Inferring biadditive models within the Bayesian paradigm

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Inferring biadditive models within the Bayesian

paradigm

Julie Josse, Jean-Baptiste Denis

To cite this version:

Julie Josse, Jean-Baptiste Denis. Inferring biadditive models within the Bayesian paradigm. [Technical

Report] R2012-4, 2012. �hal-02806657�

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Paradigm Julie Josse

1

Jean-Baptiste Denis

2

Rapport te hnique 2012-4,50 pp.

Unité Mathématiques etInformatiqueAppliquées

INRA Domaine deVilvert 78352Jouy-en-Josas edex Fran e

1

Julie.JosseAgro ampus-Ouest.Fr

2

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A rst version of this te hni al report has been prepared for a talk given atthe extraordinary MIAJ seminarof April,4.

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So alled biadditive models (most ommonly known as ammi models for additive main ee t and multipli ative intera tion) are frequently used to interpret the main traits of two ways data, for instan e for the interpretation of genotype by environment intera tions. Linked with p ate hni s, they provide e ient empiri aldes riptions of matrix stru tures.

The use ofBayesian approa hes instatisti alanalysis inin reasing formany statisti almodels due to the new omputer apa ities and the existen e of spe ialized algorithms to draw into posterior distributions.

Some work was already presented to deal with biadditivemodels ina Bayesian way. Here, we onsider the point, proposing a new solution dire tly on the overparameterized model whi h allows one the use of standard softwares, for instan e bugs implementations.

Werst giveadetailedpresentation ofourproposalandthen applyittoareal dataset oming from the litterature, fo using onthe interpretation.

In the appendix, the proposal to deal with overparameterized models is developped for any type of models.

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Lesmodèlesbiadditifs(souventappelésmodèlesammipouradditivemainee tandmultipli ative intera tion) sont fréquemment employés pour l'interprétation de tableaux de données à deux entrées, par exemple pour l'interprétationdes intera tion génotype-milieu. Pro hes des analy-ses dedonnées fa torielles,ilsreprésentent unoutil e a epourunedes riptionpar imonieuse de stru ture matri ielles.

D'autre part, l'utilisation d'appro hes bayésienne en statistique se généralise. C'est la on-séquen e de lapuissan e a ruedes al ulateurs etde ladisponibilitéd'algorithmesspé ialisés pour tirerdes é hantillonsdans lesdistributions aposteriori.

Plusieurstravauxontdéjàétéréaliséspour traiterlesmodèlesbiadditifsdemanièrebayésienne. Dans e rapport, nous proposons une nouvelleappro he qui prend dire tement en ompteune dénition surparamétrée de es modèles. Son prin ipalavantage est de permettre l'utilisation d'algorithmesgénériques, omme eux proposés dans leslogi ielsde lafamille bugs.

Un exemple de données de la litérature est traité de manière détaillée après une présentation formalisée de laproposition.

Dans l'annexe, la prise en ompte de la surparamétrisation d'un modèle dans une appro he bayésienneest traitéede manière omplètement générale.

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1 Introdu tion 7

2 Reminders 7

2.1 About Bayesian statisti s. . . 7

2.1.1 The Bayesian paradigm . . . 7

2.1.2 Advantages . . . 7

2.1.3 Drawba ks . . . 8

2.2 The biadditive model . . . 9

2.2.1 Denition . . . 9

2.2.2 Constraints . . . 9

2.2.3 Least squares estimation . . . 10

2.3 Literature proposals . . . 10

3 Dening a prior 11 3.1 Usingprior inan overparameterized framework . . . 12

3.2 Standard omplete proposal . . . 13

3.3 Prior marginaldistribution of the data . . . 14

4 A essing to the posterior distribution 15 4.1 Algebrai ally . . . 15

4.2 Graphi alpresentation: the dag. . . 15

4.3 Simulationstudy . . . 16

5 Using posterior simulations: a worked example 21 5.1 Comparingprior versus posterior . . . 21

5.2 Lookingfor a sensiblenumberof multipli ativeterms . . . 23

5.3 Interpreting the parameters . . . 23

5.4 Interpreting the genotype responses a ross the environments . . . 25

5.5 Missing values . . . 29

5.6 Ex eedinga threshold . . . 30

5.7 Randomversus xed ee ts? . . . 30

5.8 Comparisonwith Perez etal. proposal . . . 32

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A.1 Conditionalexpe tationand varian e . . . 36

A.2 First two momentsof

E

ijk

. . . 36

A.3 First two momentsof a triple produ t. . . 36

B von Mises-Fisher distribution 37 B.1 Denition . . . 37

B.2 Interpretation of the vonMises-Fisher distribution . . . 37

C Prior/posterior with overparameterized models 38 C.1 Linear tiny ase . . . 38

C.1.1 Denition . . . 38

C.1.2 Reparameterization . . . 39

C.1.3 Posterior inthe new parameterization. . . 40

C.1.4 Con lusion. . . 40 C.2 Additive ase . . . 40 C.2.1 Denition . . . 40 C.2.2 First reparameterization . . . 40 C.2.3 Se ond reparameterization . . . 42 C.3 Generalformulation. . . 42 C.3.1 Denition of overparametrization . . . 42

C.3.2 Dealingwith overparameterization . . . 43

C.3.3 Con lusion. . . 44

C.4 Biadditive ase . . . 44

D Gibbs sampling algorithms used in the literature proposals 44 D.1 Brief reminder about Gibbs sampling . . . 44

D.2 Ho's proposal . . . 45

D.3 Perez et al.'s proposal . . . 46

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Biadditive models [Denis.1992, Denis.1994, Denis.1996, Denis.1998℄ are frequently used to in-terpretthe intera tion between two fa tors. Forinstan e inplantbreeding,itis usualtostudy the intera tion between varietiesand environments for dierentpurposes su h asthe sele tion of new varieties. Biadditivemodels present several advantages ompared with the usual anal-ysis ofvarian emodels with intera tion terms: (i) simpli ationof the intera tion s heme,(ii) better estimationbe auseallresults areused for the estimationof aunique ombination

1 , (iii) thepossibilitytohaveanideaoftheintera tionandestimatethe errorvarian ewhenrepli ates are absent,even with missing values.

Insomere entpapers[Perez.2011 ,Crossa.2011℄,it anbeobservedthatthe Bayesianapproa h was proposed for biadditive models. Most of the good properties of the Bayesian approa hes are the onsequen es of a lear separation between the modelling on parameters (denition of priors) and the statisti al analysis (getting information from the data set). In this report, we willhave a look at that possibility,get some experien e on itand propose a new treatment of su h models with the Bayesian pointof view.

2 Reminders

2.1 About Bayesian statisti s

2.1.1 The Bayesian paradigm

Let

[Y

| θ]

bethe likelihooddistribution 2

ofadataset

Y

dened throughave torialparameter

θ

. In Bayesian statisti s,

θ

isnot onsidered ashavingaxed and unknown value butasbeing a random variable, the distribution of whi h hara terizes the degree of ertainty we have on the values it an take. So to omplete the des ription model, the marginal distribution of

θ

,

[θ]

,must be provided. This isthe a priori distribution, orprior for shortness.

Performing a Bayesian statisti al analysis is no more that applying twi e the so- alled Bayes theorem to get

[θ

| Y ]

denominated a posteriori distribution :

[θ

_{| Y ] =}

[Y

| θ] [θ]

[Y ]

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whi hispossiblesin e the numerator isthe joint distributionallowingthe omputationof

[Y ]

, the marginaldistribution of

Y

.

2.1.2 Advantages

More freedom about used probability distributions Thanks to sto hasti algorithms (most known are MCMC), a lot of distributions are at hand. Among the points to underline when using a BUGSfa ility(see the Jags manual[Plummer.2011℄), one an he k that

1

Whenasaturatedintera tivemodellingisusedtheestimationoftheexpe tationofonevariety-environment ombinationreliesonlyontheavailableobservationsforthis ombination.

2

Squarebra ketsstandfordensitydistributionofarandomvariable; verti albarforthe onditioning oper-ator. Then

[Y

| θ]

readsasthedensitydistributionof

Y

onditionedby

θ

.

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Censored data an be used all the same,

A large spe trumof dis rete or ontinuous, s alaror multivariatedistributions are avail-able,

Standard transformations are implemented.

No fear of non identiability When noinformationis available fromthe data,weare left atthepriorlevelofinformation(onlyforproperpriors). Forthesamereasonmissingvaluesare nomoreaproblem. Infa t,the viewpointisshifted: wearenot analyzingadataset, weare(i) deningamodelonsomephenomenum(establishingapriordistribution)and(ii)extra tingthe informationrelativetothismodelfromavailabledataset(s)(gettingthe posteriordistribution). Forthat reason, Bayesian approa hes are very onvenient for statisti al meta-analyses.

A possibility to introdu e knowledge When dening the prior distribution, one an in orporate, with mu h more exibility than when dening the likelihood of a data set, the alreadyknowledgeaboutthephenomenonunderstudy(expertknowledge,histori aldata,et .). This is why sometimes, Bayesian approa hes are linked to learning pro esses: the posterior is naturally the prior of new statisti alanalyses.

Of ourse, asequentialin orporationof datavia su h aprior/posteriors heme isequivalentto a unique globalstatisti al inferen e.

Consistent inferen e for any transformation If a orre t Bayesian inferen e is made on some set of parameters (say

θ

), a transportation of the probability distribution gives us a onsistent inferen e to any fun tion of it. No longer useful to de ide if we are interested in the standard deviation (

σ

), the varian e (

σ

2

) or the pre ision (

1 σ

2

)! Credibility intervals will orrespond exa tly.

Of ourse this isappli able to ve torialsituations.

Ease of interpretation The output (posterior distribution)is of the same nature than the input (prior distribution) so if one is able to propose a prior, he must be able to well use the posterior. Moreover, ompared to the frequentist approa h, the notion of redible intervals an beseen as easier tointerpretthan the notion of onden e interval. Indeed, in the latter, the randomness is on the data (we onsider all possible data, if we were able to do many experiments) whereas in the redibleinterval the randomness ison the parameter side.

2.1.3 Drawba ks

A priorhas tobedened Thisisthemain riti ismagaintheBayesianstatisti alpra ti e. A ordingtothestatisti iandeningtheprior,resultswillbedierent,thennotobje tive. This is true, but an be viewed as an advantage be ause good statisti ians will get better results whi his fair? Alsobad statisti ians willbe wrongas wellwhen dening the Likelihood? Also, dening the prior imply auseful onsideration of the modelinuse, whi his often a protable spent time.

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to be dened and not only the series of s alar marginals. This an be a quite tri ky issue 3

, sometimes solved with areparameterization.

Di ult toquestion the model In lassi alstatisti s,there areseveral waystoquestiona model(via the study of residuals, for example) orat least tosele t a modelwithin afamily of models. Thisis notsoeasy intheBayesianapproa h whereadi ultpointisthe denition of fairpriorsfor dierentmodels. Fairwouldbethatthe priorsbeequivalentatthe levelofevery usedobservationbut thisisrarely the asebe ausethenumberandnaturesoftheparameters

4 , is ompulsorilydierent froma modelto another.

The orre t answer would be to imagine a hierar hy of the onsidered models, giving ea h a prior probability... this is not, for the moment, easilytra table.

2.2 The biadditive model

2.2.1 Denition

This lass ofmodelsis known formany years sin e there iswasdes ribed in1923 inone of the Fisher's papers [Fisher.1923℄. Theterminologywasproposedby Denisand Gower[Denis.1992℄ and we willsti k to it, onsidering onlythe

B (m, a, b, π

Q

)

5 variety dened as

Y

ijk

= µ + α

i

+ β

j

+

Q

X

q=1

λ

q

γ

iq

δ

jq

+ E

ijk

(2)

where

Y

ijk

is the

k

th observation for the ombination of genotype

i

and environment

j

. The number of observations for ea h ombination,

K

ij

, being variable (but known) in luding zero whi hmeans missing value. Mostanalyses are arried out with

K

ij

= 1

. The number of geno-typeisdenotedwith

I

,andthenumberofenvironments

J

. Here

Q

∈ {0, ..., min (I − 1, J − 1)}

; when

Q = 0

, we are dealing with the additive model; when

Q = min (I

− 1, J − 1)

, it is the saturated intera tivemodel(no restri tiononthe intera tiveterm).

E

ijk

isthe errorterm that in ludesthe designee t, here wewillassumethat these errorsare independent, entred, with same varian e (

σ

E

) and normal distribution.

2.2.2 Constraints

This is one of the di ult pointabout biadditive models, overparameterization ispresent and there is a need for additional onstraints to determine unique values for the parameters. We would like onstraints whi hlead to aneasy interpretation of the parameters, indu ingsimple statisti alderivationsandmostoftenkeepingthesymetrybetweenthelevelsofthetwofa tors.

3

There are twoway: (i)use amultivariate distribution

[θ

1 , θ

2 , θ

3 ]

or(ii)go through onditionaldenition

[θ

1 ] [θ

2 | θ

1 ] [θ

3 | θ

1 , θ

2 ]

. 4

Thenumberofparametersisalsodi ulttodeneinaBayesianframework. However, riteriathatmimi s theAIC havebeen proposed su has theDICto sele t models. One analso talked aboutthe Bayes'Fa tor butwhi hishardto ompute.

5

Thepresen eof

m

indi atesthe onstantparameter

µ

,this of

a

thepresen eof therowmainee t,

b

for the olumn main ee t and

Q

pre ises the numberof involved multipli ative terms in the intera tion term. Whentherearenointera tiontheterm

π

vanishesandwhen

Q

= 1

,only

π

isleft.

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µ

ij

= E (Y

ij

| µ, α, β, λ, γ, δ)

= µ + α

i

+ β

j

+

Q

X

q=1

λ

q

γ

iq

δ

jq

. (3)

Itiswellknown[van_Eeuwijk.1996℄thattheparametri dimensionof

µ

ij

is

(1 + Q) ((I + J)

− (1 + Q))

meanwhile the numberof parameters in (3) is

(1 + Q) (1 + I + J)

so that

(1 + Q) (2 + Q)

ad-ditional onstraints haveto be introdu ed. Here we willuse the standard ones:

1 ′

I

α = 1

′

J

β = 0

1 ′

I

γ

q

= 1

′

J

δ

q

= 0

for

q

∈ {1, ..., Q}

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γ

′

_{γ = δ}

′

_{δ = I}

Q

λ

1 ≤ λ

2 ≤ ... ≤ λ

Q

.

where

γ

q

and

δ

q

arethe

q

th olumnsofmatri es

γ

and

δ

ofrespe tivesizes

(I

× Q)

and

(J

× Q)

. Infa t thisset of onstraintsis notsu ientsin e they don'txthe orientationsofthe

(γ

q

, δ

q

)

whi h anbesimultaneouslyinverted. Mostoften, thisindeterminationisnot onsidered. Here we will use the non standard spe i ation of orientation borrowed from [Viele.2000℄ that the

λ

sbenotrestri ted andthat

γ

1q

sand

δ

1q

s bealwayspositive. Even if itlookslikebreaking the symetry between the levelsof the fa tors out, givinga dierent role to the rst levels, wewill see later on that this is not the ase.

2.2.3 Least squares estimation

When the numberof repli ates is onstant (

K

ij

= K > 0

), with the proposed onstraints(4), the least squares estimates of the biadditive model are easy to obtain. Let

Y

be the

I

× J

matrix of the mean by ell

Y

ij

=

1 K

P

k

Y

ijk

.

µ =

₁

I

(1

′

I

1 I

)

−1

1 ′

I

Y

₁

J

(1

′

J

1 J

)

−1

1 ′

J

α = Y

1 J

(1

′

J

1 J

)

−1

1 ′

J

β = Y

′

1 I

(1

′

I

1 I

)

−1

1 ′

I

and the parameters of the intera tion terms are given by the singular ve tors and singular values for the biggest

Q

singular values of the singular value de omposition of the row and olumn entered matrix

I

− 1

I

(1

′

I

1 I

)

−1

1 ′

I

Y

I

J

− 1

J

(1

′

J

1 J

)

−1

1 ′

J

. 2.3 Literature proposals

Re entlysomeauthors[Viele.2000,Crossa.2011,Perez.2011 ℄haveusedbiadditivemodelswithin a Bayesian framework. Their motivation is that the Bayesian approa h may oer solutionsto issuesthatareoftendi ulttohandlesu has: unbalan eddata,unequal ellsize, heteros edas-ti data,thedi ult hoi eof thenumberof relevantdimensionsforthe intera tionterms,et . Moreover, aBayesian approa h alsooersdistributions ofany quantity of interest(tobe om-pare with point estimates and their onden e intervals of the maximum likelihoodapproa h)

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totakeinto a ountin the analysis previousinformationsu h ashistori aldata whi h may be very interesting inthe analysis of genotype-environment data. We an remark that ompared to the surge of interest in using Bayesian approa hes to study models of very distin s types, only few proposals have been made for the biadditive models. This an be explained by the di ultytoworkinaoverparametrizationframework(aswewillsee later). More pre isely, the problem an bequite easilyta kle forthe linearterms of the modelbut the major di ulty is to put priorson the intera tion terms taking intoa ount the onstraints.

Vieleand Srinivasan (2000)[Viele.2000℄seems tobethe rst onestopropose aBayesian treat-ment of su h models. They put uniform priors on the rst olumn of the matri es

γ

and

δ

. Sin e ea h ve tor has zero sum and unit length, it orresponds to put uniform prior on a

I

(respe tively

J

)dimensionalunit sphere. Then,they work sequentiallyand takeas onditional prior of ea h

(γ

q

)

(respe tively

(δ

q

)

) given the previous dimensions a uniform distribution on the orre t subspa e. Of ourse, due to the onstraints (the ve tors must be normalized and orthogonal to the previous ones), it is not easy to dene the supports and to sample from uniform distributions with the orre t supports. They proposed a method to do so and then used it in a spe i Gibbs sampler. On a real data set [Crossa.2011℄, their approa h seems quitepromisingand oers onden e regionshelping inthe interpretationof the results.

Spheri aluniformdistributionisaspe ial aseofvonMises-Fisher[VMF℄distributions(seeB). Su h distributions exist also for matri es. More pre isely, the set of orthonormal matri es is alledthe Stiefelmanifold[Chikuse.2002℄. The vonMises-Fisher distributionsaredistributions over this manifold. [Ho.2012, Ho.2009, Smidl.2007℄ used these distributions in a Bayesian treatment of models based on singular value de ompositions of parti ular matri es su h as models forPrin ipalComponents Analysis. From a omputationalpointof view, these models are losed to the linear-bilinear ones, the main dieren e being that the linear part is not in luded. Morepre isely,[Ho.2012,Ho.2009,Smidl.2007℄proposedtouseaspriortheuniform distributionsformatri es

γ

and

δ

,indeedaspe ial aseofvonMises-Fisherdistributions. Using su hpriors allowthemtoensurethe orthonormality onstraintsatthe posteriorlevelsin e the posterior distributions are also VMF distributions

6

. More details about the method and the asso iated algorithmisgiven in appendix D.

Very re ently, in the framework of the analysis of genotype-environement data, [Perez.2011 ℄ proposed also to use as prior distributions for matri es

γ

and

δ

spe i von Mises-Fisher dis-tributions. Their method detailledin appendix D remains to put priors onto the ells of the matrix

Y

.

3 Dening a prior

Oneofthemajordi ultytota klethe biadditivemodelistoensureapriorontheparameters taking into a ount the over-parameterization and the asso iated onstraints. As far as the onstraints are linear, it is not too mu h problemati . But for the bilinear onstraints on

γ

and

δ

matri estobe aset of

Q

orthonormal olumns,nostandard solutionisathand. In that se tion,wepropose astraightforwardsolutionhavingthe greatadvantagetobeimplementable inthe standard bugssoftwares.

6

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Intheappendix(C)wehave onsideredtheoverparameterizationdi ultywitha reparameter-ization learlyseparatingthe involved parametersintotwotransformedsubsets

(φ

1 (θ) , φ

2 (θ))

; the rst issu ienttodene the datalikelihood

[Y

| θ] = [Y | φ

1 (θ)]

and these ond isleft re-dundanton etherstistakenintoa ount

[φ

2 (θ)

| Y, φ

1 (θ)] = [φ

2 (θ)

| φ

1 (θ)]

. More,for multi-normal priors, we a hieved a lear separation getting the independen e, i.e.

[φ

1 (θ) , φ

2 (θ)] =

[φ

1 (θ)] [φ

2 (θ)]

whi h implies that the prior and posterior distributions of

φ

2 (θ)

are identi al, noinformationis given by the data ontothe

φ

2 (θ)

parameters.

It an be also argued that the overparemeterization problem is identi al to the missing value situation. Indeed, let us take the very simple example of one way fa tor without repli ate and with known varian e. Letthe parametersbe

{θ

1 , θ

2 , ..., θ

I

}

and onsider a prior-likelihood ouple as

[θ

i

]

∼ N (50, 2)

whateveris

i = 1, ..., I

independent,

[Y

i

| θ

i

]

∼ N (θ

i

, 1)

whatever is

i = 1, ..., I

independent.

There is no overparameterization in the sense that every modi ation of any non onstant fun tion of the

θ

i

modies the likelihood. But if we remove

Y

1

from the data set,

θ

1

an be modiedwithout hangeinthe likelihoodindu ingoverparameterization. Nevertheless thereis no harm for the Bayesian statisti ian, the joint distribution over parameters and data an be al ulated and the posterior dedu ed. Of ourse, it isobvious that

h

θ

1 | (Y

i

)

_i=1,...,I

i

= [θ

1 ]

, the posterior of

θ

1

is equal to itsprior, no informationhave been born onto this parameter by the data, logi al enough.

Thesame an anbeproposedforeveryoverparameterizedmodel. Thefun tionsofthe parame-tersasso iatedtothene essary onstraintsofthetype

f (θ) = 0

topreventtheindetermination does not inuen e the likelihood.

Butwe ouldimaginethatsome databeasso iatedtothem,produ ing ompleteidentiability of theparameter set. Why nottoargue thatthese dataare missing? Andthatonlytheir prior willbeidenti al totheir posterior.

Taking the ase of the

B (m, a, b, π

2 )

modelwith norepli ates:

Y

ij

∼ N (µ + α

i

+ β

j

+ λ

1 γ

i1

δ

j1

+ λ

2 γ

i2

δ

j2

, 1)

for

i = 1, ..., I

and

j = 1, ..., J

, we an say that are missingsome observations to repla ethe onstraints,for instan e

Y

10 ∼ N (µ + α

1 , 1)

Y

01 ∼ N (µ + β

1 , 1)

Y

_−1,−1

_{∼ N (λ}

1 )

Y

_i,−1

_{∼ N (λ}

1 γ

i1

, 1)

for

i = 1, ..., 2

Y

−1,j

∼ N (λ

1 δ

j1

, 1)

for

j = 1, ..., 2

Y

−2,−2

∼ N (λ

2 )

Y

_i,−2

∼ N (λ

2 γ

i2

, 1)

for

i = 1, ..., 3

Y

−2,j

∼ N (λ

2 δ

j2

, 1)

for

j = 1, ..., 3

Anditis learthatthey ensurethatallparameters(andeveryfun tionsofthem)areinformed. So in prin iple a good algorithm to produ e the posterior will work even with these values missing.

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parameters whi h are identiable, that is with a posterior dierent from the prior. The rst andidates are

µ

ij

= µ + α

i

+ β

j

+ λ

1 γ

i1

δ

j1

+ λ

2 γ

i2

δ

j2

for

i = 1, ..., I

and

j = 1, ..., J

and we will use them (see the model ode in E) and base our interpretations from fun tions of them.

3.2 Standard omplete proposal

We will use 7

the following independent normal distributions as priors plus a uniform for the error varian e.

µ

_{∼ N}

m, s

2 _µ

α

i

∼ N

0, s

2 _α

β

j

∼ N

0, s

2 β

(λ

q

)

_q=1...Q

∼

ordered sample of

Q

independent

N

0, s

2 λ

(5)

γ

1q

∼ HN (0, 1)

γ

iq

∼ N (0, 1)

for

i > 1

δ

1q

∼ HN (0, 1)

δ

jq

∼ N (0, 1)

for

j > 1

σ

E

∼ U (0, S

M E

)

where

HN (0, 1)

stands forthe half-normaldistribution (trun ationof

N (0, 1)

on the positive values). Onlyve hyparameters are used for the denition

8

;their interpretation 9

is:

m

the guessed mean value of the studied variable,

s

µ

quanties the un ertainty we believe tohave about

m

,

s

α

quanties the variability webelieve the genotypemain ee ts have got,

s

β

quanties the variabilitywe believethe environment main ee ts have got,

s

λ

quanties the variability we believe the ge intera tion is on its dierent

Q

10 ompo-nents 11 , 7

Evenifitdoesnotmatter,itwouldhavebeenmore onsistenttouse

γ

iq

∼ N 0, I

−1

and

δ

jq

∼ N 0, J

−1

tosti ktothe hoosen onstraints(4). 8

Theyaresupposedtobenumeri alvalues,soexpressedwithlatinletters. 9

In the following we give a pre ise meaning to the terms variability and un ertainty. Variability is the variationlinkedtoarandomvariableweareinterestedin. Forinstan e,ifweareinterestedintheyieldofnext year somevariabilityhas to be introdu ed as a onsequen e of the not known and inuential limate ee t. Un ertainty isrelatedto ala kof knowledgeaboutsomevalueof interest,it ouldbede reasedby a quiring additionalrelevantdata. Whenavolumeofgrains issampledto getthewetnessofthe rop,takingagreater volumewill ertainlyredu etheun ertaintywehaveaboutthehumidityofthetotal rop.

10

Thedeterminationofthenumberofmultipli ative omponentsisadi ultquestion,wewillseein Figure 7howthis anbeapproa hedfromtheposteriordistributionstoredu e theirnumber.

11

(15)

S

M E

quanties both the remainingvariability not taken intoa ountpreviously and the un ertainty of the measurement

12 .

Noti e that

we spoke about un ertainty only for the parameter

µ

sin e it represents the parameter insuringthe translation invarian eof the model(no need to have variabilityon it,it an be in orporated intothe main ee ts).

fromour pointofview,ifwewouldliketointrodu eun ertainty ontheotherparameters, either an expe tation has to be introdu ed as a random variable, and/or the standard deviation parameter has to be supposed random. Taking the example of the genotype main ee t:

α

i

| α

0i

∼ N

α

0i

, s

2 α

and

α

0i

∼ N

0, s

2 _uα

or

α

i

| ς

α

∼ N

0, ς

α

2

and

ς

α

∼ U (0, S

M α

)

or both. The distribution of the newly introdu ed

α

0i

and/or

ς

α

ould be interpreted as un ertainty.

Importantremark WhenlookingatthedenitionofthevonMises-Fisherdistribution(B), itis lear that the distributionsindu ed by our proposalontothe orthonormal matri es

γ

and

δ

are indeedvonMises-Fisher distributions: the uniform distributionsonto their spa es.

3.3 Prior marginal distribution of the data

Webelievethatitisquiteimportanttolookatwhatrepresentstheinformationputintheprior denitionstep. Inordertodothat,themorenaturalwayistolook,usingthelikelihood,atthe indu ed priordistribution onthedata set. This isdoneintegrating the joint

[θ, Y ]

distribution over

θ

.

Due to the presen e of the bilinear terms, it is not possible to obtain in lose form the distri-bution of

Y

ijk

but it is possible to expli it expressions for their rst two moments and alsoto simulate them. As dened in (2) and with the retained priors (5), one an he k that (using results presented in A.3and A.2) that

E (Y

ijk

) = m

V (Y

ijk

) = s

2 µ

+ s

2 α

+ s

2 β

+ s

2 λ

+

1

3 S

2 M E

Cov (Y

ijk

, Y

ijk

′

) = s

2 _µ

+ s

2 _α

+ s

2 _β

+ s

2 _λ

Cov (Y

ijk

, Y

ij

′

_h

) = s

2 _µ

+ s

2 _α

Cov (Y

ijk

, Y

i

′

_jh

) = s

2 _µ

+ s

2 _β

Cov (Y

ijk

, Y

i

′

_j

′

_h

) = s

2 _µ

Noti ethattheseresultsapplyequallywellwhen

i = 1

or/and

j = 1

,even ifthe omputationis slightlydierent. This iswhy wesaid in2.2.2thatthe symetrywasnotlostwiththeproposed onstraints.

12

Gammadistributionswereoftenusedforvarian eparameters,butre entworksshowthatitispossibleand moree ienttoputuniformpriors

(16)

are used forloops. Cir lednodes are random variables. Re tangular nodes are onstant. Ar s indi atethe dire trelationships between the nodes. mu is used for

µ

,al for

α

,...

4 A essing to the posterior distribution

4.1 Algebrai ally

No loseform of posterior an be imagined 13

,espe iallyinthe ase of anon equally repli ated designwhereeven LSestimators arenotknown. Nevertheless, thereisnodi ulty(inalltrials we did)to get simulatedvalues fromthe posteriorusing Jags [Plummer.2011℄ one ofthe bugs softwares.

4.2 Graphi al presentation: the dag

To better understand the me hanism of the modelling,it is of interest tobuild the underlying dire ted a y li graph,so alleddag;itispresented inFigure1. Itemphasizesthe entralrole

13

(17)

µ = 100

(α

i

) = (

−1, −1, 0, 1, 1)

(β

j

) = (

−4, −3, −2, −1, 0, 1, 2, 3, 4)

λ = 12

(γ

i

) =

2 √

10 ,

1 √

10 , 0,

−

1 √

10 ,

−

2 √

10 (δ

j

) =

1

2 ,

1

2 , 0, 0, 0, 0, 0,

−

1

2 ,

−

1

2 σ

E

=

3 ₂

Table 2: anova table of the parametervalues given in Table 1 onlybased onthe expe tation. The error varian e being equalto 2.25.

df SS MS

row ee t 4 36 9.0

olumn ee t 8 300 37.5 intera tion ee t 32 144 4.5

of

µ

ij

, the expe tation of the observed variable in a frequentist approa h. We will see in 5.7 that a Bayesian approa h allows one to play between xed and randomee t ae tation.

4.3 Simulation study

Inorderto he kthatinpra ti alsituations,aBayesianapproa hisee tiveforbiadditive mod-els, we performed a small simulation study. From a

B (m, a, b, π)

biadditive expe tation

14 we addedanormalrandomerrorwithidenti alstandarddeviations(

σ

E

)andtteda

B (m, a, b, π

2 )

model. This was done a numberof times, to es ape afavorable (orunfavorable) ase.

More pre isely, we did 49 simulations, with a posterior sample of 50 draws with a thinning oe ient of 200 to avoid auto orrelations; not be ause the omputation are lengthy: they last 140 se onds but be ause we wanted to display the results graphi ally. The introdu ed parameter values are shown inTable 1 resultingin the anovatable given inTable 2.

The results of the simulation study are given in Figures 2 (study of the suggested number of multipli ative terms), 3 (inferring the row main ee ts), 4 (inferring the row rst intera tive ee ts) and 5 (inferringthe rowse ond intera tiveee ts).

Figure 2displays the joint posterior distributionsof

(λ

1 , λ

2 )

. The values of

λ

1

are positive for allthe simulations,whereas

λ

2

takesa ontinuum ofvalues negative orpositive. This isa lear indi ationthat the redible regionof

λ

1

does not in lude the zero value, ontrary to itsof

λ

2

. Consequently, we ansay thatmostoftentheBayesian analysis orre tlyassessone multipli a-tive term(ex ept insimulation48wherethe on lusionwould betwomultipli ativeterms and anotso lear on lusionforsimulationnumber11butstillinfavorofonemultipli ativeterm). Here we an see anadvantagetoletthe singularvaluesbeing negative: itwouldhavebeenless lear if the absolute value of the singular values have been used as ommonly done.

Theprole studyof therowee ts proposed inFigures3,4and 5would bemu h learerif the bundle of the 50 proleswere repla ed by the quantileproles of a greaternumberof proles. Straightforword (but tedious) algorithms an be implemented forthat.

14

(18)

the simulated joint posterior density of

λ

1

(abs issae) and

λ

2

(ordinates) is displayed with 50 draws. The red ir leindi atesthe truevalue (

λ

1 = 12, λ

2 = 0

). Due totheindetermination of the parameterization, negative values are equivalent topositivevalues.

−10

5

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10

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0

5

10

49

Nevertheless

α

and

γ

1

parameters seemswell assessed. More heterogeneity o urs for

γ

1

where parti ularlythe alreadymentionnedsimulationnumber11hasafewoppositeproles. Looking this ase with attention, ones an noti e that the

γ

11

parameter

15

is very lose to zero and sometimesfautly;afreeattributionwouldhavegiventwosetsofprole,one similartothe true prole and the other one opposite. The errati behavior of

γ

2

proles (Figure5) is onsistent with the found unique dimension. Here too with the ex eption of simulation 48 whi h tended to propose two dimensions,where the bundle of proleshas got a systemati on ave shape.

15

(19)

joint posterior density of the

α

i

prolesis displayed with 50 draws. A prole is dened as the lines linking

(i, α

i

)

; the red lineindi atesthe true prole.

1

2

3

4

5 −4

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3

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−1

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49

(20)

simulated joint posterior density of the

γ

i1

proles is displayed with 50 draws. A prole is dened as the lines linking

(i, γ

i1

)

; the red lineindi atesthe true prole.

1

2

3

4

5 −0.5

0.5

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4

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49

(21)

the simulated joint posterior density of the

γ

i2

proles is displayed with 50 draws. A prole is dened as the broken line linking

(i, γ

i2

)

; there is no true prole sin e there is no se ond multipli ativeterm inthe proposed expe tation; the red lineis the null prole.

1

2

3

4

5 −0.5

0.5

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49

(22)

When analyzing areal genotype-environment data, ve main questions 16

arise:

Q1 What is the genotype with the best performan es a ross allthe environments?

Q2 What is the genotype with the best performan e for a spe i environment?

Q3 Are the genotypesstable a ross all the environments?

Q4 Is it possible torank the genotypes?

Q5 Could we assigna probability that a genotype willgivemore than a ertainthreshold?

Of ourse, the symetri al questions( onsidering genotype insteadof environments)are alsoof value.

In this part, we try to address these questions. Moreover, in the lassi al statisti alapproa h where estimates (and pre ision of them) of the biadditive model are obtained, it is not that easy to interpret the results. One an wonder if it would not be worst in a Bayesian ontext wheredistributionshavetobehandledinsteadofestimations? Wewouldliketoshowthatnot.

Toillustrateourpurpose, weuseaworkedexampleusingthedatasetproposedby[Crossa.1991℄ whi h omprises

I = 18

genotypes and

J = 25

environments, all ombinations, no repli ates. More pre isely we willanalyse it infour versions :

redu ed usingonlythe

40

ombinationsof therst

5

genotypeswiththe rst

8

environments, omplete the omplete data set,

redu ed_missing the redu ed data set where 4 values, randomly hoosen, were onsidered missing,

omplete_missing the ompletedatasetwhere45values,randomly hoosen,were onsidered missing.

Table 3 gives general features about the onsidered ases. From the anova tables, one an on lude that important values have been eliminated when reating a 10% missing s heme in the redu ed missing ase; less obviousfor the omplete ase.

Thestandardoutputofam m algorithmisasimulationoftheparametersfromtheirposterior distribution. We will use a supers ript

s

varying from

1

to

S

to indi ate it. For instan e

α

s

i

willbethe

s

thsimulation of the parameter

α

i

.

Be ause we are fan of them, we present the interpretation with graphs,but this also ould be done with numeri al values intables.

5.1 Comparing prior versus posterior

Asseen in2.1.3, themajor riti ismsmadetothe Bayesianapproa histhe ne essitytodene a prior onthe model parameters. Two pre autions have to be taken with this respe t.

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the same omputer;

10

5

iterations were used for the burn-in phase;

10

4

were drawn with a systemati samplingof

10 −2

giving

10

2

retained simulations.

ase anova duration(s)

redu ed df SS MS Geno 4 1.0 0.25 Envi 7 95.6 13.7 Geno:Envi 28 5.7 0.20 3 omplete Geno 17 18.1 1.1 Envi 24 2369.5 98.7 Geno:Envi 408 132.5 0.32 86 redu ed missing Geno 4 3.9 0.96 Envi 7 66.4 9.5 Geno:Envi 24 3.7 0.15 2.7 omplete missing Geno 17 16.1 0.95 Envi 24 2122.4 88.4 Geno:Envi 363 115.66 0.32 75

Table 4: Priordenitions: used onstantvaluesreferringtothe distributionsintrodu edin(5). Take are thatimpli itely allpriors are supposed to be sto hasti allyindependent.

onstants

m = 5, s

µ

= 0.8

s

α

= 0.5

s

β

= 0.5

s

λ

= 0.5

√

IJ

S

M E

= 2

(24)

porate to the priors asmu h as possible knowledge inthem. Here, we dire tly used the rather vague priors shown in Table 4.

Se ond, a areful examinationof the prior distributionshas tobe ondu ted onthe quantities ofinterest. It ano urthatprioronparametersleadtoprioronobservationoutofrelevan e!

17

This is a lear indi ationthat something is wrongand must be hanged. Also the omparison between priors and posteriors is needed to assess the level of involment of the data set in the posterior. In Figure 6 one an see that onsistently with the used priors (Table 4) no ee t is visible in the prior dimension, whi h is not the ase for the posterior dimension where three distin t lustersappear. Probably,itwouldbelogi altoredu ethevariabilityofthepriorsin e negativeyields are proposed foralmost every

µ

ij

together with uppervaluesalso not sensible.

5.2 Looking for a sensible number of multipli ative terms

Figure 7 displays the joint posterior distributions of

(λ

1 , λ

2 )

for the redu ed and omplete ases. The redu ed ase is interesting be ause we an noti e that the

S = 100

simulations are distributed in two groups: the most numerous for negative values of

λ

1

, the other one for positive values of

λ

2

. Inea h group,

λ

2

takes a ontinuum of values negativeor positive. This is a lear indi ationthat the redible region of

λ

1

does not in lude the zero value, ontraryto itsof

λ

2

. The on lusionisthat onlyone multipli ativeisneeded forthe redu ed ase. Forthe omplete ase, aglan eatthes ales showthat nodoubtisleft, both redibleintervalsarevery far from zero and at least the rst two dimensions have to be kept. As we didn't introdu e a third term, nothing an besaid about itsne essity

18 .

5.3 Interpreting the parameters

Letus on entrate our attention towards the genotypes; the same ould be made for the envi-ronment fa tor.

Themoststraightforwarduseofthe simulationsistodraws atterplotsofafa torwiththe pa-rameterdependingonit. Forinstan e

19

α

i

√

J, λ

1 γ

i,1

,but insteadofone pointforgenotypes, we will get

S

points providinga dire t view of the variability/un ertainty: Bayesian statisti s returns adistribution

20

,not apointestimate withpossibly a onden e interval. This is done inFigure8forthe omplete ase and therst ninegenotypes. They lookimpressivelydistin t. No overlapping for any pair of genotypes (this does not seem the onsequen e of the very re-du ed number of simulations: 100). Among these genotypes, the 6th looksattra tive with the strongest main ee t and limitedintera tion, then among the most stables.

16

assuggestedby Fredvan Eeuwijk (Prof. at WageningenUniversity,NL) during aworkshopon statisti al interpretationofGenotype-Environmentintera tion, 2012,Inra,JouyenJosas.

17

Stronglynegativeorexageratedyields... 18

Noti ethatin[Crossa.1991℄,threemultipli ativetermshavebeenintrodu ed,we onsideronlytwoforthe sakeof theexposure simpli ity. Neverthelessthe model ode provided in E hasgot aNQvariable whi h an takeany value. As underlined in 2.1.3,a aw of Bayesianapproa hesis the la k of toolsto sele t amodel among aset ofmodels. In aseof nestedmodels, redibilityregions anbeusedto de ideifsomeparameters takevaluesasso iatedtosomesubmodels.

19

Themultipli ationof

α

by

√

J

andof

γ

q

by

λ

q

providesthemasquarednormequalto thesumofsquares explainedbytheasso iatedtermsinthemodelwhi hseemsafairwaytorelatethem. Morepre isely

SS

Geno

=

J

P

_i

α

_b

2 _i

and

SS

G×E(1)

= λ

2

1 = λ

2

1 P

i

γ

d

i,1

2

,in theorthogonal ase. 20

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Figure 6: Prior-posterior redible boxes for the

µ

ij

for the redu ed data set.

2.5%

and

97.5%

quantiles were used to dene the boxes around the median position. The red line is the rst bisse tor.

−5

0

5

10

15

20

2

4

6

8 Crossa (5,8) 1e+05|10000|100 reduced

Prior

P

oster

ior

MU[1,1]

MU[2,1]

MU[3,1]

MU[4,1]

MU[5,1]

MU[1,2]

MU[2,2]

MU[3,2]

MU[4,2]

MU[5,2]

MU[1,3]

MU[2,3]

MU[3,3]

MU[4,3]

MU[5,3]

MU[1,4]

MU[2,4]

MU[3,4]

MU[4,4]

MU[5,4]

MU[1,5]

MU[2,5]

MU[3,5]

MU[4,5]

MU[5,5]

MU[1,6]

MU[2,6]

MU[3,6]

MU[4,6]

MU[5,6]

MU[1,7]

MU[2,7]

MU[3,7]

MU[4,7]

MU[5,7]

MU[1,8]

MU[2,8]

MU[3,8]

MU[4,8]

MU[5,8]

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Figure7: Posteriordistributions of the

λ

s for the redu ed (left)and omplete (right) ases.

−4

−3

−2

−1

0

1

2 −1

0

1

2 POSTERIOR Eigenvalues

lambda[1]

lambda[2]

17

18

19

20

21 −13

−12

−11

−10

POSTERIOR Eigenvalues

lambda[1]

lambda[2]

Takingintoa ounttheeasyinterpretationgiventotheparametersasso iatedtoea hgenotype (sin ethereisonlyonemultipli ativeterm), itispossible toanswermostofthe questionsrisen atthe beginningof this se tion. Among the ninedisplayed inFigure8,genotype6 has got the better performan ea ross theset ofallenvironments(Q1),itasalsointhe groupofthe stable genotypes{6,8,7}(Q3), ertainlyitdeservesthe rstrankmeanwhilegenotype3isonthelast position (Q4).

5.4 Interpreting the genotype responses a ross the environments

But,asthe learinterpretationoftheparametersidentiedin(2)isnotalwaysunambiguous,we suggesttoalwaysstart fromthe matrix

(µ

ij

)

dened in(3)and indi atedin thedagofFigure 1. The proposal is to show the genotype response for the spe trum of studied environments. Forthat, wethink that arandom urvefor ea h genotypealong the

E

1 I

P

i

µ

ij

is of value.

To take into a ount the un ertainty linked with the posterior distribution, we propose to representea hgenotypewithabundleof

S

urves(oneforea hsimulation)obtainedbyjoining the

J

pointsof oordinates

1 I

P

i

µ

s

ij

, µ

ij

−

1 _I

P

i

µ

s

ij

. Withthe adopted onstraints (2.2.2), it

is

µ

s

_{+ β}

s

j

, α

s

i

+

P

Q

q=1

λ

s

q

γ

iq

s

δ

js

s

. Other proposals an be think of, this one has the advantage of proposing non ommon terms between abs issae and ordinates, eliminatingnoise ee ts of systemati orrelations.

This is proposed in Figures9 and 10.

Indeed very strong dieren es appear and we retrieve the very good behavior of genotype 6. Usingthisdiagram,we analsodeterminewhi harethebest genotypesforagivenenvironment (Q2). After genotype 6 whi h is denitively the best for all environments, the rst one is the more advisable forpoorenvironement ne essarily pla edon the leftside of the abs issae.

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Figure8: Genotype ee ts (additivewith

α

i

, intera tivewith

γ

i,1

)forthe omplete ase. Ea h genotype isdrawn in a dierent plot; onlythe rst nine genotypesare shown (out of 18).