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Practical Model Selection and Multi-model Inference using R Modified from on a presentation by : Eric Stolen and Dan Hunt Theory • This is the link with science, which is about understanding how the world works Indigo Snake Habitat selection David R. Breininger, M. Rebecca Bolt, Michael L. Legare, John H. Drese, and Eric D. Stolen Source: Journal of Herpetology, 45(4):484-490. 2011. – Animal perception – Evolutionary Biology – Population Demography http://www.seaworld.org/animal-info/animal-bytes/spooky-safari/eastern-indigo-snake.htm Hypotheses • To use the Information-theoretic toolbox, we must be able to state a hypothesis as a statistical model (or more precisely an equation which allows us to calculate the maximum likelihood of the hypothesis) http://www.seaworld.org/animal-info/animal-bytes/spooky-safari/eastern-indigo-snake.htm Multiple Working Hypotheses • • We operate with a set of multiple alternative hypotheses (models) The many advantages include safeguarding objectivity, and allowing rigorous inference. Chamberlain (1890) Strong Inference - Platt (1964) Karl Popper (ca. 1960)– Bold Conjectures Deriving the model set • • • • This is the tough part (but also the creative part) much thought needed, so don’t rush collaborate, seek outside advice, read the literature, go to meetings… How and When hypotheses are better than What hypotheses (strive to predict rather than describe) Models – Indigo Snake example David R. Breininger, M. Rebecca Bolt, Michael L. Legare, John H. Drese, and Eric D. Stolen Source: Journal of Herpetology, 45(4):484-490. 2011. • • • • • • Study of indigo snake habitat use Response variable: home range size ln(ha) SEX Land cover – 2-3 levels (lC2) weeks = effort/exposure Science question: “Is there a seasonal difference in habitat use between sexes?” Models – Indigo Snake example SEX land cover type (lc2) weeks SEX + lc2 SEX + weeks llc2 + weeks SEX + lc2 + weeks SEX + lc2 + SEX * lc2 SEX + lc2 + weeks + SEX * lc2 http://www.herpnation.com/hn-blog/indigo-snake-survivaldemographics/?simple_nav_category=john-c-murphy Models – Indigo Snake example SEX land cover type (lc2) weeks SEX + lc2 SEX + weeks llc2 + weeks SEX + lc2 + weeks SEX + lc2 + SEX * lc2 SEX + lc2 + weeks + SEX * lc2 Modeling • Trade-off between precision and bias • Trying to derive knowledge / advance learning; not “fit the data” • Relationship between data (quantity and quality) and sophistication of the model Bias 2 Precision-Bias Trade-off Model Complexity – increasing umber of Parameters Bias 2 variance Precision-Bias Trade-off Model Complexity – increasing umber of Parameters Bias 2 variance Precision-Bias Trade-off Model Complexity – increasing umber of Parameters Kullback-Leibler Information • Basic concept from Information theory • The information lost when a model is used to represent full reality • Can also think of it as the distance between a model and full reality Kullback-Leibler Information Truth / reality G1 (best model in set) G2 G3 Kullback-Leibler Information Truth / reality G1 (best model in set) G2 G3 Kullback-Leibler Information Truth / reality G1 (best model in set) G2 G3 Kullback-Leibler Information Truth / reality G1 (best model in set) G2 G3 The relative difference between models is constant Akaike’s Contributions • Figured out how to estimate the relative Kullback-Leibler distance between models in a set of models • Figured out how to link maximum likelihood estimation theory with expected K-L information • An (Akaike’s) Information Criteria • AIC = -2 loge (L{modeli }| data) + 2K AICci = -2*loge (Likelihood of model i given the data) + 2*K (n/(n-K-1)) or = AIC + 2*K*(K+1)/(n-K-1) (where K = the number of parameters estimated and n = the sample size) AICcmin = AICc for the model with the lowest AICc value Di = AICci– AICcmin wi exp( 0.5 D i ) n exp( 0.5 D ) r 1 wi =Prob{gi | data} Model Probability (model probabilities) evidence ratio of model i to model j = wi / wj Least Squares Regression AIC = n loge (s2) + 2*K (n/(n-K-1)) Where s2 RSS / n Counting Parameters: K = number of parameters estimated Least Square Regression K = number of parameters + 2 (for intercept & s) Counting Parameters: K = number of parameters estimated Logistic Regression K = number of parameters + 1 (for intercept) Comparing Models Model selection based on AICc : mod4 mod7 mod1 mod5 mod2 mod6 mod3 K 4 5 3 4 3 4 3 AICc Delta_AICc AICcWt Cum.Wt LL 112.98 0.00 0.71 0.71 -51.99 114.89 1.91 0.27 0.98 -51.67 121.52 8.54 0.01 0.99 -57.47 122.27 9.29 0.01 1.00 -56.64 125.93 12.95 0.00 1.00 -59.67 128.34 15.36 0.00 1.00 -59.67 141.26 28.28 0.00 1.00 -67.34 Model 1 = “SEX ", Model 2 = "ha.ln ~ lc2", Model 3 = "ha.ln ~ weeks ", Model 4 = "ha.ln ~ SEX + lc2", Model 5 = "ha.ln ~ SEX + weeks", Model 6 = "ha.ln ~ lc2 + weeks", Model 7 = "ha.ln ~ SEX + lc2 + weeks" Model Averaging Predictions R Y wi Yi i 1 Model Averaging Predictions R Y wi Yi i 1 Model-averaged prediction Model Averaging Predictions R Y wi Yi i 1 Prediction from modeli Model Averaging Predictions R Y wi Yi i 1 Weight modeli Model Averaging Parameters R wi i i 1 Model-averaged parameter estimate Unconditional Variance Estimator var wi var i gi i i 1 R 2 Unconditional Variance Estimator SE var 95% CI 1.96 * SE