Spearman's rho - overview
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Spearman's rho | Binomial test for a single proportion |
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Variable 1 | Independent variable | |
One of ordinal level | None | |
Variable 2 | Dependent variable | |
One of ordinal level | One categorical with 2 independent groups | |
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: $\pi = \pi_0$
Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null 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 two sided: $\pi \neq \pi_0$ H1 right sided: $\pi > \pi_0$ H1 left sided: $\pi < \pi_0$ | |
Assumptions | Assumptions | |
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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. | $X$ = number of successes in the sample | |
Sampling distribution of $t$ if H0 were true | Sampling distribution of $X$ if H0 were true | |
Approximately the $t$ distribution with $N - 2$ degrees of freedom | Binomial($n$, $P$) distribution.
Here $n = N$ (total sample size), and $P = \pi_0$ (population proportion according to the null hypothesis). | |
Significant? | Significant? | |
Two sided:
| Two sided:
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Example context | Example context | |
Is there a monotonic relationship between physical health and mental health? | Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$? | |
SPSS | SPSS | |
Analyze > Correlate > Bivariate...
| Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
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Jamovi | Jamovi | |
Regression > Correlation Matrix
| Frequencies > 2 Outcomes - Binomial test
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Practice questions | Practice questions | |