Choosing the Link Scale for Saturable Hazard Effects

Why the Link Scale Matters

A saturable concentration-effect function can describe more than one kind of change in event hazard. It may specify an absolute change in hazard, or it may specify a change in the logarithm of hazard, which becomes a multiplicative change after exponentiation. These models can have the same concentration-response shape in their parameterized effect but give different predictions, especially when baseline hazards differ.

The choice is not merely a computational detail. A parameter measured in inverse time describes an absolute hazard increment or decrement. A dimensionless parameter on the log-hazard scale describes a limiting log hazard ratio. Those interpretations lead to different constraints and different scientific questions.

Definition: Let \(C\geq 0\) be concentration and let \(C_{50}>0\) be the concentration at which the unit-slope Emax function reaches half of its maximum. Define \(f(C)=C/(C_{50}+C)\). Then \(f(0)=0\), \(f(C_{50})=0.5\), and \(f(C)\) approaches 1 as \(C\) increases without bound. Let \(\lambda_0(t)>0\) denote the baseline hazard at time \(t\), and let \(\lambda(t\mid C)\) denote the hazard conditional on concentration \(C\).

The hazard has units of inverse time; for example, an event hazard may be measured in events per person-hour. Concentration and \(C_{50}\) must use the same units. The function \(f(C)\) is dimensionless. In this tutorial, the two principal models use this same function, so their differences come from the scale on which its effect is applied.

Two Emax Link Scales

An additive, or hazard-scale, Emax model adds the concentration effect directly to the baseline hazard:

\lambda(t\mid C)=\lambda_0(t)+E_{\max,\lambda}f(C).

Here \(E_{\max,\lambda}\) has units of inverse time. It is the limiting absolute hazard change as concentration becomes large. A positive value describes a saturating increase; a negative value describes a saturating decrease, provided the resulting hazard remains nonnegative.

The log-hazard Emax model applies the saturable effect to the logarithm of the hazard relative to baseline:

\log\left(\frac{\lambda(t\mid C)}{\lambda_0(t)}\right) =E_{\max,\log}f(C), \qquad \lambda(t\mid C)=\lambda_0(t)\exp\{E_{\max,\log}f(C)\}.

The parameter \(E_{\max,\log}\) is dimensionless. At high concentration it is the limiting log hazard ratio, so the limiting hazard ratio is \(\exp(E_{\max,\log})\). At \(C=C_{50}\), the hazard ratio is \(\exp(E_{\max,\log}/2)\). A negative value describes a hazard reduction; it does not make the predicted hazard negative.

Both forms return the baseline hazard at \(C=0\), and both use a saturating concentration function. However, the additive model approaches \(\lambda_0(t)+E_{\max,\lambda}\), whereas the log model approaches \(\lambda_0(t)\exp(E_{\max,\log})\). The first has a limiting absolute change that does not depend on baseline hazard. The second has a limiting relative change that does not depend on baseline hazard.

Worked Example: Comparing Predictions at the Half-Maximal Concentration

Suppose, hypothetically, that the baseline hazard is \(0.0200\ \mathrm{h}^{-1}\), \(C_{50}=2.00\ \mathrm{mg/L}\), and the concentration is \(2.00\ \mathrm{mg/L}\). For the additive model, let \(E_{\max,\lambda}=0.0300\ \mathrm{h}^{-1}\). For the log model, let \(E_{\max,\log}=\log(3)=1.0986\), corresponding to a limiting hazard ratio of 3.

At this concentration, \(f(C)=2/(2+2)=0.500\). The additive-model prediction is

\lambda(t\mid C)=0.0200\ \mathrm{h}^{-1} +(0.0300\ \mathrm{h}^{-1})(0.500) =0.0350\ \mathrm{h}^{-1}.

For the log model, the predicted hazard ratio is \(\exp\{1.0986(0.500)\}=\exp(0.5493)=1.732\). Therefore,

\lambda(t\mid C) =(0.0200\ \mathrm{h}^{-1})(1.732) =0.03464\ \mathrm{h}^{-1}.

These predictions are close at this concentration, but that agreement is not guaranteed elsewhere. At very high concentration, the additive prediction approaches \(0.0200+0.0300=0.0500\ \mathrm{h}^{-1}\), while the log-model prediction approaches \(0.0200(3)=0.0600\ \mathrm{h}^{-1}\). The parameter values describe different quantities and should not be equated just because predictions happen to be similar at one concentration.

Deriving the Scale-Specific Interpretations

The interpretations follow directly from the model equations. For the additive model, subtract the baseline hazard:

\lambda(t\mid C)-\lambda_0(t) =E_{\max,\lambda}\frac{C}{C_{50}+C}.

As \(C\) increases without bound, \(C/(C_{50}+C)\) approaches 1, so the absolute difference approaches \(E_{\max,\lambda}\). At \(C=C_{50}\), it is exactly \(E_{\max,\lambda}/2\). These are hazard differences, not relative risks.

For the log model, divide the conditional hazard by baseline and exponentiate:

\frac{\lambda(t\mid C)}{\lambda_0(t)} =\exp\left\{E_{\max,\log}\frac{C}{C_{50}+C}\right\}.

As concentration increases without bound, this ratio approaches \(\exp(E_{\max,\log})\), and at \(C=C_{50}\) it equals \(\exp(E_{\max,\log}/2)\). Thus a log-scale Emax parameter is not itself a hazard ratio. Exponentiation is required to obtain that ratio.

The distinction also appears in how predictions respond to baseline hazard. The additive model's concentration-associated difference is the same at two times with different baseline hazards, as long as its Emax parameter is constant. The log model's hazard ratio is the same at those times, while its absolute hazard difference varies with baseline. Algebraically, the log-model difference is \(\lambda_0(t)[\exp\{E_{\max,\log}f(C)\}-1]\).

Worked Example: Same Absolute Effect, Different Relative Effects

Consider a hypothetical additive model with \(E_{\max,\lambda}=0.0100\ \mathrm{h}^{-1}\), \(C_{50}=1.00\ \mathrm{mg/L}\), and \(C=1.00\ \mathrm{mg/L}\). The concentration effect is \(0.0100(0.500)=0.00500\ \mathrm{h}^{-1}\). Compare two times or groups whose baseline hazards are \(0.0100\ \mathrm{h}^{-1}\) and \(0.0400\ \mathrm{h}^{-1}\).

For the first baseline, the predicted hazard is \(0.0100+0.00500=0.0150\ \mathrm{h}^{-1}\). The hazard ratio relative to baseline is \(0.0150/0.0100=1.50\). For the second, it is \(0.0400+0.00500=0.0450\ \mathrm{h}^{-1}\), giving a hazard ratio of \(0.0450/0.0400=1.125\). The absolute increase is identical, but its relative size is not.

Now consider a log model with \(E_{\max,\log}=\log(2)=0.6931\) at the same concentration. Its hazard ratio is \(\exp\{0.6931(0.500)\}=\sqrt{2}=1.414\) for either baseline. The predicted hazards are \(0.0100(1.414)=0.01414\ \mathrm{h}^{-1}\) and \(0.0400(1.414)=0.05657\ \mathrm{h}^{-1}\). The relative increase is constant, but the absolute increases, \(0.00414\ \mathrm{h}^{-1}\) and \(0.01657\ \mathrm{h}^{-1}\), differ.

This contrast is useful when deciding what kind of effect is scientifically plausible: a fixed absolute change in hazard or a fixed relative change. It does not by itself establish which model is correct.

Assumptions and the Positivity Constraint

For either model, the chosen baseline hazard must be positive over the times at which the log model is used. The concentration effect assumes a monotone response with a single saturation level and Hill coefficient fixed at 1. It also treats the effect at a given concentration as adequately represented by \(f(C)\); it does not account for delay, hysteresis, or a separate dependence on exposure history. These are modeling assumptions, not consequences of choosing an Emax function.

The central mathematical constraint differs between scales. The exponential construction guarantees \(\lambda(t\mid C)>0\) whenever \(\lambda_0(t)>0\). The additive model does not guarantee positivity. For an inhibitory additive effect written as

\lambda(t\mid C)=\lambda_0(t)-I_{\max,\lambda}f(C), \qquad I_{\max,\lambda}\geq 0,

valid predictions require \(I_{\max,\lambda}f(C)\leq \lambda_0(t)\) at every relevant time and concentration. If \(C\) can become arbitrarily large, this requires \(I_{\max,\lambda}\leq\lambda_0(t)\) at every relevant time. A baseline hazard that changes over time can make this especially restrictive. A parameter estimate violating the constraint cannot be interpreted as a valid hazard model over that prediction domain.

Worked Example: Detecting an Invalid Additive Inhibitory Prediction

Suppose, hypothetically, that \(\lambda_0(t)=0.0200\ \mathrm{h}^{-1}\), \(I_{\max,\lambda}=0.0300\ \mathrm{h}^{-1}\), \(C_{50}=1.00\ \mathrm{mg/L}\), and \(C=3.00\ \mathrm{mg/L}\). Then \(f(C)=3/(1+3)=0.750\), and the additive prediction is

\lambda(t\mid C) =0.0200-(0.0300)(0.750) =-0.00250\ \mathrm{h}^{-1}.

A negative hazard is impossible, so this parameterization is invalid at the stated concentration. Merely describing the effect as saturable does not prevent the problem.

For comparison, use the log model with the same baseline, concentration, and \(C_{50}\), but let \(E_{\max,\log}=\log(0.25)=-1.3863\). At \(C=3.00\), its hazard ratio is \(\exp\{-1.3863(0.750)\}=\exp(-1.0397)=0.3536\). The predicted hazard is \(0.0200(0.3536)=0.007071\ \mathrm{h}^{-1}\), which is positive. Its limiting hazard ratio is 0.25, so the high-concentration hazard approaches one quarter of baseline rather than a negative value.

Model Choice, Diagnostics, and Limitations

The choice should follow the intended effect interpretation and the behavior supported by the data, not a rule that one link is universally preferable. An additive Emax effect may be appropriate when an absolute rate difference is the meaningful target and predictions remain nonnegative across the intended range. A log-hazard Emax effect is natural when proportional changes are plausible and positivity should be preserved by construction. A distinct alternative is to model a saturating hazard ratio directly as \(1+\theta f(C)\); that ratio-scale model is not the same as the log-hazard Emax model, whose ratio is \(\exp\{E_{\max,\log}f(C)\}\).

Useful checks include comparing predicted and observed event patterns across concentration ranges and across periods with different baseline hazards. Inspect predictions at \(C=0\), \(C=C_{50}\), and concentrations near the largest values used for inference. For an additive inhibitory model, explicitly calculate whether predictions remain positive over all relevant times and concentrations. For a log model, check that estimated hazard ratios and implied high-concentration limits are plausible, rather than treating positivity alone as evidence of fit.

A poorly determined \(C_{50}\) is a common warning sign. If observations cover only concentrations far below \(C_{50}\), then \(f(C)\) is approximately \(C/C_{50}\); the data may reveal a near-linear effect but provide little information about the saturation level. If observations lie only far above \(C_{50}\), \(f(C)\) is near 1 and the data may reveal the limiting effect without locating \(C_{50}\). Wide uncertainty or strong dependence between Emax and \(C_{50}\) should prompt cautious interpretation.

Neither link scale repairs a misspecified baseline hazard, poor exposure measurement, or unmodeled subject differences. A visually plausible concentration curve is not proof that a causal drug effect has been identified. Also, a constant concentration in these equations is a way to express the link-scale comparison; when exposure varies over time, predictions require a model that handles that variation appropriately.

Common pitfall: “Both models use the same Emax function, so their Emax parameters mean the same thing.” This is incorrect. \(E_{\max,\lambda}\) is an absolute hazard change with units of inverse time; \(E_{\max,\log}\) is a dimensionless limiting log hazard ratio. Compare predictions on a stated scale and at stated baseline hazards rather than comparing the raw parameter values.

Check Your Understanding

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

In an additive hazard Emax model, what does \(E_{\max,\lambda}\) represent, and what are its units?
A log-hazard model has \(E_{\max,\log}=\log(4)\). What is the limiting hazard ratio as concentration becomes large?
For an inhibitory additive model, explain why a negative predicted hazard is not an acceptable approximation.
At \(C=C_{50}\), what is the hazard ratio if \(E_{\max,\log}=\log(2)\)?
Which model gives a constant absolute hazard difference across different baseline hazards, and which gives a constant hazard ratio at a fixed concentration?

Solutions. (1) It is the limiting absolute hazard change, in inverse-time units. (2) It is \(\exp\{\log(4)\}=4\). (3) A hazard must be nonnegative; a negative model prediction signals that the additive parameterization is invalid over that prediction range. (4) It is \(\exp\{\log(2)/2\}=\sqrt{2}\approx1.414\). (5) The additive model gives a constant absolute difference; the log-hazard model gives a constant hazard ratio when concentration and its Emax parameters are fixed.