A Multivariate Methodology for Analysing Students' Performance Using Register Data
Jeanett S. Pelck, Rafael Pimentel Maia, Hildete P. Pinheiro, Rodrigo Labouriau
AA Multivariate Methodology for Analysing Students’
Performance Using Register Data
Jeanett S. Pelck , Rafael Pimentel Maia , Hildete P. Pinheiro and RodrigoLabouriau Department of Mathematics, Aarhus University, Denmark Department of Statistics, University of Campinas, Brazil *Corresponding author: Rodrigo Labouriau , [email protected]
February, 2021
Abstract
We present a new method for jointly modelling the students’ results in the univer-sity’s admission exams and their performance in subsequent courses at the university.The case considered involved all the students enrolled at the University of Camp-inas in 2014 to evening studies programs in educational branches related to exactsciences. We collected the number of attempts used for passing the university courseof geometry and the results of the admission exams of those students in seven dis-ciplines. The method introduced involved a combination of multivariate generalisedlinear mixed models (GLMM) and graphical models for representing the covariancestructure of the random components. The models we used allowed us to discussthe association of quantities of very different nature. We used Gaussian GLMM formodelling the performance in the admission exams and a frailty discrete-time Coxproportional model, represented by a GLMM, to describe the number of attemptsfor passing Geometry.The analyses were stratified into two populations: the students who received abonus giving advantages in the university’s admission process to compensate socialand racial inequalities and those who did not receive the compensation. The twopopulations presented different patterns. Using general properties of graphical mod-els, we argue that, on the one hand, the predicted performance in the admissionexam of Mathematics could solely be used as a predictor of the performance in ge-ometry for the students who received the bonus. On the other hand, the Portugueseadmission exam’s predicted performance could be used as a single predictor of theperformance in geometry for the students who did not receive the bonus.
Keywords:
Multivariate generalised linear mixed models, Graphical models, Separation prin-ciple, Global Markov property, Affirmative polices. a r X i v : . [ s t a t . A P ] F e b Introduction
In this paper, we study the admission system to a Brazilian university and the bonussystem for compensating social and racial inequalities. The data analysed below is basedon the registers of entrance and performance at the University of Campinas, Brazil (UNI-CAMP). In 2005, UNICAMP implemented an affirmative action program giving extrabonus in the final entrance examination score for students who were enrolled for theirentire high school years in the public system (with an additional bonus for those whoself declared to be African / Indigenous Brazilian descendants). See Maia et al. (2016),Pedrosa et al. (2007), Pinheiro et al. (2019, 2020) for more details.The main difficulty of the study of those registers is the multivariate nature of thecharacterisations of the object of interest, and the (unavoidable) presence of spuriousassociations. The responses observed in the data referred above are of very differentnature but can be analysed in one multivariate model as we will describe below. Theperformance at the university is measured by the number of attempts required to passthe course of geometry, which is a key course in the beginning of the university educationof the group of students we study. This response is typically right-censored, in the sensethat there might be some students that have not passed the course when the data wascollected, dropped out during the study or who’s enrolment has been cancelled. Theenrolment is cancelled if the student fails all the subjects in the first or second semester orreached the maximum number of semesters allowed without graduating (e.g. in Statisticsthis is 6 years). On the other hand, the performances at the admission exams are measuredin a standardised scale using a scoring system. Those two types of responses are modelleddifferently: the time for passing a course is modelled using a variant of the frailty Coxproportional model with discrete-time; the scores at the entrance exam are described usinga Gaussian mixed model. In both cases, the models can be represented as instances ofgeneralised linear mixed models (GLMMs). The introduced GLMMs contain two commonrandom components (one taking different values for each individual, and one taking thesame value for individuals enrolled at the same branch of study). The random componentrelated to the individuals allow us to connect the models describing the different responses,and in this way, characterise how much information each response carries on the otherresponses. This methodology differs from Pinheiro et al. (2020) where parts of the samedata were analysed but in a different context.The analyses performed were stratified into two populations: the students who receiveda bonus, and the students who did not receive the compensation. The two populationsare different and presented different patterns, justifying the stratification.This study aims to present statistical tools that allow to study the different facets ofthe type of data described above and to understand the associations between the differentresponses. These aims are fulfilled by using suitable multivariate versions of GLMMsand by using the theory of graphical models to describe the covariance structure of thecommon random component giving the variation between individuals (Pelck & Labouriau2021b). We illustrate, in this way, a process of modelling responses of very different naturein a multivariate model that arises when working with educational register data.The paper is organised as follows. Section 2 describes the data used. The multivariateGLMM are introduced in Section 3, including details on the marginal Gaussian GLMMsand the frailty discrete-time Cox Proportional model. Section 4 describes the graphicalmodel used for representing the covariance structure of the random components, andSection 5 presents and discusses the results. Appendix A presents some model control,while some details of the representation of the graphical models are given in appendix B.
The data we used contain records on all the students enrolled at the UNICAMP in2014, in evening studies programs in one of the educational branches related to exactsciences listed in Table 1. Among those students, received a bonus giving advantagesin the university’s admission process. The bonus group consists of students from a pub-lic high school and students from a public high school who are self-declared African orIndigenous Brazilian descendants.Chemical engineering Electrical engineering Economical scienceMathematics Physics Computer scienceAutomation engineering Technological chemistry Bachelor in ChemistryMedical PhysicsTable 1: Educational branches included in the population studied.The data includes eight responses recorded for each student. The first seven responsescorrespond to the student’s performance in the university admission exam in the disci-plines: Mathematics, Physics, Chemistry, Biology, History, Geography, and Portuguese.The last response was the number of attempts the students used to pass the (first year)geometry course at the university. This response was right censored (with . of cen-soring) since some students used at least the observed number of attempts, but it wasunknown whether the enrolment had been cancelled or the student passed the course afterthe data was collected. Additionally, the registers contain a range of information on eachindividual including gender and age. 3 A Multivariate Model for for Simultaneously Describ-ing the Admission Scores and the Performance in Ge-ometry
The eight responses described above were jointly analysed using a multivariate generalisedlinear mixed model as described below. We performed separate parallel analyses for thestudents who received the bonus and those who did not receive the bonus, which, weanticipate, will yield contrasting results.In each of the two separate analyses, we used a multivariate model combining sevenGaussian mixed models describing the seven admission exams, and a frailty Cox propor-tional model with discrete-time for modelling the number of attempts to pass the courseof geometry. Each of the marginal models above included two random components: oneaccounting for the variation between the different study branches, and one representingeach individual. The eight marginal models referred above were combined by assuminga joint multivariate Gaussian distribution for the random component representing theindividuals. The precise model definition for the students who received bonus is givenbelow. The models for the students that did not receive bonus are similarly defined.In order to describe the models we will use, we index the individuals by i ( i = 1 , . . . , n ,with n = 151 ), the eight responses by j ( j = 1 , . . . , ) and the educational brancheslisted in Table 1 by k ( k = 1 , . . . , ). Moreover, the educational branch of the i th student is denoted by e ( i ) . We describe below the covariance structure of the multivariategeneralised linear mixed models we want to introduce using two random components. Therandom component representing the educational branches is defined by assuming thatthere exist unobservable random variables U [ j ]1 , . . . , U [ j ]10 for each of the eight responses( j = 1 , . . . , ) corresponding to the educational branches. The random variable U [ j ] k takes the same value for the j th response for each student that is enrolled in the k th educational branch ( k = 1 , . . . , , j = 1 , . . . , ). The random components representingthe individuals is specified for the j th response ( j = 1 , . . . , ) by defining the unobservablerandom variables V [ j ]1 , . . . , V [ j ] n representing the n individuals.According to the model, for j, j (cid:48) = 1 , . . . , , the random vectors U [ j ] def = ( U [ j ]1 , . . . , U [ j ]10 ) T and V [ j (cid:48) ] def = ( V [ j (cid:48) ]1 , . . . , V [ j (cid:48) ] n ) T are independent and multivariate Gaussian distributed asgiven below, U [ j ] ∼ N (cid:0) , σ U [ j ] I (cid:1) V [ j (cid:48) ] ∼ N n (cid:0) , σ V [ j (cid:48) ] I n (cid:1) . Here I m denotes a m -dimensional identity matrix (for m ∈ N ). Furthermore, for j, j (cid:48) =1 , . . . , , with j (cid:54) = j (cid:48) , we assume that U [ j ] is independent of U [ j (cid:48) ] , and that V [ j ] i is in-dependent V [ j (cid:48) ] i (cid:48) , where i, i (cid:48) = 1 , . . . , n with i (cid:54) = i (cid:48) . Additionally, we assume that for4 = 1 , . . . , n , Cov (cid:16) V [1] i , . . . , V [8] i (cid:17) = Σ V , (1)with diag ( Σ V ) = (cid:0) σ V [1] , . . . , σ V [8] (cid:1) . Therefore, defining V T def = (cid:16) V [1] T , . . . , V [8] T (cid:17) we havethat Cov ( V ) = Σ V ⊗ I n . We will use the matrix Σ V to characterise the dependence structure of the eight responses.In particular, in Section 4, we will use a graphical model structure by imposing zeroesin the inverse of Σ V , corresponding to assume that some pairs of the random variable V i [1] , . . . , V i [8] are conditionally independent given the other random variables. For im-proving the readability of the discussion below, when relevant, we denote V i [1] , . . . , V i [8] by V i [ Math ] , . . . , V i [ Geom ] , i.e., we identify the superindices of the individual random com-ponents with a recognisable short form of the corresponding response.We formulate the marginal generalised linear mixed models for the seven responsesrelated to the admission exams by specifying the conditional expectations for the i th indi-vidual ( i = 1 , . . . , n ) in the conditional Gaussian distributions with identity link functiongiven ( U [ j ] e ( i ) , V [ j ] i ) for j = 1 , . . . , , that is, E [ Y [ j ] i | U [ j ] e ( i ) = u, V [ j ] i = v ] = x Ti β [ j ] + u + v, ∀ u, v ∈ R . In all the models above, the term x Ti β [ j ] ( j = 1 , . . . , ) adjusts for gender (female or male)and age with age divided into two groups (under 21 or 21 and above).We formulate the discrete time frailty Cox proportional hazard model describing thenumber of attempts to pass the course of geometry. According to the model, for thestudent i th ( i = 1 , . . . , n ), the discrete conditional hazard function given ( U [8] e ( i ) , V [8] i ) is λ i ( t | U [8] e ( i ) = u, V [8] i = v ) def = P ( T i = t | T i ≥ t, U [8] e ( i ) = u, V [8] i = v )= ˜ λ t exp( x Ti β [8] ) exp( u ) exp( v ) , for t = 1 , , . . . for all u, v ∈ R . Here T i is the random variable representing the number of attempts to pass the courseof geometry used by the i th individual and the term x Ti β [8] adjusts for gender (female ormale) and age with age divided into two groups (under 21 or 21 and above). The modelabove coincides with a generalised linear mixed model defined with a binomial distributionand a logarithmic link function, applied to a specially constructed data representing therisk set of the related counting process. We used a Poisson approximation for avoidingnumerical issues. See Maia et al. (2014). 5 Modelling the Covariance Structure of the RandomComponents
We complete the specification of the multivariate generalised linear mixed model intro-duced in Section 3 by defining the covariance structure of the random components rep-resenting the individual’s variation. Since the random components V [1] i , . . . , V [8] i have thesame distribution for i = 1 , . . . , n , we suppress the subindex i from the notation and write V [1] , . . . , V [8] to denote V [1] i , . . . , V [8] i for an arbitrary individual.The random components V [1] , . . . , V [8] represent the individual variation of the abilitiesof each of the n students affecting the performance in the seven admission exams and inthe course of geometry, respectively. Note that according to the model, the covariance ofthe random variables V [1] i , . . . , V [8] i is the same for all the individuals, namely Σ V , see (1).Here we will characterise this covariance structure common to all the individuals usinggraphical models, which will allow us to draw general conclusions on the interdependencebetween the eight responses studied. Before pursuing this task, we give a short accountof the theory of graphical models; for a comprehensive description of this theory seeLauritzen (1996) and Whittaker (1990).Let G = ( V , E ) denote a graph with a set of vertices, V , and edges, E ⊆ V × V . Eachvertex represents a random variable, and two vertices are connected with an edge if, andonly if, they are not conditionally independent given the remaining random variables. Wesay that there is a path between two vertices if there exist a sequence of pairs of verticesconnected with an edge connecting the two vertices.In the multivariate models described above, we consider a graph, G = ( V , E ) , with V = { V [1] , . . . , V [8] } = { V [ Math ] , . . . , V [ Geom ] } . The set of edges contains pairs of randomvariables which are not conditionally independent given the remaining random variables,and thus, they carry some information on each other that are not contained in the otherrandom variables in V . For example, suppose that there is an edge between V [ Geom ] and V [ Math ] associated to the individual responses related to the performance in the courseof geometry and the admission exam in mathematics, respectively. This means that af-ter correcting for differences in age, gender and education branch, the random variables V [ Geom ] and V [ Math ] are conditionally dependent given V \ (cid:8) V [ Geom ] , V [ Math ] (cid:9) ; therefore V [ Math ] carries information on V [ Geom ] that is not contained in the informational con-tents of the random variables V \ (cid:8) V [ Geom ] , V [ Math ] (cid:9) . Conversely, the absence of an edgeconnecting two vertices indicates that the random variables associated to those verticesare conditionally independent given the other random variables in play; therefore, theknowledge of the other random variables renders the two random variables in questionindependent.According to the theory of graphical models, a set of vertices, say S , separates two setsof vertices A and B in the graph, if, and only if, each path connecting an element of A toan element of B contains at least one element of S . A key result in the theory of graphical6odels is that if a set of vertices, S , separates two disjoint subsets of vertices A and B ,then all the variables in A are independent of the variables in B given the variables in S . This result is called the separation principle or global Markov property for undirectedgraphical models (Lauritzen 1996, page 32). For example, if the random variable V [ Math ] separates the random variable V [ Geom ] from the other vertices, then conditioning on solely V [ Math ] renders V [ Geom ] independent of the other vertices ( i.e., V \ (cid:8) V [ Geom ] , V [ Math ] (cid:9) ).Note that the graph G = ( V , E ) defined with V = { V [ Math ] , . . . , V [ Geom ] } involves theunobservable random variables associated with the individual random components, notthe observed responses. However, it is possible to extend the separation principle using themultivariate generalised linear mixed model’s properties to discuss the interdependenceof the observed responses. We explain this extended principle using an example. Supposethat the random variable V [ Math ] separates the random variable V [ Geom ] from the othervertices in V , then, according to the extended separation principle, each response corre-sponding to a vertex in V \ (cid:8) V [ Geom ] , V [ Math ] (cid:9) , say Y [ ∗ ] i , is independent of T i = T [ Geom ] i ( i.e., the number of attempts the i th individual uses to pass the course of geometry) given V [ Math ] . Here it is required to condition on the random variable V [ Math ] for obtainingindependency of Y [ ∗ ] i and T i ; conditioning on the observable responses Y [ Math ] i does notnecessarily renders the variables Y [ ∗ ] i and T i independent. See the Appendix Band Pelck& Labouriau (2021b) for a general formulation of the extended separation principle usingundirected graphical models.The graph G = ( V , E ) with vertices V = { V [1] , . . . , V [8] } was inferred using predictedvalues of the random variables to infer a graphical model that minimises the BIC. Amethod and implementation for minimising the BIC are described in Abreu et al. (2010),and an inference method for predicting values of the random components is presented inPelck & Labouriau (2021a). Notice, that this method only yields normally distributedpredictors in cases with small variance of the random components. However, in Labouriau(1998) it is shown that treating graphical models involving non Gaussian random vari-ables as being normally distributed corresponds to using optimal inferential procedure forsemiparametric models under mild regularity conditions.The survival model presented was controlled using the methods described in Maiaet al. (2014), Edwards et al. (2010). We found no indication of a lack of fit of the models.The marginal Gaussian mixed models where controlled by standard residual analyses. Seeappendix A. Figure 1 displays the graphs representing the estimated graphical models describing thecovariance structure of the individual random components for the students that receivedbonus and the students that did not. The two populations of students presented differentcovariance structures which we discuss below. We stress that each of the individuals’7andom components represents latent individual abilities affecting the performance relatedto the respective responses. The covariance structures described in Figure 1 are obtainedafter adjusting for differences in age, gender and educational branch.In the population of students that received bonus, the random component related tothe performance in the course of geometry, V [ Geom ] , is only connected to the random com-ponent related to the result in the admission exam of mathematics, V [ Math ] (see the leftpanel in Figure 1 and Figure 4). The conditional correlation between V [ Geom ] and V [ Math ] is positive. This result suggests the existence of common cognitive mechanisms associatedwith the latent abilities related to the performance in the admission exam of mathematicsand in the course of geometry. Since V [ Math ] separates V [ Geom ] from the other individ-ual random components, according to the separation principle, V [ Geom ] is conditionallyindependent of the other individual random components given V [ Math ] . This conditionalindependence indicates that the putative common cognitive mechanisms referred aboveare specific to these two disciplines and are not shared by the other disciplines’ abilities.Regarding the results of the admission exams, according to the extended separation prin-ciple, the observed performance in the course of geometry is conditionally independentof the observed results of the admission exams given the individual random component V [ Math ] . From the practical point of view, the result shows that the prediction of therandom component V [ Math ] suffices for predicting the performance of the students thatreceived the bonus in the course of geometry. After predicting V [ Math ] , both the resultsof the other admission exams and their corresponding individual random componentsbecome uninformative concerning the performance in the course of geometry.We obtain a different scenario for the population of the students that did not receive thebonus. There, the individual random component V [ P ort ] (associated with the performancein the admission exam of Portuguese) is the only random component connected with therandom component V [ Geom ] ; moreover, V [ P ort ] separates V [ Geom ] from the other individualrandom components (see the right panel of Figure 1 and Figure 3). Therefore, using asimilar rationale as above, we conclude that for the population of students that did notreceive the bonus, the prediction of V [ P ort ] suffices for predicting the performance of thestudents that received the bonus in the course of geometry.The variance of the predictions of V [ Math ] is larger in the population of studentsthat received the bonus, as compared with the variance in the group that did not receivethe bonus. A combination of two factors might cause this difference: in the populationof students who received the bonus, there might be more considerable variability in thequality of the high school teaching in Mathematics; furthermore, the students with a lowerlevel in Mathematics could enter the University by receiving a bonus. Therefore, the math-ematics skills detected in the admission exam play an essential role in the performancein geometry among the students who received the compensation. One might speculatewhether the random component V [ P ort ] is representing social-economic class which playsa key role in the performance in the course of geometry population of students that did8 [ His ] V [ Chem ] V [ Port ] V [ Geo ] V [ Math ] V [ Bio ] V [ Phys ] V [ Geom ] (a) No bonus V [ His ] V [ Chem ] V [ Port ] V [ Geo ] V [ Math ] V [ Bio ] V [ Phys ] V [ Geom ] (b) Bonus Figure 1: Independence graph representing the estimated graphical model describing thecovariance structure of the individual random components V [1] i , . . . , V [8] i for an arbitrarilychosen individual ( i = 1 , . . . , n , suppressing the index i in the graph). The estimatedconditional correlations are reported for each edge .not receive the bonus.There are also some similarities between the inferred covariance structures of the twopopulations of students studied. For example, in both populations, the individuals randomcomponent V [ Bio ] (related to the performance in the admission exam of biology) separates V [ Math ] , V [ P hys ] and V [ Chem ] from V [ His ] and V [ Geo ] . We let the reader explore furtheraspects of the results presented here. In this paper, we have exposed a new method basedon a combination of multivariate generalised linear mixed models and graphical modelsfor modelling and predicting students’ performance using responses of different nature,namely, some Gaussian responses and a discrete right-censored response. Other responsetypes can also be modelled by choosing different distributions and different link functionsfor constructing the marginal models. Acknowlegment
We acknowledge the admission committee (CONVEST) from the University of Camp-inas (UNICAMP) for giving access to the data used. The first and the last authorswere partially financed by the Applied Statistics Laboratory (aStatLab) from the De-9artment of Mathematics, Aarhus University. The third author was partially financed byConselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Grant number:310874/2018-1.
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Journal of Applied Statistics (6), 1286–1306.Maia, R. P., Pinheiro, H. P. & Pinheiro, A. (2016), ‘Academic performance of studentsfrom entrance to graduation via quasi u-statistics: a study at a brazilian researchuniversity’, Journal of Applied Statistics (1), 72–86.Pedrosa, R. H., Dachs, J. N. W., Maia, R. P., Andrade, C. Y. & Carvalho, B. S. (2007),‘Academic performance, students’ background and affirmative action at a brazilianuniversity’, Higher education management and policy (3), 1–20.Pelck, J. S. & Labouriau, R. (2021a), Multivariate generalised linear mixed models withgraphical latent covariance structure. In preparation.Pelck, J. S. & Labouriau, R. (2021b), Conditional inference for multivariate generalisedlinear mixed models. In preparation.Pinheiro, H. P., Maia, R. P., Lima-Neto, E. A. & Rodrigues-Motta, M. (2019), ‘Zero-oneaugmented beta and zero inflated discrete models with heterogeneous dispersion: anapplication to students’ academic performance’, Stat. Methods Appl. , 749–767.Pinheiro, H. P., Sen, P. K., Pinheiro, A. & Kiihl, S. F. (2020), ‘A nonparametric approachto assess undergraduate performance’, Statistica Neerlandica , 538–588.Whittaker, J. (1990), ‘Graphical models in applied multivariate analysis’, Chichester NewYork et al: John Wiley & Sons . 10 a) No bonus (b) Bonus
Figure 2: Normal QQ-plot of the responses related to the seven admission exams and ascatter plot of the observed number of events versus expected number of events for eachtime and combination of age and gender.
A Some Model Control
We briefly discuss below the validity of the models used. The residual analyses in themarginal Gaussian mixed models, representing the the responses related to the sevenadmission exams, show that there is no indication of serious lack of fit, see Figure 2.Comparing the observed and the expected number of students that passed the courseof geometry at each time for each combination of age and gender allowed us to concludethat there is no evidence of lack of global adjustment of the survival model. More precisely,the predicted number of events at time t ( t = 1 , . . . , T ), denoted n ( t ) , is calculated as thenumber of individuals "at risk" (the number of students that have not passed the courseyet and are still studying the course) times the average estimated hazard at time t . Moreprecisely, n ( t ) = | R t | (cid:88) i ∈ R t ˆ λ t exp( x Ti ˆ β ) exp(ˆ u e ( i ) ) exp(ˆ v i ) , where R t = { i ∈ { , . . . , n } : t i ≤ t } denote the set of all individuals at risk at time t and | R t | the number of individuals in R t . The results can be found in Figure 2.11 Detailed Representation of the Graphical Models In-volving the Random Components and the Responsevariables
For the reader acquaint with the theory of graphical models (see Whittaker, 1990), theextended separation principle can be formulated in general by defining an directed acyclicgraph (DAG, i.e., a graph formed by vertices and directed edges represented by arrows ob-tained by eliminating the symmetry property in the set of edges E ). Using basic propertiesof the generalised linear mixed models of the type discussed here and the factorisation ofthe joint densities of the distributions of the individual random component, it is possibleto show that the interdependence of the the observable responses and the random com-ponents related to the individuals can be represented by an acyclic graphs, where thereis an arrow from each random component pointing to the random variables represent-ing the corresponding observable responses. Additionally, the graphical representationreferred above contains an undirected edge connecting the vertices that are not condi-tionally independent in the graph representing the individual random components (seePelck & Labouriau, 2021b for the detailed construction). Noting that this acyclic graphsatisfies the Wermuth condition (see Whittaker, 1990, page 75), which implies that themoral graph obtained, in this case, by making all the edges undirected, satisfies the sepa-ration principle (see Whittaker, 1990, theorem 3.5.2 on page 76). This construction yieldsthe extended separation principle which states that "two sets of observable responses, say ˜ A and ˜ B , are conditionally independent given a set S of individual random components,provided S separates the sets of individual random components A and B corresponding tothe sets of observable responses ˜ A and ˜ B ". 12 [ His ] V [ Chem ] V [ P ort ] V [ Geo ] V [ Math ] V [ Bio ] V [ P hys ] V [ Geom ] Y [ His ] Y [ Chem ] Y [ P ort ] Y [ Geo ] Y [ Math ] Y [ Bio ] Y [ P hys ] T [ Geom ] Figure 3: Independence graph (central rectangle) representing the estimated graph-ical model describing the covariance structure of the individual random components V [1] i , . . . , V [8] i for an arbitrarily chosen individual ( i = 1 , . . . , n , suppressing the index ii
For the reader acquaint with the theory of graphical models (see Whittaker, 1990), theextended separation principle can be formulated in general by defining an directed acyclicgraph (DAG, i.e., a graph formed by vertices and directed edges represented by arrows ob-tained by eliminating the symmetry property in the set of edges E ). Using basic propertiesof the generalised linear mixed models of the type discussed here and the factorisation ofthe joint densities of the distributions of the individual random component, it is possibleto show that the interdependence of the the observable responses and the random com-ponents related to the individuals can be represented by an acyclic graphs, where thereis an arrow from each random component pointing to the random variables represent-ing the corresponding observable responses. Additionally, the graphical representationreferred above contains an undirected edge connecting the vertices that are not condi-tionally independent in the graph representing the individual random components (seePelck & Labouriau, 2021b for the detailed construction). Noting that this acyclic graphsatisfies the Wermuth condition (see Whittaker, 1990, page 75), which implies that themoral graph obtained, in this case, by making all the edges undirected, satisfies the sepa-ration principle (see Whittaker, 1990, theorem 3.5.2 on page 76). This construction yieldsthe extended separation principle which states that "two sets of observable responses, say ˜ A and ˜ B , are conditionally independent given a set S of individual random components,provided S separates the sets of individual random components A and B corresponding tothe sets of observable responses ˜ A and ˜ B ". 12 [ His ] V [ Chem ] V [ P ort ] V [ Geo ] V [ Math ] V [ Bio ] V [ P hys ] V [ Geom ] Y [ His ] Y [ Chem ] Y [ P ort ] Y [ Geo ] Y [ Math ] Y [ Bio ] Y [ P hys ] T [ Geom ] Figure 3: Independence graph (central rectangle) representing the estimated graph-ical model describing the covariance structure of the individual random components V [1] i , . . . , V [8] i for an arbitrarily chosen individual ( i = 1 , . . . , n , suppressing the index ii in the graph), for the students who did not receive bonus.13 [ His ] V [ Chem ] V [ P ort ] V [ Geo ] V [ Math ] V [ Bio ] V [ P hys ] V [ Geom ] Y [ His ] Y [ Chem ] Y [ P ort ] Y [ Geo ] Y [ Math ] Y [ Bio ] Y [ P hys ] T [ Geom ] Figure 4: Independence graph (central rectangle) representing the estimated graph-ical model describing the covariance structure of the individual random components V [1] i , . . . , V [8] i for an arbitrarily chosen individual ( i = 1 , . . . , n , suppressing the index ii