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Let $a_i$ be age in years, $g_i$ the male indicator and $m_i$ the indicator for the Messiah group. The fitted Cox proportional-hazards model iswith an unspecified common baseline hazard. Its coefficients were estimated by Cox partial likelihood, using the Breslow approximation for tied event times. There are $100$ participants, $96$ recorded completions and four censored observations.

A higher hazard function means a greater instantaneous chance of completion among those not yet completing, so it describes faster completion rather than greater mortality. Holding gender and group fixed, an extra year of age multiplies the completion hazard by $1.0066$, about a $0.66\%$ increase. The 95% confidence interval is $(0.9865,1.0270)$ and the Wald test has $p=0.5241$, so there is little evidence of an age association.

Holding age and group fixed, male participants have a hazard ratio $0.5469$ relative to female participants, about a $45.3\%$ lower completion hazard. Its 95% confidence interval is $(0.3560,0.8401)$ and $p=0.00586$. Under the fitted model this corresponds to slower completion for males. It is not a ratio of mean or median completion times.

Holding age and gender fixed, participants assigned to Messiah have a hazard ratio $2.1283$ relative to The Kingdom, with 95% confidence interval $(1.3653,3.3176)$ and $p=0.000854$. This is strong evidence of a group difference, with faster completion in the Messiah group under the fitted model. Random assignment supports interpreting a group contrast as an effect of the assigned condition, subject to the assumptions about follow-up and censoring; age and gender contrasts remain observational associations.

Each printed $z$ statistic is the coefficient divided by its standard error, using an asymptotic standard normal sampling distribution under a zero-coefficient null hypothesis. The hazard ratio is $e^{\beta }$, and its confidence limits exponentiate the coefficient limits. Proportional hazards assumes these covariate-specific hazard multipliers stay constant over follow-up, together with the specified linear age effect and independent censoring conditional on covariates.

Let $A_i(s)$ be age in years at follow-up time $s$ and let $G_i\in \{0,1\}$ indicate the genetic risk factor. An age-dependent log-linear transition intensity model isHere $e^{{\theta }_0}$ is the onset baseline hazard for a 50-year-old without the factor; $e^{{\theta }_G}$ is the genetic hazard ratio, and $e^{{\theta }_A}$ is the hazard ratio per additional year of age. A smooth function of age could replace the linear age term if a constant age hazard ratio were unsuitable.

Under the stated constant multiplicative hazard ratios, the requested extrapolation is therefore the baseline rate times both multipliers:

The untreated rates are ${\lambda }_U=0.2$ and ${\mu }_U=0.25$; treatment changes them to ${\lambda }_T=0.2\times 0.8=0.16$ and ${\mu }_T=0.25\times 2=0.5$. Their stationary distributions give symptoms fractionsConsequentlyThus the estimated long-run symptoms fraction under treatment is about 54.5% of the untreated fraction. This combines less frequent onset with faster recovery; it is not either individual hazard ratio alone.

Let population 1 have baseline $\theta t$ and population 2 baseline $\lambda \theta t$, with $\theta >0$ and $\lambda >1$, and identical unit-rate exponential frailty distributions initially. Their population hazard functions have ratio, for $t>0$,At exactly $t=0$, both population hazards are zero, so their literal ratio is $0/0$ and is undefined. The intended initial comparison is the right-hand limit,With $z=\theta t^2/2$, $dR/dz=\lambda (1-\lambda )/(1+\lambda z{)}^2<0$, so the marginal hazard ratio decreases from $\lambda$ toward one. The faster-hazard population loses highly frail individuals sooner; its survivors become less frail than the first population's survivors.

Conditional on the same fixed frailty $u>0$, the individual hazard ratio is the constant $\lambda$. Marginalizing over frailty therefore destroys proportional hazards, even though they hold at the individual level. A population-level Cox proportional-hazards model that ignores frailty would impose an incorrect constant hazard ratio here. Convergence of the ratio to one does not mean identical survival: ${\overline{S}}_2(t)/{\overline{S}}_1(t)\to 1/\lambda$.

Use a time-homogeneous continuous-time multi-state model with states $H$ (healthy), $I$ (ill), and $D$ (dead), where $D$ is absorbing. For risk-factor indicator $z\in \{0,1\}$, let the transition intensities beHere ${\lambda }_0$ is the infection rate without the risk factor, $e^{\beta }$ is the infection hazard ratio, $\gamma$ is the recovery rate, and $\delta$ is the disease-death rate. The assumption that the risk factor affects only acquisition makes $\gamma$ and $\delta$ common to both groups. The infinitesimal generator is

Writing the two functions in the question as survivor functions, $S_2(t)=S_1(t{)}^{\lambda }$. Since $H_k(t)=-\log S_k(t)$,Differentiating at times where the hazard functions exist gives $h_2(t)=\lambda h_1(t)$. Their hazard ratio is therefore the constant $\lambda$, so they form a proportional hazards family.

A proportional hazards family has hazard functions related bywhere the hazard ratio $c(z)$ is positive and independent of time. Equivalently, its cumulative hazard functions satisfy $H(t\mid z)=c(z)H_0(t)$ and its survivor functions satisfy $S(t\mid z)=S_0(t{)}^{c(z)}$.