The Information Function

Abstract

When you speak of having information, it implies that you know something about a particular object or topic. In statistics and psychometrics, the term information conveys a similar, but somewhat more technical, meaning. The statistical meaning of information is credited to Sir R.A. Fisher, who defined information as the reciprocal of the precision with which a parameter could be estimated. Thus, if you could estimate a parameter with precision, you would know more about the value of the parameter than if you had estimated it with less precision. Statistically, the precision with which a parameter is estimated is measured by the variability of the estimates around the value of the parameter. Hence, a measure of precision is the variance of the estimators, which is denoted by σ 2 . The amount of information, denoted by I, is given by the formula: [6-1] In item response theory, our interest is in estimating the value of the ability parameter for an examinee. The ability parameter is denoted by θ, and θ ^ is an estimator of θ. In the previous chapter, the standard deviation of the ability estimates about the examinee's ability parameter was computed. If this term is squared, it becomes a variance and is a measure of the precision with which a given ability level can be estimated. From equation 6-1, the amount of information at a given ability level is the reciprocal of this variance. If the amount of information is large, it means that an examinee whose true ability is at that level can be estimated with precision; i.e., all the estimates will be reasonably close to the true value. If the amount of information is small, it means that the ability cannot be estimated with precision and the estimates will be widely scattered about the true ability. Using the appropriate formula, the amount of information can be computed for each ability level on the ability scale from negative infinity to positive infinity. Because ability is a continuous variable, information will also be a continuous variable. If the amount of information is plotted against ability, the result is a graph of the information function such as that shown below.