Why Diagnose Proportional Hazards?
An exposure-hazard model may describe the direction of a drug effect well on average while misrepresenting when that effect occurs. A treatment that has little effect initially and a stronger effect later, for example, cannot generally be summarized by one constant hazard ratio. This matters for interpretation and for predictions over follow-up: an average coefficient may hide clinically relevant early and late differences.
The proportional-hazards assumption concerns the coefficient of a predictor, not whether exposure itself changes. If concentration changes over time, a model can still have a proportional effect of concentration: at each time, the same concentration contrast has the same hazard ratio, conditional on the model. The diagnostic question is whether the exposure coefficient changes with event time after accounting for the exposure values and other predictors used in the model.
This formulation allows time-varying exposure, such as a concentration that changes with dosing. It does not, by itself, model a delayed pharmacodynamic response: a delay may require a lagged or otherwise appropriate exposure representation. Nonproportionality in the coefficient and a delayed exposure-effect relationship are related possibilities, but they are not interchangeable explanations.
What Schoenfeld Residuals Show
For each observed event time, a Schoenfeld residual compares the covariate value for the individual who experienced the event with a weighted average of that covariate among individuals still at risk. Thus, it uses the event-time risk set rather than comparing an event case with the whole original sample. For a continuous exposure, a positive residual means the event individual’s exposure exceeded the model-weighted risk-set average at that event time.
Here \(t_j\) is an event time, \(E_{(j)}(t_j)\) is the event individual’s exposure, \(R(t_j)\) is the risk set just before that event, and \(\mathbf{X}\) includes the modeled predictors. This expression describes the usual no-ties form; when event times are tied, the residual calculation must use a method consistent with the model’s tie handling. A residual is not a raw difference between exposure groups and does not measure an individual treatment response.
Raw residuals can have different variability at different event times. Scaled Schoenfeld residuals adjust for the information in the fitted model so that, approximately, their trend over time reflects the departure of a time-specific coefficient from the fitted constant coefficient. For an exposure coefficient, a persistent pattern rather than random fluctuation around zero suggests that the constant-coefficient model may be inadequate. A residual plot is a diagnostic signal, not a direct estimate of a biological mechanism.
Why Scaling Can Reveal a Time Trend
The connection between residuals and a changing coefficient can be justified locally. Consider one exposure and suppose its true coefficient near event time \(t_j\) differs slightly from the fitted constant coefficient by \(\delta_j\). Under the fitted coefficient, the weighted risk-set mean is the reference. A small positive \(\delta_j\) gives greater relative event weight to individuals with larger exposure. To first order, the expected event exposure therefore shifts upward by approximately the risk-set weighted variance of exposure multiplied by \(\delta_j\).
The weighted variance is the local information about the exposure coefficient. Scaling the residual by an information-based factor puts the deviations on an approximate coefficient scale. In a multivariable model, the analogous calculation uses the information matrix and accounts for correlations among predictors. This is a first-order diagnostic approximation: it is most informative when the model is otherwise reasonable and the time-specific departure is not extreme. It does not make sparse late risk sets or strong exposure-covariate collinearity disappear.
Worked Example: Reading a Scaled Residual Trend
Suppose an illustrative diagnostic plot summarizes scaled exposure residuals at four event-time positions. The positions use a transformed time scale \(g(t)\), and the smoothed residual summaries are \((-1.5,-0.30)\), \((-0.5,-0.10)\), \((0.5,0.10)\), and \((1.5,0.30)\). The first number in each pair is \(g(t)\); the second is the scaled residual, in approximate log-hazard-coefficient units per one-unit exposure.
The summaries follow the line \(r^*(t)=0.20g(t)\). For example, the slope is \[ \frac{\sum g_jr^*_j}{\sum g_j^2} =\frac{(-1.5)(-0.30)+(-0.5)(-0.10)+(0.5)(0.10)+(1.5)(0.30)} {(-1.5)^2+(-0.5)^2+(0.5)^2+(1.5)^2} =\frac{1.00}{5.00}=0.20. \]
The increasing residual trend is consistent with a coefficient that becomes more positive as transformed time increases, relative to the fitted constant coefficient. It does not establish that a drug effect is delayed: the pattern could also reflect a changing effect for another reason, an incorrect exposure model, or chance variation. Four summaries are only a visual illustration, not an adequate basis for a formal test or a confident clinical conclusion.
Exposure-by-Time Terms
A direct way to represent nonproportionality is to allow the exposure coefficient to depend on time. For instance, choose a prespecified time function \(g(t)\) and add an exposure-by-time interaction to the linear predictor. Centering \(g(t)\) at a clinically interpretable reference time makes the main exposure coefficient interpretable at that time.
For two otherwise comparable individuals whose exposure differs by one unit at time \(t\), the modeled hazard ratio is \(\exp\{\beta_E+\gamma_E g(t)\}\). The null hypothesis of a constant exposure coefficient in this model is \(H_0:\gamma_E=0\). The choice of \(g(t)\) is substantive: \(g(t)=\log(t/t_{\mathrm{ref}})\) allows a smooth change with log time, whereas a prespecified step function allows different coefficients in defined intervals. The baseline hazard already varies with time; the interaction specifically allows the exposure effect to vary.
Worked Example: Interpreting a Log-Time Interaction
In a hypothetical model, exposure is measured in concentration units and time is measured in weeks. Let \(g(t)=\log(t/4)\), so the reference time is 4 weeks. Suppose \(\hat\beta_E=-0.60\) and \(\hat\gamma_E=0.25\) per one-unit increase in log time. The exposure coefficient at time \(t\) is \(-0.60+0.25\log(t/4)\).
At 4 weeks, \(g(4)=\log(1)=0\). The one-unit exposure hazard ratio is \[ \exp\{-0.60+0.25(0)\}=\exp(-0.60)=0.549. \] At 16 weeks, \(g(16)=\log(4)=1.386\), so the coefficient is \[ -0.60+0.25(1.386)=-0.2535, \] and the corresponding hazard ratio is \(\exp(-0.2535)=0.776\). In this hypothetical fit, the estimated association is protective at both times, but is closer to a hazard ratio of 1 at 16 weeks. That pattern is consistent with a waning protective association; it does not prove waning drug action or causation.
The interaction coefficient is defined per unit increase in \(g(t)\), not per week. Its estimate alone is not enough to assess uncertainty in the hazard ratios at each time. Those calculations require the estimated covariance of \(\hat\beta_E\) and \(\hat\gamma_E\), and conclusions should reflect the model and data support.
Worked Example: Distinguishing Possible Effect Patterns
Suppose a hypothetical event study uses an exposure contrast and a time-specific coefficient curve to summarize its fitted relationship with hazard. At early times the coefficient is near 0.00, while at later times it is near \(-0.50\). The estimated conditional hazard ratio changes from \(\exp(0.00)=1.00\) to \(\exp(-0.50)=0.607\). This is consistent with a delayed emergence of a protective association.
Now consider a curve that begins at \(-0.50\) and moves toward 0.00. The estimated hazard ratio changes from 0.607 toward 1.00, consistent with a waning protective association. If instead the coefficient changes from negative to positive, the fitted association changes direction over time. Such a reversal deserves careful scrutiny: it can be real, but it can also arise from sparse events, changes in who remains at risk, or model misspecification.
These patterns describe conditional associations on the modeled scale. A delayed concentration effect may be better represented by a biologically justified delayed exposure measure, while a genuinely time-varying coefficient can be represented by an exposure-by-time term. Comparing those explanations requires a defensible exposure model and adequate event information over time.
Diagnostics, Alternatives, and Limitations
Scaled Schoenfeld residuals are commonly examined against event time or a prespecified transformation of time. A smooth curve near zero with no persistent trend is compatible with proportionality; a systematic departure suggests a time-varying coefficient. Formal tests can assess association between scaled residuals and a chosen function of time. Their result depends on that time transformation, so a nonsignificant test does not rule out every possible form of nonproportionality. If many covariates or transformations are explored, selective interpretation of the most striking plot or test is especially risky.
A fitted exposure-by-time interaction answers a different, more explicit question: how does the estimated coefficient change according to the specified function? A linear interaction with log time may be easier to summarize than a residual plot but can miss a step change or a nonmonotonic pattern. A piecewise coefficient can represent prespecified phases, but its cut points should be justified and the event data must support estimates in each phase. These methods are complementary: residuals can indicate where to investigate, while a prespecified interaction gives a parameterized description that can be interpreted and evaluated.
| Approach | What it contributes | Main caution |
|---|---|---|
| Scaled Schoenfeld residuals | Event-time diagnostic for trends inconsistent with a constant coefficient | Patterns are noisy and do not identify their biological cause |
| Exposure-by-time interaction | Explicit estimate of how the coefficient varies with a chosen time function | Conclusions depend on the function and its scale |
| Separate time-interval coefficients | Allows distinct effects in defined follow-up periods | Can be imprecise, especially in intervals with few events |
Before interpreting a trend, check whether the exposure representation is appropriate, whether the risk sets contain enough exposure variation, and whether late follow-up is supported by enough individuals and events. A pattern can also signal a misspecified exposure-response relationship, an omitted covariate, or an unmodeled delayed exposure effect. With time-varying exposure, compare exposure values at the relevant event times; a changing concentration profile should not be mistaken automatically for a changing coefficient.
The analysis also relies on the event-time model being suitably specified, including its treatment of censoring, entry, and other design features. An apparent time trend can be distorted if these features are mishandled. Diagnostics for proportionality do not establish that censoring is noninformative, that the exposure-response scale is correct, or that the model predicts event risk adequately. They test a particular feature of a fitted model, not the model as a whole.
Common Pitfall
Incorrect reasoning: “The drug concentration falls over follow-up, and the residual plot changes over time, so the drug’s hazard effect must be waning.” This confuses the trajectory of exposure with the time-dependence of its coefficient and treats a diagnostic pattern as a causal mechanism.
Correction: State what exposure contrast the model uses at each time, then ask whether its coefficient changes after accounting for that exposure and the other modeled predictors. A residual trend motivates investigation; an exposure-by-time term estimates a specified changing coefficient. Neither alone proves why the coefficient changes. Assess uncertainty, risk-set support, exposure-model adequacy, and plausible alternative explanations before making a pharmacologic interpretation.
Check Your Understanding
Answer each question using the distinction between a changing exposure trajectory and a changing exposure coefficient.
- If concentration varies over time but the coefficient \(\beta_E\) is constant, is the exposure effect necessarily nonproportional?
- In a model with \(g(t)=\log(t/2)\), what does \(\beta_E\) represent?
- A scaled residual smooth rises from negative to positive values. What does that suggest, and what does it not establish?
- Why might a late-follow-up residual pattern be unreliable?
Solutions: (1) No. The modeled concentration can change while a one-unit concentration contrast has a constant hazard ratio \(\exp(\beta_E)\). (2) It is the exposure coefficient at 2 weeks, because \(g(2)=0\). (3) It suggests that the constant-coefficient model may miss a time-related change; it does not establish a delayed drug mechanism or causation. (4) Few individuals or events may remain at risk, producing noisy residuals and unstable estimates.