An extension of the Sensmaking-Intuition Model (SIM) by defining ethics: an answer to

Keywords: Sensmaking-Intuition Model, ethical decision model, business ethics. 1

Sonenshein based his Sensmaking-Intuition Model of ethical decision on a critical approach the rational models. In his model, sens-making for the ethical issue construction and the intuitive process of decision are the central articulations. But the model doesn't give a clear definition of ethics, suggesting that all ethical decisions can be evaluate on a scale of individual or collective behavior. But this scale doesn't represent the multiplicity of ethical orientations. I review the most important philosophical ethicals systems and expose the implications on the Sonenshein's model. I propose some modifications and suggest orientations for future researches.

In 2007 Sonenshein proposed an ethical decision model. This model presented a large number of new orientations dealing with ethical decision. They were based on a list of critical analysis of the mainstream and the use of very different theoretical approaches. This model, as most of ethical decision models presents, however, a weakness: the concept of ethics is not clearly defined and the notion of ethical decision is partial. So, making a definition of ethics based on the main philosophical ethical systems will clarify how the model comes into the ethical field of research and legitimate it. But this definition will reveal the amplitude of nuances and points of view, and the complexity of the question, overstepping the bounds of the opposition between Collective and Individual. I propose then, to introduce some modifications in the model to take account of those results and I suggest some new propositions for further researches.

The foundation of the Sensemaking-Intuition Model (SIM)

Sonenshein grounded his model (Figure 1) in a large number of critics of the most important theoretical approaches of ethical decision. The most important are the weakness to use rationality as process of decision, the impossibility to think to all consequences of the decision, and the rationalization a posteriori. This last point disables the possibility to understand the intuitive side of the decision. The second stage is the most important modification of the ethical decision approach. Sonenshein posits that ethical judgment, in opposition to rationalist approaches, is first intuitive. This is due to psychological attitudes that drive individuals judgment with emotions and feelings before other kinds of cognitive reasoning. This point of view is reinforced, according to Sonenshein, by the fact that some fundamental philosophers used depicted intuition as a central element of ethics.

The third stage occurs when an individual has to explain his decision. This justification is generally very rational because it is the best way to convince interlocutor or oneself.

The output of the model is a behavior assumed on a binary scale "collective vs individual". This represents a choice of definition of ethics, and represents a point of view witch reduce the efficiency of the Sonenshein's model. [START_REF] Jones | Ethical decision making by individuals in organizations: An issue-contingent model[END_REF] argued that most of models were not leaning against a strong definition. I agree with this point of view. There are three kinds of models with three kinds of ethics definition. First, [START_REF] Trevino | Ethical decision making in organizations: A person-situation interactionist model[END_REF] and [START_REF] Hunt | A general theory of marketing ethics[END_REF] don't define what is ethical or non ethical even though their model presents an ethical decision. Second, some models propose that ethical decision arises when it matches with the social acceptance [START_REF] Ferrell | A contingency framework for understanding ethical decision making in marketing[END_REF] or the professional best practices [START_REF] Bommer | A behavioral model of ethical and unethical decision making[END_REF].

Third, some authors state a decision is ethical when it complies with laws and if it is morally acceptable by the majority [START_REF] Jones | Ethical decision making by individuals in organizations: An issue-contingent model[END_REF][START_REF] Beu | Ethical decision-making: A multidimensional construct[END_REF]Brass, Butterfield & 4 Skaggs 1998). Thus, this represents an important weakness of those models because those definitions, when they exist, are strong simplifications of the wide range of ethical way of thinking. Then their use in order to explain or predict behaviors are restricted. Only Cavanagh, Moberg & Velasquez (1981) have touched the problem of the variety of definitions, but they did not introduce it in their model. Unfortunately, Sonenshein did not explicitly define ethics neither.

Actually he exposed his own perception of ethics in the shape of the model. The six sources of influence and the three steps of ethical decision lead to a unique scale based on an individual or a collective criteria. But this criteria is one of the multiple ways to analyze ethics. Next section will present the different forms of ethical values and the different ways of to reach the most ethical status in each form.

The philosophical sources of ethical values

The aim of this part is to demonstrate that ethical values can't be analyzed on the sole "individual vs collective" criteria. I will show the diversity of points of view by displaying three different moral streams of philosophy amongst the most important.

If the relationship between an individual and others is the ground of ethical philosophy, not all authors are thinking that thinking for himself is bad and thinking for collectivity is good. Their point of view deeply depends on, first, the nature of the reference to good and bad, and second, the way to achieve the good. For Kant, the origin of the good is God (Kant 1792: 315-318).

Humans are free to try to take the way of transcendence or to stay in their natural statement. What is good is what the religion can translate from the divine will. And the recognition of the legitimacy of others to get human respect is necessary. So the collective approach is recognized.

God, as a source of good values can be found in the radical stream of protestantism. Weber exposed in a main book (Weber 1967: 91) that the good and the bad are defined by God but that human can't have access to this definition. The reason is that God's will is unreachable for humans. So that, humans have to do what they can do best without wondering why they are doing it. From this point of view, if one knows trading very well, he or she has to do so for himself. If another has skills for social activities, he or she has to act the way to help others. An other source of good behavior is the necessity for human beings to interact with others. Morin (2004: 17-66) explains that a human being can't live without other humans. Society is a system and all negative influence on one element can lead to great negative results on other elements. Therefore, not thinking about the implications of one's own behavior on others is a kind of sociological suicide.

The collective behavior is a necessity. It can be the same approach for Levinas (1992: 93) but with a different reason, and even if his point of view associates in the same time some divine moral values. It is necessary to be careful to others because they belong to the mankind. But from his point of view, the recognition is achieved through the presence of the human face. It means then, that the collective behavior can exist only if the actor knows directly the other person. The third stream is utilitarianism. According to Mill (1843: 101-103) and [START_REF] Bentham | Déontologie ou science de la morale[END_REF], the good is to seek the utility of any action. The authors claim an individual can account for his happiness (Bentham 1834, vol 1.: 18-19). What is important is to accumulate knowledges and technologies able to keep away ill and death (Mill 1871: chapter II). This approach proposes an other reference for the good: the good is what is useful.

The fourth stream is the legal one as Rawls presented it (1971). What is good is what is legal. But this orientation can be right only if the legal system is grounded on two notions. First, moral of 6 society that represents the base of some western legal system. Thus, rules are the reflect of the good and bad behavior for humanity. Second, the ability of the system to tend to an optimum.

This last argument means that at the optimum, the rights of everyone are at their tops and that it is impossible to improve the rights of an individual without affecting the rights of others.

All those ethical systems in philosophy show that there is a multiplicity of values defined as good, and that those values are dogmatical. According to [START_REF] Schwartz | Universals in contents and structure of values: Theoretical advances and empirical tests in 20 countries[END_REF], about 56 fundamental moral values can be mapped all around the world population. They are not equally used in the different geographical groups to solve ethical dilemmas. At last, I argue that the diversity presented is a sample of the wide range of ethical systems in the world. For instance, the rational and universal ethics based on principles, is not shared by zen believers. According to them, there is no way leading to an absolute good or an absolute bad consequence. Any action has a two sides. The first (yin) is negative and the second (yang) is positive. The two sides are opposed but not exclusive. They are complementary so that it can't exist a perfect behavior. Therefor, the critics I made are reinforced by the existence of many other ethical thinkings getting the ethical values and process more complex and can't be studied through a unique scale of assumption of ethical decision.

The methods for ethical behavior

For Kant, the best way is rationalism. He assumes that rational reasoning can be used to anticipate any ethical situation. Kant suggests to think about all the action in life as if they had an universal impact. So, rationalism is the best way. From the utilitarianism point of view, the best way to reach the good behavior as an aim, is to accumulate technologies and knowledge. It 7 represents then a kind of rationalization. If the origins of good values are different, the way of taking ethical decision seems to be closed by. In this way the two main authors (Mill and Bentham) suggest to accumulate first capitals, eluding any intuition or any emotion. This fundamental proposition is logical if we consider the context of their writings. Mill and Bentham wrote their work in the first half of 18 th century. The rising of new technologies, the turn of science and knowledge in the field of biology, chemistry, physics could let hope a perfect future with no more diseases, no more fears. But in order to improve sciences one needed to accumulate capitals. This personal accumulation has to be rational, but can't be universal. They suggest to accumulate goods and capitals in order to improve the wellbeing of people in a close perimeter (family, friends...). For the others, they rely on the natural regulation by the market, to spread them the benefits of value accumulation. The market represents a mean for all to reach happiness, even if everyone can only act for his or her own very close acquaintances. So that, by using the Sonenshein's model, I would say that this kind of behavior is more individual than collective. But owing to the mobilization of the market as a regulator, it could be understood as a smooth collective approach. And the same decision and behavior can be understood differently if individuals justify their decision by a kind of lack of concern for the consequences. In this case it could really be interpreted as an individual behavior. This last point is close to the classical economical analysis. From this point of view, the unique acceptable behavior is the seeking of benefits. Nevertheless, Smith admits that intuitive judgment is used in ethical area, even though Friedman excludes it.

In a different way, Rawls suggest that ethics is a process leaded by rationality and self interest (Rawls 1971(Rawls /1997: 37-38): 37-38). This is grounded in a fundamental assumption. Legal system is able to 8 define what is good or bad, then the individual ethics is just the choice to be compliant with it.

In the Morin's ethical system, ethical behavior is the consequence of two dynamics. The first one is the contingency of an individual to the humanity. It leads to a universal recognition of others as alter egos. The second one is the assimilation of local moral rules set by the social group. The first step is the moral contingency when the group imposes moral norms. The second step is the acceptation and the adoption of this moral norms that substitute old personal moral norms. Then it becomes the new ethics of the individual.

According to Levinas (1982: 85-101), the recognition of the face of someone implies a recognition of the human part of any alter ego. This implies, for him, a universal accounting that can't be made over to any tool or medium. So, the responsibility is necessarily direct and can't be based on a legal system or an artifact witch could unburden oneself.

In others words, the same objective of ethical values can be reached by different ways, and those last are note necessary not rational. This demonstrate the weakness to scrutiny the ethical behavior through the unique prism of "individualism vs collectivism". Furthermore, some ethical decision, seen as individualist, can be interpreted as collectivism in an medium or long term as soon as a medium in summoned up to regulate social activities via some economical or legal rationalism. Then by assuming that ethical decision is first and mainly intuitive, Sonenshein puts the rational aspect of some ethical systems aside. Social reference demonstrate that one is better than another (Gregoire 1967:6-7). I agree with Sonenshein, the sens of the issue is done by the individual. But it means that the ethical issue is dependent on his ethics. So, the intput of the model should be a matrix of different values depending on the different perception of ethical good values. Actually, Sonenshein claims that issue construction is influenced by expectations, motivation drives, social anchors and representations. But Expectations and motivation drives are working like filters made of culture of the individual.

Then this ethics is an important part of individuals culture. So, the type of ethics will influence the issue construction. In other words, the individual will construct an issue closer to one's own ethical point of view, rather than using all dimensions of all ethicals systems.

The second impact is the influence of the ethics on the intuitive judgment. I showed that different ethical systems are using different way of behaving ethically. If the ethical decision process is not necessary rational, it doesn't means that rationalization is systematically excluded. So that, the ethics of an individual influences the second stage of the model. The way of judgement will be more or less intuitive or rational, depending on the ethical reference the individual uses. An individual with a non-rational based ethics will certainly construct an intuitive judgment, unlike an other individual with a rational based ethics will decide more rationally.

The third impact concerns the evolution of the ethical behavior. At the input of the model, an individual has an ethics (Ethics C for example on the figure 2). According to Sonenshein, the two stages influence this ethics and the output is the same or a different sort of ethics (Ethics F for example on the figure 2). If there is a variation of type of ethics between before and after decision, it can create a psychological discomfort. As a kind of ideology, ethics is a group of values deeply grounded in the individual [START_REF] Flament | Anatomie des pensées ordinaires[END_REF]. Therefore, a punctual 11 modification caused by local influences can create a double-bind [START_REF] Bateson | Vers une écologie de l'esprit[END_REF]. This psychological consequence and the ethical influence of the context are used by the differential association theory to explain how a individual without criminal activities and with good education can behave with no compliance with social rules if one's is progressively immersed in a deviant social context [START_REF] Sutherland | Principles of Criminology[END_REF]. More important, the individual remains the conscious of the legal and sociological expected behavior, creating a double reference to ethics.

That scheme is established for blood crimes as well as for salaries deviation [START_REF] Skinner | A social learning theory analysis of computer crime among college student[END_REF]. The salary avoids temporary his owns values and will need to rationalize his behavior to reduce the double bind.

The fourth impact is the justification process. The social reference is what the group expects from an individual in the situation. It means that it can be an individual or a collective behavior, a rule or a moral compliance. The choice of the social reference is of prime necessity and has to be not influenced by the researcher ethics. If there is a difference between social expectation and ethical behavior, the individual will use justification to suppress the gap. Because the individual understands the expectations and give them the sens, the conformity of the behavior with social expectations are not obvious, even if the individual is highly susceptible of influence at the first and second stage of decision.

The justification and the rationalization depend on the reason of the issue construction. Here, Sonenshein claims that the ethical issue is built when there is ambiguity or uncertainty. This position is coherent with Trevino's affirmation (1986) and is the origin of the Sensmaking-Intuition Model. But, I think that other situations could lead to an ethical issue construction, depending on the personal sensibility. The first is when the individual projects his own 12 understanding of what others in the situation are thinking. Goffman explained (1988: 194) that in a situation each person believe knowing what others are expecting as moral behavior. This perception is not necessary a reality (it is an interpretation) but drives the individual to behave in the way one's imagines what others think.

The second situation is when ethical values are in contradiction with the values of the organization. I mean that there is no uncertainty, but a clear conflict of values. It is the case, for example when deontological codes are driving people to behave in a different sens than they would naturally. There are numerous studies dealing with the impact of codes of ethics on the behavior. The results show that the efficiency is not evident and people react differently depending on the situation. Some codes are useless [START_REF] Ferrell | Ethical beliefs of marketing managers[END_REF][START_REF] Webley | Corporate codes of ethics: Necessary but not sufficient[END_REF], some drives salaries to a more acceptable behavior [START_REF] Mccabe | The influence of collegiate and corporate codes of conduct on ethics-related behavior in the workplace[END_REF][START_REF] Peterson | The relationship between unethical behavior and the dimensions of the ethical climate questionnaire[END_REF] and others lead to an unacceptable one [START_REF] Weaver | Outcomes of organizational ethics programs: Influences of perceived values, compliance, and distrust orientations[END_REF].

The third situation is a will of the organization to keep some "spaces" free from deontology in order to let the salary creating his or her own reference and/or let the possibility to react very quickly to an environmental modification (Frese & Fay 2006: 261-262). This lack of reference permits to realize tasks that are not anticipated but are finally necessary to reach the objective (Clot 2000: 119).

Therefor, I argue that the output of model should be the same matrix of ethical systems letting arising the influences of the two first stages. And this output should be compared to the input for rationalization and to the social expectations for the justification. 13

Figure 1

 1 Figure 1 Sensemaking-Intuition Model (SIM)

I9

  propose then a definition of ethics as follow: ethics is a group of personal values based on what an individual believes good or bad and leads him to behave.New developments for the Sonenshein's modelI will now show that a sharp definition of ethics is will permit to improve the efficiency of the decision model on three points: implication for inputs and outputs of model, implication for the issue construction and implication for judgment. This drives to a modification of the Sonenshein's model.The new modelI propose hereafter (Figure2) an adaptation of the Sonenshein's model.

Figure 2

 2 Figure 2 Sensemaking-Intuition Model adaptation

Conclusion

The aim of this paper was to propose an improvement f the Sonenshein's model. I first exposed why his model could be improved, and especially why the definition of ethics was lacking. Then I presented how the diversity of definitions could have an incidence on the three stages of the original model and their articulations. Those modifications contribute to a better connection with other theories and empirical results. This new version permits to propose some futures research orientations.

The first orientation could be to identify how each ethics (ethics A, ethics B, ethics C...) influences the different stages of the model and how each ethics is dependent on the Issue Construction factors (Expectations, motivational drives, social anchors, representations).

A second orientation could lead researchers to study witch of the ethical values or the ethical method is more susceptible to influence.

A third orientation could be to observe if ethical dependence on issue construction context could be justified or rationalized depending on the rationality or the intuitive judgment.

Those orientations would permit to better understand the and the contextual influence of sociological and instruments (codes of ethics), depending on the variety of ethics of their salaries.


Emilie Guyon
email: emilie.guyon@univmed.fr

Denys Pommeret
email: denys.pommeret@univmed.fr

Imputation by PLS regression for linear mixed models

The problem of handling missing data in linear mixed models with correlated covariates is considered when the missing mechanism concerns both the dependent variable and the design matrix. We propose an imputation algorithm combining multiple imputation and Partial Least Squares (PLS) regression. The method relies on two steps: removing random effects, fixed effects are first imputed and PLS components are constructed on the corresponding complete case. The dependent variable is then imputed inside the linear mixed model obtained by adding the random effects to PLS components. The method is applied on simulations and on real data.

Introduction

The problem of handling missing data has been extensively studied in the statistical literature. The best general reference here is [START_REF] Little | Statistical Analysis with Missing Data[END_REF], where three non-response mechanisms are described: missing completely at random (MCAR), missing at random (MAR) and missing not at random (MNAR). Suppose that the data consist of an incomplete variable Y K and a set of fully observed variables Y 1 , ..., Y K-1 . The data are MCAR if the probability P (Y K missing) is a constant that does not depend on the variables. The data are MAR if the probability of missingness may depend on the fully observed variables Y 1 , ..., Y K-1 but does not depend on the incomplete variable Y K . The MNAR mechanism assumes that the probability of missingness depends on the variable that contains missing values. This work will be concerned with the MAR mechanism. When the missing data are MAR, multiple imputation (MI) was first proposed in [START_REF] Rubin | Multiple Imputation for Non-response in Surveys[END_REF] in the context of a non-response mechanism which concerns both the response variable and covariates. This method of imputation consists of replacing each missing value by a vector of M ≥ 2 imputed values. It is shown that MI is efficient in the sense that "When the M sets of imputations are repeated random draws from the predictive distribution of the missing values under a particular model for nonresponse, the M complete-data inferences can be combined to form one inference that properly reflects uncertainty due to nonresponse under that model", [START_REF] Little | Statistical Analysis with Missing Data[END_REF].

This technique has been adapted to imputation in linear models and in generalized linear models by [START_REF] Schafer | Analysis of Incomplete Multivariate Data[END_REF] and [START_REF] Ibrahim | Incomplete data in generalized linear models[END_REF] respectively. Recently, [START_REF] Bastien | Régression PLS et données censurées[END_REF] studied the problem of missing data imputation in generalized linear models when covariates are correlated. This author combined the Partial Least Squares (PLS) regression technique [START_REF] Wold | Path models with latent variables: the non-linear iterative partial least squares (NIPALS) approach[END_REF] with the multiple imputation method, obtaining a successfull method, called Multiple Imputation with Partial Least Squares (MI-PLS). It consists of imputing the missing data on the variable of interest by a PLS regression after imputation of missing values on each explicative variable. The problem of missing data in linear mixed models has also been investigated by [START_REF] Schafer | Fitting multivariate linear mixed models with incomplete data[END_REF], who proposed a method denoted by MI-L2M (Multiple Imputation in Linear Mixed Models) using multiple imputation on covariates. However, this method can break down when covariates are linearly dependent, due to the singularity of the design matrix.

In this paper, we consider the problem of missing data in the case of a linear mixed model when the covariates are highly correlated. An algorithm is proposed which combines the PLS approach defined in [START_REF] Bastien | Régression PLS et données censurées[END_REF] with the multiple imputation method proposed by [START_REF] Schafer | Fitting multivariate linear mixed models with incomplete data[END_REF]. This method will be denoted by MI-PLS-L2M (Multiple Imputation with Partial Least Squares in Linear Mixed Models). It consists in several steps. First, omitting the random effect, a linear model with dependent errors is considered. Missing data on covariates are imputed following the method of [START_REF] Little | Statistical Analysis with Missing Data[END_REF]. Several complete dataset of covariates are obtained and transformed into PLS components. Second, random effects are reintroduced in the model and a linear mixed model is obtained in which the fixed PLS components are constructed by linear combinations of the observed regressors.The Henderson method (1959, see Appendix) is used to estimate the parameters of the model. Finally, the predictions of the final model based on the PLS components and the random components are used to reconstruct the dependent variables. Moreover, following [START_REF] Bastien | PLS generalised linear regression[END_REF], a bootstrap validation procedure allowed to test the significance of the fixed effects.

The MI-PLS-L2M method is applied on simulations and compared to two alternative procedures: the first one is the MI-PLS proposed by [START_REF] Bastien | Régression PLS et données censurées[END_REF], restricted to the linear model. The second one is the MI-L2M introduced by Schafer and Yucel (1998). This algorithm was then applied on a real dataset, namely coffee dataset from [START_REF] Vivien | Une extension multi-tabeaux de la régression PLS[END_REF]: 17 samples of coffee were evaluated by 7 judges. In [START_REF] Vivien | Une extension multi-tabeaux de la régression PLS[END_REF], for each judge, characteristics of the coffee were explained by physico-chemical properties. Here, we have chosen to model the perception of the coffee by a score associated to a linear mixed model where the fixed effects are physico-chemical variables which are correlated and with a 7-level random effect corresponding to the 7 judges.

The paper is organized as follows: in Section 2 we review some standard facts on PLS and multiple imputation methods. In Section 3 we derive the imputation algorithm by combining these two former methods. Section 4 is devoted to the simulation study and Section 5 presents the study of the coffee dataset. Section 6 contains a short discussion.

Multiple imputation and PLS background

Let Y = (Y 1 , . . . , Y n ) be a vector of n observations. We denote by X the (n × p)-matrix of the p covariate vectors X j , j = 1, ..., p, referred to as the fixed effects. The random effect is a vector u ∈ R q . Given u, we consider the linear mixed model

Y i = x i β + z i u + i , ∀i ∈ {1, ..., n}, (1)
where x i and z i are row vectors of X and Z respectively, β ∈ R p is an unknown vector of regression coefficients and Z is a known matrix associated to the random effects. We will denote by N d (a, b) the d-dimensional normal distribution with mean a and variance b. It is assumed that

u ∼ N q (0, D), i ∼ N 1 (0, R i ), Cov(u, i ) = 0, ∀i = {1, ..., n},
where D = σ 2 q A, and A is a positive-definite matrix.

Multiple Imputation

When the data are Missing At Random, Rubin (1987) (see also [START_REF] Schafer | Analysis of Incomplete Multivariate Data[END_REF] introduced the multiple imputation (MI) as the most reliable method both from accuracy and efficiency point of view. The standard scheme of the MI-algorithm consists of three steps:

• Imputation: m > 1 samples of possible values for the missing data are created.

• Analysis: each of the m complete dataset is analyzed using the standard statistical method that would be used in the absence of non-response.

• Pooling: the results of the m analysis are combined to get a single complete dataset.

In the Analysis step, we follow Honacker et al. (2010) who proposed an Expectation-Maximization (EM) algorithm [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF]Rubin, 1977, Wu, 1983) to estimate and replace the missing values. The EM part of the algorithm can be described as follows. Assume that the missing values only concern the response variable Y = (Y obs , Y mis ). We denote by θ the vector of parameters of the model for the complete dataset. The EM algorithm can be summarized in the following way:

• (1) Complete the missing data Y mis using the first estimation of θ.

• (2) Using Y obs and Y mis completed, re-estimate θ.

• (3) Using θ estimated, re-estimate Y mis .

• (4) Iterate until convergence of θ.

PLS Regression

From now on, we assume that the vectors Y , X 1 , .., X p are centered. In presence of correlation between covariates, PLS regression can replace the classical linear regression model. The idea is to take over the matrix X by a matrix T = (t 1 , . . . , t h ), for h < p, iteratively obtained by linear transformation of the columns of X [START_REF] Tenenhaus | La régression PLS: théorie et pratique[END_REF] according to the following algorithm:

• Determination of the first component t 1 Compute the linear regression of Y on each X j , for j ∈ {1, ..., p}, Y = a 1j X j + .

Normalize the estimator â1j

w 1j = â1j ||â 1 ||
, ∀j ∈ {1, ..., p}, and so

t 1 = p j=1 w 1j X j .
Compute the linear regression of Y on t 1

Y = c 1 t 1 + Y1 ,
where the residuals Y1 express the deviation between the observed and the first PLS component. Then, for j ∈ {1, ..., p}, compute the linear regression of each covariate X j on t 1 ,

X j = p 1j t 1 + X 1j ,
where p 1j is the loading vector associated to t 1 and the residuals X 1j express the deviation between the covariates and the first PLS component.

• Determination of the components t h

For h = 2, ..., H, H < p, and for j ∈ {1, ..., p}, compute the linear regression of Y on each X j as well as on the other PLS components

t h-1 , Y = a hj X (h-1)j + c 1 t 1 + ... + c h-1 t (h-1) + Y (h-1) ,
where a hj and c h are the parameters associated to X (h-1)j and t h respectively, both of them being estimated by the model.

Normalizing the estimator âhj

w hj = âhj ||â h ||
, ∀j ∈ {1, ..., p}, and using a linear regression of the matrix X on the h PLS components

X j = p 1j t 1 + ... + p (h-1)j t (h-1) + X (h)j , ∀j ∈ {1, ..., p}
where p hj is the loading vector associated to t h and X (h)j is the residual term of the model,

t h = p j=1
w hj X (h)j .

• Number of PLS components

The number h of PLS components to be retained is estimated by crossvalidation. For that task, consider the regression model of y on the h PLS components:

y = c 1 t 1 + ... + c h t h ŷh +y h . (2)
At each step h, a criterion is calculated for each new component t h :

Q 2 h = 1 - P RESS h RSS h-1 ,
where RSS h (Residual Sum of Squares) and P RESS h (PRediction Error Sum of Squares) are defined as:

RSS h = n i=1 (y i -ŷhi ) 2 and P RESS h = n i=1 (y i -ŷh(-i) ) 2
where ŷh(-i) is the prediction of y i obtained by (2) without the observation i.

For h = 1,

RSS 0 = n i=1 (y i -y i ) 2
Referring to Tenenhaus (1998), a new component is considered as significant as soon as Q 2 h ≥ 0.0975.

Adapting imputation by PLS regression to the linear mixed model

Description of the method

Let Y obs and Y mis denote the vectors of observed and missing values of Y , respectively. The proposed method can be decomposed into six consecutive steps that we describe below.

Multiple imputation on fixed effects. We first impute X 1 , ..., X p by the multiple imputation method described in 2.1. We obtain m complete dataset.

PLS regression without random effects. Temporarily eliminating the random effect and according to 2.2, we compute the PLS components T of the heteroscedastic linear model:

Y obs = βX + ,
where = + Zu. Clearly, is a vector of dependent random variable such that ∼ N n (0; Σ), with Σ = ZDZ + R. But this transformation of does not modify the PLS components, since they are completely specified by the correlation between X and Y . The selection of the appropriate number of components h is based on the cross-validation criterion Q 2 h .

Estimation of fixed and random parameters. In order to take into account the random effects, let us consider the new linear mixed model

Y obs = T C + Zu + . (3)
The fixed parameters C and the random parameter u are estimated using Henderson's method (1959, see Appendix) and we denote by Ĉ and û their respective estimators. Then, we reformulate β according to W and Ĉ, using the recovery formulas given in Bastien et al. (2005, see Appendix ).

Selection of the fixed effects. We use a bootstrap validation procedure to assess the statistical significance of explanatory variables. This selection method is inspired from the jacknife technique introduced by Westad and Martens (1999). More precisely, the bootstrap procedure consists in sampling with replacement from Y obs with their associated components T and Z. Applying (3) to the B (B fixed) bootstrap samples, we finally obtain a vector β * of B estimators of β. It allows us to calculate a bootstrapped confidence interval of the regressors and Student tests are used to retain the significant variables at a prescribed level (arbitrarily 5%).

Pooling. The last step consists of pooling the m estimates into a single one which equals the estimated parameters mean, as it is done in Little and Rubin (2004). If θ1 , ..., θm denote the m estimators of the parameter θ = (β, u), the global estimator of θ is given by

θ = (β, u) = 1 m m k=1 θk .
Moreover the pooling method takes into account the variability of the m estimates as follows: Let D1 , ..., Dm denote the m estimators of V ar(u). The variability between the m dataset is the empirical mean of the estimated variance

V = 1 m m k=1
Dk , while the variability within every dataset is given by

W = 1 m -1 m k=1 (û k -u)(û k -u) .
Therefore, the global variance of u is given by

F = V + (1 + 1 m )W .
Finally, we simulate N 1 ∼ N n (Zu, F ) and N 2 ∼ N n (0, V ). We replace the ith missing value of Y by the corresponding ith value of

Y mis = Xβ + N 1 + N 2 .

Summary of the algorithm

Hereafter, we summarize the steps of the MI-PLS-L2M algorithm.

• Step 1. Imputation of covariates to get m complete dataset.

• Step 2. PLS procedure without random effect.

• Step 3. Consideration of random effect to estimate C and u.

• Step 4. Bootstrap selection.

• Step 5. Pooling.

Simulation study

In order to evaluate the performance of the proposed algorithm, simulations were performed on several sample sizes (N = 100 and N = 500) and for different variances of the random effects (V ar(u) = 0.5 and V ar(u) = 2). For each simulation, we have computed the algorithm and the results presented were the mean of 10 simulations. The performance criterion used is the Mean Square Error for missing values defined by

M SE = Nmis i=1 ( Ŷi -Y i ) 2 V ar(Y ) ,
where N mis denotes the number of missing values.

The model

In our simulations, the design matrix X consists of a N -sample of a 15dimensional covariate vector such that the last 10 components of the covariate vector are highly correlated with the first 5 ones, as it is shown in Table 4.1. The covariates are constructed as follows: five independent normal variables X 1 ∼ N (2, 1), X 2 ∼ N (0, 1), X 3 ∼ N (0, 1), X 4 ∼ N (0, 1), X 5 ∼ N (0, 1), and ten linearly dependent variables

X 6 = X 1 + 4X 2 -X 3 + 2X 4 + 3X 5 , X 7 = 2X 1 -X 2 + 5X 3 -3X 4 -2X 5 , X 8 = X 1 + 4X 2 -X 3 + 4X 4 + 0.5X 5 , X 9 = 2X 1 -X 2 + 5X 3 + X 4 -3X 5 , X 10 = 3X 1 + 3X 2 + 4X 3 + 5X 4 + 0.5X 5 , X 11 = 3X 1 -2X 2 -5X 3 + 0.5X 4 + X 5 , X 12 = -X 1 + X 2 -X 3 + X 4 -X 5 , X 13 = X 1 -4X 2 + 4X 3 + 0.5X 4 + 0.5X 5 , X 14 = X 1 + X 2 -2X 3 -2X 4 -2X 5 , X 15 = 0.5X 1 -0.5X 2 + X 3 + X 4 + 2X 5 .
We consider an independent error term ∼ N (0, I), where I denotes the identity matrix. In a first case, for N = 100 observations, the random effect was a 3level vector u ∼ N 3 (0, 2I). In other cases, for N = 500 observations, the random effect was first a 3-level vector u ∼ N 3 (0, 2I), and second a 3-level vector u ∼ N 3 (0, 0.5I).

The output variable Y belongs to R N and the MI-PLS-L2M algorithm (as well as competitors MI-PLS and MI-L2M) was run with various percentage p of missing value on Y and X. We have chosen p ∈ {8%, 10%, 15%, 20%, 25%}. The number of multiple imputation was fixed to m = 5 following Rubin (1987).

Study of the MI-PLS-L2M performance

Using the algorithm without missing data. In order to evaluate the efficiency of the algorithm, we have first calculated estimators based on the original simulated data without missing value for the three considered scenarios. Table 2 shows the estimations of the vector β based on five retained PLS components. We can observe a bias due to the hight correlation between covariates. This bias is compensated on all the components of β. Finally a certain stability through the sample size and the variance of the random effect u is observed. Analysis of MI-PLS-L2M performance. We have studied estimations of C, the coefficient vector associated to the PLS components, and estimations of the standard errors of u and when using MI-PLS-L2M. The different estimations were obtained with p = {0%, 8%, 10%, 15%, 20%, 25%}. Tables 3-5 contain estimations for the significant PLS components at the risk level α = 0.05. The test of significance was based on B = 200 bootstrap samples. Only five components were retained for all models. It can be seen that the estimation of V ar( ) was increasing with the percentage of missing data while the estimations of V ar(u) was more stable. The MSE clearly increased with the number of missing values. Tables 3 and5 illustrate the gain of precision due to the sample size. Figure 1 illustrates the stability of the distribution of Y after multiple imputation as showed by the boxplot of Y after imputation using MI-PLS-L2M, with respect to the percentages of missing values. The sample size was N = 500 and the variance of the random effect was V ar(u) = 2. A general remark is that the distribution of the predicted data seems to be relatively close to the initial dataset with no missing values. From the left to the right : Boxplot with 0%, 8% (before and after imputation), 10% (before and after imputation), 15% and after imputation), 20% (before and after imputation), 25% (before and after imputation) of missing values

0% 08% 08%(IM) 10% 10%(IM) 15% 15%(IM) 20% 20%(IM) 25% 25%(IM) -40 -20 0

Comparing with MI-PLS and MI-L2M methods

There is no known method to deal with the problem of missing data for a linear mixed model in presence of correlation between covariates. However, to assess the contribution of the method we have confronted the result of MI-PLS-L2M with the results obtained with MI-PLS and MI-L2M. By construction, the method MI-PLS proposed by [START_REF] Bastien | Régression PLS et données censurées[END_REF] will not take into account the random effect and then for assessing the contribution of our method we have considered different variances of the random effects: it was expected that with large variance of u (V ar(u) = 2) the MI-PLS-L2M will be better, although with a small variance of u (V ar(u) = the results obtained with the two algorithms will be similar. Concerning the MI-L2M method proposed by Schafer and Yucel (1998), it was not adapted to the case of collinearity because of singularity of the design matrix, but we proposed to make the regression of Y obs on each vector of the design matrix and each component of β was obtained from a one-dimensional model. On each model, a parameter of the random effect was estimated and for the final model, the mean of the estimated parameters was calculated. The MI-PLS-L2M method took into account all variables simultaneously and we have therefore expected to obtain better results. We have used the ratio of the MSE to quantify the difference between the three methods.

The first step imputation of covariates was the same for the three algorithms. We used the method of Honacker and King (2010), that is a multiple imputation by EM algorithm of X, as defined in 2.1 and using the R package Amelia.

Table 6 presents ratio between MSE obtained for the three methods of imputation according to the proportion of missing values for N = 100 and for V ar(u) = 2. Clearly, the variance of the random effects was large but the sample size was too small to detect this effect. Then, the variability of random effect was spread within the random errors and the estimated model was close to a linear one. Hence, MI-PLS and MI-PLS-L2M gave very close results in terms of MSE. For all cases MI-L2M gave larger MSE since it was used as consecutive univariate methods. In Table 7 ratio for MSE are presented for N = 500 and V ar(u) = 2. This larger sample size allowed to well estimate the random effect and then the MI-PLS-L2M method took into account this variable. Moreover, it gave better results than MI-PLS where the random effect was omitted. Concerning MI-L2M, the conclusion was the same than in the first situation because of the collinearity. In Table 8 the ratio of MSE were obtained for N = 100 and V ar(u) = 0.5. The variance of the random effect was too small to get significant estimations with MI-PLS-L2M. Then, this situation concords with the first one where the random effect was not detected and MI-PLS-L2M gave the same results than MI-PLS. Concerning MI-L2M, the conclusion is the same as previously. In conclusion, MI-PLS-L2M performed better than the two other methods when the random effect is significantly estimated; that is, when the sample size allows to estimate its variability. When random effect was not significant (too small variance, or too small sample size), MI-PLS-L2M and MI-PLS gave similar results and they were better than MI-L2M as soon as there was a collinearity between covariates. Figure 2 summarizes this analysis by showing the associated MSE, respectively for N = 100 and V ar(u) = 2, N = 500 and V ar(u) = 0.5, and N = 500 and V ar(u) = 2. The three methods have a term of error increasing with the number of missing values. We used the data presented in [START_REF] Vivien | Une extension multi-tabeaux de la régression PLS[END_REF]: 17 coffee samples were prepared with 3 parameters which are temperature, proportion grindingwater and size, with 3, 3 and 2 modalities, respectively. 4 samples, with similar physico-chemical properties were chosen as reference sample for a comparison with the other 13 samples. In order to evaluate the different samples, 7 judges from the ENSBANA (Dijon, France) have compared 3 times each of the 13 samples to a reference sample and have answered 6 questions: Which both is the more characteristic in flavor, the more intense in flavor, the most bitter, the more acid, the most characteristic in aroma ans the most intense in aroma ? by specifying if the difference was very easy, easy, complicated, hard, or almost impossible to evaluate. In order to summary these judgements, a scale, from 1 to 5 was created where:

• 1: The reference sample is strictly greater than the sample.

• 2: The reference sample is greater than the sample.

• 3: The judge has perceived no difference.

• 4: The sample is greater than the reference sample.

• 5: The sample is strictly greater than the reference sample.

Here, we have chosen to create a mean score of these marks, called Y , which constitutes our variable of interest. The judges were considered as a random effect of 7 levels and X is a design matrix associated to 10 fixed effects corresponding to the 10 physico-chemical properties and we simulate 15% of missing values on Y and X, by random resampling without replacement. Table 9 presents the fixed effects and Table 10 provides an extract of the data.

Competitor methods

As in the simulations study, we have confronted the results of the MI-PLS-L2M method with those of MI-PLS and MI-L2M, obtained after ten compilations of the algorithm. We used the MSE to quantify the difference between the imputation of Y and its true value.

Results

Table 12 presents ratio between MSE obtained for the three methods of imputation according to the 15% of missing values. On this example, MI-PLS-L2M seems to perform better than the two other methods. We have compared the different estimations obtained with MI-PLS-L2M without missing values and with p = 15%. We estimated C, the coefficient parameter associated to the PLS components, and the standard errors of u and . Table 13 contains estimations for the significant PLS components at the risk level α = 0.05. The test of significance was based on B = 200 bootstrap samples. As expected only six components were retained for the model, as in [START_REF] Vivien | Une extension multi-tabeaux de la régression PLS[END_REF]. The loss of accuracy obtained under the missing mechanism did not appear significant. The random effect u was estimated significantly with a variance close to that of the errors . It might explain the advantage of the use of MI-PLS-L2M on this dataset. Table 14 presents estimations for the random effect. It appears that the two first judges overestimate the coffee score while the effect of the third judge seems to make decrease the score. The other judges seem to estimate in the same way the coffee score. As observed in the simulation study, the boxplots before and after imputation associated to p = 15% of missing values were very close to the initial dataset and we have omitted their representation.

Discussion

The algorithm MI-PLS-L2M was proposed to deal with the problem of missing data in a linear mixed model when covariates are correlated. It combines the multiple imputation theory developed by [START_REF] Rubin | Multiple Imputation for Non-response in Surveys[END_REF] adapted to the linear mixed models with the PLS method introduced by Wold (1975) (see also [START_REF] Tenenhaus | La régression PLS: théorie et pratique[END_REF]. It is also an adaptation of the MI-PLS algorithm proposed by [START_REF] Bastien | Régression PLS et données censurées[END_REF] and the MI-L2M initiated by Schafer and Yucel (1998), since it is dedicated to the problem of missing data in the presence of both collinearity and random effect.

Simulation studies are carried out which suggest that the proposed method is advocated as soon as a random effect is significant. When the random effect is detected, MI-PLS-L2M provides good estimations of the parameters and keeps the distribution shape of the original data before imputation. It is also shown that the MSE increases slowly with the percentage of missing values.

Moreover, the gain provided by our algorithm compared to the method used in [START_REF] Bastien | Régression PLS et données censurées[END_REF] and the method proposed in Schafer and Yucel (1998) was studied. Because of the singularity of the design matrix due to the collinearity, the MSE calculated for MI-PLS-L2M was better than that of MI-L2M. The ratio between MSE shown better performances of MI-PLS-L2M other MI-PLS when the random effects are significant. When the variance of the random effect is small, or for sample size too small to detect it, MI-PLS and MI-PLS-L2M gave similar results.

The application of our method to a real dataset showed good performance through the MSE and the estimation of the parameters. In addition, the method detected a significant random effect and then a new interpretation of the analysis of the coffe dataset. Future research will be to adapt the MI-PLS-L2M to the generalized linear mixed models. The way is to use a step of linearization of the model, adapting for instance the algorithm of [START_REF] Schall | Estimation in generalized linear models with random effects[END_REF].

Aknowledgements We are very grateful to the reviewers for their helpful comments and suggestions. We also thank the Editor and an Associate Editor for their careful readings. This work has been supported by Region Provence Alpes Cote d'Azur.

Table 2 :

 2 Values of β and its estimators for N = 100, 500 and V ar(u) = 0.5, 2 β β (N = 100; V ar(u) = 2) β (N = 500; V ar(u) = 0.5) β (N = 500; V ar(u) = 2) 1 1

  Figure1: From the left to the right : Boxplot with 0%, 8% (before and after imputation), 10% (before and after imputation), 15% and after imputation), 20% (before and after imputation), 25% (before and after imputation) of missing values

Figure 2 :

 2 Figure 2: MSE associated to the three methods with respect to the % of missing data

Table 1 :

 1 Correlation matrix between the covariates (Pearson correlation coefficients, N = 500)

		Y	X1	X2	X3	X4	X5	X6	X7	X8	X9	X10	X11	X12	X13	X14
	X1	0.336	1
	X2	0.797	0.051	1
	X3	-0.014	-0.076	-0.018	1
	X4	0.354	0.032	-0.024	-0.021	1
	X5	0.383	-0.002	0.013	-0.076	0.028	1
	X6	0.940	0.234	0.706	-0.253	0.336	0.574	1
	X7	-0.298	0.224	-0.142	0.784	-0.436	-0.385	-0.553	1
	X8	0.880	0.245	0.691	-0.225	0.650	0.130	0.848	-0.520	1
	X9	-0.160	0.247	-0.161	0.797	0.128	-0.535	-0.460	0.827	-0.175	1
	X10	0.701	0.406	0.399	0.495	0.627	0.050	0.493	0.164	0.674	0.520	1
	X11	0.010	0.507	-0.265	-0.824	0.115	0.217	0.207	-0.558	0.147	-0.536	-0.268	1
	X12	0.191	-0.399	0.438	-0.429	0.414	-0.434	-0.211	-0.558	0.542	-0.260	0.009	-0.021	1
	X13	-0.437	0.087	-0.694	0.700	0.091	0.029	-0.547	0.623	-0.529	0.682	0.203	-0.285	-0.681	1
	X14	0.183	0.489	0.475	-0.451	-0.414	-0.431	0.127	0.040	0.185	-0.130	-0.151	0.336	0.235	-0.634	1
	X15	0.350	0.168	-0.192	0.323	0.396	0.795	0.400	-0.088	0.170	0.012	0.476	0.038	-0.540	0.494	-0.685

Table 3 :

 3 Estimations associated to significant PLS component t 1 , t 2 , t 3 , t 4 , t 5 . Brackets give the standard deviations obtained by bootstrap, N = 100 and V ar(u) = 2

	Missing (%)	0%	8%	10%	15%	20%	25%
	C (sd)	7.16 (0.05) 7.37 (0.06) 7.54 (0.06) 7.48 (0.06) 7.45 (0.07) 7.35 (0.09)
		2.29 (0.06) 2.13 (0.07) 1.96 (0.07) 2.06 (0.08) 2.13 (0.09) 2.02 (0.01)
		1.12 (0.07) 0.82 (0.07) 0.79 (0.07) 0.82 (0.07) 0.83 (0.09)	0.88 (0.1)
		0.30 (0.06) 0.26 (0.07) 0.21 (0.07) 0.25 (0.08) 0.22 (0.09) 0.28 (0.12)
		0.14 (0.08) 0.09 (0.12) 0.08 (0.13) 0.11 (0.14) 0.89 (0.47) 0.15 (0.37)
	sd(u)	1.57	1.23	1.22	1.36	1.29	1.71
	sd( )	1	1.07	1.04	1.11	1.3	1.48
	MSE	0	0.21	0.19	0.36	0.4	0.51
	N observed	100	92	90	85	80	75

Table 4 :

 4 Estimations associated to significant PLS component t 1 , t 2 , t 3 , t 4 , t 5 . Brackets give the standard deviations obtained by bootstrap, N = 500 and V ar(u) = 0.5

	Missing (%)	0%	8%	10%	15%	20%	25%
	C (sd)	7.76 (0.02) 7.66 (0.02) 7.62 (0.02) 7.63 (0.03) 7.58 (0.03) 7.61 (0.04)
		2.25 (0.03) 2.04 (0.03) 2.04 (0.03) 2.07 (0.03) 2.11 (0.03) 2.02 (0.04)
		0.93 (0.03) 0.88 (0.03) 0.91 (0.04)	0.9 (0.03)	0.93 (0.03) 0.96 (0.04)
		0.26 (0.04) 0.22 (0.03) 0.22 (0.03) 0.21 (0.04) 0.22 (0.04) 0.23 (0.05)
		0.04 (0.03) 0.04 (0.03) 0.03 (0.03) 0.03 (0.04) 0.04 (0.04) 0.08 (0.07)
	sd(u)	0.60	0.58	0.53	0.56	0.54	0.69
	sd( )	0.98	0.99	0.99	1.07	1.11	1.34
	MSE	0	0.18	0.25	0.33	0.47	0.57
	N observed	500	460	450	425	400	375

Table 5 :

 5 Estimations associated to significant PLS component t 1 , t 2 , t 3 , t 4 , t 5 . Brackets give the standard deviations obtained by bootstrap, N = 500 and V ar(u) = 2

	Missing (%)	0%	8%	10%	15%	20%	25%
	C (sd)	7.47 (0.04) 7.30 (0.03) 7.48 (0.04) 7.60 (0.04) 7.36 (0.05) 7.25 (0.04)
		2.25 (0.04) 2.46 (0.04) 2.26 (0.04) 2.17 (0.05) 2.44 (0.06) 2.48 (0.05)
		1.05 (0.05) 1.12 (0.05) 1.03 (0.05) 0.86 (0.05) 1.10 (0.07) 1.12 (0.07)
		0.19 (0.04) 0.21 (0.05) 0.16 (0.05) 0.23 (0.05) 0.17 (0.07) 0.14 (0.06)
		0.10 (0.05) 0.11 (0.05) 0.11 (0.05) 0.12 (0.06) 0.13 (0.08) 0.09 (0.07)
	sd(u)	1.71	1.73	1.71	1.79	1.90	1.94
	sd( )	1.02	1.01	1.03	1.03	1.05	1.12
	MSE	0	0.15	0.22	0.31	0.46	0.83
	N observed	500	460	450	425	400	375

Table 6 :

 6 Ratio between MSE for MI-PLS and MI-L2M versus MI-PLS-L2M, N mis denoting the number of missing values, for N = 100 and var(u) = 2

	Missing values (%) N mis	MSE(MI -PLS) MSE(MI -PLS -L2M)	MSE(MI -L2M) MSE(MI -PLS -L2M)
	8	8	1.1	4.1
	10	10	1.11	5.47
	15	15	1.14	4.61
	20	20	1.08	5.05
	25	25	1.16	5.75

Table 7 :

 7 Ratio between MSE for MI-PLS and MI-L2M versus MI-PLS-L2M, N mis denoting the number of missing values, for N = 500 and var(u) = 0.5

	Missing values (%) N mis	MSE(MI -PLS) MSE(MI -PLS -L2M)	MSE(MI -L2M) MSE(MI -PLS -L2M)
	8	40	1	4.06
	10	50	1	4.82
	15	75	1.03	4.84
	20	100	1	4.60
	25	125	1.04	5.09

Table 8 :

 8 Ratio between MSE for MI-PLS and MI-L2M versus MI-PLS-L2M, N mis denoting the number of missing values, for N = 500 and var(u) = 2

	Missing values (%) N mis	MSE(MI -PLS) MSE(MI -PLS -L2M)	MSE(MI -L2M) MSE(MI -PLS -L2M)
	8	40	3.54	2.74
	10	50	3.49	4.72
	15	75	3.18	5.16
	20	100	3.19	5.18
	25	125	2.22	3.63

Table 9 :

 9 The fixed effects

	Dry extract	EXS
	Extraction rate	TEE
	PH	PHH
	Acidity	CID
	Optical density to 430n. m	DO4
	Optical density to 510n. m	DO5
	Conductance	CDT
	Caffeine	CAF
	Viscosity	VIS
	Retention of water in the milling CPR

Table 10 :

 10 Fixed effects for each coffee sampleTable11provides the correlation values of the fixed effects and the response variable. It justifies clearly the use of the PLS method.

	Cof. EXS TEE PHH CID DO4 DO5 CDT CAF VIS CPR
	1	2.5	-1.18 -0.02 1.02	0.72	0.31	3.09	0.95	0.32	0.13
	2	-1.44 -4.62 0.27 -0.66 -0.42 -0.17 -2.42	-0.2 -0.11 -0.34
	3	0.18	1.42 -0.10 0.10	0.09	0.04 -0.25 0.25 -0.09 -0.70
	4	0.85	6.59	0.05	0.35	0.56	0.28	0.69 -0.05 -0.04 -0.42
	5	0.49	4.76	0.00	0.22	0.33	0.11	0.44	0.32 -0.04 -0.07
	6	-0.08 -0.15 0.05	0.00 -0.10 -0.06 -0.06 -0.20 -0.19 0.07
	7	0.18	2.56	0.10	0.01	0.01 -0.01 0.32	0.00 -0.13 0.01
	8	-1.16 -4.12 0.05 -0.57 -0.26 -0.10 0.32	0.00 -0.13 0.01
	9	-1.07 -2.76 0.00 -0.22 -0.12 -0.04 -0.64 -0.27 -0.16 -0.09
	10	-0.45 4.76	0.00 -0.50 -0.22 -0.08 -1.68 -0.50 -0.22 0.08
	11	-1.06 -2.85 -0.05 -0.63 -0.17 -0.05 -2.08 -0.60 0.09	0.00
	12	1.89 -5.82 0.15	0.78	0.28	0.12	2.30	1.05	0.05 -0.27
	13	1.45 -6.97 0.20	0.58	0.18	0.08	2.42	.50	0.05 -0.21

Table 11 :

 11 Correlation between the response variable and the fixed effects

	Cor	Y	EXS	TEE	PHH	CID	DO4	DO5	CDT CAF	VIS	CPR
	EXS	0.624	1
	TEE 0.254	0.021	1
	PHH -0.163 0.030 -0.492	1
	CID	0.578	0.974 -0.019 0.030	1
	DO4 0.519	0.779	0.194	0.085	0.752	1
	DO5 0.644	0.859	0.283 -0.212 0.863	0.838	1
	CDT 0.400	0.755 -0.119 0.409	0.731	0.824	0.669	1
	CAF 0.401	0.862 -0.272 0.175	0.881	0.753	0.662 0.778	1
	VIS	0.429	0.681 -0.275 -0.071 0.645	0.635	0.691 0.555 0.639	1
	CPR 0.021 -0.044 -0.016 -0.087 -0.106 -0.305 -0.128 0.016 -0.158 0.084	1

Table 12 :

 12 Ratio between Mean Square Errors for MI-PLS and MI-L2M versus MI-PLS-L2M, with 15% of missing values

	MSE(MI -PLS)	MSE(MI -L2M)
	MSE(MI -PLS -L2M)	MSE(MI -PLS -L2M)
	10.516	15.671

Table 13 :

 13 Estimations associated to significant PLS component t 1 , t 2 , t 3 , t 4 , t 5 , t 6 . Brackets give the standard deviations obtained by bootstrap.

	Missing (%)	0	15
	C (sd)	0.375 (0.001)	0.366 (0.002)
		0.132 (0.097)	0.158 (0.162)
		0.066 (0.079)	0.075 (0.047)
		0.031 (0.148)	0.057 (0.158)
		0.884 (0.787)	0.629 (0.741)
		-1.283 (1.175) -1.909 (1.968)
	sd(u)	0.045	0.042
	sd( )	0.086	0.096
	MSE	0	0.221
	N observed	91	77

Table 14 :

 14 Estimations associated to the significant random effect.

	Judge	1	2	3	4	5	6	7
	û	0.42 0.49 -0.72 0.30 0.33 0.33 0.36

Appendix

The Henderson method (1959) Given the random effects, we assume that Y , X and u satisfy (1). The Henderson method (1959) defines an equation system

where β and ũ are solution of the GLS (Generalized Least Squares) and the BLUP (Best Linear Unbiased Predictor). Writing:

we obtain the following system

Expression of PLS components in terms of the original explanatory variables [START_REF] Bastien | PLS generalised linear regression[END_REF] All variables y, x 1 , ..., x j , ..., x p are assumed to be centered. The PLS regression model with h components is written as

with the constraint that the PLS components t H = p j=1 w * Hj x j are orthogonal and the parameters c H and w * Hj in (6) are to be estimated. The estimated regression equation may be then expressed in terms of the original variables x j 's: