Deming Regression

Like Passing–Bablok regression, Deming regression is a supplement to the Bland–Altman plot for characterising systematic differences between two measurement methods. It addresses the same fundamental problem — that ordinary least-squares regression cannot be used when both variables carry measurement error — but does so parametrically, using knowledge of the relative magnitude of those errors.

The Error Variance Ratio

Deming regression fits the line $y = a + bx$ by minimising the sum of squared $perpendicular$ distances from the data points to the line, weighted by the ratio of the error variances of the two methods. This ratio is denoted $\lambda$:

$$ \lambda = \frac{\sigma_x^2}{\sigma_y^2} $$

where $\sigma_x^2$ and $\sigma_y^2$ are the measurement error variances of Methods $x$ and $y$ respectively. The choice of $\lambda$ determines how the regression line is fitted:

$\lambda = 1$: Both methods have equal measurement error — the most common assumption when neither method is clearly more precise.
$\lambda < 1$: Method $x$ is more precise than Method $y$; the line is pulled closer to the $x$ values.
$\lambda > 1$: Method $y$ is more precise; the line is pulled towards the $y$ values.

When $\lambda = 1$ and data are normally distributed, Deming regression minimises the sum of squared orthogonal (perpendicular) distances, which is why it is also called orthogonal regression. The slope and intercept are estimated using the method of Linnet (1990, 1993), with confidence intervals derived from the jackknife procedure.

Weighted Deming Regression

Standard Deming regression assumes that the measurement errors are constant across the measurement range (homoscedastic errors). In many analytical systems — particularly immunoassays — the measurement error is proportional to the magnitude being measured: a result of 200 units has a larger absolute error than a result of 10 units. Weighted Deming regression accommodates this by applying weights inversely proportional to the squared measurement value, effectively assuming a constant coefficient of variation rather than a constant standard deviation. It is the appropriate choice when the Bland–Altman plot shows that the spread of differences increases with the mean.

Interpreting the Output

As in Passing–Bablok regression, the key quantities are the slope $b$ and intercept $a$:

If the 95% CI for $b$ includes 1, no proportional bias is detected.
If the 95% CI for $a$ includes 0, no constant bias is detected.

Deming vs. Passing–Bablok

The two regression methods answer the same question — is there systematic or proportional bias between the methods? — and typically give similar results when the sample is large and well-distributed. The practical difference lies in their assumptions:

Passing–Bablok is non-parametric and makes no assumption about the distribution of measurement errors or their ratio. It is robust to outliers and to non-normal error distributions. It is the default choice when the error variance ratio is unknown.
Deming regression is parametric and more efficient (narrower confidence intervals) when the normality assumption holds and $\lambda$ is correctly specified. It handles heteroscedastic errors directly through the weighted variant. It is preferred when the error variances can be estimated from replicate measurements or are known from the methods' performance specifications.

When the two methods give different conclusions — which is uncommon but possible with small samples — the discrepancy usually reflects sensitivity to distributional assumptions or outliers, and both results should be reported and discussed.

MedCalc's Deming regression procedure reports the slope and intercept with standard errors and 95% confidence intervals (jackknife method), the Pearson correlation, and a scatter diagram with the regression and identity lines. Weighted Deming regression for proportional errors is also available. For a complete method comparison analysis, the Deming regression results should be interpreted alongside the Bland–Altman plot.