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How to analyze your organism’s chance of survival? Data: Time between two distinct events, repeated among many subjects/objects/organisms. The first event is predefined while the second is typically some specific kind of transition. The time between these two events will be called the transition time Graphic representation Transition time (or survival time) 70 61 90+ time 22 It can happen that subjects exit the study for other reasons than the event of interest. This is called censored data. Transition time is more than what we have registered, but we don’t know by how much. Medical analysis – diagnosis to death by that disease: End of the study The patient wants to leave the study Death by other causes Plant survival study – start of experiment to death by environmental factors: End of study Experimenter accidentally dropped the pot on the ground Start of larval stage until transition to adult: End of the study Death of the larva Matriculation to master’s degree of students: End of study Student dropped out Death of student (hopefully not!) If we disregard censored data, we can seriously underestimate the transition time! Subject: Whatsoever the start and end events belong to. Transition: The end event. Transition time: Time between the two events of interest. Censoring: When a subject leave the study in a different way than the specified transition. Age: Time from the start event to the present time for subjects which have not been censored and which have not undergone transition. Treatment: Same as regression/dependent variable/explanation variable in ordinary regression. Finding the effect of a treatment on survival is typically the goal of survival analysis. The transition time will typically vary, so we need a statistical distribution for it. Distribution f(t) describes the probability-density of the transition times (t). A sharp peak around a value means most survival times are found around that value. A histogram of actual transition times will start to look like this distribution when we have much data. Distribution of age at death for the population of U.S.A. 2003, as derived from the histogram of ages. Survival curve, S(t) : Describes the probability of not going through a transition before a given age, t. A histogram of the age of subjects at a given time, will look like this curve. In more mathematical terms, it’s the cumulative sum (integral) of probabilities (densities) for all times larger than t. Survival curve for U.S.A. 2003, from the histogram of ages. S (t ) f (t )dt 1 F (t ) t conversely f (t ) S ' (t ) Hazard rate, h(t): The chance of going through a transition the next time interval, given that the subject has not done so earlier. Hazard rate for U.S.A. 2003, derived from the histogram of ages. P (T t t | T t ) f (t ) log(S (t ))' t 0 t S (t ) conversely h(t ) lim t S (t ) e H (t ) , where H (t ) h(u )du 0 Concept Graph Transition time distribution, f(t): Survival curve, S(t) : S (t ) f (t )dt 1 F (t ) t Hazard rate, h(t): h(t ) If we have an expression for one of these concepts, the other two can be derived. f (t ) d log(S (t )) S (t ) dt Just different ways of looking at the same thing. Form: f(t)=e-t Usage: Unstable elementary particles Radioactive isotopes Time between phone calls Life time of a particular copy of DNA for microbial organisms? PS: Conditioned on the state f(t)=e-t of the organism itself and it’s environment. Special quality - memoryless: f(t-t0 | t>t0)=f(t) S(t)=e-t If the survival probability drops to 50% in t=5, it will drop to 25% in t=10 and to 12.5% in t=15. t0 h(t)= Constant hazard. Reasonable, since the distribution is memoryless. Assume microbial survival is conditionally exponential distribution. Contribution from genetics and environment spreads out the death rate, , according to the gamma distribution, (a,b). Result: f(t)= (a/b) (1+t/b)-(a+1) f(t) for a=1, b=1 (Pareto distribution) S(t)=(1+t/b)-a h(t)=a/(b+t) Dropping hazard rate. Reasonable: •If old age => good genes and/or good env. •If young, overrepresentation of bad genes and/or bad env. Cartoon model of aging: the uniform distribution: f(t)=I(0<t<a)/a a=maximal age. All outcomes below that are equally probable. S(t)=(1-t/a)I(0<t<a) h(t)=1/(a-t) f(t) Hazard rate increases inversely proportional to the distance to a. The closer to the maximum attainable age, the more risk there is of dying. Often observed in engineering: a hazard rate that it higher for small and large times than for moderate times. Can be reasonable for complex biological organisms also. For instance humans. Possible to start with modeling hazard rates in order to make a transition time distribution. Estimated h(t) from census data 2003, U.S.A. h(t ) log(S (t ))' Increasing hazard survival-curve bends downwards on the log-scale. h' (t ) log(S (t ))'' h(t) S(t) Example: Uniform transition time (cartoon of aging) Decreasing hazard survival-curve bends upwards on the log-scale. h(t) S(t) Example: Pareto distribution (Varying genes/env. Possibly also a model for vulnerability in early life.) Kaplan-Meier is a parameter-free way of estimating the survival curve. Example: Survival plot for plant experiment – all plants Similar to histograms, in that it simply summarizes the data. Performed by first noting for which times, tj , there are transitions in the data. The number of transitions, mj , and the number of subjects “at risk”, yj , (both subjects which will transit later and subjects that are later censored), is then used. mj ˆ S ( t ) ( 1 ) Technical: yj j|t j t R code: survfit(Surv(t.event,censoring.status)) Use “plot”, to see the resulting curve. Divide your dataset into subgroups You can get a confidence interval for with different treatments and you get this estimated curve. a feel for the difference between these treatments. (In this plant study, the treatment is day length, “dlen”): R code: survfit(Surv(t.event,censoring.status)dlen, conf.type=“plain”, conf.int=0.95) R code: survfit(Surv(t.event,censoring.status), conf.type=“plain”, conf.int=0.95) PS: Note that using confidence intervals to say whether there is a difference means invoking a large number of dependent tests. Not ok. Does model comparison between grouping the data according to treatment and not grouping the data. Compares the observed number of events to that expected if the groups have equal transition time distribution. Gives a p-value for the zero-hypothesis (no effect of different treatments). R-code: survdiff(Surv(t.event,censoring)~treatment) Strength • No model assumptions => can be used for almost any dataset. • Allows categorical explanation variables. • Model-independent confidence interval and model comparison, too. Weakness • Model assumptions => stronger inference. • No inference for transition time distribution or hazard rate. • No way to stepwise add and test more explanation variables (treatments). • Explanation variables can only be added by subdividing the dataset into smaller and smaller pieces. • No way to incorporate continuous explanation variables. Addresses (almost all) the weaknesses of the KaplanMeier approach. Does so by a single model assumption: proportional hazard. Separates time dependency from variable dependency *in the hazard rate*. Hazard ratio: The hazard rate for one choice of explanation variables divided by the hazard rate of another choice. Allows for continuous explanation variables and additive effects of different categorical variables. R-code: coxph(Surv(t.event,censoring)~var1+var2+var3) Proportional hazard regression: h(t|x)=h0(t)ex or lh(t|x)=lh0(t)+x where lh=log(h). Assume one explanation variable, x: What happens if we change it from x to x+1? Log-hazard rate changes by a additive factor . lh(t|x=1)= lh(t|x=0)+0.69 h(t|x=1)= 2h(t|x=0) The hazard changes by a multiplicative factor e. Log-survival curve also changes by a multiplicative factor e. (The latter can be compared to results from the Kaplan-Meier estimator.) Survival curve changes from S(t) to S(t)exp(). (Ex: S(t|x=1)=S(t|x=0)2. Not so easy to see in a plot.) log(S(t|x=1))= 2log(S(t|x=0)) Gives a (partial) likelihood, so more model comparison techniques available. Likelihood-ratio test AIC/BIC Wald test (comparison between estimate and standard error, implemented in R). Stepwise adding/subtracting extra variables possible, either likelihood-based methods or by Wald test. Full model exploration by information criteria also possible (though prohibitively costly when the number of explanation variables is high). Implemented in R: cox.zph(…) What makes the hazard non-proportional can be viewer using the Kaplan-Meier-estimated logsurvival-curve. (Or plot(cox.zph(…))) For this plant experiment, the survival-curves for short day length and long day length seem to part company at around day 30-40. Doesn’t necessarily invalidate the Cox-regression but makes the Kaplan-Meier estimate for log-survival-curve hazard ratio an average effect.