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45a. Pénétrer le statisticisme

Ce billet résulte de la difficulté, pour de nombreux étudiants de psychologie, de comprendre ce que sont les phénomènes statistiques rapportés par de nombreuses études empiriques en psychologie. Cette difficulté est aggravée, selon moi, par le canon éditorial qui veut que tout article empirique soit structuré en quatre parties, l’introduction, la méthodologie, les résultats et la discussion (American Psychological Association, 2010, chapitre 2), dont aucune ne permet de présenter explicitement l’expérience aléatoire qui construit — aux sens de (i) concevoir la possibilité de et (ii) en saisir les traces — le phénomène statistique1. Comme il faut se livrer à une « archéologie » si on veut s’en faire une idée précise, cette norme de communication scientifique rend improbable la question de savoir quel est le support ontologique de l’expérience aléatoire — de quoi appréhende-t-on le comportement statistique ? — et permet d’admettre, à tord, que ce qui a été saisi du phénomène statistique est une description générale, c’est-à-dire valable pour tout individu (au sens générique d’entité) qui aurait pu être enrôlé dans l’étude pour enrichir l’échantillon d’unités d’observation (voir aussi Harré, 2004, Lamiell, 2013).

Des étudiants ont sélectionné Geiselman et Glenny (1977) qui, si je me fie à eux2, ont testé 30 personnes à qui ils ont présenté successivement 18 diapositives, un questionnaire et 72 mots. Les étudiants retiennent deux critères du « mode » dit « d’essai » (voix masculine vs. féminine) et trois critères de « répétition » (voix masculine, féminine, sa propre voix). Si je ne trahis pas leur analyse, ils n’ont aucune idée de ce que les chercheurs cherchent dans les données parce qu’ils n’ont aucune idée du phénomène statistique que ceux-ci ont en tête.

On lit dans le résumé de l’article « A surprise recognition test for the words indicated that the words were more likely to be recognized if they were spoken in the same Voice at test as was used to repeat them during presentation ». Le phénomène statistique en question est la reconnaissance plus probable de mots préalablement entendus dans la condition « same Voice » que dans une autre condition. Il s’agit de ne pas confondre le phénomène psychologique, qui se produit, si l’on peut dire, dans la tête de chaque sujet lorsqu’il reconnaît ou ne reconnaît pas un mot qu’il vient d’entendre, et le phénomène statistique, qui ne se produit pas nécessairement chez quelqu’un (voir aussi Falissard, 2014 ; Hanson, 1971 ; Papert, 1981). C’est tout un travail d’acculturation, pour un étudiant de psychologie, d’accepter de s’intéresser non pas au premier phénomène, pertinent parce qu’il concerne des personnes « concrètes »3 mais au second, dont la pertinence tient à sa facilité d’accès empirique — les « données »4.

1. Version 1 de l’histoire

Comment se représenter ce phénomène statistique ? Par un schéma statistique simple, qui porte sur un événement aléatoire (voir Falmagne, 2003), nommément l’occurrence d’un nombre de mots rappelés : on observe un nombre de mots qui joue le rôle du résultat d’un lancer de dé — et, on va le voir plus bas, tout le problème est de savoir ce qu’est le dé — ; de plus, on suppose que les résultats de l’expérience aléatoire dépendent de la manière dont on jette le dé — selon qu’on déclenche l’expérience aléatoire dans telle ou telle condition.

Distinguons la condition « same Voice » et la condition « not-same Voice ». Le nombre de mots qui s’observe dans chaque condition est non seulement variable d’un sujet (penser « d’un lancer ») à un autre, mais encore les chercheurs parient que la variabilité aléatoire n’est pas la même dans chaque condition. Ils s’attendent à ce qu’en moyenne, le nombre de mots soit plus grand dans la condition same Voice. Compte tenu de ce qu’ils utilisent 30 sujets (penser « 30 lancers »), ils peuvent comparer deux moyennes (une par condition), chacune étant probablement calculée à partir des 15 « lancers » de chaque condition.

En termes de tableau de données, où les colonnes sont des « variables » et les lignes les différentes occurrences de l’expérience aléatoire, on a une colonne qui indique, pour chaque occurrence (ou unité d’observation), la condition expérimentale et une colonne qui indique le nombre de mots rappelés ; et on a 30 lignes puisqu’on a 30 lancers. Le référentiel de description de cette étude est, pour la variable dite indépendante, la liste des conditions expérimentales, i.e., {same Voice, etc.5} et, pour la variable dite dépendante, {0, 1, …, n}, où n est le nombre maximum de mots pouvant être rappelés (je dirais 18) — le référentiel de l’étude est le produit cartésien de ces deux ensembles.

Concernant la population statistique qui permet de construire l’idée statistique de variable (bivariée puisqu’on a une VI et une VD), comment définir les unités statistiques qui la composent ? Du point de vue probabiliste, la population statistique se comporte, en émettant des signaux ou encore de l’information variables dans le référentiel de description de l’étude ; du point de vue empirique, et contrairement au dé comme réservoir de lancers, la population statistique n’a pas d’existence naturelle — d’où le paradoxe empiriste de ces études, qui est qu’elles sont fondées sur des données objectives qui, ensemble, décrivent le mirage d’un objet. Ce mirage résulte de la capacité des chercheurs à fabriquer un échantillon d’unités d’observation, en recrutant des sujets, en leur attribuant une condition expérimentale et en observant comment ils s’y comportent. Autrement dit, on a bien 30 unités d’observation, qui sont concrètement des couples (personne, moment de l’expérience dans la vie de cette personne), mais elles sont fabriquées au moment où les chercheurs le décident et il faut faire appel à l’imagination pour faire comme si les observations résultaient d’un échantillonnage : il n’y a pas plus de dé que d’urne qui contient à l’avance les unités statistiques dont certaines deviendraient des unités d’observation suite à un tirage au sort.

Le problème est qu’une telle façon d’étudier les gens ne permet d’apprendre d’une personne enrôlée dans l’expérience qu’un nombre : le nombre de mots qu’elle a reconnus dans telle circonstance expérimentale. Autrement dit, on ignore comment l’effet statistique mis en évidence peut intervenir chez au moins une personne et on peut même se demander si la réalité d’un tel effet expérimental a un sens au niveau individuel. On voit par là que le style de recherche qui définit la psychologie expérimentale depuis les années 30 (Danziger, 1987) consiste, pour les chercheurs, à montrer qu’ils sont capables de fabriquer des phénomènes statistiques (et ce n’est hélas pas un problème scientifique s’ils ne décrivent personne en particulier6).

2. Version 2 de l’histoire

La présentation qui précède méconnaît délibérément l’étude de Geiselman et Glenny (1977) : il s’agit, encore une fois, non pas de ce que Geiselman et Glenny racontent en fait, mais des connaissances qui sont nécessaires à leur lecteur pour qu’il obtienne de l’information quand il les lit — la compréhension d’un article empirique consiste essentiellement en une reconnaissance, laquelle n’est possible que si on connaît à l’avance ce qui y a à reconnaître. L’information essentielle dans ce type d’étude est la manière dont les chercheurs ont construit le phénomène statistique qu’ils présentent.

Il y a une autre manière possible de procéder, qui permet d’éviter l’absence de support ontologique du phénomène statistique (aucune entité empirique n’en est le siège). Il est possible que chaque sujet soit soumis à plusieurs (au moins deux) conditions expérimentales, et ceci de manière répétée, auquel cas on dispose, pour chaque sujet, du nombre moyen de mots rappelés dans chaque condition, et le phénomène statistique caractérise un sujet particulier — on a plusieurs dés.

En termes de tableau de données, il faut rajouter au tableau précédent une colonne identifiant les sujets de l’expérience. On peut alors se demander si le phénomène statistique est un fait général, c’est-à-dire si, dans le cas le plus simple, la différence de moyennes a été observée pour chacun des 30 sujets testés — ce qui est plus intéressant. Les unités d’observation sont alors, étant donné un sujet particulier, les triplets (sujet, moment auquel il a été exposé à un mot, moment auquel il l’a reconnu ou pas). Le fait de conditionner le phénomène statistique « différence de moyennes » à un sujet « concret » permet de lui donner un support empirique (où « empirique » est synonyme de « concret »).

Toutefois, dans la mesure où le référentiel est artefactuel7, la population statistique reste une fiction dont on peut se passer si on peut échapper au rite du test statistique de l’hypothèse nulle (cf. Ohlsson, 2010). Un test statistique ne dit rien sur la robustesse du phénomène statistique, parce qu’un phénomène robuste est un phénomène régulier (qui se répète) ; or, pour savoir qu’un phénomène se répète, il faut en avoir pris l’habitude ce qui implique qu’il faut recommencer l’expérience aléatoire autant de fois qu’on le souhaite jusqu’à se convaincre qu’on peut « compter dessus » — c’est une question d’induction, pas de déduction.

Pour aller plus loin : 15a. Pour des exemples de phénomène statistique simulé où on voit comment le référentiel empirique est « tapé » ou « impacté » par la répétition de l’expérience aléatoire : choix d’un film au cinéma, score de QI.

Références

American Psychological Association (2010). Publication manual (6th ed.). Washington, DC.: Author.
Danziger, K. (1987). Statistical method and the historical development of research practice in American psychology. In L. Krüger, G. Gigerenzer & M. S. Morgan (Eds.), The probabilistic revolution, Vol. 2: Ideas in the sciences (pp. 35‑47). Cambridge: MIT Press.
Falmagne, J. C. (2003). Lectures in elementary probability theory and stochastic processes. Boston: McGraw Hill.
Geiselman, R. E., & Glenny, J. (1977). Effects of imagining speaker’s voices on the retention of words presented visually. Memory & Cognition, 5, 499-504.
Lamiell, J. T. (2013). Statisticism in personality psychologists’ use of trait constructs: What is it? How was it contracted? Is there a cure? New Ideas in Psychology, 31, 65-71.

Cite this article as: Stéphane Vautier, "45a. Pénétrer le statisticisme", in Épistémologie de la psychologie, 14/01/2017, http://epistemo.hypotheses.org/4294.

  1. J’utilise le singulier parce que si une étude repose sur plusieurs expériences aléatoires, il faut les envisager une par une. []
  2. L’objet n’est pas ici de commenter Geiselman et Glenny (1977) ; l’essentiel est dans les connaissances qui sont nécessaires à leur lecteur pour qu’il comprenne ce que les chercheurs ont appris de leur étude. []
  3. Je mets des guillemets parce que quoi de plus abstrait, artificiel, illogique, ou encore contradictoire, que l’idée d’une personne, qui, dans la durée, reste identique à elle-même, reconnaissable, tout en changeant ? []
  4. Ici aussi, il faut des guillemets parce que les données ne tombent pas du ciel mais résultent d’une activité humaine compliquée sans laquelle elles n’existeraient pas — la mise en œuvre de l’expérience aléatoire. []
  5. Il faut peut être distinguer plusieurs types de condition not-same Voice. []
  6. Ici, il y aurait beaucoup à discuter. Le problème n’est pas tant le goût pour les phénomènes statistiques ; le problème est l’impérialisme culturel qui va avec (voir, par exemple, Lamiell, 2013). []
  7. Un référentiel factuel contient des modalités descriptives qui sont valables à tout instant de la vie d’une entité, tandis qu’un référentiel artefactuel ne vaut que dans une situation particulière ; par exemple, le QI des gens n’est pas une propriété factuelle, mais artefactuelle : au cinéma, Paul n’a pas de QI du tout, ni de score de reconnaissance des mots d’une liste expérimentale. []

6b. What is a Scientific Explanation?

I argue that a scientific explanation of an empirical phenomenon consists in claiming that this phenomenon obeys at least one kind of objective necessity, which satisfies the methodological criterion of falsifiability. Consequently, I reject the relevance of the notion of probabilistic explanation — which does not imply that I would argue that probalistic modeling is useless with respect to any aim (specifically non-exaplanatory aims).

This note presents two kinds of objective necessity. Let p be an empirical phenomenon.

  • a => p. If a exists such that if a occurred, then p occurred, a is a sufficient condition of p. An observation of form « a and non-p » is a falsifier of the sufficient condition a.
  • p => b. If b exists such that if p occurred, then b occurred, b is a necessary condition of p. An observation of form « p and non-b » is a falsifier of the necessary condition b.

If one knows a sufficient condition a of p, one is able to get p to occur if one is able to get a to occur (predictive explanation). To explain why p occurs consists in stating that a sufficient condition is met.

If one knows a necessary condition b of p, one is able to prevent p from occurring if one is able to prevent b from occurring (preventive explanation). To explain why p does not occur consists in stating that at least one of its necessary conditions is not met.

5b. The probability to pass the item

Psychometric item response models rest on the notion of the probability to pass the item. The present note investigates this notion from the epistemological perspective by contrast to the usual psychometric (i.e., statistical) perspective. Taking an epistemological perspective consists in asking (i) how to do to obtain some knowledge based on the probability notion, and (ii) whether this knowledge is falsifiable. The note concludes that an item response model offers no falsifiable knowledge about the quantitative differences the model invokes. If item responses were (ordinal) measurements, probabilistic item response models would be unnecessary. And if one is aware of the fact that these models are not measurement models, their usefulness remains ambiguous and should be detailed (Vautier, Veldhuis, Lacot, & Matton, 2012; Ziegler & Vautier, 2014). I would appreciate criticisms of my analysis.

1. The truth value of the statement “the probability that Paul passes this item now is 0.8”

How to know whether this statement is true or false? Let us consider any value p from ]0, 1[; the statement “the probability that Paul passes this item at this moment is p” is unfalsifiable because it entails “Paul will fail or pass”, which is tautological. Consequently, the answer to the question is: I don’t know (cf. Vautier, 2012).

If one assigns the values 0 or 1 to p, then at least one of the two statements is false. If Paul passes the item while p = 0 the statement is false; the same if Paul fails the item while p = 1.

Not so simple? Let us try again and consider the statement “the probability that Paul passes this item at this moment is 10-20”. Minuscule probability, which indicates that the possibility of passing is exceedingly small and that, as would say a friend whose psychometric scholarship is vast, one may reasonnably believe that Paul will fail the item. In other words, it would be surprising that Paul passes the item. And one would not bet that Paul will pass the item if one trusts those who claim such a probability. But… there are a lot of players in the gambling market, despite the games of chance are made in such a way that the probability to win is minuscule. The point is not scientific but philosophical: given that one ignores what governs Paul’s success, everyone is free to make his own divination.

Let us move backward to the premise. How to do to know the probability’s value? By deduction, but starting from what premises? Help, mister Valéry! By the use of relevant empirical knowledge? Only frequencies can be known, which deserves further elabotation.

2. Heads or tails, and the Binomial distribution: its works… more or less

“The probability of tails when I will flip the coin is 0.1”. Reading this satement literally, one has no available empirical knowledge since the coin has not been tossed. In other terms, one can only state ignorance; the surplus meaning of the probabilistic statement pertains not on the coin’s future behaviour but on the intention of the person who makes the probability statement (for example, “I would not bet on tails, so follow my example”).

Taking an empiricist point of view, one can invoke a reference class, that is, a set of distinct draws of the same coin, of which one postulates that they obey the same probability. Suppose the coin has been tossed n = 40 times, and one knows k = the number of tails. Can the data be used to measure the probability or to test its value?

Let us begin with the statistical test. If the number k of tails is high, one will be tempted to question this probability’s value (p = 0,1) since tails is by hypothesis very unlikely. But one should be aware of the fact that this feeling of doubt or surprise is logically invalid. The probability p = 0,1 prevents nothing: one may observe 0 tails, 1 tails, 2 tails, etc. until 40 tails in the series of the 40 draws. The Binomial distribution B(40, 0.1) indicates that the probability of observing 40 tails for example is 10-40, a very small but non-null number.

Here is the pragmatics the of null hypothesis testing in statistics: one chooses a type 1 error value (i.e., the risk of rejecting the hypothesis while it is true) of 0.05 for example, which corresponds to at least seven tails. If the series exhibits at least seven tails, the hypothesis p = 0.1 will be rejected. Warning: in this approach, rejection is not falsification because the hypothesis p = 0.1 is unfalsifiable: its consequences on the series of 40 draws prevent no logically possible events.

To sum up: the statement “the probability of tails when I will toss the coin is 0.1” is not statistically testable because the reference class of this probability contains zero element before the draw, only one element once the coin has been tossed. The number 0.1 can be used as an index (and not a measurement) of the trust one can have in the divination “tails”, but, like promises in politics, the divination commits only its believers.

Using a reference class of size n, one can test statistically the probability’s value by counting the relevant events in the series of the n draws, once a rejection policy has been adopted—but it is logically invalid since any number of tails, from 0 to n, is compatible with the hypothesis. Thus, if the divination “tails” does not succeed often enough, one has a conventional means to reject the expertise of those who stated the probability—which has few practical incidence when one considers respondents’ item responses because the experts in probability are not here; psychometricians are smart enough not to risk themselves in practical settings (see Rasch’s remark): they let scoring models’ users… use the scores.

Now let us consider the former approach, where one wants to measure the probability p by using the Binomial distribution B(40, p) and k. One refers to the probability Cnkpk(1 – p)nk of k hits in a series of n draws (see on the Internet any article on the Binomial distribution for details on the formula) and one seeks the value p that maximizes this probability. But there is a problem: if one manages to observe a new k, say k’, one can find another value and hence another measurement of p by maximizing Cnk’pk’(1 – p)nk’. Furthermore, one has to bound p by accumulating observations, instead of determining the exact value from a unique k. The issue arises whether the interval can be more accurate than ]0, 1[.

Let us reason by the limiting case. If the approximation interval could be restricted, some intervals bounded to the left by 0 or to the right by 1 would be excluded; but the probability value p that maximizes Cnkpk(1 – p)nk when k = 0 is p = 0, and the probability value p that maximizes Cnkpk(1 – p)nk when k = 40 is p = 1. As any probability lying in ]0, 1[ is compatible with k = 0 or k = n, one cannot exclude that the estimation yields the values 0 or 1. As the possibility of a counter-example precludes these extreme values, it follows that the probability that is maximized varies in ]0, 1[, depending on k. Consequently, one ignores how to measure the probability p for strictly logical reasons (see also the third section 3 of the note #4). Could anyone object to this argument?

Anyway, to estimate a probability by using the maximum likelihood requires the availability of a big-sized reference class, which stumbles over a second serious objection.

3. Passing or failling the item, and the Binomial distribution: it does not work

Let us go back to the field of scientific psychology: what reference class is available when one speaks of the probability that Paul passes the item now? It is the empty set as long as Paul has not treated the item, a singleton once he treated it. If, by a thinking experiment, one asks Paul to treat the item 40 times in order to use the number of hits to obtain a totally inaccurate measurement of his probability to pass this item at any moment, one realizes that there is a theoretical, and redhibitory impossibility: how to assume that the probability that Paul passes the item at the first trial is the same as the probability that he passes the same item at the nth trial? Paul is learning—i.e., he changes with experience. This argument seems to me sufficient to conclude that in psychology of item responses, one cannot use the notion of a reference class as an empirical basis for a probabilistic theorization of the response of anyone to any item.

4. The psychometric attempt

The preceding discussion seems not very relevant to psychometricians because they are essentially interested in the problem of how to assign a number to a m-tuple of responses. The notion of probability plays no theoretical role but an instrumental or computational role. But this role takes a meta-theoretical function: one is scoring, hence comparability is warranted.

The main idea of an item response model consists in replacing the measurement function, which models the item response with probabilities 0 or 1, by a psychometric function, which models the probability of the item response, with the assumption that the probability varies in ]0, 1[. To give an idea of what is at stake, I will refer to the Rasch model and built on Mark H. Moulton’s Rasch Estimation Demonstration Spreadsheet. As my purpose is not technical but epistemological, I will not draw the reader into the maze of the computations but rather into the analysis of the scientific meaning of the results. The objective is to understand that probabilization enables one to find out numerical values by successive trials, in such a way that these values provide necessarily, or converge toward the best possible solution. The question is: how to define the best solution? The best solution is that which minimizes a so-called residual quantity. The beauty of the art of statistical estimation rests on the existence of mathematical properties like the uniqueness of the solution that minimizes the residual. This uniqueness is a criterion for the social acceptability of the best scoring since one wants to score.

The starting point of the estimation process is a list of m-tuples as illustrated in Moulton’s spreadsheet. It can be seen that the 10-tuples are not simply ordered, which means that no measurement function exists (cf. note #3). Nevertheless, one wants to assign a numerical value to the items’ thresholds and to the 10-tuples, while admitting random ordinal defects. The random factor is modeled by the formula

Pr(Xsi = 1) = exp(βsδi)/[1 + exp(βsδi)],

where Xsi is a random, binary variable, the reference class of which is the virtual population of trials by the person s treating the item i, βs is the value to be assigned to s, and δi is the threshold of i (in the set of the real numbers instead of in [0, max]). This formula is equivalent to the logit

ln[Pr(Xsi = 1)/(1 – Pr(Xsi = 1)] = βsδi,

that defines the difference between the value of s and the item’s threshold.

This theoretical framework is nearly the same as the measurement framework, but the domain of the quantity to be estimated/measured is the set of the real numbers. However, in contrast to the scientific approach that recognizes that the quantity may not be measurable by the m-tuples if the measurement function is false, the psychometric approach assumes measurability by supplementing the theoretical framework with a probabilistic framework, the added value of which is computational, while recognizing that this framework is false (see for example Embretson and Reise’s remark—this is not redhibitory, it suffices that probabilities are useful to the scoring purpose).

Let us follow Moulton. The 10-tuples allow one to compute the proportion of hits for each of the nine respondents, and the proportion of incorrect responses for each of the 10 items, and the proportions are transformed into logits (the items’ logits being centered). The key of the estimation process  is the definition of the residual quantity of each cell in the contingency table (9 × 10). The respondents’ logits and the items’ centered logits are used as estimates of the βss and δis, which allows to estimate the probabilities to pass each item for each respondent (the probabilities are the mathematical expectations of the Xsis). The residual is the difference between the observed value (1 for success) and the expected value that has been computed.

Remark. The expectation of each Xsi is defined only if the variable’s values are treated as additive numbers. But 0 and 1 are just practical codes for « incorrect » and « correct ». The variable Xsi is not a numerical random variable; at best, it is a qualitative random variable if one admits the existence of the probability distribution that is associated with the observed response. The variable Xsi has no mathematical expectation.

Then one has to adjust the estimated logits in such a way that the residuals will be minimized, and so on until the gains become negligible. Indeed, this is great art and the algorithmicians of statistical optimisation could be admired. Finally, one obtains 10 values that order the respondents on a scale with nine rungs that have numerical values.

But the beauty of the picture should not mask this initial fact: the probability that any respondent amongst the 10 respondents pass any item amongst the nine items is a value in ]0, 1[. Even if, for pragmatic reasons, one accepts the unfalsifiability of the statements that can be derived, this probability is not statistically testable because its reference class is a singleton if the item response is known, an empty set if it is ignored.

The psychometric estimation gives the best possible values to the observed 10-tuples, that is, the best hierarchy, along with the socio-technical innovation that consists in making the experts of the optimization algorithm the sole persons who know to define exactly what is the best. The Rasch model, and more generally the psychometric item response models, are sophisticated numerical aggregation methods of partially ordered data, not measurement methods. Their development is possible just because the invoked quantities are not measurable.

4b. Classical Test Theory Is a Triple Scientific Disaster

Classical Test Theory (CTT) underpins most of psychological test validation studies but does not explain how it is possible that a test score would measure a quantity. CTT interprets merely any score as the result of a random experiment. This random experiment is modelized by a probability law, which assigns a probabilty to any possible score when the person is tested. To supply the set of the possible scores with a probability law allows one to define a numerical random variable with a mathematical expectation. This expectation is qualified as the “true score”, in such a way that thanks to this mathematical construction, any score contains a true score and a measurement error, the arithmetical residual that bridges the gap between the two terms (the test score and the true score).

If test score users consider that observed scores contain true scores, and as soon as they adopt a realist interpretation of the truth concept (what is true is what occurs in the real world), they can deduce that these scores measure something. But CTT prevents them to claim that elementary interpretations of test scores are logically valid, because of the unrestricted nature of measurement error. For example, one cannot deduce from the observation, Paul’s anxiety score today is smaller than his anxiety score yesterday, that his true score decreased. Or, Paul’s IQ is higher than Julie’s IQ; this observation does not imply that Paul’s true IQ is higher than Julie’s true IQ. Consequently, the use of CTT as a statistical framework for test validation consists in adopting the epistemological norm according to which invalid interpretations of test scores are acceptable for an assessment practice that proclaims scientificity–psychological tests are deemed to have been scientifically validated.

In the perspective of psychological science, the use of CTT is a triple disaster. Firstly, by obscuring the logical distinction between a valid argument and an invalid argument, it prevents psychologists to have a clear sense of what a scientifically grounded conclusion is. Let us consider an argument of form “if premise then conclusion”: the conclusion is valid if it logically follows from its premise; if the premise is true, then the conclusion is valid and true. For a conclusion, based on a theory and some observations, be scientifically grounded, the observations have to be objective, that is, observer-independent (who can prove that the observations are true?), and the theory has to be corroborated, that is, successfully tested (i.e., tested and not falsified).

Secondly, the use of CTT allows psychologists to ignore that measuring a quantity by observing some phenomena requires–necessary condition–that these phenomena are simply ordered. The truth of the preceding proposition can be tested (see Scoring vs. Measurement), and it is unplausible (see the Definition).

Thirdly, by establishing that numerical aggregation of qualitative facts is the normal approach to objective description in scientific psychology, CTT encourages psychological researchers to underestimate the theoretical stakes of any descriptive methdology, especially by institutionalizing the ignorance of the conceptual difference between simple and partial ordering–which, to my knowledge, is detailed in only one textbook on psychological methodology: “An introduction to the logic of psychological measurement” by Joel Michell (1990).

The present note elaborates on the preceding points by (1) detailing the true score concept, (2) deducing that the comparison of two scores viewed as measurements of true scores confines to divination, (3) highlighting that probabilistic modelization has nothing to do with theorising about measurement functions, and (4) arguing that test scores should not mask the empirical m-tuples on which they are built.

1. The true score

The concept of the true score that is contained in a test score rests on two concepts, namely a scale of possible scores, and a probability function, which assigns a probability to any possible score. To my knowledge, CTT, as any psychometric theory, does not question the existence of probabilities nor does it raise the epistemological issue of how these probabilities could be known in a testable way.

Let us consider a scale of k scores xi, i = 1, 2, …, k. CTT postulates the probability law {pi, i = 1, 2, …, k} (which entails that the sum of the probabilities is equal to unity). It postulates also a random variable the expectation of which is, by definition, the sum of the scale scores weighted by their probabilities, that is,

true score = p1 x1 + p2 x2 + … + pk xk = Σ pi xi.

2. Scores comparison is divination

The insurmountable difficulties that result from this approach are cristal clear: how to deduce anything about the true scores of two scores? For example, Paul and Julie obtained IQ scores of 120 and 130, respectively. Can one deduce that Paul’s true IQ is lower than Julie’s true score? As one ignores the probability laws that are postulated to interpret the two IQs as the result of two random experiments, one ignores the two true scores. The score viewed by CTT is to the psychologist’s desire what Saint-Exupéry’s box is to Petit Prince’s desire: a box that contains an inaccessible sheep, a statistical shell for an inaccessible number.

3. The true score measures nothing

The idealization that yields the postulation of probability laws proliferating in a metaphysical space masks the fundamental problem of measurement. Measurement rests on a metaphysics of quantity, which does not coincide with that of probabilities. Indeed, a probability is a quantity that cannot be measured.

If a probability were measurable, one would be able to invoke an increasing function, which assigns one and only one point in a set of observable values to any point from [0, 1]. I do not see what such a set could be, and it is clear that the set of frequencies (or proportions) is not suitable, since any probability from ]0, 1[ is associated with at least two observable frequencies. It suffices to consider at least two random experiments based on the same probability law, such that the results are distincts. For example, I toss up a coin and I observe heads. I toss up the same coin and I observe tails. The proportion of heads that is associated with probability p is 1 if I consider the series “heads”, and 1/2 if I consider the series “heads, heads”.

CTT obscures the concept of measurement, since it diverts attention from the measurement process that should yield the score via a measurement function, by focusing it on the true score, the value of which remains inaccessible, even by upper and lower bounding.

4. The scores are qualitative compound events

As a statistical theory, CTT occults the empirical, experimental, or instrumental meaning of the scores. Test scores are treated as data in the proper sense of the term. The psychometrician does not wish to know about their origin, but accepts them without reservations, letting to the (non-statistician) psychologist the responsibility of their making. Such a division of labour enables a fools’ game, which aims at precluding the rejection of test scores as unsuitable scientific data.

The note Scoring vs. Measurement details the making process of test scores with the help of descriptive and scoring functions. The former, which is denoted by x, assigns a m-tuple of responses to any observation unit; the latter, a, assigns a score to any m-tuple. No score if no m-tuple. By its confinement to statistical analysis of scores, CTT enables the entire community of score users not to stumble over the evidence that, in order to measure a quantity by using some observables, these observables have to be simply ordered. But psychotechnical observables are m-tuples.

How to simply order m-tuples? For example, how to order (0, 0, 1) and (1, 1, 0)? Or, how to simply order points floatting in a cube? As soon as the descriptive approach is multivariate, its descriptive space is not a simple order because a simple order is a one-dimension space, within which objects are ordinally scaled, that is, posited one after the other with possible ties.

The classical test theoretician omits to notice that “0 and 0 and 1” does not equate with “0 plus 0 plus 1”. Such omission is faulty, since probability theory, which falls in the psychometrics’ area of responsibilty, rests on the concept of event algebra. The logical conjunction “and” is defined as a product instead of a sum, which corresponds to the logical disjunction “or”.  The 3-tuple (0, 0, 1), and more generally a m-tuple is not a sum but a logical product. This is a compound event, a co-occurrence.

Conclusion. It is time that psychologists, in their efforts towards scientificity, retake control of the theoretical elaboration of their data.

3b. Psychotechnical Scoring vs. Psychotechnical Measurement

The present note should be read with patience and tenacity because some technical notions, simple but abstract, are required for a minute conceptualisation. If the need arises, the reader should not hesitate to scroll the Internet to get information on specific notions (function, function composition, increasing function, Cartesian product…).

The conceptual difference between psychotechnical scoring and genuine psychotechnical measurement can be revealed by describing both of them as the composition of two functions, namely, ax (scoring) vs. fq (ordinal measurement). It suffices to show that the two function compositions are not identical to show that scoring and measurement are not interchangeable concepts.

To determine the score of a given person at a given moment (by using a psychological test), one has to determine then to score a psychotechnical description of this person at this moment. To measure supposes that the psychotechnical description results from a measurement process. If the supposition is false and if one endorses the scientific attitude, one cannot consider that the description results from a measurement process (hence, it is tempting to score the description as a non-scientific way of compressing the psychotechnical description into a numerical description).

1. The psychotechnical description x

The psychotechnical description is a function denoted by x: Ω → D. Its domain, Ω, is a statistical population. An observation unit from Ω consists usually of a person at a given moment, where « moment » means the temporal interval required by the test administration. Any observation unit is denoted by ω. The codomain of the description function x is denoted by D (for description). The elements from D ares m-tuples, where m indicates the number of the test items. A 3-tuple is for example (0, 1, 1), where 0 and 1 denote incorrect and correct responses, respectively. The bold type of the notations x and D indicates that the description is multivariate (as opposed to onevariate)–this is a m-tuple; a score is a 1-tuple.

For example, let us consider the observation unit ω = (Paul, t). The psychotechnical description of Paul when he is tested at moment t is given by his m responses to the test. This m-tuple is denoted by x(ω). Depending on the context, the notation x may denote any description, that is, x(ω) where ω is any observation unit, or the psychotechnical description viewed as a function, that is, x: Ω → D.

The function x is the conjoint function

x1x2xm: Ω → D1×D2×…×Dm = D

(which shows that D is a Cartesian product), where any xi, i = 1, 2, …, m, assigns one and only one image to any observation unit in the scale of each item i–the scale of the item is the set of its descriptive values.

2. The scoring rule a

A psychotechnical scoring rule is a function denoted by a: DS. Its domain is the codomain of x, that is, the set of the m-tuples that are possible given the test design. The codomain of a is denoted by S; S is a set of scores–the test scale. The score of any m-tuple from D is determined by “transformation rules”, or aggregation rules (hence the notation a). For example, the score of (0, 1, 1) is 2 points if one uses the common rule, which consists in counting the correct responses in the m-tuple.

3. The psychotechnical scoring ax

To test a person at a given moment by using a given test consists in determining her responses to the m test items–x(ω)–then to assign a score to this description– a[x(ω)]. This function composition is denoted by ax. The score of the observation unit ω is

(ax)(ω) = a[x(ω)] = x(ω).

The notation x denotes a psychotechnical score but it can also denote the function ax.

4. The quantitative hypothesis q

Measurement rests on the fundamental hypothesis that a quantity does exist (one mesures a quantity or one does not measure; saying that one measures a process is an example of uncareless speaking). This hypothesis can be stated as the hypothesis of a function

q: Ω → [0, max],

which assigns one and only one amount from the segment [0, max], where « max » denotes the highest possible amount of the quantity, to any unit observation from Ω.

The statistical notion of an observation unit deserves a remark in the present context. To suppose that the person possesses an instantaneous amount of the quantity is one thing, to suppose that this amount does not vary within the temporal interval the test administration requires is another thing. The invariance convention according to which the amount to be measured during the test administration does not vary is mandatory to test the hypothesis that this amount is measurable.

5. The measurement function f

The measurement funtion

f: [0, max] → D

is defined on the domain [0, max] and takes its values in the codomain D. Let us consider a m-item test. For any item i of the test, one supposes that there exists an increasing function fi: [0, max] → Di, where Di is the descriptive scale of the item i (see Measuring with Item Responses). The function f is the conjoint function

f1f2fm: [0, max] → D1×D2×…×Dm = D

(which shows, again, that D is a Cartesian product). The measurement value of the amount q by f is a m-tuple of values to be denoted by y.

A critical property of f‘s values is that they are simply ordered because any function fi is increasing. This point is detailed in Section 6. This property of simple order, or, in other words, of comparability, implies that the proposition, according to which two specific descriptions x(ω1) and x(ω2) result from the measurement of the amounts q(ω1) and q(ω2), is falsifiable. If x(ω1) and x(ω2) are incomparable, and one writes

x(ω1) <> x(ω2),

then at least one of these descriptions is not a measurement.

6. Psychotechnical measurement fq

A test enables one to measure the amount of a psychological quantity within an observation unit ω if this amount, q(ω), determines the psychotechnical description f[q(ω)]. The psychotechnical measurement is the function composition fq.

Let ω1 and ω2 be two observation units. Their measurements are respectively (fq)(ω1) = f[q(ω1)] = y(ω1), and (fq)(ω2) = f[q(ω2)] = y(ω2), where the letter y denotes the resulting description or the function fq.

Direct product order (definition). One says that f[q(ω1)] ≥ f[q(ω2)] if and only if for all is, fi[q(ω1)] ≥ fi[q(ω2)]. This definition applies also to x.

Theorem. f is increasing.

Proof.

Let ω1 and ω2 be such that q(ω1)  ≠ q(ω2):

  • If q(ω1) > q(ω2), then for all i = 1, 2, …, mfi[q(ω1)] ≥ fi[q(ω2)], hence f[q(ω1)] ≥ f[q(ω2)].
  • If q(ω1) < q(ω2), invert the subscripts of ω and go back to the preceding case.

Comparability (simple order). The increasing property of f implies that two measurements y1 and y2 are comparable.

7. Scoring vs. measurement

It can be seen formally that scoring and measurement are distinct operations, since x = ax is not the same symbol as y = fq. Put in other words, the availability of the score x does not necessarily means that there is an increasing function f such that x = fq (f is onevariate as x is onevariate).

Let Ωn be a set of n observation units. If there is one function f such that the image of Ωn by x is the image of Ωn by fq, that is, xn) = (fq)(Ωn), then one can suppose that the psychotechnical descriptions xn) are ordinal measurements of the amounts qn). In other terms, these descriptions are simply ordered, that is, the descriptions of two distinct observation units are either equal or strictly ordered. Then the score x expresses this ordinal property of comparability.

If such a function f does not exist, one ignores how to measure the amounts qn). It is possible that our ignorance results from the fact that these amounts do not exist–in which case we suppose them wrongly–, or from our inability to determine the experimental conditions that would allow us to suppose not against the empirical evidence that f exists.

If two observations units ω1 et ω2 exist in such a way that their descriptions are incomparable, i.e., i and j exist such that xi(ω1) > xi(ω2) and xj(ω1) < xj(ω2), then f does not exist and hence one cannot assert that the descriptions result from the measurement by f of the supposed amounts.

2b. Measuring with Item Responses: Measurement Functions

To pose the measurement problem in the context of psychological measurement with item responses, it is helpful to detail the notion of a format response. The formats of responses to test items are of two types: the responses vary in a discrete or in a continuous set.

For example, if one describes the response to the item

23 + 9 = ?

as correct or incorrect, the format is discrete. If a patient is instructed to indicate her pain intensity on the visual analogue scale below,

eva_reglette

the format is continuous: the response is described as the distance between the origin, 0 cm, and the ruler’s cursor (source: SFETD).

In both cases, the problem is: how does the response depend on the quantity to be measured? Let us call mental calculus ability the quantity that inspired the mental calculus item. A correct response to the item indicates a higher ability than the quantity which is indicated by an incorrect response. Coming back to the visual analogue scale, the more the cursor moves towards the left (when one looks at the “face de mesure”), the more pain increases.

1. The continuous case

Perhaps the continuous format is more intuitive to understand the concept of a measurement function. What rationale underpins the idea according to which the more the cursor moves towards the scale’s 10, the more pain increases? The rationale is the hypothesis that the cursor’s position the patient chose depends on her amount of pain. This hypothesis can be stated as follows: any point from [0, max], that is, any amount of pain, corresponds to one and only one point in [0 cm, 10 cm], that is, the segment within which the cursor varies.

By a small visual effort, one can consider a curve, which represents the measurement function. The function is defined in the segment [0, max] (its domain), where 0 corresponds to « no pain », and max corresponds to the worst imaginable pain; the function takes its values in the segment [0 cm, 10 cm] (its codomain). This function is increasing, that is, if the amount of pain increases, the cursor’s distance increases as well. Put in mathematical symbols,

x1 > x2 => f(x1) > f(x2),

where x1, x2 denote amounts of pain, and f denotes the function that determines the corresponding distances.

One ignores this function and one has a wide choice. Linear, concave or convex curves are possible. One ignores whether a given amount of increase results in the same empirical effects (i.e., observable on the cursor’s position) depending on the initial amount of pain. For example, one ignores whether the amount of pain variation that corresponds to the cursor’s variation when it moves from 1 to 2 cm is the same as the amount of pain variation that corresponds to the cursor’s variation when it moves from 8 to 9 cm. One ignores whether the patient who indicates 4 cm suffers twice as much as when she indicates 2 cm.

The centimeter is a length unit; one admits that the measured length measures a pain amount, and that it is an ordinal measurement (that is, one speaks the language of « more or less »). As long as the function that one hypothetizes has not been discovered for a given patient, one cannot measure her pain metrically (i.e., by using a pain unit).

Let us recapitulate: « a pain scale measures pain » means that one admits that the length between the origin and the cursor depends increasingly on the amount of pain. This dependency (or causal relationship) is thought of as a function of unknown nature. This function is a measurement function (or a linking function). Our scientific ignorance implies that the measurement of pain is not metrical but ordinal. The metric nature of the response format does not imply that the psychological quantity is measuread metrically.

2. The discrete case

Coming back to the mental calculus item, what function links the mental ability (domain) to the codomain {incorrect, correct}? This time, the codomain is a discrete set. The rationale is that a correct response indicates a higher level of abiilty. The sole way of specifying the measurement function of this item is, again, to invoke an increasing function. We want

x1 > x2 => f(x1) ≥ f(x2),

where x1, x2 denote amounts of ability and f denotes the function that determines the corresponding responses. The operator « ≥ » is mandatory because the number of possible responses is smaller than the number of possible amounts of ability. The function is increasing, but not strictly increasing.

The representative curve of the function is a staircase comprising two steps, which are separated by a threshold of unknown value. The graphic below illustrates this step function.

Paliers

Consequently, the item responses do not denote amounts of ability, but intervals of amounts of ability–the responses 0 and 1 denote the intervals [0, A[ and [A, max], respectively. This is why the 0/1 numerical encoding of the responses may be misleading: these numbers are not amounts but ordinal numbers. Addition, subtraction, multiplication and division are not defined on these numbers. It would be scientifically suitable to encode the responses with letters to keep in mind that they have an ordinal meaning with respect to the quantity to be measured.

In current practice, one counts the number of correct responses. However, the number of correct responses does not measure the ability, it counts only the number of conventional units of value. In this perspective, scores are more akin to prices than to measurements. The issue of how the scores (countings) could measure the ability remains untouched.

1b. A Definition of Measurement

Definition. If observations (including responses to test items) measure a given amount of a given quantity, then their variation depends on the variation of the quantity amount, in such a way that it is possible to deduce from them that the quantity amount has increased or decreased.

Remark 1. If one admits that the variability of observations depends on other factors and if one ignores the extent to which these factors determine the observed variation, one can deduce from the observations neither an increase nor a decrease of the quantity amount, because one cannot exclude that this variation results from undesired perturbations when the quantity amount did not change. This is the fatal flaw of the psychometric approach to psychological measurement.

Remark 2. The development of measurement instruments consists, among other things, in limiting within known boundaries the effect of the disturbing factors on the indicators that are used to deduce the variations of the quantity amount to be measured.

Remark 3. The development of measurement instruments requires the experimental manipulability of the quantity amount. For example, to study the disturbing variations of a response device, one has to be sure that the quantity amount does not vary (or varies negligibly) while one observes the behavior of the response device. To study the sensitivity of the response device to the variations of the quantity amount, one has to be able to increase or decrease more and more finely the quantity amount to be measured.

Remark 4. There is nothing like this in the realm of psychometric scoring techniques: one does not know how to manipulate the amount of intelligence nor the amount of anxiety for example, like one manipulates the temperature amount of water in the cooking pot by increasing the flow of gas. But this statement deserves contradiction… Who would like to refute this argument?

Remark 5. In psychology, tests or questionnaires are called measurement instruments. This linguistic custom is misleading because if one measured some quantity amounts with tests or questionnaires, the reason would be that the respondent were the measurement instrument, as she, not the test, produces the responses. Consequently, psychologists who are scientifically responsible for their practices should stop naming tests and questionnaires “measurement instruments”. If they do not stop, they contribute to propaganda that hawks the false idea that tests enable one to measure psychological quantities. Such propaganda demonstrates to informed audience that either they have no clear idea of what measurement is, or that they do not reject its deceptive character despite the evidence.

Remark 6. The definition above does not suffice to define metric measurement usually performed by measurement instruments. The definition is suitable for qualitative indicators, which enable ordinal measurement of the quantity amount to be measured. For example, if qualitative variations fulfill the definition, in such a way that one can deduce that the quantity amount increased, one ignores the amount of the increase: one knows only that it is strictly positive. Ordinal measurement allows one to speak the language of “more or less”, not the language of  “how much”. As metric measurement supposes ordinal measurement, metric measurement is fasified if ordinal measurement is falsified.

Remark 7. As far as one describes in a given language how quantities like intelligence or anxiety manifest themselves, this language is not of the “more or less” sort. Most of observed variations do not match the “more than” nor the “less than” relationships. The scientific error of psychologists who believe in quantities like anxiety or intelligence lies in their refusal to admit that the phenomena (observables) they are studying are primarily qualitative. Instead of respecting their qualitative form, they disguise them as numbers (or scores), “counting their chickens before they are hatched”: they behave as if one could speak scientifically of anyone’s intelligence or anxiety in terms of more or less, with no problem of authenticity. But does the evidence, as exhibited in our descriptives frames, teach us that the language of “more or less” is suitable to genuinely speak of anyone’s intelligence or anxiety manifestation?

Conclusion. Either intelligence (or anxiety, or any psychological quantity) can be measured with test responses, in which case an ordering should emerge from the observations, or one cannot measure intelligence (or anxiety, or any psychological quantity) with test responses.

0b. Preparing a Précis of Methodology for Psychologists who Think Freely

I would like to contribute to a public controversy about the methodology of scientific research in psychology. I hope to draw attention of people who feel concerned on the way psychological research is conducted, in order to enable them to comment freely on ideas to be detailed and organized in a book on the methodology of psychological research.

The public debate that is possible on the web is helpful in elaborating such a book. Instead of thinking to oneself then to publish, I would prefer to formulate criticisms and propositions already submitted to critical appraisal of people who, at some point, feel concerned. I wish to encourage opinions (disputable by definition) and questions (even not well formulated).

My main criticisms can be summed up under the following four headings: (1) Responses to test items are not measurements if they are not simply ordered. (2) To save the belief that test scores are measurements, the test validation doctrine requires the epistemological norm according to which logical validity of reasoning is not mandatory in psychological science, which entails that the resulting discourse is not scientific. (3) More generally, mainstream methodology aims at saving the belief that psychological theories are explanatory and testable. (4) But such a belief grows to the detriment of the simple idea that a scientific explanation of empirical facts consists in arguing for testable, sufficient or necessary conditions to these facts.

I propose that psychological scientists should be accustomed to the ideas that (1) scientific investigation in their fields of specialization is useful to explore our objective ignorance of the laws that would govern what occurs in the currently available descriptive windows, rather than to built a fallacious sense of understanding or mastering of how people (and animals) behave– such a perspective does not preclude the discovery of general facts if they exist; it precludes the institutionalization of a form of modern obscurantism in academic psychology–; (2) psychological assessment based on psychological tests should be reframed in the the context of a philosophical, socio-cultural, economic, and politic analysis of decision making under uncertainty, as what is at stake is assigning values to people in ad hoc, non-natural evaluative spaces.