Example 91.2 Box-Cox Transformations

This example shows Box-Cox transformations with a yarn failure data set. For more information about Box-Cox transformations, including using a Box-Cox transformation in a model with no independent variable, to normalize the distribution of the data, see the section Box-Cox Transformations. In this example, a simple design was used to study the effects of different factors on the failure of a yarn manufacturing process. The design factors are as follows:

• the length of test specimens of yarn, with levels of 250, 300, and 350 mm

• the amplitude of the loading cycle, with levels of 8, 9, and 10 mmd

• the load with levels of 40, 45, and 50 grams

The measured response was time (in cycles) until failure. However, you could just as well have measured the inverse of time until failure (in other words, the failure rate). Hence, the correct metric with which to analyze the response is not apparent. You can use PROC TRANSREG to find an optimum power transformation for the analysis. The following statements create the input SAS data set:

title 'Yarn Strength';

proc format;
   value a -1 =   8 0 =   9 1 =  10;
   value l -1 = 250 0 = 300 1 = 350;
   value o -1 =  40 0 =  45 1 =  50;
run;

data yarn;
   input Fail Amplitude Length Load @@;
   format amplitude a. length l. load o.;
   label fail = 'Time in Cycles until Failure';
   datalines;
 674 -1 -1 -1    370 -1 -1  0    292 -1 -1  1    338  0 -1 -1
 266  0 -1  0    210  0 -1  1    170  1 -1 -1    118  1 -1  0
  90  1 -1  1   1414 -1  0 -1   1198 -1  0  0    634 -1  0  1
1022  0  0 -1    620  0  0  0    438  0  0  1    442  1  0 -1
 332  1  0  0    220  1  0  1   3636 -1  1 -1   3184 -1  1  0
2000 -1  1  1   1568  0  1 -1   1070  0  1  0    566  0  1  1
1140  1  1 -1    884  1  1  0    360  1  1  1
;

PROC TRANSREG is run to find the Box-Cox transformation. The lambda list is –2 TO 2 BY 0.05, which produces 81 lambdas, and a convenient lambda is requested. This many power parameters makes a nice graphical display with plenty of detail around the confidence interval. In the interest of space, only part of this table is displayed. The independent variables are designated with the QPOINT expansion. QPOINT, for quadratic point model, gets its name from PROC TRANSREG’s ideal point modeling capabilities, which process variables for a response surface analysis. What QPOINT does is create a set of independent variables consisting of the following: the original variables (Length Amplitude Load), the original variables squared (Length_2 Amplitude_2 Load_2), and the pairs of products between the variables (LengthAmplitude LengthLoad AmplitudeLoad). The following statements produce Output 91.2.1:

ods graphics on;

proc transreg details data=yarn ss2
              plots=(transformation(dependent) obp);
   model BoxCox(fail / convenient lambda=-2 to 2 by 0.05) =
         qpoint(length amplitude load);
run;

ods graphics off;

Output 91.2.1 Box-Cox Yarn Data

The optimal power parameter is –0.20, but since 0.0 is in the confidence interval, and since the CONVENIENT t-option was specified, the procedure chooses a log transformation. The plot shows in the vicinity of the optimal Box-Cox transformation, the parameters for the three original variables (Length Amplitude Load), particularly Length, are significant and the others become essentially zero.