3-Parameter Logistic Dose-Response (3PL)

Use this model when your data have been normalised to a zero baseline (e.g., 0% inhibition or 0% response at the lowest dose) and you want to estimate the dose producing 50% of the maximum effect. Common in pharmacology for potency ranking of compounds where the baseline response is known.

• The dose-response relationship follows a monotonic sigmoid curve.
• The bottom asymptote is fixed at zero (data are normalised to baseline).
• You have a sufficient range of doses to bracket the IC50/EC50 value.
• At least 4 distinct dose levels are included, with replicates at each level for reliable fitting.
• Residuals around the fitted curve are approximately normally distributed with constant variance.

• The IC50 is the most important output: it tells you the dose needed for half-maximal effect. Lower IC50 means higher potency.
• Use pIC50 (-log10 of IC50) for potency ranking. A compound with pIC50 = 7 is 10-fold more potent than one with pIC50 = 6.
• The Hill slope describes how steeply the response changes around the IC50. A slope > 1 indicates positive cooperativity; a slope < 1 indicates negative cooperativity.
• Compare 3PL and 4PL models using AIC/BIC. If the 4PL model does not substantially improve the fit (lower AIC/BIC), the simpler 3PL is preferred.
• Wide confidence intervals around IC50 suggest the data do not constrain the estimate well. Consider adding doses near the expected IC50.

• Fixing the bottom at zero is only valid if data are normalised. Using raw (unnormalised) data with a 3PL model forces an incorrect baseline, biasing the IC50.
• If data do not reach a clear plateau at high doses, the top parameter is poorly estimated, which propagates uncertainty to the IC50.
• Too few dose levels or inadequate dose range make the curve fitting unreliable. The IC50 should ideally fall within your tested dose range.
• Outliers at extreme doses can disproportionately affect the curve shape. Inspect the fitted curve visually alongside the data.

1. Define the 3PL model: Response = Top / (1 + (IC50/Dose)^HillSlope), with Bottom fixed at 0.
2. Use non-linear least squares optimisation to find the values of Top, IC50, and Hill Slope that minimise the sum of squared residuals between observed and predicted responses.
3. Compute confidence intervals for each parameter from the covariance matrix of the estimates.
4. Calculate goodness-of-fit metrics (R-squared, AIC, BIC) to evaluate model adequacy.