Goodman (1974) Exploratory Latent Structure Analysis Using Both Identifiable and Unidentifiable Models

Summary

Goodman (1974) introduces a unified maximum likelihood (ML) estimation algorithm for latent class models with $m$ polytomous manifest variables and $T$ latent classes — an iterative procedure mathematically equivalent to the expectation-maximisation (EM) algorithm, three years before Dempster-Laird-Rubin (1977) formalised the general framework. The paper also provides a practical identifiability diagnostic (a Jacobian rank test evaluated at the maximum likelihood estimate (MLE)), unifies unrestricted and restricted latent structures under a single estimation framework, and proposes an exploratory strategy that uses unidentifiable models as stepping stones toward parsimonious identifiable ones.

Key Claims

• Model: For $m$ manifest polytomous variables and a $T$-class latent variable $X$, cell probabilities factor as: $$\pi_{ijkl} = \sum_{t=1}^T \pi_t^X \pi_{it}^{AX} \pi_{jt}^{BX} \pi_{kt}^{CX} \pi_{lt}^{DX}$$ Within each latent class the manifest variables are mutually independent (local independence). Free parameters: $(I+J+K+L-3)T - 1$ for the basic set; free cell probabilities: $IJKL - 1$.

• Estimation algorithm (EM): Starting from initial trial values $\hat\pi^{(0)}$, iterate:

  • E-step: compute posterior class membership $\hat\pi_{ijklt}^{ABCD\bar X} = \hat\pi_{ijklt}^{ABCDX} / \hat\pi_{ijkl}$
  • M-step: $\hat\pi_t^X = \sum_{i,j,k,l} p_{ijkl}\, \hat\pi_{ijklt}^{ABCD\bar X}$; $\hat\pi_{it}^{AX} = (\sum_{j,k,l} p_{ijkl}\, \hat\pi_{ijklt}^{ABCD\bar X}) / \hat\pi_t^X$ (similarly for $B$, $C$, $D$)
  • Converges to a solution of the ML equations; different initialisations may reach different local maxima. A latent class is deleted if its weight tends to zero.

• Necessary condition for identifiability (counting): The number of free cell probabilities must not be less than the number of free structural parameters: $$IJKL \geq (I+J+K+L-3)T$$ When violated, parameters are structurally unidentifiable.

• Local identifiability rank test: Form the $(IJKL-1) \times [(I+J+K+L-3)T-1]$ Jacobian matrix of $\pi_{ijkl}$ with respect to the basic parameter set. Parameters are locally identifiable iff this matrix has full column rank. Goodman applies this test at $\hat\pi$ (the MLE), diagnosing whether the estimated model is identified — a practical advance over prior literature, which only checked the population $\pi$.

• Degrees of freedom: Under a locally identifiable $T$-class structure, $X^2$ (likelihood ratio) is asymptotically $\chi^2$ with $df = IJKL - (I+J+K+L-3)T$.

• Restricted latent structures: Equality constraints across latent classes reduce parameter count and can restore identifiability. Three key collapsibility conditions (classes 1 and 2 identical on all $m$, $m-1$, or $m-2$ manifest variables) cause unidentifiability. For restricted structures the Jacobian has merged columns; local identifiability requires the modified matrix to have full column rank.

• Exploratory strategy: Fit the unrestricted $T$-class model. If unidentifiable, the expected cell frequencies $\hat\pi_{ijkl}$ are still uniquely determined and can guide the choice of restrictions. Impose restrictions to obtain an equivalent identifiable model yielding the same $\hat\pi_{ijkl}$.

• Application 1 — Stouffer-Toby role conflict ($n=216$, $2^4$ table): 2-class model $H_1$ fits excellently ($X^2=2.720$, $df=6$); Stouffer-Toby originally concluded 5 classes were needed. Modal latent class ($\hat\pi_2^X = 0.721$): intrinsically particularistic except on variable $A$.

• Application 2 — Coleman schoolboys ($n=3{,}398$, $2^4$ table): One latent variable: $X^2=249.502$, $df=6$ (very poor fit). Two latent variables (crowd membership $V$ + attitude $W$, a restricted 4-class model): $X^2=1.270$, $df=4$ (excellent fit). The dramatic improvement illustrates the value of multi-latent-variable restricted structures.

Concepts Introduced or Extended

• Latent Class Model
• EM Algorithm

Entities Mentioned

• Leo A. Goodman

Quotes

"For each of the models considered here, a relatively simple method is presented for calculating the maximum likelihood estimate of the frequencies in the $m$-way contingency table expected under the model, and for determining whether the parameters in the estimated model are identifiable." (p. 215)

My Take

A foundational paper for latent class analysis that anticipates the EM algorithm three years before Dempster-Laird-Rubin (1977). The iterative procedure is derived directly from the ML equations rather than from an abstract incomplete-data framework, but the mathematics is identical. The identifiability rank test applied at the MLE is a practical innovation: it distinguishes models that are theoretically identified but empirically unidentified (flat ridges in the likelihood). The most influential methodological insight is that fitting unidentifiable models is productive — $\hat\pi_{ijkl}$ is uniquely determined regardless of identifiability, and its structure reveals the restrictions needed. The two applications are well chosen: one where parsimony wins (2 classes beat Stouffer-Toby's 5), one where structural complexity is genuinely needed (2 latent dimensions beat 1). The restricted-structure taxonomy in §§4–5 anticipates modern confirmatory latent class analysis and connects directly to the collapsibility concerns in Albert-Dodd (2004): near-identical $\hat\pi_{ijkl}$ from structurally distinct models is precisely why diagnostic accuracy estimates diverge.