Deming Regression Input
Use paired rows or separate value lists. Large screens show three columns, smaller screens show two, and mobile shows one.
Example Data Table
| Sample | Method X | Method Y | Use Case |
|---|---|---|---|
| 1 | 1.10 | 1.30 | Instrument comparison |
| 2 | 2.00 | 2.40 | Lab method validation |
| 3 | 3.20 | 3.30 | Paired assay readings |
| 4 | 4.10 | 4.40 | Calibration study |
| 5 | 5.00 | 5.10 | Replicate measurement review |
Formula Used
Deming regression fits a straight line when both X and Y contain measurement error. It is often preferred over ordinary least squares for method comparison and calibration studies.
Means: x̄ = average of X values, ȳ = average of Y values.
Sample moments: Sxx, Syy, and Sxy are the sample variance and covariance terms computed from centered values.
Slope:
b = (Syy − λSxx + √((Syy − λSxx)² + 4λSxy²)) / (2Sxy)
Intercept:
a = ȳ − b x̄
Fitted line:
ŷ = a + bx
Variance ratio: λ represents measurement-error variance in X divided by measurement-error variance in Y.
How to Use This Calculator
- Enter data as paired lines or separate X and Y lists.
- Set the variance ratio λ. Use 1 when both methods have similar error variance.
- Choose the number of decimal places for reporting.
- Optionally enter one X value to estimate a fitted Y value.
- Press the calculate button to display the regression result above the form.
- Review the slope, intercept, residual table, and diagnostics.
- Use the CSV or PDF buttons to export the visible result summary.
Frequently Asked Questions
1. What is Deming regression used for?
It is used when both variables contain measurement error. Common examples include comparing laboratory instruments, calibrating sensors, and evaluating agreement between paired methods.
2. How is it different from ordinary least squares?
Ordinary least squares assumes X is measured without error. Deming regression allows error in both X and Y, which can produce less biased line estimates in method comparison studies.
3. What does the variance ratio λ mean?
λ is the error variance of X divided by the error variance of Y. When both methods have similar error variance, analysts often start with λ = 1.
4. Can I use separate X and Y lists?
Yes. Select separate list mode and enter matching counts. Each X value must correspond to the Y value in the same position.
5. What happens if covariance is near zero?
The fitted line becomes unstable because the paired variables do not vary together enough. The calculator stops and shows an error instead of returning a misleading result.
6. Why are orthogonal residuals shown?
Orthogonal residuals measure shortest distance from each point to the fitted line. They help assess fit when error exists in both variables rather than only vertically in Y.
7. Is R² always the main quality measure here?
Not always. R² is convenient, but orthogonal residual size, variance ratio choice, and study context are also important when evaluating Deming regression performance.
8. When should I keep λ equal to one?
Keep λ at one when both measurement systems are believed to have similar precision. Change it only when external error-variance evidence supports another value.