4-Parameter Logistic Dose-Response (4PL)
PharmacologyFits a four-parameter sigmoid dose-response curve estimating both the bottom and top asymptotes, the IC50/EC50, and the Hill slope. The standard model for dose-response analysis.
When to Use
Use this model when both the baseline and maximum response are unknown and need to be estimated from the data. This is the default dose-response model in most pharmacological applications, suitable for both normalised and unnormalised data.
Assumptions
- The dose-response relationship follows a monotonic sigmoid curve.
- Both the bottom and top asymptotes are finite and estimable from the data.
- Sufficient dose range to capture the full sigmoid transition from bottom to top.
- At least 4 distinct dose levels (ideally 6-8) with replicates.
- Residuals are approximately normally distributed with constant variance.
Required Inputs
| Input | Type | Notes |
|---|---|---|
| Dose / Concentration | Numeric | X-axis values, must be positive |
| Response | Numeric | Y-axis values (measured response) |
Output Metrics
| Metric | What it means |
|---|---|
| Bottom | Minimum response (lower asymptote). With SE and 95% CI. |
| Top | Maximum response (upper asymptote). With SE and 95% CI. |
| IC50 / EC50 | Dose producing 50% of the response range (midpoint between Bottom and Top). With SE and 95% CI. |
| Hill Slope | Steepness of the curve at the inflection point. With SE and 95% CI. |
| pIC50 | Negative log10 of IC50 for potency comparisons. |
| R-Squared | Proportion of variance explained by the model. |
| Adj R-Squared | R-squared adjusted for the number of parameters (4). |
| RMSE | Root mean squared error of the residuals. |
| AIC | Akaike Information Criterion (lower is better). |
| BIC | Bayesian Information Criterion (lower is better). |
Interpretation
- The 4PL is the standard sigmoid model. The response window is defined by Bottom (baseline) and Top (maximum), with IC50 at the midpoint.
- The Hill slope controls curve steepness. Slopes near 1 are typical for simple receptor-ligand binding. Very steep slopes (> 3) may indicate cooperative binding or an all-or-none response.
- Compare to the 3PL model using AIC/BIC. If the bottom is genuinely zero (normalised data), the 3PL may be preferred for parsimony.
- Confidence intervals for the parameters indicate estimation precision. Parameters with very wide CIs suggest insufficient data to constrain the model.
- An R-squared well below 1.0 may indicate poor model fit, systematic deviations, or that the data do not follow a simple sigmoid.
Common Pitfalls
- If the data do not span the full response range (doses are all above or below the IC50), the model will extrapolate the missing asymptote, producing unreliable parameter estimates.
- Fitting a 4PL to data that are already normalised to 0-100% can result in redundant Bottom/Top estimates close to 0 and 100. In this case, a 3PL may be more appropriate.
- The Hill slope and IC50 are correlated parameters. Poorly constrained data can produce large covariance between them, making individual estimates unreliable even when the overall curve looks reasonable.
How It Works
- Define the 4PL model: Response = Bottom + (Top - Bottom) / (1 + (IC50/Dose)^HillSlope).
- Use non-linear least squares to estimate Bottom, Top, IC50, and Hill Slope by minimising the sum of squared residuals.
- Compute the parameter covariance matrix and derive standard errors and confidence intervals for each parameter.
- Calculate goodness-of-fit statistics (R-squared, AIC, BIC) and generate predicted values for the fitted curve.
Citations
References
- Hill, A. V. (1910). The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. Journal of Physiology, 40, iv-vii.
- DeLean, A., Munson, P. J., & Rodbard, D. (1978). Simultaneous analysis of families of sigmoidal curves. American Journal of Physiology, 235(2), E97-E102.