In the previous article, I briefly discussed the purpose of analysis of variance (ANOVA) and how it relates to the factorial design. In this article, I will expand the discussion by using an orthodontic example to show how a 2-way ANOVA works and how to interpret the results.
In this example of 2-way ANOVA without interaction, we would like to assess the effect on the generated forces of 2 factors (wire type and bracket type). As a reminder, if the objective were to assess only 1 factor, we would use 1-way ANOVA.
The statistical package output is shown in Table I , and we can see that the type of wire is a significant predictor of the generated forces, whereas type of bracket did not reach statistical significance at the 0.05 alpha level.
| Source | Partial SS | df | MS | F | P value |
|---|---|---|---|---|---|
| Model | 1.84 | 3 | 0.61 | 14.50 | <0.001 |
| Wire type | 1.70 | 2 | 0.85 | 20.09 | <0.001 |
| Bracket type | 0.14 | 1 | 0.14 | 3.33 | 0.08 |
| Residual | 1.86 | 44 | 0.04 | ||
| Total (Model + Residual) | 3.70 | 47 | 0.08 |
Table II shows the same analysis but also includes an interaction test to assess whether there is an interaction between the type of wire and the type of bracket. We can see that the interaction test is nonsignificant at the 0.05 alpha level; hence, we can use the simpler analysis shown in Table I .
| Source | Partial sum of squares | Degrees of freedom | Mean squares | F statistic | P value |
|---|---|---|---|---|---|
| Model | 1.84 | 5 | 0.37 | 8.35 | <0.001 |
| Wire type | 1.70 | 2 | 0.85 | 19.23 | <0.001 |
| Bracket type | 0.14 | 1 | 0.14 | 3.19 | 0.08 |
| Wire type#bracket type | 0.01 | 2 | 0.003 | 0.06 | 0.94 |
| Residual | 1.85 | 42 | 0.04 | ||
| Total (Model + Residual) | 3.70 | 47 | 0.08 |
Table III is generic representation of a 2-way ANOVA table with interaction. If you substitute in Table III the numbers from the first 2 columns in Table II , you should be able to calculate the values in the last 2 columns.
| Source | Sum of squares | Degrees of freedom | Mean squares | F statistic |
|---|---|---|---|---|
| Model | SS M (= SS A + SS B + SS AB ) | Ab − 1 | MS M = SS/df M | M S M M S ( w i t h i n ) |
| Main effect A | SS A | a − 1 | MS A = SS/df A | M S ( A b e t w e e n ) M S ( w i t h i n ) |
| Main effect B | SS B | b − 1 | MS B = SS/df B | M S ( B b e t w e e n ) M S ( w i t h i n ) |
| Interaction effect | SS AB | (a − 1) (b − 1) | MS A*B = SS/df A*B | M S ( A ∗ B b e t w e e n ) M S ( w i t h i n ) |
| Within | SS within | ab(n −1) | MS within = SSw/df within | – |
| Total | Sum of SSm + SSwithin | abn − 1 | – | – |
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