Bland and Altman agreement method: to plot differences against means or differences against standard? An endless tale?

Appendix: Simulation study of Hopkins paper [9]

Hopkins [9] firstly simulated a “True Value” (V₁, say) Gaussian distributed as: G∼( μ V = 50 , σ V 2 = 169 with the variance expressing only the biological variability). Then, by adding a measurement error, he calculated:

1. -Y₁ = β _0Y⋅V₁ + α _0Y + ε _0Y, Gaussian distributed as: G ∼ ( β 0 Y ⋅ μ V + α 0 Y ; β 0 Y 2 σ V 2 + σ 0 Y 2 ) , with β 0 Y = 1 , α 0 Y = 0 , a n d   σ 0 Y 2 = 9 ; and

2. -X₁ = β _0X⋅V₁ + α _0X + ε _0X, Gaussian distributed as G∼( β 0 X ⋅ μ V + α 0 X ; β 0 X 2 σ V 2 + σ 0 X 2 with β 0 X = 30 , α 0 X = 100 , a n d   σ 0 X 2 = 40,000 ).

It has to be noted that we above used Y₁ and X₁ instead of Hopkins’ weird notation of Y-Criterion (first step) and X-Practical, respectively [9].

Then, Hopkins [9] made a not standard “calibration study” between Y₁ and X₁, by fitting an OLS regression, instead of the appropriate Deming’s regression [11], and obtained the OLS slope (b₁, say) and intercept (a₁, say). It has to be noted that the expected value of b₁ is given by the covariance between Y₁ and X₁ equal to β 0 Y ⋅ β 0 X ⋅ σ V 2 divided by the variance of X₁ given by σ 0 X 2 ; then, the expected value of the intercept (a₁) can be calculated.

At the second step, Hopkins [9] simulated a new sample of “True Value” (V₂, say) and calculated from V₂ a second couple of variables (Y₂, and X₂, say for avoiding the Hopkins’s definition [9] of “Criterion Y”, already used in Step 1, and of “Hidden X” in Excel^® spreadsheet) following the previously reported method. Then, by using the coefficients (b₁ and a₁) of the calibration straight line previously obtained, Hopkins [9] calculated a variable, defined “Practical-Y”, as: “Practical-Y”=b₁⋅X₂ + a₁, distributed as G∼( b 1 ⋅ μ X + a 1 ; b 1 2 σ X 2 ) that shows, as expected, a perfect regression (slope=1, and intercept=0) with Y₁, used in the calibration regression. Furthermore, it has to be stressed that “Practical-Y” has an expected lower variance than Y₁ (about 3/4, from Hopkins’ simulation parameters [9]) leading to the surprising result of having obtained a more precise measurement method than the one used for calculating the calibration equation and, in addition, an expected regression slope (correlation coefficient) of the DM analysis different from zero.

Actually, from the parameters of Hopkins’ simulation [9] it is possible to calculate an expected correlation coefficient between Y₂ and “Practical Y” of 0.867030 that with the expected values of their variances of 178 and 133.784606, respectively, gives for the DM approach a correlation coefficient of −0.2763283 and a regression coefficient of −0.152629. Otherwise, for the DS approach the values are 0.000165 and 0.000949, respectively.

However, it must be noted that the values selected by Hopkins [9] allow to obtain always progressively lower slope and correlation coefficient for the DS approach with the consequent claim that DS approach gives less biased results than the B&A DM proposal [1, 2].

It has to be noted that the final steps of Hopkins’s paper [9] are in according to the sentence in Krouwer’s paper [8]: “Then, in order to gain regulatory approval, the manufacturer will repeat the study but this time as a method comparison study (e.g. using the commercial assay with its built in calibration equations).” and to some requirements from regulatory and scientific authorities reported also in the paper of Stevens et al. [17].