Your ability is not merely the number of your successes. If it was, the longer the test, the more successes you would have, and so the greater your ability!
If your ability was (number of successes - number of failures) and you were succeeding more than half the time, then the longer the test, the greater your ability!
So we define your ability as the "number of your successes / the number of your failures".
We are comparing counts of successes and failures. These are the manifestations of probabilities of success and failure. So
Your ability = Prob(your success on a standard item) / Prob(your failure on a standard item)
Similarly
Item's difficulty = Prob (standard person fails on item)/ Prob(standard person succeeds on item)So when we want to know how likely you (a particular member of the sample) are to succeed on a particular item (belonging to the test), we combine these these two probability ratios multiplicatively. This is how Georg Rasch originally did it. For convenience we say that a standard person has a .5 probability of succeeding on a standard item.
(Probability of your success an a particular item / Probability of failure) = (your ability) / (difficulty of particular item)
But we quickly discover that multiplication is much more difficult to organize and explain than addition. And the method of replacing multiplication and division by addition and subtraction is to take the logarithms. Then
log_{e}((Probability of success / Probability of failure)) = log_{e}(ability) - log_{e}(difficulty) = Ability - Difficulty
This is the Rasch model!
Test performance can be viewed in two ways:
1) Person ability is indicated by the count of items on which the person
succeeds, the person's raw score, R (for "right direction").
2) Test difficulty is indicated by the number of items which the person
fails to pass, W (for "wrong direction").
As a first approximation, R follows a ratio scale in which "0" corresponds to an "absolute zero" of person ability. "1" is one ratio unit of ability away from this absolute zero. W also follows a ratio scale, in which "0" corresponds to an "absolute zero" of test difficulty. Ratio scales convert to a linear form by logarithms. Thus, log_{e}(R) linearizes the ability count of the person and log_{e}(W) linearizes the difficulty count of the test. This relationship between ratio and linear scales can be put to work by representing a person's measure on a linear scale of ability as B = log_{e}(R) and a test's difficulty on the same linear scale as D = log_{e}(W). [This parallels Georg Rasch's use of the Poisson distribution, "Probabilistic Models", p. xvii]
Person ability, B, is only observed when the person encounters a test with difficulty, D. The useful relationship for a measurement model is the difference (B-D) between person ability, B, and the test difficulty, D. This parametric difference is realized empirically by the difference between their corresponding linearized manifestations, log_{e}(R) and log_{e}(W). Thus,
B - D = log_{e}(R) - log_{e}(W) = log_{e}(R/W)
This simple formulation opens the way to a general measurement model. First, the situation-specific statements about actual values of particular data, R and W, are reformulated into a more general statement about probabilities, the probabilities of whatever success may be earned by whatever persons on whatever items the test may contain. To do this, person score R is replaced by the probability of success, Pni, by particular person n on particular item i. Test score W is replaced by the probability, 1-Pni, of failure by that person n on that item i:
log_{e}[Pni/(1-Pni)] = Bn - Di
The outcome of these simple steps is the necessary and sufficient measurement model for converting raw scores based on dichotomous, "pass/fail", observations into measures, as Georg Rasch discovered.
Rasch Model derived from Ratio-Scale Counts, B Wright … Rasch Measurement Transactions, 1992, 6:2 p. 219
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