Nesting Monte Carlo EM for high-dimensional item factor analysis

Abstract

The item factor analysis model for investigating multidimensional latent spaces has proved to be useful. Parameter estimation in this model requires computationally demanding high-dimensional integrations. While several approaches to approximate such integrations have been proposed, they suffer various computational difficulties. This paper proposes a Nesting Monte Carlo Expectation-Maximization (MCEM) algorithm for item factor analysis with binary data. Simulation studies and a real data example suggest that the Nesting MCEM approach can significantly improve computational efficiency while also enjoying the good properties of stable convergence and easy implementation.

1. Introduction

All major psychological and educational tests are built, nowadays, with item response theory (IRT), since the methodology can significantly improve measurement accuracy, reliability, and validity while providing potential significant reductions in time and effort in assessment, especially via computerized adaptive testing [CAT; ¹]. IRT was originally developed with the assumption that the latent space is unidimensional. Although the uni-dimensional IRT model has proved to be useful, the limitations caused by the uni-dimensional assumption have been widely recognized [2, p62]. In recent years, IRT models have also become popular in other areas, such as health behavior [3-5], biology[6-8] and engineering [9, 10]. These new applications often require multidimensional latent spaces to accurately capture the latent structure of the measurement domains. Exploratory item factor analysis [11, 12] has proved to be very useful for these new applications.

While estimation methods for the unidimensional IRT model are well developed, efficient and easy to use estimation methods for the high dimensional item factor analysis model are still under development. The biggest difficulty is caused by intractable high dimensional integrations involved in the marginal likelihood. Existing solutions for high dimensional item factor analysis can be divided into three categories based on how they approximate the integration: (1) Adaptive Gauss-Hermite quadrature based EM [13]; (2)Monte Carlo based EM (MCEM)[14]; (3) Monte Carlo based stochastic approximation [15]. While adaptive G-H quadrature [16] can significantly improve computational efficiency beyond fixed points and weighted G-H quadrature, it is still computationally infeasible for high dimensional problems because the number of quadrature points grows exponentially as the number of latent factors increases. As compared with G-H quadrature based methods, MCEM and stochastic approximation are computationally more efficient for high dimensional problems. An advantage that stochastic approximation has over MCEM is that it converges without increasing the required Monte Carlo sampling, but its convergence diagnosis is more difficult [17]. The most important advantages of the MCEM algorithm are that it can be implemented easily without sophisticated numerical analysis skills, and that it can be extended smoothly to handle other modeling issues such as missing data.

For item factor analysis models, the corresponding conditional distributions involved in MCEM and stochastic approximation are often very complicated. Directly sampling from these distributions is often impossible. Computationally expensive Markov Chain Monte Carlo (MCMC) algorithms are often used instead. The use of MCMC algorithms will greatly reduce computational efficiency. During the last two decades, improving the convergence rate of EM-type algorithms has been a focus of research. Many accelerating techniques have been proposed. Using these accelerating techniques with MCEM algorithms can significantly improve computational efficiency while the resulting algorithms still enjoy the advantages of EM algorithms [18, page 28]. The objective of this paper is to improve the computational efficiency of the MCEM algorithm for item factor analysis by using the nesting strategy proposed by [19]. The focus and contribution of this paper are within the MCEM framework. The relative advantage of the nesting strategy is illustrated by comparing it with the MCEM algorithm. While investigating the relative computational efficiency of the nesting MCEM algorithm as compared with G-H quadrature based EM and stochastic approximation is of great value, this topic will be left for future research because the effort required for a fair and comprehensive investigation would lead us too far afield from the narrow thrust of this paper.

The rest of this paper is organized as follows. In section two we briefly review the full information item factor analysis model and the corresponding MCEM algorithm. In section three, the Nesting Monte Carlo EM algorithm is introduced and its specific form for full information item factor analysis is elaborated. The convergence properties of Nesting MCEM are investigated with two simulation studies and a real data example in section four. In the last section, the paper concludes with a discussion of potential future research in this area.

2. Item Factor Analysis Model

Let y_ij be the binary response from subject i, i = 1, …, n to item j, j = 1, …, J, and x_ij be a continuous latent variable underlying y_ij. An item factor analysis model with d factors can be expressed by the following equations,

$$x_{ij}=b_j+a_{j1}{z}_{i1}+\dots +a_{jd}{z}_{id}+{\epsilon }_{ij}=b_j+z_i^T{a}_j+{∊}_{ij},$$ (1)

$$y_{ij}=\{\begin{array}{cc}1,\hfill & if{x}_{ij}>0\hfill \\ 0,\hfill & if{x}_{ij}<0\hfill \end{array}\phantom{\}},$$ (2)

where A = {a_jk, j = 1,…, J, k = 1, … ,d} is the slope parameter matrix, b = {b_j, j = 1, … , J} is a vector of difficulty parameters, and z_ik, k = 1, … , d are latent factors. The model assumes that z_i ~ N_d(0, I), ε_i N_J (0, I), and z_i⊥ε_i. Let θ = {A, b} represent the set of model parameters. Equation (1) can also be parameterized under the factor analysis framework as follows

$$x_{ij}=b_j+z_i^T{\lambda }_j+{∊}_{ij},$$ (3)

where Λ = {λ_jk, j = 1, … , J, k = 1, … ,d} is the factor loading matrix, and ε_i = {ε_i1, … , ε_iJ} is assumed to follow independent normal distributions with mean 0 and variance

$${\sigma }_j^2=1-\sum _{k=1}^d{\lambda }_{jk}^2.$$ (4)

For the purpose of comparing different approaches, in the following analysis we may transfer the slop parameters to factor loadings,

$${\lambda }_{jk}=\frac{a_{jk}}{\sqrt{1+{\Sigma }_{l=1}^d{a}_{jl}^2}},$$ (5)

and apply some rotations [20] to the resulting factor loading matrix.

Following the suggestion of [11], when there are many subjects sharing the same response pattern, response patterns will be sorted, and the number of subjects, s_i, sharing the same response pattern will be recorded. Now let i index different response patterns, and the total number of different response patterns be n₀. Other notation used in this paper is as follows: (1) a_j is a d-element column vector, and its elements equal the jth row of A; (2) z_i is a d-element column vector of latent scores for subjects with response pattern i, and its elements equal the ith row of Z; (3) x_i is a J-element column vector corresponding to the ith row of X, a n₀ by J matrix.

The probability that subject i with factor scores $z_i^T=(z_{i1},\dots ,z_{id})$ answers correctly on item j is given by

$$Pr(y_{ij}=1\mid z_i,a_j,b_j)=Pr(x_{ij}\ge 0\mid z_i,a_j,b_j)=\Phi (z_i^T{a}_j+b_j),$$ (6)

where φ is the cumulative distribution function for the standard normal distribution.

Expectation maximization (EM) algorithms have been widely used to fit various models. For latent variable models with complex latent structures, such as item factor analysis and hierarchical models, the E step usually cannot be solved analytically. When closed form solutions are not available for the E step, the Monte Carlo EM algorithm o ers a stable and easily to implement approach for many complex models [e.g, ^21-23]. For models with categorical responses, direct samples may be very difficult to generate. In such a situation, computationally intensive simulation methods, such as the Gibbs sampler, are often used in the E step of the MCEM algorithm. From equation (1) we can see that once X and Z are observed, the item factor analysis model reduces to a multivariate linear model and closed form solutions based on sufficient statistics become available. Inspired by this fact, [14] augmented the observed data {Y} to complete data {Y, X, Z} [also see 24], and proposed a Gibbs sampler based MCEM algorithm for parameter estimation. With initial value θ⁽⁰⁾, at iteration (t + 1) the MCEM algorithm works as follows: E step: Given θ^(t), and z_i from N_d(0, I)

1. For j = 1, …, J

   $$x_{ij}\mid z_i,y,{\theta }^{\left(t\right)}~iid\{\begin{array}{cc}N(z_i^T{a}_j^{\left(t\right)}+b_j^{\left(t\right)},1)\text{truncated at left by 0}\hfill & \text{if}{y}_{ij}=1\hfill \\ N(z_i^T{a}_j^{\left(t\right)}+b_j^{\left(t\right)},1)\text{truncated at right by 0}\hfill & \text{if}{y}_{ij}=0\hfill \end{array}\phantom{\}}$$ (7)
2. $$z_i\mid x_i,y,{\theta }^{\left(t\right)}~iid{N}_d(V^{\left(t\right)}{A}^{T\left(t\right)}(x_i-b^{\left(t\right)}),V^{\left(t\right)}),i=1,\dots ,N,$$ (8)

where V = (I +A^T A)⁻¹. The above two full conditional distributions form a Gibbs sampler with the joint conditional distribution of x_i and z_i, f(x_i, z_i|y_i, θ^(t)), as its stationary distribution.

Suppose $\{x_i^{\left(k\right)},z_i^{\left(k\right)},k=1,\dots ,K\}$ are random samples generated by the Gibbs sampler from the joint conditional distribution p(x_i, z_i|y_i, θ^(t)), i = 1, … , n₀. Then the conditional expectations of the complete data sufficient statistics, {Z^T Z, Z^T X, 1^T X, 1^T Z}, can be approximated by corresponding sample means, or by equation (9) to (11) in [14] which use only $\{X_k^{\left(t\right)},k=1,\dots ,K\}$.

M step: Update θ^(t) to θ^(t+1)based on corresponding sufficient statistics as follows

$$A^{T,(t+1)}={[E\left[Z^{T}Z\right]-E{\left[1^{T}Z\right]}^{T}E\left[1^{T}Z\right]∕n]}^{-1}[E\left[Z^{T}X\right]-E{\left[1^{T}Z\right]}^{T}E\left[1^{T}X\right]∕n]$$ (9)

$$b^{T,(t+1)}=\left[E\left[1^{T}X\right]-E\left[1^{T}Z\right]A^{(t+1)}\right]∕n$$ (10)

These two steps are repeated iteratively until convergence. As shown by equation (7), at each iteration we need to draw n₀ × J random samples from truncated normal distributions, which is known to be nontrivial. Thus it is reasonable to expect that the above MCEM algorithm will be computationally very intensive when the number of response patterns and number of items is large.

3. Nesting MCEM for Full Information Binary Factor Analysis

Nesting MCEM has been proposed to improve the computational efficiency of MCEM when the E-step is computationally expensive but the M-step is relatively much cheaper, conditional on part of the augmented data [19, p. 206]. With almost no extra programming effort beyond MCEM, Nesting MCEM can maintain the stability of EM while increasing computational efficiency. In the last section, we noted that the E-step of the MCEM algorithm for item factor analysis is computationally intensive. From equation (1) we can also observe that if X were observed, the model can be reduced to a traditional factor analysis model for which deterministic EM [25] can be used. To make full use of the computationally intensive E-step, random samples {X_k, k = 1, …, K} from the conditional distribution of X, f(X|Y, θ^(t)) can be fixed, and several iterations taken of the deterministic EM conditional on {X_k, k = 1, …, K}. Since closed form solutions are available for the deterministic EM, computationally this approach will be much cheaper.

The above results suggest that the nesting strategy can be easily adopted for the item factor analysis model using the following two nested data augmentation schemes, Y_augu1 = {Y, X} and Y_aug2 = Y, X, Z. More specifically, with initial value θ⁽⁰⁾, at iteration (t + 1) the Nesting{MCEM}algorithm for this model with L inner iterations works as follows:

Outer E step: Given θ^(t), random numbers $\{X_k^{\left(t\right)},Z_k^{\left(t\right)},k=1,\dots ,K\}$ are drawn from the joint conditional distribution p(X, Z|Y, θ^(t)), as in the E-step of MCEM.

Inner Deterministic E-step: at (l + 1) inner iteration, given $\{X_k^{\left(t\right)},k=1,\dots ,K\}$, and θ^(t,l), conditional expectations of sufficient statistics {Z^T Z, Z^T X, 1^T X, 1^T Z} are calculated by equations (9) to (11) in [14].

Inner Deterministic M-step: Update θ^(t,l) to θ^{(t), (l+1)} based on corresponding sufficient statistics. When l = L, θ^(t) is updated to θ^(t+1) by θ^{(t), (L)} and the algorithm returns to the outer E-step.

The benefit of this nesting strategy can be explained by its intuitive connection with the blocked Gibbs sampler. Please refer to [19, p. 204] for an excellent discussion regarding this connection. In general, as L increases, the Nesting MCEM converges within fewer iterations although the computation time for each iteration also increases. Thus a key to the success of the Nesting MCEM is to wisely choose the number of inner iterations. One optimal way would be to automatically adjust the number of inner iterations by monitoring the convergence of the inner EM algorithm. However, computations involved in monitoring the inner EM would often defeat the advantage of Nesting MCEM. [19, p. 212] recommended that moderate values of L, say between 2 and 10, typically can make considerable improvement in computational efficiency.

4. Simulation Studies and Real Data Examples

In this section, we empirically investigate the relative computational properties of the Nesting MCEM algorithm with two simulation studies and one real data example. For all the analyses, initial values are obtained by factor extraction based on the sample tetrachoric correlation matrix. Rotational indeterminacy is fixed by letting A_ij = 0, for j ≥ i. Our experience suggests that the Gibbs sampler for this model mixes very fast with autocorrelation usually decreasing to near 0 after 15 iterations. To be more conservative in our implementation, the first 100 iterations are discarded, and samples are chosen every 20th iteration until the required number of samples is obtained. Nesting MCEM with L=3, 5, 7 and 9 inner iterations is used to examine how the number of inner iterations affects the convergence rate. Because closed form solutions are available for the inner EM iterations, the computational time of each iteration is roughly the same for MCEM and Nesting MCEM with a small number of inner iterations. A graphic approach is adopted to monitor the convergence of MCEM algorithms by plotting the observed data log likelihood against iterations, in which the log likelihoods are calculated directly using Gauss Hermite quadrature. Raw CPU computing time for each iteration is also recorded. All algorithms are implemented in Fortran 90. Computations are conducted on a PC with a 2GHz CPU and 1GB of RAM.

4.1. Example 1: A Two-Factor Simulation

In this simulation study, 1000 responses to a 24-item simple structure 2-factor test were simulated and 1000 distinct response patterns are obtained. difficulty parameters are randomly generated from U(−2, 2). True model parameters are shown in Table 1. In the E step, the number of random samples generated by the Gibbs sampler from p(x_i, z_i|y, θ^(t))), i = 1, … , 1000 starts with 20 and is then increased by 3 after each iteration. This means the total number of random samples in the E-step starts with 20000 and then increases by 3000 after each iteration.

Table 1.

True model parameters and estimates for simulation 1. NMCEM refers to the Nesting MCEM with 3 inner iterations.

	Difficulties			Factor 1			Factor 2
Items	True	MCEM	NMCEM	True	MCEM	NMCEM	True	MCEM	NMCEM
1	0.24	0.32	0.31	1.68	1.70	1.70	0.00	0.00	0.00
2	−0.31	−0.35	−0.35	2.17	2.15	2.05	0.00	−0.05	−0.04
3	−0.99	−1.11	−1.12	1.43	1.53	1.54	0.00	0.10	0.11
4	−1.64	−1.65	−1.65	0.89	0.84	0.83	0.00	0.10	0.10
5	0.33	0.34	0.34	0.91	0.99	0.99	0.00	−0.03	−0.03
6	−0.10	−0.04	−0.04	1.84	1.95	1.95	0.00	0.04	0.05
7	1.60	1.36	1.36	1.78	1.57	1.56	0.00	−0.01	−0.01
8	0.88	0.81	0.76	2.25	2.37	2.21	0.00	−0.03	−0.02
9	0.68	0.63	0.64	1.47	1.32	1.32	0.00	−0.06	−0.06
10	1.36	1.48	1.38	2.13	2.41	2.20	0.00	0.00	0.01
11	0.63	0.70	0.70	0.89	0.95	0.94	0.00	0.07	0.07
12	−0.14	−0.22	−0.22	1.45	1.35	1.37	0.00	−0.00	0.00
13	−0.46	−0.41	−0.41	0.00	0.12	0.11	1.72	1.78	1.78
14	0.19	0.23	0.24	0.00	0.03	0.02	0.83	0.86	0.85
15	0.92	0.94	0.94	0.00	0.12	0.11	0.92	0.95	0.96
16	−0.36	−0.33	−0.31	0.00	0.07	0.06	2.18	2.40	2.24
17	−1.67	−1.59	−1.57	0.00	0.09	0.08	1.65	1.49	1.46
18	0.64	0.70	0.70	0.00	0.04	0.03	0.98	0.99	1.00
19	1.45	1.59	1.51	0.00	0.09	0.08	2.23	2.09	1.96
20	1.52	1.62	1.61	0.00	0.05	0.05	1.33	1.41	1.41
21	0.01	0.14	0.14	0.00	0.17	0.16	1.51	1.45	1.47
22	1.43	1.41	1.37	0.00	0.06	0.05	1.76	1.70	1.65
23	0.06	0.03	0.03	0.00	0.22	0.21	2.19	2.11	2.06
24	−0.08	−0.03	−0.04	0.00	0.12	0.12	1.58	1.57	1.57
RMSE		0.087	0.085		0.121	0.105		0.080	0.090

Convergence diagnosis plots are shown in Figures 1 and 2, which plot the observed data log likelihood against iterations. Figure 1 shows that the log likelihood increases faster for Nesting MCEM than for MCEM, suggesting that Nesting MCEM converges faster than MCEM. Figure 2 is a close-up plot showing iterations 10 to 150. Figure 2 shows that the MCEM algorithm converges at around 120 iterations, while all four Nesting MCEM algorithms converge at around 40 iterations. Because the number of random samples increases after each iteration, the computational time also increases after each iteration. For this simulation, the Nesting MCEM algorithms converge faster than MCEM by a factor of 3 in the number of iterations and a factor of 7 in total computation time. These plots also suggest that the choice of inner iteration, L, is not crucial, as the convergence rates of the four Nesting MCEM algorithms are roughly equivalent.

Figure 1.

Convergence Diagnosis Plot for Simulation 1

Figure 2.

Close-up Convergence Diagnosis Plot for Simulation 1

Parameter estimates by MCEM and Nesting MCEM with 3 inner iterations are provided in Table 1. Accuracy is measured by root mean square error (RMSE) as listed in the last row of Table 1. As Table 1 shows, MCEM and Nesting MCEM can reproduce the true parameters satisfactorily. Their differences are very small and can be ignored.

4.2. Example 2: A Five-Factor Simulation

To examine the relative computational properties of the Nesting MCEM algorithm in a high dimensional situation, 1000 responses to a 26-item 5-factor test with quasi-simple structure were simulated and 1000 different response patterns were obtained. The true parameter values are shown in Table 2. The number of random samples in the E-step starts with 30 and then increases by 15 after each iteration.

Table 2.

True model parameters for simulation 2.

		Slop Parameters
Items	Difficulties	F1	F2	F3	F4	F5
1	−0.89	1.26	0.00	0.00	0.00	0.00
2	−0.55	0.99	0.00	0.00	0.00	0.00
3	−0.08	1.31	0.00	0.00	0.00	0.00
4	−0.79	2.12	0.00	0.00	0.00	0.00
5	0.43	1.87	0.00	0.00	0.00	0.00
6	−0.27	1.16	0.65	0.00	0.00	0.00
7	−1.65	0.00	1.76	0.00	0.00	0.00
8	0.63	0.00	1.26	0.00	0.00	0.00
9	1.16	0.00	2.13	0.00	0.00	0.00
10	0.34	0.00	0.92	0.00	0.00	0.00
11	1.13	0.00	1.41	0.62	0.00	0.00
12	−0.29	0.00	0.00	1.83	0.00	0.00
13	1.51	0.00	0.00	1.18	0.00	0.00
14	0.88	0.00	0.00	0.91	0.00	0.00
15	−1.08	0.00	0.00	1.54	0.59	0.00
16	1.52	0.00	0.00	0.67	1.45	0.00
17	−1.73	0.00	0.00	0.00	2.14	0.00
18	−0.38	0.00	0.00	0.00	1.78	0.00
19	0.77	0.00	0.00	0.00	0.89	0.00
20	1.00	0.00	0.00	0.00	0.63	1.31
21	−0.68	0.00	0.00	0.00	0.71	1.01
22	0.57	0.00	0.00	0.00	0.00	0.76
23	−1.54	0.00	0.00	0.00	0.00	1.71
24	−0.76	0.00	0.00	0.00	0.00	2.12
25	1.38	0.00	0.00	0.00	0.00	1.28
26	0.52	0.00	0.00	0.00	0.00	0.97

Results similar to those in simulation 1 are observed for this example. Estimation accuracy is roughly the same for MCEM and Nesting MCEM. Convergence diagnosis plots are shown in Figures 3 and 4. Figure 3 shows that the log likelihood increases faster for Nesting MCEM than for MCEM. As Figure 4 suggests, MCEM converged at around 70 iterations while Nesting MCEM converged at around 35 iterations. Again, the number of inner iterations does not significantly affect the convergence rate of Nesting MCEM. Here Nesting MCEM improved computational efficiency by a factor of 3 in total computation time.

Figure 3.

Convergence Diagnosis Plot for Simulation 1

Figure 4.

Close-up Convergence Diagnosis Plot for Simulation 1

4.3. Example 3: The 1978 Quality of American Life Survey

To investigate the relative performance of the Nesting MCEM compared with MCEM in practice, we examine the same data as used by [14]: The 1978 Quality of American Life Survey. In this survey, subjects were asked to rate their satisfaction on a 7 point scale with the following aspects: (1) Community; (2) Neighborhood; (3) Housing; (4) Life in U.S.; (5) Amount of education received; (6) Own health; (7) Job; (8) How spare time is spent; (9) Friendships; (10) Family life; (11) Standard of living; (12) Saving and investment; (13) Life as a whole; and (14) Self as person. Score 1 indicates being completely satisfied, and score 7 indicates being completely unsatisfied. For illustration purposes, these ratings are dichotomized as follows: values above 3 are dichotomized as 0, and 1 otherwise.

Following [14], we fit a five factor model to the dichotomized data. The number of random samples in the E-step starts with 30 and then increases by 15 after each iteration. Parameter estimates by MCEM and Nesting MCEM are quite close and thus lead to the same factor loadings for this real dataset. The varimax rotated factor loading estimates are given in Table 3. Convergence of MCEM and Nesting MCEM can be determined from Figures 5 and 6. These diagnostic plots indicate that Nesting MCEM converged at around 40 iterations, while MCEM barely converged with 100 iterations. Overall, for this real data, Nesting MCEM improved computational efficiency by a factor of 5 in total computation time.

Table 3.

Varimax rotated factor loadings for example 3.

	Factor Loadings
Items	F1	F2	F3	F4	F5
1	0.928	0.282	0.143	0.194	0.276
2	1.455	0.172	0.333	0.000	0.284
3	0.755	0.108	0.515	0.000	0.269
4	0.465	0.586	0.144	0.273	0.284
5	0.207	0.264	0.305	0.000	0.284
6	0.124	0.797	0.189	0.000	0.339
7	0.368	0.159	0.418	0.511	0.515
8	0.369	0.378	0.318	0.000	0.972
9	0.245	0.000	0.239	0.134	0.703
10	0.184	0.308	0.226	0.380	0.924
11	0.588	0.251	1.309	0.000	0.649
12	0.282	0.311	1.309	0.226	0.391
13	0.480	0.711	0.560	0.143	1.219
14	0.199	0.668	0.425	−0.257	1.167

Figure 5.

Convergence Diagnosis Plot for Real Data Example

Figure 6.

Close-up Convergence Diagnosis Plot for Real Data Example

5. Discussion

Results from our simulation studies and the real data example suggest that Nesting MCEM can considerably improve the computational efficiency of MCEM in item factor analysis. Typically a small number of deterministic inner iterations, say 3, would significantly improve efficiency. Further increasing L does not lead to any major improvement in the convergence rate while at the same time defeating overall computational efficiency. As pointed out by [19], the Monte Carlo sampling approach used in the E step can notably affect the relative performance of the nesting strategy. In this paper, we tried different increments to the number of random samples in the E step. We observed that this number can affect the relative performance of the nesting strategy. As the increment becomes larger, however, its effect becomes less significant.

It can be expected that extra efficiency could be obtained by combining Nesting MCEM with other acceleration techniques, such as Expectation/Conditional Maximization [ECM; ²⁶] and Parameter expansion EM [PX-EM; ²⁷]. Recent research on these acceleration techniques has mainly focused on the random effects model and its generalizations in the fields of statistics and biostatistics. Like the item factor analysis model, traditional latent variable models and their extensions often contain a larger fraction of missing information than the random effects model, and hence the corresponding EM-type algorithms are often slower. Thus further research on improving EM-type algorithms for traditional latent variable models by the use of various acceleration techniques would be worthwhile.