Where Does The Response Stop Rising? Fit The Curve, Get The EC50

How it works

A four-parameter logistic curve is fitted by ordinary nonlinear least squares (base R nls) on log10 dose: response = low plateau + (high plateau - low plateau) / (1 + 10^((log10(EC50) - log10(dose)) x hill)). Starting values are derived from the data — the plateaus from the mean response at the lowest and highest tested dose levels, and the Hill slope and midpoint from a least-squares line through the logit-transformed normalised response — because nls fails loudly on poor starts. Each parameter carries a Wald 95% interval; the midpoint is estimated on the log scale and back-transformed so its interval is correctly asymmetric. A baseline-constrained 3PL and a log-linear model are fitted alongside and compared on AIC, with a nested F test for the 4PL against the 3PL. Residual diagnostics include a pure-error lack-of-fit test wherever replicate doses exist. With a grouping column, each group gets its own curve and an extra-sum-of-squares F test compares separate midpoints against a shared one.

Use it whenever a response is expected to saturate rather than grow without limit, and you need the half-maximal point: concentration-response assays, drug potency, marketing spend against conversions, exposure or contact frequency against uptake, intensity against effect.

Not for a response that keeps climbing with no ceiling in sight (there is no plateau to estimate, and the fit will be refused), not for a bell-shaped or hormetic response that reverses at high dose, not for time-series trends (use a trend or forecasting tool), and not for a response with fewer than four distinct dose levels — four parameters cannot be determined from three points.

What data do you need?

One dose and one response per row, ideally with replicates at each dose level. A zero-dose control row is optional but useful. For example, an assay plate:

Minimum 12 rows · Best with 6-12 dose levels with 3 or more replicates each, spanning both plateaus

What's in the report?

Standard-library analysis: how does response change with dose or intensity? Map a dose column and a response column and get a four-parameter logistic (4PL) curve fitted by nonlinear least squares — the low-dose plateau, the high-dose plateau, the Hill slope, and the EC50 (or IC50 when the response falls) as the headline, each with a 95% confidence interval. The fitted curve is drawn over the observed points on a properly handled log-dose axis with zero-dose controls shown separately, backed by residual diagnostics including a replicate-based lack-of-fit test, and compared against a baseline-constrained 3PL and a plain log-linear model so a sparse design is not locked into an over-parameterised curve. If the curve cannot be fitted, the analysis says so with a diagnosis instead of substituting a straight line.