Normal Ogive Model Research Articles

THE USE OF THE BRITISH ABILITY SCALES WITH A GROUP OF AUSTRALIAN SCHOOLCHILDREN — SOME PRELIMINARY DATA

Summary. In a preliminary study, the four core tests of the British Ability Scales were administered to a sample of 208 Australian schoolchildren aged 9.6 to 10.5 years. In general the results obtained were comparable with the UK standardisation group although there was a tendency towards higher scores by the Australian sample. Analysis of the scale characteristics of the items were carried out using the Rasch method and a normal ogive model (NOHARM). The Rasch scaling appears to be robust with this population. The NOHARM analysis produced strikingly similar results. Sample estimates of reliability and validity were obtained and were found to be acceptable.

The dimensionality of tests and items

An explication is offered for the notion of dimensionality both for tests and items. A set of n tests or of n binary items is unidimensional if and only if the tests or the items fit a common factor model, generally non‐linear, with one common factor, that is, one latent trait. Both test scores and item responses in general contain stable specific factors as well as errors of retest measurement. The two‐parameter normal ogive model can be obtained from a joint space which in general is of n + 1 dimensions. One of these is the latent trait continuum while the remaining n are dimensions of unique (specific and error) variation. If and only if the items fit the perfect scale the n + 1 dimensions collapse into one dimension. Proposals to regard coefficient alpha as a coefficient measuring homogeneity, internal consistency, or generalizability, do not appear to be well founded.

The Urry Method of Approximating the Item Parameters of Latent Trait Theory

This note demonstrates that the convenient and inexpensive procedure suggested by Urry (1974a) for approximating test models (i.e., the normal ogive and logistic models) tends to systematically underestimate ai (item discriminatory power) and overestimate /b i/ (item difficulty). A simple correction for error in estimated ability (θ) is presented which serves to eliminate these biases. Implications for item screening and for item parameterization via maximum likelihood methods and via Urry's more recently developed estimation procedure (1974b) are discussed.

Effects of Individual Optimization in Setting the Boundaries of Dichotomous Items on Accuracy of Estimation

Applying the normal ogive model of latent trait theory, two sets of data, simulated and empirical, were analyzed. The objective was to determine how much accuracy of estimation of the subjects' latent ability can be maintained by tailoring for each testee the order of presentation of the items and the border of dichotomization for each item. This was compared to the information provided by the ori ginal graded test items. Results indicated that tailored testing is promising especially when the number of items is not too small, and that a graded item can effectively be used as the initial item in tailored testing because of its branching effect.

A Simple Technique for Estimating Latent Trait Mental Test Parameters

A very simple and economical method for estimating initial parameter values for the normal ogive or logistic latent trait mental test model is outlined. The accuracy of the method in comparison with maximum likelihood estimation is investigated through the use of Monte-Carlo data. The study yields a number of observations concerning how sample size and true item parameter values of the data influence both estimation methods.

Normal ogive model on the continuous response level in the multidimensional latent space

The homogeneous case of the continuous response model is expanded to the multi-dimensional latent space, and the normal ogive model is presented. The operating density characteristic of the continuous item response and the vector of basic functions are developed. It is found out that there is a vector of sufficient statistics for estimating the subject's vector of latent traits, given the item parameter vectors. The relationship between the model and the linear factor analysis is observed. The matrix of item response information functions is introduced. Some additional observations are also made.

A comparison of computer routines for the calculation of the tetrachoric correlation coefficient

In calculations of the discriminating-power parameter of the normal ogive model, Bock and Lieberman compared estimates derived from their maximum-likelihood solution with those derived from the heuristic solution. The two sets of estimates were in excellent agreement provided the heuristic solution used accurate tetrachoric correlation coefficients. Three computer methods for the calculation of the tetrachoric correlation were examined for accuracy and speed. The routine by Saunders was identified as an acceptably accurate method for calculating the tetrachoric correlation coefficient.

Fitting a response model forn dichotomously scored items

A method of estimating the parameters of the normal ogive model for dichotomously scored item-responses by maximum likelihood is demonstrated. Although the procedure requires numerical integration in order to evaluate the likelihood equations, a computer implemented Newton-Raphson solution is shown to be straightforward in other respects. Empirical tests of the procedure show that the resulting estimates are very similar to those based on a conventional analysis of item “difficulties” and first factor loadings obtained from the matrix of tetrachoric correlation coefficients. Problems of testing the fit of the model, and of obtaining invariant parameters are discussed.