Fitting straight lines when both variables are subject to error.
Tóm tắt
Usual methods for fitting a straight line, Y =α + βX,to data fail if the “independent” variable Xis subject to error. The problem is further complicated if there is no strong reason for selecting one of the two variables as independent; neither of the two lines may be correct. This review article discusses the maximum likelihood estimators of α and β under functional and structural models. These models involve differing assumptions about the statistical distributions of the dependent and independent variables. In addition, least-squares procedures are also considered. All these methods lead to the same result, a quadratic equation which can be solved to give an estimate of β. This result requires knowledge of the ratio of the error variances, λ = φ
2/τ2, where φ2 is the variance of the Yresiduals and τ
2
is the variance of the X
residuals. If φ
2 and τ2
are unknown, estimates of λ can be difficult to obtain. If replicate sampling was employed, estimates of the variances can be made, and then of λ.
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