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.

Table I

Statistical package output for 2-way ANOVA assessing the effect of wire type and bracket type, assuming no interaction between the 2 factors: ie, the generated force by the type of wire is independent of the type of bracket, or the generated force by the type of bracket is independent of the type of wire

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

Number of observations, 48; R ^{2 }= 0.50; root mean square error = 0.21; adjusted R ^{2 }= 0.46.

SS , Sum of squares; MS , mean squares.

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 .

Table II

Statistical package output for 2-way ANOVA assessing the effect of wire type and bracket type with interaction between the 2 factors

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

Number of observations, 48; R ^{2 }= 0.50; root mean square error = 0.210159; adjusted R ^{2 }= 0.44.

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.

Table III

Two-way analysis of variance with interaction

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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