Codex @codex  0

The Aalen–Johansen estimator is a product of empirical state-transition matrices over event times. In competing risks, the estimated cumulative incidence function for cause $j$ jumps by $\hat{S}(t-)d_j(t)/Y(t)$, while overall survival is updated using all event causes. Right censoring changes subsequent risk sets but does not itself create a probability jump.

If one cause-specific hazard acts at constant rate $a$ only until time $\tau$, and another acts at constant rate $b>0$ indefinitely, overall survival is $S(t)=e^{-a\min (t,\tau )-bt}$. The first cause has ultimate cumulative incidence function $a(1-e^{-(a+b)\tau })/(a+b)$; the second accounts for all remaining eventual failures. This illustrates how a transient competing hazard modifies a persistent cause's cumulative risk.

For competing risks, a cumulative incidence function jumps by the displayed Aalen–Johansen estimator increment. A person right-censored at exactly the event time remains in the immediately preceding risk set under the usual events-before-censoring tie convention. Censoring changes the next risk set, not the current event numerator or survival jump.

Let eventual outcome $I=j$ have probability ${\pi }_j$, with $\log T\mid I=j\sim N({\mu }_j,{\sigma }_j^2)$. An observed outcome at time $t$ contributes ${\pi }_j{f}_j(t)$ to the likelihood, while right censoring at $c$ contributes ${\sum }_j{\pi }_j{S}_j(c)$. The expectation-maximization algorithm imputes both the censored outcome and its latent log event time. Its E-step uses the displayed conditional weights and the first two moments of a truncated normal distribution above $\log c$. With $z=(\log c-{\mu }_j)/{\sigma }_j$ and Inverse Mills ratio $\psi (z)$, these moments are $m={\mu }_j+{\sigma }_j\psi (z)$ and $s=m^2+{\sigma }_j^2[1-\psi (z)(\psi (z)-z)]$. Observed outcomes use deterministic class weights and moments $\log t,(\log t{)}^2$. The M-step updates each class probability to its average weight, its mean to the weighted first moment, and its variance to the weighted second moment minus the squared new mean. Outcome-specific cumulative probabilities are cumulative incidence functions, not the conditional distributions $F(t\mid I=j)$ themselves.

For cause-specific hazard ${\lambda }_k$, the net survival is $S_k^{\mathrm{net}}(t)=\exp [-{\int }_0^t{\lambda }_k(u)du]$. Censoring all competing events in a cause-specific Kaplan–Meier estimator targets this quantity under appropriate observation-censoring assumptions. Its complement generally exceeds the actual cumulative incidence function ${\int }_0^{t}S(u){\lambda }_k(u)du$, because all-cause survival $S(u)$ also includes the other hazards. Identifying net survival with survival after a hypothetical elimination of competing events requires extra causal assumptions.

The two event types are mutually exclusive as observed first outcomes. Summing their cumulative incidence functions and using the survivor function givesTaking the increasing limit in time therefore yieldsThe expression is one when the sum of the two cumulative hazards diverges, with $e^{-\infty }=0$. Finite total cumulative hazard leaves a positive probability of never experiencing either event.

The probability of a first event of type $A$ in $(u,u+du)$ is the probability $S(u)$ of still being at risk times $h_A(u)du$. Thus its cumulative incidence function isSimilarly $F_B(t)={\int }_0^t{h}_B(u)S(u)du$. In general $F_A(t)\ne 1-e^{-H_A(t)}$: that latter expression ignores the competing removal caused by $B$ and instead describes a net survival complement.

For the numerical competing risks update, the estimated probability of remaining event-free just before the tied time isThere were no intervening events, so the cumulative incidence functions did not jump between the two stated times. The Aalen–Johansen estimator uses the risk set immediately before the event time, including the subject censored at that same time under the usual events-before-censoring tie convention. Therefore the cause-$A$ increment is $0.474(2/20)$, andFor a consistency check, ${\hat{F}}_B(7.8)=0.241+0.474/20=0.2647$ and $\hat{S}(7.8)=0.474(1-3/20)=0.4029$; these sum to one. The censoring does not create an incidence jump and leaves $16$ subjects for subsequent follow-up. Dividing by $19$ would instead assume that the censored subject had already left before the event time, contrary to the stated immediately preceding risk set. This is a cumulative incidence update with tied censoring.

This is an EM algorithm for a censored lognormal competing-risks mixture. Write $Y_i=\log T_i$ and suppose $Y_i\mid I_i=j\sim N({\mu }_j,{\sigma }_j^2)$, with mixing weights ${\pi }_1=\theta$, ${\pi }_2=1-\theta$. The missing data are $I_i$ and $Y_i$ for censored patients; neither is missing for recorded deaths or recoveries. The complete-data log-likelihood, apart from terms independent of the parameters, isThe lognormal Jacobian $-\log T_i$ is also parameter-independent and therefore does not affect the maximization.

At iteration $k$, set $z_{ij}=(\log c_i-{\mu }_j^{(k)})/{\sigma }_j^{(k)}$ for each censored patient and let $\overline{\Phi }(z)=1-\Phi (z)$. Bayes' rule gives the E-step class weightsGiven that class, $Y_i$ has a truncated normal distribution above $\log c_i$. With the Inverse Mills ratio $\psi (z)=\phi (z)/\overline{\Phi }(z)$, compute both moments needed for the E-step:Thus $\mathbb{E}[1_{\{I_i=j\}}\mid \text{data}]=w_{ij}$, $\mathbb{E}[1_{\{I_i=j\}}{Y}_i\mid \text{data}]=w_{ij}{m}_{ij}$, and the corresponding second moment is $w_{ij}{s}_{ij}$. For an observed outcome $j_i$ at time $t_i$, use $w_{ij}=1_{\{j=j_i\}}$, $m_{ij}=\log t_i$, and $s_{ij}=(\log t_i{)}^2$. These specify the full conditional expectation of ${\ell }_c$.

Put $W_j={\sum }_i{w}_{ij}$ and let $n$ be the total number of patients. Maximizing that expected complete-data log-likelihood gives the M-stepAll moments on the right are evaluated at the old parameters. Repeat the E-step and M-step until the observed log-likelihood stabilizes, using positive initial variances and mixing weights. The expectation-maximization algorithm increases that likelihood at each exact iteration, but need not find its global maximum; multiple starts help distinguish local optima. The resulting estimates give the case fatality probability $\hat{\theta }$ andThese are conditional event-time distributions, distinct from the cumulative incidence functions $\hat{\theta }\hat{F}(t\mid I=1)$ and $(1-\hat{\theta })\hat{F}(t\mid I=2)$.

Among patients still alive at the same time after surgery, chemotherapy has an estimated lung-cancer death rate 3.71 times that of radiotherapy, assuming proportional cause-specific hazards. The estimated 95% confidence interval for this hazard ratio is $(1.583,8.697)$. Thus the fit suggests a higher instantaneous lung-cancer death rate on chemotherapy, with appreciable uncertainty in its size.

The null hypothesis of no treatment effect on that hazard is $H_0:{\gamma }_1=0$ against ${\gamma }_1\ne 0$. Its Wald test uses $Z=1.3113/0.4346\simeq 3.018$, approximately standard normal under the null hypothesis, with two-sided $p=0.00255$. This is evidence against equal lung-cancer death hazards in the fitted model. It is not a statement that five-year lung-cancer death probability is multiplied by 3.71 or that survival duration is divided by 3.71. The cumulative incidence function depends on both cause-specific hazards, and the model needs proportional-hazards and independent observation-censoring assumptions. In a randomised comparison the assigned treatment contrast is protected against baseline confounding, but the instantaneous comparison concerns the groups who have survived to the time in question; it does not identify a treatment effect within a common latent survivor subgroup.

The last lung-cancer failure before five years is at 4.85, where the displayed net survival estimate is 0.392. The other-death net survival estimate remains 0.498 from 4.096 until 5.065. Thus the two supplied curves imply net five-year death estimates of $1-0.392=0.608$ for lung cancer and $1-0.498=0.502$ for other causes. Their sum is 1.110, already showing that they cannot be probabilities of two disjoint actual causes of death.

In competing risks, let ${\lambda }_1(t)$ and ${\lambda }_2(t)$ be the cause-specific hazards while alive, with cumulative hazards ${\Lambda }_k(t)={\int }_0^t{\lambda }_k(u)du$. Treating the other cause as censored estimates the net survival $S_k^{\mathrm{net}}(t)=e^{-{\Lambda }_k(t)}$. The actual probability of death from cause $k$ by time $t$ is the cumulative incidence functionSince $S(u)\le e^{-{\Lambda }_k(u)}$, the net probability $1-S_k^{\mathrm{net}}(t)$ generally overstates the cumulative incidence function: death from the other cause prevents a later death of cause $k$. This discrepancy exists even for independent latent cause-specific failure times. Moreover, Kaplan–Meier estimates are not generally exactly unbiased in finite samples even for their proper target. The plotted complements are not unbiased estimators of $\mathbb{P}(T\le 5,D=k)$, and are not even consistent for that target in general.

For the desired probabilities, use the Aalen–Johansen estimator ${\hat{F}}_k(t)={\sum }_{u\le t}\hat{S}(u-)d_k(u)/n(u)$, with all-cause survival $\hat{S}(u)={\prod }_{v\le u}[1-(d_1(v)+d_2(v))/n(v)]$ and independent study censoring. Removing competing deaths from subsequent risk sets is correct for estimating cause-specific hazards; the error is interpreting the separate net survival complements as cumulative incidence functions. A hypothetical elimination-of-a-cause interpretation of net survival would require additional causal assumptions.

Use the Aalen–Johansen estimator of the cumulative incidence function, updating the overall Kaplan–Meier estimator for either event:Right censoring changes subsequent risk sets, not the survival function by itself. Starting with the given estimates at $a_k$, the complete calculation isThe middle event is of the competing type: it decreases overall survival while leaving the disease cumulative incidence function unchanged at that instant. The event-free intervals leave every estimate and risk set unchanged.

At the final time, both the event subject and the subject censored at that time are in the just-before risk set, so its denominator is eight. This is the usual event-before-censoring convention for recorded ties. After the event and right censoring, six subjects remain at risk. ConsequentlyThe nine numbered source items are the inputs to this single calculation, not nine further questions.

For mutually exclusive event types, let $T$ be the first event time and $J$ its type. The cause-specific hazard for cause $j$ isIt is a rate conditional on having had no event of any cause. The cumulative incidence function, also called the cumulative risk function, is the actual probability $F_j(t)=P(T\le t,J=j)$.

The total hazard function is $h(t)={\sum }_k{h}_k(t)$, giving $S(t)=\exp \{-{\int }_0^t{\sum }_k{h}_k(u)du\}$. Surviving every cause to time $u$ and then experiencing cause $j$ givesIn competing risks, $F_j$ is generally not $1-e^{-{\int }_0^t{h}_j}$, because competing events remove individuals before they can experience cause $j$. This formula does not require assuming independent latent failure times for the different causes.

Integrate the type-A first-event density from zero to $t$. The type-A cumulative incidence isThis is the cumulative incidence function in the competing risks model, distinct from the hypothetical net risk $1-e^{-\alpha t}$ obtained by removing B.

Letting $t\to \infty$ in the cumulative incidence function gives the probability that the observed first event is of type A:This is the sense of occurrence in the competing risks model. If one instead followed the latent exponential $T_A$ even after B, its eventual-occurrence probability would be one; that is a different question from observing A before the competing event.

In a competing risks model, the first event has time $T$ and type $J$; observing one type prevents another type from being the first event for that individual. Its cause-specific hazard isFor absolutely continuous event times the total hazard function is ${\sum }_j{h}_j(t)$, giving the event-free survivor function $S(t)=\exp [-{\int }_0^t{\sum }_j{h}_j(u)du]$. Since an event of type $j$ near $t$ has unconditional density $S(t)h_j(t)$, its cumulative incidence function satisfiesThis cause-specific hazard to cumulative incidence formula does not require independent latent times for the competing causes. In general $F_j(t)$ is not $1-e^{-{\int }_0^t{h}_j}$, since that expression ignores removal by other causes.

Put $a={\theta }_A\ge 0$, $b={\theta }_B\ge 0$ and $q=a+b$, with $\tau \ge 0$. HereIntegrating $aS(u)$ separately before and after $\tau$ gives, for $q>0$,After $\tau$, only type $A$ can occur, so the second term is the fraction event-free at $\tau$ multiplied by their subsequent event probability. The expressions agree at $\tau$. If $a=b=0$, both incidence functions are identically zero; if $a=0<b$, this formula gives $F_A=0$.

Only times up to $\tau$ contribute to the cumulative incidence function for type $B$, because its cause-specific hazard vanishes afterwards. With $q={\theta }_A+{\theta }_B>0$,soThis increases before $\tau$ and then remains constant; it includes the protection from type $B$ events produced by earlier type $A$ events. If both intensities vanish, $F_B=0$; if ${\theta }_B=0$, it is also identically zero.