practical model selection and multi-model inference using r
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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. - PowerPoint PPT PresentationTRANSCRIPT
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 selectionDavid 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 exampleDavid 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 exampleSEXland cover type (lc2)weeksSEX + lc2SEX + weeksllc2 + weeksSEX + lc2 + weeksSEX + lc2 + SEX * lc2 SEX + lc2 + weeks + SEX * lc2
http://www.herpnation.com/hn-blog/indigo-snake-survival-demographics/?simple_nav_category=john-c-murphy
SEXland cover type (lc2)weeksSEX + lc2SEX + weeksllc2 + weeksSEX + lc2 + weeksSEX + lc2 + SEX * lc2 SEX + lc2 + weeks + SEX * lc2
Models – Indigo Snake example
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
Precision-Bias Trade-offB
ias
2
Model Complexity – increasing umber of Parameters
Precision-Bias Trade-offB
ias
2
varia
nce
Model Complexity – increasing umber of Parameters
Precision-Bias Trade-offB
ias
2
varia
nce
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
G3The 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
i = AICci– AICcmin
wi =Prob{gi | data} Model Probability (model probabilities)
evidence ratio of model i to model j = wi / wj
n
r
iiw
1
)5.0exp(
)5.0exp(
Least Squares Regression
AIC = n loge () + 2*K (n/(n-K-1))
Where RSS / n
Counting Parameters:
K = number of parameters estimated
Least Square Regression K = number of parameters + 2 (for intercept &
Counting Parameters:
K = number of parameters estimated
Logistic Regression K = number of parameters + 1 (for intercept
Comparing Models
Model selection based on AICc :
K AICc Delta_AICc AICcWt Cum.Wt LLmod4 4 112.98 0.00 0.71 0.71 -51.99mod7 5 114.89 1.91 0.27 0.98 -51.67mod1 3 121.52 8.54 0.01 0.99 -57.47mod5 4 122.27 9.29 0.01 1.00 -56.64mod2 3 125.93 12.95 0.00 1.00 -59.67mod6 4 128.34 15.36 0.00 1.00 -59.67mod3 3 141.26 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
iiiYwY
1
R
iiiYwY
1
Model-averaged prediction
Model Averaging Predictions
R
iiiYwY
1
Prediction from modeli
Model Averaging Predictions
R
iiiYwY
1
Weight modeli
Model Averaging Predictions
R
i
iiw1
Model-averaged parameter estimate
Model Averaging Parameters
Unconditional Variance Estimator
2
1
varvar i
R
iiii gw
varSE
SECI *96.1%95
Unconditional Variance Estimator