Measurement in Psychology: Historical Facts and Statistical Invariance

Measurement in Psychology: Historical Facts

The Problem of Measuring in Psychology

The problem of measurement in Psychology is strictly related to the nature of the phenomena that we measure. While in other fields the measurement is conducted on extensive quantities like length, duration or weight, in Psychology we deal with psychological constructs that are never directly measurable since they are typically theoretical objects whose definition may vary from one theory to another (e.g. intelligence). This situation introduces the need for two main sets of cautions:

1. the clear identification of the measurement properties of the investigated variables;
2. the reliability and validity of the measure in representing the quantity to be evaluated.

When we measure, our main objective is to reproduce in a numerical form the specific relations that exist in the empirical world. Therefore, the important point is not only the numerical value which is assigned to a certain empirical quantity, but also the capability of such a numerical value to be coherent with the empirical relations among the objects to be measured. Thus, if two objects have, for instance, two different amounts of a given empirical quantity (e.g., length, or intelligence), the numerical representation must preserve such relation between them: the shorter must be assigned the lower numerical value.

Historical Facts: The Pioneers

Measurement is a fundamental aspect of science. Therefore, the interest for measurement in Psychology rose together with the interest of scientifically studying the link between stimulus and perception. The empirical research of Weber and the development of Psychophysics (Fechner,1860) represented the stepping stone for the study and formalization on the one hand of Psychology as a science and, on the other hand, of the problem of measuring in this discipline.

Two fundamental laws about the relation between the real modification in a stimulus and its perceived magnitude were formalized by Fechner (1860) and are known as the Weber-Fechner Laws. These laws represent the first real attempt to derive a mathematical function to predict a mental process, namely the perception of changes in the magnitude of a physical stimulus. The important aspect of these fundamental laws is that they paved the way to the possibility of representing and studying mental phenomena as mathematical functions.

Historical Facts: From Spearman to the Crisis

The works of Weber and Fechner showed the possibility of having a measurement in psychology. This idea was then applied to measure other psychological phenomena like intelligence (Binet, 1905), and to develop methods to represent psychological constructs. Among them, a particular mention is deserved by the Factor Analysis by Spearman (1904). This method was based on the study of the unexpected covariance among different scores which led him to postulate the existence of a common factor (called g factor) able to explain such relation. This procedure was then refined by other researchers like Cattell and Thurstone. This last, in 1927 developed a fundamental method for the pairwise comparison of stimuli which led to the disentanglement of psychological constructs measurement and physical measure. The disentanglement paved the way to the further development of Signal Detection Theory.

Despite the wide application of psychological measurement, a fundamental shortcoming was highlighted in the Fechnerian approach: the assumption, without experimental support, of the quantitative nature of psychological variables. This problem was at the basis of the results of the so- called Ferguson Committee.

Historical Facts: From the Ferguson Committee to Stevens

In 1932 the British Association for the Advancement of Science established the Ferguson Committee in order to evaluate the quantitative nature of the sensorial phenomena (such as perception). Committee's conclusions, included in two reports (dated 1938 and 1940), established that sensorial variables are not measurable given the lack of a measurement unit (like meters, grams, miles, et.c), and additivity, which is the possibility of summing two stimuli and obtaining a stimulus which is the result of such operation (as it happens for length: if we put two object one after the other and measure the resulting length, we will obtain as result the sum of the two lengths).

Therefore psychology was excluded from the group of disciplines for which a "true" measurement was conceived as possible. Nonetheless, the conclusions of the Ferguson Committee was the trigger to activate a strong revolution in the conceptualization and application of measurement in Psychology. This revolution started with the change in measurement paradigm introduced by Stevens.

Historical Facts: The Contribution of Stevens

In 1946 Stevens introduced for the first time an operational conceptualization of measurement in psychology based on two fundamental points:

1. the introduction of the direct methods of measurement;
2. the definition of four fundamental measurement scales.

The former, differently from the classical approach adopted in psychophysics, allowed individuals to directly assign a numerical value to the perceived sensation. This possibility was granted by the definition of measurement according to Stevens, that is: "Measurement, in its broadest sense, is defined as the assignment of numerals to objects or events according to rules". This extremely operational definition led to a strong differentiation betweenpsychology and quantitative sciences. In fact, according to it, the test of the actual quantitative nature of the variables to be measured was no longer important for psychologists. Nonetheless, this definition was able to establish the basis for a further axiomatization of measurement starting from the definition of the four different levels of measurement which correspond to four different kinds of measurement scales: nominal, ordinal, interval and ratio.

Modern Measurement in Psychology

The operational definition of the four levels of measurement resulting in the four main kinds of measurement scales represented one the most important contributions of Stevens to measurement theory. These tools represent still today the most important reference point for the classification of variables with respect to their measurement properties and to the different statistical indexes that can be measured on such variables.

The idea of Stevens represented also the basis for a number of following developments in the measurement theory within and outside psychology. Some remarkable approaches that rose from the more mathematical and formal realboration of Stevens' contribution are:

• the Axiomatic Measurement Theory by the giants like Patrick Suppes, Joseph Zinnes, Robert Duncan Luce, Louis Narens and Johannes Pfazangl;
• the Item Response Theory and the Rasch Model by Georg Rasch, Frederic M. Lord and Paul Lazarsfeld (among the others);
• Conjoint Measurement by Debrau, Luce and Tuckey;
• and the Prospect Theory by Daniel Kahneman (who won a Nobel Prize in Economics for this theory) and Amos Tversky.

Non-Numerical Measurement in Psychology

All the theories introduced so far are based on the fundamental assumption that measurement consists in relating empirical objects to numbers according to some specific rules able to preserve the relations among these last with respect to the amount of the measured characteristic (e.g., weight, intelligence or depression).

In recent years some theories disputed this approach by proposing that some psychological variables such as knowledge, could be better evaluated according to a non-numerical approach. In these approaches the level of knowledge is not measured by a numerical score, but by the collection of problems that an individual masters, or the collection of skills that an individual proves to have. These theories are the Knowledge Space Theory by Jean-Claude Falmagne and Jean-Paul Doignon (1985) and the Cognitive Diagnostic Models by Kikuma K. Tatsuoka and Maurice M. Tatsuoka (1985).

Even more recently, this kind of non-numerical approach has also been applied to the assessment of psychological disorders by the Formal Psychological Assessment by Andrea Spoto, Giulio Vidotto and Luca Stefanutti.

These new theories represent an important step towards a more comprehensive approach to the 160 years-old problem of measurement in psychology.

Different Scales to Measure Different Variables

Preliminaries: The Concept of a Variable

A variable in the scientific framework is a quantity or quality that can assume different values. The measurement of variables and their modification under different conditions is exactly the procedure used in experimental research to test the existence and consistency of relationships between/ among variables. Such relations should be related to some theory able to explain and interpret the meaning of the measured evidence.

There are many different "objects" that can be defined as variables. Some of them represent a quantity, that is something whose variation refers to a change in the amount of that specific characteristic. Examples of such variables are physical quantities like length and weight, but also psychological variables conceived in the classical approach such as sensation about stimuli, intelligence (e.g., measured by the I.Q.), or even anxiety (when measured, for instance, through a self-report questionnaire). Those that are subject to measurement as typically intended.

Another fundamental class of variables contains those characteristics whose modification does not result in the change of a quantity, but in that of a quality. Examples of such variables are gender and nationality, but also psychological diagnosis or the group of an experimental procedure which a participant belongs to.

Relational Systems

The presence of different levels of variables in different objects define among them a relationship with respect to that characteristic. For instance, if we measure the height of a group of individuals, we can obtain the relation "to be taller than" defined among the members of this group. If we test this relation between all the possible pairs of individuals in the group we would end up with an order of the individuals from the shortest one(s) to the tallest one(s).