Nonlinear Concentration Effects on Poisson Count Rates

Why Model Concentration Nonlinearly?

A pharmacodynamic count can rise or fall with drug concentration, but the relationship need not be a straight line. A response may change steadily over the observed range, approach a ceiling, or show a shallow start followed by a steep transition. For count outcomes, the concentration-response function is commonly placed in the Poisson log-mean, so it changes the expected count rate while preserving a positive predicted mean.

The earlier material on Poisson regression established the log link for counts, and the earlier material on exposure time offsets established how to account for different observation durations. Here, the main task is to specify the concentration function and translate its fitted values into contrasts that have a direct rate or count interpretation.

Definition: Let \(Y_i\) be the count observed for individual or interval \(i\), \(C_i\) its concentration (in a specified unit), and \(t_i\) its observation time. A Poisson log-rate model specifies \(Y_i\mid C_i,t_i\sim\operatorname{Poisson}(\mu_i)\), with \(\mu_i=t_i\lambda(C_i)\) and \(\log\{\lambda(C_i)\}=\eta(C_i)\). Thus \(\log(\mu_i)=\log(t_i)+\eta(C_i)\), where \(\log(t_i)\) is an offset and \(\lambda(C_i)=\exp\{\eta(C_i)\}\) is the expected count rate.

The basic assumptions are that counts are conditionally Poisson given the modeled predictors, observations are independent conditional on those predictors, and the observation time is correctly represented. In particular, an offset assumes expected counts are proportional to time when the rate is held fixed. If concentration varies substantially during an interval, a single concentration value may not represent the whole interval: the expected count is then more appropriately based on \(\int_0^{t_i}\lambda\{C_i(s)\}\,ds\), where \(C_i(s)\) is the concentration over time.

Three Concentration Functions on the Log-Rate Scale

A linear concentration effect is the simplest form. With concentration measured in mg/L, for example, \(\beta_1\) is the change in log rate per mg/L. The model does not say that the rate itself changes linearly; exponentiating the log-rate gives a positive, exponentially changing rate.

\eta(C)=\beta_0+\beta_1 C,\qquad \lambda(C)=\exp(\beta_0+\beta_1 C)

A saturable effect lets concentration changes have diminishing impact on the log-rate as concentration rises. The parameter \(E_{\max}\) is the limiting change in log-rate from the value at \(C=0\), and \(EC_{50}\) is the concentration at which half that limiting change has occurred. The sign of \(E_{\max}\) determines the direction of the effect.

\eta(C)=\beta_0+E_{\max}\frac{C}{EC_{50}+C},\qquad \lambda(C)=\exp\left(\beta_0+E_{\max}\frac{C}{EC_{50}+C}\right)

A sigmoid, or Hill, function adds a positive shape parameter \(h\). When \(h=1\), it reduces to the saturable function. Larger \(h\) generally gives a sharper transition around the half-maximal concentration. \(EC_{50}\) still gives the concentration at which the modeled effect has reached one-half of \(E_{\max}\).

\eta(C)=\beta_0+E_{\max}\frac{C^h}{EC_{50}^h+C^h},\qquad \lambda(C)=\exp\left(\beta_0+E_{\max}\frac{C^h}{EC_{50}^h+C^h}\right)

For the saturable and sigmoid forms, the rate at zero concentration is \(\exp(\beta_0)\), and the limiting rate as concentration increases without bound is \(\exp(\beta_0+E_{\max})\). These are limits on the log-rate effect; exponentiation means the rate ratio between those limits is \(\exp(E_{\max})\). The \(EC_{50}\) and \(E_{\max}\) parameters must be interpreted in the model’s chosen concentration and log-rate units.

Deriving Concentration Contrasts

A useful result follows directly by comparing the model-predicted rates at two concentrations. For any specified function \(\eta(C)\), the rate ratio is the exponential of the difference in log-rates. The rate difference is the difference between their exponentiated log-rates. For equal observation times, the ratio of expected counts equals the rate ratio; with unequal times, the expected-count ratio also includes the ratio of durations.

\frac{\lambda(C_2)}{\lambda(C_1)} =\frac{\exp\{\eta(C_2)\}}{\exp\{\eta(C_1)\}} =\exp\{\eta(C_2)-\eta(C_1)\}

\lambda(C_2)-\lambda(C_1) =\exp\{\eta(C_2)\}-\exp\{\eta(C_1)\},\qquad \frac{\mu_2}{\mu_1} =\frac{t_2}{t_1}\exp\{\eta(C_2)-\eta(C_1)\}

For the linear model, \(\eta(C_2)-\eta(C_1)=\beta_1(C_2-C_1)\), so a one-unit increase in concentration multiplies the rate by \(\exp(\beta_1)\). For nonlinear models, the rate ratio depends on both concentrations, not just their difference. This dependence is why reporting a single rate ratio for an entire nonlinear curve can be misleading.

Worked Example: A Linear Concentration Effect

Suppose a hypothetical model has \(\beta_0=\log(2.00)\) and \(\beta_1=-0.0800\) per mg/L. Calculate rates at 2.00 and 8.00 mg/L, then compare their predicted rates and expected counts over 6.00 hours.

At 2.00 mg/L, the log-rate is \(\log(2.00)-0.0800(2.00)\). Therefore,

\lambda(2)=2.00\exp(-0.160)=1.704\ \text{counts/hour}

At 8.00 mg/L, the log-rate is \(\log(2.00)-0.0800(8.00)\). Evaluating without rounding the exponential first gives

\lambda(8)=2.00\exp(-0.640)=1.0545849\approx1.055\ \text{counts/hour}

The rate ratio for 8.00 versus 2.00 mg/L is \(\exp\{-0.0800(8.00-2.00)\}=\exp(-0.480)=0.6188\). Under this model, the predicted rate at 8.00 mg/L is about 0.619 times the rate at 2.00 mg/L, or about 38.1% lower. The rate difference is \(1.0545849-1.7042876=-0.6497\) counts/hour.

Over 6.00 hours, the expected counts are \(6.00(1.7042876)=10.2257\approx10.23\) and \(6.00(1.0545849)=6.3275\approx6.328\) counts. The expected-count difference, high concentration minus low concentration, is approximately \(6.3275-10.2257=-3.898\) counts over 6.00 hours. These are model-based expectations, not observed counts.

Worked Example: A Saturable Effect

Consider a hypothetical increasing saturable effect with baseline rate \(\exp(\beta_0)=1.50\) counts/hour, \(E_{\max}=0.900\), and \(EC_{50}=5.00\) mg/L. Calculate predicted rates at 2.00 and 10.0 mg/L and expected counts over equal 4.00-hour intervals.

At 2.00 mg/L, the fraction of the maximal log-rate effect is \(2.00/(5.00+2.00)=0.285714\). Its log-rate increment is \(0.900(0.285714)=0.257143\), so

\lambda(2)=1.50\exp(0.257143)=1.940\ \text{counts/hour}

At 10.0 mg/L, the fraction is \(10.0/(5.00+10.0)=0.666667\), and the increment is \(0.900(0.666667)=0.600000\). Hence

\lambda(10)=1.50\exp(0.600000)=2.733\ \text{counts/hour}

The rate ratio is \(\exp(0.600000-0.257143)=\exp(0.342857)=1.409\). The model therefore predicts a rate about 1.41 times as high at 10.0 mg/L as at 2.00 mg/L. The rate difference is approximately \(2.733-1.940=0.793\) counts/hour. For 4.00 hours, the expected counts are \(4.00(1.940)=7.760\) and \(4.00(2.733)=10.93\), a difference of approximately 3.17 counts. The equal durations make the expected-count ratio equal to the rate ratio.

Worked Example: A Sigmoid Inhibitory Effect

Suppose a hypothetical sigmoid model has baseline rate 4.00 counts/hour, \(E_{\max}=-1.20\), \(EC_{50}=6.00\) mg/L, and \(h=3.00\). Compare predictions at 3.00 and 12.0 mg/L, including expected counts over 5.00 hours.

At 3.00 mg/L, the sigmoid fraction is \(3.00^3/(6.00^3+3.00^3)=27/243=1/9\). The log-rate effect is \(-1.20(1/9)=-0.133333\), giving

\lambda(3)=4.00\exp(-0.133333)=3.501\ \text{counts/hour}

At 12.0 mg/L, the fraction is \(12.0^3/(6.00^3+12.0^3)=1728/1944=8/9\). The log-rate effect is \(-1.20(8/9)=-1.066667\), so

\lambda(12)=4.00\exp(-1.066667)=1.377\ \text{counts/hour}

The rate ratio for 12.0 versus 3.00 mg/L is \(\exp\{-1.066667-(-0.133333)\}=\exp(-0.933334)\approx0.3932\). The predicted rate is therefore about 60.7% lower at 12.0 mg/L. The rate difference is approximately \(1.377-3.501=-2.124\) counts/hour. Over 5.00 hours, the expected counts are about 17.50 and 6.883, respectively; their difference is approximately \(-10.62\) counts. The steepness is controlled by \(h\), not by the sign of the effect, which is set here by the negative \(E_{\max}\).

Choosing a Shape and Checking Its Plausibility

The concentration function should be selected to describe a plausible pattern over the data-supported range, not merely because it can produce a desired prediction. A linear effect is parsimonious and may be adequate over a narrow range. A saturable effect is useful when a plateau is scientifically plausible and supported by observations. A sigmoid form can represent a sharper transition, but estimating its additional shape parameter generally requires enough information across low, transition, and high concentrations.

Form	Pattern on the log-rate scale	Main consideration
Linear	Constant change per concentration unit	Simple, but extrapolates without a plateau
Saturable	Change diminishes toward a limiting effect	Needs information to distinguish the plateau and \(EC_{50}\)
Sigmoid	Saturating effect with adjustable transition steepness	More flexible, but \(h\), \(EC_{50}\), and \(E_{\max}\) may be difficult to estimate separately

A direct Emax model on the rate scale is a different specification: it models \(\lambda(C)\) itself as a baseline plus a saturable increment. Here the function is on the log-rate scale; consequently, the asymptotic rate is \(\exp(\beta_0+E_{\max})\), not \(\exp(\beta_0)+E_{\max}\). Keeping the scale explicit prevents a common interpretive error.

Signs and units also matter. \(E_{\max}\) is a change in log rate and is dimensionless when the rate ratio is formed; \(EC_{50}\) has the same concentration units as \(C\); and \(h\) is dimensionless. A positive \(E_{\max}\) raises the rate toward its plateau, while a negative value lowers it. The baseline rate at \(C=0\) is meaningful only if zero concentration is a relevant reference for the modeled population and setting.

Failure Modes and Practical Limits

Reading a log-rate coefficient as a rate difference. In the linear model, \(\beta_1\) is a log-rate change per concentration unit. The corresponding rate multiplier for a one-unit increase is \(\exp(\beta_1)\), not a change of \(\beta_1\) counts/hour.
Assuming a plateau is established without high-concentration data. If observations do not cover the flattening region, \(E_{\max}\) and \(EC_{50}\) can be weakly identified and different parameter combinations may fit similarly.
Overinterpreting a sigmoid fit. The extra parameter \(h\) can make the curve flexible, but sparse data around the transition may yield unstable parameter estimates and uncertain predictions.
Extrapolating a linear curve too far. Its log-rate changes at a constant rate indefinitely, so the predicted count rate can become implausibly large or approach zero outside the measured range.
Substituting a single concentration for a changing exposure without justification. In general, the rate at an average concentration is not the average of the rates, because exponentiation and nonlinear functions do not commute with averaging.

These are questions about the exposure-response function and the information in the concentration range. They are separate from checking whether the Poisson variance assumption is adequate; count-model diagnostics address that issue directly.

Key takeaway: Specify the concentration effect in the Poisson log-rate, exponentiate to obtain rates, and compare concentrations through \(\exp\{\eta(C_2)-\eta(C_1)\}\). State the time interval whenever translating rates into expected counts.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In a linear log-rate model, what rate ratio corresponds to a concentration increase of 3 mg/L when \(\beta_1=0.10\) per mg/L?
For a saturable or sigmoid effect, what is the modeled log-rate increment at \(C=EC_{50}\)?
At equal observation times, how are the expected-count ratio and the rate ratio related? What changes if the observation times differ?
Why can a sigmoid model be hard to estimate when measurements are concentrated well below \(EC_{50}\)?

Solutions. 1. The rate ratio is \(\exp\{0.10(3)\}=\exp(0.30)=1.350\), so the predicted rate is 1.35 times as high. 2. The fractional effect is one-half, so the log-rate increment is \(E_{\max}/2\). 3. With equal times the ratios are equal; otherwise the expected-count ratio is the rate ratio multiplied by \(t_2/t_1\). 4. Measurements far below \(EC_{50}\) may show little information about the transition or plateau, making \(EC_{50}\), \(E_{\max}\), and \(h\) difficult to distinguish.