Box-Tidwell power transformation

This macro executes the Box-Tidwell procedure to determine appropriate predictor variable power transformations for a regression model linear in the transformed predictors. It is important to note that this procedure can be numerically unstable resulting in error conditions for some data sets.

Download the Macro

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Important

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Required Inputs

  • One or more columns of predictor variables
  • A column of response values
  • The power transformation parameter used on the response variable

Running the Macro

Suppose you have two columns of predictor variables in C1 and C2, with the response variable in C3. The power transformation parameter was 0 (natural log).

  1. Click anywhere in the Session window and choose Editor > Show Command Line.
  2. At the command prompt (MTB>), type the following:
    %BTTRANS
  3. Press Enter.
  4. Respond to the command prompts like this:
     Executing from file: C:\Mtb14\MACROS\BTtrans.MAC 
    Please enter the number of predictor variables in the analysis... 
    DATA> 2 
    Please enter column number of predictor variable... 
    DATA> 1 
    Please enter column number of predictor variable... 
    DATA> 2 
    Please enter column number of response variable... 
    DATA> 3 
    Please enter response power transformation parameter value... 
    DATA> 0 <-- natural log transformation of response specified 

More Information

This macro applies the Box-Tidwell procedure to estimate appropriate predictor variable power transformations in a regression model of the form

As shown in the examples below, the macro prompts the user to specify a value for λ, the response power transformation parameter. Typically, one chooses . With a starting value of unity for each α, updated estimates for each are iteratively determined and are output by the macro. The default number of iterations is 3 and can be changed by opening the macro file (with MS Notepad say) and changing 3 in "do k174 = 1:3" to the desired number. While the procedure generally converges quickly, experience indicates that it can exhibit numerical instability resulting in error conditions for some data sets.

The first example data set is the surgical services data from Myers (1990) and may be used to verify output. Using the predictor x (surgical cases) and the response y (man-hours per month), one can verify that the macro estimates agree with those given in the reference.

Surgical Services Data

x y
230 1275
235 1350
250 1650
277 2000
522 3750
545 4222
625 5018
713 6125
735 6200
820 8150
992 9975
1322 12200
1900 12750
2022 13014
2155 13275
Results for: Surgical Services Data 
MTB > %BTtrans 
Executing from file: C:\Mtb14\MACROS\BTtrans.MAC 
Please enter the number of predictor variables in the analysis... 
DATA> 1 
Please enter column number of predictor variable... 
DATA> 1 
Please enter column number of response variable... 
DATA> 2 
Please enter response power transformation parameter value... 
DATA> 1 <-- no transformation of response specified 
BOX-TIDWELL POWER TRANSFORMATION PROCEDURE ... 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 
ITERATION... 
1 
ESTIMATED ALPHA(S)... 
0.0220992 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 
ITERATION... 
2 
ESTIMATED ALPHA(S)... 
0.300677 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 
ITERATION... 
3 
ESTIMATED ALPHA(S)... 

The second example data set is from Delozier (2004) and represents a portion of a metal cutting experiment where the objective was to develop a tool life response surface model for a prototype tool in terms of two important machining productivity predictor variables, cutting speed and feed rate, to compare performance with competitor tools.

Tool Life Data

Speed Feed ToolLife
600 0.007 53.5
600 0.007 68.0
1200 0.007 3.0
1200 0.007 5.3
600 0.019 11.8
600 0.019 14.0
1200 0.019 0.9
1200 0.019 0.5
476 0.013 86.5
1324 0.013 0.4
900 0.005 20.0
900 0.021 2.9
900 0.013 4.0
900 0.013 2.2
900 0.013 3.2
900 0.013 4.0
900 0.013 3.0
900 0.013 3.2
900 0.013 4.0
900 0.013 3.5
Results for: Tool Life Data 
MTB > %BTtrans 
Executing from file: C:\Mtb14\MACROS\BTtrans.MAC 
Please enter the number of predictor variables in the analysis... 
DATA> 2 
Please enter column number of predictor variable... 
DATA> 1 
Please enter column number of predictor variable... 
DATA> 2 
Please enter column number of response variable... 
DATA> 3 
Please enter response power transformation parameter value... 
DATA> 0 <-- natural log transformation of response specified 
BOX-TIDWELL POWER TRANSFORMATION PROCEDURE ... 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 
ITERATION... 
1
ESTIMATED ALPHA(S)... 
-0.07764 
-1.25327 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 
ITERATION... 
2 
ESTIMATED ALPHA(S)... 
-0.246083 
-0.739007 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 
ITERATION... 
3 
ESTIMATED ALPHA(S)... 
-0.274498 
-0.867107 

It is interesting to note in this second example that, considering power transformations of the response only, conventional first and second order response surface models exhibit appreciable to moderate lack of fit. However, applying the Box-Tidwell results above, it is easy to verify that

provides an adequate approximation model for tool life with no significant lack of fit. The predictor power transformations of -0.25 and -1 were selected for simplicity while recognizing that the transformation estimates provided by the procedure are subject to uncertainty (estimates of which are not calculated by the macro). By using the natural log of tool life as the response and transforming the predictors in this way, the costs of augmenting the experimental design to accommodate higher-order models were avoided.

REFERENCES

1. Box G. E. P. and Tidwell, P. W. (1962), "Transformation of the Independent Variables," Technometrics, 4, 531-550.

2. Delozier, M. R. (2004), Introduction to Applied Industrial Statistics, Industrial Short-Course Participant Manual.

3. Myers, R. H. (1990), Classical and Modern Regression with Applications, Second Edition, Duxbury Press (PWS-KENT Publishing Company), 307-309.

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