Double Classification Analysis of Variance

 

Variance due to Subjects

In the previous chapter, we were discussing partitioning of the total component of variance into the column component of variance and the residual component. In this chapter, we will add another variance component, capturing variance due to the row marginal referents of a data matrix. Within the research in the social sciences, the row component of variance typically captures variance among subjects. Using the same data sets, you may observe that as the total and column variance components remain the same, the variance captured by the row variance component diminishes the residual component of variance.

One-Way ANOVA (Single-Factor ANOVA)

One-way ANOVA involves only one independent variable with two or more levels. There are two forms of one-way ANOVA: one-way independent measures ANOVA and one-way repeated measures ANOVA.

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Single Classification ANOVA

Purpose: Test the differences among two or more independent samples

Worksheet Method

In the context, single classification ANOVA means the classification is taking place in the columns dimension. Thus, single classification ANOVA is corresponding to one-way independent measures ANOVA. There is one independent variable (or factor) with several levels (columns).

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Double Classification ANOVA

Purpose: Test the differences among two or more related samples

Worksheet Method

In the context, double classification ANOVA means the classification is taking place in both rows and columns dimensions. Each row represents one subject and the same subject participated in all the experimental conditions. The double classification ANOVA is corresponding to one-way repeated measures ANOVA. 

Idealized Repeated Measures Design

Before the Experiment

An idealized arrangement of subjects prior to the onset of an experiment is outlined as

 

 

Y0

Y1

(Y0+Y1)/2

Y

Allen

1

1

1

1

Becky

2

2

2

2

Cathy

3

3

3

3

Allen

 

   

1

Becky

 

   

2

Cathy

 

   

3

M

2

2

2

2

s2

.67

.67

.67

.67

          

Within the framework of the repeated measures design, initially, we have no reason to assume that the means and variances of columns, rows, and the total group will differ. Notice that the scores in the above table do not simulate the actual scores, but are, instead, hypothetical assumptions about the scores that could be expected in the absence of the experimental treatment.

Experimental Conditions

Using the same brand of wine, a control group coded as 0 will be given a glass of non-alcoholic wine while an experimental group coded as 1 will be given a glass of wine containing alcohol. Reaction time measurements in seconds are taken one hour after the wine was consumed.

Repeated Measures Design

In repeated measures designs, all subjects are subjected to all conditions of the experiment. There are 3 subjects in each conditions. The total number of subjects is still 3. However, the total number of scores is 6.

O: Non-Alcoholic                 1: Alcohol

Allen                                      Allen     

Becky                                    Becky     

Cathy                                     Cathy  



Why do we use the repeated measures design?

The main purpose is to remove the variance due to subjects from the residual term. Thus, the residual term will be reduced. As the error decreases, the value of a test statistic (e.g., z, t, and F) increases. Larger z, t , or F values are more likely to find the relationships to be statistically significant. 

Repeated Measures Factor (Within-Subjects Factor)

In the context of the repeated measures design, the variable manipulated by the researcher is called the repeated measures factor or within-subjects factor. The repeated measures factor in this study is alcohol consumption with two levels: placebo (non-alcoholic wine) and wine containing alcohol.

Placebo Group

  O: Non-Alcoholic

Allen

Becky

Cathy

 

Experimental Group

  1:  Alcohol

Allen

Becky

Cathy


Dependent Measures

The dependent variable is reaction time in seconds. Reaction time is measured at each level of the within-subjects factor for every subject. 

          Y0                                 Y1

Allen    ? seconds     Allen   ? seconds

Becky ? seconds   Becky  ? seconds

Cathy   ? seconds   Cathy   ? seconds

 

After the Experiment

Following some type of intervention induced by the experiment, the means and variances are computed below  

 

Y0

Y1

(Y0 +Y1)/2

Y

Allen

1

3

2

1

Becky

2

2

2

2

Cathy

3

4

3.5

3

Allen

 

   

3

Becky

 

   

2

Cathy

 

   

4

M

2

3

2.5

2.5

s2

.67

.67

.50

.92

 

Source of Variance

Variance Due to the Changed Column Means

Let us illustrate this conjecture, as shown in the following table, containing the means of the two groups.

 

Column M ean

Y0

2

Y1

3

M

2.5

s2

.25

 

The variance between group means equals .25. 

Variance Due to the Changed Row Means

Let us illustrate this conjecture, as shown in the following table, containing the means of the rows.

 

 

Y0

Y1

(Y0 +Y1)/2

Allen

1

3

2

Becky

2

2

2

Cathy

3

4

3.5

M

2

3

2.5

s2

.67

.67

.50

 

The variance due to changed row means equals .50.

Variance Not Accounted For

By finding variance corresponding to the row means, we can partition the total variance of the data into its column, row, and remaining components as  .92 = .25 + .50 + .17.  You may also compute the error variance as

 

 

Y0

Y1

(Y0 -Y1)/2

Allen

1

3

-1

Becky

2

2

0

Cathy

3

4

-.5

M

2

3

-.50

s2

.67

.67

 .17

 

In Sum of Squares Terms

To transform true variance components to sum of squares, multiple the true variance components by the total number of scores, 6. Thus,

5.5 = 1.5 + 3.0 + 1.0

Let us see whether the double classification analysis of variance will support this conjecture.

 

Double Classification Analysis of Variance within the Microsoft Excel Framework

The Column Component

For the example, the spreadsheet for the single classification analysis of variance, described in the previous chapter is   

 

Data

Sums

Squares

Corrections

 

Data

1     3

2     2

3     4

 

 

 

Sums

6     9

15

43

 

Squares

36   81

5.5

255

 

Corrections

12   27

39

1.5

37.5

   

The Row Component

Adding sums for the rows of the data matrix can further partition the variance of the data

 

Data

Sums

Squares

Corrections

 

Data

1     3

2     2

3     4

4

4

7

 

 

Sums

6     9

15

43

 

Squares

36     81

5.5

225

 

Corrections

12     27

39

1.5

37.5

 

and their squares 

 

Data

1     3

2     2

3     4

4

4

7

16

16

49

 

Sums

6     9

15

43

 

Squares

36     81

5.5

225

 

Corrections

12     27

39

1.5

37.5

 

Dividing the squared row sums by their respective ns, for the example (16 / 2), (16 / 2), and (49 / 2), results in

 

Data

1     3

2     2

3     4

4

4

7

16

16

49

8

8

24.5

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