Least squares estimation for latent variables with dichotomous item response data

A new method for estimating the ability parameters in the statistical models based on item response theory (IRT) was presented in this study. A brief review of existing parameter estimation methods suggested the utility of a parameter estimation method which requires no specific assumption. By re-examining the performance of maximizing the ln $L$ of the joint maximum likelihood estimation (JMLE; Lord, 1980), and by applying theorems taken from Rockafellar (1970), it was proved that ln $L$ has a unique global maximum point. The performance of maximizing ln $L$ and the problem caused by maximizing ln $L$ were analyzed mathematically. In order to avoid the problem of extreme values and to have a more plausible result, this study introduced a new objective function, $Q\sb1$, to be minimized. The relationship between maximizing ln $L$ and minimizing $Q\sb1$ was also analyzed mathematically. The close-form solution of minimizing $Q\sb1$ was derived algebraically. A new set of ability parameter estimators were found and called least squares estimators (LSE). The new estimator, LSE, requires no specific assumption except the general assumptions of the IRT models. The asymptotic equivalence between JMLE and LSE was briefly discussed. Some possible applications of LSE were proposed. For example, by treating the LSE's as the true values of ability parameters, the JMLE for item parameters can be carried out. Then, two simulation studies were carried out to validate LSE and its applications. The result of the first simulation study suggested that the empirical standard error of LSE is less than the asymptotic standard error of JMLE, and the empirical bias of LSE is less than those of JMLE and Bayesian estimators. The result of the second simulation study suggested that the accuracy of LSE and its applications is comparable to that of BILOG, one of the most popular and recommended IRT parameter estimation computer programs. Some possible extensions of this study were also briefly discussed.