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In a proportional hazards model $h(t\mid z)=h_0(t)e^{{\beta }^{T}z}$, the baseline survival function is $S_0(t)=\exp \{-{\int }_0^t{h}_0(u)du\}$, corresponding to the reference covariate vector $z=0$. Integrating the hazard function gives $S(t\mid z)=S_0(t{)}^{e^{{\beta }^{T}z}}$. Estimated hazard ratios alone therefore do not determine absolute survival probabilities: the baseline hazard or baseline survival function must also be estimated.

Assume independent individual event times with positive time-constant hazard multipliers and a common baseline hazard whose cumulative hazard tends to infinity. Transforming each event time by that cumulative hazard then gives independent exponential event times with corresponding rates, so all event orders exist. Their complete-order probabilities select each next label proportionally to its remaining multiplier. Marginalizing over the position of a deleted label leaves the order law of the other exponential times, hence the same sequential formula without that label. This proves the deletion identity used when unobserved ranks are summed out. It does not supply a likelihood function for censoring times; ignoring censoring in observed survival analysis still needs independent censoring.

A cure model mixes an individual class with identically zero hazard function and a susceptible class with survivor function $S_{\ast }$. For $0\le \pi <1$, its representation as a proportional frailty model with $\mathbb{E}U=1$ is $U=0$ with probability $\pi$ and $U=1/(1-\pi )$ otherwise, with baseline hazard $h_0=(1-\pi )h_{\ast }$. The frailty law is a mixture of two Dirac measures, rather than an ordinary density. Its Laplace transform evaluated at $H_0$ gives the displayed survivor function. The limiting survival equals $\pi$ when $S_{\ast }(t)\to 0$; otherwise the plateau also includes susceptible individuals who never experience the event.

Separate first-event episodes from all later episodes with a stratum indicator. A Stratified Cox model permits one baseline hazard for first events and another for recurrent events while retaining a common treatment coefficient. Later episodes can share their baseline; stratifying each recurrence number separately is a stronger model than the first-versus-later distinction alone.

A nuisance parameter is a statistical parameter required to specify the distribution of observations but not itself the target of inference. A profile likelihood maximizes over it; Bayesian model evidence integrates over it using a proper prior distribution. These operations differ and need not give the same inference. A baseline hazard in a Cox proportional-hazards model is an infinite-dimensional example.

In a frailty model, an individual's unobserved susceptibility to an event is represented by a nonnegative frailty random variable multiplying that individual's hazard function. Conditional on its value, subjects with the same measured covariates can therefore have different event rates. The frailty is a persistent latent characteristic of the individual, not newly sampled noise at each time.

Here the conditional hazard function is $u{e}^{\beta z}{h}_0(t)$, with baseline hazard $h_0$. A multiplier $u>1$ raises susceptibility and $u<1$ lowers it. Normalization $\mathbb{E}U=1$ fixes the frailty scale: without it, multiplying every frailty by a constant and dividing the baseline hazard by that constant would give the same individual hazards. The variance of $U$ measures unobserved heterogeneity. An atom at $U=0$, if allowed, would give a permanently zero conditional hazard.

The hazard ratio between groups is $e^{\beta }$ for individuals compared at the same frailty. It need not equal the population hazard ratio, because survival selection changes the frailty distribution among those still alive or event-free.

A frailty model represents unobserved individual heterogeneity by a random effect in the hazard function. In a proportional frailty model, an individual with frailty random variable $U=u$ has hazard $u{h}_0(t)$. The normalization $\mathbb{E}U=1$ identifies the scale of the baseline hazard: without it, multiplying all frailties by a constant and dividing $h_0$ by that constant would leave all individual hazards unchanged.

Let $H_0(t)={\int }_0^t{h}_0(v)dv$. Conditional survival follows by integrating the hazard function:Taking the expectation over the frailty law yieldsThis is the Laplace transform of the frailty density evaluated at the cumulative hazard. More generally, the same calculation integrates against a probability measure, allowing atoms. It averages conditional survival rather than substituting $\mathbb{E}U$ into its exponent. Indeed, the frailty distribution among survivors is proportional to $e^{-u{H}_0(t)}g(u)$, giving population hazardSurvival progressively selects smaller frailties, so the population hazard need not be the baseline or even retain the individual's proportional structure.

For the cure model, first take $0\le \pi <1$. The necessary mean-one frailty law and baseline hazard areThen $\mathbb{E}U=\pi \cdot 0+(1-\pi )/(1-\pi )=1$. A cured person's conditional hazard is zero, while a susceptible person's conditional hazard is $h_0/(1-\pi )=h_{\ast }$. Taking $U$ to be zero or one while leaving $h_0=h_{\ast }$ would instead have mean frailty $1-\pi$ and violate the required normalization.

The frailty law is the measurewhere the $\delta$ terms are Dirac measures. This is not an ordinary Lebesgue density; the cure construction requires permitting a discrete frailty law. Its Laplace transform isWriting $H_{\ast }(t)={\int }_0^t{h}_{\ast }(v)dv$, we have $H_0=(1-\pi )H_{\ast }$, and thereforeIf $H_{\ast }(t)\to \infty$, the susceptible class eventually experiences the event and $S(t)\to \pi$. If $H_{\ast }(\infty )<\infty$, the limiting survival is larger than $\pi$ because some susceptible subjects also never experience the event. At $\pi =1$, everyone is cured: set $h_0\equiv 0$ and, for example, $U\equiv 1$ to retain mean one, giving $S\equiv 1$ without using the singular formula $1/(1-\pi )$.

Use the right-continuous Breslow estimator, so the increase from the value at $x_{m-1}$ to the value at $x_m$ means the interval $(x_{m-1},x_m]$. There is no event between these consecutive exit times, and at $x_m$ the risk set contains only subject $m$. Henceand the fitted individual cumulative hazard increases byIt is the subject's integrated hazard, rather than the reference baseline increment, that equals the event indicator.

For the final unheaded request, a fitted martingale residual at a subject's exit isThe revised observation leaves the last subject at risk until a later event time, still after every other subject has exited. Every earlier risk set is exactly the same as before, and the new final event again contributes the constant factor one. Thus part a gives unchanged coefficients, and every earlier Breslow estimator increment is unchanged. No additional events occur during the extended interval because all other subjects have already left observation.

For $i<m$, the event indicator and fitted cumulative hazard at $x_i$ remain unchanged, hence so does ${\hat{M}}_i$. For the terminal subject, let $C$ denote its old fitted integrated hazard at its censored exit. That old residual is $-C$. Part b shows that the new final event adds one to its fitted integrated hazard, giving new residualTherefore all fitted martingale residuals remain unchanged. Their invariance does not assert that the fitted survivor function at later times or the terminal event record is unchanged: the new late baseline hazard jump is real. This is the terminal-observation invariance of Cox martingale residuals, which relies on a unique final subject and unchanged covariate history.

Use the usual semiparametric Cox proportional-hazards model with observed covariates $z_i$:The covariates are implicit in the question's regression fit, even though the displayed data notation only lists exit times and event indicators. Assume entry at a common origin, time-constant covariates, distinct observed times and independent censoring. Write $R_i=\{j:x_j\ge x_i\}$ and $w_j(\beta )=e^{{\beta }^T{z}_j}$. Conditional on a single event at $x_i$ in its risk set, the chance that individual $i$ supplies it isThe baseline hazard cancels. Consequently estimate $\beta$ by maximizing Cox partial likelihoodFor example, the score and information of Cox partial likelihood follow by differentiation. Define $S_i^{(0)}={\sum }_{j\in R_i}{w}_j$, $S_i^{(1)}={\sum }_{j\in R_i}{w}_j{z}_j$ and $S_i^{(2)}={\sum }_{j\in R_i}{w}_j{z}_j{z}_j^T$. ThenWhen a finite identifiable maximum exists, iterate $\beta \leftarrow \beta +I(\beta {)}^{-1}U(\beta )$ until convergence. The observed Fisher information also gives the usual asymptotic coefficient uncertainty. Censored subjects enter the relevant risk sets, even though they supply no numerator event factor.

After fitting $\hat{\beta }$, estimate the cumulative hazard of the reference subject by the Breslow estimatorIts jump at an event is the observed event count divided by the total fitted hazard multiplier at risk. The estimate is an integrated baseline hazard, with hazard represented by a measure placing these jumps at observed event times. If a smooth ordinary hazard function is wanted, smooth these increments with a chosen kernel and bandwidth; differentiating the step function as an ordinary function would incorrectly give zero away from the jumps. The fitted individual cumulative hazard is ${\hat{H}}_i(t)=e^{{\hat{\beta }}^T{z}_i}{\hat{H}}_0(t)$ over that individual's follow-up. The invariances below concern this Cox partial likelihood fit and this Breslow estimator; they need not hold for a fully parametric proportional-hazards fit or an arbitrarily smoothed hazard estimator.

Converting size to a regression factor allows two unrestricted contrasts rather than imposing a linear trend on numerical size codes. The command relevel with a numeric second argument selects the second existing factor level as its reference level. For sorted codes $0,1,2$ this is medium, coded $1$. However, the displayed coefficient labels size1 and size3 instead correspond to codes $1,2,3$ with code $2$ as reference. The printed coding and coefficient labels are inconsistent. The actual factor levels and labels must be checked. Under the intended medium-size reference, the two reported contrasts compare small and large businesses with medium ones; with literal $0,1,2$ coding their printed names would be size0 and size2.

Using B's delayed-entry survival object, the Cox proportional-hazards model isHere $h_0$ is the unspecified baseline hazard for a medium-size village business and $c$ is the Cambridge indicator. The fitted hazard ratios are conditional on the other covariates and constant over business age under the proportional hazards assumption. The function fits regression coefficients through partial likelihood; its survival object must actually be B's version, rather than A's earlier object with the same name.

The location coefficient $-0.803$, with standard error $0.344$, gives $z=-2.335$ and a two-sided Wald test p-value about $0.02$. Its reported hazard ratio is $0.448$, with 95% confidence interval $(0.228,0.879)$. At the same age and size, a Cambridge business therefore has an estimated closure hazard about 55.2% lower than a village business. The reciprocal ratio is $2.23$. A hazard ratio is not a survival-probability ratio or a proportional increase in lifetime.

The intended small-versus-medium contrast is $-0.322$, with standard error $0.482$, $z=-0.669$ and p-value $0.50$. Its hazard ratio is $0.724$, with 95% confidence interval $(0.282,1.863)$. The intended large-versus-medium contrast is $-0.224$, with standard error $0.375$, $z=-0.598$ and p-value $0.55$; its hazard ratio is $0.799$, with interval $(0.383,1.666)$. Neither size contrast provides convincing evidence of an effect. The wide intervals also show why these results do not establish equality of size-specific hazards.

The sample has $60$ businesses. The likelihood-ratio test, Wald test and score test jointly test all three coefficients being zero; their statistics are $5.45$, $5.76$ and $6.01$ on three degrees of freedom, with p-values $0.142$, $0.124$ and $0.111$. None rejects at 5%. This is compatible with the significant one-coefficient location test: testing a specified one-dimensional contrast and testing three coefficients together are different questions. The location result offers some adjusted evidence, but the joint output does not demonstrate a strong overall improvement, and interpretation should acknowledge the uncertainty and the number of comparisons.

The reported $R^2=0.087$ is a Cox–Snell likelihood pseudo-R-squared, here $1-e^{-5.45/60}\approx 0.087$. It measures likelihood improvement rather than a literal fraction of variance in lifetimes explained. Its attainable maximum is reported as $0.979$. It does not validate the proportional hazards assumption or establish useful predictive performance. Adjusted survival predictions would require estimating the baseline survival function as well: $S(t\mid z)=S_0(t{)}^{e^{{\beta }^{T}z}}$. The coefficient summary alone does not supply adjusted five- or ten-year survival probabilities.

Before reporting results, check the size coding discrepancy, the precise sampling dates and the survival records. The stated January 2000–December 2005 span is six calendar years, despite being called five years. Check nonnegative ages, entry no later than exit, correctly recorded closure indicators and the same opening-based clock throughout. Establish whether inclusion and censoring are plausibly independent of lifetime given the covariates, and whether different opening cohorts or calendar environments can reasonably share the fitted lifetime model. Simply including delayed entry does not cure informative selection, omitted confounders or dependence between businesses.

Assess proportional hazards using Schoenfeld residuals against time and a proportional hazards assumption test, supplemented by groupwise log-minus-log survival plots. Investigate influential observations with coefficient-deletion diagnostics and deviance residuals, and inspect overall fit using Cox–Snell residuals. Consider a city-by-size interaction or time-varying effects if supported by the available events, rather than relying on the displayed three main effects. Finally, report the uncertainty and declining late risk sets—only six remain at the final listed event—and distinguish a conditional association from a causal explanation or a prediction for all future businesses.

A proportional hazards family has a common baseline hazard $h_0$ and positive constants $a_z$ such that $h_z(t)=a_z{h}_0(t)$. Thus its hazard ratios are constant in time, and its survivor functions satisfy $S_z(t)=S_0(t{)}^{a_z}$. An accelerated life family instead has positive constants $c_z$ such that $T_z$ has the distribution of $c_z{T}_0$. Equivalently, $S_z(t)=S_0(t/c_z)$: all time quantiles are multiplied by the same factor. These two properties are generally different.

Integrating the printed Weibull distribution density, or differentiating the following expression and checking its initial value, givesHere $\lambda$ is an inverse-time parameter: the coefficient of $t^p$ in the cumulative hazard is ${\lambda }^p$, not $\lambda$. This matters when comparing different parameterizations of a Weibull distribution.

With common shape $p$, the hazard ratio isIt does not depend on $t$, proving membership in one proportional hazards family. For the other property,Consequently they also belong to one accelerated life family. A larger inverse-time parameter gives shorter survival times and larger hazards, consistently in both descriptions. This is the common-shape case of Weibull accelerated-life and proportional-hazards families.

Write the Cox proportional-hazards model as $h_i(t)=h_0(t)u_i$, where $u_i=\exp ({\beta }^T{z}_i)$ is the hazard multiplier and $h_0$ is the unspecified baseline hazard. At an untied event time $t_j$, let $R_j$ be the risk set immediately before the event and $i_j$ its failing individual. With independent censoring conditional on the covariates, the conditional probability that individual $i$ supplies the next event, given an event at that time and the risk set, is obtained by dividing its instantaneous event rate by the aggregate rate:More precisely this is the limit of the conditional probability for one event in a short interval; simultaneous events have smaller-order probability in the continuous model. Multiplication of these successive conditional contributions gives the Cox partial likelihoodCensored individuals remain in each risk set until their censoring time but supply no numerator. Conditioning removes the baseline hazard; this is a partial likelihood for $\beta$, rather than a complete likelihood for event and censoring times.

Ties require an observation model or an approximation. Suppose a recorded tie hides the order of $d$ continuous events, with no intervening entry or censoring. Put $D$ for the tied event set and $S={\sum }_{i\in R}{u}_i$. An exact marginal contribution for the coarsened ranks is the sum of the ordinary rank contributions over every possible order:For example, if $A,B$ fail together while $C$ remains at risk, the two possible orders giveThe risk set is depleted after the first of the two events. This expression concerns the missing event order; it is not the full likelihood of an interval in which the events occurred.

Convenient alternatives are the Breslow approximation for tied event times and the Efron approximation for tied event times:The first keeps the initial denominator at every event; the second removes an average share of the tied-event multipliers successively. In the example their denominators are $S^2$ and $S[S-(u_A+u_B)/2]$. These conventional partial-likelihood factors omit multiplicities independent of $\beta$, which do not affect its estimate; they should not be mistaken for normalized probabilities of the unordered event set.

For genuinely discrete event times there is another exact construction. If individual event odds are proportional to $u_i$, conditioning on exactly $d$ events gives the exact tied-set conditional likelihoodFor the same three individuals this is $u_A{u}_B/(u_A{u}_B+u_A{u}_C+u_B{u}_C)$. It is not generally equal to the continuous-time rank sum: the assumptions differ.

Large tie groups make naive summation over $d!$ orders or $\left(\genfrac{}{}{0ex}{}{|R|}{d}\right)$ subsets impractical. Recursion helps with the subset denominator, and Breslow or Efron avoids exhaustive enumeration. However, extensive ties can also indicate that time is measured too coarsely for an exact continuous-time ordering to be a useful target.

For example, if patients are assessed only at annual visits, use one row per patient-year still at risk and model the probability $p_{ij}$ of an event in that interval. A grouped proportional-hazards model follows directly by integrating the hazard over the interval, with covariates constant there:Fit the resulting person-period likelihood, with Bernoulli distributions for the interval event indicators and interval-specific intercepts. Many events in the same year are then ordinary observations rather than a combinatorial ordering problem. This requires a compatible interval-observation scheme and appropriate independent censoring; it avoids pretending that unobserved within-year event times are known.

Let $R(t^{\ast })$ be the risk set immediately before $t^{\ast }$, conditional on the preceding history. Only its members can experience the next event. In a short interval $[t^{\ast },t^{\ast }+dt)$, the conditional probability that subject $j$ fails is $h_0(t^{\ast })e^{\beta z_j}dt+o(dt)$, whereas the probability of one event from the risk set is $h_0(t^{\ast }){\sum }_{i\in R(t^{\ast })}{e}^{\beta z_i}dt+o(dt)$. Dividing and taking the small-interval limit givesIt is zero outside the risk set. The common baseline hazard cancels. This is the conditional event-label probability underlying the Cox partial likelihood. Conditioning at an exact continuous event time is understood by this limiting conditional-intensity argument, rather than as division by the zero unconditional probability of an event at a fixed time. We use the standard simple event process with no simultaneous events and independent censoring.

A frailty model uses an unobserved nonnegative random hazard multiplier to represent variation in individual risk. In a proportional frailty model, conditional on $U=u$ the cumulative hazard function is $u{H}_0(t)$, giving survivor $e^{-u{H}_0(t)}$. Averaging over the frailty random variable by the law of total probability yieldsthe Laplace transform of the frailty distribution evaluated at the baseline integrated hazard. This also applies to atoms by integration against the probability measure; the printed Dirac expression is not an ordinary continuous density.

In the given two-atom case, the Dirac delta function notation assigns probabilities $1/4$ and $3/4$ to $U=4/7$ and $U=8/7$. These have mean one, fixing the frailty scale relative to the baseline hazard. With $H_0(t)=7t/4$, the two conditional integrated hazards are $t$ and $2t$. ThereforeThe frailty distribution among survivors shifts toward the smaller multiplier $4/7$ over time, reproducing the declining population hazard and limit one without changing either individual's constant conditional hazard.

Add a stratum indicator that is “First” for the first episode and “Later” for every subsequent episode, including the final censored episode:

A Stratified Cox model then uses distinct baseline hazards $h_{0,\mathrm{first}}$ and $h_{0,\mathrm{later}}$, with a common treatment coefficient $\beta$ unless an interaction is explicitly desired. Form separate risk sets within the two strata and multiply their partial likelihoods. There is no need to force every later event number to have its own baseline when the stated aim is only first versus subsequent headaches. This is a first-event versus recurrent-event baseline stratification.

At each event time, a Schoenfeld residual is the observed event covariate minus its risk-set weighted mean:Under a correct constant-coefficient model, these event-time residuals have no systematic time trend. Plot scaled Schoenfeld residuals against time or a prespecified transformation such as log time, with smooth curves and uncertainty bands. Depending on the plotting convention, adding the fitted coefficient produces an estimate of ${\beta }_k(t)$; a horizontal curve then supports a constant coefficient, while a clear trend suggests violation. Examine sparse late follow-up with its wider uncertainty.

Formal tests can use residual-time association or a proportional-hazards time interaction extension ${\beta }_k(t)={\beta }_k+{\gamma }_{k}g(t)$, testing ${\gamma }_k=0$ jointly for a multi-parameter factor and globally across covariates. Evaluate $z_{k}g(t)$ at each current event/risk-set time: multiplying a baseline variable by that person's eventual observed follow-up time would use outcome information and would not be the intended time-dependent model. A nonsignificant test with limited information does not establish proportionality.

For categorical groups, approximately parallel empirical log-minus-log survival curves provide another check, because proportional hazards imply $\log [-\log S(t\mid z)]=z^{\mathsf{T}}\beta +\log H_0(t)$. Use observed group curves, not fitted proportional-hazards curves that are parallel by construction, and recognize possible confounding by other variables. If a violation is real, consider stratifying on a categorical nuisance variable, adding an explicitly time-varying effect, or choosing another survival-model family. Stratification allows separate baseline hazards but sacrifices a single estimated hazard ratio for that stratification variable. Do not force a constant coefficient merely to preserve the original model label; describe the time pattern or limit the interpretation of the constant summary.

Begin by defining the scientific target: prediction for a new patient, or estimation of adjusted covariate effects. Check event definitions, follow-up origins, right-censoring codes and any delayed entry. The analysis assumes independent patients and censoring that is noninformative conditional on modeled covariates. For fixed explanatory variables $z_i$, the Cox proportional-hazards model iswith an unrestricted baseline hazard and coefficients constant over time. Report $\exp ({\beta }_k)$ as the conditional hazard ratio for a one-unit change in the coded variable, not as a risk ratio or automatically a causal effect.

For untied event times, fit the coefficients by maximizing the Cox partial likelihoodThe baseline cancels within each event risk set. Use an appropriate tied-event method, such as an Efron approximation or an exact method for a genuinely discrete event scale; the Breslow approximation for tied event times is another explicit approximation. Numerical score/information methods fit the model, and inverse information gives conventional covariance estimates for independent subjects. Report coefficient estimates, hazard ratios, uncertainty and the coding that makes their interpretation meaningful. After fitting, the Breslow estimator gives ${\hat{H}}_0(t)={\sum }_{t_j\le t}{d}_j/{\sum }_{i\in R(t_j)}{e}^{z_i^{\mathsf{T}}\hat{\beta }}$, from which $\hat{S}(t\mid z)=\exp [-{\hat{H}}_0(t)e^{z^{\mathsf{T}}\hat{\beta }}]$ provides predicted survival.

Adequacy is broader than significance of coefficients. Inspect influential observations, deviance residuals and data errors. Cox–Snell residuals ${\hat{H}}_0(x_i)e^{z_i^{\mathsf{T}}\hat{\beta }}$ should have approximately unit-exponential survival under a well-fitting model, retaining the original censoring indicators; a cumulative-hazard plot of these residuals should be near the 45-degree line. This is an overall diagnostic, not a substitute for the functional-form and proportionality checks below. For prediction, assess held-out discrimination and survival prediction calibration at prespecified time horizons with methods accounting for censoring. A high concordance index alone does not establish good calibration or correct hazard structure.

Use bootstrap or held-out cross-validation that repeats the complete modeling procedure, including imputation, variable selection and tuning. Validation of only the final fitted coefficients understates overfitting from earlier decisions. Report any substantive lack of fit rather than presenting a single time-independent hazard ratio when the data do not support that representation.

Conditional on an event and the immediately preceding history, probabilities are proportional to the three instantaneous hazards. Put $w=e^{{\beta }_0}$. The two zero-covariate subjects each have weight one, and the one-covariate subject has weight $w$. The common baseline hazard cancels. ThusEach individual with zero covariate has probability $1/(2+w)$; the first boxed probability is their combined probability. Conditioning on an event at a specified continuous time can be understood by the limiting conditional event probabilities in a short interval.

The hazard-weighted covariate mean is $w/(2+w)$, so the Schoenfeld function at the true coefficient isMultiplying by the two conditional probabilities givesThis verifies the score-centering property directly for this risk set.

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.

$Y_{j,k}$ is the number in group $k$ still observed and event-free immediately before event time $a_j$: its group-specific risk set size. Set $Y_j=Y_{j,0}+Y_{j,1}$. Under the common hazard function and conditional on exactly one event at that time, each at-risk individual has the same infinitesimal event probability. ThereforeThe indicator $g_{\pi (j)}$ records whether the event came from group 1, while $p(j,1)$ is its conditional expectation under equal hazards. Thus $z_j$ is the observed-minus-expected group-1 event count at that time. Under the null these are martingale differences, with conditional varianceThe log-rank statistic $z={\sum }_j{z}_j$ has null mean zero. Its predictable variance estimate is $V={\sum }_j{p}_j(1-p_j)$, so the equal-hazards test usesUse a two-sided large-sample log-rank test if no direction was prespecified; $V>0$ and adequate information are needed. The variance is the no-ties form. A positive $z$ indicates more group-1 events than expected. Independent censoring conditional on the model is required for these at-risk event intensities to describe the data.

Under the proportional hazards model, the total instantaneous group-$k$ rate is $Y_{j,k}{e}^{\beta k}{h}_0(a_j)$. Conditioning on the next event cancels the baseline hazard, givingKeep $z_j$ defined using its null probability $p(j,1)$, as in the question. Its conditional expectation under the alternative isIt is zero at $\beta =0$, as required, and has the sign of $\beta$ when both groups remain at risk. Replacing $p(j,1)$ inside the definition of $z_j$ with the alternative probability would give a different residual, whose mean would instead be zero under that alternative.