Spearman's rho overview

This page offers structured overviews of one or more selected methods. Add additional methods for comparisons (max. of 3) by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table

Spearman's rho	Chi-squared test for the relationship between two categorical variables	Sign test
Variable 1	Independent /column variable	Independent variable
One of ordinal level	One categorical with $I$ independent groups ($I \geqslant 2$)	2 paired groups
Variable 2	Dependent /row variable	Dependent variable
One of ordinal level	One categorical with $J$ independent groups ($J \geqslant 2$)	One of ordinal level
Null hypothesis	Null hypothesis	Null hypothesis
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.	H₀: there is no association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable: H₀: the distribution of the dependent variable is the same in each of the $I$ populations If there is one random sample of size $N$ from the total population: H₀: the row and column variables are independent	H₀: P(first score of a pair exceeds second score of a pair) = P(second score of a pair exceeds first score of a pair) If the dependent variable is measured on a continuous scale, this can also be formulated as: H₀: the population median of the difference scores is equal to zero A difference score is the difference between the first score of a pair and the second score of a pair.
Alternative hypothesis	Alternative hypothesis	Alternative hypothesis
H₁ two sided: $\rho_s \neq 0$ H₁ right sided: $\rho_s > 0$ H₁ left sided: $\rho_s < 0$	H₁: there is an association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable: H₁: the distribution of the dependent variable is not the same in all of the $I$ populations If there is one random sample of size $N$ from the total population: H₁: the row and column variables are dependent	H₁ two sided: P(first score of a pair exceeds second score of a pair) $\neq$ P(second score of a pair exceeds first score of a pair) H₁ right sided: P(first score of a pair exceeds second score of a pair) > P(second score of a pair exceeds first score of a pair) H₁ left sided: P(first score of a pair exceeds second score of a pair) < P(second score of a pair exceeds first score of a pair) If the dependent variable is measured on a continuous scale, this can also be formulated as: H₁ two sided: the population median of the difference scores is different from zero H₁ right sided: the population median of the difference scores is larger than zero H₁ left sided: the population median of the difference scores is smaller than zero
Assumptions	Assumptions	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.	Sample size is large enough for $X^2$ to be approximately chi-squared distributed under the null hypothesis. Rule of thumb: 2 $\times$ 2 table: all four expected cell counts are 5 or more Larger than 2 $\times$ 2 tables: average of the expected cell counts is 5 or more, smallest expected cell count is 1 or more There are $I$ independent simple random samples from each of $I$ populations defined by the independent variable, or there is one simple random sample from the total population	Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
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.	$X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$ Here for each cell, the expected cell count = $\dfrac{\mbox{row total} \times \mbox{column total}}{\mbox{total sample size}}$, the observed cell count is the observed sample count in that same cell, and the sum is over all $I \times J$ cells.	$W = $ number of difference scores that is larger than 0
Sampling distribution of $t$ if H₀ were true	Sampling distribution of $X^2$ if H₀ were true	Sampling distribution of $W$ if H₀ were true
Approximately the $t$ distribution with $N - 2$ degrees of freedom	Approximately the chi-squared distribution with $(I - 1) \times (J - 1)$ degrees of freedom	The exact distribution of $W$ under the null hypothesis is the Binomial($n$, $P$) distribution, with $n =$ number of positive differences $+$ number of negative differences, and $P = 0.5$. If $n$ is large, $W$ is approximately normally distributed under the null hypothesis, with mean $nP = n \times 0.5$ and standard deviation $\sqrt{nP(1-P)} = \sqrt{n \times 0.5(1 - 0.5)}$. Hence, if $n$ is large, the standardized test statistic $$z = \frac{W - n \times 0.5}{\sqrt{n \times 0.5(1 - 0.5)}}$$ follows approximately the standard normal distribution if the null hypothesis were true.
Significant?	Significant?	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 $\alpha$	Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$	If $n$ is small, the table for the binomial distribution should be used: Two sided: Check if $W$ observed in sample is in the rejection region or Find two sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $W$ observed in sample is in the rejection region or Find right sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $W$ observed in sample is in the rejection region or Find left sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$ If $n$ is large, the table for standard normal probabilities can be used: Two sided: Check if $z$ observed in sample is at least as extreme as critical value $z^$ or Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $z$ observed in sample is equal to or larger than critical value $z^$ or Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
n.a.	n.a.	Equivalent to
-	-	Two sided sign test is equivalent to McNemar's test: $W = b$, and $z^2 = X^2$ Friedman test with two related groups
Example context	Example context	Example context
Is there a monotonic relationship between physical health and mental health?	Is there an association between economic class and gender? Is the distribution of economic class different between men and women?	Do people tend to score higher on mental health after a mindfulness course?
SPSS	SPSS	SPSS
Analyze > Correlate > Bivariate... Put your two variables in the box below Variables Under Correlation Coefficients, select Spearman	Analyze > Descriptive Statistics > Crosstabs... Put one of your two categorical variables in the box below Row(s), and the other categorical variable in the box below Column(s) Click the Statistics... button, and click on the square in front of Chi-square Continue and click OK	Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples... Put the two paired variables in the boxes below Variable 1 and Variable 2 Under Test Type, select the Sign test
Jamovi	Jamovi	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	Frequencies > Independent Samples - $\chi^2$ test of association Put one of your two categorical variables in the box below Rows, and the other categorical variable in the box below Columns	Jamovi does not have a specific option for the sign test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the two sided $p$ value that would have resulted from the sign test. Go to: ANOVA > Repeated Measures ANOVA - Friedman Put the two paired variables in the box below Measures
Practice questions	Practice questions	Practice questions

Spearman's rho - overview