More on Survival Analysis1. Dependent variable or response is the waiting time until theoccurrence of an event.2. Observations are censored, in the sense that for some units theevent of interest has not occurred at the time the data are ana-lyzed.3. There are predictors or explanatory variables whose effect onthe waiting time we wish to assess.Definition Review:• T = waiting time until occurrence of an event• f(t) = probability density function of T• F (t) = Pr(T ≤ t) = prob event occurs prior to t• λ(t) = limdt→0Pr{t<T ≤t+dt | T >t}dt=f(t)S(t)= −ddtlog S(t) =hazard function• S(t) = 1 − F (t) =R∞tf(x)dx = exp−Rt0λ(t)dx= Survivalfunction• Λ(t) =Rt0λ(x)dx = Cumulative hazard function• Survival and hazard functions provide alternative but equivalentcharacterizations of the distribution of T .• Given the survival function, we can always differentiate to obtainthe density and then calculate the hazard.• Given the hazard, we can always integrate to obtain the cu-mulative hazard and then exponentiate to obtain the survivalfunction.Example: The simplest possible survival distribution is obtained byassuming a constant risk over time:λ(t) = λ for all t.The corresponding survival function isS(t) = expZt0λds= exp { − λt},which is the exponential distribution with parameter λ.The density is obtained by multiplying the survival function by thehazard to obtainf(t) = λ exp { − λt},which has mean 1/λ. This distribution is very important in survivalanalysis.Expected LifetimeLetting µ denote the mean or expe cted value of T , we haveµ =Z∞0t f(t) dt.Integrating by parts, and noting that −f(t) is the derivative of S(t),which has S(0) = 1 and S(∞) = 0, we haveµ =Z∞0S(t) dt,which implies that the mean is simply the integral of the survivalfunction.• Note that we have assumed that the event will occur if we waitlong enough, with probability 1 (i.e., S(∞) = 0)• This conditional implies that the cumulative hazard must diverge(i.e., Λ(∞) = ∞).• Clearly, in many applications, there are events which are notcertain to occur.Cancer Clinical Trials Example• In many cancer clinical trials, tumors are removed surgically andthe patients are treated with chemotherapy• Often, the time to reoccurrence of the cancer is the responsevariable of interest.• Standard survival models assume that all patients will eventuallyget the cancer again.• However, some patients may actually be cured.Cure Rate Models• If some proportion, π, of the patients are cured and are thereforeno longer at risk of getting the tumor, we havelimt→∞S(t) = S(∞) = π• The density, f(t), then consists of a mixture of a point mass at∞ and a proper density,f(t) = π1(t=∞)+ (1 − π)f∗(t).• Letting f∗(t) denote the density for those subjects who are notcured, we havef∗(t) =f(t)1 − S(∞)=f(t)1 − πand λ∗(t) =f∗(t)S∗(t)=f(t)S(t) − S(∞)• We can implement this idea by assuming each subject has alatent binary random variable, ξ, with ξ = 1 if the subject iscured and ξ = 0 otherwise.More on Non-informative Censoring• Type I censoring: a sample of n units is followed for a fixedtime τ. The number of units experiencing the event is random,but the study duration is fixed.• Fixed censoring: Each unit has a potential maximum observa-tion time τi, for i = 1, . . . , n which may be different for differentsubjects, but is fixed in advance.• Type II censoring: a sample of n units is followed as long asnecessary until d deaths have occurred. Number of events isfixed, but study duration is random.• Random censoring: Each unit has a potential censoring time Ciand a potential lifetime Ti, which are assumed to be independentrandom variables. We observe Yi= min{Ci, Ti} and an indicatorδitelling us the type of event (censored or death).• All of these censoring mechanisms are non-informative and theylead to essentially the same likelihood function.• Censoring of an observation should not provide any informationregarding the prospects of survival of that particular unit beyondthe censoring time.• Assumption: All we know for an observation censored at dura-tion t is that the lifetime exceeds t.Illustrative Example: Rodent Tumorigenicity Studies• Scientific interest focuses on whether a test chemical increasestumor incidence, which is defined as the hazard rate of tumoronset for tumor free animals.• Animals are randomized to dose groups for treatment with thetest agent for their lifetimes, with animals surviving to 2 yearskilled in a terminal sacrifice.• Many animals die prior to 2 years, due to toxicity associatedwith the test agent, to t umor-related causes, or to other naturalcauses (also in some cases sick animals are killed in a moribundsacrifice for humane reasons).Data Available:• Dose of exposure, xi.• Time of death, ti.• Type of death, δi= 1 if natural causes and δi= 0 if sacrifice• Occurrence of tumors, ∆i= 1 if tumor at death and ∆i= 0otherwise.We care about inferences on increases with dose in tumor incidenceHomework Exercise:• How do we analyze these data (Conceptually)?• Is informative censoring an issue?Likelihood Function for Censored DataSuppose we have n units, with unit i observed for a time ti. If theunit died at ti, its contribution to the likelihood function (undernon-informative censoring) isLi= f(ti) = S(ti)λ(ti)If the unit is still alive at ti, all we know under non-informativecensoring is that the lifetime excee ds ti. The probability of thisevent isLi= S(ti),which becomes the contribution of a censored observation to thelikelihood.Letting dibe a death indicator, we haveL =nYi=1Li=nYi=1λ(ti)diS(ti).Taking logs, we havelog L =nXi=1{dilog λ(ti) − Λ(ti)}.Suppose we have exponentially distributed survival times, so thatλ(t) = λ for all t. Then, the log like lihood follows the formlog L =nXi=1{dilog λ − λti}.Letting D =Pdibe the total number of deaths and T =Ptibe thetotal time at risk, we havelog L = D log λ − λT.Differentiating this with respect to λ, the score function isu(λ) = D/λ − T.Setting this equal to 0, we obtain the maximum likelihood estimatebλ = D/T,which is simply the total number of deaths divided by the time atrisk.The observed information is minus the second derivative of the score,which is I(λ) = D/λ2. Taking the inverse and plugging in the mle,we havedvar(bλ) = D/T2.A useful observation is that the log-likelihood for exponential