It currently seems to me that the heart of mental test theory is the concept of true score.

The trouble starts as soon as it is realized that the actual test score obtained by a particular individual might just as well have been some numerical value other than the one actually observed. The individual examinee might have guessed differently, might have been less nervous, or might have slept better the night before. The testing conditions might have been different–lighting, facilities for use of paper and pencil, presence or absence of distraction. Finally, any one of many different but equally appropriate psychological tests might have been constructed and administered.

Mental test theory must deal with all these kinds of disturbing influences. However, we are not really interested in each of the different test scores that an examinee might obtain under all sorts of conditions. We are interested in something approximated by these scores. This something may be called the true score on the test.

A theoretical approach to the understanding of human behavior in uncertain outcome situations is suggested, an approach which draws upon utility theory, decision-making theory, and statistical association theory. Experimental evidence supporting this approach as opposed to alternative approaches is summarized. Three different formalizations are presented, and a variety of experimental tests is suggested.

To explain changes in the sociometric configuration of a group through time, a problem arises of the extent to which such changes may be viewed as the aggregation of part-processes occurring at the level of two-person choice structures. A possible model is a Markov chain in which three possible states are mutual choice, one-way choice, and indifference, one realization for each pair of choosing individuals in the group. Choice data for an eighth-grade classroom are fitted to this model and are used to answer questions of constancy of transition probabilities, order of the chain, and sex differences.

A quantitative approach to the psychology of pattern recognition requires knowledge of the number of possible variants of any particular pattern. The general solution for the number [p/m2] of pattern variants that p counters can form on a square network of m2 positions is obtained by elementary group theory. The exact solution is given in terms of the different types (symmetric, asymmetric, etc.) of patterns possible and an approximate formula for the total number of patterns is also developed.

A battery of 18 tests of intermediate algebra and 20 reference tests was administered to two successive second-year algebra classes. Each battery was separately factor analyzed by Thurstone methods, and the two analyses were synthesized by the Tucker method. The five congruent factors obtained were identified as: Verbal Comprehension, Deductive Reasoning, Algebraic Manipulative Skill, Number Ability, and Adaptability to a New Task.

The problem of combining abilities of group members to maximize the performance of the group as a whole is examined in terms of redundancy in task assignments. In particular, ways of distributing a given number of items of information among a given number of individuals to obtain the maximum probability of each item being recalled by at least one individual are studied. It is shown that there exists an optimal distribution scheme which is independent of the amount of material originally given, the size of the group, and individual differences in ability.

Mood's likelihood ratio test is generally considered an unreliable x2 approximation in 2 × 2 contingency tables containing expected cell frequencies less than five. Probability values were computed for 60 such tables as part of an item analysis for two 30-item alternate forms of a measure. The rank orders of the items, from best to worst differentiators, as determined separately by Mood's test and by Fisher's exact test correlated .97 for one form and .96 for the other.