# Validation and updating of a laboratory conversion formula (OLS, Passing-Bablok, Deming, Bland-Altman)

Hello to all.

Haven’t seen this discussed here before, so perhaps others will find helpful too.

• There are two biomarkers A and B.
• They are related (one comes from the other) and address the same underlying physiology.
• They are metabolized differently such that patient characteristics (e.g., weight, kidney function) affect the relationship between the concentration of A and concentration of B.
• Studies have been done that show you can convert between A and B using formulas fit with multiple OLS regression where A was regressed on B + covariates.

We want to test one of the formulas in an independent sample and to recalibrate/extend it as necessary.

The issue that seems to have been ignored in numerous studies that have developed and validated these formulas is that there is measurement error in observed A and predicted A (from B + covariates), violating an assumption of OLS. If we evaluate one of these formulas with OLS on our sample I suspect the slope will be underestimated. Methods like Bland-Altman analysis, Deming regression, and Passing-Bablok regression address this.We can use Deming or Passing-Bablok regression to estimate the calibration slope/intercept of observed A vs A predicted from the original formula (based on B + covariates).

To improve the existing formula with additional covariates or updated intercepts on individual covariates is a challenge. Deming and Passing-Bablok regression do not allow for covariates, just a single independent variable. Is there any sense in first fitting the new model with OLS and then calculating a recalibration slope/intercept on the predicted value with Passing-Bablok or Deming? I wonder if could still bootstrap to evaluate optimism-corrected performance.

Alternatively, does anyone have experience with Partial Least Squares or Structural Equation Modelling, which I understand can recover the Deming slope in a multivariable setting?

Thanks very much

One easy solution is getting a bootstrap confidence interval on the mean absolute discrepency. See Biostatistics for Biomedical Research - 16  Analysis of Observer Variability and Measurement Agreement

Great chapter! I worked through the examples.

If there is a proportional bias, say, 20%, then discrepancies between measured and predicted values of the biomarker A will be larger in absolute terms at the higher end of the biomarker than at the lower end. It seems then that to calculate a correction factor I would need a relative measure (like the slope of a regression line). The nice thing about a regression method is the ability to decompose the proportional error (slope) and the constant error (intercept).

I wonder if I first fit via OLS, then use the predictions as the single independent variable in the Passing-Bablok regression, I would get a slope and intercept that I can use to recalibrate the OLS fit so it would generalize better. This I can also do through bootstrapping and use the mean of the bootstrapped slope and intercept. What do you think?

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Good question. And there might me some useful thoughts from Eugene Harris’ Clinical Chemistry book.

Thank you for referring me to this book. Much related fundamental material but so far nothing in my read of it that informs the issue directly.

Interestingly, I have found nothing in the literature so far that addresses this scenario. The PB-recalibrated OLS idea I think warrants some empiric evaluation.

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