spearman's rho in SPSS:

Analyze > Correlate > Bivariate...

Put your two variables in the box below Variables
Under Correlation Coefficients, select Spearman

Jamovi

How to compute Spearman's rho in 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