Ann Arbor ASA Chapter 2022-03-22

FDA CDER Office of Biostatistics Expanded 2023-06-21, 2023-07-11

- How many have asked sponsors submitting \(t\)-tests and linear regression analysis to verify normality and parallelism (on \(\Phi^{-1}(F(y))\) scale) assumptions (no relationship of an \(X\) and \(\text{var}(Y|X)\))?
- How many have asked sponsors submitting linear mixed model analysis to provide a variogram to demonstrate agreement between the assumed and actual within-patient correlation patterns?

- Do you know of anyone who has worried about the proportional hazards assumption and recommended the sponsor provide a logrank test instead?
- Do you know of anyone who has worried about the proportional odds assumption and recommended the sponsor provide a Wilcoxon test instead?

- 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
- competing risks
- Wilcoxon, Kruskal-Wallis, logrank tests
- zero inflation adaptations of Poisson and negative binomial models

- Have you stopped to consider
- whether random effects are the first line of defense against within-patient correlation
- whether the emphasis given by random effects to patient-level outcome estimation is an important goal for treatment comparisons?

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

- 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

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

- Peterson & Harrell, 1990

- 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

- Other marginal models (Lee & Daniels, Schildcrout); direct \(\Pr(\)state occupancy\()\)
- don’t explicitly handle absorbing states

- 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 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

- 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 model’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

- Univariate proportional odds model resources: fharrell.com/post/rpo
- Attributes of good outcome measures
- Full case study for discrete ordinal \(Y\) with code: hbiostat.org/rmsc/markov.html
- Full case study for continuous \(Y\) analyzed ordinally with code: hbiostat.org/rmsc/long.html#sec-long-bayes-markov

- See hbiostat.org/rmsc/long.html#modeling-within-subject-dependence for a table comparing repeated measures ANOVA, GEE, mixed effects, GLS, Markov, LOCF, and summary statistic approaches to longitudinal data
- Natural tendency for reviewers to be more accepting of familiar methods
- Many statisticians wrongly believe CLT protects finite sample size analyses
- sample size for CLT to kick in may easily exceed 50,000
- CLT applies to \(\alpha\), not \(\beta\)

- Gaussian methods have far more assumptions that semiparametric models such as PO
- Mixed effects model for longitudinal data
- \(Y|X\) is Gaussian
- constant variance (exact analogy of PO assumption)
- shape of \(X\)-relationships with \(E(Y|X)\)
- interactions among \(X\) and with treatment

- Mixed effects model
- within-patient correlation of \(Y\) at times \(a\) and \(b\) (\(a \neq b\)) is constant regardless of \(|a - b|\)
- unrealistic for long-duration follow-up
- failure to properly model how strongly repeated measurements are connected to each other \(\rightarrow\) wrong SE(treatment effect), \(\alpha\)

- Data from earlier trial or OS
- check that assumed model structure can reproduce
- first-order properties
- second-order properties (e.g., variogram)
- raw data

- check that assumed model structure can reproduce
- Within-trial assumption checking

- Secondary analyses
- Add random effects and estimate their variance
- Compare correlation vs. lag induced by model to that in raw data
- Add lag 2 (second order Markov)
- Impact of PO assumption for treatment
- Usual linearity and interaction assessments

- What assumptions are needed for longitudinal ordinal first-order Markov models?
- usual \(X\)-transformation and interaction assumptions
- distribution of \(Y\) given all previous \(Y\) is adequately captured by the previous \(Y\)
- distribution of \(Y\) given previous \(Y\) and \(X_1\) is a simple shift of the shape given previous \(Y\) and \(X_2\) (PO assumption; analogy of equal variance assumption)

- Note: distribution of \(Y\) given previous \(Y\) and \(X\) can be
**any**shape

- Are these assumptions plausible?
- the data format used respects typical “current status” data generating processes
- excellent fits to previous datasets in similar disease/treatment situations
- direct modeling of correlation structure as covariates adds great flexibility and leads to simple diagnostics

- Compare with huge assumptions made in simple analyses
- MACE: patient dying a year after nonfatal MI \(\rightarrow\) death completely ignored
- Nonfatal MI counted equally bad as death
- Recurrent hospitalizations are ignored
- PH for each endpoint may lead to non-PH for time to first endpoint

- Highly visible gross violations of assumptions about
**data** - Possible violation of assumptions about the data model

- Does the output of these models produce clinically meaningful results, and how do clinical colleagues interpret such results?
- extensive experience in using these models with COVID-19 DSMBs
- daily stacked bar charts by treatment with covariate-adjusted outcome tendencies accepted immediately
- similar for posterior densities of covariate-specific differences between treatments in mean times in various states (esp. time with no or minimal symptoms)

- Corresponding point estimates, e.g., “best guess of reduction in days unwell by treatment B is 0.5” (also include uncertainty interval)

- How to capture in a label
- Physicians and patients understand time more than risks
- Example for long-term CV study: treatment B provided an additional 0.5 years free of hospitalization, MI, or death
- As with any absolute quantity, the estimate will depend on baseline severity etc.
- \(\Pr(OR < 1) = 0.97\): evidence for benefit; posterior probability is independent of baseline severity unless treatment interacts with a covariate

- How to implement in a sequential design?
- Frequentist: simulate \(\alpha\) based on look
**intentions** - Bayesian: trivial, since current posterior probability is self-contained

- Frequentist: simulate \(\alpha\) based on look

- A comparison of partitioned survival analysis and state transition multi-state modelling approaches using a case study in oncology
- Nonparametric estimation of the survival function for ordered multivariate failure time data: A comparative study
- Direct modeling of regression effects for transition probabilities in the progressive illness–death model
- Estimation in the progressive illness-death model: A nonexhaustive review

- Duration of and time to response in oncology clinical trials from the perspective of the estimand framework
- Statistical analysis of complex survival data: new contributions in statistical inference, software development and biomedical applications
- Estimating lengths-of-stay of hospitalized COVID-19 patients using a non-parametric model
- Tutorial in biostatistics: Competing risks and multi-state models