# ANOVA repeated measures -formula and example

### ANOVA repeated measures - design

#### What are ANOVA repeated measures statistical designs?

ANOVA repeated measures statistical designs employ one group of subjects and give them the treatments of the experiments in succession. For example, a researcher that wants to test four different drugs for their possible effect on body temperature uses a singler group of subjects rather than four groups. He will give them Drug 1, wait for a week, then give Drug 2, wait for a week, give Drug 3, wait for a week, then give Drug 4.The difference of repeated measures designs as compared to independent groups designs can be best illustrated by giving the layout of each.

#### The layout of independent groups ANOVA

Treatment 1 Treatment 2 Treatment 3 Treatment 4
Subject 1
Subject 2
Subject 3
Subject 4
Subject 5
Subject 6
Subject 7
Subject 8
Subject 9
Subject 10
Subject 11
Subject 12
Subject 13
Subject 14
Subject 15
Subject 16
Subject 17
Subject 18
Subject 19
Subject 20

#### The layout of repeated measures ANOVA

Subjects Treatment 1 Treatment 2 Treatment 3 Treatment 4
1
2
3
4
5
6
7
8
9
10
X Subject 1
X Subject 2
X Subject 3
X Subject 4
X Subject 5
X Subject 6
X Subject 7
X Subject 8
X Subject 9
X Subject 10
X Subject 1
X Subject 2
X Subject 3
X Subject 4
X Subject 5
X Subject 6
X Subject 7
X Subject 8
X Subject 9
X Subject 10
X Subject 1
X Subject 2
X Subject 3
X Subject 4
X Subject 5
X Subject 6
X Subject 7
X Subject 8
X Subject 9
X Subject 10
X Subject 1
X Subject 2
X Subject 3
X Subject 4
X Subject 5
X Subject 6
X Subject 7
X Subject 8
X Subject 9
X Subject 10

#### FORMAT OF REPEATED MEASURES ANOVA SUMMARY TABLE

SOURCE1 SS df MS F p
Between Treatments
Between Rows
Error
Totalr

#### ANOVA repeated measures - formula

The formula for Anova repeated measures experiments is the formula for variance The final step in the analysis of our data is the F ratio, which is variance of the data in particular ways. $$F={{MS_{between}} \over {MS_{within}}}$$

#### ANOVA repeated measures- practice example

An experimenter wanted to test four new Drugs for their effect on body temperature. He randomly selected six subjects and gave them Drug 1. He waited for one hour and then he measured the temperature of each subject using a Celsius thermometer. He recorded the data. On week later he gave these subjects Drug 2. He waited for one hour and measured the temperature of each subject. A week later he gave these subjects Drug 3 One hour later he measured their temperature One week later he gave these same subjects Drug 4. He waited for one hour and then he measured their temperature. The data are presented in the table below.
Subjects Treatment 1 Treatment 2 Treatment 3 Treatment 4
1
2
3
4
5
6
36.5
36.8
35.4
35.6
36.5
36.4
36.2
36.4
36.6
36.4
36.3
36.2
36.9
36.8
37.7
37.3
37.9
37.1
36.9
37.8
37.7
37.5
37.9
37.6

#### REPEATED MEASURES ANOVA SUMMARY TABLE

SOURCE SS df MS F p
Between Treatments 8.2433 3 2.7478 15.84241 <0.05
Between Rows 3.2967 5 3.2967
Error 2.6017 15 0.1734
Totalr 23

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