Errors-in-variables models

Illustration of regression dilution (or attenuation bias) by a range of regression estimates in errors-in-variables models. Two regression lines (red) bound the range of linear regression possibilities. The shallow slope is obtained when the independent variable (or predictor) is on the x-axis. The steeper slope is obtained when the independent variable is on the y-axis. By convention, with the independent variable on the x-axis, the shallower slope is obtained. Green reference lines are averages within arbitrary bins along each axis. Note that the steeper green and red regression estimates are more consistent with smaller errors in the y-axis variable.
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Background

In statistics, errors-in-variables models or measurement error models are regression models that account 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.[citation needed]

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.[1][2][3]

  1. ^ Griliches, Zvi; Ringstad, Vidar (1970). "Errors-in-the-variables bias in nonlinear contexts". Econometrica. 38 (2): 368–370. doi:10.2307/1913020. JSTOR 1913020.
  2. ^ Chesher, Andrew (1991). "The effect of measurement error". Biometrika. 78 (3): 451–462. doi:10.1093/biomet/78.3.451. JSTOR 2337015.
  3. ^ Carroll, Raymond J.; Ruppert, David; Stefanski, Leonard A.; Crainiceanu, Ciprian (2006). Measurement Error in Nonlinear Models: A Modern Perspective (Second ed.). ISBN 978-1-58488-633-4.