Marques-Vidal P, Rakotovao R, Ducimetière P
Project MONICA, INSERM U326/C.H.U. Purpan, Toulouse.
Rev Epidemiol Sante Publique. 1994;42(1):58-67.
The consequences of a measurement error of known variance in the explanatory variable of a linear regression were assessed. On the average, the ordinary least squares (OLS) method underestimated the regression slope, the bias increasing with the variance of the measurement error and the strength of the relationship. Simulation results showed that the corrected-for-the-error estimate slightly overestimated the slope, the bias increasing with the variance of the measurement error and the strength of the relationship, but rapidly decreasing when the number of observations increased. In all cases the corrected estimate has a larger variance than the OLS estimate, Nevertheless, the mean square deviation of the corrected estimate to the "true" slope value can be smaller than the OLS one, even for a relatively small number of observations (< or = 100). In those conditions, the corrected estimate might be preferred when a "good estimation" of the regression slope is needed. Whereas a measurement error in the dependent variable, does not bias the slope estimator, when it is independent of the error in the explanatory variable, this is not the case when both measurement errors are correlated. An example of the need to correct for such a correlation is given.
评估了线性回归解释变量中已知方差的测量误差的后果。平均而言，普通最小二乘法（OLS）低估了回归斜率，偏差随着测量误差的方差和关系强度的增加而增大。模拟结果表明，误差校正估计略微高估了斜率，偏差随着测量误差的方差和关系强度的增加而增大，但随着观测值数量的增加而迅速减小。在所有情况下，校正估计的方差都比OLS估计的方差大。然而，即使对于相对较少的观测值（≤100），校正估计与“真实”斜率值的均方偏差也可能小于OLS估计的均方偏差。在这些条件下，当需要对回归斜率进行“良好估计”时，校正估计可能更受青睐。虽然当因变量中的测量误差与解释变量中的误差无关时，它不会使斜率估计器产生偏差，但当两个测量误差相关时情况并非如此。给出了一个需要校正这种相关性的例子。
