Power Considerations in Interactive Models

The text shows that the power of the test of the interaction or product variable is unaffected by deviation transformation of its components, as long as those components are included as predictors in the model. However, the redundancy between the predictors and their product term is greatly changed when transformations of the predictors takes place.

When the predictor is a product of other predictors, then various transformations can radically affect its redundancy with the other predictors. Yet, such transformations have no effect on the power of the test of the product's regression coefficient, as long as the component variables are included as predictors in the model.

Why is this important?
Many computer programs will not allow you to do regressions when the redundancy between variables is high. MINITAB is one of those programs. If you were doing interactions this way, it would be impossible. It then makes sense to simply transform the component variables to mean deviation form to reduce the redundancy and continue with the analysis.

A General Procedure for the Derivation of "SIMPLE" Relationships

In this section your text develops a general procedure for the derivation of "Simple" relationships. If you know calculus you are in like Jake. If not it's not too difficult.

It is easiest to interpret regression results if you can express the equation in a "Simple" form.

So far it has been easy to express our results in a "Simple" form. We have just put together the terms that do and do not contain the single predictor variable. The terms that do not include the predictor are grouped together to form the intercept and the terms that do include the predictor form the slope of the "Simple" equation.

For more complex relationships, the procedure we used will not always work, but this new procedure will work. First take the derivative of the regression equation with respect to the predictor variable you are interested in. This derivative times the predictor variable are then subtracted from and added to the original equation. This produces the simple equation. Because you have added and subtracted this product to the equation, you have not changed it.

The "Simple" intercept will be the original equation minus the product of the derivative and the predictor and the "Simple" slope is the product of the derivative and the predictor.

You know from calculus that the slope of any function at a given point is provided by the derivative or partial derivative. To derive derivatives, you also must remember these rules.

1. The derivative of a sum equals the sum of the derivatives of the components of the sum.
2. The derivative of aXm equals amXm-1, where a can either be a constant or another variable with which X is multiplied.
3. The derivative of a component that does not include X equals zero.

Usually students have difficults with the subscripts so I'll do the same three equations but not use subscripts.

Looking at the Interaction of a Variable with Itself

or

Powers of Predictor Variables.

Suppose that we hypothesize that the relationship between Time and Miles depends on levels of Miles. We suspect that when one does not train much, the relationship between training and race times might be a lot stronger than it is when one trains a great deal. This problem is discussed extensively in the text. Let me turn the table and ask a school psych related question.

Does the relationship between Read and VIQ change with levels of VIQ? Doesn't it make sense that VIQ would have a different relationship to academic achievement when children had difficulty in this area than when they were quite gifted in the area?

Let's take a look.

DEP VAR:    READ      N:     200   MULTIPLE R: 0.574  SQUARED MULTIPLE R: 0.329
ADJUSTED SQUARED MULTIPLE R: 0.323     STANDARD ERROR OF ESTIMATE:       12.033

  VARIABLE    COEFFICIENT    STD ERROR     STD COEF TOLERANCE    T    P(2 TAIL)

CONSTANT          104.401       19.844        0.000      .       5.261    0.000
     VIQ           -0.992        0.474       -1.015     0.014   -2.093    0.038
  VIQVIQ            0.009        0.003        1.569     0.014    3.234    0.001


                             ANALYSIS OF VARIANCE

   SOURCE   SUM-OF-SQUARES    DF  MEAN-SQUARE     F-RATIO       P

 REGRESSION      14017.019     2     7008.509      48.400       0.000
   RESIDUAL      28526.481   197      144.804

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Let' try a more complex model

DEP VAR:    READ      N:     200   MULTIPLE R: 0.576  SQUARED MULTIPLE R: 0.331
ADJUSTED SQUARED MULTIPLE R: 0.318     STANDARD ERROR OF ESTIMATE:       12.079

  VARIABLE    COEFFICIENT    STD ERROR     STD COEF TOLERANCE    T    P(2 TAIL)

CONSTANT           99.115       21.274        0.000      .       4.659    0.000
     VIQ           -1.225        0.578       -1.253     0.010   -2.119    0.035
     PIQ            0.335        0.467        0.356     0.014    0.718    0.474
  VIQPIQ           -0.004        0.005       -0.619     0.004   -0.708    0.480
  VIQVIQ            0.012        0.005        2.136     0.004    2.293    0.023


                             ANALYSIS OF VARIANCE

   SOURCE   SUM-OF-SQUARES    DF  MEAN-SQUARE     F-RATIO       P

 REGRESSION      14092.270     4     3523.068      24.147       0.000
   RESIDUAL      28451.230   195      145.904

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