Survival Analysis {Chapter 12} Time Specific Life Tables Age Specific Method Point Survival Non-parametric Comparing Curves Parametric

Life tables – a way of summarizing statistics on mortality and survivorship. Cohort – a group of individuals born at the same time. There are two different types of life tables: Time specific – information is collected at one time on the age of death of individuals. 2. Age specific – constructed by following a cohort through time and keeping track of the age of death of individuals.

Terms in a life table: x = age 2. nx = number of individuals alive at the start of age x. 3. lx = survivorship = the number of individuals surviving at time x = nx/n0. 4. dx = mortality = the number of individuals dying at time x. 5. qx = mortality rate = the rate at which individuals are dying at time x = dx/nx. 6. Lx = average number of individuals alive between x and x + 1, = (nx + Nx+1)/2. 7. ex = life expectancy = how long the average individual can expect to live at time x = ex = Tx/nx where Tx = xLx

II. Point Estimators Uses mark-and-recapture to estimate survivorship associated with a particular sample period. Generally based on some variation of Jolly-Seber. The most powerful, and potentially user-friendly programs are MARK and SURGE.

III. Comparison of Curves Survival data is often both right and left censured, i.e., how long individuals had been alive prior to sampling, and how long they survive following the termination of sampling, is not known. Parametric: Assumes the distribution of the survival data can be fit to some known distribution: normal, logistic, (or if log transformed) exponential, Weibull, lognormal, loglogistic, or gamma. Non-parametric: Calculates what is known as a survival distribution function {SDF}. The SDF is evaluated at each time t and is the probability that an individual from the population will survive past t. [S(t)=Prob(T>t)]. Only about 5% less powerful than the parametric approach.

Do not confuse with “classic” Type I, II, & III survivorship curves the apply to biological populations.

Example of SAS Statistical Output Parameter Standard Hazard 95% Hazard Ratio Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio Confidence Limits drug 1 0.77919 0.24226 10.3451 0.0013 2.180 1.356 3.504 Percent Total Event Censored Censored 605 489 116 19.17

title ‘Learning Survival Analysis is No Fun’; data inverts; input years plot weight @@; delete weight; censored = (years < 0); years = abs(years); if plot 1 or plot 2 then treat = 1; else treat = 2; cards; 15 1 55 18 1 97 31 4 45 -25 4 45 27 4 61 24 4 36 26 4 78 25 4 45 -25 4 73 28 4 88 ; proc lifetest plots=(s, ls); time years*censored strata treat; Run;

ARIMA: Auto Regressive Integrated Moving Average To describe populations dynamics requires resampling the population, resulting in a lack of independence. Autocorrelation = serial correlation = temporal correlation. White noise – random population dynamics Nonstationarity – when there is a trend in the data.

White Noise:

Negative Autocorrelation:

Positive Autocorrelation

Trending Time Series