Inverse Prediction One use of a regression model E ( Y ) = 0 + 1 x - - PowerPoint PPT Presentation

ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II

Inverse Prediction

One use of a regression model E(Y ) = β0 + β1x is to predict Y for a new x, x0. Sometimes, instead, we observe a new y0, and want to make an inference about the new x0. Often x is expensive to measure, but Y is cheap; the relationship is determined from a calibration dataset.

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ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II

Because y0 = β0 + β1x0 + ǫ0, we can solve for x0: x0 = y0 − β0 − ǫ0 β1 . We do not observe ǫ0, but we know that E(ǫ0) = 0. Similarly, we do not know β0 and β1, but we have estimates ˆ β0 and ˆ β1.

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ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II

This suggests the estimate ˆ x0 = y0 − ˆ β0 ˆ β1 . This is known as inverse prediction. An approximate 100(1 − α)% prediction interval for x0 is: ˆ x0 ± tα/2 × s ˆ β1 ×

• 1 + 1

n + (ˆ x − ¯ x)2 SSxx .

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ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II

An alternative approach is to fit the inverse regression: x = γ0 + γ1y + ǫ. Then use the standard prediction interval ˆ x0 ± tα/2 × sx|y ×

• 1 + 1

n + (y0 − ¯ y)2 SSyy where ˆ x0 = ˆ γ0 + ˆ γ1y0. This is not supported by the standard theory, because, in the calibration data, x is fixed and y is random. But it has been shown to work well in practice.

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