Calibration or inverse regression

Calculates the point and interval estimate for a new value of X, the independent predictor variable in a simple regression equation, for a specified new determination of Y, the dependent (response) variable. This approach, sometimes known as "inverse regression," or statistical calibration has technical applications in the validation of new instruments or the assessment of sample "unknowns" against a set of standard values.

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Important

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Required Inputs

• Column of dependent values (Y)
• Column of independent values (X)
• Column of future/new values for the dependent Y

Optional Inputs

CLEVEL K
The default confidence level for the confidence interval is 95%, but you can change the confidence level by using the subcommand CLEVEL followed by the confidence level you wish to use (1-99).

Running the Macro

Suppose your Y is in C1, X is in C2, and the new values of Y are in C3, and you want a confidence level of 99%. Choose Edit > Command Line Editor and type:
```%CALIB C1 C2 C3;
CLEVEL 99.```

Click Submit Commands.

A large-sample approximation for a confidence interval around a point estimate of X is given on pp. 172-174 of Neter, Wasserman, and Kutner's 1985 text, Applied Linear Statistical Models. The Minitab macro CALIB.MAC performs this analysis for you. (Problem 5.24 on page 180 of Neter, Wasserman and Kutner is used to demonstrate the macro. This problem refers to a data set from Problem 2.18 on page 55. The data is given below the problem statement.)

The dependent response variable, Y, is the hardness of items molded from plastic (measured in Brinell units) and the independent predictor variable, X, is elapsed time measured in hours since the end of the molding process. The set of 12 paired observations form a straight-line functional relationship, with the linear regression solution Y = 153.9 + 2.42X. The problem then asks to calculate a 99% confidence interval about the estimated number of hours (X) associated with an item with a hardness (Y) of 298. In the example, Y values of 200, 250, 298, 325, and 350 were added to show that the macro could manage multiple Y values at the same time.

Enter the following 3 columns of data into C1, C2, and C3.

Y X New
230 32 200
262 48 250
323 72 298
298 64 325
255 48 350
199 16
248 40
279 48
267 48
214 24
359 80
305 56

To run the macro, choose Edit > Command Line Editor and type the command

```%CALIB C1 C2 C3;
CLEVEL 99.```

Click Submit Commands. The output will look like this:

`The regression equation is Y = 154 + 2.42 X Predictor Coef SE Coef T P Constant 153.917 8.067 19.08 0.000 X 2.4167 0.1575 15.35 0.000 S = 9.75833 R-Sq = 95.9% R-Sq(adj) = 95.5% Analysis of Variance Source DF SS MS F P Regression 1 22427 22427 235.51 0.000 Residual Error 10 952 95 Total 11 23379 The 95.00% Confidence Interval(s) for the Predicted Values of X Row Y_New CI_Low X_Hat CI_High Width 1 200 8.8056 19.0690 29.3323 20.5268 2 250 30.3180 39.7586 49.1992 18.8812 3 298 50.1055 59.6207 69.1359 19.0304 4 325 60.8611 70.7931 80.7251 19.8640 5 350 70.6098 81.1379 91.6660 21.0562 The correction factor is 0.0210800, which is less than 0.1 indicating that the interval(s) above are probably good approximations. `
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