SciStat

# Two-way ANOVA

## Description

The two-way analysis of variance is an extension to the one-way analysis of variance. There are two qualitative factors (A and B) on one dependent variable Y.

Three null hypotheses are tested in this procedure:

• factor A does not influence variable Y
• factor B does not influence variable Y
• the effect of factor A on variable Y does not depend on factor B (i.e. there is no interaction of factors A and B).

If the calculated P-values for the two main effects 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 the alternative hypothesis that there is indeed an influence, must be accepted.

Two-way analysis of variance requires that there are data for each combination of the two qualitative factors A and B.

## 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

## Results

The two-way analysis of variance is an extension to the one-way analysis of variance. There are two qualitative factors (A and B) on one dependent variable Y.

### Levene's test for equality of variances

Prior to the ANOVA 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 ANOVA are not met.

### 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 is indeed a difference between groups.

When the 2-factor interaction 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 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.

## Literature

• Altman DG (1991) Practical statistics for medical research. London: Chapman and Hall.
• Armitage P, Berry G, Matthews JNS (2002) Statistical methods in medical research. 4th ed. Blackwell Science.
• Neter J, Kutner MH, Nachtsheim CJ, Wasserman W (1996) Applied linear statistical models. 4th ed. McGraw-Hill.