Classic vs. Coded Experimental Designs

Select (Designs, Regression Models)

and click the Accept command.

Select (Operations, Add variables) and compose variable Y. Select (Analysis I, Regression Analysis), select variables X and Y and add variables Y' and Y^ to the vector display. Click on the Descriptive Statistics and (Standardize Variance, Select Standardizing Variable Y). At this point, the vector display should look as

Variance of Y0 and Y1 variables is related to the variance of the variables Y' and Y^ as

 

Variance of Y'

Select (Transfers, Descriptors to the scalar module), check Arithmetic Mean and True Variance, select variables Y0 and Y1 and click on the Accept command. Click on the Right Justify command to shift indices on the scalar display toward the right.  Select (Analysis I, Variance) and click on the PQ Components of Binary Variables. Select the parent vector X and  np, nq, and n.

 Click on the Left Justify command to compact the scalar display. Rename the np and nq indices as n1 and n0 and using the above formula, compute variance of variable Y' (1.5).

Variance of Y^

Compute the variance of the Y^ variable (.500) as

The ns, means, and variances of the Y0 and Y1 variables are the building blocks of formulae for finding the statistical significance of differences between the control and experimental groups.

 

Standard Variance of Y' and Y^

 Inspecting the output panel associated with the regression analysis of control and experimental groups, indexed by the parent vector X, you may notice that the standard variance of Y' (.750) and Y^  (.250) is equal to the values of coefficients of Determination (.750) and Alienation (.250).

as also indicated by the output panel for the coded independent measures of the same data. The F ratio is defined as a ratio of coefficients of determination and alienation, multiplied by the degrees of freedom (1.5 / .5)(3.0), equal to 9.00. The t-ratio is the square root of this value (3.00).

The pq Notation for Binary Parent Vectors

Conceptualization of variance of the Y' and Y^ variables in terms of group ns, means, and variances can be simplified by adopting the pq notation, where p equals the number of 1 values divided by n, q equals 1-p, and the variance of a binary variable equals pq. Let us reconsider the previous example

with variances of the Y' and Y^ variables defined this time as

Select (Transfers, Descriptors - Scalars), and select Arithmetic Mean and True Variance for variables Y0 and Y1. Select (Analysis I, Variance, PQ Components of Binary Variables)

and the parent vector X.

On the scalar module shift the scalars to the right by clicking the Right Justify command and press the Accept command on the PQ Components panel. At this point, the scalar module should look as

Using the pq notation, compute variance of the Y' (1.50) and Y^ (.50) variables as

The pq notation simplifies the computation, but cannot compete with the coded regression analysis of the computer age, in which the F-ratio is computed as a ratio of coefficients of determination (.750) and alienation (.250),

multiplied by the degrees of freedom and the t-ratio as its square root. The components of standard variance

are equivalent to coefficients of determination and alienation, and thus

For the example the F ratio equals (.750 / .250) 3 = 9.00. The t-ratio equals 3.00. These values can be obtained directly by selecting (Data,  Bivariate Prototypes) and (Analysis I, Regression Analysis)

and agree with values obtained by selecting (Analysis II, Coded Independent Measures Design) of the same data. For the F-ratio

Click on the Variance Ratio command, to change the F ratio to the t-ratio

These results can be complemented by selecting (Tables, t-Distribution with Symmetric Critical Areas) and dragging the obtained t and df values to its t and df  input boxes,

 

or by selecting (Tables, t-Distribution with Asymmetric Critical Area)

and dragging the obtained probability (.029) back into the display panel

obtained by coded regression analysis.