Woods and Lin (2009) proposed a unidimensional item response theory (IRT) model where the distribution of the latent variables is estimated using a semi-nonparametric (SNP, Gallant & Nychka, 1987) density. Estimation of the latent variable density can reduce bias in parameter estimates that results from misspecifying the form of the density (Woods & Thissen, 2006; Woods & Lin, 2009). However, application of the Woods and Lin (2009) model is restricted to the unidimensional setting. To address this limitation, the present research generalizes the Woods and Lin (2009) model to multidimensional IRT (MIRT). The resulting model, the SNP-MIRT model, may also be considered a generalization of the "standard" MIRT model, which specifies a normal density for the latent variables.

A secondary focus of this research concerns a new proposal for calculating student growth percentiles (SGP, Betebenner, 2009). In Betebenner (2009), quantile regression (QR, Koenker & Bassett, 1978; Koenker, 2005) is used to estimate the SGPs. However, a shortcoming of the original methodology is that measurement error in the score estimates, which always exists in practice, leads to bias in the SGP estimates (Shang, 2012). One way to address this issue is to estimate the SGPs using a modeling framework that can directly account for the measurement error. MIRT is one such framework, and the one utilized here. To maximize the generality of the approach, as well as guard against misspecification of the latent variable density, the SNP-MIRT model is used. SNP-MIRT estimates, in turn, are used with the calibrated projection linking methodology (Thissen, Varni, et al., 2011; Thissen, Liu, Magnus, & Quinn, 2014; Cai, in press-a, in press-b) to produce SGP estimates.

Preliminary simulation studies are conducted to investigate the fidelity of the SNP-MIRT model and SGP estimation implementations. The simulation study for the SNP-MIRT model focuses on recovery of the shape of the data-generating latent variable density. The simulation study for the proposed SGP method focuses on comparing the accuracy of the QR, standard MIRT, and SNP-MIRT approaches. Finally, empirical applications are provided to illustrate the new methods.

In Ramsay curve item response theory (RC-IRT, Woods & Thissen, 2006), the shape of the latent trait distribution is estimated simultaneously with the item parameters. In its original implementation, RC-IRT is estimated via Bock and Aitkin's (1981) EM algorithm, which yields maximum marginal likelihood estimates. This method, however, does not produce the parameter covariance matrix as an automatic byproduct upon convergence. In turn, researchers are limited in when they can employ RC-IRT, as the covariance matrix is needed for many statistical inference procedures. The present research remedies this problem by estimating the RC-IRT model parameters by the Metropolis-Hastings Robbins-Monro (MH-RM, Cai, 2010) algorithm. An attractive feature of MH-RM is that the structure of the algorithm makes estimation of the covariance matrix quite convenient. Additionally, MH-RM is ideally suited for multidimensional IRT, whereas EM is limited by the "curse of dimensionality." When RC-IRT is further generalized to multiple latent dimensions, MH-RM would appear to be the logical choice for estimation.