SURVIVAL ANALYSIS - About alawing/materials/ESSM689/SurvivalAnalysis.pdf · PDF fileSurvival Analysis –What is it? Analysis of time duration until one or more events happen •What

Jul 11, 2018

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• SURVIVAL ANALYSIS

By

Danielle Walkup

Cyrenea Piper

And Thomas Huff

• Outline Todays Presentation

What IS survival analysis

What kind of data do you need

What are the basic assumptions

Why useful in ecology

Specific example: Population modeling

Estimating survival for population or groups within a population Cormack-Jolly-Seber Model (CJS)

Using survival rates for population analysis the Matrix Population Model (MPM)

• Survival Analysis What is it?

Analysis of time duration until one or more events happen

What proportion of a population will survive past a certain time?

Of those surviving, at what rate will they die or fail?

Can multiple causes of death or failure be taken into account?

How do particular circumstances or characteristics increase or decrease the probability of survival?

Useful tool in a predictive capacity

• Survival Analysis - Aliases

Reliability Theory/Reliability Analysis - Engineering

Duration Analysis/Duration Modeling - Economics

Event History Analysis - Sociology

• Survival Analysis The Data

Dependent variable Time to event

Event status (did the event of interest occur)

Data is hands on often in periodic resampling of the population (i.e. mark recapture)

Time can be measured in days, weeks, years, etc

Optimum to have data from birth to death but often have censored data

Right trending data (data missing actual termination date lost individuals or study ends before they die)

• Survival Analysis what it does

Estimate the survival and hazard functions

Survival function for a given time, the probability of surviving up to that time

Hazard function the potential that the event will occur, per time unit, given an individual has survived up to that specified time

Incorporates information from censored and uncensored observations

Can also include covariates

• Survival Analysis - Approaches

Parametric

***Interested in description of the distribution of survival times and the change in their distribution as a function of the predictors***

Assumes underlying distribution follows a probability distribution (i.e. exponential, Weibull, lognormal)

Model parameters estimate by maximum likelihood

• Survival Analysis - Approaches

Non-parametric

***Estimate and graph survival probabilities as a function of time; obtain univariate descriptive statistics for survival data***

Assume nonlinear relationship between hazard function and predictors

Kaplan Meier method

• Survival Analysis - Approaches

Semi-parametric

***Differences in survival times of two or more groups of interest (can include covariates)***

No assumption about the shape of the hazard functions BUT proportional hazards assumption (the hazard ratio comparing any two observations is constant over time where predictor variables do not vary over time

Cox proportional hazard regression models

• Survival Analysis - Examples

For more on Survival Analysis and Hazard Functions see:

http://rpubs.com/daspringate/survival

www.ms.uky.edu/~mai/Rsurv.pdf

www.stat.ucdavis.edu/.../R_tutorial

http://rpubs.com/daspringate/survivalhttp://www.ms.uky.edu/~mai/Rsurv.pdf

• Example from Ecology

What will happen to a population in the long term?

One form of survival analysis using Capture-Recapture methods

Using estimated survival rates along with reproduction rates in Matrix Population Models

• Capture Mark Recapture Methods

At the first trapping session, we capture N individuals

• Capture Mark Recapture Methods

http://www.rrbo.org/conservation-science/research/bird-banding/find-a-banded-bird/

• Capture Mark Recapture Methods

And (ideally) recapture them in the future

• Cormack Jolly Seber Model

Given: a set of encounter histories

frequency of encounter histories

We can estimate the probabilities that give rise to these frequencies

GLM (Rcapture) or Maximum Likelihood (RMark)

• CJS Model Parameters

1 (phi) apparent survival,model doesnt differentiate between survival and permanent emigration (apparent survival true survival)Its

alive!

• CJS Model Parameters

p2

p Recapture probability,Given that the organism is in the survey area, what is the chance that we will see it again?

And we saw it!

• CJS Model Parameters

1 2

p2 p3With multiple trapping sessions, we add parameters and also start to build capture histories for individuals

• CJS - Assumptions

However there are a few things we need to be aware of when working with these models.

• CJS Assumptions

1. every marked animal present in the population at time(i) has the same probability of recapture = (pi)

2. every marked animal in the population immediately after time (i) has the same probability of surviving to time (i+1) = (phii)

3. marks are not lost or missed.

4. all samples are instantaneous, relative to the interval between occasion (i) and (i+1), and each release is made immediately after the sample.

• In practice, we know that most animal populations arent that easy to constrain and so numerous

models have been developed to account for unmet assumptions and various modeling issues

encountered

CJS You know what they say about when you assume

• Side Note - Some Alternate Models

Joint Live and Dead Encounters Includes r(i) reporting rates and

F(i) fidelity rates

Known Fate Model Used with radio-tracking studies;

p(i)=1

Closed Capture Models Allows estimates of N population

size

Robust Design Models Assumes multiple open periods

that include closed trapping sessions

Multi-state Models Allows animals to move between

states w/ transition probabilities

Nest Survival Model Allows estimation of daily nest

survival rates as a function of season and age of nest

Occupancy Models Estimates proportion of sites

occupied, incorporating detection

Mark-Resight Models Estimates of population size when

marks are only applied once

Jolly-Seber Models Extend CJS models to include

recruitment, can estimate N and lambda rate of population change

Band Recovery Model Includes r(i) = band reporting rate

• Side Note - A Model For Everything (Almost)!

What if your animals arent individually marked?! DISTANCE sampling, transects, etc.

R Package unmarked

Want to include spatial data?

Hair traps, camera traps

R Package scrbook (follows Spatial Capture-Recapture (2013) by Royle et al.)

• CJS Model Building Encounter Histories

Encounter histories are just a series of

1s the individual was captured

So we know the individual is alive and well

0s the individual was not captured

Meaning the individual: Was not encountered (1-p)

Died or permanently emigrated (1-)

• CJS Model Building Encounter Histories

Encounter history

N(frequency)

1 55

In this example we start with 55 individuals captured and marked in the first trap session animals-pics.com

http://animals-pics.com/bird-jam-software/43/lazuli-bunting/

• CJS Model Building Encounter Histories

Encounter history

N(frequency)

11 20

10 35

At the second trapping session, marked individuals are either recaptured (add a 1) or not recaptured (add a 0). animals-pics.com

http://animals-pics.com/bird-jam-software/43/lazuli-bunting/

• CJS Model Building Encounter Histories

Encounter history

N(frequency)

111 7

110 13

101 6

100 29

In the third session, again individuals are either recaptured or not. animals-pics.com

http://animals-pics.com/bird-jam-software/43/lazuli-bunting/

• CJS Model Probabilities

Encounter history

N(frequency)

Probability of Encounter History

111 7

110 13

101 6

100 29

1 2

p2 p3

• CJS Model Encounter Histories?!

Encounter history

N(frequency)

Probability of Encounter History

111 7 1

110 13

101 6

100 29

1 2

p2 p3

• CJS Model Encounter Histories?!

Encounter history

N(frequency)

Probability of Encounter History

111 7 1p2

110 13

101 6

100 29

1 2

p2 p3

Individuals recaptured in time 2 multiply 1 by p2

• CJS Model Encounter Histories?!

Encounter history

N(frequency)

Probability of Encounter History

111 7 1p22

110 13

101 6

100 29

1 2

p2 p3

Individuals survive from time 2 to time 3 multiply 1p2 by 2

• CJS Model Encounter Histories?!

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