Analysis of variance Two-Way ANOVA

T. Dhasaratharaman*

Statistician, Kauvery Hospitals, India

The two-way ANOVA is an extension of the one-way ANOVA. The “two-way” comes because each item is classified in two ways, as opposed to one way. For example, one-way classifications might be gender, political party, religion, or race. Two-way classifications might be by gender and political party, gender, and race, or religion and race.

Each classification variable is called a factor and so there are two factors, each having several levels within that factor. The factors are called the “row factor” and the “column factor” because the data is usually arranged in table format. Each combination of a row-level and a column-level is called a treatment.

The two-way ANOVA that we are going to discuss requires a balanced design. The balanced design is where each treatment has the same sample size.

Example 1

A pharmaceutical company is testing a new drug to see if it helps reduce the time to recover from a fever. They decide to test the drug on three different races (Caucasian, African American, and Hispanic) and both genders (male and female). This makes six treatments (3 races × 2 genders = 6 treatments). They randomly select five test subjects from each of those six treatments, so all together, they have 3 × 2 × 5 = 30 test subjects. The response variable is the time in minutes after taking the medicine before the fever is reduced. The data might look something like this.

Data

	Male	Female
Caucasian	54, 49, 59, 39, 55	25, 29, 47, 26, 28
African American	53, 72, 43, 56, 52	46, 51, 33, 47, 41
Hispanic	33, 30, 26, 25, 29	18, 21, 34, 40, 24

Example 2

The following data represent clotting times (mins) of plasma from eight subjects treated in four different ways. The eight subjects (blocks) were allocated at random to each of the four treatment groups.

Treatment 1	Treatment 2	Treatment 3	Treatment 4
8.4	9.4	9.8	12.2
12.8	15.2	12.9	14.4
9.6	9.1	11.2	9.8
9.8	8.8	9.9	12
8.4	8.2	8.5	8.5
8.6	9.9	9.8	10.9
8.9	9	9.2	10.4
7.9	8.1	8.2	10

Solution

Variables: Treatment 1, Treatment 2, Treatment 3, Treatment 4

Source of Variation	Sum Squares	Mean Square
Between blocks (rows)	78.98	11.28
Between treatments (columns)	13.01	4.33
Residual (error)	13.77	0.655
Corrected total	105.77

F (VR between blocks) = 17.20 P < 0.0001

F (VR between treatments) = 6.615 P = 0.0025

Here we can see that there was a statistically highly significant difference between mean clotting times across the groups. The difference between subjects is of no particular interest here

Kauverian Bookshelf

Applied Medical Statistics

Analysis of variance Two-Way ANOVA