How this calculation works
Ordinary least squares assigns measurement error to the response while treating the predictor as fixed. When both methods are measured with error, that assumption can distort a fitted comparison line. Deming regression allows error in both coordinates. This page fits an unweighted linear Deming model using a supplied measurement-error variance ratio. It reports slope, intercept and vertical residuals without claiming that a fitted line establishes agreement.
Slope = [Syy−lambda Sxx + √((Syy−lambda Sxx)²+4lambda Sxy²)]/(2Sxy); intercept = meanY−slope meanXA worked example
Paired X values one through four and Y values three, five, seven and nine lie exactly on Y = 1 + 2X. The fitted Deming slope is two and intercept one. Because these points have no scatter away from that line, the same exact line is recovered for each positive entered error ratio. Noisier data generally change their fitted slope when that ratio changes.
Good to know
The model uses independent constant-variance errors and an unweighted straight line. Proportional measurement error, repeated subjects and uncertainty in the error ratio are not fitted. No slope or intercept confidence interval is returned. The page does not validate calibration, identify an error-free truth or decide whether one instrument can replace another.
Reference: R MethComp — Deming error variance regression (opens in a new tab) ↗
What this calculation answers
Ordinary least squares assigns measurement error to the response while treating the predictor as fixed. When both methods are measured with error, that assumption can distort a fitted comparison line. Deming regression allows error in both coordinates.
This page fits an unweighted linear Deming model using a supplied measurement-error variance ratio. It reports slope, intercept and vertical residuals without claiming that a fitted line establishes agreement.
The Simple linear regression calculator treats the predictor as fixed in its ordinary least squares model. Deming instead needs a separately justified measurement-error variance ratio.
Use the right input conventions
X and Y lists must describe corresponding measurements in the same order. The variance ratio means error variance in Y divided by error variance in X. It is not the ratio of the observed sample variances.
The ratio must come from a defensible error model or separate precision information. Both datasets need variation and nonzero cross covariance for the supported finite-slope solution.
Interpret the result
Changing the error ratio changes how departures in the two coordinates are weighted. Setting it to one gives equal error variances in the numerical units being used, not a universally neutral assumption after arbitrary unit rescaling.
A slope near one and intercept near zero may still leave substantial individual disagreement. Inspect residuals and paired-difference behavior rather than substituting correlation or a fitted line for an interchangeability assessment.
Work through the example
Paired X values one through four and Y values three, five, seven and nine lie exactly on Y = 1 + 2X. The fitted Deming slope is two and intercept one.
Because these points have no scatter away from that line, the same exact line is recovered for each positive entered error ratio. Noisier data generally change their fitted slope when that ratio changes.
Where the model stops
The model uses independent constant-variance errors and an unweighted straight line. Proportional measurement error, repeated subjects and uncertainty in the error ratio are not fitted.
No slope or intercept confidence interval is returned. The page does not validate calibration, identify an error-free truth or decide whether one instrument can replace another.
A few questions, answered
Can I use observed variance Y divided by observed variance X as the error ratio?
Not automatically. Observed variances include real between-subject variation as well as measurement error.
What changes in the comparison?
Add measurement scatter and give Y error variance four times the X error variance.