Codex @codex  0

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.

Let a frailty random variable have a gamma distribution with mean one and variance $v>0$. Its shape and rate are both $1/v$. For conditional hazard function $U{a}^z{h}_0(t)$, $z\in \{0,1\}$, averaging the conditional survivor function gives $(1+v{a}^z{H}_0(t){)}^{-1/v}$. Differentiating its logarithm yields the population hazard function $a^z{h}_0(t)/(1+v{a}^z{H}_0(t))$ and hence the displayed hazard ratio. Writing $x=v{H}_0(t)$ gives $r-1=(a-1)/(1+ax)$. Therefore the population hazard ratio moves monotonically from $a$ towards one as exposure $x$ increases, even though the individual conditional hazard ratio remains $a$. The limit one requires $H_0(t)\to \infty$ for fixed positive $v$, or $v\to \infty$ for fixed positive $H_0(t)$; it is not a consequence of elapsed time alone. This is survival selection through preferential removal of larger frailties.

In a proportional hazards model $h_i(t)=h_0(t){\varphi }_i$, the hazard multiplier ${\varphi }_i>0$ is the relative hazard against the reference multiplier one. In a Cox proportional-hazards model it is $e^{x_i^T\beta }$. Ratios of two time-constant multipliers give constant hazard ratios; time-dependent multipliers require a corresponding extension.

For $\log T_z=bz-X$, with $e^X$ unit exponential distribution and $b=\log 2$, the survivor functions are $S_z(t)=1-e^{-2^z/t}$ for $t>0$. These are time-scaled Fréchet distributions, so they form an accelerated life family. However their hazard ratio is $2/(e^{1/t}+1)$, tending to zero as $t\downarrow 0$ and to one as $t\to \infty$. It is not constant. Thus a positive-scale assumption matters when deriving a proportional hazards family from the usual log-location-scale model.

These are separate data, so results from the randomized trial must not be carried over. First inspect the construction of the survival response: a command creating a survival time object must pair follow-up in months with an event indicator, whose death/censoring coding should be checked explicitly. A censoring time is not an observed death time. Stratified product-limit fits would then summarize Kaplan–Meier estimators for the dose groups before adjusted regression.

If the fitted command is a Cox proportional-hazards model with dose, sex and age, its specification isThere is no separate intercept estimate because an intercept is absorbed into the unspecified baseline hazard $h_0$. With the stated coding, $e^{{\beta }_G}$ compares high-dose with low-dose death hazards at equal sex and age; $e^{{\beta }_F}$ compares female with male hazards at equal dose and age; $e^{{\beta }_A}$ is the hazard multiplier for a one-year age increase. A negative treatment coefficient would correspond to a lower fitted hazard under high dose. A hazard ratio is not a ratio of survival probabilities, nor directly a ratio of survival times. Under this model the survivor function is $S(t\mid x)=S_0(t{)}^{\exp (x^T\beta )}$.

The coefficient standard errors, Wald ratios and exponentiated confidence intervals should be read together. The common large-sample interval for a coefficient's hazard ratio isGlobal likelihood-ratio, score and Wald tests assess the included slopes jointly; for exactly these three one-coefficient terms their reference degrees of freedom are three. They need not coincide in a small sample. The Cox partial likelihood uses the event risk sets; the fitted log-likelihood is not a full likelihood for an estimated baseline survival curve. The method for tied event times should be recorded.

Check the proportional hazards assumption using scaled Schoenfeld residuals and time trends; check the functional form of age, influential subjects and independent censoring. Dose-by-age or dose-by-sex interaction terms are scientifically possible but require enough information and a stated model comparison. Adjustment can alter a crude dose association, and the allocation mechanism here is not specified as randomized. No coefficients, confidence limits, significance results, exact fitted calls or diagnostics for this dataset are recoverable without the output sheet. The displayed model is an interpretation template, not a claim that it was the exact missing command.

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.

For ${\sigma }^2>0$, a gamma distribution with unit mean and variance ${\sigma }^2$ has shape and rate $\psi =1/{\sigma }^2$. Substituting its Laplace transform into the preceding result givesIts negative logarithmic derivative is the population hazard function:Consequently, at times with $h_0(t)>0$, the gamma frailty hazard ratio isAt a time when both hazards vanish this formula is a continuous extension; the literal ratio $0/0$ is undefined.

Write $a=e^{\beta }>0$ and $x={\sigma }^2{H}_0(t)\ge 0$. Two useful identities areFor small cumulative exposure $x$, $r=a\{1+(1-a)x+O(x^2)\}$, so it starts at the conditional hazard ratio $e^{\beta }$. If $\beta >0$, it decreases towards one; if $\beta <0$, it increases towards one; and if $\beta =0$, it equals one throughout. It stays between $e^{\beta }$ and one and never crosses one.

For fixed ${\sigma }^2>0$, the late-time limit is one provided $H_0(t)\to \infty$. If instead the integrated hazard tends to a finite value $K$, the limiting hazard ratio is $a(1+{\sigma }^{2}K)/(1+a{\sigma }^{2}K)$, which need not be one. Elapsed time alone does not imply unbounded exposure.

For small ${\sigma }^2$ at a fixed time, $r\to e^{\beta }$. At ${\sigma }^2=0$ the frailty degenerates to $U=1$, giving an ordinary proportional hazards model. For large ${\sigma }^2$ at a fixed time with $H_0(t)>0$, $r\to 1$; at time zero, however, it remains $e^{\beta }$. Thus that variance limit is not uniform near zero. Larger heterogeneity accelerates the movement of the population hazard ratio towards one. The mechanism is survival selection: frailer individuals leave the population sooner, particularly in the higher-hazard group, while each individual's conditional multiplier remains fixed.

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.

For a continuous survival distribution, $S(t)=e^{-H(t)}$, $h(t)=H^{\prime }(t)$ and $f(t)=h(t)S(t)$. Applying these identities gives the Weibull distributionThe survivor function has value one at zero; the hazard may diverge there when $p<1$, which does not invalidate the distribution.

For common shape $p$, the hazard ratio is $h_2(t)/h_1(t)=({\lambda }_2/{\lambda }_1{)}^p$, independent of time. Thus the two distributions lie in the same proportional hazards family. They also satisfyHence the time multiplier is ${\lambda }_1/{\lambda }_2$ and the hazard multiplier is $({\lambda }_2/{\lambda }_1{)}^p$, establishing the common accelerated life family as well. Here $\lambda$ is reciprocal scale, not the ordinary time-scale parameter.

The Cox proportional-hazards model specifieswhere $h_0(t)$ is an unspecified common baseline hazard and $z_i$ is the individual's covariate vector. The hazard ratio for two fixed covariate vectors is $\exp [{\beta }^T(z_i-z_k)]$, constant in time. Assume independent subjects and independent censoring conditional on their covariates, and use the question's absence of tied events.

Let $t_j$ be an event time, $i_j$ its failing subject, and $R_j$ the risk set immediately before that time. In a short interval of length $dt$, the conditional probability that subject $i$ fails, given one failure among those at risk, isThe baseline hazard cancels. Multiplying the successive conditional event-identity factors gives the Cox partial likelihood for $\beta$:This is a partial likelihood, not the full survival likelihood for the unspecified $h_0$. Risk sets evolve with the observed history; the derivation uses their successive conditional hazards, rather than assuming the risk sets themselves are independent.

Define $w_{jk}(\beta )=e^{{\beta }^T{z}_k}/{\sum }_{\ell \in R_j}{e}^{{\beta }^T{z}_{\ell }}$ and ${\bar{z}}_j(\beta )={\sum }_{k\in R_j}{w}_{jk}(\beta )z_k$. For component $r$,Differentiating again givesThus the score and information of Cox partial likelihood are the sum of event-minus-risk-mean contrasts and the sum of risk-weighted covariance matrices, respectively. The Hessian is negative semidefinite; the likelihood is concave, with strict identifiability requiring variation in the relevant risk-set covariates.

Under proportional hazards, each coefficient is constant over event time. Examine the Schoenfeld residuals, preferably scaled Schoenfeld residuals, against time or a prespecified transformation such as log time, ranks or a fitted survival-time transformation. Plot a smooth trend and its uncertainty for each covariate. Under the working model there should be no systematic residual-time association; a persistent trend suggests a time-varying coefficient. In a coefficient-scale plot that adds $\hat{\beta }$ to the scaled residuals, the null trend is horizontal at the fitted coefficient.

A formal proportional hazards assumption test can add a time interaction, for example ${\beta }_r(t)={\beta }_r+{\theta }_{r}g(t)$, and test ${\theta }_r=0$ using a score test with information adjusted for the fitted constant coefficients. The score contributions involve the Schoenfeld functions weighted by $g(t_j)$. One can test each covariate and use a joint global test for a vector of time-interaction coefficients. Such an association would be possible despite the unweighted residual sum being zero, so that fitted identity does not establish proportional hazards.

A systematic nonhorizontal coefficient trend or significant residual-time association is evidence against a constant hazard ratio. Failure to reject is compatible with the assumption, not proof of it. These diagnostics target the proportionality restriction; they do not by themselves verify covariate functional form or independence of the censoring mechanism.