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Model Identification & Model Selection

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Model Identification & Model Selection

With focus on Mark/Recapture Studies

- 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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- 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

All models are false, but some are useful.

George Box

- Quantitative measures of the validity and utility of models
- Social control on the behavior of scientists

- Illuminating
- Communicable
- Defensible
- Transferable

- All statistical methods exist in the mind only, but some are useful.
- Mark Taper

- Frequentist Statistics - Bayesian Statistics
- Error Statistics – Evidential Stats – Bayesian Stats

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

- Single model only
- P-value= Prob of discrepancy at least as great as observed by chance.
- Not terribly useful for model selection

- 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.

- 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.

In conclusion framework, multiple statistical metrics not “incompatible”

All are tools for aiding scientific thought

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

- Likelihood ratios
- Differences in information criteria
- Others available
- E.g. Log(Jackknife prediction likelihood ratio)

- 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.

- Estimation framework rather than testing framework
- Confidence intervals rather than testing
- Rejection of “true model formalism”

- 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

Empirical Distribution - “Truth”

Model 1

Potentially better model

Model adequacy measure

Evidence measure

- Badness of fit measures & goodness of fit tests
- Comparison of model to a nonparametric estimate of true distribution.
- G2-Statistic
- Helinger Distance
- Pearson χ2
- Neymanχ2

- Badness of fit is the scope for improvement
- Evidence for one model relative to another model is the difference of badness of fit.

- 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.

Note cutoff are arbitrary and vary with scale

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

- Mark/Recapture based on multinomial likelihoods
- Observation is a capture history not a session

- 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

- χ2 goodness of fit test for most general model
- If reject H0 estimate variance inflation
- c^ = χ2 /df
- Correct fit component of IC & redo selection

- 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!

- Explicitly include random component in model
- Then redo model selection

- Bootstrapped median c^
- Model selection with Jackknifed prediction likelihood

- 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

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

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

'

- M subset size, P= # of possible terms

- 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

Akaike Weights & Model Averaging

Beware, there be dragons here!

- “Relative likelihood of model i given the data and model set”
- “Weight of evidence that model i most appropriate given data and model set”

- “Conditional” Variance
- Conditional on selected model

- “Unconditional” Variance.
- Actually conditional on entire model set

- 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

- 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

Model adequacy

Model adequacy

Model dimension

Model dimension

- Bootstrap Data
- Fit model set & select best model
- Estimate derived parameter θ from best model
- Accumulate θ

Repeat

Within

Time

Constraints

Mean or median θ with percentile confidence intervals