The four parameters describe two asymptotes, a midpoint, and slope
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.
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.
Worked example: what to inspect before accepting 18.0 as an EC50
Imagine an eight-level reference dilution series with replicates. The fit returns lower 72, upper 1,710, midpoint 18.0 ng/mL, and slope 1.24. The midpoint is not interpretable just because the optimizer converged.
First confirm that observed concentrations bracket 18.0 and responses approach both fitted asymptotes. Then inspect well-level residuals for curvature, changing spread, and isolated positions. Review uncertainty and whether any parameter is against a bound. Finally, compare the result with configured reference-performance and replicate-variability criteria.
Restrictions and weighting answer different problems
Fixing an asymptote or sharing a slope can stabilize a fit or encode a relative-potency model, but it changes what is estimated. The scientific rationale and exact restriction should be visible in development comparisons and routine reports.
Weighting addresses nonconstant response variance. It should be selected from assay behavior and evaluated across relevant runs, not introduced because one difficult curve looks better. Preserve the weight function and whether weights depend on observed or fitted response.
Use diagnostics that can reveal different failure modes
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.
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.