A theory of consistency of ordering generalizable to tailored testing

Measures of consistency and completeness of order relations derived from test-type data are proposed. The measures are generalized to apply to incomplete data such as from tailored testing. The measures are based on consideration of the items-plus-persons by items-plus-persons matrix as an adjacency matrix in which a 1 means that the row element, whether item or person, dominated the column element. From this the number of item-item and person-person dominance relations can be derived. Indices of consistency are developed which compare the number of dominance relations in a matrix to the number that would occur if it were perfectly consistent and to the number in a random matrix; relations to other indices are noted.The research reported here was supported by the Office of Naval Research, Contract N00014-75-C-0684, NR150-373. The author wishes to thank Dr. Thomas J. Reynolds for a number of helpful discussions of this material.     

Estimation of other coefficients of correlation from the phi coefficient

Formulas are developed for estimating a point-biserialr or a tetrachoricr from an obtained phi coefficient. The estimate of a tetrachoricr, which is calledr _φ, is shown to be equivalent to that obtained from first-order use of the tetrachoricr series. A tabulation is made of corrections needed to maker _φ equivalent numerically to the tetrachoricr. In spite of its greater generality than estimates of tetrachoricr by previous methods, there are limitations, which are pointed out.     

Some applications of attribute prediction theory to psychophysics

The following development is a coordination of certain aspects of mental test methods and psychophysics. The biserial and triserial prediction equations of the Katzell-Cureton theory are utilized to reformulate the determination of absolute and differential limens from data produced by the constant method.     

The reliability of a point biserial coefficient of correlation

An attempt is made to elaborate upon Lev's results concerning the reliability of the point biserial coefficient of correlation in a manner that will be helpful to the psychological statistician. Procedures required in the use of the non-centralt tables prepared by Johnson and Welch are described as they relate to the determination of the fiducial limits for a point biserial coefficient. A normal approximation technique for the estimation of fiducial limits is also suggested. Numerical evidence is presented which shows that relative to a given level of significance the width of the fiducial interval estimated from a point biserial coefficient of any size is smaller than that of the fiducial interval corresponding to an ordinary Pearsonian coefficient of the same magnitude.     

Bisection of loudness

In a loudness bisection task, subjects varied one sound to lie halfway between two given sounds in terms of loudness. The two given sounds were varied from 30 to 90 dB in a 4 by 9 factorial design. Functional measurement methods based on monotone analysis provided good support for the bisection model, and yielded a loudness scale with an exponent of about .3, except for a falloff at lower intensities. Two other tasks, judging average loudness and difference in loudness of the two given sounds, yielded mixed results. In Experiment 2, in particular, the differencing judgments were not additive, even under monotone transformation. These analyses also indicated that previous applications of monotone analysis have typically lacked adequate power to allow any conclusion about the operative model. Overall, the present bisection scale agrees with Garner’s lambda scale, and the present theoretical approach agrees with that of Garner in its emphasis on algebraic models as a foundation for psychological measurement.