Ecological Archives E087-063-A1

Graham E. Forrester and Rachel J. Finley. 2006. Parasitism and a shortage of refuges jointly mediate the strength of density dependence in a reef fish. Ecology 87:1110–1115.

Appendix A. A description of the initial modeling of resighting and survival probabilities from mark–recapture data.

 

Developing a starting model for survival and resighting probability

As a starting point, we selected the general Cormack-Jolly-Seber (CJS) model constructed separately by infection status (parasitism). This model is denoted f(parasitism*t)P(parasitism*t), where t indicates time and * represents an interaction between variables (Lebreton et al. 1992). The CJS model assumes that censuses are instantaneous relative to the interval between censuses. The fact that f is generally high in our study means that our violation of this assumption is likely to have a minor effect on estimates of f and P (Hargrove and Borland 1994). We tested whether the data met other assumptions of the CJS model using goodness-of-fit (GOF) tests implemented with the programs RELEASE v. 3.0 and U-CARE v. 2.02 (Burnham et al. 1987). The four components of the overall GOF test (Test3.SR, Test3.SM, Test2.CT and Test2.CM) evaluate different violations of the CJS assumptions that all marked individuals have equal probabilities of being resighted and surviving.

Goodness-of-fit testing for the mark–recapture model

The overall GOF statistic indicated a lack of fit of the CJS model (c2 = 49.78, df = 15, P < 0.00001). The results of Test3.SR led us to reject the hypothesis of equal survival probability for both parasitized and unparasitized fish (unparasitized fish: χ2 = 17.98, df = 3, P = 0.0004; parasitized fish: χ2 = 15.93, df = 3, P = 0.001). Inspection of the resighting histories used to calculate this test revealed lower survival in the interval following marking than at any time later (Cooch et al. 1996) and our observations suggest that this occurs because gobies are more vulnerable to predators immediately after being captured, marked and released back to the reef. A GOF test accounting for the effect of marking on survival still indicated possible lack of fit (χ 2 =17.363, df = 9, P = 0.043) due to a possible failure of the assumption of equal resighting probability (χ 2 =10.10, df = 4, P = 0.038, test performed on data pooled across parasitism to provide adequate sample sizes). Inspection of the contingency tables used to calculate this test and a directional log odds ratio test (LOR statistic = -2.16, df = 3, P = 0.030) suggests a slight increase in our chance of resighting a fish if it had been seen (or marked) in the preceding census, an effect equivalent to short-term “trap happiness”.

Because there was evidence that the data failed to meet CJS assumptions, we next tested various alternative models to account for these violations as simply as possible, using the software program MARK v. 4.2. To account for reduced survival immediately after marking we made survival a function of time since marking (TSM) (Pradel et al. 1997). TSM was a categorical variable with two groups: one for the interval after the fish was marked and another for all subsequent intervals. Short-term “trap-happiness” is best modeled by making the resighting rate a function of whether the fish was sighted during the preceding census (Pradel 1993). Since the resighting bias in our study was slight, we made a less precise but much simpler correction for this dependence, which took advantage of the fact that most resightings of gobies (341 out of 389) were made in the first census after they were marked. We made P a function of the categorical variable TSM: with one group for the interval immediately after marking and a second for all subsequent intervals.

Modeling f as a function of both TSM and time, led to a marked improvement in model fit, and a more modest improvement was also obtained when P was a function of TSM and time (Table A1). Making both f and P a function of TSM and time yielded a further modest improvement in model fit (model f( parasitism *t*TSM)P(parasitism *t*TSM), Table A1). Several of the parameters in the time-dependent models were, however, not estimable (e.g., 12 of 36 parameters in the model f( parasitism *t*TSM)P(parasitism *t*TSM)), probably because of our small data set. A starting model so complex that several parameters could not be estimated would allow little scope for the addition of yet more parameters to test our main hypothesis about effects of density, parasitism, and refuges. In an attempt to simplify our starting model, we tested models from which time-dependence of f and/or P was eliminated. A model lacking time dependence in both f and P (f(parasitism*TSM)P(parasitism*TSM) fit less well than the fully time-dependent model (f(parasitism*t*TSM)P(parasitism*t*TSM)). It was still, however, a marked improvement over the CJS model (Table A1). We thus selected f(parasitism*TSM)P(parasitism*TSM) as our starting model, because it was by far the simplest model that fit the data.

Effects of covariates on resighting probability

We next compared a set of simple models in which the effect of each covariate (size, density, or refuges) on P was added to the starting model (Table A2). We also tested whether simplifying the starting model by removing the effect of parasitism on P yielded a better overall fit (Table A2). We adopted the strategy of analyzing and setting parameters for P before detailed modeling of f (Lebreton et al. 1992) because, although accurately specifying P is important (MacKenzie and Kendall 2002), these parameters are of less ecological interest than f. Once recapture probabilities are set, they thus cancel out when comparing alternative survival models.

Models in which P was a function of either size or refuges yielded no improvement of fit over the starting model (Table A2) so resighting was modeled without their effects. Removing the effect of parasitism on P from the starting model reduced model fit significantly (Table A2), so we retained separate resighting probabilities for fish with and without parasites. Adding a term for the effect of density allowed us to better predict resighting probability (Table A2), so we used model f(parasitism*TSM)P(parasitism *TSM+density) as our starting model from which to test effects of the covariates on survival.

We tested whether survival was a function of body size by adding a term for this covariate to the starting model. We also made a second check for effects of body size by adding a term for size to the final model selected after examination of the other covariates. The fit of neither the starting model, nor the final model was improved by making survival a function of size (Table A3).

LITERATURE CITED

Burnham, K. P., D. R. Anderson, G. C. White, C. Brownie, and K. H. Pollock. 1987. Design and analysis methods for fish survival experiments based on release-recapture. American Fisheries Society Monograph 5.

Cooch E. G., R. Pradel, and N. Nur. 1996. A practical guide to mark-recapture analysis using SURGE. Centre d'Ecologie Fonctionnelle et Evolutive - CNRS, Montpellier, France.

Hargrove, J. W., and C. H. Borland. 1994. Pooled population parameter estimates from mark-recapture data. Biometrics 50:1129–1141.

Lebreton, J. D., K. P. Burnham, J. Clobert, and D. R. Anderson. 1992. Modeling survival and testing biological hypotheses using marked animals: a unified approach with case studies. Ecological Monographs 62:67–118.

MacKenzie, D. I., and W. L. Kendall. 2002. How should detection probability be incorporated into estimates of relative abundance? Ecology 83:3532.

Pradel, R. 1993. Flexibility in survival analysis from recapture data: handling trap-dependence. Pages 29–37 in J. D. Lebreton, and P. M. North, editors. Marked individuals in the study of bird populations. Birkhaüser Verlag, Basel, Switzerland.

Pradel, R., J. E. Hines, J. D. Lebreton, and J. D. Nichols. 1997. Capture-recapture survival models taking account of transients. Biometrics 53:60–72.

 

TABLE A1. Candidate mark–recapture models screened initially for goodness-of-fit.

Model

AICc

ΔAICc

wi

np

1§

f(parasitism*t*TSM)P(parasitism*t*TSM)

957.3

0.0

0.801

36

2

f(parasitism*t*TSM)P(parasitism*t)

961.0

3.8

0.120

28

3

f(parasitism*t*TSM)P(parasitism*TSM)

962.7

5.4

0.054

28

4

f(parasitism*TSM)P(parasitism*TSM)

964.3

7.0

0.024

8

5

f(parasitism*TSM)P(parasitism*t*TSM)

970.1

12.9

0.001

28

6

f(parasitism*t)P(parasitism*t*TSM)

983.5

26.3

<0.0001

28

7

f(parasitism*t)P(parasitism*t)

989.0

31.8

<0.0001

20

   Notes: ΔAICc is the difference in AICc between the lowest AICc model and AICc of model i; np is the number of parameters; * indicates an interaction between variables. § Indicates the model selected based on AICc and wi/wj.

 

TABLE A2. Candidate mark–recapture models screened for effects of covariates on resighting probability.

Model

AICc

ΔAICc

wi

np

1§

P(parasitism*TSM + density)

958.4

0.0

0.912

9

2

P(parasitism*TSM)

964.3

5.9

0.048

8

3

P(parasitism*TSM + refuges)

965.8

7.4

0.023

9

4

P(parasitism*TSM + size)

966.3

7.9

0.018

9

5

P(TSM)

974.0

15.6

0.000

6

   Notes: For all models, survival is modeled as f(parasitism*TSM). * and + indicate models with and without interactions respectively. For other notation see Table A1.

 

TABLE A3. Mark–recapture models to check for effects of body size on survival. For all models, resighting probability is modeled as P(parasitism*TSM + density). For other notation see Tables A1 and A2.

 

AICc

ΔAICc

wi

np

Models for first check

       

1§

f(parasitism*TSM)

958.4

0.0

0.587

8

2

f(parasitism*TSM + size)

959.1

0.7

0.413

9

         

Models for second check

       

1§

f(parasitism + TSM + parasitism*TSM + density + refuges + density*refuges + parasitism*density + parasitism*refuges + parasitism*density*refuges)

933.6

0.0

0.634

14

2

f(parasitism + TSM + parasitism*TSM + density + refuges + density*refuges +  parasitism*density + parasitism*refuges + parasitism*density*refuges + size)

934.8

1.1

0.366

15



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