The study compares the extent to which naturally structured data and artificial, relatively random data (both with the same basic parameters) produce simple structure factors which are uniquely determined. Two examples of unstructured matrices were compared with the ball problem matrix. The results show that an oblique position of maximum hyperplane count in the structured data differs from that in the unstructured by reaching a significantly more unique position in terms of the exactitude with which it is re-discoverable when starting from different positions, and by reaching (at the maximum) a significantly higher hyperplane count.

The Fisher-Yates “Exact Test” has become widely known to researchers in psychology as the significance test of choice for comparing pairs of proportions in 2 × 2 contingency tables, particularly when the expected frequency of one or more of the cells is small. Less widely recognized are some readily available tabular aids for calculating the “Exact Test” in a substantial number of the cases in which it might be applied. Thus, Finney [1] has tabled probability levels at or beyond the .05 level for those 2 × 2 tables in which neither of the two row marginals exceeds 15. Since it is a matter of arbitrary choice as to which pair of marginals of an obtained table is designated the row pair and which the column pair, Finney’s table has broad applicability. It has been incorporated into the Biometrika Tables for Statisticians [3], which also present extensive discussion of its applications and extensions. Finney’s table has been extended by Latscha [2] to accommodate those contingency tables in which neither of the two row marginals exceeds 20. The Biometrika Tables provide additional, if less direct, help in their tables of n! for values of n up to 250, and of log10n! for values of n up to 1000.

Improved estimation procedures are suggested and a more complete distribution theory is presented for Restle's Cue Learning Model.

A two-variable generating function is described which yields the sampling probabilities of the Shannon-Wiener information measure. Expansion and collection of terms in like powers of the first variable imposes the restriction that the sum of the k category frequencies equal n; collection of terms in like powers of the second variable then produces terms whose coefficients are the required probabilities. The method may be used with either equal or unequal category probabilities for any finite n and k, and thus represents a general solution to the small sample problem. Tables of sampling probabilities are presented.

The powered vector method of factor analysis yields directly and without rotation factors satisfying objectives of parsimony, orthogonality, and meaningfulness. The method is objective, computationally efficient, and easily programmed for digital computers. The computational procedure is described. Illustrative analyses are presented. Results of applications of the powered vector method are compared with results obtained using the principal axes solution followed by orthogonal rotation.

In this method of analyzing learning data, entire learning curves are described quantitatively by single numbers which are used in a statistical test to determine whether two or more groups of learning curves are significantly different. The method has some logical advantages over prevailing methods in that it avoids the use of average learning curves and of arbitrary measures of slope and asymptote. Its disadvantage is computational. Since it involves the use of factor analytic procedures, it may be tedious to apply unless computation is carried out on a high-speed computer.

The problem of detecting emergent social structure in small groups is examined in terms of establishing a baseline or expected structure. The standard proposed here is equalitarianism; that is to say, every member is equally likely to interact with every other member. Using information theory, a formal statement of the equalitarian argument is possible. In addition to detecting a deviation by a single group, the structural tendency of several groups assigned to some experimental condition is discussed. This technique is designed for the preliminary analysis of structure; it deals only with the general character of the total structure.

On the assumption that a partitioning can be found such that three mutually exclusive test vector configurations span the same factor space, a procedure is derived whereby symmetric parts of the correlation matrix are estimated from functions of asymmetric parts treated symmetrically. This yields an explicit matrix formula for communality estimation which generalizes earlier work by Albert. Conventional factoring methods, with all their computational and fitting advantages, can be applied once the symmetric portions of the correlation matrix have been estimated. Extension to four subgroups of test vectors allows for a matrix generalization of the old tetrad difference criterion to the multiple-factor case.

Brogden's coefficient of selective efficiency and Clemans' λ are more efficient than biserial r when the correlation is high. Some empirical sampling studies are reported comparing the standard errors of these statistics.

Solutions for simple structure obtained from empirical data through the use of the Oblimax, Quartimax, and Varimax procedures are matched with independently determined graphic solutions for the same data. The relative merits of the two analytic orthogonal solutions (Quartimax and Varimax) are discussed. Factors of employee morale are interpreted.

This paper presents a way of factoring the Guttman circumplex to get factor weights whose pattern is close to that postulated by Guttman for the circumplex. An empirical example, including a crude procedure for diagonal estimation, is shown. Other sources of nonuniqueness are discussed, and the limitation to odd-numbered complexity for the tests is indicated. An orthodox factor analysis is applied to the example, and this provides a basis for comparison of the two opposing models.

The biserial correlation between an item and the total test of which the item is a part tends to be misleadingly high when used in item analysis, since the item is included in the total test. Two formulas with correction for this overlap are derived and compared with Zubin's and Guilford's formulas. One of the new coefficients is invariant to test length.

It is shown how results of Prokhovnik on the number of pattern variants that may be formed by k markers on a square network of m2 positions may be derived more simply by means of a combinatorial theorem of Pólya's, which may also be used to solve systematically many other problems of this type.

The purpose of this note is to reconsider the Kelley-Cureton definition of optimal extreme groups for estimating item-criterion correlations. Optimal tail per cents are derived, using the criterion of minimum sampling variance of the tetrachoric correlation coefficient, and the findings are related to earlier work of Mosteller. It is shown that upper and lower 27 per cent groups yield the most precise estimate of the tetrachoric coefficient only when the population correlation is close to zero. When the population value is .4, extreme 20 per cent groups provide estimates with the smallest sampling error variance. It is further shown, however, that 27 per cent extremes yield highly efficient estimates. Thus no change is recommended in traditional item analysis procedures.