Andersen (1982) Latent Structure Analysis: A Survey

Summary

Andersen surveys the latent structure analysis literature from Lazarsfeld's (1950) founding through early 1980s developments. The unifying framework: manifest variables are conditionally independent given a latent variable $\theta$, so cell probabilities factorize as $\pi_{ijkl} = \int \pi_i^A(\theta)\pi_j^B(\theta)\pi_k^C(\theta)\pi_l^D(\theta)\varphi(\theta)d\theta$. The paper covers the discrete case (latent class model, expectation-maximization (EM) algorithm), the continuous case with parametric item characteristic curves (Rasch/Birnbaum-Lord models), latent profile analysis for continuous variables, multi-population comparison models, longitudinal models with correlated latent variables, and latent factor analysis.

Key Claims

• Local independence is the defining assumption: given $\theta$, manifest variables A, B, C, D are mutually independent. The structure is simple given $\theta$ but complex in the marginal — this gap is "crucial to the understanding of latent structure models."
• Latent class model (Lazarsfeld 1950): $\pi_{ijkl}=\sum_{v=1}^m\pi_{iv}^A\pi_{jv}^B\pi_{kv}^C\pi_{lv}^D\varphi_v$ with $5m-1$ free parameters; classical estimation via accounting equations (Lazarsfeld) and the determinantal method (T.W. Anderson 1954); asymptotic properties established only in Anderson (1968).
• EM algorithm (Goodman 1974a; Dempster-Laird-Rubin 1977): treat latent class membership as missing data — the augmented table $X_{ijklv}$ forms an exponential family. E-step computes posterior class memberships $\pi^*_{ijklv}$; M-step updates $\pi_{iv}^A, \varphi_v$. Converges to a stationary point; global maximum not guaranteed. EM is slower than Newton-Raphson but avoids Hessian inversion.
• Model flexibility (Goodman): restricting/equating parameters simply merges EM equations — scaling models and multi-latent-variable models derived by addition, not re-derivation.
• Item characteristic curves (from item response theory, IRT): Lawley-Lord normal ogive $\Phi((\theta-\alpha_A)\beta_A)$ (1943/1952) and Birnbaum-Rasch logistic $\exp\{(\theta-\alpha_A)\beta_A\}/[1+\exp\{(\theta-\alpha_A)\beta_A\}]$ (1957/1960). Under the pure Rasch model ($\beta_A=\beta_B=\beta_C=\beta_D=1$) the total score $Y=\sum X_{ij}$ is sufficient for $\theta$; population parameters $(\mu_\theta,\sigma_\theta^2)$ estimated by marginal maximum likelihood (ML) with one-dimensional integrals via Newton-Raphson (Andersen-Madsen 1977) or EM (Sanathanan-Blumenthal 1978).
• Latent profile analysis (Gibson 1959): accounting equations are formally identical for continuous variables — means $\mu_i^{(v)}$ and within-class zero covariances replace conditional probabilities; determinantal method estimates class weights $\varphi_v$ and class means $\mu_i^{(v)}$.
• Multi-population models: $\theta\sim N(\mu_l,\sigma_l^2)$ per population $l$; Rasch item characteristic curves; hypothesis $\mu_1=\cdots=\mu_L$ tests whether populations share the same latent level.
• Longitudinal extension: bivariate normal $(\theta_1,\theta_2)$ at two time points leads to double integrals — "no satisfactory numerical method is available" as of 1982; programmatic vision of autoregressive (AR) time-series models for latent autocorrelations $\varrho_t=c^t\varrho_0$ or AR($r$) at multiple time points.
• Latent factor analysis: Christoffersen-Muthén factor structure $\theta=\Gamma x+\varepsilon$ in latent space; Bartholomew (1980) general class $G(\pi^A(\theta))=\alpha_{A0}+\sum_t\alpha_{At}H(\theta_t)$; Rasch/normal model is a special case with $G=\Psi^{-1}$, $H=\Phi^{-1}$.

Concepts Introduced or Extended

• Latent Class Model — foundational local independence framework; accounting equations; EM algorithm derivation; identifiability
• EM Algorithm — derives the EM for latent class as an exponential family missing-data problem; Sundberg/DLR theoretical framework; E-step and M-step mechanics
• Rasch Model — item characteristic curves; sufficiency of total score; marginal ML for population distribution $(\mu_\theta,\sigma_\theta^2)$

Entities Mentioned

• Erling B. Andersen

Quotes

"The difference between [the conditional model given $\theta$ and the marginal model] is crucial in the understanding of latent structure models."

"No satisfactory numerical method is available at present for dealing with the likelihood equations [involving double integrals for the longitudinal latent variable model]."

My Take

A clean pedagogical survey that earns its place as the reference entry point for latent structure analysis. The principal contributions for this wiki are: (1) the canonical local-independence model unifying the entire family; (2) the EM algorithm derivation for latent class as an exponential family missing-data problem (predating or parallel to Dempster-Laird-Rubin 1977 per Goodman 1974); (3) the Rasch model sufficiency result and marginal ML formulation. The longitudinal section (Section 9) is more programmatic than substantive. The latent factor analysis section is dense. From a modern perspective, the paper's most lasting contribution is showing how the EM algorithm transforms the estimation of all latent structure models — a theme that runs through mixture-of-normals, hidden Markov model (HMM), and hierarchical Bayes methods that the wiki covers extensively elsewhere.