ANCOVA: overview

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Spearman's rho
Independent variablesIndependent variable
One or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates)One categorical of ordinal level
Dependent variableDependent variable
One quantitative of interval or ratio levelOne categorical of ordinal level
-$\rho_s = 0$
$\rho_s$ is the unknown Spearman correlation in the population.

In words:
there is no monotonic relationship between the two variables in the population
n.a.Alternative hypothesis
-Two sided: $\rho_s \neq 0$
Right sided: $\rho_s > 0$
Left sided: $\rho_s < 0$
-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.
n.a.Test statistic
-$t = \dfrac{r_s \times \sqrt{N - 2}}{\sqrt{1 - r_s^2}} $
where $r_s$ 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.
n.a.Sampling distribution of $t$ if H0 were true
-Approximately a $t$ distribution with $N - 2$ degrees of freedom
-Two sided: Right sided: Left sided:
n.a.Example context
-Is there a monotonic relationship between physical health and mental health?
-Analyze > Correlate > Bivariate...
  • Put your two variables in the box below Variables
  • Under Correlation Coefficients, select Spearman
-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
Practice questionsPractice questions