Non-linear Regression and IC50/4PL
How to fit dose-response curves and sigmoidal models in Licklider, including the 4PL model and IC50 estimation, with explicit guidance on model choice, log-dose handling, and the current support boundary.
Non-linear regression fits a mathematical model whose relationship between the predictor and outcome is not a straight line. In life sciences research, this is most commonly used for dose-response analysis, where the response follows a sigmoidal curve as dose increases.
Available models
Licklider supports three sigmoidal models:
4PL (four-parameter logistic)
The most flexible model. Estimates four parameters: the bottom asymptote, the top asymptote, the IC50 (or EC50), and the Hill slope. Appropriate when neither the minimum nor the maximum response is known in advance.
Use 4PL when you want the model to learn both plateaus from the data. This is the default scientific starting point when the assay does not justify fixing either asymptote in advance, but it also asks the most of the data because all four parameters must be estimated together.
3PL (three-parameter logistic)
A constrained version of the 4PL where the bottom asymptote is fixed at zero. Appropriate when the baseline response is known to be zero — for example, when the outcome is an inhibition percentage.
Use 3PL when the lower plateau is not just expected to be near zero, but is fixed by assay design or normalization. The constraint reduces model flexibility and can stabilize IC50 estimation when the experiment does not contain enough information to estimate a free bottom asymptote reliably.
Hill model
Both the bottom (0) and the top (100) are fixed. Appropriate when the response is expressed as a percentage of the maximum possible effect and the full range is known.
Use the Hill model only when both plateaus are scientifically fixed before fitting. The extra constraints make the fit more stable, but they are only defensible when the assay scale truly anchors the response at 0 and 100 rather than merely suggesting those values.
How to request it
Describe the analysis in the Chat. For example:
- "Fit a dose-response curve"
- "Calculate the IC50"
- "Fit a 4PL model to this data"
- "Show the sigmoidal fit"
Licklider will fit the appropriate model and overlay the fitted curve on the scatter plot.
This automation is meant to reduce routine setup work, not to replace assay judgment. Licklider can fit a sigmoidal curve from the data you provide, but it cannot know whether your chosen normalization, control definition, or concentration range is scientifically appropriate unless that is already reflected in the dataset.
What the results include
Parameter estimates
Each fitted parameter is reported with its estimate and 95% confidence interval:
- Bottom — the lower asymptote
- Top — the upper asymptote
- IC50 / EC50 — the dose at which the response is halfway between the bottom and top
- Hill slope — the steepness of the sigmoidal transition (also called the Hill coefficient)
Model fit statistics
- R² — variance explained
- RMSE — root mean square error
- AIC — Akaike information criterion
Convergence
Whether the fitting algorithm converged to a stable solution. Non-convergence typically indicates that the data does not support the model at the chosen parameter constraints, or that the dose range does not span the full sigmoidal transition.
Convergence is necessary, but not sufficient, for trust. A converged fit can still be scientifically weak if the dose range is too narrow, the plateaus are not observed, or a few influential points dominate the curve.
Dose on a log scale
Dose-response analysis is typically performed with dose on a log<sub>10</sub> scale because sigmoidal transitions are often easier to model and interpret when equal multiplicative dose steps are spaced evenly on the x-axis. Licklider applies a log<sub>10</sub> transformation to the dose variable automatically when the data pattern suggests a log-linear dose range. The IC50 is reported on the original scale.
Important: Licklider does not automatically know whether a log dose scale is scientifically appropriate for every assay. If your dose column contains zeros or negative values, if the spacing was designed on a linear rather than multiplicative scale, or if the zero-dose control should be handled separately from the fitted curve, you must review that design choice before trusting the fit.
Minimum data requirements
| Model | Minimum observations |
|---|---|
| 4PL | 5 |
| 3PL | 4 |
| Hill | 3 |
These minima are practical lower bounds, not quality guarantees. Each model needs more observations than free parameters to estimate the curve at all, and fits near the minimum are usually fragile. For reliable parameter estimation, the dose range should ideally span from below the lower asymptote to above the upper asymptote. If the dose range does not cover the full transition, the IC50 estimate will be extrapolated and its confidence interval will be wide.
Licklider does not automatically detect whether your assay has truly captured both plateaus. If the response never approaches the lower or upper asymptote, the model can still converge while producing an IC50 that is driven more by extrapolation than by observed data.
Design rationale
Licklider separates 4PL, 3PL, and Hill fits because the right constraint pattern depends on what the assay already fixes before model fitting. A free 4PL is preferred when both asymptotes are unknown; 3PL and Hill are provided for cases where one or both plateaus are known from assay design and should be fixed rather than re-estimated.
Licklider keeps IC50 on the original dose scale even when the fit uses log<sub>10</sub> dose internally, because researchers usually interpret potency in the native concentration units used in the experiment. The log transformation helps the optimizer and the visual display; it should not force the reported potency into a less readable unit system.
The minimum-observation table is intentionally conservative. A model with too few observations relative to its free parameters may return a numerical solution, but that solution is often unstable and gives a misleading sense of precision. More doses across the full transition are generally more valuable than many replicates at only one side of the curve.
Design rationale and references
The four-parameter logistic family is the standard workhorse for dose-response analysis because it directly represents lower plateau, upper plateau, midpoint potency, and slope in one interpretable model [1, 2]. Constrained variants such as 3PL or fixed-top-and-bottom Hill fits are appropriate only when the assay scale justifies those constraints in advance [1].
Log-dose modeling is standard because concentration-response relationships are typically organized over multiplicative dose changes, and the central transition is easier to estimate and visualize on a logarithmic dose axis [1, 3]. At the same time, log scaling requires positive doses and does not rescue a poorly chosen dose range [1].
IC50 estimates become unstable when the data do not bracket the transition well. That is why Licklider emphasizes plateau coverage, confidence intervals, and convergence together rather than treating any one of them as sufficient on its own [1, 2].
Methodological foundations
Sebaugh, J. L. (2011). Guidelines for accurate EC50/IC50 estimation. Pharmaceutical Statistics, 10(2), 128-134. -> Practical guidance for choosing sigmoidal dose-response models, placing doses across the transition, and interpreting unstable potency estimates.
Ritz, C., Baty, F., Streibig, J. C., & Gerhard, D. (2015). Dose-response analysis using R. PLoS ONE, 10(12), e0146021. -> Describes standard dose-response model families, including constrained and unconstrained logistic fits, and the importance of matching model form to assay design.
Hill, A. V. (1910). The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. The Journal of Physiology, 40(Suppl), iv-vii. -> Origin of the Hill-style sigmoidal formulation that underlies slope-based interpretation in concentration-response modeling.
What this page does not cover
- Linear relationships between dose and response → see Linear Regression (OLS)
- Comparing IC50 values between groups is not currently supported as a statistical test
- Dose-response workflow → see Dose-response Curves
- Repeated measures or clustered dose-response data → see Repeated Measures and Mixed Models
Current support boundary
- Licklider can fit and visualize sigmoidal dose-response curves, but it does not automatically verify that your normalization scheme makes
0and100scientifically valid fixed bounds. - Licklider does not automatically decide whether zero-dose controls should be excluded from a log-dose fit, modeled separately, or represented through a transformed placeholder value.
- Licklider does not automatically detect outliers, influential points, plate effects, or hidden clustering that can distort a non-linear fit while still allowing convergence.
- A converged model is not proof that the selected curve form is correct. If the dose range does not span the transition, the fit may rely heavily on extrapolation.
- Comparing potency parameters such as IC50 across groups is not currently documented here as an inferential test; use this page for single-curve fitting and interpretation, not for between-group hypothesis testing.