# Spearman's rho - overview

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Spearman's rho | Kruskal-Wallis test | One sample $t$ test for the mean |
You cannot compare more than 3 methods |
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Variable 1 | Independent/grouping variable | Independent variable | |

One of ordinal level | One categorical with $I$ independent groups ($I \geqslant 2$) | None | |

Variable 2 | Dependent variable | Dependent variable | |

One of ordinal level | One of ordinal level | One quantitative of interval or ratio level | |

Null hypothesis | Null hypothesis | Null hypothesis | |

H_{0}: $\rho_s = 0$
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: H _{0}: 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:
- H
_{0}: the population medians for the $I$ groups are equal
Formulation 1: - H
_{0}: 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
- H
_{0}: 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.
| H_{0}: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis. | |

Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |

H_{1} two sided: $\rho_s \neq 0$H _{1} right sided: $\rho_s > 0$H _{1} 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:
- H
_{1}: not all of the population medians for the $I$ groups are equal
Formulation 1: - H
_{1}: the poplation scores in some groups are systematically higher or lower than the population scores in other groups
- H
_{1}: 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 $g$)
| H_{1} two sided: $\mu \neq \mu_0$H _{1} right sided: $\mu > \mu_0$H _{1} left sided: $\mu < \mu_0$
| |

Assumptions | Assumptions | Assumptions | |

- Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
| - 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
| - Scores are normally distributed in the population
- Sample is a simple random sample from the population. That is, observations 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. | $H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$ | $t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $s$ is the sample standard deviation, and $N$ is the sample size. The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ is removed from $\mu_0$. | |

Sampling distribution of $t$ if H_{0} were true | Sampling distribution of $H$ if H_{0} were true | Sampling distribution of $t$ if H_{0} 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. | $t$ distribution with $N - 1$ degrees of freedom | |

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 $\alpha$
- 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 $\alpha$
- 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 $\alpha$
| 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 $\alpha$
- 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 $\alpha$
- 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 $\alpha$
| |

n.a. | n.a. | $C\%$ confidence interval for $\mu$ | |

- | - | $\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
where the critical value $t^*$ is the value under the $t_{N-1}$ distribution with the area $C / 100$ between $-t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20). The confidence interval for $\mu$ can also be used as significance test. | |

n.a. | n.a. | Effect size | |

- | - | Cohen's $d$:Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean $\bar{y}$ is removed from $\mu_0.$ | |

n.a. | n.a. | Visual representation | |

- | - | ||

Example context | 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? | Is the average mental health score of office workers different from $\mu_0 = 50$? | |

SPSS | 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
| Analyze > Compare Means > One-Sample T Test...
- Put your variable in the box below Test Variable(s)
- Fill in the value for $\mu_0$ in the box next to Test Value
| |

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
| 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
| T-Tests > One Sample T-Test
- Put your variable in the box below Dependent Variables
- Under Hypothesis, fill in the value for $\mu_0$ in the box next to Test Value, and select your alternative hypothesis
| |

Practice questions | Practice questions | Practice questions | |