Regression dilution in the context of "Linear regression"

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⭐ Core Definition: Regression dilution

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.

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Regression dilution in the context of Errors-in-variables models

In statistics, an errors-in-variables model or a measurement error model is a regression model that accounts for measurement errors in the independent variables. In contrast, standard regression models assume that those regressors have been measured exactly, or observed without error; as such, those models account only for errors in the dependent variables, or responses.

In the case when some regressors have been measured with errors, estimation based on the standard assumption leads to inconsistent estimates, meaning that the parameter estimates do not tend to the true values even in very large samples. For simple linear regression the effect is an underestimate of the coefficient, known as the attenuation bias. In non-linear models the direction of the bias is likely to be more complicated.

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← Errors-in-variables models
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Errors-in-variables models → Bias (statistics) → Linear regression → Regression slope → Independent variable → Uncertainty →