ANCOVA overview

This page offers structured overviews of one or more selected methods. Add additional methods for comparisons (max. of 3) by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table

ANCOVA	Spearman's rho	Marginal Homogeneity test / Stuart-Maxwell test
Independent variables	Variable 1	Independent variable
One or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates)	One of ordinal level	2 paired groups
Dependent variable	Variable 2	Dependent variable
One quantitative of interval or ratio level	One of ordinal level	One categorical with $J$ independent groups ($J \geqslant 2$)
THIS TABLE IS YET TO BE COMPLETED	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: for each category $j$ of the dependent variable, $\pi_j$ for the first paired group = $\pi_j$ for the second paired group. Here $\pi_j$ is the population proportion in category $j.$
n.a.	Alternative hypothesis	Alternative hypothesis
-	H₁ two sided: $\rho_s \neq 0$ H₁ right sided: $\rho_s > 0$ H₁ left sided: $\rho_s < 0$	H₁: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group.
n.a.	Assumptions	Assumptions
-	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.	Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
n.a.	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.	Computing the test statistic is a bit complicated and involves matrix algebra. Unless you are following a technical course, you probably won't need to calculate it by hand.
n.a.	Sampling distribution of $t$ if H₀ were true	Sampling distribution of the test statistic if H₀ were true
-	Approximately the $t$ distribution with $N - 2$ degrees of freedom	Approximately the chi-squared distribution with $J - 1$ degrees of freedom
n.a.	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 $\alpha$ Right sided: 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 $\alpha$ Left sided: 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 $\alpha$	If we denote the test statistic as $X^2$: 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 $\alpha$
n.a.	Example context	Example context
-	Is there a monotonic relationship between physical health and mental health?	Subjects are asked to taste three different types of mayonnaise, and to indicate which of the three types of mayonnaise they like best. They then have to drink a glass of beer, and taste and rate the three types of mayonnaise again. Does drinking a beer change which type of mayonnaise people like best?
n.a.	SPSS	SPSS
-	Analyze > Correlate > Bivariate... Put your two variables in the box below Variables Under Correlation Coefficients, select Spearman	Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples... Put the two paired variables in the boxes below Variable 1 and Variable 2 Under Test Type, select the Marginal Homogeneity test
n.a.	Jamovi	n.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	-
Practice questions	Practice questions	Practice questions

ANCOVA - overview