1. Read the four parameters in the equation
One common increasing-response parameterization is shown below. The lower and upper parameters describe fitted asymptotes, the midpoint is the concentration halfway between those fitted responses, and the slope controls the transition. Other packages may reverse the response direction, use a signed Hill slope, or fit log concentration directly.
Because conventions vary, a report should identify the exact equation and software convention rather than list “4PL” alone. Units are especially important for the midpoint: 2 in µg/mL is not the same result as 2 ng/mL.
response = lower + (upper − lower) ÷ [1 + (midpoint ÷ concentration)^slope]| Parameter | Interpretation | Warning sign |
|---|---|---|
| lower | Fitted response at the low asymptote | Few low-response observations |
| upper | Fitted response at the high asymptote | Curve never approaches a plateau |
| midpoint | Concentration halfway between fitted asymptotes | Estimate falls outside tested range |
| slope | Steepness and direction of transition | Extreme value or high uncertainty |
See what midpoint and slope change
Move the controls to see how two fitted parameters affect an increasing 4PL curve. This diagram explains the equation; it does not fit experimental observations.
y = 75 + (1900 − 75) / (1 + (EC50 / x)slope)When fitting real data, confirm that the tested concentrations support the midpoint and both plateaus, then inspect residuals and parameter uncertainty.
2. Prepare concentrations and responses without erasing the plate
Keep raw wells, sample identity, concentration, replicate, plate position, and controls. If the method normalizes to positive and negative controls, retain both raw and normalized values and record the formula. Decide in advance whether the model uses individual wells or a summary at each concentration.
A logarithmic concentration axis is useful for visualization and optimization, but zero cannot be logged. Zero-dose controls can remain part of assay assessment while being handled separately from the positive-concentration model domain.
- Verify concentration units and dilution direction.
- Plot every observation before fitting.
- Confirm points cover low response, transition, and high response where the intended estimate requires them.
- Apply only method-defined transformations and exclusions.
- Record whether replicate wells are fitted individually or summarized.
3. Fit and inspect the worked concentration series
Use the eight illustrative observations below as a transparent fitting exercise. Create four parameter cells, calculate one predicted response per concentration with the stated equation, calculate residual = observed minus predicted, square each residual, and minimize their sum by changing lower, upper, midpoint, and slope. Use a nonlinear least-squares optimizer and retain its starting values, bounds, convergence status, and final objective value.
For these exact synthetic observations, the optimum is lower 75, upper 1,900, midpoint 20.0 ng/mL, and slope 1.20. At 20.0 ng/mL the fitted response is halfway between the two asymptotes: 987.5 response units. Real observations will not normally land exactly on the curve, so the residuals and parameter uncertainty—not agreement with these synthetic numbers—determine whether the fit is supported.
- Place concentration and observed response in separate columns.
- Choose starting values from the low and high responses, a midpoint near the center of the transition, and a moderate positive slope for this increasing convention.
- Calculate predicted response, residual, squared residual, and total residual sum of squares.
- Minimize total squared residual by changing the four parameter cells while enforcing only scientifically justified bounds.
- Confirm convergence, inspect the observed-versus-fitted plot and residuals, and repeat from materially different starting values to detect an unstable solution.
| Concentration (ng/mL) | Observed response |
|---|---|
| 1 | 123.8 |
| 2.5 | 214.0 |
| 5 | 365.7 |
| 10 | 628.5 |
| 20 | 987.5 |
| 40 | 1,346.5 |
| 80 | 1,609.3 |
| 160 | 1,761.0 |
4. Decide whether the fitted midpoint is supported
No single statistic establishes fit quality. R² can be displayed, but it should sit beside evidence that tests coverage, residual behavior, parameter stability, and the method’s intended use.
- Observed and fitted values on an appropriate concentration scale.
- Residuals by fitted response, concentration, and plate position.
- Convergence state, bounds, starting-value behavior, and warnings.
- Parameter standard errors or confidence intervals where supported.
- Midpoint location relative to the tested range.
- Replicate mean, standard deviation, and CV with individual wells available.
- Repeated-run performance on reference or control materials.
5. Report the result with its evidence
Report the exact equation and convention, concentration and response units, unrounded parameter estimates, uncertainty method, included observations, fitted and residual plots, convergence and bound state, exclusions, suitability outcomes, software version, and controlled analysis settings.
Keep the fitted midpoint distinct from the biological or reportable decision. A converged 20.0 ng/mL estimate is still unsupported when the tested range, residual behavior, uncertainty, or predefined suitability rule fails.
Limits, restrictions, and weighting
Fixing an asymptote or sharing a slope can stabilize a fit or encode a relative-potency model, but it changes what is estimated. Weighting addresses nonconstant response variance. Either choice requires representative evidence and an exact, controlled definition before routine use.
A linear fit, 5PL, different slope convention, or another software parameterization answers a different modeling question. Compare alternatives during development; do not switch them after seeing one routine result. Provenarium’s supported relative-potency calculations are limited to its exact parallel-line log-log, common-shape 4PL, and common-shape 5PL contracts; this single-curve teaching example is not a claim of broader model support.
Frequently asked questions
Is a 4PL always better than linear interpolation?
No. The model should match the response and intended estimate. A linear model may be appropriate over a justified local range; a full sigmoid is useful when the design supports its asymptotes and transition.
Why can two programs return different 4PL parameters?
They may use different equations, slope signs, log transformations, weights, bounds, starting values, replicate handling, or fitting tolerances. Preserve the complete model convention and analysis settings.
Can I choose between 4PL and 5PL for every routine run?
Routine model selection is generally more reproducible when defined in the method. Compare models during development and validation, then control the approved choice unless a procedure explicitly permits alternatives.