Proportional Hazards Assumptions in PK/PD

Why Proportional Hazards Is an Assumption About Time

In a time-to-event PK/PD analysis, an exposure effect may be expressed as a relative change in the instantaneous event rate among subjects who have not yet experienced the event. A proportional-hazards model makes a particular claim about that relative change: for a specified contrast in exposure, the hazard ratio is the same at every event time. The baseline hazard may rise, fall, or vary irregularly; it is the exposure-associated ratio, not the hazard itself, that is assumed constant.

This distinction matters clinically. If the underlying event hazard changes as subjects progress through follow-up, a constant hazard ratio allows the absolute difference in hazards to change too. It does not assert that exposure has the same absolute effect at every time, nor that event risks remain in a fixed ratio.

Definition: Let $T$ be event time, $X$ a fixed exposure measure, and $Z$ a vector of other covariates. A proportional-hazards model is $$ h(t\mid X,Z)=h_0(t)\exp\{\beta X+\boldsymbol{\gamma}^{\mathsf T}Z\}, $$ where $h(t\mid X,Z)$ is the hazard at time $t$ conditional on being event-free just before $t$, $h_0(t)$ is the baseline hazard, and $\beta$ and $\boldsymbol{\gamma}$ are regression coefficients. For two subjects with the same $Z$ and exposures $x_1$ and $x_0$, the hazard ratio is $$ \frac{h(t\mid x_1,Z)}{h(t\mid x_0,Z)} =\exp\{\beta(x_1-x_0)\}, $$ which does not depend on $t$. This time-invariance is the proportional-hazards assumption.

Here $X$ is a specified, fixed exposure measure, such as a baseline exposure summary. The model does not by itself determine which exposure summary is scientifically appropriate. In particular, inserting a changing concentration profile as though it were a fixed $X$ would change the model specification and its interpretation. The exposure scale and units must be stated: $\beta$ is the log hazard ratio per one-unit increase in $X$, conditional on the other covariates.

What the Assumption Does—and Does Not—Require

The baseline hazard $h_0(t)$ describes how the hazard changes over time for the reference covariate values. It need not be constant or follow a particular parametric curve. The model separates this time pattern from the covariate multiplier. For a fixed exposure contrast, the multiplier remains the same while the baseline hazard can vary.

The assumption concerns conditional hazards among subjects who remain event-free at each time. It is not a statement that the exposed and unexposed groups have the same composition among survivors as follow-up proceeds. Nor does a hazard ratio directly give the ratio of cumulative event probabilities. Those probabilities depend on the baseline hazard and follow-up time as well as on the exposure coefficient.

Worked Example: A Constant Ratio With Changing Hazards

Suppose a hypothetical model has $\beta=-0.0400$ per mg·h/L of exposure. Compare two otherwise similar subjects whose exposure measures differ by 10 mg·h/L, with $x_1$ the higher exposure. Suppose the baseline hazard is 0.0200 per day at day 7 and 0.00600 per day at day 30.

The exposure-associated hazard ratio is $$ \mathrm{HR}=\exp\{\beta(x_1-x_0)\} =\exp\{(-0.0400)(10\ \mathrm{mg\cdot h/L})\} =\exp(-0.400)=0.6703. $$

At day 7, the higher-exposure subject's modeled hazard is $0.0200\times0.6703=0.01341$ per day. At day 30, it is $0.00600\times0.6703=0.004022$ per day. The ratio is 0.6703 at both times, even though both hazards are lower at day 30. Under this hypothetical model, the higher exposure is associated with a 32.97% lower instantaneous hazard at each event time, conditional on the modeled covariates. This is a model-based association, not by itself evidence that increasing exposure causes the hazard to fall.

Why the Hazard Ratio Is Constant

The proportionality follows directly from the model form. For two exposure values and the same values of $Z$, the common baseline hazard and covariate multiplier cancel in the ratio. Taking logarithms makes the time-invariant contrast especially clear.

\begin{aligned} \log h(t\mid x_1,Z)-\log h(t\mid x_0,Z) &=\log h_0(t)+\beta x_1+\boldsymbol{\gamma}^{\mathsf T}Z \\ &\quad-\{\log h_0(t)+\beta x_0+\boldsymbol{\gamma}^{\mathsf T}Z\}\\ &=\beta(x_1-x_0). \end{aligned}

The result depends on both subjects sharing the same baseline hazard and on the exposure coefficient being constant in time. If the exposure effect changes with $t$, the log-hazard contrast will generally change too, and the model above no longer describes that feature.

A related result helps prevent a common misinterpretation. For fixed covariates, define the cumulative baseline hazard $H_0(t)=\int_0^t h_0(u)\,du$. Integrating the model gives $H(t\mid X,Z)=H_0(t)\exp\{\beta X+\boldsymbol{\gamma}^{\mathsf T}Z\}$. Since survival is $S(t\mid X,Z)=\exp\{-H(t\mid X,Z)\}$, the hazard ratio is constant but the survival ratio generally is not.

Worked Example: Constant Hazard Ratio, Different Risk Ratio

Consider a hypothetical comparison in which the higher-exposure group has a hazard ratio of 1.50 relative to the lower-exposure group, with the same other covariates. Suppose the lower-exposure group's cumulative hazard by day 30 is 0.600.

The higher-exposure group's cumulative hazard is $1.50\times0.600=0.900$. Their survival probabilities at day 30 are therefore $$ S_0(30)=\exp(-0.600)=0.5488,\qquad S_1(30)=\exp(-0.900)=0.4066. $$

The corresponding event risks by day 30 are $1-0.5488=0.4512$ and $1-0.4066=0.5934$. The event-risk ratio is $0.5934/0.4512=1.315$, not 1.50. The hazard ratio compares instantaneous hazards in the event-free risk sets; the risk ratio compares cumulative event probabilities over a specified period. They are different quantities.

Assumptions and Diagnostics

A proportional-hazards analysis requires more than a coefficient and a baseline hazard. The event and time origin must be defined consistently; the exposure and other covariates must be measured and represented appropriately; and subjects being compared must have meaningful overlap in their covariate values. If the coefficient is intended to describe a causal effect, additional design and modeling conditions are needed to address confounding. A fitted hazard ratio alone does not establish causality.

Right censoring also matters. For standard survival likelihood or partial-likelihood analyses, censoring should be independent of the event process conditional on the modeled information, unless the analysis explicitly accounts for dependence. Informative loss to follow-up can distort estimated exposure effects even if the proportional-hazards form is correct.

Proportionality cannot be confirmed simply because a model converges or a coefficient is statistically significant. Diagnostics can reveal patterns inconsistent with the assumption, although no single diagnostic proves it true. For an exposure grouped for display, approximately parallel curves of log cumulative hazard against time are compatible with proportional hazards. A systematic divergence or convergence is a warning, though such plots can be unstable with sparse data.

A residual-based diagnostic uses scaled Schoenfeld residuals, which are computed at event times for covariates in a fitted proportional-hazards model. Under proportionality, their systematic association with event time should be absent. A smooth trend or a test for a time trend can flag a time-dependent coefficient. The test has limited power with few events, and a small p-value can also arise from model misspecification or chance; interpretation should combine plots, clinical knowledge, and model checks.

Worked Example: A Possible Change in the Exposure Ratio

In a hypothetical study, crude event rates are tabulated over two follow-up intervals for lower- and higher-exposure groups. For illustration, divide events by person-time within each interval. These rates are not adjusted for other covariates.

In the first interval, the lower-exposure group has 8 events over 400 person-days, or $8/400=0.0200$ events per person-day. The higher-exposure group has 12 events over 400 person-days, or $12/400=0.0300$. The crude rate ratio is $0.0300/0.0200=1.50$.

In the second interval, the lower-exposure group has 12 events over 300 person-days, or $12/300=0.0400$ events per person-day. The higher-exposure group has 6 events over 150 person-days, or $6/150=0.0400$. The crude rate ratio is $0.0400/0.0400=1.00$.

The interval-specific crude ratios differ, which is a potential warning that the relative hazard may change over time. They are not proof of nonproportionality: the counts are small, the groups may differ in other risk factors, and the tabulation does not adjust for risk-set composition. A fitted-model diagnostic, with uncertainty considered, would be needed before drawing a conclusion.

When Proportionality Is Not Plausible

If diagnostics and subject-matter reasoning suggest that the exposure-associated hazard ratio changes over time, a single constant coefficient can obscure clinically important patterns. For example, it may average a strong early association and a weak later association into a value that represents neither period well. The model may still produce a numerical estimate, but that estimate should not be interpreted as a time-invariant effect if the assumption is implausible.

Possible responses include specifying an exposure-by-time effect, stratifying on a covariate whose effect is not of primary interest, or choosing a model with a different interpretation, such as an accelerated failure time model. Each approach makes different assumptions and answers a different question; the choice should follow the scientific aim and data support. A more flexible baseline hazard alone does not fix a changing exposure coefficient, because baseline flexibility changes the common time pattern, not the exposure-associated ratio.

Common pitfall: “The model is proportional hazards, so the exposed group has 1.5 times the event risk at every time.” This confuses hazard with risk. The model states that the conditional instantaneous hazard ratio is 1.5 at every time for the specified covariate contrast. Cumulative risk ratios generally vary with follow-up time and need not equal 1.5.

Check Your Understanding

Answer each question using the distinction between hazard, survival, and cumulative event risk.

In a proportional-hazards model, what part may vary freely over time, and what is assumed constant for a fixed exposure contrast?
If $\beta=0.0300$ per mg·h/L, what is the hazard ratio for a 5 mg·h/L increase in exposure?
Does a constant hazard ratio imply a constant ratio of event risks by day 30 and day 60? Explain.
Name one diagnostic signal that could suggest nonproportionality, and state why it is not conclusive by itself.

Solutions. (1) The baseline hazard may vary over time; the hazard ratio for a fixed exposure contrast is constant. (2) $\exp\{0.0300\times5\}=\exp(0.150)=1.162$, so the modeled hazard is about 1.162 times as large per 5 mg·h/L increase, conditional on other covariates. (3) No. Cumulative risks depend on the baseline hazard and follow-up time, so their ratio generally changes. (4) A systematic trend in scaled Schoenfeld residuals with event time is a warning; sparse events, chance variation, or other model misspecification can also produce apparent patterns.