Deming regression

What is Deming regression?

Deming (1943) developed a regression that allows comparing two measurement methods X and Y. Deming regression assumes that measurement error is present in both X and Y. It overcomes the assumptions of the classical linear regression that are inappropriate for this application. As a reminder the assumptions of the Ordinary Least Squares regression are:

• The explanatory variable X in the model y(i)=a+b.x(i)+e(i) is deterministic (no measurement error),
• The dependent variable Y follows a normal distribution with expectation aX
• The variance of the measurement error is constant.

Furthermore, extreme values can highly influence the model.

Deming proposed a method which overcomes these assumptions: the two variables are assumed to have a random part (representing the measurement). The distribution has to be normal. We then define:

• y(i)=y(i)*+e(i)
• x(i)=x(i)*+ η(i)

Assume that the available data (yi, xi) are mismeasured observations of the “true” values (y(i)*, x(i)*) where errors ε and η are independent. The ratio of their variances is assumed to be known:

• d=s²(η)/s²(e)

XLSTAT allows you to define variances of error measurement on both X and Y.

We seek to find the line of “best fit” y* = a + b x*, such that the weighted sum of squared residuals of the model is minimized.

The Deming method allows computing the a and b coefficients as well as a confidence interval around these values. The study of these values helps comparing the methods. If they are very close, then b is close to 1 and a is close to 0.

The Deming regression has two forms:

• Simple Deming regression: The error terms are constant and the estimation of the parameters a and b is very simple using a direct formula (Deming, 1943).
• Weighted Deming regression: In this case, we suppose that the error terms are not constant but only proportional. Parameters a and b are estimated using an iterative method (Linnet, 1990).

Confidence intervals for the intercept and slope coefficients are complex to compute. XLSTAT uses a jackknife approach to compute confidence intervals, as proposed in Linnet (1993).