# "Logits"?

"Logit" is a contraction of "Log-Odds Unit" (pronounced "low-jit"). It is no more obscure a measurement unit of an underlying and invisible variable than an "Ampère" is of invisible electric current. The essential ingredient of Amps and logits is that they be additive.

Real apples are not additive. One Apple + One Apple = Two Apples. But Two Apples are twice as much as One Apple only when the Two Apples are perfectly identical. Real apples are not perfectly identical. When we say One Amp + One Amp = Two Amps, we say "all Amps are identical," wherever they appear on the Ammeter. Logits form an equal interval linear scale, just like Amps. When any pair of logit measurements have been made with respect to the same origin on the same scale, the difference between them is obtained merely by subtraction and is also in Logits. This is how Amps work.

Like an Ammeter, the logit scale is unaffected by variations in the distribution of measures that have been previously made, or by which items (resistances) may have been used to construct and calibrate the scale. The logit scale can be made entirely independent of the particular group of items that happen to be included in a test this time, or the particular samplings of persons that happen to have been used to calibrate these items.

We construct a logit scale in the same way that we construct an Amp scale. We deduce a theory that produces equal interval, linear measures and derive a method for applying that theory. In the case of qualitative ordered observations (right/wrong, present/absent, none/some/all), the necessary and sufficient theory is the Rasch model, and the method of application is numerous administrations of similar agents (test items) to relevant objects (persons).

The theory is This is a "linear" model because all elements can be represented as fixed positions along one straight line. In games of chance, the (Probability of Success)/(Probability of Failure) is called the "odds of success". "Loge[(Probability of Success)/(Probability of Failure)]" is called log-odds. The units of measurement constructed by this theory are called "log-odds units" or "logits".

Not all numbers represent equal interval scales, no matter how equally spaced their values appear. Rank orders are counted with equally spacing, but rank order numbers do not specify whether the distance between 1 and 2 is equal, greater or less than the distance between 2 and 3.

How do we know that logits are equal interval? By observing that when data fit the theory, the specification that a one logit positive difference between any person and any item anywhere on the scale always has the same stochastic consequence.

When data fit, the interval specification of the theory is realized in the data. For these data, the interval scale is established. The implication is that for similar data the scale will continue. This implication, however, is always tested when fit is analyzed for each new application, just as wise use of an Ammeter requires that it too be continually checked. When in the new application, the fit criteria are met, then the linear scale continues - the logit unit is maintained.

Benjamin D. Wright

Logits? Wright BD. … 1993, 7:2 p.288

Logits? Wright BD. … Rasch Measurement Transactions, 1993, 1993, 7:2 p.288

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
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