Spearman's rho overview

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Spearman's rho	One way MANOVA	Binomial test for a single proportion $z$ test for a single proportion $z$ test for the difference between two proportions Goodness of fit test Chi-squared test for the relationship between two categorical variables One sample $z$ test for the mean One sample $t$ test for the mean Paired sample $t$ test Two sample $z$ test Two sample $t$ test - equal variances not assumed Two sample $t$ test - equal variances assumed One way ANOVA Two way ANOVA Pearson correlation Regression (OLS)Logistic regression Mann-Whitney-Wilcoxon test Kruskal-Wallis test Sign test McNemar's test Cochran's Q test Marginal Homogeneity test / Stuart-Maxwell test Friedman test Wilcoxon signed-rank test One sample Wilcoxon signed-rank test Spearman's rho
Variable 1	Independent/grouping variable
One of ordinal level	One categorical with $I$ independent groups ($I \geqslant 2$)
Variable 2	Dependent variables
One of ordinal level	Two or more quantitative of interval or ratio level
Null hypothesis	THIS TABLE IS YET TO BE COMPLETED
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	n.a.
H₁ two sided: $\rho_s \neq 0$ H₁ right sided: $\rho_s > 0$ H₁ left sided: $\rho_s < 0$	-
Assumptions	n.a.
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	n.a.
$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 of $t$ if H₀ were true	n.a.
Approximately the $t$ distribution with $N - 2$ degrees of freedom	-
Significant?	n.a.
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	n.a.
Is there a monotonic relationship between physical health and mental health?	-
SPSS	n.a.
Analyze > Correlate > Bivariate... Put your two variables in the box below Variables Under Correlation Coefficients, select Spearman	-
Jamovi	n.a.
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	Practice questions

Spearman's rho - overview