Semiparametric Factor Analysis for Item-Level Response Time Data

Abstract

Item-level response time (RT) data can be conveniently collected from computer-based test/survey delivery platforms and have been demonstrated to bear a close relation to a miscellany of cognitive processes and test-taking behaviors. Individual differences in general processing speed can be inferred from item-level RT data using factor analysis. Conventional linear normal factor models make strong parametric assumptions, which sacrifices modeling flexibility for interpretability, and thus are not ideal for describing complex associations between observed RT and the latent speed. In this paper, we propose a semiparametric factor model with minimal parametric assumptions. Specifically, we adopt a functional analysis of variance representation for the log conditional densities of the manifest variables, in which the main effect and interaction functions are approximated by cubic splines. Penalized maximum likelihood estimation of the spline coefficients can be performed by an Expectation-Maximization algorithm, and the penalty weight can be empirically determined by cross-validation. In a simulation study, we compare the semiparametric model with incorrectly and correctly specified parametric factor models with regard to the recovery of data generating mechanism. A real data example is also presented to demonstrate the advantages of the proposed method.

Derivatives for M-Step Optimization

Taking logarithm of the conditional density (Eq. 4) yields

$$\begin{aligned} \log f_j(y|x) = g_j(x, y) - \log \int _0^1\exp \left( g_j(x, y') \right) \mathrm{d}y'. \end{aligned}$$

(30)

The first and second derivatives of Eq. 30 with respect to the reduced coefficients $\varvec{\theta }_j$ are

$$\begin{aligned} \frac{\partial \log f_j}{\partial \varvec{\theta }_j}(x, y) = \frac{\partial g_j}{\partial \varvec{\theta }_j}(x, y) - \frac{\int _0^1 \frac{\partial g_j}{\partial \varvec{\theta }_j}(x, y') \exp \left( g_j(x, y') \right) \mathrm{d}y'}{\int _0^1\exp \left( g_j(x, y') \right) \mathrm{d}y'}, \end{aligned}$$

(31)

and

$$\begin{aligned} \frac{\partial ^2\log f_j}{\partial \varvec{\theta }_j\partial \varvec{\theta }_j^\top }(x, y) =&- \frac{\int _0^1 \frac{\partial g_j}{\partial \varvec{\theta }_j}(x, y')\frac{\partial g_j}{\partial \varvec{\theta }_j^\top }(x, y') \exp \left( g_j(x, y') \right) \mathrm{d}y'}{\int _0^1\exp \left( g_j(x, y') \right) \mathrm{d}y'} \nonumber \\&+ \frac{\left( \int _0^1 \frac{\partial g_j}{\partial \varvec{\theta }_j}(x, y') \exp \left( g_j(x, y') \right) \mathrm{d}y'\right) \left( \int _0^1 \frac{\partial g_j}{\partial \varvec{\theta }_j^\top }(x, y') \exp \left( g_j(x, y') \right) \mathrm{d}y'\right) }{\left( \int _0^1\exp \left( g_j(x, y') \right) \mathrm{d}y'\right) ^2},\quad \end{aligned}$$

(32)

respectively. Because the spline coefficients for the main effect and interaction functions are separable, we have

$$\begin{aligned} \frac{\partial g_j}{\partial \varvec{\theta }_j}(x, y) = \left( \frac{\partial g_j^y}{\partial \varvec{\alpha }_j^\top }(y),\ \frac{\partial g_j^{xy}}{\partial \mathrm {vec}(\mathbf {B}_j)^\top }(x, y) \right) ^\top , \end{aligned}$$

(33)

in which

$$\begin{aligned} \frac{\partial g_j^y}{\partial \varvec{\alpha }}(y) = \mathbf {N}^\top \varvec{\psi }(y), \end{aligned}$$

(34)

and

$$\begin{aligned} \frac{\partial g_j^{xy}}{\partial \mathbf {B}}(x, y) = \mathbf {N}^\top \varvec{\psi }(y)\varvec{\psi }(x)^\top \mathbf {N}. \end{aligned}$$

(35)

The penalty terms (Eqs. 15 and 16) are quadratic forms in the spline coefficients. Let $\mathbf {Q}= \mathbf {D}_2\mathbf {N}$. The corresponding derivatives are obtained as follows:

$$\begin{aligned}&\frac{\partial p_1}{\partial \varvec{\alpha }_j}(\varvec{\alpha }_j; \uplambda ) = \uplambda \mathbf {Q}^\top \mathbf {Q}^{}\varvec{\alpha }_j, \end{aligned}$$

(36)

$$\begin{aligned}&\frac{\partial ^2 p_1}{\partial \varvec{\alpha }_j\partial \varvec{\alpha }_j^\top }(\uplambda ) = \uplambda \mathbf {Q}^\top \mathbf {Q}^{}, \end{aligned}$$

(37)

$$\begin{aligned}&\frac{\partial p_2}{\partial \mathrm {vec}(\mathbf {B}_j)}(\mathbf {B}_j; \uplambda ) = \uplambda \left( \mathbf {I}\otimes \mathbf {Q}^\top \mathbf {Q}+ \mathbf {Q}^\top \mathbf {Q}\otimes \mathbf {I}\right) \mathrm {vec}(\mathbf {B}_j), \end{aligned}$$

(38)

and

$$\begin{aligned} \frac{\partial ^2 p_2}{\partial \mathrm {vec}(\mathbf {B}_j)\partial \mathrm {vec}(\mathbf {B}_j)^\top }(\uplambda ) = \uplambda \left( \mathbf {I}\otimes \mathbf {Q}^\top \mathbf {Q}+ \mathbf {Q}^\top \mathbf {Q}\otimes \mathbf {I}\right) . \end{aligned}$$

(39)