Ann Arbor ASA Chapter
2022-03-22

Proportional Odds Model

Univariate Proportional Odds Model

  • \(\Pr(Y \geq y | X) = \mathrm{expit}(\alpha_{y} + X\beta)\)
  • \(Y\)-transformation invariant, does not use \(Y\) spacings
  • Handles arbitrarily heavy ties or continuous \(Y\), bi-modality, …
  • Direct competitor of the linear model

PO Model, continued

  • Wilcoxon-Mann-Whitney test: concordance probability \(c \approx \frac{OR^{0.66}}{1 + OR^{0.66}}\) whether or not PO holds
  • Kruskal-Wallis test: same; \(\beta\) for \(X=j\) vs. \(X=i\) reflects \(c\) for \(X\) vs. \(Y\) computed on observations with \(X \in \{i, j\}\)
  • See fharrell.com/post/wpo

Extensions to Allow Non-PO

  • Partial PO model
  • Constrained partial PO model
    • analogous to unequal variances in linear models
  • Peterson & Harrell, 1990

Longitudinal Models

Full Likelihood Extensions

  • Random intercepts/slopes
    • assumed correlation structures may be unrealistic
    • can’t have absorbing states
  • Markov models
    • most flexible, fastest, easiest to program
    • require unconditioning on previous \(Y\) to get marginal distributions
  • Other marginal models (Lee & Daniels, Schildcrout); direct \(\Pr(\)state occupancy\()\)

First-Order Discrete Time Markov Proportional Odds Model

  • Current state depends only on covariates, previous state, gap
  • Let measurement times be \(t_{1}, t_{2}, \dots, t_{m}\), and the measurement for a patient at time \(t\) be denoted \(Y(t)\) \[\Pr(Y(t_{i}) \geq y | X, Y(t_{i-1})) =\] \[\mathrm{expit}(\alpha_{y} + X\beta + g(Y(t_{i-1}), t_{i}, t_{i} - t_{i-1}))\]

Examples of Parametric Modeling Previous State in \(g\):

  • Linear in numeric codes for \(Y\)
  • Single binary indicator for a specific state such as the lowest or highest one
  • Discontinuous bi-linear relationship where there is a slope for in-hospital outcome severity, a separate slope for outpatient outcome severity, and an intercept jump at the transition from inpatient to outpatient (or vice versa).

Most Important Effects to Include

  • Previous state
  • Flexible function of time \(t\) since randomization
    • No time effect \(\sim\) constant hazard rate
  • Non-proportional odds effects for \(t\)
    • Mix of events can change over time, e.g., early ventilator use, late death
  • \(t \times\) previous state interaction
    • Example: Hospitalized patients more stable over time = increasing effect of previous state

Effects to Possibly Include, continued

  • Flexible function of gap times (if gap times and absolute time are virtually collinear, one of these may be omitted)
  • Interaction between previous state and gap time if gaps are very non-constant
  • Interaction between time and treatment if treatment effect is delayed, etc.

Unifying Approach

Markov PO Model: A Unified Approach

  • Time to terminating event
    • transition probability = discrete hazard rate
    • OR \(\approx\) HR when time intervals small
    • easily handles time-dependent covariates, left-truncation
  • Recurrent binary events
  • Recurrent binary events + a terminal event

Unified Approach, continued

  • Competing risks
    • death explicitly handled as a bad outcome
    • easier to interpret than competing risk models
  • Serial current status data
    • events of different severities
    • no need to judge whether an early heart attack is worse than a late death
  • Missing data and interval-censored \(Y\)

Unified Approach, continued

  • Standard longitudinal continuous \(Y\)
  • Longitudinal continuous or ordinal \(Y\) interrupted by clinical events
  • Easily handles multiple absorbing states

Examples of Longitudinal Ordinal Outcomes

  • 0=alive 1=dead
    • censored at 3w: 000
    • death at 2w: 01
  • 0=at home 1=hospitalized 2=MI 3=dead
    • hospitalized at 3w, rehosp at 7w, MI at 8w & stays in hosp, f/u ends at 10w: 0010001211

Examples, continued

  • 0-6 QOL excellent–poor, 7=MI 8=stroke 9=dead
    • QOL varies, not assessed in 3w but pt event free, stroke at 8w, death 9w: 12[0-6]334589
    • MI status unknown at 7w: 12[0-6]334[5,7]89
  • Can make first 200 levels be a continuous response variable and the remaining values represent clinical event overrides

Applications

Quality of Fit in a Critical Illness RCT

  • VIOLET trial from the PETAL network, NHLBI
  • Vitamin D in critically ill adults
  • Daily assessments for 28d on 4-level ordinal \(Y\): home, hospitalized, vent/ARDS, dead
  • See here for details and code

Second-Order Fit

  • Add random effects: negligible (indicates conditional independence)
  • Variogram

From Transition Probabilities to State Occupancy Probabilities

Unconditioning on Previous States

  • For equal time spacing:
    \(\Pr(Y(t)=y | X) =\)
    \(\sum_{j=1}^{k}\Pr(Y(t)=y | X, Y(t-1) = j) \times\)
    \(\Pr(Y(t-1) = j | X)\)
  • Use this recursively
  • Yields a semiparametric unconditional (except for \(X\)) distribution of \(Y\) at each \(t\) (SOPs)
  • soprobMarkovOrd* functions in the R Hmisc package make this easy for frequentist and Bayesian models

Estimands

  • Transition odds ratios (original parameters)
  • Prior state and covariate-specific transition probabilities
  • Covariate-specific SOPs
    • \(\Pr(\)stroke in week 4 or death in or before week 4\()\); \(\Pr(\)stroke and alive\()\)
  • Time in state \(Y=y\) (like RMST)
  • Time in states \(Y \geq y\) (e.g., mean time unwell)
  • Differences in mean time in state between treatments

Computing

Special Advantage of Bayesian Models and MCMC Posterior Sampling

  • SOPs involve complex derived parameters for which frequentist CLs are very hard to derive
  • Bayesian posterior distribution and uncertainty intervals derived from it are trivial to compute
  • Example: 4,000 posterior draws from transition model’s basic parameters; compute 4,000 values of each derived parameter (SOP; mean time in state, etc.)

Software

  • Frequentist: R VGAM package
  • Bayesian: R rmsb package

More Information

More Information, continued