The Line and the Cloud

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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 whose work spanned heredity, meteorology and statistics. He measured the heights of parents and children, tabulated their frequencies in a grid and found elliptical contours.

The ellipse's shape became his measure of association: a narrow ellipse around a line indicates a strong linear relationship when the axis scales are held fixed. This 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 — Figure 2. Plate X from *Regression towards Mediocrity in Hereditary Stature* (1886).

Galton's original diagram from 1886. The printed numerals are Galton’s rough interpolations of a smoothed frequency surface, not raw cell counts (original article, p. 263). The optional colour overlay interpolates the transcribed printed values, including the two obscured entries confirmed as 13 and 2. Blank entries are left missing, not treated as observed zeros. The tilted ellipse is the correlation: narrowness about a line shows strong linear covariation when the axis scales are held fixed. Steepness alone does not say how strong the relationship is. Pearson formalised this as r barely a decade later (1895–96). Regression toward mediocrity: the children's heights regress towards the mean. "Locus of horizontal tangential points" is the regression line.

Correlation

What is r?

Figure 3. r describes the shape of the cloud, not a line through it.

Correlation

Restriction of range

Figure 4. Correlation depends on the range of values we examine.

Agreement

Correlation ≠ agreement

Figure 5. High correlation does not mean that the methods give the same results.

Agreement

Is a mean bias of 16 the whole story?

Figure 6. The difference between the methods increases with concentration.

Regression models

OLS: the invisible default

Figure 7. OLS minimises the sum of squared vertical residuals.

Regression models

Deming: the principled default

Figure 8. Deming accounts for measurement error in both methods.

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 replicate measurements 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 using 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

Figure 9. Passing–Bablok is based on ranked pairwise slopes.

Non-parametric: built on ranked pairwise slopes. Classical Passing–Bablok uses a shifted median and special rules for certain slopes; the animation illustrates the principle. 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 and no error model against which to assess the residuals. Confidence intervals for slope and intercept are computed analytically (EP09c app. I). Confidence intervals for bias at a decision level require bootstrapping (EP09c app. K2); the jackknife gives unrealistically narrow intervals because a single point rarely moves the median. Suitable when there are 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

Figure 10. Measurement error in x attenuates the OLS slope.

Regression models

Model convergence

Figure 11. OLS, Deming and Passing–Bablok with adjustable correlation.

Engines 1 and 2. Slider for r from 0.80 to 0.99, three lines. When the random measurement error is small relative to the clinically relevant measuring interval, the models often converge. Increase r to 0.99 to reveal the question mark. Click it to zoom into the twenty central observations in a narrower x interval. Once the animation finishes, all three models are fitted to those pairs and the correlation within that range is shown below the plot. The slider still shows r for the full cloud. The return arrow restores the full cloud. Do not say that the choice of model is irrelevant above r = 0.99. That undermines the point about restriction of range, since r depends on the sampled range and not only on the precision of the method. Say instead: at low measurement error the models converge, but model choice depends on the error structure, not the value of r.

Regression

The role of regression

Slope ≠ 1: proportional bias

Intercept ≠ 0: constant bias

R² = 1 − SS_res / SS_tot

SS_res: the sum of the squared vertical distances from the points to the regression line — the residual variation.

SS_tot: the sum of the squared deviations of the y values from their mean — the total variation.

R² = 0.90 means that the residual sum of squares is 10% of the total sum of squares.

r measures linear association. R² = r² holds for simple OLS regression with an intercept.

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 = Σ(y_i − ȳ)² is the total sum of squares about the mean. SS_res = Σ(y_i − ŷ_i)² is the residual sum of squares about the fitted line. These are sums of squares, not variances as such. Compare the line with the simple prediction that every y value equals the mean. If SS_tot = 100 and SS_res = 10, R² = 0.90. The coefficient of determination describes fit, not causal explanation or agreement between methods. 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 the coefficient of determination coincide algebraically. For Deming and Passing–Bablok the same ratio can be computed, but it does not have the OLS model's variance decomposition and should not be used as a general measure of model adequacy. 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?

Figure 12. Anscombe’s quartet: four different patterns with the same summary statistics.

Visual inspection

Regression models have no eyes

Figure 13. Different point patterns with the same summary statistics.

Model philosophy

Ptolemy's geocentric model with epicycles — Figure 14. 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."

Error structure

Constant SD or constant CV?

Constant SD

Roughly the same absolute scatter.

Constant CV

Scatter grows with concentration.

Figure 15. Illustrative residuals around a correctly specified linear relationship: the same noise pattern on different scales.

A closer look · error structure

Three approaches, different assumptions

Weighted Deming

Models measurement error in both methods. With constant CV, the model uses concentration-dependent variances.

Imprecision data inform the error model. High concentrations should not dominate simply because their absolute variance is larger.

Log scale

Can stabilise proportional scatter when values are positive, but also changes the form of the relationship.

log y = α + β log x
y = e^αx^β

Passing–Bablok

Built on ranked pairwise slopes. A robust alternative, with assumptions of its own.

Robustness is no substitute for examining outliers, linearity and error structure.

may be consideredwith outliers

Weighted Deming: under constant CV the variance is proportional to the square of the concentration; the imprecision of both methods and the estimated concentrations determine the weights. Log–log regression becomes a power-law relationship after back-transformation, not y = a + bx in general. The relationship refers to the fitted value back-transformed from the log scale; predicting the mean on the original scale may need further attention to the error model. Passing–Bablok uses a shifted median and special rules for pairwise slopes. This slide can be skipped during the talk. CLSI EP09c, 3rd edition, 2018: original PDF in Zotero, parent 7W9U3GEN, attachment PIC3B5C7. Full-text sections read: 6.2.1–6.2.4, Appendices B and J. Section 6.2.1 recommends constant-SD Deming as default; 6.2.2 recommends constant-CV Deming and cautions against comparing transformed slopes with original-scale acceptance criteria. Sections 6.2.3–6.2.4 describe Passing–Bablok for mixed variability and aberrant results. Known precision profiles can support profile-weighted Deming (6.2.3 and Appendix J). PDF pages 25–27, 34–35 and 56–57 in the 63-page eCLIPSE printout. This summarises CLSI without implying that Passing–Bablok has no assumptions of its own. NIST Passing Bablok Fit.

Summary

Why the choice of model matters

r measures linear association, not agreement, bias or interchangeability.
OLS ignores measurement error in x. Classical, independent measurement error in x attenuates the slope.
Deming accounts for error in both methods, as required when comparing two measurement procedures.
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 model choice depends on the error structure, not the value of r.

Back in the laboratory

From scatter plots to method decisions

Plot the data. Examine the relationship, differences, outliers and measuring interval.
Assess the error structure. SD, CV and imprecision in both methods.
Choose the model. Check linearity and the relevant assumptions.
Estimate bias and uncertainty at clinically relevant levels.
Compare with predefined requirements. Is the evidence sufficient?

Acceptance criteria are set before the study.

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.
Bland JM, Altman DG. Measuring agreement in method comparison studies. Stat Methods Med Res 1999;8:135–160. NIST Dataplot: Passing Bablok Fit.
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)

Filip Landgren

filip.landgren@gmail.com
https://klinkemipedia.se

View this presentation on a computer

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

Is a mean bias of 16 the whole story?

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

Constant SD or constant CV?

Constant SD

Constant CV

Three approaches, different assumptions

Weighted Deming

Log scale

Passing–Bablok

Why the choice of model matters

From scatter plots to method decisions

References

A visual guide to regression models
in clinical chemistry