# Analysis of covariance (ANCOVA)

## Description

Analysis of covariance allows to compare one variable in 2 or more groups taking into account (or to correct for) variability of other variables, called covariates.

Analysis of covariance combines one-way or two-way analysis of variance with linear regression (General Linear Model, GLM).

## Required input

- Dependent data: select a continuous variable.
- Factor A and B: select categorical or qualitative variables. These variables may either contain character or numeric codes. These codes are used to break-up the data into a two-way classification table.
- Filter: an optional filter to include a subset of cases.
- Options
- Residuals: you can select a Tests for Normal distribution of the residuals.

## Results

### Levene's test for equality of variances

Prior to the ANCOVA test, Levene's test for equality of variances is performed. If the Levene test is positive (P<0.05) then the variances in the groups are different (the groups are not homogeneous), and therefore the assumptions for ANCOVA are not met.

### Homogeneity of regression slopes

The interpretation of ANCOVA and the associated adjusted means relies on the assumption of homogeneous regression slopes for the various groups (Huitema, 1980). If this assumption is not met (P<0.05) the ANCOVA results are unreliable.

### Tests of Between-Subjects Effects

If the calculated P-values for the two main factors A and B, or for the 2-factor interaction is less than the conventional 0.05 (5%), then the corresponding null hypothesis is rejected, and you accept the alternative hypothesis that there are indeed differences among groups.

When the 2-factor interaction (FactorA*FactorB) is significant the effect of factor A is dependent on the level of factor B, and it is not recommended to interpret the means and differences between means (see below) of the main factors.

### Estimated marginal means

In the following tables, the marginal means (sometimes referred to as "corrected means") with standard error and 95% Confidence Interval are given for all levels of the two factors. Also, differences between groups, with Standard Error, and Bonferroni corrected P-value and 95% Confidence Interval of the differences are reported.

### General Linear Model

Since this ANCOVA procedure is an implementation of the General Linear Model (GLM), the procedure:

- reverts to one-way ANOVA when you do not specify covariates and only one factor
- reverts to a 2-way ANOVA when you specify 2 factors but no covariates
- reverts to multiple regression when you do not specify factors.

### Analysis of residuals

## Literature

- Glantz SA, Slinker BK (2001) Primer of applied regression & analysis of variance. 2
^{nd}ed. McGraw-Hill. - Huitema BE (1980) The analysis of covariance and alternatives. Wiley-Interscience.
- Neter J, Kutner MH, Nachtsheim CJ, Wasserman W (1996) Applied linear statistical models. 4
^{th}ed. McGraw-Hill. - Wildt AR, Ahtola OT (1978) Analysis of covariance. Sage Publications.