Taking the dipper example, how much of the annual variation in apparent survival is explained by the flood covariate? You might fit three nested models: a global model with time-dependence Phi(t), a covariate model for flood Phi(flood), and a constant model Phi(.). The dipper model has 7 occcasions and 6 transitions. The flood model has flood years in the second and third transitions (see page 6-19 of the manual). Detection rates are set as p(t) for all models, but just as an example. That gives the following model selection table:
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Delta AICc Model
Model AICc AICc Weight Likelihood #Par Deviance
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{Phi(flood) p(t)} 674.034 0.00 0.86436 1.0000 8.0000 75.211
{Phi(.) p(t)} 678.748 4.71 0.08185 0.0947 7.0000 82.003
{Phi(t) p(t)} 679.588 5.55 0.05378 0.0622 11.000 74.473
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Selecting the menu options for Tests | ANODEV | Select All | OK produces the following summary table:
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Source df Dev Mean Dev F P
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Uncorrected Total 11 82.003
Grand Mean 7 74.473
Corrected Total 4 7.530
Total Covariate 1 6.792 6.792 27.5959 0.0134
Error 3 0.738 0.246
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The calculations in the ANODEV table follow the basics of the Likelihood ratio test for nested models where differences in deviance are a chi-square statistic, and differences in K are the df (see page 3-14 of the manual). The Uncorrected Total is the deviance of Phi(.), the Grand Mean is the deviance of Phi(t), and the Corrected Total is the difference between the two deviances: 82.003-74.473 = 7.530. The Total Covariate is the difference between the deviances of Phi(.) and Phi(flood): 82.003-75.211 = 6.792. The percentage of annual variation in apparent survival explained by flooding can be calculated as: 6.792/7.530 = 90.2%. The Error is the leftover unexplained variation or 0.738/7.530 = 9.8%. The Mean Dev values are calculated as Dev/df, and the F-Statistic is then the ratio of the two Mean Dev values: 6.792/0.246 = 27.5959. Flooding explains a significant amount of the annual variation in apparent survival (F1,3 = 27.6, P = 0.013). The above models are based on fixed effects, but Appendix D of the Mark manual addresses random effects. Corrections welcome if I've made any errors.
Regards,
Brett K. Sandercock
Norwegian Institute for Nature Research