- Do you wonder why we have so many special cases in statistics that require seemingly different methods?
- time to first event
- recurrent events
- recurrent events with an absorbing state
- cure models (two absorbing states)
- competing risks

ICSA 2024, 2024-06-17

- Do you wonder why we have so many special cases in statistics that require seemingly different methods?
- time to first event
- recurrent events
- recurrent events with an absorbing state
- cure models (two absorbing states)
- competing risks

- More special cases
- Wilcoxon, Kruskal-Wallis, logrank tests
- zero inflation adaptations of Poisson and negative binomial models
- longitudinal data analysis

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

- Random intercepts
- massive lack of fit for correlation structure
- implies there is interest in individual pt-level outcomes vs. group level (treatment level)
- absorbing states destroy the correlation pattern
- typically assumes that > 6 observations per patient do not increase power

- massive lack of fit for correlation structure

- Random intercepts and slopes
- more flexible correlation structure but still may not fit
- too many parameters to estimate

- can’t have absorbing states

- more flexible correlation structure but still may not fit
- Markov models
- most flexible, fastest, easiest to program
- trivial to implement with ML (until you want state occupancy probabilities)

- most flexible, fastest, easiest to program

- Markov models apply to all \(Y\)
- binary
- unordered categorical
- ordinal categorical
- ordinal continuous
- ordinal mixed continuous and categorical
- continuous
- left, right, and interval censored
- require unconditioning on previous \(Y\) to get marginal distributions

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

- 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

- 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 \(Y\)

- Standard longitudinal continuous \(Y\)
- Longitudinal continuous or ordinal \(Y\) interrupted by clinical events
- Easily handles multiple absorbing states
- Serial correlation: condition on previous outcome
- Random intercepts (compound symmetry correlation): condition on average of all previous outcomes

- 0=alive 1=dead
- censored at 3w:
**000** - death at 2w:
**01**

- censored at 3w:
- 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**

- hospitalized at 3w, rehosp at 7w, MI at 8w & stays in hosp, f/u ends at 10w:

- 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**

- QOL varies, not assessed in 3w but pt event free, stroke at 8w, death 9w:
- Can make first 200 levels be a continuous response variable and the remaining values represent clinical event overrides

- 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

- Transition odds ratios (original parameters)
- Prior state and covariate-specific transition probabilities
- State occupancy probabilities (SOPs; marginalize over time)
- 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
- Contiuous \(Y\) example: mean time with SBP < 130mmHg
- Nice way to handle treatment \(\times\) time interaction
- No categorization of SBP

Bayesian transition models for ordinal longitudinal outcomes

MD Rohde, B French, TG Stewart, FE Harrell Jr.

*Statistics in Medicine* 2024-06-09

DOI: 10.1002/sim.10133

See also fharrell.com/talk/cmstat