# 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, 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. 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.

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