The ENTROPY Procedure (Experimental)

Example 13.2 Unreplicated Factorial Experiments

Factorial experiments are useful for studying the effects of various factors on a response. For the practitioner constrained to the use of OLS regression, there must be replication to estimate all of the possible main and interaction effects in a factorial experiment. Using OLS regression to analyze unreplicated experimental data results in zero degrees of freedom for error in the ANOVA table, since there are as many parameters as observations. This situation leaves the experimenter unable to compute confidence intervals or perform hypothesis testing on the parameter estimates.

Several options are available when replication is impossible. The higher-order interactions can be assumed to have negligible effects, and their degrees of freedom can be pooled to create the error degrees of freedom used to perform inference on the lower-order estimates. Or, if a preliminary experiment is being run, a normal probability plot of all effects can provide insight as to which effects are significant, and therefore focused, in a later, more complete experiment.

The following example illustrates the probability plot methodology and the alternative by using PROC ENTROPY. Consider a $2^{4}$ factorial model with no replication. The data are taken from Myers and Montgomery (1995).

data rate;
   do a=-1,1; do b=-1,1; do c=-1,1; do d=-1,1;
      input y @@;
      ab=a*b; ac=a*c; ad=a*d; bc=b*c; bd=b*d; cd=c*d;
      abc=a*b*c; abd=a*b*d; acd=a*c*d; bcd=b*c*d;
   end; end; end; end;
   45 71 48 65 68 60 80 65 43 100 45 104 75 86 70 96

Analyze the data by using PROC REG, then output the resulting estimates.

proc reg data=rate outest=regout;
   model y=a b c d ab ac ad bc bd cd abc abd acd bcd abcd;

proc transpose data=regout out=ploteff name=effect prefix=est;
   var a b c d ab ac ad bc bd cd abc abd acd bcd abcd;

Now the normal scores for the estimates can be computed with the rank procedure as follows:

proc rank data=ploteff normal=blom out=qqplot;
   var est1;
   ranks normalq;

To create the probability plot, simply plot the estimates versus their normal scores by using PROC SGPLOT as follows:

title "Unreplicated Factorial Experiments";
proc sgplot data=qqplot;
   scatter x=est1 y=normalq / markerchar=effect
   xaxis label="Estimate";
   yaxis label="Normal Quantile";

Output 13.2.1: Normal Probability Plot of Effects

Normal Probability Plot of Effects

The plot shown in Output 13.2.1 displays evidence that the a, b, d, ad, and bd estimates do not fit into the purely random normal model, which suggests that they may have some significant effect on the response variable. To verify this, fit a reduced model that contains only these effects.

proc reg data=rate;
   model y=a b d ad bd;

The estimates for the reduced model are shown in Output 13.2.2.

Output 13.2.2: Reduced Model OLS Estimates

Unreplicated Factorial Experiments

The REG Procedure
Model: MODEL1
Dependent Variable: y

Parameter Estimates
Variable DF Parameter
t Value Pr > |t|
Intercept 1 70.06250 1.10432 63.44 <.0001
a 1 7.31250 1.10432 6.62 <.0001
b 1 4.93750 1.10432 4.47 0.0012
d 1 10.81250 1.10432 9.79 <.0001
ad 1 8.31250 1.10432 7.53 <.0001
bd 1 -9.06250 1.10432 -8.21 <.0001

These results support the probability plot methodology.

PROC ENTROPY can directly estimate the full model without having to rely upon the probability plot for insight into which effects can be significant. To illustrate this, PROC ENTROPY is run by using default parameter and error supports in the following statements:

proc entropy data=rate;
   model y=a b c d ab ac ad bc bd cd abc abd acd bcd abcd;

The resulting GME estimates are shown in Output 13.2.3. Note that the parameter estimates associated with the a, b, d, ad, and bd effects are all significant.

Output 13.2.3: Full Model Entropy Results

Unreplicated Factorial Experiments

The ENTROPY Procedure

GME Variable Estimates
Variable Estimate Approx Std Err t Value Approx
Pr > |t|
a 5.688414 0.7911 7.19 <.0001
b 2.988032 0.5464 5.47 <.0001
c 0.234331 0.1379 1.70 0.1086
d 9.627308 0.9765 9.86 <.0001
ab -0.01386 0.0270 -0.51 0.6149
ac -0.00054 0.00325 -0.16 0.8712
ad 6.833076 0.8627 7.92 <.0001
bc 0.113908 0.0941 1.21 0.2435
bd -7.68105 0.9053 -8.48 <.0001
cd 0.00002 0.000364 0.05 0.9569
abc -0.14876 0.1087 -1.37 0.1900
abd -0.0399 0.0516 -0.77 0.4509
acd 0.466938 0.1961 2.38 0.0300
bcd 0.059581 0.0654 0.91 0.3756
abcd 0.024785 0.0387 0.64 0.5312
Intercept 69.87294 1.1403 61.28 <.0001