Survival Data survival time examples: –time a cancer patient is in remission –time til a disease-free person has a heart attack –time til death of a healthy mouse –time til a computer component fails –time til a paroled prisoner gets rearrested –time til death of a liver transplant patient –time til a cell phone customer switches carrier –time til recovery after surgery all are "time til some event occurs" - longer times are better in all but the last…

estimate the survival function compare survival functions (e.g., across levels of a categorical variable - treatment vs. placebo) understand the relationship of the survival function to explanatory variables ( e.g., is survival time different for various values of an explanatory variable?) Three goals of survival analysis

The survival function S(y)=P(Y>y) can be estimated by the empirical survival function, which essentially gets the relative frequency of the number of Y’s > y… Look at Definition 1.3 on p.5: Y 1, …,Y n are i.i.d. (independent and identically distributed) survival variables. Then S n (y) =empirical survival function at y = (# of the Y’s > y)/n = estimate of S(y). Note that where I is the indicator function…

Review of Bernoulli & Binomial RVs: –Show that the expected value of a Bernoulli rv Z with parameter p (i.e., P(Z=1)=p) is p and that the variance of Z is p(1-p) –Then knowing that the sum of n iid (independent and identically distributed) Bernoullis is a Binomial rv with parameters n and p, show on the next slide that the empirical survivor function S n (y) is an unbiased estimator of S(y)

Note that and as such nS n has B(n,p) where p=P(Y>y)=S(y). Also note that for a fixed y * so S n is unbiased as an estimator of S What is the Var(S n )? (see 1.6 and on p.6 where the confidence interval is computed…) Try this for Example 1.3, p.6

Example 1.4 on page 8 shows that it is sometimes difficult to compare survival curves since they can cross each other… (what makes one survival curve “better” than another?) One way of comparing two survival curves is by comparing their MTTF (mean time til failure) values. Let’s try to use R to draw the two curves given in Ex. 1.4: S 1 (y)=exp(-y/2) and S 2 (y)=exp(-y 2 /4)… see the handout R#1.

Note that the MTTF of a survival rv Y is just its expected value E(Y). We can also show (Theorem 1.2) that (Math & Stat majors: Show this is true using integration by parts and l’Hospital’s rule…!) So suppose we have an exponential survival function: (btw, can you show this satisfies the properties of a survival function?)

Then the MTTF for this variable is  - show this… And for any two such survival functions, S 1 (y)=exp(-y/    and S 2 (y)=exp(-y/    one is “better” than the other if the corresponding beta is “better”… HW: Use R to plot on the same axes at least two such survival functions with different values of beta and show this result.

The hazard function The hazard function gives the so-called “instantaneous” risk of death (or failure) at time t. Recall that for continuous rvs, the probability of occurrence at time t is 0 for all t. So we think about the probability in a “small” interval around t, given that we’ve survived to t, and then let the small interval go to zero (in the limit). The result is given on page 9 as the hazard rate or hazard function…

Definition of hazard function: notes –the hazard function is conditional on the individual having already survived to time y –the numerator is a non-decreasing function of  y (it is more likely that Y will occur in a longer interval) so we divide by the length of the interval to compensate –we take the limit as the length of the interval gets smaller to get the risk at exactly y - “instantaneous risk”

–we can show (see p.9) that the hazard function is equal to –use f(y)=-d/dy(S(y)) and the above to show that –so all three of f, h, and S are representations that can be found from the others and are used in various situations…

more notes on the hazard function: –hazard is in the form of a rate - hazard is not a probability because it can be >1, but the hazard must be > 0; so the graph of h(y) does not have to look at all like that of a survivor function –in order to understand the hazard function, it must be estimated. –think of the hazard h(y) as the instantaneous risk the event will occur per unit time, given that the event has not occurred up to time y. –note that for given y, a larger S(y) corresponds to a smaller h(y) and vice versa…

life expectancy at age t: –if Y=survival time and we know that Y>t, then Y-t=residual lifetime at age t and the mean residual lifetime at age t is the conditional expectation E(Y-t|Y>t) = r(t) –it can be shown that –note that when t=0, r(0)=MTTF –we define the mean life expectancy at age t as E(Y|Y>t) = t +r(t) –go over Example 1.6 on page 11…