# Two-way ANOVA

Two-way ANOVA (analysis of variance) tests the null hypotheses that several univariate samples have the same mean across each of two factors A and B, and that there are no dependencies (interactions) between factors. The samples are assumed to be close to normally distributed and have similar variances. If the sample sizes are equal, these two assumptions are not critical. The default analysis is a fixed-effect crossed design (the most usual case). There is also an option for random-effect crossed two-way ANOVA, and fixed-effect and random-effect, nested two-way ANOVA (menu at bottom right). There is no interaction term for the nested design.

Three columns are needed: A group column (set data type to Group with ‘Column attributes’) with the levels for factor A, a group column with the levels for factor B, and a column of the corresponding measured values.

The algorithm uses weighted means for unbalanced designs. See the Past manual for mathematical details.

### Crossed, fixed effects

For the fixed-effects ANOVA, the *F* ratios are

*F _{A}* = MS

_{A}/ MS

_{wg}

*F*= MS

_{B}_{B}/ MS

_{wg}

*F*= MS

_{AxB}_{AxB}/ MS

_{wg}

### Crossed, random effects

The random-effects ANOVA is computed as fixed effects ANOVA, except

*F _{A}* = MS

_{A}/ MS

_{AxB}

*F*= MS

_{B}_{B}/ MS

_{AB}

For random-effects ANOVA, the components of variance are also computed (only for balanced designs).

### Nested

For the fixed-effects model, *F _{A}* is calculated as above, while for random-effects it is

*F*= MS

_{A}_{A}/ MS

_{B}. The nested factor (factor B) is always random-effect.

### Graph of means

The graph of means is a simple graphical device, traditionally used to see the effects of factors and their interaction for a two-way ANOVA. The means are shown with either the A levels or the B levels on the x axis, and the other levels as separate lines.

### Tukey’s post-hoc tests

Tukey’s post-hoc tests are available for the two main factors, and for all unconfounded interactions. For the main factors, the Studentized Range Statistic *Q* is given in the lower left triangle of the array, and the probabilities *p*(equal) in the upper right. The interaction test uses the “adjusted k” value, accounting for the number of unconfounded comparisons, for up to 7 levels in each factor. For larger number of levels the k value is not adjusted.

**Missing values**: Rows with missing values are deleted.