Sequential Ordinal Model

Definition

The sequential ordinal model generates an ordinal response $Y_i \in \{1,\ldots,J\}$ via a sequence of $J-1$ binary decisions: subject $i$ "exits" at level $j$ (i.e., $Y_i=j$ ) with conditional probability $\Phi(\gamma_j - \mathbf{x}_i'\boldsymbol{\delta})$ , given that it has "survived" through levels $1,\ldots,j-1$ . This discrete-time hazard interpretation makes the sequential model the natural Bayesian framework for ordinal outcomes arising from survival or duration processes, and is equivalent to the continuation-ratio logit model of Agresti (1990) and Cox (1972).

Key Ideas

Stage-by-stage binary decisions: unlike the cumulative ordinal model (Albert-Chib 1993b), which posits a single latent variable crossing $J-1$ ordered cutpoints, the sequential model factorizes into $J-1$ independent binary probits, one per level.
Unordered cutpoints: $\gamma_1,\ldots,\gamma_{J-1}$ are not constrained to be increasing; they are absorbed into an unrestricted parameter vector $\boldsymbol{\beta}=(\gamma_1,\ldots,\gamma_{J-1},\boldsymbol{\delta}')'$ , simplifying posterior sampling.
Right censoring: handled natively — a right-censored observation at level $j$ generates only the "passed through" $\mathcal{TN}_{(0,\infty)}$ draws for levels $1,\ldots,j-1$ , with no exit draw.
Extensions: (i) interaction model with category-specific covariate effects $\boldsymbol{\delta}_j$ ; (ii) polynomial baseline $\gamma_j=\phi_0+\phi_1 j+\phi_2 j^2$ (3 parameters vs. $J-1$ ).
Equivalence: identical to the continuation-ratio model (Agresti 1990; Cox 1972; Ten Have-Uttal 1994) and to a discrete-time proportional hazard model (Kalbfleisch-Prentice 1980; Fahrmeir-Tutz 1994).

How It Works

Data Augmentation and Gibbs Sampler (Albert-Chib 2001, Algorithm 1)

For subject $i$ with outcome $Y_i=j$ (uncensored), introduce $j$ latent variables $z_{i1},\ldots,z_{ij}$ via $z_{ik}=w_{ik}-\gamma_k$ , $w_{ik}\sim\mathcal{N}(\mathbf{x}_{ik}'\boldsymbol{\beta},1)$ . The observation rule is: $Y_i = j \iff z_{i1}>0,\ldots,z_{i,j-1}>0,\;z_{ij}\leq 0$ $Y_i = J \iff z_{i1}>0,\ldots,z_{i,J-1}>0$

Two-block Gibbs sampler:

Block 1 — $\mathbf{z}|\mathbf{y},\boldsymbol{\beta}$ : Draw each $z_{ik}$ as an independent truncated normal:

$z_{ik}\sim\mathcal{TN}_{(0,\infty)}(\mathbf{x}_{ik}'\boldsymbol{\beta},1)$ for $k=1,\ldots,j-1$ ("passed through" levels)
$z_{ij}\sim\mathcal{TN}_{(-\infty,0]}(\mathbf{x}_{ij}'\boldsymbol{\beta},1)$ at the exit level
For $Y_i=J$ : all $J-1$ draws from $\mathcal{TN}_{(0,\infty)}$

Block 2 — $\boldsymbol{\beta}|\mathbf{y},\mathbf{z}$ : Stack all $z_{ik}$ into a single response vector and the corresponding design rows into $\mathbf{X}$ . The full conditional is: $\boldsymbol{\beta}|\mathbf{y},\mathbf{z} \sim \mathcal{N}_k\!\left(\hat{\boldsymbol{\beta}},\;(\mathbf{B}_0^{-1}+\mathbf{X}'\mathbf{X})^{-1}\right)$

Hierarchical Sequential Model (Albert-Chib 1998, §3.2)

Bridges the free-cutpoint model ( $\mathcal{M}_2$ ) and the quadratic-polynomial model ( $\mathcal{M}_4$ ) by placing a hierarchical prior on the cutpoints: $\gamma_j = \phi_0 + \phi_1 j + \phi_2 j^2 + \varepsilon_j, \quad \varepsilon_j \overset{iid}{\sim} \mathcal{N}(0, \tau^2)$ with $\tau^2 \sim \mathrm{IG}(c, d)$ . As $\tau^2 \to 0$ , the model collapses to $\mathcal{M}_4$ (quadratic restriction); as $\tau^2 \to \infty$ , to $\mathcal{M}_2$ (free cutpoints). The four-block Gibbs sampler (Algorithm 2 of the 1998 manuscript) iterates over $(z, \beta, \beta_0, b)$ with the standard conjugate updates for the hierarchical Normal–Normal structure:

$\hat{\beta}_0 = (A'B(b)A)^{-1}(A'B(b)\beta), \qquad \hat{\beta} = (B(b) + \textstyle\sum X_i'X_i)^{-1}(B(b)A\beta_0 + \textstyle\sum X_i'z_i)$

Hospital stay application: hierarchical cutpoint posterior means fall between $\mathcal{M}_2$ and $\mathcal{M}_4$ values, with $\tau^2$ posterior concentrating near values intermediate between the two extremes, confirming data-adaptive shrinkage toward the quadratic curve.

Marginal Likelihood via Chib (1995)

The Chib (1995) identity evaluates at a high-density point $\boldsymbol{\beta}^*$ : $\ln m(\mathbf{y}) = \ln f(\mathbf{y}|\boldsymbol{\beta}^*) + \ln p(\boldsymbol{\beta}^*) - \ln p(\boldsymbol{\beta}^*|\mathbf{y})$

The posterior ordinate $p(\boldsymbol{\beta}^*|\mathbf{y})$ is Rao-Blackwellised: since $\boldsymbol{\beta}|\mathbf{z}$ is Normal with known parameters, average $p(\boldsymbol{\beta}^*|\mathbf{z}^{(g)})$ over the $G$ Gibbs draws of $\mathbf{z}$ .

A training-sample prior is used: fit the model on an initial $n_0=200$ observations with a flat prior; the resulting posterior $(\hat{\boldsymbol{\beta}}_0, \hat{\mathbf{B}}_0)$ becomes the prior $(\boldsymbol{\beta}_0, \mathbf{B}_0)$ for the full-data analysis.

Hospital Length-of-Stay Application ( $N=1000$ , $J=12$ )

Model	Description	$\ln m$
$\mathcal{M}_1$	Cumulative probit	−2146.7
$\mathcal{M}_2$	Sequential basic	−2098.6
$\mathcal{M}_3$	Sequential + COMORB interaction	−2117.0
$\mathcal{M}_4$	Sequential + quadratic baseline	−2094.0
$\mathcal{M}_5$	Weibull	−2191.2
$\mathcal{M}_6$	Log-logistic	−2223.2
$\mathcal{M}_4^*$	Best reduced (5 covariates)	−2092.1

Why It Matters

Provides a natural Bayesian framework for discrete-time survival analysis: the sequential structure matches the data-generating process when ordinal levels represent ordered duration categories.
Frees cutpoints from the ordering constraint of the cumulative model, simplifying the Gibbs sampler.
The data-augmentation + Chib (1995) machinery transfers from binary probit with almost no modification, demonstrating the composability of Bayesian latent variable methods.
Handles right censoring organically within the same sampler, with no additional algorithmic machinery.

Open Questions

Efficiency degrades when $J$ is large: the number of augmented latent variables grows linearly with $J$ , inflating the stacked design matrix.
The basic model assumes stage-invariant covariate effects; the interaction extension multiplies parameters as $J$ grows.
Prior sensitivity of marginal likelihood estimates to the training-sample prior choice ( $n_0$ ).