Nonlinear Concentration Effects on the Logit Scale

Why the Scale of the Concentration Effect Matters

For a binary pharmacodynamic endpoint, a logistic model relates the probability of an event to a predictor through the logit link. The predictor can include concentration in different ways. A linear concentration term is one possibility, but it is not the only one: a saturable or sigmoid function may better represent how the event probability changes as exposure increases. Choosing a function is a substantive modeling decision, not simply a choice of how to draw a curve.

A second distinction is equally important. The concentration function describes change on the logit scale; the logistic transformation maps that predictor to a probability. Even when the predictor changes linearly with concentration, the resulting probability curve is generally curved. The shape of the probability curve therefore cannot, by itself, tell us whether the concentration effect on the logit scale is nonlinear.

Definition: Let $Y_i$ indicate whether subject or observation $i$ has the event of interest, and let $C_i$ be the corresponding drug concentration. A logistic concentration-response model specifies $$ Y_i\mid C_i\sim\operatorname{Bernoulli}(p_i),\qquad \operatorname{logit}(p_i)=\log\left(\frac{p_i}{1-p_i}\right)=\eta_i=\beta_0+f(C_i), $$ so that $p_i=\exp(\eta_i)/(1+\exp(\eta_i))$. Here $f(C_i)$ is the chosen concentration effect on the logit scale.

This formulation assumes that the binary outcomes are conditionally independent given the modeled predictors, that the Bernoulli likelihood is appropriate, and that the concentration values are treated as measured without error. It also assumes that important systematic predictors or dependencies have not been left out. For repeated or clustered observations, conditional independence may fail unless the dependence is handled in the model. Concentration must be expressed in a stated unit, because the units of the model parameters depend on that choice.

Three Concentration Functions on the Logit Scale

A linear function is $f(C)=\beta_C C$. Its slope $\beta_C$ is measured in logit units per concentration unit. It imposes a constant change in the logit for each additional concentration unit. The intercept $\beta_0$ is the logit when $C=0$, provided the model contains no other predictors.

A saturable function allows the logit effect to approach a finite limit:

f(C)=E_{\max}\frac{C}{EC_{50}+C},\qquad C\geq 0,\quad EC_{50}>0.

Here $E_{\max}$ is the limiting change in logit as concentration increases, and $EC_{50}$ is the concentration at which the function has reached half of that change. $E_{\max}$ is in logit units, while $EC_{50}$ has the same units as $C$. The sign of $E_{\max}$ determines the direction of the logit change. In this model, “half maximum” refers to half the logit effect, not half the probability of the event.

A sigmoid function adds a shape parameter:

f(C)=E_{\max}\frac{C^h}{EC_{50}^h+C^h},\qquad C\geq 0,\quad EC_{50}>0,\quad h>0.

The exponent $h$ controls how sharply the logit effect rises around $EC_{50}$. When $h=1$, this reduces to the saturable function. For all positive $h$, the function reaches half of $E_{\max}$ at $C=EC_{50}$. Larger values of $h$ produce a sharper transition near that concentration; they do not change the meaning of $E_{\max}$ as the limiting logit change.

Function	Logit-scale shape	Important limitation
Linear	Constant slope over concentration	Does not saturate and may extrapolate implausibly
Saturable	Rises or falls toward a finite limit	Requires enough exposure range to identify the approach to the limit
Sigmoid	Saturates, with adjustable steepness	Requires data to support the extra shape parameter

Worked Predictions From the Three Forms

Worked Example: A Linear Logit Effect With a Curved Probability Response

Suppose a hypothetical model for an event uses concentration $C$ in mg/L and has $\beta_0=-2.0$ and $\beta_C=0.080$ per mg/L. The predictor is $\eta=-2.0+0.080C$. Calculate the event probabilities at $C=0$, $10$, and $30$ mg/L.

At $C=0$, $\eta=-2.0+0.080(0)=-2.0$, so $p=\exp(-2.0)/(1+\exp(-2.0))=0.1192$. At $C=10$, $\eta=-2.0+0.080(10)=-1.2$, so $p=\exp(-1.2)/(1+\exp(-1.2))=0.2315$. At $C=30$, $\eta=-2.0+0.080(30)=0.400$, so $p=\exp(0.400)/(1+\exp(0.400))=0.5987$.

The predictor is a straight line in $C$, but the probabilities do not increase by a constant amount for each 10 mg/L increase. The changes are $0.2315-0.1192=0.1123$ and $0.5987-0.2315=0.3672$. That difference in probability-scale increments is not evidence, by itself, that the logit-scale concentration function is nonlinear.

Worked Example: A Saturable Logit Effect

Consider a hypothetical model with $\beta_0=-2.2$, $E_{\max}=2.4$ logit units, and $EC_{50}=12$ mg/L. Its predictor is $\eta=-2.2+2.4C/(12+C)$. Calculate predictions at $C=0$, $12$, and $36$ mg/L.

At $C=0$, the concentration function is zero, so $\eta=-2.2$ and $p=0.09975$. At $C=12$, the fraction is $12/(12+12)=0.5$. Thus $f(C)=2.4(0.5)=1.2$, $\eta=-2.2+1.2=-1.0$, and $p=\exp(-1)/(1+\exp(-1))=0.2689$.

At $C=36$, the fraction is $36/(12+36)=0.75$. Therefore $f(C)=2.4(0.75)=1.8$, $\eta=-2.2+1.8=-0.400$, and $p=\exp(-0.400)/(1+\exp(-0.400))=0.4013$. As concentration increases, the logit effect approaches 2.4, but the probability approaches $\operatorname{expit}(-2.2+2.4)$, about 0.5498, rather than 2.4 or 1.

Worked Example: A Sigmoid Logit Effect

Suppose a hypothetical model has $\beta_0=-3.0$, $E_{\max}=4.0$ logit units, $EC_{50}=20$ mg/L, and $h=2$. Its concentration function is $4C^2/(20^2+C^2)$. Calculate predictions at $C=10$, $20$, and $40$ mg/L.

At $10$ mg/L, the fraction is $10^2/(20^2+10^2)=100/500=0.2$. The logit effect is $0.800$, so $\eta=-3.0+0.800=-2.2$ and $p=0.09975$. At $20$ mg/L, the fraction is $400/800=0.5$; hence $\eta=-3.0+2.0=-1.0$ and $p=0.2689$.

At $40$ mg/L, the fraction is $1600/(400+1600)=0.8$; hence the logit effect is $3.2$, $\eta=-3.0+3.2=0.200$, and $p=\exp(0.200)/(1+\exp(0.200))=0.5498$. The concentration of 20 mg/L is the midpoint of the logit effect, not the midpoint between the baseline event probability and its limiting probability.

Why a Linear Logit Can Look Nonlinear in Probability

The distinction follows directly from differentiation. Since $p=\exp(\eta)/(1+\exp(\eta))$, differentiation with respect to concentration gives:

\frac{dp}{dC}=p(1-p)\frac{d\eta}{dC}.

Differentiating again gives the probability-scale curvature:

\frac{d^2p}{dC^2} =p(1-p)\left[\frac{d^2\eta}{dC^2}+(1-2p)\left(\frac{d\eta}{dC}\right)^2\right].

To see the consequence, set $\eta=\beta_0+\beta_C C$. Then $d^2\eta/dC^2=0$, but $d^2p/dC^2=p(1-p)(1-2p)\beta_C^2$, which is generally nonzero. If $0<p<0.5$, the probability curve is convex for a nonzero linear slope; if $0.5<p<1$, it is concave. Thus the probability curve can change curvature at $p=0.5$ even though the logit-scale effect remains exactly linear.

For a saturable function, the logit-scale derivatives are $d f/dC=E_{\max}EC_{50}/(EC_{50}+C)^2$ and $d^2 f/dC^2=-2E_{\max}EC_{50}/(EC_{50}+C)^3$. In that case, the probability-scale curvature includes both the function’s own curvature and the curvature induced by the logistic transformation. Looking only at a plot of probabilities does not separate those contributions.

Fitting, Diagnostics, and Failure Modes

For each proposed function, the model assigns a probability $p_i$ to every observed binary outcome. Parameters can be estimated by maximizing the Bernoulli likelihood, as in logistic regression. The key comparison is between plausible specifications for $f(C)$, not between a straight probability line and a curved probability line. When comparing functions, use a common analysis data set and a stated model-comparison strategy; do not choose the most flexible curve solely because it follows every observed fluctuation.

Check whether the concentration range contains information about the shape being estimated. If observations are concentrated far below $EC_{50}$, a saturable function may look nearly linear. If nearly all observations are far above $EC_{50}$, they may reveal a plateau but provide little information about where the midpoint lies. Estimating $E_{\max}$, $EC_{50}$, and $h$ together is particularly difficult when the data do not span the transition region. A fitted sigmoid may then have a steepness estimate that is unstable or very sensitive to a few observations.

Compare predictions with the observed outcome pattern across the exposure range, while remembering that individual binary observations are 0 or 1 rather than smooth response values. A grouped plot of observed event proportions can help reveal gross lack of fit, but groups discard exposure detail and proportions based on small groups are noisy. A diagnostic should not be treated as proof that one functional form is correct. Also examine whether a proposed curve makes plausible predictions across the concentration range where it will be used; predictions beyond observed concentrations are extrapolations.

An alternative is to use a flexible basis, such as a spline, when a prespecified linear or saturable form is not credible. Flexibility can reduce dependence on a particular parametric shape, but it also requires enough data across the exposure range and may yield unstable tail behavior. A linear logit model is simpler and often easier to estimate; saturable and sigmoid forms encode a plateau, with the sigmoid adding flexibility in steepness. None is automatically preferable without regard to the scientific question and available exposure coverage.

Common Mistake and Practical Interpretation

Common mistake: “The observed event probabilities bend upward, so concentration must have a nonlinear effect on the logit scale.” This conclusion does not follow: the logistic transformation itself creates probability-scale curvature from a linear logit predictor. First specify or estimate the logit-scale function, then distinguish its curvature from the curvature introduced by mapping logits to probabilities.

A related error is to call $E_{\max}$ the maximum probability change. In these models, $E_{\max}$ is the limiting change in the predictor, measured in logit units. The limiting probability also depends on the baseline predictor and is obtained by applying the inverse-logit transformation. Saturation on the logit scale therefore does not mean that the event probability increases by $E_{\max}$, or that the limiting probability equals $E_{\max}$.

The choice of concentration function affects estimated exposure-response shape, predictions, and extrapolation. A linear form can be a reasonable local approximation over a limited range; saturable and sigmoid forms may be preferable when a plateau or transition is scientifically plausible and supported by the data. The distinction between predictor scale and probability scale should be kept explicit throughout model construction and interpretation.

Check Your Understanding

Answer each question, then compare with the solutions below.

In a saturable logit function, what does $EC_{50}$ identify, and what units does it have?
Can a model with a linear concentration effect on the logit scale produce a curved probability response? Explain briefly.
For a sigmoid function with $h=1$, which standard concentration function does it become?
Why might $h$ be difficult to estimate when the observed concentrations are all much lower than $EC_{50}$?

Solutions. (1) At $C=EC_{50}$, the concentration function has reached half of $E_{\max}$; $EC_{50}$ is measured in the same units as concentration. (2) Yes. The logistic transformation makes $d^2p/dC^2=p(1-p)(1-2p)\beta_C^2$ for a linear predictor, which is generally nonzero. (3) It becomes the saturable function $E_{\max}C/(EC_{50}+C)$. (4) At concentrations far below $EC_{50}$, the sigmoid may appear nearly linear, so the data provide little information about the steepness of its transition near $EC_{50}$.