Regression dilution

Regression dilution is a statistical phenomenon also known as "attenuation".

Consider fitting a straight line (linear regression) for the relationship of an outcome variable y to a predictor variable x, and estimating the gradient (slope) of the line. Statistical variability, measurement error or random noise in the y variable causes imprecision in the estimated gradient, but not bias: on average, the procedure calculates the right gradient. However, variability, measurement error or random noise in the x variable causes bias in the estimated gradient (as well as imprecision). In a nutshell: the more variability is in the x measurement, the closer the estimated gradient gets to 0 instead of the true gradient. This 'dilution' of the gradient towards 0 is referred to as "regression dilution," "attenuation," or "attenuation bias."

It may seem counter-intuitive that noise in the predictor variable x induces a bias, but noise in the outcome variable y does not. Recall that linear regression is not symmetric: the line of best fit for predicting y from x (the usual linear regression) is not the same as the line of best fit for predicting x from y (see, for example, Draper & Smith, "Applied Regression Analysis"; page 5 of the 1966 edition).

The case of a randomly distributed x variable
The case that the x variable arises randomly is known as the structural model or structural relationship. For example, in a medical study patients are recruited as a sample from a population, and their characteristics such as blood pressure may be viewed as arising from a random sample.

Under certain assumptions (typically, normal distribution assumptions) there is a known ratio between the true gradient, and the expected estimated gradient. Frost and Thompson (2000) review several methods for estimating this ratio and hence correcting the estimated gradient. The term regression dilution ratio (beware - not defined in quite the same way by all authors) is used of this general approach, in which the usual linear regression is fitted, and then a correction applied. The reply to Frost & Thompson by Longford (2001) refers the reader to other methods, expanding the regression model to acknowledge the variability in the x variable, so that no bias arises. The book by Fuller (1987) is one of the standard references for assessing and correcting for regression dilution.

Hughes (1993) shows that the regression dilution ratio methods apply approximately in survival models. Rosner (1992) shows that the ratio methods apply approximately to logistic regression models. The book by Carroll et al. (1995) gives more detail on regression dilution in nonlinear models. Carroll et al. present the regression dilution ratio methods as the simplest case of regression calibration methods, in which additional covariates may also be incorporated.

In general, methods for the structural model require some estimate of the variability of the x variable. This will require repeated measurements of the x variable in the same individuals, either in a sub-study of the main data set, or in a separate data set. Without this information it will not be possible to make a correction.

The case of a fixed x variable
The case that x is fixed, but measured with noise, is known as the functional model or functional relationship. See, for example, Riggs et al. (1978).

Multiple x variables
The case of multiple predictor variables (possibly correlated) subject to variability (possibly correlated) has been well-studied for linear regression, and for some non-linear regression models (Fuller, 1987; Carroll, 1995). Other non-linear models, such as proportional hazards models for survival analysis, have been considered only with a single predictor subject to variability (Hughes, 1993).

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 (2000) 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. Stevens (2001) discusses this case (earlier references may exist) when developing a regression equation 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.

Caveats
All of these results can be shown mathematically, in the case of simple linear regression assuming normal distributions throughout (the framework of Frost & Thompson). However, Davey-Smith and Philips (1996) observe that a poorly executed correction for regression dilution may do more damage to an estimate than no correction.