Wars and Whales: Extensions and Applications of Confidence ... · Overview Four papers in total,...

Wars and Whales: Extensions and Applications ofConfidence Curves and Focused Model Selection

Celine Cunen

Department of Mathematics, University of Oslo

07/12/2018

1/20

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

γ

conf

iden

ce c

urve

s ●

●

●

●

●

●

0.00235 0.00240 0.00245 0.00250

−0.

0076

−0.

0074

−0.

0072

−0.

0070

−0.

0068

FIC

µ

M0

M1

M2

M3

M4

M5

2/20

Overview

Four papers in total, ranging from primarily methodological toprimarily applied.

Confidence curves (and confidence distributions) appear in allthe papers.

The Focused information criterion (FIC) appears in twopapers.

Change-points play a role in two (even three) papers.

Real data problems are treated in all papers, for instance:Wars;Whales;Tirant lo Blanch.

3/20

Confidence distributions (CDs)

The goal of statistical inference is to obtain data-dependentstatements about unknown parameters, usually with anattached uncertainty measurement.Statements? Point-estimates; tests; confidence intervals;densities (usually for Bayesians only);...

0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

µ

dens

ity

4/20


CD ≈ a posterior without having to specify a priora sample-dependent distribution function on the parameter spacecan be used for inference (for example for constructing confidenceintervals of all levels)

0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

µ

conf

iden

ce d

istr

ibut

ion

0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

µ

conf

iden

ce d

ensi

ty

0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

µ

conf

iden

ce c

urve

5/20


CD ≈ a posterior without having to specify a priora sample-dependent distribution function on the parameter spacecan be used for inference (for example for constructing confidenceintervals of all levels)

0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

µ

conf

iden

ce d

istr

ibut

ion

0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

µ

conf

iden

ce d

ensi

ty

0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

µco

nfid

ence

cur

ve

5/20

Requirements for CDs

DefinitionA function C(θ,Y ) is called a confidence distribution for aparameter θ if:

C(θ,Y ) is a cumulative distribution function on theparameter spaceat the true parameter value θ = θ0, C(θ0,Y ) as a function ofthe random sample Y follows the uniform distribution U[0,1]

The second requirement ensures that all confidence intervalshave the correct coverage.

We will typically construct CDs and confidence curves byexact or approximate pivots.

Note that any method producing confidence intervals fulfillingthese requirements can be used to make CDs (no matter theunderlying paradigm).

6/20

PapersPaper ICunen, Hermansen, and Hjort (2018). Confidence distributions forchange-points and regime shifts. Journal of Statistical Planning andInference 195, 14–34.

Paper IICunen, Hjort, and Nygard (2018). Statistical Sightings of Better Angels:Analysing the Distribution of Battle Deaths in Interstate Conflict overTime. Invited to submit a revision to Journal of Peace Research.

Paper IIICunen, Walløe, and Hjort (2018). Focused model selection for linearmixed models, with an application to whale ecology. Invited to resubmitto Annals of Applied Statistics.

Paper IVCunen and Hjort (2018). Combining information across diverse sources:The II-CC-FF paradigm. Submitted for publication.

7/20

I: Change-point problemsWe have a sequence of observations,

and we would like to find thepoint of maximal change - and quantify the degree of change

.

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●●

●

●

●●

●

●●

●

●

●●●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●●

●

●

●

●

●●

●

●

●

●●

●

●

●●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●

●●

●●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●●

●

●

●●●

●

●

●●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●

●●●

●

●●

●●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●●

●●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●●●

●

●●

●●

●●●●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

3.8

4.0

4.2

4.4

4.6

4.8

chapter

aver

age

wor

d le

ngth

0 175 350

and we assume the following model,yi ∼ N(µL, σ

2L/mi ) for i ≤ τ,

yi ∼ N(µR , σ2R/mi ) for i ≥ τ + 1.

8/20

I: Change-point problems

We have a sequence of observations, and we would like to find thepoint of maximal change - and quantify the degree of change.

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●●

●

●

●●

●

●●

●

●

●●●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●●

●

●

●

●

●●

●

●

●

●●

●

●

●●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●

●●

●●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●●

●

●

●●●

●

●

●●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●

●●●

●

●●

●●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●●

●●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●●●

●

●●

●●

●●●●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

3.8

4.0

4.2

4.4

4.6

4.8

chapter

aver

age

wor

d le

ngth

µL = 3.96

σL = 3.46

µR = 4.13

σR = 4.40

0 175 350

●

0.0

0.2

0.4

0.6

0.8

1.0

τ

conf

iden

ce c

urve

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ● ●

● ● ●

● ● ●

● ● ●

● ● ●

● ● ● ●

● ● ● ●

● ● ● ●

● ● ● ●

● ● ● ●

● ● ● ●

● ● ● ●

● ● ● ● ●

● ● ● ● ●

● ● ● ● ●

● ● ● ● ●

● ● ● ● ●

● ● ● ● ●

● ● ● ● ●

● ● ● ● ● ●

● ● ● ● ● ●

● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

τ = 345

329 350 371

8/20

I: Two methods

We assume

y1, . . . , yτ ∼ f (y , θL), yτ+1, . . . , yn ∼ f (y , θR).

Method A relies on testing the homogeneity of the sequences(y1, . . . , yτ ) and (yτ+1, . . . , yn) at each potential change-point τ .Say we have a test statistic Za,b for the sequence (ya, . . . , yb) withnull distribution Ga,b. For each level α the confidence set is givenby

R(τ) = {τ : Z1,τ ≤ G−11,τ (√α) and Zτ+1,n ≤ G−1

τ+1,n(√α)}.

Flexible. Makes no a priori assumptions on the existence of achange.

9/20

I: Two methodsWe assume

y1, . . . , yτ ∼ f (y , θL), yτ+1, . . . , yn ∼ f (y , θR).

Method B uses the log-likelihood profile

`n,prof(τ) = max{`1,τ (θL) + `τ+1,n(θR) : all θL, θR}= `1,τ (θL) + `τ+1,n(θR).

This leads (i) to the ML estimate τ ; (ii) to the deviance functionD(τ, y) = 2{`n,prof(τ)− `n,prof(τ)}. We use

cc(τ, yobs) = Prτ{D(τ,Y ) < D(τ, yobs)} = Kτ (D(τ, yobs)),

with Kτ (x) the c.d.f. of the random D(τ,Y ). This requiressimulations for each candidate τ . We can also compute aconfidence curve for the degree of change ρ = d(θL, θR) by similarmethods.More powerful. Explicitly assumes the existence of a change.

9/20

II: The long peace?Current debate: has there been a (significant) decrease in the sizesof wars?

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●●

●

●●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

● ●

●

●

●●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

1850 1900 1950 2000

68

1012

1416

year

war

siz

es (

log

battl

e de

aths

)

We investigate the related question: when is the point of maximalchange in this sequence? And what is the magnitude of thischange?

10/20

II: The long peace?Current debate: has there been a (significant) decrease in the sizesof wars?

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●●

●

●●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

● ●

●

●

●●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

1850 1900 1950 2000

68

1012

1416

year

war

siz

es (

log

battl

e de

aths

)

We investigate the related question: when is the point of maximalchange in this sequence? And what is the magnitude of thischange? 10/20

II: Change-point analysis with method BAfter careful modelling of the war size distribution, and usingmethod B from the previous paper, we find:

a point estimate in τ = 1950.483→ the Korean War.

●●●●●●

1867 1881 1895 1909 1923 1937 1951 1965 1979 1993

0.0

0.2

0.4

0.6

0.8

1.0

year

conf

iden

ce s

ets

for

the

chan

gepo

int

●●●●●●

●●●●●●

●●●●●●

●●●●●●

●●●●●●

●●●●●●

●●●●●●

●●●●●●

●●●●●●

●●●●●●

●●●●●●

●● ●●●●●●●●●

●● ●●●●●●●●●

●● ●●●●●●●●●

●● ●●●●●●●●●

●● ●●●●●●●●● ●●●

●● ●●●●●●●●● ●●●

●●●●●●●●●●●● ●●●●●●●●● ●●●

●●●●●●●●●●●● ●●●●●●●●● ●●●●

●●●●●●●●●●●● ●●●●●●●●● ●●●●

●●●●●●●●●●●● ●●●●●●●●● ●●●● ●

●●●●●●●●●●●● ●●●●●●●●● ●●●● ●●●●●●●

●●●●●●●●●●●● ●●●●●●●●● ●●●●●● ●●●●●●●

●●●●●●●●●●●●● ●●●●●●●●● ●●●●●● ●●●●●●●

●●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●● ●●●●●●●

●●●●●●●●●●●●● ●●●●●●●●●●●● ●●●●●● ●●●●●●●

●●●●●●●●●●●●● ●●●●●●●●●●●● ●●●●●● ●●●●●●●

●●●●●●●●●●●●● ●●●●●●●●●●●● ●●●●●● ●●●●●●●

●●●●●●●●●●●●● ●●●●●●●●●●●● ●●●●●● ●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●●

● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●

● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●

● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●

● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●

● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●

● ●●●●●●● ●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●

●● ●●●●●●● ●●● ●●●● ●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●

●● ●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●● ●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●

●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●● ●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●

●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

11/20

II: The degree of changeLet ρ1 = φ0.50,L/φ0.50,R (dashed) and ρ2 = φ0.75,L/φ0.75,R (fullydrawn),

2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

ρ

conf

iden

ce

At ρ = 1 we are testing H0: the world is constant.12/20

II: Other efforts

Modelling of the full sequence of war sizes

Modelling only the largest wars

Model selection (FIC)

Including covariates in the change-point analysis (democracyscores)

Analysing the timings between wars

13/20

III: Focused model selection for linear mixed modelsMotivation: are Antarctic Minke whales becoming thinner?

●●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●●●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●●●●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●●

●

●

●

●

●

●

● ●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

5 10 15

1.0

1.5

2.0

2.5

year

fat w

eigh

t

Models:

M0 : Y ∼Year + BodyLength + Sex + Diatom + Date + Date2

+ Latitude + Sex ∗ FetusLength + Sex ∗ Diatom + Diatom ∗ Date+ Diatom ∗ Date2 + Latitude ∗ Date + Latitude ∗ Date2

+ BodyLength ∗ Date ∗ Sex + (1 + Date + Date2|Year).

. . .

M5 : Y ∼ Year + BodyLength + Sex + Date + (1 + Date |Year).

Which model gives the most precise estimates of βyear? FIC offersan anwer to this question, in particular FIC ranks models accordingto their estimated mean-squared error FIC(M) = mseM ,

mseM = E (βM,year − βyear)2 = Var βM,year + (E βM,year − βyear)2.

14/20

III: Focused model selection for linear mixed modelsMotivation: are Antarctic Minke whales becoming thinner?

Models:

M0 : Y ∼Year + BodyLength + Sex + Diatom + Date + Date2

+ Latitude + Sex ∗ FetusLength + Sex ∗ Diatom + Diatom ∗ Date+ Diatom ∗ Date2 + Latitude ∗ Date + Latitude ∗ Date2

+ BodyLength ∗ Date ∗ Sex + (1 + Date + Date2|Year).

. . .

M5 : Y ∼ Year + BodyLength + Sex + Date + (1 + Date |Year).

Which model gives the most precise estimates of βyear? FIC offersan anwer to this question, in particular FIC ranks models accordingto their estimated mean-squared error FIC(M) = mseM ,

mseM = E (βM,year − βyear)2 = Var βM,year + (E βM,year − βyear)2.

14/20

III: Whale results

●

●

●

●

●

●

0.00235 0.00240 0.00245 0.00250

−0.

0076

−0.

0074

−0.

0072

−0.

0070

−0.

0068

FIC

β

M0

M1

M2

M3

M4

M5

Model M5 is considered the best! (note that the scale is tons)15/20

III: Some detailsThe whale models are linear mixed effect models (LMEs):

yi ∼ Nmi

(Xiβ, σ

2(I + ZiDZ ti )),

with i = 1, . . . , n natural groups of mi observations. We have afocus parameter µ = µ(β, σ,D) and estimate it by µ = µ(β, σ, D).

In the candidate models yi ∼ Nmi

(XM,iβM , σ

2M(I + ZM,iDMZ t

M,i )),

we have µM = µ(βM , σM , DM).

The expectations and variances in the FIC formulas are computedwrt the wide model, using the following joint approximatedistribution, ( √

n(µ− µtrue)√n(µM − µM,0)

)≈d N2

(0,(νwide νM,cνM,c νM

)),

with νwide = ctJ−1n c, νM,c = ctJ−1

n CM,nJ−1M,ncM , νM = ct

MJ−1M,nKM,nJ−1

M,ncM .We have explicit formulas for all the necessary quantities.

16/20

IV: Confidence curves for combination of informationWe would like to combine independent sources of informationi = 1, . . . , k.

They might inform on some common parameter φ(meta-analysis);or on different parameters which inform on some overallparameter φ = φ(ψ1, . . . , ψk).From sources we might have the full dataset, from others wemight have summaries only.The sources may be of different quality.

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

γ

conf

iden

ce c

urve

s

17/20

IV: II-CC-FF – a 3-step recipeCombining information, for inference about a focus parameterφ = φ(ψ1, . . . , ψk):II: Independent Inspection: From data source yi to estimates andintervals, in the form of a confidence distribution/curve:

yi =⇒ Ci (ψi )

CC: Confidence Conversion: From the confidence distribution to aconfidence log-likelihood,

Ci (ψi ) =⇒ `c,i (ψi )

FF: Focused Fusion: Use the combined confidence log-likelihood`f (ψ1, . . . , ψk) =

∑ki=1 `c,i (ψi ) to construct a CD for the given

focus φ = φ(ψ1, . . . , ψk), often via profiling:

`f (ψ1, . . . , ψk) =⇒ Cfusion(φ)

18/20

IV: Some other whalesSource 1 informs on ψ1 the North Atlantic humpback whaleabundance in 1995.Source 2 informs on ψ2 the North Atlantic humpback whaleabundance in 2001.Both sources only report non-sufficient summaries (3 quantiles).

What can we learn about the annual growth rate?ρ = (ψ2 − ψ1)/(6ψ1)

0 5000 15000 25000

0.0

0.2

0.4

0.6

0.8

1.0

whale abundance

conf

iden

ce c

urve

s

−0.1 0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

ρ19/20

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

γ

conf

iden

ce c

urve

s ●

●

●

●

●

●

0.00235 0.00240 0.00245 0.00250

−0.

0076

−0.

0074

−0.

0072

−0.

0070

−0.

0068

FIC

µ

M0

M1

M2

M3

M4

M5

20/20

Wars and Whales: Extensions and Applications of Confidence ... · Overview Four papers in total,...

Documents

Transcript of Wars and Whales: Extensions and Applications of Confidence ... · Overview Four papers in total,...