Log-Linear Concentration Effects on Hazard

Why Model Concentration on the Log-Hazard Scale?

In a time-to-event PK/PD analysis, the event hazard is the instantaneous event rate among individuals who have not yet experienced the event. A concentration effect on hazard describes how that instantaneous rate differs across concentrations, conditional on the model and the information included in it. The log-linear model is useful when a constant change in concentration is expected to multiply the hazard by a constant factor. It is a model for the log hazard, not a claim that the hazard itself changes by a constant additive amount.

This tutorial considers a concentration $C$ that is treated as fixed for the hazard comparison being made. It does not develop a time-varying concentration model. The baseline hazard can vary over time, while the concentration effect multiplies it.

Definition: A log-linear concentration effect on hazard is specified by $$ h(t\mid C)=h_0(t)\exp\{\beta(C-C_{\mathrm{ref}})\}, $$ where $h(t\mid C)$ is the hazard at time $t$ for concentration $C$, $h_0(t)$ is the baseline hazard at reference concentration $C_{\mathrm{ref}}$, and $\beta$ is the concentration coefficient. If concentration is measured in mg/L, $\beta$ has units of (mg/L)$^{-1}$. The model assumes that the log hazard is linear in concentration over the range to which the model is applied.

Interpreting the Concentration Coefficient

For two concentrations $C_a$ and $C_b$, the ratio of their hazards at the same time is

\frac{h(t\mid C_b)}{h(t\mid C_a)} =\exp\{\beta(C_b-C_a)\}.

The baseline hazard cancels, so this ratio does not depend on the value of $t$ under the model. A positive $\beta$ means that higher concentration is associated with higher hazard; a negative $\beta$ means higher concentration is associated with lower hazard. For a one-unit concentration increase, the hazard ratio is $\exp(\beta)$. The coefficient is not itself a percentage change in hazard: the multiplicative change is obtained by exponentiating it.

The unit matters. If $C$ is in mg/L, then a one-unit increase means 1 mg/L. Rescaling concentration changes the numerical coefficient but not the modeled hazard ratios when the rescaling is done consistently. Centering at $C_{\mathrm{ref}}$ makes the baseline hazard correspond to a meaningful concentration; it does not change the slope $\beta$.

Worked Example: A Negative Coefficient and a Concentration Contrast

Suppose a hypothetical fitted model has $\beta=-0.12$ per mg/L and $C_{\mathrm{ref}}=0$ mg/L. Compare the hazard at 5 mg/L with the hazard at 2 mg/L.

The contrast is $5-2=3$ mg/L. Substituting in the hazard-ratio formula gives

\frac{h(t\mid 5)}{h(t\mid 2)} =\exp\{-0.12(5-2)\} =\exp(-0.36) =0.6977.

At the same time, the modeled hazard at 5 mg/L is about 0.698 times the hazard at 2 mg/L, or about 30.2% lower, because $1-0.6977=0.3023$. This is a conditional comparison of instantaneous hazards, not a statement that the probability of an event by a particular time is 30.2% lower.

For an absolute prediction, suppose the baseline hazard at a selected time is hypothetically $0.040$ events per person-day at 0 mg/L. At 5 mg/L,

h(t\mid 5)=0.040\exp\{-0.12(5-0)\} =0.040\exp(-0.60) =0.02195\ \text{events per person-day}.

The units remain events per person-day: the exponential multiplier is dimensionless. The numerical prediction depends on the specified baseline hazard as well as on $\beta$.

Fitting and Using the Model

With right-censored event times, the concentration coefficient can be estimated within a proportional-hazards model using the event-time likelihood or, when the baseline hazard is left unspecified, the Cox partial likelihood. The partial likelihood compares the concentration of the individual who experiences an event with the concentrations of individuals still at risk at that event time. Censored individuals contribute to the risk sets up to their censoring times but are not treated as having experienced the event. Ties require an appropriate tie-handling method. The particular estimator does not change the interpretation of $\beta$.

To use the fitted model for a selected concentration, first identify the reference concentration and the baseline hazard at the time of interest, then multiply the baseline hazard by the exponential concentration factor. If the baseline hazard is not estimated or specified, the model still gives relative hazards between concentrations, but it does not by itself give absolute hazards.

Worked Example: Predicted Hazards at Selected Concentrations

Consider a hypothetical model with $C_{\mathrm{ref}}=1$ mg/L, $\beta=0.25$ per mg/L, and a baseline hazard of $0.020$ events per person-day at the time of interest. Calculate the hazard at 1, 3, and 5 mg/L.

At 1 mg/L, the concentration contrast from the reference is zero:

h(t\mid 1)=0.020\exp\{0.25(1-1)\} =0.020\ \text{events per person-day}.

At 3 mg/L, the contrast is 2 mg/L:

h(t\mid 3)=0.020\exp\{0.25(3-1)\} =0.020\exp(0.50) =0.03297\ \text{events per person-day}.

At 5 mg/L, the contrast is 4 mg/L:

h(t\mid 5)=0.020\exp\{0.25(5-1)\} =0.020\exp(1.00) =0.05437\ \text{events per person-day}.

Each 1 mg/L increase multiplies the hazard by $\exp(0.25)=1.284$, or about 28.4%. A 2 mg/L increase multiplies it by $\exp(0.50)=1.649$. The hazards rise multiplicatively, rather than by a fixed additive amount.

Derivation: Why the Coefficient Gives a Hazard Ratio

Take the natural logarithm of the model:

\log h(t\mid C)=\log h_0(t)+\beta(C-C_{\mathrm{ref}}).

For two concentrations at the same time, subtract the log hazards. The common baseline term and reference-centering term cancel:

\log h(t\mid C_b)-\log h(t\mid C_a) =\beta\{(C_b-C_{\mathrm{ref}})-(C_a-C_{\mathrm{ref}})\} =\beta(C_b-C_a).

Exponentiating both sides yields $h(t\mid C_b)/h(t\mid C_a)=\exp\{\beta(C_b-C_a)\}$. This also shows why the coefficient is a log hazard ratio per concentration unit. For a small concentration change $\Delta C$, the exact relative change is $\exp(\beta\Delta C)-1$, not $\beta\Delta C$; the latter is only an approximation when $|\beta\Delta C|$ is small.

Uncertainty in a Fitted Effect

A coefficient estimate should be interpreted with its uncertainty. A Wald confidence interval for $\beta$, when its estimated sampling distribution is adequately approximated by a normal distribution, is $\hat{\beta}\pm z_{1-\alpha/2}SE$. Exponentiating the coefficient and both interval limits gives the hazard ratio and its interval for a one-unit increase. A Wald test of $H_0:\beta=0$ uses $z=\hat{\beta}/SE$; under the null and the approximation, the two-sided p-value is the probability of a standard normal result at least as extreme in absolute value.

Worked Example: Interpreting a Fitted Coefficient and Its Uncertainty

In a hypothetical analysis of 120 individuals with 46 observed events, suppose the fitted concentration coefficient is $\hat{\beta}=-0.080$ per mg/L with $SE=0.030$ per mg/L. Assume the model’s proportional-hazards and log-linear concentration assumptions are adequate for this interpretation.

The estimated hazard ratio for a 1 mg/L increase is

\exp(\hat{\beta})=\exp(-0.080)=0.9231.

Thus, the fitted model associates a 1 mg/L increase with a 7.69% lower hazard, since $1-0.9231=0.0769$. The two-sided Wald statistic for $H_0:\beta=0$ is

z=\frac{-0.080}{0.030}=-2.667.

The corresponding two-sided p-value is approximately 0.0077. At a 0.05 significance level, this is evidence against a zero concentration coefficient under the model and test assumptions; it is not the probability that the null hypothesis is true.

A 95% Wald interval for the coefficient is

-0.080\pm1.96(0.030) =(-0.1388,\,-0.0212)\ \text{per mg/L}.

Exponentiating the endpoints gives the 95% interval for the hazard ratio per 1 mg/L increase:

\bigl(\exp(-0.1388),\exp(-0.0212)\bigr) =(0.8704,\,0.9790).

Under repeated sampling and the stated approximation, the interval procedure would cover the true hazard ratio in 95% of repetitions. In this analysis, the interval is consistent with a reduction in hazard per 1 mg/L increase, conditional on the model.

Assumptions, Diagnostics, and Alternatives

The concentration coefficient has the interpretation above only if the model is a reasonable description of the data and the analysis is appropriate. Key conditions include:

Log-linearity over the analyzed range. The log hazard changes linearly with concentration. A strong bend in the concentration–effect relationship can make a single slope misleading.
Proportional hazards. The concentration hazard ratio is constant over time. If the relative effect changes with time, one constant $\beta$ does not summarize it adequately.
Suitable event-time and censoring information. The event and censoring times must be recorded and handled according to the likelihood assumptions; censoring should be non-informative conditional on modeled information.
Informative concentration comparisons. There must be enough concentration variation among individuals at risk for the effect to be estimated with useful precision. The estimate is conditional on included covariates and does not by itself establish causation.

Useful diagnostics include plotting or tabulating observed event patterns across concentration ranges, inspecting uncertainty and influential observations, and checking whether the concentration effect appears to vary over time. Sparse events at one end of the concentration range can produce unstable estimates. A fitted model should not be trusted to predict far beyond the concentrations represented in the data: the assumed straight line on the log-hazard scale may fail outside that range.

The log-linear model is parsimonious and gives a simple per-unit hazard ratio. A model with no concentration effect is a useful nested comparison when the scientific question is whether the slope is distinguishable from zero. Categorizing concentration avoids imposing a straight-line effect but loses information and depends on cut points. A saturable concentration-effect model is an alternative when a plateau is scientifically plausible; choosing and fitting that form requires different assumptions and is treated separately in the material on Emax concentration effects on event hazard.

Common pitfall: It is incorrect to say that $\beta=-0.12$ means the hazard falls by 12% for every 1 mg/L increase. The coefficient is a change in log hazard per mg/L. The corresponding hazard ratio is $\exp(-0.12)=0.8869$, a reduction of about 11.3% per 1 mg/L. For a 3 mg/L contrast, use $\exp(-0.12\times3)$, not three times the percentage reduction.

Check Your Understanding

Use the model and definitions above. Answers are provided so you can check each calculation and interpretation.

A fitted coefficient is $0.10$ per mg/L. What is the hazard ratio for a 2 mg/L increase, and what is its direction?
For $\beta=-0.20$ per mg/L, compare the hazard at 4 mg/L with the hazard at 1 mg/L.
With $C_{\mathrm{ref}}=2$ mg/L, $\beta=0.15$ per mg/L, and baseline hazard $0.030$ events per person-day, calculate the hazard at 4 mg/L.
A 95% confidence interval for $\beta$ is $(-0.04,0.02)$ per mg/L. Does the corresponding interval for the one-unit hazard ratio include 1? What does that imply about evidence for a nonzero coefficient at the two-sided 0.05 level using this interval?

Solutions. (1) $\exp(0.10\times2)=\exp(0.20)=1.221$; the model predicts a 22.1% higher hazard for the 2 mg/L increase. (2) $\exp\{-0.20(4-1)\}=\exp(-0.60)=0.5488$, so the hazard at 4 mg/L is about 45.1% lower than at 1 mg/L. (3) $0.030\exp\{0.15(4-2)\}=0.030\exp(0.30)=0.04050$ events per person-day. (4) Exponentiating the limits gives $(\exp(-0.04),\exp(0.02))=(0.9608,1.020)$, which includes 1. The interval therefore does not exclude a zero coefficient at the two-sided 0.05 level under the Wald interpretation.