Resultados del Diagnóstico Estratégico en el hotel Paradisus los Cayos

Informative censoring

An important assumption in the analysis of censored survival data is that the ac- tual survival time of an individual, t, is independent of any mechanism that causes

an individual's survival time to be censored at time c, where c < t. This means

that if we consider a group of individuals, all of whom have the same values of relevant prognostic variables, an individual whose survival time is censored will be representative of those at risk at the censoring time if the censoring process oper- ates randomly. When survival data are to be analysed at a predetermined point in calendar time, or at a xed interval of time after the origin for each patient, the prognosis for individuals who are still alive can be taken to be independent of the censoring, so long as the time of analysis is specied before the data are examined. Informative censoring is the case where the censoring is related to the survival time, such as if patients in a trial comparing treatments were withdrawn from one arm of the study because that treatment caused life-threatening side eects. The survival

rates for that treatment would appear larger than they were, leading to an incorrect estimate of the treatment dierence. In such a case, sensitivity analysis is performed to compare the original data analysis to analysis which rst assumes the censored patients were high risk and experienced the event right after censoring, and then assumes they were low risk and survived the longest. If informative censoring begins only after a signicant period of time, survival up to onset of censoring cause can be analysed. The assumption of uninformative censoring can be examined by plotting observed survival times against the values of explanatory variables, distinguishing censored from uncensored survival times. If there is a greater proportion of censored survival times in patients with a particular range of values of explanatory variables, there is evidence of informative censoring.

Kaplan-Meier estimate of the survivor function

The rst step in the analysis of censored survival data is often to calculate the Kaplan-Meier estimate of the survivor function, _S(t)ˆ _{, from the data. It can be}

shown that the Kaplan-Meier estimator maximizes the generalized likelihood over the space of all distributions so its evaluation on large data sets gives a good qual- itative description of the true survival function (Eleuteri et al. [2003]). To obtain a Kaplan-Meier estimate, a series of time-intervals is created, each containing a single death time. Each death event is assumed to have occurred at the beginning of the time interval. The Kaplan-Meier estimate of the survivor function is based on the assumption that the r death times of the n individuals in the sample occur inde-

pendently of one another. If nj individuals are alive just before dj deaths occur at time tj then the estimated probability of survival through the time interval from

tj −δ to tj, where δ is small and contains only one death, is (nj

−dj)

nj . Then, the

estimated survivor function at any time,t, in thekth constructed time interval from t(k) tot(k+1) ,k= 1,2, . . . , r,where t(r+1) is dened to be∞, will be the estimated

probability of surviving beyond t(k). This is actually the probability of surviving

through the interval fromt_(k) to t_(k+1) and all the preceding intervals, and leads to

the Kaplan-Meier estimate of the survivor function, which is given by:

ˆ S(t) = k Y j=1 (nj−dj) nj (4.9) fort(k)≤t < t(k+1).

Basic equations of survival modelling

Survivor function is the probability that the survival time T is greater than some

valuet:

S(t) =P(T ≥t). (4.10)

F(t) is the cumulative density function of T, and f(t) is the probability density

function ofT: F(t) =P(T < t) = Z t 0 f(u)du= 1−S(t), (4.11) so f(t) = dF(t) dt =−S 0_{(t). (4.12)}

The hazard function is the risk or hazard of death at timet, and is derived from the

probability that an individual dies at timetconditional on their having survived up

to that time. This conditional probability is expressed as probability per unit time by dividing by the time intervalδtto give a rate (sometimes called the hazard rate,

the force of mortality, or the instantaneous death rate), and the hazard function,

h(t), is the limiting value as δttends to zero. This leads to: h(t) = f(t)

S(t) =− d

dt(logS(t)). (4.13)

The cumulative hazard,H(t) is H(t) = Z t 0 h(u)du=−logS(t), (4.14) so that S(t) =exp(−H(t)). (4.15)

When tting survival models to data, estimates of the unknown parameters are found by maximising the logarithm of the likelihood.

The likelihood function for randomly censored data

When the censoring times are not informative, the likelihood function is derived (Collett [2003]) as follows:

The likelihood function for randomly censored data of nindividuals, with observed

timeti forith individual,i= 1,2, . . . , n, and event indicator∆which takes the value

∆ = 1if ti is an event time and ∆ = 0if the event of interest is censored.

Ci be the random variable associated with time to censoring. Then the value ti is an observation on the random variable τi = min(Ti, Ci). The density function of

Ti is fT i(t) and the survivor function ST i(t). Similarly, the the random variable associated with censoring timeCi has density function fCi(t) and survivor function

SCi(t).

The probability density distribution for the pair (τi,∆i) for censored observations, so that∆ = 0is described by

p(τi =t,∆i= 0) =P(Ci=t, Ti > t).

This joint probability is a mixture of continuous and discrete components but to simplify the presentation, P(Ti = t), for example, will be understood to be the probability density of function of Ti. Assuming independence of the event time distributionTi from censoring timeCi,

P(Ci =t, Ti> t) =P(Ci=t)P(Ti> t)

=fCi(t)STi(t)

,

and so

P(τi=t,∆i = 0) =fCi(t)STi(t).

Similarly, if the observations are not censored, so that∆ = 1 P(τi =t,∆i = 1) =P(Ti=t, Ci > t)

=P(Ti=t)P(Ci> t)

=fTi(t)SCi(t)

,

again assuming that the distributions ofCi andTi are independent.

Combining these results, the joint probability, or likelihood of the n observations, t1, t2, . . . , tn, is n Y i=1 (fTi(ti)SCi(ti)) ∆i_(f Ci(ti)STi(ti)) (1−∆i)

which can be written as n Y i=1 (fCi(ti) (1−∆i)_S Ci(ti)) ∆i _× n Y i=1 (fTi(ti) ∆i_S Ti(ti)) (1−∆i)_.

Under the conditions of non-informative censoring, the rst product will not contain any parameters that are relevant to the distribution of survival times, and so can be regarded a constant. The likelihood of the observed data is therefore proportional

to the second product: L= n Y i=1 (fTi(ti) ∆i_S Ti(ti)) (1−∆i)_. (4.16) Log-likelihood is log L= n X i=1 [∆i log(f(ti)) + (1−∆i)log(S(ti))] (4.17) which can be maximised to t a survival model to data.