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

2. 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.

3. 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

4. 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.

6. 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.

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

12. Example of SAS Statistical Output
Parameter Standard Hazard 95% Hazard Ratio
Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio Confidence Limits
drug
Percent
Total Event Censored Censored

13. 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; ;
proc lifetest plots=(s, ls);
time years*censored
strata treat;
Run;

14. 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.

15. White Noise:

16. Negative Autocorrelation:

17. Positive Autocorrelation

18. Trending Time Series