Analyse Formelle des Concepts https://fr.wikipedia.org/wiki/Analyse_formelle_de_concepts

https://en.wikipedia.org/wiki/Formal_concept_analysis

Applications dans le domaine de la psychologie clinique

Formal Psychological Assessment

• Les travaux de Rudolf WILLE et son « école de Darmstadt »

• & ceux du CAMS (le Centre d’Analyse et de Mathématique Sociales) de l’EHESS de Paris (Marc Barbut et ses élèves)

Ouvrages mathématiques

• Une relation d’ordre dans un ensemble est une relation binaire dans cet ensemble qui permet de comparer ses éléments entre eux de manière cohérente.

• Un ensemble muni d’une relation d’ordre est un ensemble ordonné.

• On dit aussi que la relation définit sur cet ensemble une structure d'ordre

relation d'ordre

• Soit E un ensemble ; une relation interne R sur

E est une relation d'ordre si pour tous x, y et z éléments de E :

• x R x (réflexivité)

• x R y et y R x sont incompatibles si x ≠ y

(antisymétrie)

• (x R y et y R z) ⇒ x R z (transitivité)

Ordre strict vs Ordre large

On parle :

- d’ordre strict lorsque la réflexivité n'est pas satisfaite,

- et d’ordre large lorsque la réflexivité

est satisfaite

Préordre, ordres total et partiel

Un préordre est une relation binaire réflexive

et transitive

Un ordre total est une relation d’ordre pour laquelle deux éléments de E sont toujours comparables. S’il n’est pas total, il est partiel.

Graphe et sa réduction transitive

Diagramme de Hasse de l’Inclusion, sur l’ensemble des parties d’un ensemble de 3 éléments

Revenons à l’ Analyse Formelle des Concepts

et ses Applications dans le domaine de la psychologie clinique

Formal Psychological Assessment

CONCEPT ?

• En logique, l’intension (« intent » en anglais ou « compréhension » chez les auteurs les plus anciens) et l’extension (« extent » en anglais) sont deux façons de définir un concept.

• L'intension d'un concept est sa définition, ce sont ses attributs. Par exemple, l'intension de « chat » est : « animal à quatre pattes de la famille des félins ».

• L'extension est l'ensemble des objets auxquels l'intension (la définition) s'applique. Par exemple : mon chat, le chat de mon voisin, les chats siamois, etc.

Représentation sous forme d’un tableau de la relation dite d’incidence

nombre composé pair impair premier carré

1 x x

2 x x

3 x x

4 x x x

5 x x

6 x x

7 x x

8 x x

9 x x x

10 x x

Nous avons donc

• un ensemble d'objets (Gegenstände) : les nombres entiers de 1 à 10

G = {1,2,3,4,5,6,7,8,9,10}

• et un ensemble d'attributs (Merkmale): des propriétés mathématiques

M = {pair, impair, composé, premier, carré}

21/01/2002 Tutoriel TG-ECBD / Conf EGC'02

Treillis de Galois - Définition • Contexte = triplet (O, A, I) tel que:

– O : ensemble fini d’objets

– A : ensemble fini d’attributs

– I : relation binaire entre O et A , (I OxA) (dite relation d’incidence)

2 autres exemples :

O\A a b c

1 1 1 1

2 1 1

3 1 1

a b c d e f g h

1 1 1 1 1 1 1 1

2 1 1 1 1 1 1 1

3 1 1 1 1 1 1 1

4 1 1 1 1 1

5 1 1 1 1 1

6 1 1 1 1

7 1 1 1

Correspondance de Galois • Correspondance de Galois: Exemple

– O1 = {6, 7} f(O1) = {a, c}

– A1 = {a, c} g(A1) = {1, 2, 3, 4, 6, 7}

Remarque: h(O1) = g • f(O1) = g (A1) ≠ O1

En revanche, h{1,2,3,4,6,7} = {1,2,3,4,6,7}

a b c d e f g h

1 1 1 1 1 1 1 1

2 1 1 1 1 1 1 1

3 1 1 1 1 1 1 1

4 1 1 1 1 1

5 1 1 1 1 1

6 1 1 1 1

7 1 1 1

nombre composé pair impair premier carré

1 x x

2 x x

3 x x

4 x x x

5 x x

6 x x

7 x x

8 x x

9 x x x

10 x x

On a f({3}) = {impair, premier} et g({impair, premier }) = {3,5,7}} Donc ({3,5,7}, {impair, premier}) est un concept formel dont l’extension est {3,5,7} et l’intension est {impair, premier}

• Relation d’ordre ( ≤ ) sur l’ensemble des concepts : Sous-concept / Sur-concept (spécialisation /

généralisation)

Rappel : Diagramme de Hasse de l’Inclusion, sur l’ensemble des parties d’un ensemble par exemple de 3 éléments

Un sous-concept d’un concept est défini par

(A1, B1) ≤ (A2, B2) ⇔ A1 ⊆ A2

ou de manière équivalente

(A1, B1) ≤ (A2, B2) ⇔ B1 ⊇ B2

Les concepts (A,B) ordonnés par inclusion sur leurs premiers membres A - ou dualement sur leurs deuxièmes membres B - forment un treillis appelé treillis de Galois ou treillis de concepts.

• Triadic concept analysis replaces the binary relation between objects and attributes by a ternary relation between objects (g), attributes (m), and conditions (c).

• (g,m,c) then expresses that the object g has the attribute m under the condition c.

• Although triadic concepts can be defined in analogy to the formal concepts above, the theory of the trilattices formed by them is much less developed than that of concept lattices, and seems to be difficult.[Voutsadakis has studied the n-ary case

(Voutsadakis G. "Polyadic Concept Analysis". Order19(3), 295–304, 2002.

• The Formal Psychological Assessment is the formal conjunction and clinical application of two theories of mathematical psychology :

- the Knowledge Space Theory (KST) (Doignon, Falmagne, ..) - and the Formal Concept Analysis (FCA) (écoles allemande et française citées là-haut) https://link.springer.com/content/pdf/10.3758%2FBRM.42.1.342.pdf

RESEARCH ARTICLE (2015) Formal Psychological Assessment in Evaluating Depression: A New Methodology to Build Exhaustive and Irredundant Adaptive Questionnaires Francesca Serra, Andrea Spoto, Marta Ghisi & Giulio Vidotto Department of General Psychology, University of Padua, Padua, Italy

• Psychological Assessment can be defined as a complex procedure of information collection, analysis and processing.

• Formal Psychological Assessment (FPA) tries to improve this procedure by providing a formal framework to build assessment tools.

• In this paper, FPA is applied to depression. Seven questionnaires widely used for the self-evaluation of depression were selected. Diagnostic criteria for major depressive disorder were derived from the DSM-5, literature and Seligman’s and Beck’s theories.

• A Boolean matrix was built, including 266 items from the questionnaires in the rows and 20 selected attributes, obtained through diagnostic criteria decomposition, in the columns.

• In the matrix, a 1 in a cell meant that the corresponding item investigated the specific attribute.

• It was thus possible to analyze the relationships between items and attributes and among items.

Results • While none of the considered questionnaires could

alone cover all the criteria for the evaluation of depressive symptoms, we observed that a set of 30 items contained the same information that was obtained redundantly with 266 items.

• Another result highlighted by the matrix regards the relations among items. FPA allows in-depth analysis of currently used questionnaires based on the presence/absence of clinical elements.

• FPA allows for going beyond the mere score by differentiating the patients according to symptomatology. Furthermore, it allows for computerized adaptive assessment.

Le problème posé par le « score »

For instance, if a nine dichotomous items scale is administered to a patient and the clinical cut-off score of the scale is 7, there are 46 different clinically significant response patterns (one pattern with score 9, nine patterns with score 8, and 36 patterns with score 7).

Each of these patterns may convey clinically different information about the patient.

• By highlighting the specific clinical elements investigated by each single item of a questionnaire, FPA highlights the differences among patients that would otherwise be hidden by the simple score.

• From a clinical perspective, it allows for an idiographic and nomothetic diagnosis.

Nomothetic versus Idiographic

• Nomothetic is based on what Kant described as a

tendency to generalize, and is typical for the natural sciences. It describes the effort to derive laws that explain types or categories of objective phenomena, in general.

• Idiographic is based on what Kant described as a tendency to specify, and is typical for the humanities. It describes the effort to understand the meaning of contingent, unique, and often cultural or subjective phenomena.

Autre recherche : MOCQ

• The relations between the items of the form of the Maudsley Obsessive-Compulsive Questionnaire (Sanavio and Vidotto (MOCQ-R; 1985), and the diagnostic criteria for Obsessive-Compulsive Disorder (OCD) included in the Diagnostic and Statistical Manual of Mental Disorders IV-TR (DSM-IV-TR; American Psychiatric Association [APA], 1995).

MOCQ-R • The short form is composed of 21 dichotomous

items (True-False) investigating the main characteristics of OCD.

• The questionnaire is subdivided into three sub-scales investigating three of the main specifications of the disorder :

- Checking (8 items)

- Cleaning (9 items)

- Doubting-Ruminating (4 items)

Exemples d’items

Checking

• item 4 “I must check many times some particular things (e.g. gas or water taps, doors, etc.)”,

• or item 14 “I usually check things more than once”

Exemples d’items

Cleaning

• item 3 “When I touch an animal I feel contaminated”,

• or item 17 “Every morning I spend a lot of time in washing myself completely”

Exemples d’items

Doubting-Ruminating

• item 2 “I frequently have disagreeable thoughts and I cannot get rid of them”.

• ORIGINAL RESEARCH ARTICLE

• Front. Psychol., 06 July 2018 https://doi.org/10.3389/fpsyg.2018.01101

• New Perspectives in the Adaptive Assessment of Depression: The ATS-PD Version of the QuEDS

• Andrea Spoto, Francesca Serra, Ivan Donadello, Umberto Granziol and Giulio Vidotto

Summary 1/3

• Measurement is a crucial issue in psychological assessment. In this paper a contribution to this task is provided by means of the implementation of an adaptive algorithm for the assessment of depression.

• More specifically, the Adaptive Testing System for Psychological Disorders (ATS-PD) version of the Qualitative-Quantitative Evaluation of Depressive Symptomatology questionnaire (QuEDS) is introduced.

Summary 2/3

• Such implementation refers to the theoretical background of Formal Psychological Assessment (FPA) with respect to both its deterministic and probabilistic issues.

• Three models (one for each sub-scale of the QuEDS) are fitted on a sample of 383 individuals. The obtained estimates are then used to calibrate the adaptive procedure whose performance is tested in terms of both efficiency and accuracy by means of a simulation study.

Summary 3/3

• Results indicate that the ATS-PD version of the QuEDS allows for both obtaining an accurate description of the patient in terms of symptomatology, and reducing the number of items asked by 40%.

• Further developments of the adaptive procedure are then discussed.

Des questions ?

• pas toutes et tous à la fois …