4.6.3.3. Transformations to Improve Fit

4. Process Modeling
4.6. Case Studies in Process Modeling
4.6.3. Ultrasonic Reference Block Study

4.6.3.3. Transformations to Improve Fit

Tranformations

One approach to the problem of non-homogeneous variances is to apply transformations to the data.

Plot of Common Transformations to Obtain Homogeneous Variances

The first step is to try transformations of the response variable that will result in homogeneous variances. In practice, the square root, log, and reciprocal transformations often work well for this purpose. We will try these first.

In examining these plots, we are looking for the plot that shows the most constant variability across the horizontal rqnge of the plot.

This plot indicates that the square root transformation is a good candidate model for achieving the most homogeneous residuals.

Plot of Common Transformations to Predictor Variable

After transforming the response variable, it is often helpful to transform the predictor variable as well. In practice, the square root, log, and reciprocal transformations often work well for this purpose. We will try these first.

This plot shows that none of the porposed transformations offers an improvement over using the raw predictor variable.

Square Root Fit

Based on the above plots, we choose to fit a model with a square root transformation for the response variable and no transformation for the predictor variable. Dataplot generated the following output for this model (it is edited slightly for display).

  
LEAST SQUARES NON-LINEAR FIT
SAMPLE SIZE N =      214
MODEL--YTEMP =EXP(-B1*XTEMP)/(B2+B3*XTEMP)
REPLICATION CASE
REPLICATION STANDARD DEVIATION =     0.2927381992D+00
REPLICATION DEGREES OF FREEDOM =         192
NUMBER OF DISTINCT SUBSETS     =          22
  
         FINAL PARAMETER ESTIMATES           (APPROX. ST. DEV.)    T VALUE
        1  B1                 -0.154326E-01   (0.8593E-02)         -1.8
        2  B2                  0.806714E-01   (0.1524E-02)          53.
        3  B3                  0.638590E-01   (0.2900E-02)          22.
  
RESIDUAL    STANDARD DEVIATION =         0.2971503735
RESIDUAL    DEGREES OF FREEDOM =         211
REPLICATION STANDARD DEVIATION =         0.2927381992
REPLICATION DEGREES OF FREEDOM =         192
LACK OF FIT F RATIO =       1.3373 = THE  83.6085% POINT OF THE
F DISTRIBUTION WITH     19 AND    192 DEGREES OF FREEDOM

Although the residual standard deviation is lower than it was for the original fit, we cannot compare them directly since the fits were performed on different scales.

The fitted model is

SQRT(Y) = EXP(-0.0154*X)/(0.0807+0.0639*X)

Plot of Predicted Values

The plot of the predicted values with the transformed data indicates a good fit.

6-Plot of Fit

6-plot indicates fit assumptions satisfied

Since we transformed the data, we need to validate that all of the fit assumptions are still satisfied.

The 6-plot of the model indicates no obvious violations of the fit assumptions.

Plot of Residuals

In order to see more detail, we generate a full size version of the residuals versus predictor variable plot. This plot shows that the residuals now satisfy the assumption of homogeneous variances.