Ann Arbor ASA Chapter  2022-03-22
FDA CDER Office of Biostatistics  Expanded 2023-06-21, 2023-07-11

Background Questions

Background Questions

  • 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?

Background Questions, continued

  • 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?

Background Questions, continued

  • 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

Background Questions, continued

  • 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?

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

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
    • 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

Full Likelihood Extensions, continued

  • Random intercepts and slopes
    • more flexible correlation structure but still may not fit
      • too many parameters to estimate
    • can’t have absorbing states
  • Markov models
    • most flexible, fastest, easiest to program
      • trivial to implement with ML (until you want state occupancy probabilities)

Full Likelihood Extensions, continued

  • 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

Full Likelihood Extensions, continued

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

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


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


  • 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


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.)


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

More Information

More Information, continued


Afterthoughts and Q&A

  • See 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

Afterthoughts, continued

  • 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

Afterthoughts, continued

  • 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), \(\alpha\)

Assumptions of Longitudinal Models: Strategies

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

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)

Q&A, continued

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

Q&A, continued

  • 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

Plausibility of Assumptions, continued

  • 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

Keep Things in Context

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

Q&A, continued

  • 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)

Q&A, continued

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

Q&A, ‘r cont’

  • 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

Q&A, continued

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

Additional References Provided by Alexei Ionan

References, continued

References, continued