1. Wildlife Population Analysis What are those βs anyway?
Odds ratios, link functions, & design matrix

2. References
Hosmer, D.W., and S. Lemeshow Applied logistic regression. John Wiley & Sons, Inc. New York.
Blackduck example:
C:\Program Files\MARK\Examples\BlckDuck
Conroy M.J., Costanzo, G.R., and Stotts, D.B Winter Survival of Female American Black Ducks on the Atlantic Coast. J. Wildlife Management 53:

3. Odds ratios
Gambling on survival

4. Odds ratios
When using the logit link coefficients (βi) are inverse of natural log of odds ratios
Odds of success is the ratio of the probability of success to the probability of failure, or

5. Estimating odds Example of known fate data 50 individuals
Under observation for 1 time periods
40 individuals survive (fate – 1) the first time period.
Fate 1
1 (survived) 40
0 (died) 10
At risk 50

6. Effects on odds
The change in odds ratios between periods 1 and 2 is then

7. Effects on odds - example
Example of known fate data
50 individuals
Under observation for 2 time periods
40 individuals survive (fate – 1) the first time period,
30 survive the second time period.
Time period
Fate 1 2
1 (survived) 40 30
0 (died) 10
Total 50

8. Effects on odds - calculation
Survival for period 1 is 40/50 = 0.80,
OR = 40/10 = 0.80/0.20 = 4/1
Survival in period 2 is 30/40 = 0.75.
OR = 30/10= 0.75/0.25=3/1

9. Change in odds between periods
Odds survival for period 1 versus period 2:
Thus, individuals were 1.33 times as likely to survive period 1 as they were to have survived period 2.
The log of the odds ratio is
ln(1.33)=

10. Pre-defined model S(t) logit link B1 S Intercept Parm B2 S t1 1 1:S
/* Known fate data from lab 05 Odds Ratio spreadsheet 1 groups 2 occasions */
known fate group=1;
50 10;
40 10;

11. Estimated Parameters Estimated Parameter Estimate Odds (exp(β) β1
1.0986
3.0
0.75 β2
0.2877
1.33 --
β1+β2
1.3863
4.0
0.80

12. Black Duck example Age (juvenile and adult)
0 or 1 - indicator or dummy variable.
May be treated as groups in MARK
Sex – indicator variable
Survival could be related to weather events
Changes in harvest effort
Natural mortality
MIN<0 – continuous variable
/* Conroy black duck radiotracking data,
Encounter occasions=8, groups=1, individual covariates=4,
individual covariate names = Age (0=subadult, 1=adult),
Weight (kg), Wing Length (cm), and Condition Index. */
/* 01 */ ;
/* 04 */ ;
/* 05 */ ;
/* 06 */ ;
/* 07 */ ;
/* 08 */ ;
/* 09 */ ;
/* 10 */ ;
/* 11 */ ;
/* 12 */ ;
/* 13 */ ;
/* 14 */ ;
/* 15 */ ;
/* 16 */ ;
/* 17 */ ;
/* 18 */ ;

13. S(.) Intercept only – constant survival
Black Duck Radio-tracking Data
LOGIT Link Function Parameters of {S(.)}
95% Confidence Interval
Parameter Beta Standard Error Lower Upper
1:Intercept
B1 S Int
Parm 1
1:S
2:S
3:S
4:S
5:S
6:S
7:S
8:S
Black Duck Radio-tracking Data
Real Function Parameters of {S(.)}
95% Confidence Interval
Parameter Estimate Standard Error Lower Upper
1:S
2:S
3:S
4:S
5:S
6:S
7:S
8:S

14. S(t) Identity coding B1 S t1 B2 S t2 B3 S t3 Parm B4 S t4 B5 S t5
B1 S t1
B2 S t2
B3 S t3
Parm
B4 S t4
B5 S t5
B6 S t6
B7 S t7
B8 S t8 1
1:S
2:S
3:S
4:S
5:S
6:S
7:S
8:S

15. S(t) Intercept coding 1 B1 S Int B2 S t1 B3 S t2 B4 S t3 Parm B5 S t4
B1 S Int
B2 S t1
B3 S t2
B4 S t3
Parm
B5 S t4 B6
S t5
B7 S t6
B8 S t7 1
1:S
2:S
3:S
4:S
5:S
6:S
7:S
8:S

16. Intercept versus identity coding
Results identical
Estimates of the real parameters
Deviance
Likelihood,
AICc.

17. Intercept coding

18. Continuous dependent covariates
βi associated with continuous dependent variables are interpreted as changes in the log odds per unit change in the value of the variable while holding all other variables constant.

19. Example – continuous covariates
black duck model S(min<0).
min<0 - number of days below freezing during intervals between occasions
Group covariates in the design matrix.

20. Example – continuous covariates
Design Matrix and the associated logits

21. Example – continuous covariates
Estimates of the βi
Logits
Odds ratios are the inverse log of difference between the logits
Estimated Parameter
Estimate
Odds β1
6.449
632.1 β2
-0.608
0.544
min<0 (x1)
Logit
Odds ratio 3
4.6239
0.9903 4
4.0154
0.9823
0.544 5
3.4070
0.9679 6
2.7985
0.9426 7
2.1901
0.8993 8
1.5816
0.8294

22. Graphing

23. Interpretation
Our interpretation should be that the odds of survival each week are lowered 0.54x (e-0.608) for each additional (one) day below freezing during that week.

24. Multivariable additive models
When more than one variable is included in the model the interpretation of the odds must include the differences in the distributions of the values for each variable.

25. Multivariable additive models
Consider the Black Duck example again, and the model S(min0+age), where age = 1 for juveniles and age = 0 for adults. The design matrix and the logits are:

26. Multivariable additive models
Estimated parameters

27. Multivariable additive models
Real parameters

28. Multivariable additive models
Real parameters

29. Multivariable additive models
Graphing

30. Multivariable additive models
Interpretation
Logits are parallel,
Graphs of survival are not
Odds ratios remain constant.
Thus, when comparing survival rates for models with more than one variable the values of the parameters at which the relationships are evaluated must be specified explicitly.
At any given min<0 the odds of juveniles surviving the week are reduced (0.85x)
For both ages, the odds of surviving are reduced 0.54x for each additional day min<0
When min<0 = 7, survival is of juveniles is 1.5% lower

31. Multivariable models interactions
When interactions are estimated in the model even more care must be taken with the interpretation.
Consider the Black Duck example again, and the model S(min0*age), where age = 1 for juveniles and age = 0 for adults.

32. Multivariable models interactions
The design matrix and the logits are:

33. Multivariable models interactions
Estimated parameters
Interaction
Allows for age effects to differ across temperatures

34. Multivariable models interactions
Real parameters
Change in odds with respect to temperature constant within ages
Change in odds between ages differs across temperature

35. Multivariable models interactions
Logits are NOT parallel (i.e., they have different slopes).
Change in odds between ages is not constant
Change in odds as min<0 increases is constant within ages

36. Multivariable models interactions
Interpretation
Comparison of survival values of the is valid only for specific values of min<0 and age.
For adults survival odds decrease each week 0.71x for each additional night below freezing.
For juveniles survival odds decrease each week 0.45x for each additional night below freezing.
Survival odds decrease 1.6x more rapidly for adult as min<0 increases
The reason MARK uses a specific value for covariates in results for “real” parameters.

37. Summary So what are those s ? Parameters estimated by MARK
If using logit link, log of the odds ratios
Inverse log (ex) of differences between logits is the difference in odds between the “real” parameters

38. Wildlife Population Analysis
Other link functions

39. Sine
Default in MARK
Constrained 0-1

40. Logit
Default in MARK w/design matrix
Constrained 0-1
Robust

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

42. Complimentary LogLog 𝑆 =1− 𝑒 −𝑒 𝑋𝛽 CLogLog in MARK Constrained 0-1
May optimize better as S1
𝑆 =1− 𝑒 −𝑒 𝑋𝛽

43. Log
Constrained 0-inf
Covariates additive

44. Less commonly used Identity Parameter specific Multinomial logit
Don’t confuse with identity matrix
NOT Constrained
Parameter specific
Different link for each row in design matrix
Estimating some parameters as rates (e.g., detection rates, survival) and others that are not (e.g., abundance).
Multinomial logit
All but one parameter 0-1 (rates)
Last parameter 1-(others)

45. Less commonly used Cumulative logit link Forced link function
Parameters monotonically increase
𝑆 1 < 𝑆 2 < 𝑆 3
Forced link function
Some parameters in some estimators have forced link functions
Estimator won’t work without forced link.
and f parameters of Pradel data types and Link-Barker data type of are set to a log link function, population estimates (N)

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