Deming regression

It is obvious that there is error in both the x and the y values. Calculation of the least-squares slope and intercept by "standard" methods is clearly not valid. In the custom function that follows, the method of Deming is used to calculate the regression parameters for the straight line y = mx + b. The Deming regression calculation assumes Gaussian distribution of errors in both x and y values and uses duplicate measurements of x values (and of y values) to estimate the standard errors. A portion of a data table is shown in Figure 17-1. 

Figure 17-1. Portion of data table to calculate Deming regression parameters.

Calculates Deming regression parameters for Y = mX + b Equations from Cornbleet Gochman, Clin. Chem. 1979, 25, 432. Copyright 1997 by E. J. Billo Begun 5/10/97. Last modified 10/3/00... 

Near the end of the code some calculations that are specific to the Deming equations are performed. The function returns an array of two values, the slope and intercept of the Deming regression. Since an array is returned, the user must press CONTROL+SHIFT+ENTER (Windows) or CONTROL+SHIFT+RETURN (Macintosh). 

Figure 14-18 Simulated example with positive and negative differences in the low and high range, respectively A, Bland-Altman plot. B, An x-y plot with diagonal (dotted straight line) and estimated Deming regression line fso//d //ne) with 95% confidence curves (dashed lines).

Deming Regression Analysis and Ordinary Least-Squares Regression Analysis (OLR) (Constant SDs)... 

Figure 14-22 In OLR, the sum of squared deviations from the line is minimized in the vertical direction. In Deming regression analysis, the sum of squared deviations is minimized at an angle to the line depending on the random error ratio. Here the symmetric case is displayed with orthogonal deviations. (From Linnet K.The performance of Deming regression analysis in case of a misspecified analytical error ratio. Clin Cbem 1998 44 1024-1031 [Figure I]).

To compute the slope in Deming regression analysis, the ratio between the standard deviations of the random errors of xl and x2 is necessary, i.e.,... 

For the OLR procedure, given that the assumptions are fulfilled (which they almost never are), these SEs are exact. For the Deming procedure, they apply approximately. More correct formulas have been given or they can be conveniently derived by a computerized method (see later in this chapter). For both OLR and Deming regression analysis, the correct interpretation of SDji is important, but quite often it is misinterpreted (see the next section). 

The estimated slope and intercept provide an estimate of the systematic difference or error between two methods over the analytical measurement range. Additionally an estimate of the random error is important. As mentioned above, it is commonplace to consider the dispersion around the line in the vertical direction, which is quantified as SD, ( (here denoted SD21). SD21 has originally been introduced in the context of OLR, but it may as well be considered in relation to Deming regression analysis. 

Figure 14-26 A, A scatter plot with the Deming regression line (solid line) with an outlier (filled point). The dotted straight line is the diagonal, and the curved dashed lines demarcate the 95% confidence region. B, Standardized residuals plot with indication of the outlier.

Figure 14-29 Distances from data points to the line in weighted Deming regression assuming proportional random errors in xl and x2.The symmetric case is illustrated with equal random errors and a slope of unity yielding orthogonal projections onto the line. (From Linnet K. Necessary sample size for method comparison studies based on regression analysis. Clin Chem 1999 45 882-94.)...

Figure 14-30 An example of application of weighted Deming regression analysis.The solid line is the estimated regression line, and the dotted line is the line of identity.The estimated 95% confidence bands obtained by the jackknife approach are the curved dashed lines.

TABLE 14-12 Results of Weighted Deming Regression Analysis for the Comparison of Drug Assays Example, N - 65 (xl,x2) Measurements... 

Deming regression analysis given proportional SDs (i.e., constant coefficients of variation) CV. - CV. 

Deming regression analysis is performed with a significance level (Type I error) of 5% and a statistical power of 90%. To get the necessary sample size, we consult Table 14-13 and look under a range ratio of 2 and a standardized slope deviation of 3 and find the sample size N = 70. 

Table 14-14 covers application of weighted Deming regression analysis in situations with proportional SDs (constant CVs) and includes intervals with range ratios extending from 2 to 100. The standardized slope deviation is here ... 

Most published method evaluations fail to apply regression analysis in a rigorous fashion. This section considers both the use of OLR instead of Deming regression and the use of unweighted analysis in the setting of proportional random errors. 

Sample size for weighted Deming regression analysis Equivalent sample size for unweighted Deming regression analysis for slope testing... 

Linnet K. The performance of Deming regression analysis in case of a misspecified analytical error ratio. Clin Chem 1998 44 1024-31. 

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