Edward Lee Thorndike (1904) criticizes a pioneering 1895 spelling test of Dr. Joseph M. Rice, because its words are of unequal difficulty (p. 8, see below). "In arranging a scale of measurement one must as far as possible, (1) keep free of individual opinion, ... (2) call equal only those things which can be interchanged without making any difference to the issue involved." (p. 14-15, italics Thorndike's). This suggests that, at the time, Thorndike considered that the measurement of ability could only be performed by ascertaining the level of success on a set of equally difficult items. Though Thorndike's later writings relax this requirement, let us derive a measurement model from his 1904 criteria.
Imagine that Dr. Rice compiles a set {i} of spelling-word items all of equal difficulty d_{i}, and Prof. Thorndike another set {j} all of equal difficulty d_{j}. Both sets are administered to examinee n of ability b_{n} and to examinee m of ability b_{m}.
We observe that examinee n has F_{ni} successes and F_{in} failures on item set {i}. So the empirical odds of success for examinee n on item set {i} are F_{ni}/F_{in}. From this, we infer that examinees of ability b_{n}, such as examinee n, have P_{ni}/P_{in} odds of success on items of difficulty d_{i}, such as item set {i}, where
and f is the relationship between b_{n} and d_{i} yielding these odds.
Similarly, when examinee m encounters item set {i}:
The ability difference between m and n is indicated by their relative odds of success on item set {i}:
and g is the relationship between b_{n}, b_{m} and d_{i} yielding these odds.
The same holds true when examinees n and m encounter item set {j}:
and their relative odds of success are
Thorndike's first criterion is that measures must be free of individual opinion, i.e., of who chooses the spelling-word test items. Accordingly, for invariance in measuring ability differences, the ability difference between b_{n} and b_{m} must be independent of the item set. Therefore it must be the same for items sets {i} and {j}. Consequently,
Rearranging,
We can choose item set {j} to have any difficulty, in particular that of item "0" at the local origin of the scale, i.e., of difficulty d_{j}=d_{0}. Also the ability of examinee n must be independent of the particularly ability of examine m, so we can also choose m to be examinee "0" whose ability is at the local origin of the scale, b_{j}=b_{0}. The local origins of the ability and difficulty scales can be constructed to coincide, so that the probability of success of examinee "0" on item "0" becomes 0.5.
so that
P_{n0}/P_{0n} is the odds of success of an examinee of ability of b_{n} on a standard item at the local origin of the scale. Let this be the definition of b_{n}. Similarly, (P_{i0}/P_{0i}) is the odds of failure on an item of difficulty d_{i} by a standard examinee at the local origin of the scale. Let this be the definition of d_{i}. Then
f(b_{n},d_{i}) = (P_{ni}/P_{in}) = (P_{n0}/P_{0n}) / (P_{i0}/P_{0i}) = b_{n} / d_{i}
yielding the ratio form of the Rasch model:
P_{ni}/P_{in} = b_{n} / d_{i}
and
g(b_{n},b_{m}) = g(b_{n},b_{m},d_{i}) = (P_{ni}/P_{in})/(P_{mi}/P_{im})
= (b_{n}/d_{i})/(b_{m}/d_{i}) = b_{n}/b_{m}
In a ratio metric, the odds of success of examinee n on item i is the ability of examinee n divided by the difficulty of item i. Expressing f(b_{n},d_{i}) in a more convenient additive metric, and reparameterizing, we obtain the familiar form of the dichotomous Rasch model:
where B_{n} and D_{i} are ability and difficulty expressed in an additive metric. Thus, E. L. Thorndike's 1904 criteria lead to the Rasch model.
And the difference in ability between m and n is B_{m} - B_{n} ≈ log(F_{mi}/F_{im}) - log(F_{ni}/F_{in}) ≈ log(F_{mj}/F_{jm}) - log(F_{nj}/F_{jn})
John Michael Linacre
Thorndike, E.L. (1904). An introduction to the theory of mental and social measurements. New York: Teacher's College.
Excerpts from Thorndike, E. (1904). Introduction to the theory of mental measurement. New York: Science Press (pp. 5-6):
The Special Problems of Mental Measurements
In the mental sciences, as in the physical, we have to measure things, differences, changes and relations. The psychologist thus measures the acuity of vision, the changes in it due to age, and the relation between acuity of vision and ability to learn to spell. The economist thus measures the wealth of a community, the changes due to certain inventions and perhaps the dependence or the wealth of communities upon their tariff laws or labor laws or poor laws. Such measurements, which involve human capacities and acts, are subject to certain special difficulties, due chiefly to the absence or imperfection of units in which to measure, the lack of constancy in the facts measured, and the extreme complexity of the measurements to be made.
lf, for instance, one attempts to measure even so simple a fact as the spelling ability of ten-year-old boys, one is hampered at the start by the fact that there exist no units in which to measure. One may, of course, arbitrarily make up a list of 10 or 50 or 100 words and measure ability by the number spelled correctly. But if one examines such a list, for instance the one used by Dr. J. M. Rice in his measurements of the spelling ability of 18,000 children, one is or should be at once struck by the inequality of the units. Is 'to spell certainly correctly' equal to 'to spell because correctly'? In point of fact, I find that of a group of about 120 children, 30 missed the former and only one the latter. All of Dr. Rice's results which are based on the equality of any one of his 50 words with any other of the 50 are necessarily inaccurate, ....
The problem for a quantitative study of the mental sciences is thus to devise means of measuring things, differences, changes and relationships for which standard units of amount are often not at hand, which are variable, and so unexpressible in any case by a single figure, and which are so complex that to represent any one of them a long statement in terms of different sorts of quantities is commonly needed. This last difficulty of mental measurements is not, however, one which demands any form of statistical procedure essentially different from that used in science in general.
The Rasch Model derived from E. L. Thorndike's 1904 Criteria. Thorndike, E.L.; Linacre, J.M. … Rasch Measurement Transactions, 2000, 14:3 p.763
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