Albert-Chib (1998) Sequential Ordinal Modeling with Applications to Survival Data

Summary

This is the May 1998 working paper manuscript that eventually appeared as Albert and Chib (2001) in Biometrics. Same title, same hospital length-of-stay application ( $N=1{,}000$ , $J=12$ ), but substantially longer (28 pages vs. 8) and containing material cut from the published version. Key additions over the published paper are: (i) a hierarchical sequential model (Algorithm 2) that shrinks the free cutpoints toward a quadratic polynomial surface; (ii) a 6-model comparison table with Weibull and log-logistic continuous-time survival models as additional alternatives; and (iii) full derivations of the cumulative ordinal Markov chain Monte Carlo (MCMC) algorithm (Algorithm 3) with the log-spacing reparameterization.

Key Claims

Basic sequential model (Algorithm 1): Two-block Gibbs sampler identical to Albert-Chib (2001); the re-parameterized design vector $\mathbf{x}'_{ij} = (0,\ldots,-1,0,\ldots,\mathbf{x}'_i)$ absorbs cutpoints into $\boldsymbol{\beta}=(\gamma_1,\ldots,\gamma_{J-1},\boldsymbol{\delta}')'$ ; censored observations contribute only the "passed through" truncated-normal draws.
Interaction model: Category-specific covariates $\mathbf{x}_{i(1)}$ may have time-varying effects $\boldsymbol{\delta}_j$ ; their coefficients lie on a low-order polynomial $\delta_{jl} = \eta_{l0} + \eta_{l1}j + \eta_{l2}j^2$ via the Kronecker construction $A_j = I_m \otimes (1\; j\; j^2)$ ; the full model fits as Algorithm 1 with an expanded design matrix.
Quadratic polynomial baseline ( $M_4$ ): Restricts $\gamma_j = \phi_0 + \phi_1 j + \phi_2 j^2$ ; parameters $\boldsymbol{\phi}=(\phi_0,\phi_1,\phi_2)$ replace the $J-1$ free cutpoints; still fits via Algorithm 1 with modified $X_i$ .
Hierarchical sequential model (Algorithm 2, not in published paper): Cutpoints follow $\gamma_j = \phi_0 + \phi_1 j + \phi_2 j^2 + \varepsilon_j$ with $\varepsilon_j \overset{iid}{\sim} \mathcal{N}(0, \tau^2)$ ; $\tau^2 \sim \mathrm{IG}(c,d)$ (Inverse-Gamma); as $\tau^2 \to 0$ collapses to $M_4$ , as $\tau^2 \to \infty$ to basic $M_2$ ; four-block Gibbs ( $z$ , $\beta$ , $\beta_0$ , $b$ ); hierarchical posterior cutpoints interpolate between $M_2$ and $M_4$ values.
Cumulative ordinal model (Algorithm 3): Applies the log-spacing reparameterization $\alpha_1 = \log\gamma^c_1$ , $\alpha_j = \log(\gamma^c_j - \gamma^c_{j-1})$ (Eq. 10) from Albert-Chib (1997b) to lift ordered cut-points to an unrestricted vector; multivariate- $t$ Metropolis-Hastings (MH) proposal centred at mode $\hat{\alpha}$ with dispersion $\tau^2 V$ .
Weibull and log-logistic models (§4.2): $h(t) = \alpha\lambda(\lambda t)^{\alpha-1}$ , $S(t) = \exp(-(\lambda t)^\alpha)$ for Weibull; $h(t) = \alpha\lambda(\lambda t)^{\alpha-1}/[1+(\lambda t)^\alpha]$ , $S(t) = 1/[1+(\lambda t)^\alpha]$ for log-logistic; $\lambda_i = \exp(-x_i'\delta)$ ; posterior sampled via MH (Chib-Greenberg 1995).
6-model comparison (Table 2, training sample $n_0=200$ , evaluation $n=800$ ): $M_1$ cumulative ( $\ln m = -2172.8$ ), $M_2$ basic sequential ( $-2127.2$ ), $M_3$ sequential+comorbidity index (COMORB) interaction ( $-2145.0$ ), $M_4$ sequential+quadratic baseline ( $-2121.2$ ), $M_5$ Weibull ( $-2230.4$ ), $M_6$ log-logistic ( $-2255.6$ ); Bayes factor $\mathrm{BF}_{21} = \exp(45.6) = 6.4 \times 10^{19}$ ; best model $M_4$ reduced — race (RACE), private insurance (PRIVINS), COMORB, coronary artery bypass graft (CABG), percutaneous transluminal coronary angioplasty (PTCA) — $\ln m = -2117.4$ .
Note on numbers: The manuscript's log marginal likelihoods differ from the published Albert-Chib (2001) values (e.g., $M_4$ full: $-2121.2$ here vs. $-2094.0$ published) because the training/evaluation split differs between manuscript and published versions.
Prior elicitation: Training sample of size $n_0 = 200$ used to estimate hyperparameters $(\beta_0, B_0)$ of the multivariate normal prior; remainder ( $n=800$ for evaluation) used in model comparison; same training-sample approach as Albert-Chib (1997b).

Concepts Introduced or Extended

Entities Mentioned

Quotes

"The sequential model is useful in the analysis of discrete-time survival data." (p. 2)

"Although this was not discussed in the paper, [these methods] produced MCMC output that mixed remarkably well." (p. 26)

My Take

The main substantive addition over the published 2001 paper is the hierarchical sequential model of Section 3.2, which elegantly bridges the free-cutpoint model ( $M_2$ ) and the quadratic-polynomial model ( $M_4$ ) via a single variance hyperparameter $\tau^2$ . This is genuinely useful when neither extreme seems right: the hierarchical shrinkage produces cutpoints that are smoother than $M_2$ but not forced onto a rigid polynomial. The hierarchical posterior means in Table 4 confirm the interpolation is working as designed. The 6-model comparison table in the manuscript also gives clearer context than the published paper — seeing all six log marginal likelihoods side by side makes the hierarchy $M_4 > M_2 \gg M_1 > M_3 \gg M_5 > M_6$ immediately legible. The differences in marginal likelihood values between the manuscript and published paper (due to different training/evaluation splits) are worth keeping in mind: the manuscript uses $n_0=200$ out of 1,000, while the published paper apparently uses a different split.