5.5.9.9.6. Motivation: Why is the 1/2 in the Model?

Presence of 1/2 term does not affect predictive quality of model

The leading 1/2 is a multiplicative constant that we have chosen to include in our expression of the linear model. Some authors and software prefer to "simplify" the model by omitting this leading 1/2. It is our preference to include the 1/2. This follows a hint given on page 334 of Box, Hunter, and Hunter (1978) where they note that the coefficients that appear in the equations are half the estimated effects.

The presence or absence of the arbitrary 1/2 term does not affect the predictive quality of the model after least squares fitting. Clearly, if we choose to exclude the 1/2, then the least squares fitting process will simply yield estimated values of the coefficients that are twice the size of the coefficients that would result if we included the 1/2.

Included so least squares coefficient estimate equal to estimated effect

We recommend the inclusion of the 1/2 because of an additional property that we would like to impose on the model; namely, we desire that:

the value of the least squares estimated coefficient B for a given factor (or interaction) be visually identical to the estimated effect E for that factor (or interaction).

For a given factor, say X2, the estimated least squares coefficient B2 and the estimated effect E2 are not in general identical in either value or concept.

Effect

For factor X2, the effect E2 is defined as the change in the mean response as we proceed from the "-" setting of the factor to the "+" setting of the factor. Symbolically:

Note that the estimated effect E2 value does not involve a model per se, and is definitionally invariant to any other factors and interactions that may affect the response. We examined and derived the factor effects E in the previous steps of the general DEX analysis procedure.

On the other hand, the estimated coefficient B2 in a model is defined as the value that results when we place the model into the least squares fitting process (regression). The value that returns for B2 depends, in general, on the form of the model, on what other terms are included in the model, and on the experimental design that was run. The least squares estimate for B2 is mildly complicated since it involves a behind-the-scenes design matrix multiplication and inversion. The coefficient values B that result are generally obscured by the mathematics to make the coefficients have the collective property that the fitted model as a whole yield a minimum sum of squared deviations ("least squares").

Orthogonality

Rather remarkably, these two concepts and values:

1. factor and interaction effect estimates E, and
2. least squares coefficient estimates B

merge for the class of experimental designs for which this 10-step procedure was developed, namely, 2-level full and fractional designs that are orthogonal. Orthogonality has been promoted and chosen because of its desirable design properties. That is, every factor is balanced (every level of a factor occurs an equal number of times) and every 2-factor cross-product is balanced. But to boot, orthogonality has 2 extraordinary properties on the data analysis side:

1. For the above linear models, the usual matrix solution for the least squares estimates for the coefficients B reduce to a computationally trivial and familiar form, namely,
   

2. The usual general modeling property that the least squares estimate for a factor coefficient changes depending on what other factors have been included in or excluded from the model is now moot. With orthogonal designs, the coefficient estimates are invariant in the sense that the estimate (e.g., B2) for a given factor (e.g., X2) will not change as other factors and interactions are included in or excluded from the model. That is, the estimate of the factor 2 effect (B2) remains the same regardless of what other factors are included in the model.

The net effect of the above two properties is that a factor effect can be computed once, and that value will hold for any linear model involving that term regardless of how simple or complicated the model is, provided that the design is orthogonal. This process greatly simplifies the model-building process because the need to recalculate all of the model coefficients for each new model is eliminated.

Why is 1/2 the appropriate multiplicative term in these orthogonal models?

Given the computational simplicity of orthogonal designs, why then is 1/2 the appropriate multiplicative constant? Why not 1/3, 1/4, etc.? To answer this, we revisit our specified desire that

when we view the final fitted model and look at the coefficient associated with X2, say, we want the value of the coefficient B2 to reflect identically the expected total change Y in the response Y as we proceed from the "-" setting of X2 to the "+" setting of X2 (that is, we would like the estimated coefficient B2 to be identical to the estimated effect E2 for factor X2).

Thus in glancing at the final model with this form, the coefficients B of the model will immediately reflect not only the relative importance of the coefficients, but will also reflect (absolutely) the effect of the associated term (main effect or interaction) on the response.

In general, the least squares estimate of a coefficient in a linear model will yield a coefficient that is essentially a slope:

= (change in response)/(change in factor levels)

associated with a given factor X. Thus in order to achieve the desired interpretation of the coefficients B as being the raw change in the Y (Y), we must account for and remove the change in X (X).

What is the X? In our design descriptions, we have chosen the notation of Box, Hunter and Hunter (1978) and set each (coded) factor to levels of "-" and "+". This "-" and "+" is a shorthand notation for -1 and +1. The advantage of this notation is that 2-factor interactions (and any higher-order interactions) also uniformly take on the closed values of -1 and +1, since

   -1*-1 = +1
   -1*+1 = -1
   +1*-1 = -1
   +1*+1 = +1

and hence the set of values that the 2-factor interactions (and all interactions) take on are in the closed set {-1,+1}. This -1 and +1 notation is superior in its consistency to the (1,2) notation of Taguchi in which the interaction, say X1*X2, would take on the values

   1*1 = 1
   1*2 = 2
   2*1 = 2
   2*2 = 4

which yields the set {1,2,4}. To circumvent this, we would need to replace multiplication with modular multiplication (see page 440 of Ryan (2000)). Hence, with the -1,+1 values for the main factors, we also have -1,+1 values for all interactions which in turn yields (for all terms) a consistent X of

X = (+1) - (-1) = +2

In summary then,

   B = ()
     = (Y) / 2
     = (1/2) * (Y)

and so to achieve our goal of having the final coefficients reflect Y only, we simply gather up all of the 2's in the denominator and create a leading multiplicative constant of 1 with denominator 2, that is, 1/2.

Example for k = 1 case

For example, for the trivial k = 1 case, the obvious model

Y = intercept + slope*X1
Y = c + ()*X1

becomes

Y = c + (1/X) * (Y)*X1

or simply

Y = c + (1/2) * (Y)*X1
Y = c + (1/2)*(factor 1 effect)*X1
Y = c + (1/2)*(B^*)*X1, with B^* = 2B = E

This k = 1 factor result is easily seen to extend to the general k-factor case.