Least squares, Deming or Passing-Bablok: choosing and checking a method comparison regression

What the line is for

A method comparison runs the same patient samples on the current and the new measurement procedure and estimates the bias between them [CLSI EP09]. A straight line of new on old splits that bias into two parts: a constant difference (the intercept) and a difference that grows with concentration (the slope). From the line you can read the expected difference at any concentration, which is what a laboratory needs at its decision limits.

Which line, though? The three in common use make different assumptions about measurement error. When the assumptions fail, the slope and intercept are wrong, and so is every bias read from them. The guide to bias at clinical decision limits takes a fitted line and counts the patients who cross each limit; this guide is about getting the line right first.

Ordinary least squares and the error in x

Ordinary least squares assumes the x values, here the current method’s results, are measured without error. In a method comparison they never are. The error in x pulls the slope towards zero and pushes the intercept the other way, so least squares reports a proportional difference that is not there. One early study found the error becomes significant when the standard deviation of a single x measurement is more than 0.2 of the standard deviation of the x values across the data set [Cornbleet and Gochman, 1979]. In other words, least squares is safe only when the range of results is wide compared with the method’s imprecision.

A simulation study later found that least-squares point estimates of bias were reliable when the correlation coefficient exceeded 0.975, but that its confidence intervals for bias were unreliable in every case studied [Martin, 2000]. So a high correlation can rescue the estimate, but not the interval.

Deming regression: say how noisy each method is

Deming regression allows error in both methods. It needs one extra input: the ratio of the two methods’ error variances. Many analyses set it to one by default. With a short range of values relative to the measurement error, a wrong ratio can produce a bias of up to two thirds of the least-squares bias, and misleading standard errors; the ratio should come from duplicate measurements or from quality-control data. Even with a misspecified ratio, Deming still did better than least squares [Linnet, 1998].

Simple Deming assumes each method’s standard deviation is constant across the range. Most methods have an error closer to a constant coefficient of variation, so the standard deviation grows with concentration. There, simple Deming’s intervals for bias become unreliable, and only an iteratively reweighted form produced unbiased estimates with reliable intervals in every case simulated [Martin, 2000]. Weighting also saves samples: at a range ratio of 10, a weighted approach cut the number needed by more than 50% [Linnet, 1999].

Passing-Bablok regression: robust, with conditions

Passing-Bablok regression makes no assumption about the distribution of the samples or of the errors, and gives the same answer whichever method is put on the x axis [Passing and Bablok, 1983]. It is robust to outliers, which is why laboratories reach for it. It does have conditions: continuously distributed data and a linear relationship between the two methods. A cumulative sum (cusum) test checks linearity, a residual plot shows outliers and curvature, and data that are not linear are not suitable for concluding that methods agree [Bilić-Zulle, 2011]. The original procedure tests linearity before it gives intervals for slope and intercept [Passing and Bablok, 1983].

Checks that matter more than the model

Comparing all of these models on real data, one group concluded that the quality of the analytical input data matters more than the choice of model [Stöckl et al., 1998]. When a fit looks poor, investigate the data before reaching for a different regression. The checks below start from their list:

• Range. Spread the samples across the whole range the laboratory reports, with extra samples near each decision limit. The range ratio (highest over lowest result) drives power more than any other factor. To detect one standardised slope deviation at a range ratio of 2 took 544 samples; at a range ratio of 10, 64. The conventional 40 to 100 samples often need to be reconsidered, and very narrow ranges such as electrolytes need very large studies [Linnet, 1999].
• Scatter against imprecision. Compare the standard deviation of the residuals with the two methods’ combined analytical imprecision. If the scatter is much larger, something sample-related, such as an interference or a matrix effect, is at work, and no regression will remove it [Stöckl et al., 1998].
• Linearity. Look at the residual plot. A curve means one straight line cannot describe the relationship, and the bias at each limit should be estimated locally instead.
• Outliers. Investigate them, do not just drop them. One rule removes points further than 4 times the standard error of the estimate from the line [Cornbleet and Gochman, 1979]; whatever rule you use, fix it before the data are seen and report what was excluded.
• Correlation is not agreement. A high correlation coefficient says the range is wide relative to the scatter, not that the methods agree [Bland and Altman, 1986]. It is a screening check on whether a least-squares fit is usable [Stöckl et al., 1998], not a verdict on the change.

Reading the line at a decision limit

“Slope interval includes one, intercept interval includes zero” is a statistical statement about the whole line. It is not the same as “the bias at our decision limits is acceptable”. A small intercept matters a lot at a low limit; a slope a few per cent from one matters most at a high one. Read the line at each limit.

A published example shows why. In one hospital laboratory’s analyser change, the Deming line for serum creatinine was new = 0.944 × old + 0.165 mg/dL, from 44 patient samples averaging about 5 mg/dL [Bush et al., 2020]. Reading that line at three concentrations: