Application of statistical methods to mental test data

Abstract

The papers of which this thesis is composed are mainly concerned with the application of statistical methods to certain problems of mental testing. Educational psychologists have on the whole been rather slow to make use of modern statistical technique, and factorial analysts in particular appear to have overlooked any consideration of the efficiency of the methods which they have employed in factor estimation. In the first paper the method of maximum likelihood was therefore applied to the problem of estimating factor loadings, and equations were obtained which, when solved, gave efficient estimates cf these quantities. In addition a criterion was found for deciding, in large samples, how many factors could be regarded as statistically significant.The process of solving the maximum likelihood equations was however somewhat laborious, and in a second paper an easier method of solution was derived. This was also compared with another method of factor estimation based on different assumptions and leading to different results. The third paper, although non-mathematical in form, has been included for the purpose of snowing how the maximum likelihood method can be reduced to a simple arithmetical process and applied to actual data.The fourth paper represents an attempt to provide a mathematical basis for problems connected with the construction of tests composed of a large number of questions each of which is scored 1 or 0 according as the response is correct or incorrect. The treatment is developed further in a fifth, and as yet unpublished, article. A new method is given for analysing factorially a battery of such multiple item tests, a method which appears to be specially applicable when the tests differ widely in difficulty.The last parer concerning Karl Pearson' s selection formulae is really of a general statistical nature, as these formulae have been made use of in many fields of research. They are however particularly applicable in many educational and psychological investigations. The precise conditions under which the formulae held true have teen established.