Open AccessUNIVARIATE AND BIVARIATE LOGLINEAR MODELS FOR DISCRETE TEST SCORE DISTRIBUTIONS
Author(s) -
Holland Paul W.,
Thayer Dorothy T.
Publication year - 1998
Publication title -
ets research report series
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.235
H-Index - 5
ISSN - 2330-8516
DOI - 10.1002/j.2333-8504.1998.tb01776.x
Subject(s) - bivariate analysis , univariate , mathematics , bivariate data , statistics , parametric statistics , smoothing , grouped data , exponential family , histogram , parametric model , computation , econometrics , multivariate statistics , computer science , algorithm , artificial intelligence , image (mathematics)
The well‐developed theory of exponential families of distributions is applied to the problem of fitting the univariate histograms and discrete bivariate frequency distributions that often arise in the analysis of test scores. These models are powerful tools for many forms of parametric data smoothing and are particularly well‐suited to problems where there is little or no theory to guide a choice of probability models, e.g. smoothing a distribution to eliminate roughness and zero frequencies in order to equate scores from different tests. Attention is given to efficient computation of the maximum likelihood estimates of the parameters using Newton's Method and to computationally efficient methods for obtaining the asymptotic standard errors of the fitted frequencies and proportions. We discuss tools that can be used to diagnose the quality of the fitted frequencies for both the univariate and the bivariate cases. Five examples, using real data, are used to illustrate the methods of this paper.