Dose-Response Analysis in R: drc Package for Sigmoidal Curves

Dose-response analysis models how a biological or chemical response, like plant growth, cell mortality, or enzyme activity, changes with increasing dose. The drc package in R fits sigmoidal curves (log-logistic, Weibull, and more), estimates effective doses such as ED50 or EC50, and compares curves between treatment groups, all through one consistent interface built around drm().

How do you fit a dose-response curve with drc?

A single call to drm() fits a sigmoidal curve to your dose-response data. You pass a formula, the data frame, and a model family (the log-logistic family is the default workhorse), and drm() returns a fitted object that feeds every downstream step: parameter estimation, effective-dose computation, plotting, and group comparison. The bundled ryegrass dataset, ferulic-acid concentrations versus ryegrass root lengths, is the canonical worked example from the drc authors.

Four parameters describe the curve. d is the upper asymptote (7.79 cm of root growth at zero dose), c is the lower asymptote (0.48 cm at very high dose), e is the ED50 (3.06 mM of ferulic acid halves root length), and b is the slope at ED50. All four estimates are highly significant, meaning the four-parameter curve fits better than flat or simpler alternatives. The residual standard error of 0.53 cm is small relative to the 7.3 cm response range, so the fit is tight.

Figure 1: A typical drc workflow. One drm() fit feeds summary(), ED(), plot(), and compParm().

How do you estimate ED50, EC50, and other effective doses?

The effective dose EDp is the dose producing a response p percent of the way between the two asymptotes. ED50 is the halfway dose, ED10 is an early-effect marker, ED90 is near the plateau. In pharmacology the same quantity is called EC50 (effective concentration) or IC50 (inhibitory concentration); in toxicology it is LC50 (lethal concentration). All four names refer to the same value, the e parameter in a log-logistic fit. The ED() function gives you every level you ask for, with delta-method confidence intervals.

ED50 is 3.06 mM, matching the e parameter from the summary. Its 95% confidence interval is narrow (2.66 to 3.46), which makes sense because the data cluster densely around the inflection point. ED90 has a much wider CI (4.73 to 8.06), because fewer data points anchor the high-dose plateau. Always report ED estimates with their CIs, a point estimate without uncertainty is a half-answer.

If you are curious about the math, the four-parameter log-logistic curve has the form:

$$f(x) = c + \frac{d - c}{1 + \exp\{b \cdot [\log(x) - \log(e)]\}}$$

Where:

• $x$ is the dose
• $c$ is the lower asymptote (response at infinite dose)
• $d$ is the upper asymptote (response at zero dose)
• $b$ is the slope coefficient at ED50 (negative for decreasing response, as in root growth)
• $e$ is the ED50 itself, embedded directly in the model

If you're not interested in the math, skip to the next section, the practical code above is all you need.

Which sigmoidal model should you choose?

LL.4 is the safe default, but drc ships with a family of alternatives for curves that are not symmetric, or that need one extra degree of flexibility. Here are the most useful ones:

The fastest way to pick is mselect(), which refits the data under every candidate and prints information criteria side by side.

Read the table two ways. The IC column (smaller is better) ranks W2.4 first and LL.4 second, separated by a 1.6-unit gap, which is a modest preference for W2.4. The Lack of fit column is a p-value from a goodness-of-fit test; all five models are above 0.05, so none are rejected outright. When two models are close in IC and both pass lack-of-fit, either choice is defensible; prefer the simpler model (fewer parameters) unless you have a biological reason to pick an asymmetric shape.

Figure 2: Decision guide for picking among LL.4, LL.3, LL.5, and Weibull families.

modelFit() runs the same lack-of-fit test as a standalone check once you have settled on a model.

A p-value of 0.28 means we have no evidence the Weibull type-2 model systematically misfits the data, good news.

How do you compare dose-response curves between groups?

Real experiments rarely involve one curve. You compare two herbicides, four cell lines, or three drug formulations, and the question is whether their curves differ. The curveid argument to drm() fits a separate curve per group in one model, and compParm() runs pairwise tests on any single parameter. The bundled S.alba dataset shows biomass of Sinapis alba (white mustard) versus dose, compared across two herbicides, bentazone and glyphosate.

Eight parameters, four per herbicide. Bentazone's ED50 is 33.2 g/ha, glyphosate's is 0.072 g/ha, a 460-fold potency ratio in favour of glyphosate. But is that difference statistically significant, or a sampling artefact? compParm() answers directly.

The "-" operator tests the difference e_Bentazone - e_Glyphosate. The 33.1 g/ha estimated difference is overwhelmingly significant (p < 0.001), so glyphosate is genuinely more potent. Use "/" instead to test the ratio, which is often more interpretable for dose-response data.

How do you visualize dose-response fits?

A dose-response plot lives or dies by its x-axis: doses span orders of magnitude, so the x-axis must be on a log scale, always. drc's built-in plot() method handles that automatically, draws a smoothed curve, and can overlay confidence bands. For publication figures, extract predictions with predict() and build the plot yourself in ggplot2.

The first plot() call draws the fitted curve plus a 95% confidence ribbon (type = "confidence"). The second call, with add = TRUE, overlays the raw data points (type = "all"). Two calls, one figure. You can see the glyphosate curve shifted roughly three orders of magnitude to the left of bentazone, the visual version of the 462-fold ED50 ratio.

For a ggplot2 version, generate prediction data on a fine dose grid, ask predict() for a confidence interval, then layer geom_ribbon() + geom_line() on top of the raw points.

expand.grid() builds every combination of doses and herbicides, predict() returns a matrix with prediction, lower, and upper bounds, and ggplot2 layers the ribbon (confidence band), line (fit), and points (raw data). Swap colours, themes, and scales freely; the shape of the plot stays the same.

Practice Exercises

Exercise 1: Species comparison on selenium toxicity

The selenium dataset (bundled with drc) records mortality of four species of Daphnia at increasing selenium concentrations. Fit a LL.4 model with curveid = type, print the summary, and identify the species with the lowest ED50 (most sensitive to selenium). Save the fit to my_selenium_model.

Exercise 2: Model selection and prediction

Using the spinach dataset (spinach biomass vs dose, compared across two curve types), fit a LL.4 model with curveid = CURVE, compare it to W2.4 with mselect(), and use predict() to estimate the response at dose 5 under the lower-IC model. Save the prediction to my_spinach_pred.

Complete Example

An end-to-end S. alba bioassay workflow, tying every section together.

Step 1 fits, step 2 justifies the model family, steps 3 and 4 answer the research question (how potent, and do they differ), step 5 communicates the result. This five-step template works for most dose-response comparisons: fit, validate, estimate, test, visualize.

Summary

The consistent pattern across the package: every function takes a fitted drm object and operates on it, so once you have a good fit, every downstream analysis is one line of code.

References

1. Ritz, C., Baty, F., Streibig, J.C., Gerhard, D. (2015). Dose-Response Analysis Using R. PLOS ONE 10(12): e0146021. Link
2. Ritz, C., Streibig, J.C. (2005). Bioassay Analysis using R. Journal of Statistical Software 12(5). Link
3. drc package on CRAN, reference manual and vignettes. Link
4. Seefeldt, S.S., Jensen, J.E., Fuerst, E.P. (1995). Log-logistic analysis of herbicide dose-response relationships. Weed Technology 9(2): 218-227.
5. Motulsky, H., Christopoulos, A. (2004). Fitting Models to Biological Data Using Linear and Nonlinear Regression. Oxford University Press.
6. Finney, D.J. (1971). Probit Analysis (3rd ed.). Cambridge University Press.

Continue Learning

• Factorial Experiments in R - the parent tutorial covers full-factorial designs where dose is one of several factors.
• Logistic Regression in R - the log-logistic curve is the continuous-response cousin of logistic regression's binary-response S-curve.
• Linear Regression in R - sigmoidal fits reduce to linear fits at low or high dose; linear regression is always the first tool to reach for.