Regression Dilution - Is Correction Necessary?

Is Correction Necessary?

In many (perhaps most) applications, correction is neither necessary nor appropriate. To understand this, consider the measurement error as follows. Let y be the outcome variable, x be the true predictor variable, and w be an approximate observation of x. Frost and Thompson suggest, for example, that x may be the true, long-term blood pressure of a patient, and w may be the blood pressure observed on one particular clinic visit. Regression dilution arises if we are interested in the relationship between y and x, but estimate the relationship between y and w. Because w is measured with variability, the gradient of a regression line of y on w is less than the regression line of y on x.

Does this matter? In predictive modelling, no. Standard methods can fit a regression of y on w without bias. There is bias only if we then use the regression of y on w as an approximation to the regression of y on x. In the example, assuming that blood pressure measurements are similarly variable in future patients, our regression line of y on w (observed blood pressure) gives unbiased predictions.

An example of a circumstance in which correction is desired is prediction of change. Suppose the change in x is known under some new circumstance: to estimate the likely change in an outcome variable y, the gradient of the regression of y on x is needed, not y on w. This arises in epidemiology. To continue the example in which x denotes blood pressure, perhaps a large clinical trial has provided an estimate of the change in blood pressure under a new treatment; then the possible effect on y, under the new treatment, should be estimated from the gradient in the regression of y on x.

Another circumstance is predictive modelling in which future observations are also variable, but not (in the phrase used above) "similarly variable". For example, if the current data set includes blood pressure measured with greater precision than is common in clinical practice. One specific example of this arose when developing a regression equation based on a clinical trial, in which blood pressure was the average of six measurements, for use in clinical practice, where blood pressure is usually a single measurement.