# Reverse transformation after box-cox

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• #34589

Pallab
Participant

Hi,
I have a very typical kind of a problem. In one of the GB projects, we had a non normal data set for cycle time. I transformed the data using box cox but in the process of doing it, I got my USL less than LSL in the transformed data space, the reason is quite obvious, the optimal lamda is negative.
Now, with LSL > USL, I cant use minitab to calculate process sigma, who can help me in this to find a work around with a logical explaination.

Regards.
Pallab B.
MBB, TCS.

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#95538

Hemant Gham
Participant

Hi Pallab,
Since I do not have your original data I am compelled to create a hypothetical case using statistical analysis software (Minitab).
I took a hypothetical example generating Random data in Minitab 12.
Columns given below show Original data (Non-normal with p=0.001) and Transformed data (Normal with p= 0.098). I have taken negative lambdaopt ( – 0.1 ) just to simulate the scenario described by you in your posting since it would have good to have actual values that you had in your original data.
I will have to post my reply in parts since this site is not allowing me to post the tables and picture at one time.
to be continued…

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#95539

Hemant Gham
Participant

conitnued from part I

Original       Transformed
8.48            0.81
2.73            0.90
4.57            0.86
1.33            0.97
3.52            0.88
7.61            0.82
0.11            1.25
2.76            0.90
3.68            0.88
2.29            0.92
0.73            1.03
0.46            1.08
0.40            1.10
0.07            1.30
0.50            1.07
5.59            0.84
4.04            0.87
1.39            0.97
0.25            1.15
10.80            0.79
0.50            1.07
5.31            0.85
0.17            1.20
1.27            0.98
2.70            0.91
2.20            0.92
0.61            1.05
1.13            0.99
3.62            0.88
0.98            1.00

LSLorig            0.05            1.35            LSLtrans
USLorig            9.00            0.80            USLtrans
-0.1            lambda-opt
to be continued…

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#95540

Hemant Gham
Participant

conitnued from part II
In Minitab go to Capability Analysis (Normal Distribution) and use Box-Cox transformation with process capability commands, i.e., select data arranged as Original as is given in the table above, subgroup size = 1, Lower Limit = 0.05 (chosen by me from original data values), Upper Limit = 9.00 (chosen by me from original data values). Then go to option, check  Box Cox power transformations,  check other and enter the value of optimum lambda. In our hypothetical case it is  0.1.
Using this Box-Cox power transformation in Minitab you will get a process capability plot that displays a capability histogram for the transformed data. You may also see a small histogram of the original data in the upper left side of the plot. By looking at the normal curve included in the capability histogram you can determine whether the transformation was successful in making your original data more normal.  In this case it is perfect to solve our purpose.
to be continued…

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#95541

Hemant Gham
Participant

continued from part III

to be continued…

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#95542

Hemant Gham
Participant

continued from part IV
Also, this method transforms the LSL and USL and target automatically, so that all the data are on the same scale. Process parameters (mean, short-term standard deviation, and long-term standard deviation) and capability statistics (both long-term and short-term) are calculated using the transformed data and specification limits. The transformed statistics display with an ” * ” just next to their names in the table Process Data” of the plot in part IV.
So I think you can still get process sigma from the transformed data with LSLtrans > USLtrans Hope this helps. You may get more inputs from experts participating in this forum.
All the Best !
Hemant

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#96401

Pallab
Participant

Thanks,that really helps. As far as Minitab is concerned this is OK, but how much correct that histogram is with LSL at right of USL. If you would like to calculate Yield due to LSL(only), then do you take area at right or left ??

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#96454

Hemant Gham
Participant

The LSL and USL are transformed automatically, so that all the data are on the same scale. Since mean, standard deviation and capability statistics are calculated using the transformed data and specification limits we need to take the area as they appear on histogram of transformed data.
For simplicity and understanding purpose I have attempted to sort the data, and it appears like this.
Orig-sorted            Trans-sorted
0.07                           1.30
0.11                           1.25
0.17                           1.20
0.25                           1.15
0.40                           1.10
0.46                           1.08
0.50                           1.07
0.50                           1.07
0.61                           1.05
0.73                           1.03
0.98                           1.00
1.13                           0.99
1.27                           0.98
1.33                           0.97
1.39                           0.97
2.20                           0.92
2.29                          0.92
2.70                           0.91
2.73                           0.90
2.76                           0.90
3.52                           0.88
3.62                           0.88
3.68                           0.88
4.04                           0.87
4.57                           0.86
5.31                           0.85
5.59                           0.84
7.61                           0.82
8.48                           0.81
10.80                         0.79
The relationship is obvious.
Then plotted histograms with changed intervals for transformed data.

It is clear that for yield calculations with USL, we need to take area to the left. And for LSL, to the right.
In Minitab confusion is bound to happen since capability statistics show PPM SL without the asterisk sign, that is ‘ * ‘. With ‘ * ‘ signs reverse and become PPM > or < SL*.
Hope I am making it clear.
Hemant

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