Coded Multiple Regression

Coded regression analysis is used to partition variance within the context of various experimental designs where the coded predictor variables index the membership of subjects in various conditions of the experiment. Whenever possible, the conditions of the experiment should be coded by using orthogonal predictor variables. The values of the orthogonal coding vectors should sum to zero, as should their products. One of algorithms to generate orthogonal  variables is Helmert's procedure.

Helmert's Procedure

A convenient way to obtain a set of mutually orthogonal variables is to use Helmert's procedure. If you need k orthogonal variables, outline a matrix with k columns and k+1 rows and enter the elements column-wise. Place k as the first element of the first column and fill the rest of the column by -1s. Place 0 as the first element of the second column, k-1 as its second element, and fill the rest of the column with -1s. Create the third column by entering two 0's, k-2, filling the rest of the elements with -1s. Continue in this fashion, entering 0s, decrementing k and entering -1s until all columns are filled. The last column will be filled with 0s but for its last two elements, 1 and -1. Outline of this algorithm is

For example, when you need four orthogonal variables, select ( Codes, Helmert Orthogonal Codes ), fill-out the input panel,

and click on the Accept command.

Notice that every column of the above matrix sums to zero. To illustrate that the sum of sums or products of any variable with any other variable also equals zero,  select ( Transfers, Launch Matrix Module, Transfers, Vectors to Matrix ) and Transpose the transferred data. Click on the X + Y command

and on the X * Y command

Select ( Analysis i, Correlation, Matrix of Correlations ) to verify

that the correlation matrix, associated with Helmert's coefficients, is an identity matrix.

For three orthogonal variables, the Helmert's contrasts are

and for two variables

  Coding Experiments by Helmert Contrasts

Let us consider a simple experiment. One of the symptoms of Korsakoff psychosis, a disease of chronic alcoholics, is a marked loss of short-term memory and this experiment was designed to answer the question whether a new experimental drug, physostigmine, will improve the impaired memory of subjects suffering from Korsakoff psychosis. 

Experimental Conditions

In the course of this experiment, the control group was given placebo, the first experimental group was given dexfermetrazine, a stimulant. The second experimental group was given physostigmine, a drug we thought will improve memory of patients suffering from Korsakoff psychosis. The reason one group of patients was given dexfermetrazine was to differentiate the effect of physostigmine from a likely memory improvement due to a general arousal of subjects.

Dependent Variable

The criterion variable, within the context of experiments often called the dependent variable, was the number of nonsense syllabi subjects remembered. The dependent variable, measured for the control group C and both experimental groups, designated by a subscripted letter E, was recorded for nine subjects, randomly assigned to each condition of the experiment, as

The group means and unbiased standard deviations were computed as shown below

 

Orthogonal Codes

The above planned group comparisons can be directly tested by using Helmert's contrasts. There are three groups. To code this experiment, we need two orthogonal vectors, one less than the number of groups coded.

Using Helmert's contrasts for two variables, the experiment can be coded as

Contrasts

The first orthogonal vector X1 describes a contrast between the placebo group and both experimental groups that received either Dexfermetrazine or Physostigmine. The second orthogonal vector X2 describes a contrast between the two experimental groups that received either Dexfermetrazine or Physostigmine. The table of the contrasts is

A .05 Significance Level

A potential problem in performing multiple tests is that some comparisons may not be independent of other comparisons. Therefore, the researcher needs to set the alpha level for individual comparisons at a lower rate so that the experiment-wise error is not greater than .05.

For our example, the comparisons are planned before the study starts and the two comparisons are not correlated. Thus, the researcher does not need to adjust the overall alpha level for the number of comparisons being performed. In general, planned comparisons are usually evaluated at an uncorrected significance level.

Correlation Supermatrix

The matrix of coefficients of correlation for the above data set is

Coefficients of Determination

The associated matrix of coefficients of determination is

 

 

Contribution of Each Predictor

The multiple regression when the predictor variables are not correlated is simple, elegant, and interpretable. Approximately 67.5% of the variance in the dependent variable can be explained by the first contrast and about 22.5% of the variance can be explained by the second contrast.

Coefficients of Multiple Determination and Alienation

Since the predictors are orthogonal, the variance contribution of each predictor is additive (.675 + .225 = .90). The coefficient of multiple determination is .90 and the coefficient of multiple alienation is .10 (1-.90=.10).

ANOVA Summary Table

The analysis of variance summary table can be constructed as

 

The F ratio for the first contrast can be computed as

 

 

The F ratio for the second contrast can be computed as

 

Interpretations

There was a significant difference in the number of nonsense syllabi subjects remembered between the placebo group and both experimental groups, F(1,6) = 40.50, p < .05. There was also a significant difference between the two experimental groups, F(1,6) = 13.50, p < .05.

Physostigmine is a promising drug in therapy of the memory loss suffered by patients with symptoms of Korsakoff psychosis. Overall, about 90% of variance in the dependent variable is accounted for by the treatments.   

Multiple Regression Analysis with Coded Predictor Variables

Use a computer program to run a multiple regression analysis. The results can be shown below.

Many computer programs require that the subjects participating in different conditions of an experiment are indexed by an arbitrary coding vector X, for our example defined as [1 1 1 2 2 2 3 3 3].

Then the computer programs generate the analysis of variance summary table as

The obtained F ratio pertains to the over-all significance of the experiment. To find out which particular differences between groups involved in the analysis are significant, one can enter orthogonal codes, for the example

Notice that when comparing two means, the F equals t².

True Variance

To compare this solution with the solution we outlined in the preceding sections, multiply the standard variance components by the variance of the variable Y, for the example equal to 6.67. The analysis of variance summary table, using true variance components, will be as the one shown below.

Sums of Squares

By multiplying the true variance components by the number of observations, n, for the example equal to 9, the analysis of variance summary table, using sums of squares, can be constructed as

 

Both described approaches provide identical results and each approach has certain advantages.

To use the non-orthogonal codes, index group membership by an arbitrary number. Any number will do, however, the simplest and most frequently used numbers are one and zero (dummy coding). The control group is often assigned zeroes throughout. Our example of the study of short-term memory can be coded by non-orthogonal coding vectors as

The coding vector X1 consisted of 1's for subjects in the dexfermetrazine group, 0`s for all others. The coding vector X2 consisted of 1's for subjects in the Physostigmine group, 0`s for all others. Notice that the control group (the reference group) is assigned zeroes by both coding vectors. 

Since the predictors are correlated, the variance contributions of each coding vector contains also the covariance term and thus the variance contributions [.30 1.20] of the coding vectors are not additive, i.e., do not sum to the standard variance component of the predicted variable.