CFE-CMStatistics 2021

2021-12-18

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

- 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.Â \(X=i\) reflects \(c\) for \(X\) vs.Â \(Y\) computed on observations with \(X \in \{i, j\}\)
- See fharrell.com/post/wpo

- Partial PO model
- Constrained partial PO model
- analogous to unequal variances in linear models

- Peterson & Harrell, 1990

- Random intercepts/slopes
- assumed correlation structures may be unrealistic
- canâ€™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\()\)

- 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 \(Y(t)\) \[\Pr(Y(t_{i}) \geq y | X, Y(t_{i-1})) =\] \[\mathrm{expit}(\alpha_{y} + X\beta + g(Y(t_{i-1}), t_{i}, t_{i} - t_{i-1}))\]

- 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*).

- 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

- 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.

- 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

- 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 excellentâ€“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

- 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

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

- 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
- 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

- 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 modelâ€™s basic parameters; compute 4,000 values of each derived parameter (SOP; mean time in state, etc.)

- Frequentist: R
`VGAM`

package - Bayesian: R
`rmsb`

package

- COVID-19 statistical resources: hbiostat.org/proj/covid19 including detailed analyses of the VIOLET, ORCHID, and ACTT-1 studies plus an examination of the handling of irregular measurement times in a Markov model
- Markov modeling references: hbiostat.org/bib/markov.html
- Longitudinal ordinal analysis references: hbiostat.org/bib/ordSerial.html
- General references on longitudinal data analysis: hbiostat.org/bib/serial.html

- Introduction to the proportional odds model: hbiostat.org/bbr/md chapter 7
- More about the proportional odds model: hbiostat.org/rms
- Attributes of good outcome measures