Question

1. What is meant by single-factor or one-way ANOVA? Please provide an example.
2. What are factorial design ANOVAs? Can you give an example of a 2 × 2 design? What about a 3 × 2 design?

Answer 1

Solution:

a) What is meant by single-factor or one-way ANOVA? Please provide an example.

The one-way ANOVA is a type of statistical test that determines whether there are any statistically significant differences between the means of three or more independent groups.

Specifically, it tests the null hypothesis:

Ho: ===..........=

H1: At least one is different

where =group mean,=population mean of the ith group and k=number of groups.

If the result is statistically significant, we accept the alternative hypothesis, which is atleast two group means that are statistically significantly different from each other.

Example:A firm wishes to compare four programs for training workers to perform a certain manual task. Twenty new employees are randomly assigned to the training programs, with 5 in each program. At the end of the training period, a test is conducted to see how quickly trainees can perform the task. The number of times the task is performed per minute is recorded for each trainee.

Program 1	Program 2	Program 3	Program 4
9	10	12	9
12	6	14	8
14	9	11	11
11	9	13	7
13	10	11	8

One -way ANOVA is as follows:

SUMMARY
Groups	Count	Sum	Average	Variance
Program 1	5	59	11.8	3.7
Program 2	5	44	8.8	2.7
Program 3	5	61	12.2	1.7
Program 4	5	43	8.6	2.3

ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	54.95	3	18.31667	7.044872	0.003113	3.238872
Within Groups	41.6	16	2.6
Total	96.55	19

From the above table we can see that F observed value=7.044872 is greater than F crit value=3.238872. Hence, we reject null hypothesis.

b) What are factorial design ANOVAs? Can you give an example of a 2 × 2 design? What about a 3 × 2 design?

A factorial ANOVA compares means of two or more independent variables. The factorial ANOVA has two or more independent variables that split the sample in four or more groups. The simplest case of a factorial ANOVA uses two binary variables as independent variables, thus creating four groups within the sample.

In the two factor design, we have two independent variable, each of which has two values.This is known as two by two(2x2) factorial design.