Calculator Inputs
Enter assigned exposure, Berkson uncertainty, regression terms, confidence level, and a reference exposure for comparison.
Example Data Table
| Assigned Exposure | Berkson SD | β0 | β1 | Residual SD | n | Confidence | Expected Outcome | Outcome SD |
|---|---|---|---|---|---|---|---|---|
| 12.0000 | 1.8000 | 5.0000 | 2.4000 | 3.2000 | 50 | 95.00% | 33.8000 | 5.3761 |
The example uses the same default values loaded into the form so visitors can reproduce the displayed outputs immediately.
Formula Used
X = W + U, with E(U) = 0 and Var(U) = σu2
Y = β0 + β1X + ε, with E(ε) = 0 and Var(ε) = σe2
E[X|W] = W
Var(X|W) = σu2
Exposure interval = W ± z × σu
E[Y|W] = β0 + β1W
Var(Y|W) = β12σu2 + σe2
Outcome interval = E[Y|W] ± z × √(Var(Y|W))
SE(mean X) = σu/√n
SE(mean Y) = √(Var(Y|W))/√n
Mean CI = estimate ± z × SE
These formulas suit a simple additive Berkson setting with one assigned exposure and a linear response model. More complex multivariable or nonlinear cases may need specialized adjustment methods.
How to Use This Calculator
- Enter the assigned exposure or predicted value used in your study design.
- Provide the Berkson error standard deviation that reflects spread around the assigned value.
- Enter the regression intercept and slope from your linear model.
- Add the residual standard deviation for unexplained outcome noise.
- Set sample size and your preferred confidence level.
- Supply a reference exposure if you want a side-by-side outcome comparison.
- Click the calculate button to show the result above the form.
- Use the export buttons to save the result as CSV or PDF.
Frequently Asked Questions
1) What does Berkson error mean?
Berkson error means the assigned or measured value is treated as fixed, while the true value varies around it. This often appears with group averages, monitor assignments, or values produced by prediction equations.
2) How is Berkson error different from classical error?
Classical error adds noise to the observed value around the truth. Berkson error flips that relationship: the true value varies around the assigned value. Their impact on regression bias and uncertainty can be very different.
3) Does Berkson error always leave slope estimates unbiased?
Not always. In simple linear settings with independence assumptions, slope estimates can remain unbiased. In multivariable, correlated, or nonlinear models, Berkson error can still introduce bias and invalidate standard inference.
4) What does the outcome interval represent?
It summarizes the modeled spread of individual outcomes after combining Berkson exposure uncertainty with ordinary residual noise. It is wider when the slope, Berkson spread, or residual standard deviation becomes larger.
5) Why does sample size affect the mean confidence intervals?
Sample size does not change the underlying individual-level variability. It changes how precisely the average exposure or average outcome can be estimated. Larger samples shrink the standard error of the mean.
6) What should I enter for the reference exposure?
Use a baseline, policy limit, control value, historic average, or any comparison point meaningful to your analysis. The calculator uses that reference to compute expected outcome differences and percentage change.
7) Can I use this tool for logistic or nonlinear models?
This page is designed for a simple linear response model. Nonlinear models such as logistic regression often require simulation, regression calibration, SIMEX, Bayesian modeling, or other specialized correction strategies.
8) What is the main takeaway from the variance share metric?
It estimates how much of modeled outcome variability comes from Berkson uncertainty rather than ordinary residual noise. A larger percentage means assigned-exposure uncertainty is playing a stronger role in total spread.