The Line and the Cloud

A visual guide to regression models
in clinical chemistry

Filip Landgren

Historical background

Sir Francis Galton

1822–1911

Darwin's half-cousin and a Victorian polymath in heredity, meteorology and statistics. He measured the heights of parents and children, tabulated the frequencies on a grid and saw that the contours formed ellipses.

The shape of the ellipse became his measure of covariation: narrow and steep means a strong relationship. That was correlation, barely a decade before Pearson formalised r (1895–96).

Historical background

Galton's ellipse

Galton 1886, Plate X: ellipse diagram of the heights of parents and children

Plate X from Regression towards Mediocrity in Hereditary Stature (1886)

Correlation

What is r?

r describes the shape of the cloud, not a line through it.

Correlation

Restriction of range

Agreement

Correlation ≠ agreement

Agreement

Bland–Altman

Regression models

OLS: the invisible default

Regression models

Deming: the principled default

Deming minimises weighted orthogonal distances and accounts for measurement error in both methods. λ is the ratio of the methods' measurement error variances, λ = σ²(reference)/σ²(test), the convention used by Linnet (Clin Chem 1998) and the R package mcr. High λ: the reference method has more measurement error and the residuals tilt towards the horizontal. Low λ: the test method has more measurement error and the residuals approach OLS. λ = 1 (orthogonal regression) is a common assumption when imprecision data are lacking, but in practice it is an assumption of equal measurement error. λ must be based on imprecision data, not on the scatter about the regression line. The best source is duplicates within the comparison study itself: within-sample replicates are independent of the x–y relationship and give λ directly (Linnet, Clin Chem 1998; CLSI EP09c 6.2.2). Without replicates, use imprecision from EP05 studies or QC data. Without those, set λ = 1 and run a sensitivity analysis over other values (EP09c 6.2.2). Estimating λ from the residual scatter of single measurements is circular. Show that the angle of the residuals changes with λ. CLSI EP09c (3rd ed., 2018) recommends Deming as the default, but which variant depends on the error structure: constant-SD Deming when SD is constant across the measuring interval (6.2.1), weighted constant-CV Deming when CV is constant (6.2.2). Passing–Bablok is listed as viable under constant CV and as the first choice when the distribution is non-normal or skewed (6.2.4).

Regression models

Passing–Bablok: the robust alternative

Non-parametric: the median of all pairwise slopes. Animate: show all slopes, then highlight the median. Robust to outliers and does not require normality. The method is not assumption-free, however. Passing–Bablok implicitly assumes that the ratio of the methods' measurement error SDs equals the slope β (variance ratio β²), that is, that the errors are proportional to each method's own scale (EP09c app. I; Baumdicker and Hölker, Stat Probab Lett 2020, remark 2.5). With β close to 1 this is unproblematic; with large differences in scale the assumption is not self-evident. The residuals (perpendicular distances to the line) are used in the method's own CUSUM linearity test but have no parametric interpretation: no standard errors, no error model to diagnose against. Confidence intervals for slope and intercept are computed analytically (EP09c app. I). Confidence intervals for bias at a decision level require bootstrap (EP09c app. K2); the jackknife gives unrealistically narrow intervals because a single point rarely moves the median. Suited to outliers, pronounced heteroscedasticity and non-normal or skewed distributions of the differences. Not better than Deming, but more robust when the assumptions are uncertain.

Regression models

Regression dilution

Regression models

Model convergence

Regression

The role of regression

Slope ≠ 1: proportional bias

Intercept ≠ 0: constant bias

r measures covariation between two variables and is independent of the regression model.
R² = 1 − SS_res / SS_tot is the proportion of variance the model explains. That R² = r² is a property of OLS, not a general truth.

r says almost nothing about agreement once r exceeds 0.98. It still has a supporting role: EP09c app. G uses r ≥ 0.975 as a check that the x range is wide enough for OLS to be defensible, not as a measure of agreement. Recalibrating one method against another requires the equation. R² is the coefficient of determination, 1 − (residual sum of squares / total sum of squares). SS_tot is the total variance of y about its mean, SS_res the variance the model does not capture. R² answers how much of the variation in y the model explains. Pearson's r measures linear covariation between x and y, a descriptive measure that is independent of whichever model is fitted. R² = r² holds only for OLS, where minimising vertical residuals makes correlation and explained variance coincide algebraically. Deming and Passing–Bablok do not minimise the sum of squares in the y direction, so R² in the traditional sense is not defined. Do not go deeper into R². The message is that r and R² are different measures answering different questions.

Visual inspection

Can you see a difference?

Visual inspection

Regression models have no eyes

Model philosophy

Ptolemy's geocentric model with epicycles

All models are geocentric

Identical statistics can hide entirely different realities, and a model with the wrong assumptions can still predict well.

Ptolemy's epicycles predicted planetary positions with high precision. The ontology was wrong.

"All regression models are geocentric, but not equally geocentric. The assumptions about error structure decide which slope you get."

Regression models

Heteroscedasticity

Weighted Deming (constant-CV Deming): weight = 1/(analytical variance at that concentration), in practice 1/(estimated concentration)², so that every concentration level contributes equally. The recommended default under constant CV per CLSI EP09c 6.2.2.

Log transformation: take logarithms first so that the scatter becomes constant, fit Deming on the log scale and back-transform.

Passing–Bablok: ranks slopes and needs no weights. Viable under constant CV per EP09c, but requires more samples than parametric methods for the same precision.

Left: constant scatter (homoscedastic); Deming works well. Right: scatter increasing with concentration (heteroscedastic), the so-called trumpet or fan shape. Proportional noise is common in clinical chemistry (immunoassays, mass spectrometry). Under pronounced heteroscedasticity there are two routes. Passing–Bablok ranks slopes and is insensitive to the scatter structure. Weighted Deming (constant-CV Deming) uses the weight 1/(estimated concentration)², inversely proportional to the analytical variance at each level (EP09c app. B, formula B4), so that every level has equal influence. Do not use the observed x value alone as the weight; it carries measurement error. The mean of the estimated concentrations of x and y is standard. CLSI EP09c (2018, superseding EP09-A3) recommends constant-CV Deming when the difference plot shows constant CV (6.2.2) and constant-SD Deming when it shows constant SD (6.2.1). Assess the error structure in the difference plot first; it governs the choice. Do not go deeper into weighted regression. The principle: if the scatter is proportional to the concentration, the weights should compensate for it.

Summary

Why the choice of model matters

r measures covariation, not agreement, bias or interchangeability
OLS assumes x is error-free. It never is in a method comparison, and the slope is attenuated.
Deming models the error in both axes, which is the correct assumption when two measurement procedures are compared
Passing–Bablok is robust, but robustness is not always an advantage. The method does not use imprecision data and rests on a hidden assumption about the error ratio.
When the measurement error is small relative to the measuring interval, the models converge, but the error structure decides, not the value of r

"Choose the model by its assumptions, not by convenience."

References

CLSI EP09c, 3rd ed. 2018. Measurement Procedure Comparison and Bias Estimation Using Patient Samples. Ch. 6.2 (choice of regression), app. B (weighted Deming), app. G (r and the x range), app. I (Passing–Bablok), app. K (CIs for bias).
Linnet K. Performance of Deming regression analysis in case of misspecified analytical error ratio. Clin Chem 1998;44:1024–31.
Linnet K. Necessary sample size for method comparison studies based on regression analysis. Clin Chem 1999;45:882–94.
Stöckl D, Dewitte K, Thienpont LM. Validity of linear regression in method comparison studies. Clin Chem 1998;44:2340–6.
Passing H, Bablok W. J Clin Chem Clin Biochem 1983;21:709–20 (Part I) and 1984;22:431–45 (Part II).
Baumdicker F, Hölker U. Stat Probab Lett 2020;164:108801.
Galton F. Regression towards mediocrity in hereditary stature. J Anthropol Inst 1886;15:246–63.
Anscombe FJ. Am Stat 1973;27:17–21. Matejka J, Fitzmaurice G. Same stats, different graphs. CHI 2017.
McElreath R. Statistical Rethinking, 2nd ed. 2020, ch. 4.

"The task is not so much to see what no one has yet seen, but to think what nobody has yet thought about that which everybody sees."

Arthur Schopenhauer, Parerga und Paralipomena (1851)

Thank you

Filip Landgren

The Line and the Cloud

A visual guide to regression modelsin clinical chemistry

Sir Francis Galton

Galton's ellipse

What is r?

Restriction of range

Correlation ≠ agreement

Bland–Altman

OLS: the invisible default

Deming: the principled default

Passing–Bablok: the robust alternative

Regression dilution

Model convergence

The role of regression

Can you see a difference?

Regression models have no eyes

All models are geocentric

Heteroscedasticity

Why the choice of model matters

References

Thank you

A visual guide to regression models
in clinical chemistry