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Statistical bias in linear regressions

Regression dilution, also known as regression attenuation, is the biasing of the linear regression slope towards zero (the underestimation of its absolute value), caused by errors in the independent variable. Consider fitting a straight line for the relationship of an outcome variable y to a predictor variable x, and estimating the slope of the line. Statistical variability, measurement error or random noise in the y variable causes uncertainty in the estimated slope, but not bias: on average, the procedure calculates the right slope. However, variability, measurement error or random noise in the x variable causes bias in the estimated slope (as well as imprecision). The greater the variance in the x measurement, the closer the estimated slope must approach zero instead of the true value. 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.

Regression slope and other regression coefficients can be disattenuated as follows.

The case that x is fixed, but measured with noise, is known as the functional model or functional relationship. It can be corrected using total least squares and errors in variables models in general.

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 slope, and the expected estimated slope. Frost and Thompson (2000) review several methods for estimating this ratio and hence correcting the estimated slope. The term regression dilution ratio, although not defined in quite the same way by all authors, is used for 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. 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. Carroll et al. (1995) give more detail on regression dilution in nonlinear models, presenting 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 multiple predictor variables subject to variability (possibly correlated) has been well studied for linear regression, and for some non linear regression models. i. e., to "rid a correlation coefficient from the weakening effect of measurement error". In measurement and statistics, the procedure is also called correlation disattenuation or the disattenuation of correlation. The correction assures that the Pearson correlation coefficient across data units (for example, people) between two sets of variables is estimated in a manner that accounts for error contained within the measurement of those variables.

A correction for regression dilution is necessary in statistical inference based on regression coefficients. However, in predictive modelling applications, correction is neither necessary nor appropriate. In change detection, correction is necessary. 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. All of these results can be shown mathematically, in the case of simple linear regression assuming normal distributions throughout (the framework of Frost & Thompson). It has been discussed that a poorly executed correction for regression dilution, in particular when performed without checking for the underlying assumptions, may do more damage to an estimate than no correction.