In the One Way ANOVA example the five replications within each row were all taken under the same test conditions. Consider the example where three treatments are evaluated on four different patients:
Therapy

Andrew

Belinda

Chris

Dave

Relaxed 
110

140

100

130

Normal 
115

150

105

135

High Intensity 
117

155

100

135

The hypothesis is:
H_{0} The therapies all give the same result
H_{1} At least one of the therapies gives a different response
Use a level of significance of 0.05.
The patients are all different, and a 'One Way ANOVA' would not cause the null hypothesis to be rejected. However a two way ANOVA separates the variation due to the therapy from that due to different patient characteristics:
Source of Variation

Sum of Squares

Degrees of Freedom

Mean Square

F_{0}

pvalue 
Therapy

113.17

2

56.58

5.40

0.046

Patient

3832.67

3

1277.56

121.99

0

Error

62.83

6

10.47



Total

4008.67

11




The null hypothesis is rejected, the therapies give different results at the 0.05 level of significance.
