# Two-way analysis of variance: Part 1

## Two-way analysis of variance: Part 1

American Journal of Orthodontics and Dentofacial Orthopedics, 2015-12-01, Volume 148, Issue 6, Pages 1078-1079, Copyright © 2015 American Association of Orthodontists

In the previous article, I discussed 1-way analysis of variance (ANOVA) when we are interested in comparing the generated forces from 3 types of wires. In that example, we considered only 1 factor, which was the type of wire. If we want to compare simultaneously the effect of 2 factors on the outcome of interest, then we can use 2-way ANOVA. The 2 independent variables in a 2-way ANOVA are called factors, indicating 2 variables that affect the dependent variable, and each factor has 2 or more levels in it. There are 3 sets of hypotheses with the 2-way ANOVA.

• 1.

The sample means across the levels of the first factor are equal, just like the 1-way ANOVA (row factor).

• 2.

The sample means across the levels of the second factor are equal, just like the 1-way ANOVA (column factor).

• 3.

There is no interaction between the 2 factors. No interaction means that the first factor affects the dependent variable on average similarly in each level of the second factor, and vice versa.

The first 2 hypotheses assess the main effects. The main effects involve the independent variables one at a time, ignoring any interaction. Just the rows or just the columns are used; this part is similar to the 1-way ANOVA. The third hypothesis assesses the interaction effect: ie, the effect of one factor on the other factor.

## Example

Research question: Compare the forces generated (continuous outcome) for the 3 types of wires, A, B, and C, and also account for another parameter such as type of bracket—conventional or self-ligating. We can make this assessment by using 2-way ANOVA, since we have 2 factors: wire type (3 levels: A, B, and C) and bracket type (2 levels: labial and lingual). This design is a 3 × 2 factorial design, and it is efficient, in the absence of interaction, because it allows testing of 2 factors in a study by using the same sample size.

The Figure is a diagrammatic presentation of the 3 × 2 factorial design to assess simultaneously the effect of wire type and bracket mesh type on the generated forces. Diagrammatic representation of a 3 × 2 factorial design.

The forces generated by conventional brackets can be compared with those generated by lingual self-ligating brackets irrespective of wire type (y1 + y2 + y3 vs y4 + y5 + y6) and similarly for wire type irrespective of bracket type (y1 + y4 vs y2 + y5, y1 + y4 vs y3 + y6, y2 + y5 vs y3 + y6) as shown in the Table .

Table
A factorial 3 × 2 table: the first factor is the type of wire (A, B, or C) and the second is the type of bracket (conventional or self-ligating)
Wire A Wire B Wire C
Bracket type
Conventional y1 y2 y3
Self-ligating y4 y5 y6

Interaction is a tricky term that is sometimes difficult to understand intuitively. In this example, interaction means that the generated forces differ significantly between the 2 bracket types, and vice versa. The comparisons across rows or columns are valid only in the absence of interaction.

Many pair-wise comparisons are possible in each row (eg, y1 vs y2, y1 vs y3, y2 vs y3, y4 vs y5, y4 vs y6, and y5 vs y6) or column (y1 vs y4, y2 vs y5, and y3 vs y6). However, as the number of comparisons increases, so does the chance for false positives. Additionally, testing on subsamples reduces precision and, hence, the statistical power of the results. In general, it is best to prespecify the number of post hoc comparisons, use them mainly on an exploratory basis, and be cautious about making inferences from them. Irrational post hoc multiple testing after looking at the results has been termed “data dredging” and can be misleading if only significant results are reported.

In the next article, I will give examples of using 2-way-ANOVA with and with an interaction.

## Reference

• 1. Pandis N., Walsh T., Polychronopoulou A., Katsaros C., Eliades T.: Factorial designs: an overview with applications to orthodontic clinical trials. Eur J Orthod 2014; 36: pp. 314-320.