Model Identification & Model Selection

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Model Identification & Model Selection

With focus on Mark/Recapture Studies

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Overview

• Basic inference from an evidentialist perspective

• Model selection tools for mark/recapture– AICc & SIC/BIC– Overdispersed data– Model set size– Multimodel inference

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DATA /* 01 */ 1100000000000000 1 1 1.16 27.7 4.19; /* 04 */ 1011000000000000 1 0 1.16 26.4 4.39; /* 05 */ 1011000000000000 1 1 1.08 26.7 4.04; /* 06 */ 1010000000000000 1 0 1.12 26.2 4.27; /* 07 */ 1010000000000000 1 1 1.14 27.7 4.11; /* 08 */ 1010110000000000 1 1 1.20 28.3 4.24; /* 09 */ 1010000000000000 1 1 1.10 26.4 4.17; /* 10 */ 1010110000000000 1 1 1.42 27.0 5.26; /* 11 */ 1010000000000000 1 1 1.12 27.2 4.12; /* 12 */ 1010101100000000 1 1 1.11 27.1 4.10; /* 13 */ 1010101100000000 1 0 1.07 26.8 3.99; /* 14 */ 1010101100000000 1 0 0.94 25.2 3.73; /* 15 */ 1010101100000000 1 0 1.24 27.1 4.58; /* 16 */ 1010101100000000 1 0 1.12 26.5 4.23; /* 17 */ 1010101000000000 1 1 1.34 27.5 4.87; /* 18 */ 1010101011000000 1 0 1.01 27.2 3.71; /* 19 */ 1010101011000000 1 0 1.04 27.0 3.85; /* 20 */ 1010101000000000 1 1 1.25 27.6 4.53; /* 21 */ 1010101011000000 1 0 1.20 27.6 4.35; /* 22 */ 1010101011000000 1 0 1.28 27.0 4.74; /* 23 */ 1010101010110000 1 0 1.25 27.2 4.59; /* 24 */ 1010101010110000 1 0 1.09 27.5 3.96; /* 25 */ 1010101010110000 1 1 1.05 27.5 3.82; /* 26 */ 1010101010101100 1 0 1.04 25.5 4.08; /* 27 */ 1010101010101010 1 0 1.13 26.8 4.22; /* 28 */ 1010101010101010 1 1 1.32 28.5 4.63; /* 29 */ 1010101010101010 1 0 1.18 25.9 4.56; /* 30 */ 1010101010101010 1 0 1.07 26.7 4.01; /* 31 */ 1010101010101010 1 1 1.26 26.9 4.68; /* 32 */ 1010101010101010 1 0 1.27 27.6 4.60; /* 33 */ 1010101010101010 1 0 1.08 26.0 4.15; /* 34 */ 1010101010101010 1 1 1.11 27.0 4.11; /* 35 */ 1010101010101010 1 0 1.15 27.1 4.24; /* 36 */ 1010101010101010 1 0 1.03 26.5 3.89; /* 37 */ 1010101010101010 1 0 1.16 27.5 4.22;

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Models carry the meaning in science

• Model

– Organized thought

• Parameterized Model

– Organized thought connected to reality

Science is a cyclic process of model reconstruction and model reevaluation

• Comparison of predictions with observations/data

• Relative comparisons are evidence

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All models are false, but some are useful.

George Box

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Statistical Inferences

• Quantitative measures of the validity and utility of models

• Social control on the behavior of scientists

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Scientific Model Selection Criteria

• Illuminating• Communicable• Defensible• Transferable

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Common Information Criteria

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Statistical Methods are Tools

• All statistical methods exist in the mind only, but some are useful.

– Mark Taper

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Classes of Inference

• Frequentist Statistics - Bayesian Statistics

• Error Statistics – Evidential Stats – Bayesian Stats

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Two key frequencies in frequentist statistics

• Frequency definition of probability• Frequency of error in a decision rule

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Null H tests with Fisherian P-values

• Single model only• P-value= Prob of discrepancy at least as great

as observed by chance.• Not terribly useful for model selection

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Neyman-Pearson Tests

• 2 models• Null model test along a maximally sensitive axis.• Binary response: Accept Null or reject Null• Size of test (α) describes frequency of rejecting null in

error.– Not about the data, it is about the test.– You support your decision because you have made it with a

reliable procedure.• N-P test tell you very little about relative support for

alternative models.

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Decisions vs. Conclusions

• Decision based inference reasonable within a regulatory framework. – Not so appropriate for science

• John Tukey (1960) advocated seeking to reach conclusions not making decisions.– Accumulate evidence until a conclusion is very

strongly supported.– Treat as true.– Revise if new evidence contradicts.

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In conclusion framework, multiple statistical metrics not “incompatible”

All are tools for aiding scientific thought

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Statistical Evidence

• Data based estimate of the relative distance between two models and “truth”

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Common Evidence Functions

• Likelihood ratios• Differences in information criteria• Others available

– E.g. Log(Jackknife prediction likelihood ratio)

Model Adequacy

• Bruce Lindsay

• The discrepancy of a model from truth• Truth represented by an empirical

distribution function, • A model is “adequate” if the estimated

discrepancy is less than some arbitrary but meaningful level.

Model Adequacy and Goodness of Fit

• Estimation framework rather than testing framework

• Confidence intervals rather than testing• Rejection of “true model formalism”

Model Adequacy, Goodness of Fit, and Evidence

• Adequacy does not explicitly compare models

• Implicit comparison

• Model adequacy interpretable as bound on strength of evidence for any better model

• Unifies Model Adequacy and Evidence in a common framework

Model adequacy interpreted as a bound on evidence for a possibly better model

Empirical Distribution - “Truth”

Model 1

Potentially better model

Model adequacy measure

Evidence measure

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Goodness of fit misnomer

• Badness of fit measures & goodness of fit tests• Comparison of model to a nonparametric

estimate of true distribution.– G2-Statistic– Helinger Distance– Pearson χ2

– Neyman χ2

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Points of interest

• Badness of fit is the scope for improvement• Evidence for one model relative to another

model is the difference of badness of fit.

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ΔIC estimates differences of Kullback-Leibler Discrepancies

• ΔIC = log(likelihood ratio) when # of parameters are equal

• Complexity penalty is a bias correction to adjust of increase in apparent precision with an increase in # parameters.

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Evidence Scales

L/R Log2 ln Log10

Weak <8 <3 <2 <1

Strong 8 - <32 3 - <5 2 - <7 1 - <2

Very Strong > 32 > 5 > 7 > 2

Note cutoff are arbitrary and vary with scale

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Which Information Criterion?

• AIC? AICc ? SIC/BIC?• Don’t use AIC• 5.9 of one versus 6.1 of the other

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What is sample size for complexity penalty?

• Mark/Recapture based on multinomial likelihoods

• Observation is a capture history not a session

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To Q or not to Q?

• IC based model selection assumes a good model in set.

• Over-dispersion is common in Mark/Recapture data– Don’t have a good model in set– Due to lack of independence of observations– Parameter estimate bias generally not influenced– But fit will appear too good!– Model selection will choose more highly

parameterized models than appropriate

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Quasi likelihood approach

1) χ2 goodness of fit test for most general model

2) If reject H0 estimate variance inflation3) c^ = χ2 /df4) Correct fit component of IC & redo selection

c

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QICs

Knc

yLQBIC

KnKKK

cyLQAICc

e

e

lnˆ|ˆln2

1122

ˆ|ˆln2

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Problems with Quasilikelihood correction

• C^ is essentially a variance estimate.– Variance estimates unstable without a lot of data

• lnL/c^ is a ratio statistic– Ratio statistics highly unstable if the uncertainty in

the denominator is not trivial• Unlike AICc, bias correction is estimated.

– Estimating a bias correction inflates variance!

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Fixes

• Explicitly include random component in model– Then redo model selection

• Bootstrapped median c^• Model selection with Jackknifed prediction

likelihood

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Large or small model sets?

• Problem: Model Selection Bias– When # of models large relative to data size some

models will have a good fit just by chance• Small

– Burnham & Anderson strongly advocate small model sets representing well thought out science

– Large model sets = “data dredging”• Large

– The science may not be mature– Small model sets may risk missing important factors

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Model Selection from Many Candidates Taper(2004)

SIC(x) = -2In(L) + (In(n) + x)k.

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Performance of SIC(X) with small data set.

N=50, true covariates=10, spurious covariates=30, all models of order <=20, 1.141 X 1014 candidate

models'

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Chen & Chen 2009

• M subset size, P= # of possible terms

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Explicit Tradeoff

• Small model sets– Allows exploration of fine structure and small effects– Risks missing unanticipated large effects

• Large model sets– Will catch unknown large effects– Will miss fine structure

• Large or small model sets is a principled choice that data analysts should make based on their background knowledge and needs

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Akaike Weights & Model Averaging

Beware, there be dragons here!

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Akaike Weights

• “Relative likelihood of model i given the data and model set”

• “Weight of evidence that model i most appropriate given data and model set”

R

m

m

i

iw

12exp

2exp

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Model Averaging

• “Conditional” Variance– Conditional on

selected model

• “Unconditional” Variance.– Actually conditional

on entire model set

2

2

1

1

ˆˆ|ˆˆ

ˆˆ

ii

R

ii

i

R

ii

imVarwVar

w

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Good impulse with Huge Problems

• I do not recommend Akaike weights• I do not recommend model averaging in this

fashion• Importance of good models is diminished by

adding bad models• Location of average influenced by adding

redundant models

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Model Redundancy

• Model Space is not filled uniformly• Models tend to be developed in highly

redundant clusters.• Some points in model space allow few models• Some points allow many

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Redundant models do not add much information

Model dimension

Mod

el a

dequ

acy

Model dimension

Mod

el a

dequ

acy

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A more reasonable approach

1) Bootstrap Data2) Fit model set & select best

model3) Estimate derived parameter θ

from best model4) Accumulate θ

RepeatWithinTime Constraints

Mean or median θ with percentile confidence intervals