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Two-way ANOVA Example. Analgesic study with three treatments crossed with gender. Data on pain and factor levels. male C 12.4 female A 7.69 male C 14.0 female A 9.69 male C 11.6 female A 8.89 female A 6.94 female A 2.13 female A 7.26 female A 5.87 male B 12.9 female C 12.2

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two way anova example

Two-way ANOVA Example

Analgesic study with three treatments crossed with gender.

data on pain and factor levels
Data on pain and factor levels

male C 12.4

female A 7.69

male C 14.0

female A 9.69

male C 11.6

female A 8.89

female A 6.94

female A 2.13

female A 7.26

female A 5.87

male B 12.9

female C 12.2

female A 7.20

male C 13.9

male A 8.18

male B 16.6

female C 9.41

male C 11.2

female B 8.35

male A 7.24

female A 6.81

male B 9.81

female A 6.67

female A 6.98

female A 7.07

female C 2.40

male B 7.84

female B 3.84

male B 9.42

male A 7.00

male A 7.00

female A 5.00

male A 8.00

factors and response variable
Factors and Response Variable
  • Response variable, Y, is Pain index
  • Gender is one factor (sometimes called independent variable)
  • Drug (type of analgesic agent) is the other factor
  • All factor combinations are considered
statistical model
Statistical Model

So that Pain level is modeled as a linear function of Factor levels.

anova table
ANOVA table

Source DF Sum of Squares F Ratio Prob> F

gender 1 73.8082 12.6378 0.0014*

drug251.05914.37130.0227*

drug*gender 2 30.5427 2.6148 0.0916

what if we had dropped the interaction from the model since it was not significant
What if we had dropped the interaction from the model since it was not significant?

The Drug Effect is now close to being not significant. Why?

Because we have inflated our MSE. In ANOVA, the model you

originally fit is generally the model you use for reporting significance.

range test on main effect of drug
Range Test on Main Effect of Drug

Level Least Sq Mean

C A 10.316622

B A B 8.710767

A B 7.134856

Levels not connected by same letter are significantly different.

what did interaction term test
What did Interaction term test?

The interaction tests whether an “Additive Model” is adequate. The Additive model only contains Main Effects and has the form:

Another way to think about Interaction is to look at Interaction Plots.

goodness of fit
Goodness of Fit

Goodness-of-Fit Test

Shapiro-Wilk W Test

W Prob<W

0.970483 0.4936

Note: Ho = The data is from the Normal distribution. Small p-values reject Ho.