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R. A. Bilonick, Z. Tai, R. W. Hertle, D. Yang, G. Wollstein; Using a Structural Equation Model to Calibrate Measurement Methods When Replicate Errors Are Correlated. Invest. Ophthalmol. Vis. Sci. 2010;51(13):5372.
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© ARVO (1962-2015); The Authors (2016-present)
Determining the calibration equation (CE) that relates methods of measurement requires estimating the systematic error (bias) and random error (imprecision) for each method. When only two methods are compared, each method should be replicated so that the bias and the imprecision can be simultaneously estimated. In some cases the random errors for replicates are correlated. A structural equation model (SEM) describing the measurement error (ME) is necessary so that the proper model is fitted to the covariance structure of the observed measurements. This method is illustrated using visual acuity (VA) measurements from patients suffering from nystagmus.
Automated nystagmus acuity function (ANAF) was compared to the expanded nystagmus acuity function (NAFX). A SEM was used to estimate the bias (αs and βs) and imprecision (σA and σN). The diagram shows the relationship between measurements and the true (unknown) VA (µ). The errors between replicates were allowed to be correlated (ρ). The model was fitted to the natural log transformed data. The CE for methods is: A=exp(αA-αNβA/βN)NβA/ βN and N=exp(αN-αAβN/βA)AβN/βA where A denotes ANAF and N denotes the NAFX. Mx SEM software was used to estimate the parameters using maximum likelihood (ML). Starting values were estimated using the ncb.od function from the merror package for the R statistical programming language.
The CE for methods was: A=0.9448N0.9395 and N=1.0584A1.0644. Methods had similar imprecision. No bias was detected between replicates and the ratio of the scale-adjusted imprecisions replicate 1 to 2 was 0.5776 (95% CI: 0.4385 to 0.7618). ρ was 0.634.
SEMs for ME provide a flexible method for correctly determining bias and imprecision. ML provides optimum estimates for any function of the parameters along with confidence intervals allowing statistical tests for any hypothesis of interest.
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