1. Wildlife Population Analysis
Lecture 03 - Kaplan-Meier Procedure and Known Fate Analyses

2. Motivation
Describe the Kaplan Meier procedure for estimating survival rates
Describe an analogous method based on the likelihood for examining factors related to survival

3. Kaplan-Meier or Product Limit Procedure
Kaplan-Meier procedure developed in the late 50's for use in medicine and engineering
First suggested for wildlife survival studies in mid- to late-80's by Pollock (1984) and Pollock et. al (1989).
Early approaches assumed that survival of every animal was independent (i.e., constant survival probability over all animals and periods)

4. Kaplan-Meier or Product Limit Procedure
Used when relocation is certain or unrelated to mortality
Radio-tagged animals
Plants or sessile creatures
It is a nonparametric approach that does not assume any distribution; e.g., the exponential or weibull

5. Terms and definitions
Interval survival rates – the proportion of animals surviving some discrete period of time
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. Thus the survival rate for the interval can never be greater than the lowest estimate of survival during any intermediate period.
Survival function – equation describing the probability of an animal in a population surviving t units of time from the beginning of a study.

6. Terms and definitions
At-risk – having some probability of detectable mortality.
Thus, in radio-telemetry studies, transmitter failure or emigration beyond the study area boundaries is grounds for removal
Right censor – to remove an animal from the at-risk group for some reason unrelated to mortality
Staggered Entry — the addition of additional subjects to the at-risk group during the course of the study.

7. Study design
Individuals marked or uniquely identifiable
radio tags
sessile individuals
Subjects can be relocated without failure
All individuals relocated at the end of each survival interval (e.g., at each aj)

8. Identification of time origin is crucial
Study design
Identification of time origin is crucial
in medical and engineering studies – treatment time
in wildlife studies
for example studies in winter may have a different survival rate that studies in summer;
survival rates may vary greatly after the start of a hunting season or other event

9. Assumptions
Animals in the population of interest have been sampled randomly – sex, age, location, etc.
Survival times are independent for different animals
Time of death is known exactly
Capturing, marking, and observing does not influence survival (but may condition to eliminate potential effects)
Censoring is random (i.e., unrelated to fate – either survival or mortality)
May be useful to assume that marker failure is zero (e.g. studies of emigration time)
Survival probabilities are same within groups

10. Staggered entry
Assumes that newly tagged animals have the same survival function as those previously tagged
Alternatively condition data
Model the assumption

11. Basic K-M calculations
a1, a2, … ag – discrete time points when deaths (failures) occur (alternatively use regular sampling periods e.g. days, weeks, etc.)
r1, r2, … rg – number of animals at risk at these time points
d1, d2, … dg – number of deaths at these time points

12. Basic K-M calculations
the probability of surviving the interval from ai-1 to ai and
Proportion dying (mortality)
the probability of surviving the interval from ai-1 to ai+1

13. Basic K-M calculations
the probability of surviving the interval from ai-1 to ai and
Read
“product”

14. Censoring and staggered entry
When individuals are right censored, the number in the at risk group ri is reduced at time ai.
Likewise when individuals are entered into the at risk sample the value ri is increased at time ai.
Alternatively, newly marked individuals can be "conditioned" to allow for marking or handling effects by delaying their addition to the at risk group for some predetermined time period.

15. Example K-M calculations
Week
No. risk
No. deaths Si
Survival
No. censored
No. added 1 7
1.0000 2 6 5 3 11
0.9091 4 10 16
0.9375
0.8523 15
0.9333
0.7955 8 14 9
0.7857
0.6250
Total

16. Example K-M calculations
Table 3-2 from Pollock et al. 1989: Bobwhite survival in fall of 1986
Week
No. risk
No. deaths
Survival
No. censored
No. added
Var(S)
LCL
UCL 1 7
1.0000 2 6 5 3 11
0.9091
0.0068
0.8957
0.9225 4 10
0.0075
0.8944
0.9238 16
0.8523
0.0067
0.8391
0.8654 15
0.0072
0.8383
0.8663
0.7955
0.0086
0.7785
0.8124 8 14
0.0092
0.7773
0.8136 9
0.6250
0.0105
0.6045
0.6455
Total

17. Basic K-M calculations
Estimating standard deviation and SE
95% Confidence intervals or

18. K-M Example

19. Known fate models

20. Any good statistical text on logistic regression
Known Fate Analysis
Resources:
White et al.:
Known Fate models
Link function
Any good statistical text on logistic regression
Hosmer, D.W. and Lemeshow, S Applied logistic regression. 2nd Ed. John Wiley & Sons, Inc. New York.

21. Known Fate Models
Based on binomial probability models
i.e., survived or died
Subjects can be relocated and fate determined without failure.
Same assumptions as K-M survival estimator
Much greater flexibility
Allows covariates for modeling heterogeity via a link function
Based on MLE  model selection

22. Survival rate (probability)
Kaplan-Meier estimate: where di was the number dying and ri was the number at risk during i th interval of some longer time period t.
Using slightly different symbology: where yi is the number surviving and ni is the number at risk

23. Survival rate
Likelihood for this equation is:  is the survival model for the m periods,
ni is the number of individuals at risk during each period,
yi is the number surviving each period
Si is the MLE of survival during each period.

24. Survival rate
Product of the probability of the individual observations: or the sum of the log-likelihoods where Sij is the probability of individual i surviving the jth interval.

25. Covariates
Assumption is that heterogeneity can be modeled
Survival probabilities are same within groups
Frequently more interested in the differences among survival rates or the effects of environment, treatment, or condition of individuals.
Factors that vary with parameters of interest (e.g., survival probability)

26. Covariates
Can be discrete classes or continuous measurements
Can be used to model group effects at the individual level
Sex, location, time, length, color, cohort
Can vary over time
Temperature, weight, age, date

27. Link functions
“Link” the covariates, the data (X), with the response variable (i.e., fate of the marked individual)

28. Link functions
Recall that the likelihood for the n individuals over the m weeks in our known fate analysis was
In order to estimate the survival rate as a function of some variable (e.g., age or weight) MARK uses a link function.
Usually, this is done by way of the logit link: where x1i is the first covariate (data) in the ith interval.

29. Link functions
Since the data in the link can be coded to represent time periods or other discrete groups (more on how we do this later), we can drop the product of the likelihoods and substituting the logit link into the likelihood:
(ignoring the binomial coefficient), thus the log-likelihood is: where n is the number of individuals in the sample.

30. Link function
More than one covariate can be included
Extend the logit (linear equation).
bs are the estimated parameters (effects);
estimated for each period or group
constrained to be equal using the data (xij).

31. Link function
The survival rates or real parameters of interest are calculated from the s as in this equation.
HUGE concept and applicable to EVERY estimator we examine.
Survival probabilities are replaced by the link function submodel of the covariate(s).
Conceivably every animal has a different survival probability that is related to the value of the covariates xij .

32. Example
Fictitious critter
All are captured and radio-marked at the same time,
Mass determined at the time of capture
Classified as belonging to two discrete groups (e.g., gender)
At the end of the first time period (0), #5 is discovered to have died. By the end of the second period mouse #4 also is dead.

33. Mass at capture (g) (x3i)
Known fate example
Covariates
Obs
Individual (i)
Fate (Survived) (yi)
Time (x1i)
Group (x1i)
Mass at capture (g) (x3i) 1 26 2 28 3 25 4 22 5 20 6 7 8 9

34. Model 1: all survival probabilities equal (null).
0 estimates the survival rate of all individuals.

35. Initial values
In the table below the column labeled 0 is the logit for this model, S is from equation 1, and ln(L) is from equation 2.
The ln(L) for the observations based on initial values for i = 0.5.
Obs
Individual
Fate
Time
Group b0
ln(L)
(i)
(Survived)
(x1i)
(x2i)
(yi) 1
0.5
1.25
-0.25 2 3 4 5
-1.50 6 7 8 9 

36. Solution ln(L) is maximized when 0 = 1.25.
Note that because of the model specification the are equal within groups and time periods.
Obs
Individual
Fate
Time
Group b0
ln(L)
(i)
(Survived)
(x1i)
(x2i)
(yi) 1
1.25
0.778
-0.25 2 3 4 5
-1.50 6 7 8 9 S
-4.77

37. Model 2: Survival rates vary between times
0 - survival rate of individuals in group 0 during time 0;
1 - difference in survival between time periods 0 and 1.
Obs
Individual
(i)
Fate (Survived)
(yi)
Time
(x1i) 1 2 3 4 5 6 7 8 9
Thus, for individual 1 in time period 1:
And, for individual 1 in time period 2:

38. Initial values
0+x1i1 is the linear equation for this model, S is from equation 1, and ln(L) is from equation 2. The ln(L) for the observations is based on initial values for i = 0.5, 2 = 0.5.
Obs
Individual
Fate
Time
Group
b0+x1ib1
ln(L)
(i)
(Survived)
(x1i)
(x2i)
(yi) 1
0.50
0.622
-0.47 2 3 4 5
-0.97 6
1.00
0.731
-0.31 7 8 9
-1.31 S b1
0.5 b2

39. Solution MLE for 0= -1.39 and 1 = 0.29.
Note that because of the model specification the survival probabilities are equal for all individuals in each time period.
Obs
Individual
Fate
Time
Group
b0+x1ib1
ln(L)
(i)
(Survived)
(x1i)
(x2i)
(yi) 1
1.39
0.800
-0.22 2 3 4 5
-1.61 6
1.10
0.750
-0.29 7 8 9
-1.39 S
-4.75

40. Model 3: Survival varies by group and time.
0 equal the survival rate of individuals in group 0 during time 0 ignoring the effect of mass on survival.
1 equal the difference in survival between time periods 0 and 1;
2 equal the difference in survival between groups 0 and 1.
Thus, it follows from Equation 1 that for group 1 at time 1:

41. Initial values
In the table below the column labeled 0+x1i1+x2i2 is the linear equation for this model, is from equation 1, and ln(L) is from equation 2. The ln(L) for the observations is based on initial values i = 0.5.
Obs
Individual
Fate
Time
Group
b0+x1ib1+x2ib2
ln(L)
(i)
(Survived)
(x1i)
(x2i)
(yi) 1
0.50
0.622
-0.47 2 3
1.00
0.731
-0.31 4 5
-1.31 6 7 8
1.50
0.818
-0.20 9
-1.70 S
-5.42

42. Solution
ln(L) is maximized when 0 = , 1 = , and 2 =
Note that because of the model specification they are equal within groups and time periods.
Obs
Individual
Fate
Time
Group
b0+x1ib1+x2ib2
ln(L)
(i)
(Survived)
(x1i)
(x2i)
(yi) 1
16.01
1.000
0.00 2 3
0.69
0.667
-0.41 4 5
0.333
-1.10 6
15.32 7 8
0.500
-0.69 9 S
-3.30

43. Wildlife Population Analysis
Lecture Other link functions

44. Default in MARK with PIMs Constrained 0-1
Sine
Default in MARK with PIMs
Constrained 0-1

45. Default in MARK w/Design Matrix Constrained 0-1 Robust
Logit
Default in MARK w/Design Matrix
Constrained 0-1
Robust

46. May optimize better as S0
LogLog
Constrained 0-1
May optimize better as S0

47. May optimize better as S1
Complimentary LogLog
CLogLog in MARK
Constrained 0-1
May optimize better as S1

48. Log
NOT Constrained
Covariates additive

49. Don’t confuse with identity matrix NOT Constrained

50. Remember
Logit, sin, loglog, and complimentary loglog
Constrain estimates to the interval [0,1].
The default in MARK is the sine function.
Log and identity
Do not constrain to the interval [0, 1],
Useful when estimating things other than rates

51. Other
More complex link functions are beyond the scope of this class
see the program MARK documentation.

52. End of Lecture
Lab 03
30-45 min – Introduction to Program Mark
Kaplan-Meier & Known Fate Analyses

53. End of Lecture Material

54. Introduction to Program Mark
Download MARK and documentation:
Cooch and White:
Windows-based program
Estimation of parameters based on re-encounters of marked individuals.
Patch occupancy analysis
Virtual population analysis
MLE – by numerical techniques
Model selection by information theoretic methods (AIC)

55. Model Specification

56. PIMs

57. Design Matrix Rows correspond to “real” parameters
Columns correspond to βs “estimated parameters”

58. Matrix terminology Rank – dimensions
Vector – matrix with either single column or single row
For multiplication:
do not have to be of the same rank;
must have the same “inner” dimension;
dimensions are specified as rows x columns.
Thus a 4x3 matrix can be multiplied by a 3x1 matrix,
But a 1x3 matrix cannot be multiplied by a 4x3 matrix
The product of matrices has the dimension of “outer” dimensions of the multiplicands
Thus a 4x3 x 3x1 yields a 4x1

59. The identity matrix.
For any square matrix, the identity is a diagonal matrix of equal rank with all of the diagonal elements = 1.

60. Matrix multiplication
D – Design matrix
Elements are data (covariates)
B – Betas from logit
Estimated parameters
DxB = logits for all survival estimates in model

61. Matrix multiplication
Logits

62. Design Matrix

63. Link Functions

64. Link function