Why do some decreasing curves look symmetric while others fall sharply and then decay slowly? Two useful mathematical models—the logistic function and the Burr type XII survival function—provide different answers. Move the sliders below to see why.
What Is the Logistic (Sigmoid) Function?
The decreasing logistic curve is I(x) = 100 / [1 + exp(k(x − x₀))], where k > 0 sets the steepness and x₀ sets the midpoint. The value at x = x₀ is exactly 50%. Its shape is symmetric around (x₀, 50%).
What Is the Burr XII Function?
The Burr XII survival curve is I(x) = 100[1 + (x/λ)β]−α, for x ≥ 0, λ > 0, β > 1, and α > 0. Here, λ sets the x-scale, β and α control the curve's shape. It starts at 100% when x = 0 and approaches zero for large x. In this article we display 100 times the survival function, not the probability density.
Its halfway point is x₅₀ = λ(21/α − 1)1/β. Importantly, λ is not generally the halfway point. A Burr curve can be asymmetric, but not every choice of parameters produces a sigmoid-shaped profile.
Mathematical Constraints vs. Simulator Limits
The multiplier 100 is a normalization choice, not a universal property of either model. More generally, replace 100 by an upper scale L > 0. This simulator fixes L = 100, so the vertical axis is a normalized percentage. The horizontal interval 0–100 is only a display window, not the end of either model's domain.
| Quantity | Mathematical condition | Simulator setting |
|---|---|---|
| Logistic k | k > 0 for a decreasing curve | 0.02–0.60 |
| Logistic x₀ | Any real value | 5–95 |
| Burr λ | λ > 0 | 5–95 |
| Burr β | β > 0 | 1.1–45 (restricted to an S-like decline) |
| Burr α | α > 0 | 0.05–4 |
| Input x | Logistic: any real x; Burr: x ≥ 0 | Internal graph coordinate 0–100 |
| Response | Logistic: 0 < I < 100; Burr: 0 < I ≤ 100 | Y-axis 0–100% |
The logistic curve approaches 100 as x → −∞ and 0 as x → +∞; Burr XII equals 100 at x = 0 and approaches 0 as x → +∞. For Burr XII, β > 1 ensures an interior inflection point at xinf = λ[(β − 1)/(αβ + 1)]1/β. The two models have different tails: exponential for logistic versus a power law proportional to x−αβ for Burr XII.
Interactive Sigmoid vs. Burr XII Simulator
Choose an illustrative profile or adjust all five parameters. The black points are synthetic teaching examples—not experimental measurements. Choose a biological application to change the meaning of the axes and the synthetic demonstration data. The two fitted-model parameters in each preset are illustrative settings, not reported research estimates.
Axes: x = generic input (arbitrary units); y = normalized response (%). All points are synthetic teaching examples.
Logistic · two parameters
Increasing k makes the transition steeper; changing x₀ shifts it left or right without changing its symmetry.
Burr XII · three parameters
The effects of α and β interact. At large x, Burr XII decays approximately as x−αβ; neither parameter controls only one feature.
Live equations and midpoint
Logistic x₅₀ = — · Burr XII x₅₀ = —
Difference from synthetic example points
| Model | RMSE | MaxAE | R² |
|---|---|---|---|
| Logistic | — | — | — |
| Burr XII | — | — | — |
RMSE = √[Σ(y − ŷ)²/n]; MaxAE = max|y − ŷ|; R² = 1 − Σ(y − ŷ)²/Σ(y − ȳ)². Metrics update whenever a slider changes. Models are not automatically refitted.
CSV exports curve predictions for x = 0–100 and synthetic observations when available. PNG exports the plotted canvas.
Understanding Symmetric and Asymmetric Curves
A logistic curve is symmetric about its midpoint: equal distances to the left and right give complementary response values. A Burr XII survival curve can place a steep transition beside a longer tail. Try increasing β while keeping α fixed, then vary α: the transition and tail both respond, but not independently. A third parameter adds flexibility; it does not guarantee better predictions on unseen data.
When Should You Use Each Function?
Logistic curves are useful when a symmetric transition on the x-axis is a reasonable approximation. Burr XII survival curves are useful for positive-domain data that need a flexible, possibly skewed decline. Model choice should consider domain, parameter identifiability, residuals, and predictive validation—not RMSE alone.
Applications in Biology, Biochemistry, and Pharmacology
Select a biological application above to see how the very same mathematical curves can describe different kinds of measurements. Every displayed observation is synthetic; these examples do not use patient data, cell experiments, or laboratory measurements. The slider parameters are illustrative and do not automatically refit the points.
1. Pharmacology: Drug Dose–Response Curves and IC₅₀
In a cell-viability assay, the fraction of surviving cells may decrease as drug concentration increases. An S-shaped curve often becomes apparent when concentration is shown on a logarithmic scale. Here the horizontal plotting coordinate x = 0–100 is mapped to log₁₀(C/M) = −9 + 0.06x, corresponding to concentrations from 1 nM to 1 mM. The normalized logistic model therefore gives a teaching-example IC₅₀ = 10−9 + 0.06x₀ M if the upper and lower response levels are exactly 100% and 0%.
In experimental pharmacology, the four-parameter logistic (4PL) model is more common because the measured top and bottom plateaus need not equal 100% and 0%. The simulated two-parameter formula is deliberately simpler. Also, IC₅₀ is an operational assay-dependent estimate, not automatically a binding affinity or mechanistic constant.
2. Cell Biology: Survival Time and the Burr XII Distribution
Suppose individual cells have different lifetimes under a defined condition. A survival function S(t) = P(T > t) describes the fraction expected to remain alive past time t. In the cell-survival example, the graph's horizontal variable is time (hours), from 0 to 100, and the response is 100 × S(t). The Burr XII model is S(t) = [1 + (t/λ)β]−α, with median survival time t₅₀ = λ(21/α − 1)1/β.
Burr XII is a possible parametric distribution, not a standard mechanistic law of cell death. Real survival experiments may include censored observations; statistical fitting must account for censoring rather than simply treating every point as an ordinary percentage.
3. Biochemistry: Protein Degradation and Model Selection
For simple first-order degradation of a protein, the starting model is often P(t) = 100e−kt, where k is a rate constant and t is time. This is an exponential curve, not a logistic curve. The protein-degradation example shows synthetic exponential-like observations, with an exponential reference (dashed green) to remind the reader that the biologically appropriate model need not be either of the two highlighted sigmoid families.
Comparing Logistic and Burr XII with a first-order benchmark encourages students to check mechanisms and residuals before choosing a flexible fit. An empirical fit alone does not establish a biochemical mechanism.
What Should You Learn from the Three Examples?
The same percent-response scale can describe dose response, survival probabilities, or remaining protein, but their horizontal axes, assumptions, and interpretation of the halfway point differ. Logistic symmetry is a useful simplification; Burr XII can accommodate a long power-law tail. Whether either is an appropriate model must be decided from the scientific question, experimental design, and data—not curve appearance alone.
Frequently Asked Questions
Is Burr XII always an asymmetric sigmoid?
No. This survival function is decreasing, but its visible shape depends on β and α. For some parameters it has an interior inflection point; for others it is simply curved without an S-like transition.
Is λ the 50% point?
No. The correct halfway point is λ(21/α − 1)1/β.
Does a lower RMSE prove that a model is better?
No. Extra parameters can improve the fit to existing observations. Consider generalization to new data and uncertainty in the fitted parameters.
Can I use the interactive plot in a presentation?
Yes: open this article in a browser during your presentation and manipulate the controls live. Interactive embedding within a slide depends on the presentation software and any web-view add-in.
References
Burr, I. W. (1942). Cumulative frequency functions. Annals of Mathematical Statistics, 13(2), 215–232. DOI: 10.1214/aoms/1177731607.
MathWorks. Burr Type XII Distribution (definition, survival function, and parameterization). SciPy. scipy.stats.burr12.