Deming-Regression

Deming regression is used to compare two measurement metods. Run Deming regression in Excel using the XLSTAT add-on statistical software.

What is Deming regression?

Deming (1943) developed a regression that allows comparing two measurement methods (for example, two techniques for measuring concentration of an analyte), which supposes 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 OLS 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(e)/s2(h)

In practice, the variance of the x and y is often unknown which complicates the estimate of d but when the measurement methods for x and y are the same they are likely to be equal so that d=1 for this case. XLSTAT-Life allows you to define d.

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.

Where h and ε follow a normal distribution. The Deming method allows calculating 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 ratio between variances has to be chosen (with default value being 1). The estimation is very simple using a direct formula (Deming, 1943).
  • Weighted Deming regression: In the case where replicates of the experiments are present, the weighted Deming regression supposes that the error terms are not constant but only proportional. Within each replication, you can take into account the mean or the first experiment to estimate the coefficients. In that case, a direct estimation is not possible. An iterative method is used (Linnet, 1990).

Confidence interval of the intercept and slope coefficient are complex to compute. XLSTAT uses a jackknife approach to compute confidence intervals, as stated in Linnet (1993).