Spearman's rho overview

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Spearman's rho	Kruskal-Wallis test	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 variable
One of ordinal level	One of ordinal level
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.	If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations: H₀: the population medians for the $I$ groups are equal Else: Formulation 1: H₀: the population scores in any of the $I$ groups are not systematically higher or lower than the population scores in any of the other groups Formulation 2: H₀: P(an observation from population $g$ exceeds an observation from population $h$) = P(an observation from population $h$ exceeds an observation from population $g$), for each pair of groups. Several different formulations of the null hypothesis can be found in the literature, and we do not agree with all of them. Make sure you (also) learn the one that is given in your text book or by your teacher.
Alternative hypothesis	Alternative hypothesis
H₁ two sided: $\rho_s \neq 0$ H₁ right sided: $\rho_s > 0$ H₁ left sided: $\rho_s < 0$	If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations: H₁: not all of the population medians for the $I$ groups are equal Else: Formulation 1: H₁: the poplation scores in some groups are systematically higher or lower than the population scores in other groups Formulation 2: H₁: for at least one pair of groups: P(an observation from population $g$ exceeds an observation from population $h$) $\neq$ P(an observation from population $h$ exceeds an observation from population $g$)
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.	Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2, $\ldots$, group $I$ sample is an independent SRS from population $I$. That is, within and between groups, observations are independent of one another
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.	$H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$ Here $N$ is the total sample size, $R_i$ is the sum of ranks in group $i$, and $n_i$ is the sample size of group $i$. Remember that multiplication precedes addition, so first compute $\frac{12}{N (N + 1)} \times \sum \frac{R^2_i}{n_i}$ and then subtract $3(N + 1)$. Note: if ties are present in the data, the formula for $H$ is more complicated.
Sampling distribution of $t$ if H₀ were true	Sampling distribution of $H$ if H₀ were true
Approximately the $t$ distribution with $N - 2$ degrees of freedom	For large samples, approximately the chi-squared distribution with $I - 1$ degrees of freedom. For small samples, the exact distribution of $H$ should be used.
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 $\alpha$	For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$: 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 $\alpha$
Example context	Example context
Is there a monotonic relationship between physical health and mental health?	Do people from different religions tend to score differently on social economic status?
SPSS	SPSS
Analyze > Correlate > Bivariate... Put your two variables in the box below Variables Under Correlation Coefficients, select Spearman	Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples... Put your dependent variable in the box below Test Variable List and your independent (grouping) variable in the box below Grouping Variable Click on the Define Range... button. If you can't click on it, first click on the grouping variable so its background turns yellow Fill in the smallest value you have used to indicate your groups in the box next to Minimum, and the largest value you have used to indicate your groups in the box next to Maximum Continue and click OK
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	ANOVA > One Way ANOVA - Kruskal-Wallis Put your dependent variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
Practice questions	Practice questions

Spearman's rho - overview