Cormack-Jolly-Seber survival estimates in RMark
Scott Creel
Monday, October 13, 2014
This is an example initially provided by Jeff Laake with the RMark package, that I have modified a bit so that it just provides a worked example of using the Cormack-Jolly-Seber (CJS) method to estimate survival rates. The data set is for European dippers (Cinclus cinclus) - small birds that are famous for their ability to walk around underwater while foraging. This dataself itself is also pretty famous, because it is the example provided for the CJS model in program MARK. The data originally come from a monograph by Jean-Dominique Lebreton:
Lebreton, J.-D., K. P. Burnham, J. Clobert, and D. R. Anderson. 1992. Modeling survival and testing biological hypotheses using marked animals: case studies and recent advances. Ecol. Monogr. 62:67-118
As described in Chapter 4 of CWP and discussed in class, the CJS model estimates survival (phi) while correcting for detection probability (p), using logic similar to the Lincoln-Peterson index, but applicable to sampling on many occasions.
Phi is actually a measure of ‘apparent survival’ because it cannot distinguish between death and permanent emigration. If the study site is ‘geographically open’ so that an appreciable fraction of marked animals leave the site during the period of study, then phi will be less than the true survival rate. Consequently, it is important to assess the evidence for geographic closure when using a CJS model to estimate survival.
First, install the RMark package if you have not done so. RMark is installed on the machines in 407. Load RMark, read the dipper dataset (which comes with the RMark package), and inspect the first 10 rows of the data.
Each row has a capture history for seven occasions. The first variable in the data set must be the capture history, and it must be named ch. In this example, the only covariate is sex.
library(RMark)## This is RMark 2.1.8data(dipper)
head(dipper,10)## ch sex
## 1 0000001 Female
## 2 0000001 Female
## 3 0000001 Female
## 4 0000001 Female
## 5 0000001 Female
## 6 0000001 Female
## 7 0000001 Female
## 8 0000001 Female
## 9 0000001 Female
## 10 0000001 FemaleThen fit the simplest possible CJS model
dipper.model=mark(dipper)##
## Output summary for CJS model
## Name : Phi(~1)p(~1)
##
## Npar : 2
## -2lnL: 666.8
## AICc : 670.9
##
## Beta
## estimate se lcl ucl
## Phi:(Intercept) 0.2421 0.1020 0.0422 0.4421
## p:(Intercept) 2.2263 0.3251 1.5891 2.8635
##
##
## Real Parameter Phi
##
## 1 2 3 4 5 6
## 1 0.5602 0.5602 0.5602 0.5602 0.5602 0.5602
## 2 0.5602 0.5602 0.5602 0.5602 0.5602
## 3 0.5602 0.5602 0.5602 0.5602
## 4 0.5602 0.5602 0.5602
## 5 0.5602 0.5602
## 6 0.5602
##
##
## Real Parameter p
##
## 2 3 4 5 6 7
## 1 0.9026 0.9026 0.9026 0.9026 0.9026 0.9026
## 2 0.9026 0.9026 0.9026 0.9026 0.9026
## 3 0.9026 0.9026 0.9026 0.9026
## 4 0.9026 0.9026 0.9026
## 5 0.9026 0.9026
## 6 0.9026As noted in the output, the model above is is an intercept-only (~1) model for both survival (phi) and detection (p), It estimates one overall mean detection probability (0.9026) and one overall mean survival rate (0.5602).
Because there are 7 sampling occasions, there are estimates of survival (phi) for the transition from time 1 to time 2, from time 2 to time 3, …. from time 6 to time 7.
And there are estimates of p for every occasion after the first one.
In the model output, the Beta values are effects of each variable on survival (phi) or detection (p), on the logit scale. The Real Parameters are estimates of survival rate and detection probability, backtransformed to be on a regular linear scale from 0 to 1.
The model above is better than an analysis that ignores detection, but it is likely that survival is affected by other variables. With the data set we have here, we can fit models that estimate the effects of sex and time (sampling period) on survival, detection, or both. Below, we’ll add another covariate covariate, Flood, that describes whether a year was unusually wet, which might reasonably affect detection, survival, or both.
Process the data in the manner required by RMark
dipper.processed=process.data(dipper,groups=("sex"))
dipper.ddl=make.design.data(dipper.processed)Add Flood covariates for Phi and p. Flood = 0 means no flooding, Flood = 1 means there was flooding. Note that flooding is defined differently for Phi and for p, so times 2 and 3 are flood years with respect to survival (Phi) but only time 3 is a flood year with respect to detection (p)
dipper.ddl$Phi$Flood=0
dipper.ddl$Phi$Flood[dipper.ddl$Phi$time==2 | dipper.ddl$Phi$time==3]=1
dipper.ddl$p$Flood=0
dipper.ddl$p$Flood[dipper.ddl$p$time==3]=1Define several competing models for survival (phi)
Phidot=list(formula=~1)
Phitime=list(formula=~time)
Phisex=list(formula=~sex)
Phisextime=list(formula=~sex+time)
Phisex.time=list(formula=~sex*time)
# the model just above includes an interaction between sex and time
# meaning that differences among years don't have to be the
# same for males and females
PhiFlood=list(formula=~Flood)And define several competing models for detection (p)
pdot=list(formula=~1)
ptime=list(formula=~time)
psex=list(formula=~sex)
psextime=list(formula=~sex+time)
psex.time=list(formula=~sex*time)
pFlood=list(formula=~Flood)Run a set of CJS models that use the definitions just above. The default setting is model = “CJS” in the mark() function. Use F1 to get help on the mark() function to understand it better.
dipper.phitime.ptime =mark(dipper.processed,dipper.ddl,
model.parameters=list(Phi=Phitime, p=ptime))##
## Note: only 11 parameters counted of 12 specified parameters
##
## AICc and parameter count have been adjusted upward##
## Output summary for CJS model
## Name : Phi(~time)p(~time)
##
## Npar : 12 (unadjusted=11)
## -2lnL: 657
## AICc : 681.7 (unadjusted=679.58789)
##
## Beta
## estimate se lcl ucl
## Phi:(Intercept) 0.93546 0.7685 -0.5708 2.4418
## Phi:time2 -1.19828 0.8707 -2.9048 0.5082
## Phi:time3 -1.02283 0.8049 -2.6005 0.5548
## Phi:time4 -0.41986 0.8091 -2.0058 1.1661
## Phi:time5 -0.53610 0.8031 -2.1103 1.0381
## Phi:time6 0.24814 256.8861 -503.2485 503.7448
## p:(Intercept) 0.82928 0.7837 -0.7068 2.3654
## p:time3 1.65563 1.2914 -0.8755 4.1867
## p:time4 1.52210 1.0729 -0.5808 3.6250
## p:time5 1.37674 0.9885 -0.5607 3.3142
## p:time6 1.79509 1.0689 -0.2999 3.8901
## p:time7 -0.01476 196.1878 -384.5428 384.5133
##
##
## Real Parameter Phi
## Group:sexFemale
## 1 2 3 4 5 6
## 1 0.7182 0.4347 0.4782 0.6261 0.5985 0.7656
## 2 0.4347 0.4782 0.6261 0.5985 0.7656
## 3 0.4782 0.6261 0.5985 0.7656
## 4 0.6261 0.5985 0.7656
## 5 0.5985 0.7656
## 6 0.7656
##
## Group:sexMale
## 1 2 3 4 5 6
## 1 0.7182 0.4347 0.4782 0.6261 0.5985 0.7656
## 2 0.4347 0.4782 0.6261 0.5985 0.7656
## 3 0.4782 0.6261 0.5985 0.7656
## 4 0.6261 0.5985 0.7656
## 5 0.5985 0.7656
## 6 0.7656
##
##
## Real Parameter p
## Group:sexFemale
## 2 3 4 5 6 7
## 1 0.6962 0.9231 0.913 0.9008 0.9324 0.6931
## 2 0.9231 0.913 0.9008 0.9324 0.6931
## 3 0.913 0.9008 0.9324 0.6931
## 4 0.9008 0.9324 0.6931
## 5 0.9324 0.6931
## 6 0.6931
##
## Group:sexMale
## 2 3 4 5 6 7
## 1 0.6962 0.9231 0.913 0.9008 0.9324 0.6931
## 2 0.9231 0.913 0.9008 0.9324 0.6931
## 3 0.913 0.9008 0.9324 0.6931
## 4 0.9008 0.9324 0.6931
## 5 0.9324 0.6931
## 6 0.6931 dipper.phitime.pdot =mark(dipper.processed,dipper.ddl,
model.parameters=list(Phi=Phitime,p=pdot))##
## Output summary for CJS model
## Name : Phi(~time)p(~1)
##
## Npar : 7
## -2lnL: 659.7
## AICc : 674
##
## Beta
## estimate se lcl ucl
## Phi:(Intercept) 0.514391 0.4768 -0.4201 1.4489
## Phi:time2 -0.698141 0.5537 -1.7834 0.3871
## Phi:time3 -0.600936 0.5301 -1.6399 0.4381
## Phi:time4 -0.006106 0.5335 -1.0517 1.0395
## Phi:time5 -0.075712 0.5276 -1.1099 0.9585
## Phi:time6 -0.178063 0.5266 -1.2101 0.8540
## p:(Intercept) 2.220395 0.3289 1.5758 2.8650
##
##
## Real Parameter Phi
## Group:sexFemale
## 1 2 3 4 5 6
## 1 0.6258 0.4542 0.4784 0.6244 0.6079 0.5833
## 2 0.4542 0.4784 0.6244 0.6079 0.5833
## 3 0.4784 0.6244 0.6079 0.5833
## 4 0.6244 0.6079 0.5833
## 5 0.6079 0.5833
## 6 0.5833
##
## Group:sexMale
## 1 2 3 4 5 6
## 1 0.6258 0.4542 0.4784 0.6244 0.6079 0.5833
## 2 0.4542 0.4784 0.6244 0.6079 0.5833
## 3 0.4784 0.6244 0.6079 0.5833
## 4 0.6244 0.6079 0.5833
## 5 0.6079 0.5833
## 6 0.5833
##
##
## Real Parameter p
## Group:sexFemale
## 2 3 4 5 6 7
## 1 0.9021 0.9021 0.9021 0.9021 0.9021 0.9021
## 2 0.9021 0.9021 0.9021 0.9021 0.9021
## 3 0.9021 0.9021 0.9021 0.9021
## 4 0.9021 0.9021 0.9021
## 5 0.9021 0.9021
## 6 0.9021
##
## Group:sexMale
## 2 3 4 5 6 7
## 1 0.9021 0.9021 0.9021 0.9021 0.9021 0.9021
## 2 0.9021 0.9021 0.9021 0.9021 0.9021
## 3 0.9021 0.9021 0.9021 0.9021
## 4 0.9021 0.9021 0.9021
## 5 0.9021 0.9021
## 6 0.9021 dipper.phitime.psex =mark(dipper.processed,dipper.ddl,
model.parameters=list(Phi=Phitime,p=psex))##
## Output summary for CJS model
## Name : Phi(~time)p(~sex)
##
## Npar : 8
## -2lnL: 659.2
## AICc : 675.5
##
## Beta
## estimate se lcl ucl
## Phi:(Intercept) 0.510202 0.4753 -0.4213 1.4417
## Phi:time2 -0.689982 0.5524 -1.7728 0.3928
## Phi:time3 -0.595817 0.5289 -1.6324 0.4407
## Phi:time4 -0.002544 0.5322 -1.0456 1.0405
## Phi:time5 -0.073789 0.5262 -1.1051 0.9575
## Phi:time6 -0.166511 0.5259 -1.1972 0.8642
## p:(Intercept) 2.002146 0.4148 1.1892 2.8151
## p:sexMale 0.478379 0.6400 -0.7761 1.7328
##
##
## Real Parameter Phi
## Group:sexFemale
## 1 2 3 4 5 6
## 1 0.6249 0.4552 0.4786 0.6243 0.6074 0.5851
## 2 0.4552 0.4786 0.6243 0.6074 0.5851
## 3 0.4786 0.6243 0.6074 0.5851
## 4 0.6243 0.6074 0.5851
## 5 0.6074 0.5851
## 6 0.5851
##
## Group:sexMale
## 1 2 3 4 5 6
## 1 0.6249 0.4552 0.4786 0.6243 0.6074 0.5851
## 2 0.4552 0.4786 0.6243 0.6074 0.5851
## 3 0.4786 0.6243 0.6074 0.5851
## 4 0.6243 0.6074 0.5851
## 5 0.6074 0.5851
## 6 0.5851
##
##
## Real Parameter p
## Group:sexFemale
## 2 3 4 5 6 7
## 1 0.881 0.881 0.881 0.881 0.881 0.881
## 2 0.881 0.881 0.881 0.881 0.881
## 3 0.881 0.881 0.881 0.881
## 4 0.881 0.881 0.881
## 5 0.881 0.881
## 6 0.881
##
## Group:sexMale
## 2 3 4 5 6 7
## 1 0.9228 0.9228 0.9228 0.9228 0.9228 0.9228
## 2 0.9228 0.9228 0.9228 0.9228 0.9228
## 3 0.9228 0.9228 0.9228 0.9228
## 4 0.9228 0.9228 0.9228
## 5 0.9228 0.9228
## 6 0.9228 dipper.phitime.psex.time =mark(dipper.processed,dipper.ddl,
model.parameters=list(Phi=Phitime,p=psex.time))##
## Note: only 16 parameters counted of 18 specified parameters
##
## AICc and parameter count have been adjusted upward##
## Output summary for CJS model
## Name : Phi(~time)p(~sex * time)
##
## Npar : 18 (unadjusted=16)
## -2lnL: 654.5
## AICc : 692.2 (unadjusted=687.86359)
##
## Beta
## estimate se lcl ucl
## Phi:(Intercept) 0.92419 0.7593 -0.5640 2.4124
## Phi:time2 -1.19133 0.8611 -2.8790 0.4964
## Phi:time3 -1.00874 0.7955 -2.5679 0.5505
## Phi:time4 -0.41172 0.8001 -1.9799 1.1565
## Phi:time5 -0.52484 0.7941 -2.0814 1.0317
## Phi:time6 0.51430 169.4091 -331.5276 332.5562
## p:(Intercept) 0.78503 1.0156 -1.2056 2.7757
## p:sexMale 0.10285 1.2979 -2.4411 2.6468
## p:time3 1.12355 1.4320 -1.6832 3.9303
## p:time4 1.52566 1.4402 -1.2972 4.3485
## p:time5 1.19192 1.2498 -1.2578 3.6416
## p:time6 1.88209 1.4321 -0.9249 4.6891
## p:time7 -0.39505 80.4802 -158.1363 157.3462
## p:sexMale:time3 15.70081 3159.1599 -6176.2528 6207.6544
## p:sexMale:time4 -0.02558 1.9286 -3.8056 3.7544
## p:sexMale:time5 0.46805 1.7839 -3.0283 3.9644
## p:sexMale:time6 -0.17282 1.9174 -3.9309 3.5853
## p:sexMale:time7 0.44969 35.4249 -68.9832 69.8826
##
##
## Real Parameter Phi
## Group:sexFemale
## 1 2 3 4 5 6
## 1 0.7159 0.4336 0.4789 0.6254 0.5985 0.8082
## 2 0.4336 0.4789 0.6254 0.5985 0.8082
## 3 0.4789 0.6254 0.5985 0.8082
## 4 0.6254 0.5985 0.8082
## 5 0.5985 0.8082
## 6 0.8082
##
## Group:sexMale
## 1 2 3 4 5 6
## 1 0.7159 0.4336 0.4789 0.6254 0.5985 0.8082
## 2 0.4336 0.4789 0.6254 0.5985 0.8082
## 3 0.4789 0.6254 0.5985 0.8082
## 4 0.6254 0.5985 0.8082
## 5 0.5985 0.8082
## 6 0.8082
##
##
## Real Parameter p
## Group:sexFemale
## 2 3 4 5 6 7
## 1 0.6868 0.8709 0.9098 0.8784 0.9351 0.5963
## 2 0.8709 0.9098 0.8784 0.9351 0.5963
## 3 0.9098 0.8784 0.9351 0.5963
## 4 0.8784 0.9351 0.5963
## 5 0.9351 0.5963
## 6 0.5963
##
## Group:sexMale
## 2 3 4 5 6 7
## 1 0.7085 1 0.9159 0.9274 0.9307 0.7196
## 2 1 0.9159 0.9274 0.9307 0.7196
## 3 0.9159 0.9274 0.9307 0.7196
## 4 0.9274 0.9307 0.7196
## 5 0.9307 0.7196
## 6 0.7196 dipper.phiFlood.pFlood =mark(dipper.processed,dipper.ddl,
model.parameters=list(Phi=PhiFlood, p=pFlood))##
## Output summary for CJS model
## Name : Phi(~Flood)p(~Flood)
##
## Npar : 4
## -2lnL: 660.1
## AICc : 668.2
##
## Beta
## estimate se lcl ucl
## Phi:(Intercept) 0.4377 0.1307 0.1816 0.6938
## Phi:Flood -0.5658 0.2184 -0.9939 -0.1377
## p:(Intercept) 2.1718 0.3412 1.5030 2.8406
## p:Flood 0.2036 1.0218 -1.7992 2.2063
##
##
## Real Parameter Phi
## Group:sexFemale
## 1 2 3 4 5 6
## 1 0.6077 0.468 0.468 0.6077 0.6077 0.6077
## 2 0.468 0.468 0.6077 0.6077 0.6077
## 3 0.468 0.6077 0.6077 0.6077
## 4 0.6077 0.6077 0.6077
## 5 0.6077 0.6077
## 6 0.6077
##
## Group:sexMale
## 1 2 3 4 5 6
## 1 0.6077 0.468 0.468 0.6077 0.6077 0.6077
## 2 0.468 0.468 0.6077 0.6077 0.6077
## 3 0.468 0.6077 0.6077 0.6077
## 4 0.6077 0.6077 0.6077
## 5 0.6077 0.6077
## 6 0.6077
##
##
## Real Parameter p
## Group:sexFemale
## 2 3 4 5 6 7
## 1 0.8977 0.9149 0.8977 0.8977 0.8977 0.8977
## 2 0.9149 0.8977 0.8977 0.8977 0.8977
## 3 0.8977 0.8977 0.8977 0.8977
## 4 0.8977 0.8977 0.8977
## 5 0.8977 0.8977
## 6 0.8977
##
## Group:sexMale
## 2 3 4 5 6 7
## 1 0.8977 0.9149 0.8977 0.8977 0.8977 0.8977
## 2 0.9149 0.8977 0.8977 0.8977 0.8977
## 3 0.8977 0.8977 0.8977 0.8977
## 4 0.8977 0.8977 0.8977
## 5 0.8977 0.8977
## 6 0.8977 dipper.phisex.pdot =mark(dipper.processed,dipper.ddl,
model.parameters=list(Phi=Phisex,p=pdot))##
## Output summary for CJS model
## Name : Phi(~sex)p(~1)
##
## Npar : 3
## -2lnL: 666.7
## AICc : 672.7
##
## Beta
## estimate se lcl ucl
## Phi:(Intercept) 0.20364 0.1397 -0.07021 0.4775
## Phi:sexMale 0.07929 0.1973 -0.30750 0.4661
## p:(Intercept) 2.22749 0.3252 1.59013 2.8648
##
##
## Real Parameter Phi
## Group:sexFemale
## 1 2 3 4 5 6
## 1 0.5507 0.5507 0.5507 0.5507 0.5507 0.5507
## 2 0.5507 0.5507 0.5507 0.5507 0.5507
## 3 0.5507 0.5507 0.5507 0.5507
## 4 0.5507 0.5507 0.5507
## 5 0.5507 0.5507
## 6 0.5507
##
## Group:sexMale
## 1 2 3 4 5 6
## 1 0.5703 0.5703 0.5703 0.5703 0.5703 0.5703
## 2 0.5703 0.5703 0.5703 0.5703 0.5703
## 3 0.5703 0.5703 0.5703 0.5703
## 4 0.5703 0.5703 0.5703
## 5 0.5703 0.5703
## 6 0.5703
##
##
## Real Parameter p
## Group:sexFemale
## 2 3 4 5 6 7
## 1 0.9027 0.9027 0.9027 0.9027 0.9027 0.9027
## 2 0.9027 0.9027 0.9027 0.9027 0.9027
## 3 0.9027 0.9027 0.9027 0.9027
## 4 0.9027 0.9027 0.9027
## 5 0.9027 0.9027
## 6 0.9027
##
## Group:sexMale
## 2 3 4 5 6 7
## 1 0.9027 0.9027 0.9027 0.9027 0.9027 0.9027
## 2 0.9027 0.9027 0.9027 0.9027 0.9027
## 3 0.9027 0.9027 0.9027 0.9027
## 4 0.9027 0.9027 0.9027
## 5 0.9027 0.9027
## 6 0.9027 dipper.phisex.psex =mark(dipper.processed,dipper.ddl,
model.parameters=list(Phi=Phisex,p=psex))##
## Output summary for CJS model
## Name : Phi(~sex)p(~sex)
##
## Npar : 4
## -2lnL: 666.2
## AICc : 674.2
##
## Beta
## estimate se lcl ucl
## Phi:(Intercept) 0.22320 0.1441 -0.05922 0.5056
## Phi:sexMale 0.04163 0.2042 -0.35857 0.4418
## p:(Intercept) 2.01113 0.4211 1.18574 2.8365
## p:sexMale 0.47517 0.6630 -0.82432 1.7746
##
##
## Real Parameter Phi
## Group:sexFemale
## 1 2 3 4 5 6
## 1 0.5556 0.5556 0.5556 0.5556 0.5556 0.5556
## 2 0.5556 0.5556 0.5556 0.5556 0.5556
## 3 0.5556 0.5556 0.5556 0.5556
## 4 0.5556 0.5556 0.5556
## 5 0.5556 0.5556
## 6 0.5556
##
## Group:sexMale
## 1 2 3 4 5 6
## 1 0.5658 0.5658 0.5658 0.5658 0.5658 0.5658
## 2 0.5658 0.5658 0.5658 0.5658 0.5658
## 3 0.5658 0.5658 0.5658 0.5658
## 4 0.5658 0.5658 0.5658
## 5 0.5658 0.5658
## 6 0.5658
##
##
## Real Parameter p
## Group:sexFemale
## 2 3 4 5 6 7
## 1 0.882 0.882 0.882 0.882 0.882 0.882
## 2 0.882 0.882 0.882 0.882 0.882
## 3 0.882 0.882 0.882 0.882
## 4 0.882 0.882 0.882
## 5 0.882 0.882
## 6 0.882
##
## Group:sexMale
## 2 3 4 5 6 7
## 1 0.9232 0.9232 0.9232 0.9232 0.9232 0.9232
## 2 0.9232 0.9232 0.9232 0.9232 0.9232
## 3 0.9232 0.9232 0.9232 0.9232
## 4 0.9232 0.9232 0.9232
## 5 0.9232 0.9232
## 6 0.9232 dipper.phisex.psex.time =mark(dipper.processed,dipper.ddl,
model.parameters=list(Phi=Phisex,p=psex.time))##
## Note: only 12 parameters counted of 14 specified parameters
##
## AICc and parameter count have been adjusted upward##
## Output summary for CJS model
## Name : Phi(~sex)p(~sex * time)
##
## Npar : 14 (unadjusted=12)
## -2lnL: 662.2
## AICc : 691.3 (unadjusted=687.00512)
##
## Beta
## estimate se lcl ucl
## Phi:(Intercept) 0.232679 0.1564 -7.385e-02 0.5392
## Phi:sexMale -0.007414 0.2092 -4.173e-01 0.4025
## p:(Intercept) 1.205677 1.0582 -8.685e-01 3.2798
## p:sexMale 0.180164 1.4889 -2.738e+00 3.0983
## p:time3 0.351600 1.3761 -2.346e+00 3.0488
## p:time4 0.617893 1.4023 -2.131e+00 3.3664
## p:time5 0.835420 1.2797 -1.673e+00 3.3437
## p:time6 1.584664 1.4604 -1.278e+00 4.4470
## p:time7 0.672948 1.6062 -2.475e+00 3.8210
## p:sexMale:time3 21.379527 0.0000 2.138e+01 21.3795
## p:sexMale:time4 0.058614 2.0022 -3.866e+00 3.9830
## p:sexMale:time5 0.425786 1.9350 -3.367e+00 4.2184
## p:sexMale:time6 -0.287232 2.0568 -4.319e+00 3.7441
## p:sexMale:time7 14.316968 2171.8461 -4.243e+03 4271.1354
##
##
## Real Parameter Phi
## Group:sexFemale
## 1 2 3 4 5 6
## 1 0.5579 0.5579 0.5579 0.5579 0.5579 0.5579
## 2 0.5579 0.5579 0.5579 0.5579 0.5579
## 3 0.5579 0.5579 0.5579 0.5579
## 4 0.5579 0.5579 0.5579
## 5 0.5579 0.5579
## 6 0.5579
##
## Group:sexMale
## 1 2 3 4 5 6
## 1 0.5561 0.5561 0.5561 0.5561 0.5561 0.5561
## 2 0.5561 0.5561 0.5561 0.5561 0.5561
## 3 0.5561 0.5561 0.5561 0.5561
## 4 0.5561 0.5561 0.5561
## 5 0.5561 0.5561
## 6 0.5561
##
##
## Real Parameter p
## Group:sexFemale
## 2 3 4 5 6 7
## 1 0.7695 0.826 0.861 0.885 0.9422 0.8675
## 2 0.826 0.861 0.885 0.9422 0.8675
## 3 0.861 0.885 0.9422 0.8675
## 4 0.885 0.9422 0.8675
## 5 0.9422 0.8675
## 6 0.8675
##
## Group:sexMale
## 2 3 4 5 6 7
## 1 0.7999 1 0.8872 0.9338 0.936 1
## 2 1 0.8872 0.9338 0.936 1
## 3 0.8872 0.9338 0.936 1
## 4 0.9338 0.936 1
## 5 0.936 1
## 6 1 dipper.phisex.ptime =mark(dipper.processed,dipper.ddl,
model.parameters=list(Phi=Phisex,p=ptime))##
## Output summary for CJS model
## Name : Phi(~sex)p(~time)
##
## Npar : 8
## -2lnL: 664.3
## AICc : 680.6
##
## Beta
## estimate se lcl ucl
## Phi:(Intercept) 0.17283 0.1472 -0.1156 0.4613
## Phi:sexMale 0.08149 0.1943 -0.2992 0.4622
## p:(Intercept) 1.29472 0.7436 -0.1628 2.7522
## p:time3 0.80519 1.1642 -1.4767 3.0870
## p:time4 0.65429 1.0020 -1.3096 2.6182
## p:time5 0.99873 0.9454 -0.8542 2.8516
## p:time6 1.46686 1.0304 -0.5527 3.4864
## p:time7 2.04060 3.2087 -4.2484 8.3296
##
##
## Real Parameter Phi
## Group:sexFemale
## 1 2 3 4 5 6
## 1 0.5431 0.5431 0.5431 0.5431 0.5431 0.5431
## 2 0.5431 0.5431 0.5431 0.5431 0.5431
## 3 0.5431 0.5431 0.5431 0.5431
## 4 0.5431 0.5431 0.5431
## 5 0.5431 0.5431
## 6 0.5431
##
## Group:sexMale
## 1 2 3 4 5 6
## 1 0.5632 0.5632 0.5632 0.5632 0.5632 0.5632
## 2 0.5632 0.5632 0.5632 0.5632 0.5632
## 3 0.5632 0.5632 0.5632 0.5632
## 4 0.5632 0.5632 0.5632
## 5 0.5632 0.5632
## 6 0.5632
##
##
## Real Parameter p
## Group:sexFemale
## 2 3 4 5 6 7
## 1 0.7849 0.8909 0.8753 0.9083 0.9406 0.9656
## 2 0.8909 0.8753 0.9083 0.9406 0.9656
## 3 0.8753 0.9083 0.9406 0.9656
## 4 0.9083 0.9406 0.9656
## 5 0.9406 0.9656
## 6 0.9656
##
## Group:sexMale
## 2 3 4 5 6 7
## 1 0.7849 0.8909 0.8753 0.9083 0.9406 0.9656
## 2 0.8909 0.8753 0.9083 0.9406 0.9656
## 3 0.8753 0.9083 0.9406 0.9656
## 4 0.9083 0.9406 0.9656
## 5 0.9406 0.9656
## 6 0.9656Make an AIC table
dipper.table <- collect.models(type = "CJS")
dipper.table## model npar AICc DeltaAICc weight Deviance
## 2 Phi(~Flood)p(~Flood) 4 668.2 0.000 6.715e-01 77.58
## 1 Phi(~1)p(~1) 2 670.9 2.710 1.732e-01 58.16
## 3 Phi(~sex)p(~1) 3 672.7 4.577 6.809e-02 84.20
## 7 Phi(~time)p(~1) 7 674.0 5.842 3.617e-02 77.25
## 4 Phi(~sex)p(~sex) 4 674.2 6.091 3.194e-02 83.67
## 8 Phi(~time)p(~sex) 8 675.5 7.348 1.704e-02 76.68
## 6 Phi(~sex)p(~time) 8 680.6 12.494 1.300e-03 81.83
## 10 Phi(~time)p(~time) 12 681.7 13.550 7.668e-04 74.47
## 5 Phi(~sex)p(~sex * time) 14 691.3 23.116 6.419e-06 79.77
## 9 Phi(~time)p(~sex * time) 18 692.2 24.058 4.007e-06 72.06Flood model wins by an appreciable margin. None of the others are as good as the model with one mean phi and one mean p.