Bandeen-Roche-Miglioretti-Zeger-Rathouz (1997) Latent Variable Regression for Multiple Discrete Outcomes

Summary

The paper develops the concomitant latent class model (a.k.a. latent class regression, LCR) for settings where a construct — here physical disability in gerontology — is measured by multiple discrete indicators rather than a single number, and the goal is to relate that latent construct to covariates. It gives practical theory for reducing and identifying such models, standard-error fitting that parallels ordinary latent class analysis, and diagnostics, thereby supplementing the Dayton-Macready (1988) proposal. The key structural insight is that the LCR model reduces to a standard finite mixture, so its fitting and identifiability are nearly as tractable as plain latent class analysis.

Key Claims

• Measurement model = standard latent class. For $M$ dichotomous indicators and $J$ classes with prevalences $\pi_j$, local independence gives $\Pr[Y_i=y]=\sum_{j=1}^J \pi_j\prod_{m=1}^M \eta_{mj}^{y_m}(1-\eta_{mj})^{1-y_m}$ — the categorical analogue of factor analysis (the $\eta_{mj}$ are class-conditional item probabilities).
• The LCR / concomitant extension. Let class membership depend on covariates: $\pi_j(x_i)=\Pr[S_i=j\mid x_i]$ is a latent polytomous (multinomial) regression of the unobserved class $S_i$ on $x_i$, typically $\pi_j(x_i)=\pi_j(x_i'\delta)$ in a GLM/link-function form — exactly the Dayton-Macready (1988) model.
• Nondifferential measurement condition. $\Pr[Y_{im}=y_m\mid S_i,x_i]=\Pr[Y_{im}=y_m\mid S_i]$: the indicators depend on the covariates only through the latent class, i.e. $Y_i \perp x_i \mid S_i$. This is what makes $S_i$ a legitimate summary construct — covariate effects on outcomes are entirely mediated by latent status.
• Posterior class membership is available as $\theta_{ij}=\Pr[S_i=j\mid Y_i,x_i]$, combining the covariate-driven prior $\pi_j(x_i)$ with the measurement likelihood.
• Reduction to a mixture ⇒ tractable fitting and identifiability. Marginalizing the latent regression shows LCR is a standard mixture model, so inferring the number of classes, EM parameter estimation, standard errors, and identifiability analysis carry over from ordinary latent class analysis — the covariates do not fundamentally complicate matters.
• Diagnostics + application. Proposes diagnostic strategies for the fitted model and illustrates on physical-disability data (difficulty with activities of daily living) from an ongoing gerontologic study, where the aim is the association of disability with covariates.

Concepts Introduced or Extended

• Latent Class Model

Entities Mentioned

• Karen Bandeen-Roche
• Diana L. Miglioretti
• Scott L. Zeger
• Paul J. Rathouz

Quotes

"In this article we study the concomitant latent class model for analyzing such multivariate categorical outcome data. We develop practical theory for reducing and identifying such models."

"This condition [nondifferential measurement] is not merely convenient, but together with [local independence] also defines the sense in which the $S_i$ serve as summary constructs — namely, that $Y_i$ and $x_i$ are independent given $S_i$."

My Take

The paper's real service is demystification: a "latent variable regression for discrete outcomes" sounds like it needs its own machinery, but Bandeen-Roche et al. show it is just a finite mixture whose mixing weights are a multinomial regression on covariates, so everything you know about latent class analysis — EM, class-count selection, identifiability caveats — transfers. The nondifferential measurement assumption is the conceptually load-bearing part and worth flagging for the wiki: it is exactly the condition that lets you interpret covariate effects as operating "through" the latent construct, and it is also the assumption most likely to fail (direct effects of a covariate on an indicator, net of class, break it — the same "local dependence" concern the bivariate-residual diagnostics in latent class work target). It complements the model-typology view of Magidson-Vermunt and the identifiability cautions of Albert-Dodd (2004): this is the regression face of latent class analysis, giving the concept its covariate-driven, epidemiology-facing form.