Why Model Time to Response?
A binary endpoint records whether a patient has responded by a chosen visit. A time-to-response endpoint also uses when the response first occurs. That timing can be clinically informative: two groups may have similar response proportions at the end of follow-up while one group responds earlier. Conversely, an apparent difference in response timing does not by itself establish a difference in the probability of ever responding.
Let \(T\) be the time from a prespecified origin, such as treatment initiation, to the first occurrence of a prespecified efficacy response. The response definition, time origin, assessment schedule, and handling of treatment discontinuation must be fixed before analysis. If response is only known to have occurred between visits, the event time is interval-censored rather than exactly observed; methods that account for that interval may be needed.
For a patient with no recorded response before the end of observation, the response time is right-censored: we know that \(T\) exceeds the observed follow-up time, but not its eventual value. Such a patient contributes information up to censoring and should not simply be discarded or treated as if response occurred at censoring.
Exposure Models and Their Assumptions
The exposure metric and time origin should match the clinical question. Here, \(x_i\) denotes a patient’s exposure on a chosen scale, such as average exposure during a prespecified period, optionally centered and scaled so that one unit has a useful clinical meaning. Earlier material on choosing exposure metrics and causal assumptions is relevant: an exposure coefficient from an observational exposure-response model is not automatically a causal treatment effect.
A parametric model specifies the hazard or survival distribution directly. For example, the exponential model assumes a constant response hazard over time for each patient. A Cox proportional-hazards model leaves the baseline hazard unspecified and models how covariates multiply it:
Here \(h_0(t)\) is the baseline hazard, \(\beta\) is the exposure coefficient, and \(\mathbf{z}_i\) contains other modeled covariates with coefficients \(\boldsymbol{\gamma}\). The model assumes proportional hazards: for fixed covariates, the hazard ratio comparing two patients does not change over time. It also requires a correctly defined event and, for standard right-censoring methods, censoring that is independent of the unobserved response time conditional on modeled covariates. If, for example, patients leave follow-up because their unobserved response prospects are worsening, ordinary censoring methods may be biased.
For a one-unit increase in exposure, holding the other modeled covariates fixed, the Cox hazard ratio is \(\exp(\beta)\). A ratio above one means a higher instantaneous response rate among patients who have not yet responded; it does not mean a corresponding increase in the probability of response at any fixed time. In a parametric model, the chosen distribution adds assumptions about how the hazard changes over time. Those assumptions can enable estimates of absolute response probabilities and median response times, but can also make extrapolation more sensitive to model misspecification.
Worked Example: Exposure and Response Under an Exponential Model
Worked Example: Exposure and Response Under an Exponential Model
Hypothetical data. Suppose two exposure groups are followed until response or censoring. In the lower-exposure group, there are 8 responses over 160 total person-weeks at risk. In the higher-exposure group, there are 12 responses over 150 total person-weeks at risk. Assume an exponential response time within each group, with independent censoring.
Under a constant hazard, the maximum-likelihood estimate of the group hazard is the number of observed responses divided by total observed time at risk. Thus:
The estimated hazard ratio is \(0.0800/0.0500=1.60\). Under the exponential model, the median response time is \(\ln(2)/\lambda\), so the estimated medians are \(0.6931/0.0500=13.86\) weeks for lower exposure and \(0.6931/0.0800=8.664\) weeks for higher exposure.
At 12 weeks, the exponential model gives \(F(t)=1-\exp(-\lambda t)\). Therefore:
Under these assumptions, the estimated response probability by week 12 is 45.1% in the lower-exposure group and 61.7% in the higher-exposure group. These are model-based probabilities, not direct consequences of the hazard ratio alone. The example also assumes a constant hazard, so it would be inappropriate to report these predictions without considering whether that assumption is plausible.
Why the Cox Partial Likelihood Works
The Cox model estimates exposure effects without specifying the shape of \(h_0(t)\). Its partial likelihood uses the covariate values of the patient who responds at each event time relative to those of all patients still at risk at that time. For clarity, suppose event times are distinct and there are no ties. At event time \(t_j\), let \(i(j)\) be the patient who responds and \(R(t_j)\) the risk set: patients who are still under observation and have not yet responded immediately before \(t_j\).
This conditional probability follows by writing each at-risk patient’s hazard as \(h_0(t_j)\exp(\eta_k)\), where \(\eta_k=\beta x_k+\boldsymbol{\gamma}^{\mathsf T}\mathbf{z}_k\), and dividing the event patient’s hazard by the sum of hazards in the risk set. The common baseline hazard cancels. Multiplying these conditional contributions over event times gives the partial likelihood:
A patient who is censored contributes to risk sets up to the censoring time, but does not contribute an event numerator. With tied event times, software uses an appropriate tie method, such as Efron’s approximation or the Breslow method. The derivation explains both the Cox model’s flexibility about the baseline hazard and its reliance on valid risk sets and the proportional-hazards assumption.
Worked Example: Interpreting a Cox Exposure Effect
Worked Example: Interpreting a Cox Exposure Effect
Hypothetical model output. Let one exposure unit represent a 10 mg/L increase in the selected exposure metric. Suppose a Cox model adjusted for prespecified covariates estimates \(\hat{\beta}=0.400\), with \(SE=0.153\). Assume the model’s proportional-hazards and censoring assumptions are plausible for this illustration.
The estimated hazard ratio per 10 mg/L increase is:
A Wald 95% confidence interval for the coefficient is \(0.400\pm1.96(0.153)\), or \(0.100\) to \(0.700\). Exponentiating both endpoints gives a hazard-ratio interval of \(\exp(0.100)=1.105\) to \(\exp(0.700)=2.014\). The Wald statistic is \(z=0.400/0.153=2.614\), giving a two-sided \(p\)-value of approximately 0.0089.
In this hypothetical adjusted model, each 10 mg/L increase in exposure is associated with an estimated 49.2% higher instantaneous response hazard among patients who have not yet responded; the 95% confidence interval corresponds to approximately 10.5% to 101.4% higher hazard. The \(p\)-value describes how unusual a statistic at least this extreme would be under the null hypothesis of a zero coefficient, assuming the model and its standard-error calculation are valid. Neither the interval nor the \(p\)-value establishes causation. The hazard ratio alone also does not provide the response probability by a particular week; that requires the baseline survival or hazard as well as the covariate effect.
Assessing Proportional Hazards
Proportional hazards is a substantive assumption, not merely a software setting. If the exposure effect changes over follow-up, a single hazard ratio can conceal clinically important differences in the early and late response periods. Assessment should combine graphical and model-based evidence with clinical understanding; no single diagnostic proves the assumption.
- Scaled Schoenfeld residuals: examine whether residuals for exposure show a systematic trend with time. A trend suggests that the exposure coefficient may vary over time. A formal test can flag departures, but its result depends on sample size and should not replace inspection.
- Log-minus-log survival plots: for categorical exposure groups, roughly parallel curves can support proportional hazards. Strongly changing separation or crossing curves raises concern. This plot is less straightforward for continuous exposure and does not establish the assumption.
- Time-specific estimates: compare estimated effects across clinically meaningful follow-up periods. Marked variation may indicate non-proportionality, though estimates can be imprecise when few responses occur in a period.
If proportional hazards is not credible, report that the single Cox hazard ratio is inadequate as a summary. Depending on the scientific question, alternatives include a prespecified time-varying exposure coefficient, a stratified Cox model for a covariate whose effect is not of interest, or a parametric model that allows the exposure effect to vary with time. These approaches change the question or add assumptions; they should not be selected solely because they yield a preferred result.
Worked Example: A Changing Exposure Effect Over Time
Hypothetical diagnostic pattern. Suppose an analysis estimates an exposure hazard ratio of 2.0 during weeks 0–8 and 0.70 during weeks 8–20, with adequate responses in both periods. These are period-specific estimates, not a single constant effect. The reversal in direction suggests that a constant exposure hazard ratio may be misleading, although sampling variability must also be considered.
A single Cox estimate could average these opposing patterns and appear near 1, masking a possible early acceleration and later reduction in response rate. The next step is to inspect residual plots and uncertainty, verify event and risk-set definitions, and consider a clinically justified time-varying effect. It would be incorrect to claim that the exposure has no effect merely because a pooled estimate is near 1, or to assert a true reversal without considering uncertainty and model assumptions.
Common Pitfalls and Model Choice
| Approach | Strength | Main limitation |
|---|---|---|
| Exponential parametric model | Directly yields response probabilities and median times under a simple model. | Assumes a constant hazard within modeled groups or covariate profiles. |
| Other parametric survival models | Can represent changing hazard shapes and support absolute predictions. | Results depend on the selected distribution and its adequacy over the follow-up period. |
| Cox proportional-hazards model | Estimates covariate hazard ratios without specifying the baseline hazard. | Assumes proportional hazards for a time-constant coefficient and does not by itself give absolute probabilities. |
A common mistake is to interpret a hazard ratio of 1.5 as a 50% higher probability of response by a given date. The correction is that it describes the relative instantaneous response rate among patients still awaiting response. The probability by a date depends on the baseline hazard and elapsed time, as the exponential example demonstrates. Another mistake is to count a censored patient as a nonresponder through the end of all follow-up; the correct record is that the patient was response-free only through the observed censoring time.
These models also have limitations beyond their mathematical assumptions. Response assessment schedules can make event times imprecise, and choice of response definition can change the endpoint. Unmeasured confounding, exposure measurement error, or model misspecification can distort an exposure association. Parametric extrapolation beyond observed follow-up is particularly dependent on the assumed hazard shape. Report the exposure scale, event definition, time origin, censoring assumptions, model form, uncertainty, and relevant diagnostics alongside the estimate.
Check Your Understanding
Answer each question, then compare with the solutions below.
- In a time-to-response analysis, a patient is followed for 10 weeks without a recorded response and then lost to follow-up. What is known about that patient’s response time?
- A Cox exposure coefficient is \(-0.20\) per exposure unit. Calculate and interpret the hazard ratio, assuming proportional hazards.
- Under an exponential model with a response hazard of 0.04 per week, calculate the probability of response by 10 weeks.
- Why can a hazard ratio not, by itself, determine the probability of response by week 12?
Solutions. (1) The response time is right-censored at 10 weeks: the patient is known to have remained response-free through that time, but the later response time is unknown. (2) \(\exp(-0.20)=0.819\). A one-unit exposure increase is associated with an estimated 18.1% lower instantaneous response hazard among patients still awaiting response, conditional on modeled covariates. (3) \(F(10)=1-\exp(-0.04\times10)=1-\exp(-0.40)=0.3297\), or about 33.0%. (4) The hazard ratio is relative and does not specify the baseline hazard or survival. Absolute response probability at week 12 requires the baseline survival information and a model that supports that prediction.