Spearman's rho - overview

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Spearman's rho
MANCOVA
Variable 1Independent variables
One of ordinal levelOne or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates)
Variable 2Dependent variables
One of ordinal levelTwo or more quantitative of interval or ratio level
Null hypothesisTHIS 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.
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Alternative hypothesisn.a.
H1 two sided: $\rho_s \neq 0$
H1 right sided: $\rho_s > 0$
H1 left sided: $\rho_s < 0$
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Assumptionsn.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.
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Test statisticn.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.
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Sampling distribution of $t$ if H0 were truen.a.
Approximately the $t$ distribution with $N - 2$ degrees of freedom-
Significant?n.a.
Two sided: Right sided: Left sided: -
Example contextn.a.
Is there a monotonic relationship between physical health and mental health?-
SPSSn.a.
Analyze > Correlate > Bivariate...
  • Put your two variables in the box below Variables
  • Under Correlation Coefficients, select Spearman
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Jamovin.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
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Practice questionsPractice questions