Double Classification Analysis of Variance

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.

Idealized Repeated Measures Design

Another prototypical scientific experiment involves subjecting a group of subjects to two types of experimental treatment. All subjects are subjected to all conditions of the experiment. An idealized arrangement of subjects prior to the onset of an experiment is outlined as

 

       As within the framework of the independent groups design, initially, we have no reason to assume that the means and variances of both groups will differ. Following some type of intervention induced by the experiment,

 

 

ideally, the variances of the control and experimental groups should not change (assumption of homoscedascity) and the change of the total group's variance should be due to the variance between the changed mean of one of the groups , where the mean of 2 changed to a mean equal to 3.

Variance Due to the Changed Column Means

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

 

 

 Note that the variance of any three consecutive integers equals .67 and the variance of any two consecutive integers equals .25. Thus, the variance of the means for our example (2 and 3) equals .25, and we can partition the variance of the total variable as .92 = .67 + .25.

Variance Due to Changed Row Means

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

 

 

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 + .18. Let us see whether the two-way analysis of variance will support this conjecture.

The Microsoft Excel Framework

For the example, the spreadsheet for the one-way 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

 

Row Component of Variance

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
Sums	6     9	15	43
Squares	36     81	5.5	225
Corrections	12     27	39	1.5	37.5

 

Summing the corrected values of the row sums, for the example 8 + 8 +24.5, and entering this sum into the spreadsheet, fills the last but one empty filed of the spreadsheet, as

 

Data	1     3 2     2 3     4	4 4 7	16 16 49	8 8 24.5
Sums	6     9	15	43	40.5
Squares	36     81	5.5	225
Corrections	12     27	39	1.5	37.5

 

 To obtain the row variance component, subtract the correction term from this sum (40.5 - 37.5), as

 

Data	1     3 2     2 3     4	4 4 7	16 16 49	8 8 24.5
Sums	6     9	15	43	40.5
Squares	36     81	5.5	225	3
Corrections	12     27	39	1.5	37.5

 

In tabular representation,

 

Source of Variance	Extended Variance Components	Standard Variance Components	Variance Components
Columns	1.5	.27	.25
Rows	3.0	.55	.50
Residual	1.0	.18	.17
Total	5.5	1.00	.92

 

Note that the amount of information accounted for increased. The column variance component, due to the experimental treatment, remained the same. The residual component decreased by the row component of variance, due to variability among subjects.

Summary Tables

In the course of the previous analysis, we were searching for the sums of squares to be entered to a table such as outlined in the table below.

 

Source of Variance	Degrees of Freedom	Sums of Squares	Mean Square	F	Probability
Columns	k-1	SS ?	SS / df	MS / MSRES	p ?
Rows	n-1	SS ?	SS / df	MS / MSRES	p ?
Residual	(k-1)(n-1)	SS ?	SS / df
Total	nk-1	SS ?

 

For the example, the traditional summary table of the analysis of variance is

 

Source of Variance	Degrees of Freedom	Sums of Squares	Mean Square	F	Probability
Columns	1	1.5	1.5	3.0	.1440
Rows	2	3.0	1.5	3.0
Residual	2	1.0	.5
Total	5	5.5

 

Using the standard components of variance, the above table can be conceptualized

 

Source of Variance	df	Standard Variance Components	Unbiased Variance Components	F	Probability
Columns	1	.27	.27	3.0	.1440
Rows	2	.55	.27	3.0
Residual	2	.18	.09
Total	5	1.00

 

into a form that, as in the case of the one way analysis of variance, is more informative and easier to interpret. As compared with the summary table for the single classification of variance where the residual source of variance consists of the residual plus row sources of variance, the double-classification analysis of variance is more likely to find the relationships to be statistically significant.

Repeated Measures t-Tests as a Special Case of Two Way Analysis of Variance

As in the case of the independent measures t-test being a special case of the one way analysis of variance, the t-test for the repeated measures design is a special case of the two-way analysis of variance. For our example, the conceptual framework of the correlated t-test is shown in the following table

 

	Y1	Y2	Y1+ Y2
S1	1	3	4
S2	2	2	4
S3	3	4	7
M	2	3	5
2	.67	.67	2
2 /k2y2			.55

 

Conceptualized within the correlational framework, this formula is

 

 

 

In the above table, the k is a constant equal to the number of groups. For the example, the r_xy equals .253 and the k²σ_y² expression equals 2²(12.80).

Summary

Le us recall the idealized form of a repeated measures design

 

 

with variances due to changed column means computed as

 

 

 

and the variances due to changed row means computed as

 

 

We can also directly find the residual component of variance by subtracting the Y₀ and Y₁ variables

 

 

In the above tables, the variances of the means of sums and means of the differences could have been computed directly from the variances of sums and differences. To do that recall that variance of a variable divided by a constant equals to the variance of that variable divided by a square of that constant. Thus the variance components of a two variables can be found as

 

 

 

For the example

 

 

 

 

summarized as

 

 

the standard variance components of two variables as can be computed as

 

 

For the example

 

 

the standard variance components can be summarized as

 

 

The sequential presentation of the independent-measures and repeated-measures t-tests leads within the one way and two way analysis of variance methods leads directly to computation of both the strength and significance of the observed relationships and provides for gradual transition to the more general analysis of variance methods.