In a clinical trial, the Cox model estimates a hazard ratio of 0.6 for the treated group versus controls. At month 12, the control group's hazard is 0.08 events/month. Under the proportional hazards assumption, what is the treated group's hazard at month 12?
A0.6 events/month — the hazard ratio directly gives the treated group's hazard rate
B0.048 events/month — the hazard ratio multiplies the baseline hazard: 0.6 × 0.08
C0.048 events/month, but only if the baseline hazard is constant over time
DCannot be determined without fitting a parametric survival model to specify the baseline hazard
The Cox model specifies h(t|X) = h₀(t) × exp(βX). For the treated group, HR = exp(β) = 0.6, so h_treated(t) = 0.6 × h₀(t) at every time point t. At month 12: h_treated = 0.6 × 0.08 = 0.048 events/month. This is the proportional hazards assumption in action — the hazard ratio is constant across time, so you can compute the treated group's hazard at any time point by multiplying the control group's hazard by 0.6. The baseline hazard h₀(t) does NOT need to be constant over time — it can be any function of t. The Cox model leaves it completely unspecified.
Question 2 Multiple Choice
After fitting a Cox model, the Kaplan-Meier survival curves for the treated and control groups cross at month 18. Why is this a concern?
ACrossing curves indicate a data entry error in the recorded event times
BCrossing curves suggest the proportional hazards assumption is violated — the hazard ratio appears to change direction over time
CCrossing curves indicate the treatment is definitively harmful after month 18 and the trial should be stopped
DCrossing curves mean the log-rank test statistic cannot be computed
If the proportional hazards assumption holds, survival curves for two groups should diverge (or stay proportionally apart) without crossing — one group should consistently have lower hazard than the other. When KM curves cross, it means that one group has higher hazard early and lower hazard late (or vice versa), implying the hazard ratio changes sign over time. This directly violates the proportional hazards assumption. The formal test uses Schoenfeld residuals, but visually crossing KM curves are a warning sign that Cox regression may produce misleading results without modification (e.g., adding a time-interaction term).
Question 3 True / False
The proportional hazards assumption in the Cox model requires that the hazard rate for each group remains constant over time.
TTrue
FFalse
Answer: False
This is the most common misstatement of the PH assumption. The assumption is NOT that hazards are constant over time — that would be an exponential distribution assumption. The assumption is that the *ratio* of any two groups' hazards is constant over time. Both the treated and control hazards can vary freely (and typically do — hazard often increases with age or disease duration), as long as they vary proportionally. This is why Cox is called semi-parametric: the baseline hazard h₀(t) is left completely unspecified and can have any shape, but the ratio between groups is fixed at exp(β).
Question 4 True / False
The Cox model's semi-parametric nature means it can estimate adjusted hazard ratios for multiple covariates without requiring the analyst to specify the shape of the underlying survival time distribution.
TTrue
FFalse
Answer: True
This is the key practical advantage of Cox regression over parametric survival models (exponential, Weibull, log-normal). The 'semi' in semi-parametric refers to this split: the covariate effects are fully parametrized (βs estimated from partial likelihood), but the baseline hazard h₀(t) is left entirely non-parametric — it is never modeled or estimated directly. The partial likelihood cleverly conditions on who is at risk at each event time, extracting information about the βs without ever needing to specify or estimate h₀(t). This gives Cox regression the flexibility of non-parametric methods with the confounder-adjustment power of regression.
Question 5 Short Answer
Why is the Cox proportional hazards model preferred over a simple comparison of Kaplan-Meier curves when analyzing an observational study comparing treated and untreated patients?
Think about your answer, then reveal below.
Model answer: Kaplan-Meier curves describe survival patterns for groups as observed, without adjusting for any differences in patient characteristics. In an observational study, treated and untreated patients may differ systematically in age, disease severity, comorbidities, or other confounders — making a raw KM comparison misleading. The Cox model is a regression framework that can simultaneously adjust for multiple covariates, estimating the treatment hazard ratio 'holding all other measured covariates equal.' This confounder adjustment is essential for causal inference in observational data. Additionally, the Cox model can handle continuous covariates, interactions, and time-varying covariates — capabilities KM cannot provide.
The parallel to linear regression is useful: just as a t-test comparing group means is replaced by multiple regression when confounders are present, the log-rank test comparing KM curves is replaced by Cox regression. Both pairs of methods address the same limitation — unadjusted group comparisons conflate the effect of interest with confounding. Cox regression is to survival outcomes what linear regression is to continuous outcomes.