CIMAT Taller de Modelos de Capture y Recaptura Known Fate Survival Analysis

Transcription

1 CIMAT Taller de Modelos de Capture y Recaptura 2010 Known Fate urvival Analysis

2 B D BALANCE MODEL implest population model N = λ t+ 1 N t Deeper understanding of dynamics can be gained by identifying variation in mechanisms: N ( β δ) = t N t

3 URVIVAL ANALYI Wide variety of ways to estimate survival, depending on the data structure (Murray & Patterson 2006) With proper analysis, one can examine how survival varies across space, time, and over life as organisms develop and age

4 KNOWN FATE TUDIE

5 KNOWN FATE FRAMEWORK Individuals marked / uniquely identifiable Individuals can be relocated without failure Can be censored from study All individuals monitored continuously, at regular intervals (e.g., weeks, months, years, etc.), or even irregular intervals 5

6 TERMINOLOGY Known Fate urvival Analysis if monitoring is continuous, it might also be called time to event analysis, failure time analysis At risk number of individuals exposed to detectable mortality in a study (emigration out of study area is grounds for removal from the at risk group) Right censor to remove an individual from the at risk group for some reason unrelated to mortality (individual remains in at risk group until it is censored) taggered Entry the addition of subects to the at risk group during the course of the study (a.k.a. left truncation)

7 MONITORING TIME TO EVENT Individual Observation Times Event taggered entry death death death censor death death Time

8 FUNCTION Density function f(t) PDF of failure times or mortality times; F(t) is the associated CDF

9 FUNCTION urvival function equation describing the probability of an individual in population surviving at least t units of time from the beginning of a study (or the beginning of life) = 1 F( t) = Pr( T t) t Where T is lifetime

10 FUNCTION Hazard function risk of dying at a point in time h t = lim Δt 0 Pr( t T < t + Δt T t) Δt

11 FUNCTION IDENTITIE h t t Cumulative Hazard function cumulated risk of dying from beginning of life up to a point in time t Continuous time function t f () t = h = f() t t h t t [ ln ] ' = H t = 0 h dx H t t t t = e & H = ln x

12 MONITORING OVER INTERVAL Interval survival rates the proportion of individuals surviving some discrete time interval (e.g., weeks): Time intervals () Product limit refers to the fact that survival rate during any interval can be calculated as the product of the survival rates during shorter intermediate intervals: t

13 WORD OF CAUTION In large study areas, mobile individuals may go missing then found later Do not reconstruct unobserved observation times! Censor from data set then re enter under staggered entry Or if common, use capture mark recapture analysis If individuals are separately monitored at irregular intervals Use Nest urvival models to estimate survival (next lecture) A special type of known fate analysis Commonly needed in large scale studies where logistics are difficult

14 KAPLAN MEIER MODEL The Kaplan Meier (K M) model was the earliest statistical estimator for time or age varying survival Originally developed for use in Medicine & Engineering (1958) Pollock (1984, 1989, 1995) first suggested its use in wildlife telemetry studies Used when fate of each individual is certain Perfect observation in medical and engineering trials Radio telemetry Plants, sessile animals, a few other special situations

15 K M VARIABLE a0, a1,, a discrete time points when deaths (failures) occur (alternatively use regular sampling periods e.g., weeks, months, etc.) r0, r1,, r d0, d1,, d number of individuals at risk at these time points number of deaths recorded at these time points a r d censored added Bobwhite survival in fall 1986 (Pollock 1989)

16 K M URVIVAL FUNCTION Probability of surviving interval ˆ ( ) 1 d a = r Probability of surviving interval a ˆ d d + 1 ( a+ 1) = 1 1 r r to Probability of surviving up to time t is a product limit ˆ t = 1 t d r a to a a 1 + 1

17 K M URVIVAL a r d () (t) censored added Bobwhite survival in fall 1986 (Pollock 1989)

18 K M URVIVAL FUNCTION Cox and Oakes (1984) formula for K M variance var ˆ t = ˆ 2 t se.. = var ˆ t 1 r t ˆ t

19 K M URVIVAL a r d () (t) censored added Var LCL UCL Bobwhite survival in fall 1986 (Pollock 1989)

20 LOG RANK TET Used to test whether two (or more) survival functions are equal For two groups, the test is based on the summed (over time) observed minus expected number of deaths for a given group and its variance Compute either a Z or χ 2 statistic and associated p value

21 K M URVIVAL AUMPTION Individuals randomly sampled within groups (age, sex, location) Time or interval of death is known (unless censored) Detection probability = 1 Individual fates are independent of one another Capture/marking does not influence survival Censoring is random (independent of fate) taggered entry assumes that newly tagged individuals have the same survival function as previously tagged

22 KNOWN FATE URVIVAL MODEL K M is a nonparametric method Can test for group effects (log rank test) Cannot incorporate covariates to control for confounding variables

23 KNOWN FATE URVIVAL MODEL emi parametric proportional hazard models Cox Proportional Hazards Useful for analysis of cause specific mortality (Heisey and Patterson 2006) Useful for examining effects of time varying covariates h = h () t ie t 0 ln h = ln h ( t) + t 0 βx t βx t

24 KNOWN FATE URVIVAL MODEL emi parametric proportional hazard models (Cox PH) Useful when fates in clustered groups of individuals are not independent Can account for unobserved individual heterogeneity (Frailty) Heterogeneous ub-groups Observed Marginal Hazard Heterogeneity at the beginning of life has important effect on mortality estimates at later ages

25 Case tudy: Black Legged Kittiwakes Jean Joachim 25 Lise Aubry

26 tudy Area tudy started in 1979 Jean Yves Monnat (Univ. Bretagne Occidentale) Emmanuelle Cam (Univ. Toulouse III) Field site: Cap izun 26

27 Follow Up 5 color bands: 8 possible colors (W, N, B, J, 0, R,V, P) 2 are specific to the banding cohort 3 other bands in the other leg 1 metal band (Museum of Natural History, Paris) After first reproduction, all birds sighted every year until mortality WOPBV Lise Aubry 27

28 KITTIWAKE URVIVAL Cox proportional hazard models With time varying covariates & Frailty term Aubry et al. submitted t P x = = e H t t t 1

29 KNOWN FATE URVIVAL MODEL Parametric hazard models (Gompertz, Weibull, iler etc.) Can account for Frailty Useful for examining rates of aging Gompertz accelerated failure time model: h t = bt ln h = a+ bi t t ai e Exponential acceleration in mortality (b) can be replaced with more complicated parameterization that incorporates covariates h t

30 KNOWN FATE URVIVAL MODEL Generalized Linear (Additive) Models Amenable to discrete interval data Easily incorporate covariates Known Fate model in Program Mark

31 GLM APPROACH K M model ˆ t = 1 t Write things in terms of survival (instead of death) ˆ y t = n t d r Where y is the number surviving interval and n is the number at risk during interval

32 GLM APPROACH Known fate survival ˆ y t = n t Likelihood model ( ) y, ( 1 ) θ L y n = K = 1 Where θ is the survival model for the K periods n y

33 MLE for GLM APPROACH a n d y? L ( ) 0 ( ) 0 ( ) 1 ( ) 0 ( ) 1 ( ) 0 ( ) 1 ( ) 0 ( ) 3 = 1 Multiply likelihood over all intervals y = ( 1 ) Bobwhite survival in fall 1986 (Pollock 1989) Optimize to find values of that maximize: K n y K = 1 y ( 1 ) n y

34 MLE for GLM APPROACH a n d y (t) L ( ) 0 ( ) 0 ( ) 1 ( ) 0 ( ) 1 ( ) 0 ( ) 1 ( ) 0 ( ) 3 Multiply likelihood over all intervals = Optimize to find values of that maximize: (Note: Easier to optimize log likelihood) Bobwhite survival in fall 1986 (Pollock 1989) K = 1 y ( 1 ) n y

35 MLE for GLM APPROACH Instead of writing likelihood for data summarized across individuals ( ) y, ( 1 ) θ L y n = K = 1 Can also write likelihood for individually based data K Where ν = 1 for survived interval and ν = 0 for died n y ( ) ν θ, = ( 1 ) 1 L y n i i = 1 i= 1 n ν

36 MLE for GLM APPROACH a id f? L ( ) 0 i ( ) 0 i ( ) 0 i ( ) 0 i ( ) 0 i ( ) 1 i ( ) 1 i ( ) 0 i 1 i 1 1 i 1 1 i 1 1 i 1 1 i 1 0 i 1 0 i 1 1 i 1 = 1 i= 1 Multiply likelihood over all rows (observations) Optimize to find values of i that maximize: Or sum & optimize log likelihood: K n = K n Bobwhite survival in fall 1986 (Pollock 1989) ( 1 ) 1 i ν ν i K n = 1 i= 1 ( θ ) ν ( i) ν ( i) ln L y, n = ln + (1 ) 1 = 1i= 1 ( 1 ) 1 i ν ν i

37 MLE for GLM APPROACH MLE estimation of a GLM for survival over intervals is an extension of the binomial likelihood (day 1 of course) MLE of a GLM provides greater flexibility to model variation in survival Constant survival (.) K M can only model (t) Categorical factors (groups; (g)) such as sex, habitat, etc. Can be implemented as categorical covariates Continuous covariates (body mass, temporal trends, etc.)

38 LOGITIC REGREION In the family of Generalized Linear Models Instead of linear regression Y = β + β X Logistic regression utilizes the Logit Link function: Y β0+ β1x1+ e = + 1 e β + β X Why is this useful? Constrains Y between 0 and 1 Useful for modeling survival probabilities as function of covariates and factors

39 LOGITIC REGREION Uses Logit Link function to connect Y to the regression string e Pr ( Y = 1 β ) 0, β 1, X1 = = β0+ β1x1 1+ e π Where Y = 1 indicates success and Y = 0 indicates failure Method assumes errors in Y are binomially distributed Variance in errors = π(1 π) β + β X

40 LOGITIC REGREION o what s this Logit Link function all about? The log odds of our probability of success is linearly related to the regression string ln π = β + β X 1 π This is nice, but it s for the linearized transformation of the response probability of interest π

41 LOGITIC REGREION Relationship between π and regression string is shaped or Logistic π e π = 1 + e β + β X β + β X X 1

42 LOGITIC REGREION Logistic regression fitted to 0 1 data (failure and success) π X 1 X 2

43 GENERALIZED LINEAR MODEL traightforward to apply Generalized Linear Models to Known Fate survival analysis because survival () is ust a probability of success (π) Other link functions might be more appropriate for certain types of data logit log log log complimentary log log, and more

44 GLM APPROACH How do we formally incorporate concepts of Link Functions into interval based survival analysis? Recall the likelihood for known fate survival over n individuals and K intervals K ( ) ν θ, = ( 1 ) 1 L y n i i = 1 i= 1 n uppose we want to model survival as a function of some variable(s) like age, body mass, location, etc. ν

45 GLM APPROACH We can do this using the Logit Link, or another link i e = 1 + e β + β X This equation can be implemented into the individually based survival likelihood ( β β ) β + β X K n β0+ β1x1 β0+ β1x1 e e L, y, n = 1 0 1, β + β X β + β X = 1 i= 1 1+ e 1+ e ν ν

46 MLE for GLM APPROACH Estimates of regression intercepts and slopes (Betas: β) can be estimated with MLE

47 GLM PARAMETER Conceivably, every individual has a different survival probability depending on their covariate values X i Age 1 Age 2 Age 3 e = 1 + e β + β X + β X i β + β X + β X Body ize

48 GLM PARAMETER Note on interpretation of Betas: The parameter β 0 determines the baseline intercept The parameter β x (x = 1,2,.) determines the rate of increase or decrease of the shaped curve for covariate x relative to β 0 The sign (±) of β x indicates whether the curve ascends or descends

49 Note: GLM MODELING More than 1 covariate can be examined structure of regression string in the logit link depends on your hypotheses; for example, does survival depend on: Age ex Body mass Age, body mass, and sex Is it constant across time and individuals Advanced approaches allow one to control for fatedependence within clusters (pairs or families), unobserved individual heterogeneity with random effects

50 GLM MODELING Can compute Maximum Likelihoods for any model Compare models with any IC n AICc = 2ln ( L) + 2K n K 1 or even Likelihood Ratio Tests for nested models LRT L ( ˆ simple θ ) = 2ln L ( ˆ general θ )

51 KNOWN FATE IN MARK In program MARK, the individual fates should be read in as a Live Dead (LD) encounter history As a row LDLDLDLD Individual made it to 4 th interval and died during 4 th interval

52 KNOWN FATE MODEL IN MARK o how can we model survival in Program MARK?

53 KNOWN FATE MODEL IN MARK

54 KNOWN FATE MODEL IN MARK A variety of pre defined models can be examined

55 KNOWN FATE MODEL IN MARK Run models using Logit Link

56 KNOWN FATE MODEL IN MARK Individual covariates in Design Matrix