# Spearman's rho - overview

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Spearman's rho | Kruskal-Wallis test | One sample Wilcoxon signed-rank test |
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 of ordinal 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}: $m = m_0$
Here $m$ is the population median, and $m_0$ is the population median 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: $m \neq m_0$H _{1} right sided: $m > m_0$H _{1} left sided: $m < m_0$
| |

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
| - The population distribution of the scores is symmetric
- 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. | $H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$ | Two different types of test statistics can be used, but both will result in the same test outcome. We will denote the first option the $W_1$ statistic (also known as the $T$ statistic), and the second option the $W_2$ statistic.
In order to compute each of the test statistics, follow the steps below:
- For each subject, compute the sign of the difference score $\mbox{sign}_d = \mbox{sgn}(\mbox{score} - m_0)$. The sign is 1 if the difference is larger than zero, -1 if the diffence is smaller than zero, and 0 if the difference is equal to zero.
- For each subject, compute the absolute value of the difference score $|\mbox{score} - m_0|$.
- Exclude subjects with a difference score of zero. This leaves us with a remaining number of difference scores equal to $N_r$.
- Assign ranks $R_d$ to the $N_r$ remaining
*absolute*difference scores. The smallest absolute difference score corresponds to a rank score of 1, and the largest absolute difference score corresponds to a rank score of $N_r$. If there are ties, assign them the average of the ranks they occupy.
- $W_1 = \sum\, R_d^{+}$
or $W_1 = \sum\, R_d^{-}$ That is, sum all ranks corresponding to a positive difference or sum all ranks corresponding to a negative difference. Theoratically, both definitions will result in the same test outcome. However:- Tables with critical values for $W_1$ are usually based on the smaller of $\sum\, R_d^{+}$ and $\sum\, R_d^{-}$. So if you are using such a table, pick the smaller one.
- If you are using the normal approximation to find the $p$ value, it makes things most straightforward if you use $W_1 = \sum\, R_d^{+}$ (if you use $W_1 = \sum\, R_d^{-}$, the right and left sided alternative hypotheses 'flip').
- $W_2 = \sum\, \mbox{sign}_d \times R_d$
That is, for each remaining difference score, multiply the rank of the absolute difference score by the sign of the difference score, and then sum all of the products.
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Sampling distribution of $t$ if H_{0} were true | Sampling distribution of $H$ if H_{0} were true | Sampling distribution of $W_1$ and of $W_2$ 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. | Sampling distribution of $W_1$:
If $N_r$ is large, $W_1$ is approximately normally distributed with mean $\mu_{W_1}$ and standard deviation $\sigma_{W_1}$ if the null hypothesis were true. Here $$\mu_{W_1} = \frac{N_r(N_r + 1)}{4}$$ $$\sigma_{W_1} = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{24}}$$ Hence, if $N_r$ is large, the standardized test statistic $$z = \frac{W_1 - \mu_{W_1}}{\sigma_{W_1}}$$ follows approximately the standard normal distribution if the null hypothesis were true. Sampling distribution of $W_2$: If $N_r$ is large, $W_2$ is approximately normally distributed with mean $0$ and standard deviation $\sigma_{W_2}$ if the null hypothesis were true. Here $$\sigma_{W_2} = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{6}}$$ Hence, if $N_r$ is large, the standardized test statistic $$z = \frac{W_2}{\sigma_{W_2}}$$ follows approximately the standard normal distribution if the null hypothesis were true. If $N_r$ is small, the exact distribution of $W_1$ or $W_2$ should be used. Note: if ties are present in the data, the formula for the standard deviations $\sigma_{W_1}$ and $\sigma_{W_2}$ is more complicated. | |

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 $\alpha$
| For large samples, the table for standard normal probabilities can be used: Two sided: - Check if $z$ observed in sample is at least as extreme as critical value $z^*$ or
- Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
- Check if $z$ observed in sample is equal to or larger than critical value $z^*$ or
- Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
- Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or
- Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
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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 median mental health score of office workers different from $m_0 = 50$? | |

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
| Specify the measurement level of your variable on the Variable View tab, in the column named Measure. Then go to:
Analyze > Nonparametric Tests > One Sample... - On the Objective tab, choose Customize Analysis
- On the Fields tab, specify the variable for which you want to compute the Wilcoxon signed-rank test
- On the Settings tab, choose Customize tests and check the box for 'Compare median to hypothesized (Wilcoxon signed-rank test)'. Fill in your $m_0$ in the box next to Hypothesized median
- Click Run
- Double click on the output table to see the full results
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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 Tests, select Wilcoxon rank
- Under Hypothesis, fill in the value for $m_0$ in the box next to Test Value, and select your alternative hypothesis
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Practice questions | Practice questions | Practice questions | |