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
MANCOVA
Chi-squared test for the relationship between two categorical variables
 You cannot compare more than 3 methods
Independent variablesIndependent variablesIndependent /column variable
One or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates)One or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates)One categorical with $I$ independent groups ($I \geqslant 2$)
Dependent variableDependent variablesDependent /row variable
One quantitative of interval or ratio levelTwo or more quantitative of interval or ratio levelOne categorical with $J$ independent groups ($J \geqslant 2$)
THIS TABLE IS YET TO BE COMPLETEDTHIS TABLE IS YET TO BE COMPLETEDNull hypothesis
--H0: there is no association between the row and column variable

More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
  • H0: the distribution of the dependent variable is the same in each of the $I$ populations
If there is one random sample of size $N$ from the total population:
  • H0: the row and column variables are independent
n.a.n.a.Alternative hypothesis
--H1: there is an association between the row and column variable

More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
  • H1: the distribution of the dependent variable is not the same in all of the $I$ populations
If there is one random sample of size $N$ from the total population:
  • H1: the row and column variables are dependent
n.a.n.a.Assumptions
--
  • Sample size is large enough for $X^2$ to be approximately chi-squared distributed under the null hypothesis. Rule of thumb:
    • 2 $\times$ 2 table: all four expected cell counts are 5 or more
    • Larger than 2 $\times$ 2 tables: average of the expected cell counts is 5 or more, smallest expected cell count is 1 or more
  • There are $I$ independent simple random samples from each of $I$ populations defined by the independent variable, or there is one simple random sample from the total population
n.a.n.a.Test statistic
--$X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here for each cell, the expected cell count = $\dfrac{\mbox{row total} \times \mbox{column total}}{\mbox{total sample size}}$, the observed cell count is the observed sample count in that same cell, and the sum is over all $I \times J$ cells.
n.a.n.a.Sampling distribution of $X^2$ if H0 were true
--Approximately the chi-squared distribution with $(I - 1) \times (J - 1)$ degrees of freedom
n.a.n.a.Significant?
--
  • 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.n.a.Example context
--Is there an association between economic class and gender? Is the distribution of economic class different between men and women?
n.a.n.a.SPSS
--Analyze > Descriptive Statistics > Crosstabs...
  • Put one of your two categorical variables in the box below Row(s), and the other categorical variable in the box below Column(s)
  • Click the Statistics... button, and click on the square in front of Chi-square
  • Continue and click OK
n.a.n.a.Jamovi
--Frequencies > Independent Samples - $\chi^2$ test of association
  • Put one of your two categorical variables in the box below Rows, and the other categorical variable in the box below Columns
Practice questionsPractice questionsPractice questions