# Conduct and Interpret a Factorial ANCOVA

**What is the Factorial ANCOVA?**

ANCOVA is short for *An**alysis of Covariance*. The factorial analysis of covariance is a combination of a factorial ANOVA and a regression analysis.

In basic terms, the ANCOVA looks at the influence of two or more independent variables on a dependent variable while removing the effect of the covariate factor. ANCOVA first conducts a regression of the independent variables (the covariate) on the dependent variable. The residuals (the unexplained variance in the regression model) are then subject to an ANOVA. Thus the ANCOVA tests whether the independent variables still influence the dependent variable after the influence of the covariate(s) has been removed.

The factorial ANCOVA includes more than one independent variable and the factorial ANCOVA can include more than one covariate, SPSS handles up to ten. If the ANCOVA model has more than one covariate it is possible to run the factorial ANCOVA with contrasts and post hoc tests just like the one-way ANCOVA or the ANOVA to identify the influence of each covariate.

The factorial ANCOVA is most useful in two ways: 1) it explains a factorial ANOVA’s within-group variance, and 2) it controls confounding factors.

First, the analysis of variance splits the total variance of the dependent variable into:

- Variance explained by each of the independent variables (also called between-groups variance of the main effect)
- Variance explained by all of the independent variables together (also called the interaction effect)
- Unexplained variance (also called within-group variance)

The factorial ANCOVA looks at the unexplained variance and tries to explain some of it with the covariate(s). Thus it increases the power of the factorial ANOVA by explaining more variability in the model. *[Note that just like in regression analysis and all linear models, over-fitting might occur. That is, the more covariates you enter into the factorial ANCOVA the more variance it will explain, but the fewer degrees of freedom the model has. Thus entering a weak covariate into the factorial ANCOVA decreases the statistical power of the analysis instead of increasing it.]*

Secondly, the factorial ANCOVA eliminates the covariates effect on the relationship between independent variables and the dependent variable which is tested with a factorial ANOVA. The concept is very similar to the partial correlation analysis. Technically it is a semi-partial regression and correlation.

The factorial ANCOVA needs at least four variables (the simplest case with two factors is called two-way ANCOVA):

- Two or more independent variables, which group the cases into four or more groups. The independent variable has to be at least of nominal scale.
- The dependent variable, which is influenced by the independent variable. It has to be of continuous-level scale (interval or ratio data). Also, it needs to be homoscedastic and multivariate normal
- The covariate, also referred to as the confounding factor, or concomitant variable, is the variable that moderates the impact of the independent on the dependent variable. The covariate needs to be a continuous-level variable (interval or ratio data). The ANCOVA covariate is often a pre-test value or a baseline.

Typical questions the factorial ANCOVA answers are as follows:

**Medicine**– Does a drug work? Does the average life expectancy significantly differ between the three groups that received the drug versus the established product versus the control and accounting for the dose (high/low)? This question can be answered with a factorial ANOVA. The factorial ANCOVA allows additional control of covariates that might influence the outcome but have nothing to do with the drug, for example healthiness of lifestyle, risk taking activities, age.**Sociology**– Are rich people living in the countryside happier? Do different income classes report a significantly different satisfaction with life when looking where they live (urban, suburban, rural)? This question can be answered with a factorial ANOVA. Additionally the factorial ANCOVA controls for confounding factors that might influence satisfaction with life, e.g., marital status, job satisfaction, social support system.**Management Studies**– Which brands from the BCG matrix have a higher customer loyalty? The BCG matrix measures brands in a brand portfolio with their business growth rate (high/low) and their market share (high/ low). A factorial ANOVA answers the question to which brand are customers more loyal – stars, cash cows, dogs, or question marks? And a factorial ANCOVA can control for confounding factors, like satisfaction with the brand or appeal to the customer.

*The Factorial ANCOVA in SPSS*

The Factorial ANCOVA is part of the General Linear Models in SPSS. The GLM procedures in SPSS contain the ability to include 1-10 covariates into an ANCOVA model. Without a covariate the GLM procedure calculates the same results as the Factorial ANOVA. The levels of measurement need to be defined in SPSS in order for the GLM procedure to work correctly.

We return to the research question from the chapter on the factorial ANOVA. This time we want to know if gender and the outcome of the final exam (pass /fail) have an influence on the math score when we control for the reading ability as measured by the score of the standardized reading test.

The Factorial ANCOVA can be found in *Analyze/General Linear Model/Univariate…*

This opens the GLM dialog, which allows us to specify any linear model. To answer our research question we need to add the independent variables (*Exam *and* Gender*) to the list of fixed factors.

*Remember that the factor is fixed if it is deliberately manipulated and not just randomly drawn from a population. In our ANCOVA example this is the case. This also makes the ANCOVA the model of choice when analyzing semi-partial correlations in an experiment, instead of the partial correlation analysis with requires random data.*]

The Dependent Variable is the *Math Test*, and the Covariate is the *Reading** Test*.

In the dialog box *Model… *we leave all settings on the default. The default for all GLM (including the Factorial ANCOVA) is the full factorial model. The field *post hocs* is disabled when one or more covariates are entered into the analysis. If we want to include a group comparison into our factorial ANCOVA we can add contrast to the analysis.

If we want to compare all groups against a specific group we need to select *Simple* as the Contrast Method, and we also need to specify which group (either the first or the last) should be compared against all other groups. However, since in this example both of our fixed factors only have two factor levels (male/female and pass/fail) we do not really need contrasts to answer the research question.