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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

    www.allaboutbirds.org

  • Capture Mark Recapture Methods

    And assign individuals unique, permanent marks

    www.allaboutbirds.org ; www.rrbo.org

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

  • Capture Mark Recapture Methods

    And (ideally) recapture them in the future

    www.allaboutbirds.org; www.rrbo.org

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

  • 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!

    www.allaboutbirds.org

  • 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!

    www.allaboutbirds.org

  • CJS Model Parameters

    1 2

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

    www.allaboutbirds.org ; mspaforums.com

    http://mspaforums.com/showthread.php?58186-TWIG

  • 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

    Lets start with a simple example using the 111 capture history

  • 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

    Individuals survive from time 1 to time 2 start with 1

  • 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?!