AN EXPERIMENTAL INVESTIGATION OF A MATHEMATICAL LEARNING MODEL1, 2
Authors: Best Phillip J. 
Published: August 2014

When mathematical learning models are applied to experimental data, frequently the adequacy of the model is assessed solely on the basis of the agreement between the model and data. Most of the existing learning models have been found to fit the data very well, in spite of the fact that many of the models are based on different underlying assumptions.

The criterion of goodness of fit is therefore inadequate, and should be supplemented by another consideration such as the usefulness of the parameters. A scientifically useful model should contain parameters which could be used in prediction. If simple functional relationships are found between the parameters for a group of subjects on a set of problems, then these parameters could be used to predict the performance of these subjects on new problems or of new subjects on the original problems.

The purpose of the present study is to determine the scientific usefulness of a special case of a model derived by Audley and Jonkheere (1956). This special case of the model is a direct mathematical formulation of Thorndike's Law of Effect and is isomorphic to a model proposed by Gulliksen (1934). The model has three parameters. One parameter (po) represents the initial probability of a correct response. The other two parameters are learning parameters. They represent the effect (α) of a reinforcement of a correct response on the strength of a correct response and the effect (δ) of nonreinforcement of an incorrect response on the strength of an incorrect response.

The model was applied to data from two experiments in which cats were trained on visual discrimination problems. In the first experiment, six cats were trained on a series of seven problems. The best fitting parameter values were factor analyzed to determine if a simple relationship existed between the parameters. A two‐factor solution was obtained for the learning parameters, and a one‐factor solution was obtained for the initial performance parameter.

In the second experiment, nine “split brain” cats were trained on a visual discrimination problem, first with one eye occluded then with the other eye occluded. Previous split brain research has demonstrated that under the conditions used in this experiment no interocular transfer occurs. The learning parameters (α and δ) that were obtained for the first and second training had no significant differences. The reliability of the learning parameters (α and δ) was very high (.992 and .952 respectively). The reliability of the initial probability parameter (po) was very low (.09), and the difference between the parameter for the first to the second training was significant at the .10 level.

In both experiments, no significant differences were found between the model and the data. The model was therefore considered to fit the data well.

The high reliability and presence of a two‐factor solution for the learning parameters (α and δ) indicate that these parameters follow simple functional transformations from one problem to another and from one cat to another. The presence of simple functional relationships indicates that the parameters obtained from the performance of a new cat on any one problem can give a good approximation of his performance on the other problems.

Even though the initial performance parameters (po) could be accounted for by a one‐factor solution, the low reliability indicates that this parameter may not be useful as a predictor across cats or across problems. However, the initial performance parameter is useful within any single cat and problem, because it allows the differences in initial performances to be partialed out when the problem difficulty or cat performances are studied.