A Cox model of mortality after cardiac surgery includes age, sex, and ejection fraction. The hazard ratio for female sex is 0.75. What does this mean?
AFemales have a 75% higher hazard of death compared to males
BFemales have a 25% lower instantaneous rate of death at any time point compared to males with the same age and ejection fraction
C75% of females survive the surgery
DThe median survival for females is 0.75 times the median survival for males
A hazard ratio of 0.75 means the hazard (instantaneous event rate) for females is 75% of the hazard for males, adjusted for age and ejection fraction — equivalently, a 25% lower hazard. Under the proportional hazards assumption, this ratio holds at every time point. It does not directly translate to a difference in median survival or a survival probability without additional information about the baseline hazard.
Question 2 Multiple Choice
The Cox model's semi-parametric nature means it does not require specifying the baseline hazard function h_0(t). Why is this considered an advantage over fully parametric survival models?
AIt makes the model faster to compute
BIt avoids the need to assume a particular distributional form for event times (exponential, Weibull, etc.), making the model robust to misspecification of the time dependence
CIt eliminates the need for the proportional hazards assumption
DIt allows the model to handle continuous outcomes, not just time-to-event data
Parametric models (exponential, Weibull, log-normal) require specifying the mathematical form of the baseline hazard — if this specification is wrong, the covariate effect estimates may be biased. The Cox model leaves h_0(t) completely unspecified and estimates covariate effects using only the relative ordering of event times (partial likelihood). This semi-parametric approach sacrifices some efficiency when the parametric form is correct but gains robustness when it is not — a favorable tradeoff in most applied settings where the true hazard shape is unknown.
Question 3 True / False
Schoenfeld residuals plotted against time show a clear upward trend for a covariate in a Cox model. This indicates the proportional hazards assumption holds for that covariate.
TTrue
FFalse
Answer: False
Schoenfeld residuals that trend with time indicate a violation of the proportional hazards assumption — the effect of that covariate is changing over time rather than remaining constant. Under proportional hazards, Schoenfeld residuals should show no systematic pattern over time (random scatter around zero). A trend suggests the hazard ratio increases or decreases as time progresses, and the model may need a time-covariate interaction, stratification by that variable, or a time-varying coefficient approach.
Question 4 Short Answer
Explain how partial likelihood estimation allows the Cox model to estimate covariate effects without specifying the baseline hazard.
Think about your answer, then reveal below.
Model answer: At each event time, partial likelihood considers only which subject experienced the event relative to all subjects still at risk. The probability that the specific subject who died is the one who died, given that one death occurred, depends only on the relative hazard values exp(Xβ) across subjects in the risk set — the baseline hazard h_0(t) cancels because it multiplies both the numerator and denominator equally. The partial likelihood is the product of these conditional probabilities across all event times, and maximizing it yields estimates of β without ever estimating h_0(t).
Cox's key insight was that the ordering of events contains sufficient information to estimate covariate effects. At each event time, the partial likelihood asks: given the risk set (everyone who could have had an event), what is the probability that the subject with covariates X_i is the one who did? This depends on exp(X_i β) / sum of exp(X_j β) over all j at risk — which is structurally identical to a conditional logistic regression. The baseline hazard determines when events tend to happen but not who they happen to, so the covariate effects are identifiable without it.