22 Semiparametric Ordinal Longitudinal Models

Key reference: Rohde et al. (2024); 2024 Statistics in Medicine tutorial

22.1 Longitudinal Ordinal Models as Unifying Concepts

This material in this section is taken from hbiostat.org/talks/rcteff.html. See also hbiostat.org/proj/covid19/ordmarkov.html and hbiostat.org/endpoint and Ordinal state transition models as a unifying risk prediction framework.

22.1.1 General Outcome Attributes

Timing and severity of outcomes
Handle
- terminal events (death)
- non-terminal events (MI, stroke)
- recurrent events (hospitalization)
Break the ties; the more levels of Y the better: fharrell.com/post/ordinal-info
- Maximum power when there is only one patient at each level (continuous Y)

22.1.2 What is a Fundamental Outcome Assessment?

In a given week or day what is the severity of the worst thing that happened to the patient?
Expert clinician consensus of outcome ranks
Spacing of outcome categories irrelevant
Avoids defining additive weights for multiple events on same week
Events can be graded & can code common co-occurring events as worse event
Can translate an ordinal longitudinal model to obtain a variety of estimates
- time until a condition
- expected time in state
Bayesian partial proportional odds model can compute the probability that the treatment affects mortality differently than it affects nonfatal outcomes
Model also elegantly handles partial information: at each day/week the ordinal Y can be left, right, or interval censored when a range of the scale was not measured

22.1.3 Examples of Longitudinal Ordinal Outcomes

0=alive 1=dead
- censored at 3w: 000
- death at 2w: 01
- longitudinal binary logistic model OR \(\approx\) HR
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
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]89¹
- Can make first 200 levels be a continuous response variable and the remaining values represent clinical event overrides

¹ Better: treat the outcome as being in one of two non-contiguous values {5,7} instead of [5-7] but no software is currently available for this

22.1.4 Statistical Model

Proportional odds ordinal logistic model with covariate adjustment
Patient random effects (intercepts) handle intra-patient correlation
Better fitting: Markov model
- handles absorbing states, extremely high day-to-day correlations within subject
- faster, flexible, uses standard software
- state transition probabilities
- after fit translate to unconditional state occupancy probabilities
- use these to estimate expected time in a set of states (e.g., on ventilator or dead); restricted mean survival time without assuming PH
Extension of binary logistic model
Generalization of Wilcoxon-Mann-Whitney Two-Sample Test
No assumption about Y distribution for a given patient type
Does not use the numeric Y codes

22.2 Case Studies

Random effects model for continuous Y: Section 7.8.3
Markov model for continuous Y: Section 7.8.4
Multiple detailed case studies for discrete ordinal Y: hbiostat.org/proj/covid19.
- ORCHID: hydroxychloroquine for treatment of COVID-19 with patient assessment on select days
- VIOLET: vitamin D for serious respiratory illness with assessment on 28 consecutive days
  - Large power gain demonstrated over time to recovery or ordinal status at a given day
  - Loosely speaking serial assessments for each 5 day period had the same statistical information as a new patient assessed once
- ACTT-1: NIH-NIAID Remdesivir study for treatment of COVID-19 with daily assessment while in hospital, select days after that with interval censoring
  - Assesses time-varying effect of remdesivir
  - Handles death explicitly, unlike per-patient time to recovery
- Other: Bayesian and frequentist power simulation, exploration of unequal time gaps, etc.

22.3 Case Study For 4-Level Ordinal Longitudinal Outcome

VIOLET: randomized clinical trial of seriously ill adults in ICUs to study the effectiveness of vitamin D vs. placebo
Daily ordinal outcomes assessed for 28 days with very little missing data
Original paper: DOI:10.1056/NEJMoa1911124 focused on 1078 patients with confirmed baseline vitamin D deficiency
Focus on 1352 of the original 1360 randomized patients
Extensive re-analyses:
Fitted a frequentist first-order Markov partial proportional odds (PO) model to 1352 VIOLET patients using the R VGAM package to simulate 250,000 patient longitudinal records with daily assessments up to 28d: hbiostat.org/data/repo/simlongord.html
Simulation inserted an odds ratio of 0.75 for tx=1 : tx=0 (log OR = -0.288)
Case study uses the first 500 simulated patients
- 13203 records
- average of 26.4 records per patient out of a maximum of 28, due to deaths
- full 250,00 and 500-patient datasets available at hbiostat.org/data
4-level outcomes:
- patient at home
- in hospital or other health facility
- on ventilator or diagnosed with acute respiratory distress syndrome (ARDS)
- dead
Death is an absorbing state
- only possible previous states are the first 3
- at baseline no one was at home
- a patient who dies has Dead as the status on their final record, with no “deaths carried forward”
- later we will carry deaths forward just to be able to look at empirical state occupancy probabilities (SOPs) vs. model estimates
Frequentist modeling using the VGAM package allows us to use the unconstrained partial PO (PPO) model with regard to time, but does not allow us to compute uncertainty intervals for derived parameters (e.g., SOPs and mean time in states)
Can use the bootstrap to obtain approximate confidence limits (as below)
Bayesian analysis using the rmsb package provides exact uncertainty intervals for derived parameters but at present rmsb only implements the constrained PPO model when getting predicted values
PPO for time allows mix of outcomes to change over time (which occurred in the real data)
Model specification:
- For day \(t\) let \(Y(t)\) denote the ordinal outcome for a patient
  \(\Pr(Y \geq y | X, Y(t-1)) = \text{expit}(\alpha_{y} + X\beta + g(Y(t-1), t))\)
- \(g\) contains regression coefficients for the previous state \(Y(t-1)\) effect, the absolute time \(t\) effect, and any \(y\)-dependency on the \(t\) effect (non-PO for \(t\))
- Baseline covariates: age, SOFA score (a measure of organ function), treatment (tx)
- Time-dependent covariate: previous state (yprev, 3 levels)
- Time trend: linear spline with knot at day 2 (handles exception at day 1 when almost no one was sent home)
- Changing mix of outcomes over time
  - effect of time on transition ORs for different cutoffs of Y
  - 2 time components (one slope change) \(\times\) 3 Y cutoffs = 6 parameters related to day
Reverse coding of Y so that higher levels are worse

22.3.1 Descriptives

all state transitions from one day to the next
SOPs estimated by proportions (need to carry death forward)

Code

require(rms)
require(data.table)
require(VGAM)
getHdata(simlongord500)
d <- simlongord500
setDT(d, key='id')
d[, y     := factor(y,     levels=rev(levels(y    )))]
d[, yprev := factor(yprev, levels=rev(levels(yprev)))]
setnames(d, 'time', 'day')
# Show descriptive statistics for baseline data
describe(d[day == 1, .(yprev, age, sofa, tx)], 'Baseline Variables')

Baseline Variables Descriptives

Baseline Variables

4 Variables 500 Observations

yprev

n	missing	distinct
500	0	2

 Value      In Hospital/Facility            Vent/ARDS
 Frequency                   340                  160
 Proportion                 0.68                 0.32

age

n	missing	distinct	Info	Mean	pMedian	Gmd	.05	.10	.25	.50	.75	.90	.95
500	0	73	0.999	56.08	56.5	17.49	28.95	36.00	46.00	58.00	67.00	75.10	79.05

lowest : 19 20 21 22 23 , highest: 89 90 91 92 94

sofa

n	missing	distinct	Info	Mean	pMedian	Gmd	.05	.10	.25	.50	.75	.90	.95
500	0	18	0.992	5.274	5	3.901	0.00	1.00	3.00	5.00	7.25	10.00	11.00

 Value          0     1     2     3     4     5     6     7     8     9    10    11
 Frequency     38    38    41    51    59    53    58    37    24    32    32    17
 Proportion 0.076 0.076 0.082 0.102 0.118 0.106 0.116 0.074 0.048 0.064 0.064 0.034
                                               
 Value         12    13    14    15    17    18
 Frequency      8     4     4     2     1     1
 Proportion 0.016 0.008 0.008 0.004 0.002 0.002

For the frequency table, variable is rounded to the nearest 0

n	missing	distinct	Info	Sum	Mean
500	0	2	0.75	256	0.512

Code

# Check that death can only occur on the last day
d[, .(ddif=if(any(y == 'Dead')) min(day[y == 'Dead']) -
                                max(day) else NA_integer_),
          by=id][, table(ddif)]

ddif
 0 
51

Code

require(ggplot2)
propsTrans(y ~ day + id, data=d, maxsize=4, arrow='->') +
    theme(axis.text.x=element_text(angle=90, hjust=1))

Figure 22.1: Transition proportions from data simulated from VIOLET

Show state occupancy proportions by creating a data table with death carried forward.

Code

w <- d[day < 28 & y == 'Dead', ]
w[, if(.N > 1) stop('Error: more than one death record'), by=id]

Empty data.table (0 rows and 1 cols): id

Code

w <- w[, .(day = (day + 1) : 28, y = y, tx=tx), by=id]
u <- rbind(d, w, fill=TRUE)
setkey(u, id)
u[, Tx := paste0('tx=', tx)]
propsPO(y ~ day + Tx, data=u) +
  guides(fill=guide_legend(title='Status')) +
  theme(legend.position='bottom', axis.text.x=element_text(angle=90, hjust=1))

Figure 22.2: State occupancy proportions from simulated VIOLET data with death carried forward

22.3.2 Model Fitting

Fit the PPO first-order Markov model without assuming PO for the time effect
Also fit a model that has linear splines with more knots to add flexibility in how time and baseline covariates are transformed
Disregard the terrible statistical practice of using asterisks to denote “significant” results

Code

f <- vglm(ordered(y) ~ yprev + lsp(day, 2) + age + sofa + tx,
          cumulative(reverse=TRUE, parallel=FALSE ~ lsp(day, 2)), data=d)
summary(f)

Call:
vglm(formula = ordered(y) ~ yprev + lsp(day, 2) + age + sofa + 
    tx, family = cumulative(reverse = TRUE, parallel = FALSE ~ 
    lsp(day, 2)), data = d)

Coefficients: 
                            Estimate Std. Error z value Pr(>|z|)    
(Intercept):1              -5.029733   0.629002  -7.996 1.28e-15 ***
(Intercept):2             -13.192829   0.577954 -22.827  < 2e-16 ***
(Intercept):3             -19.579818   0.936205 -20.914  < 2e-16 ***
yprevIn Hospital/Facility   8.773118   0.265786  33.008  < 2e-16 ***
yprevVent/ARDS             15.138338   0.321944  47.022  < 2e-16 ***
lsp(day, 2)day:1           -1.431455   0.284257  -5.036 4.76e-07 ***
lsp(day, 2)day:2           -0.707384   0.257032  -2.752  0.00592 ** 
lsp(day, 2)day:3            0.137447   0.468675   0.293  0.76932    
lsp(day, 2)day':1           1.484425   0.285982   5.191 2.10e-07 ***
lsp(day, 2)day':2           0.746015   0.261124   2.857  0.00428 ** 
lsp(day, 2)day':3          -0.119694   0.475903  -0.252  0.80142    
age                         0.011307   0.002789   4.054 5.03e-05 ***
sofa                        0.064084   0.012596   5.088 3.63e-07 ***
tx                         -0.111407   0.084573  -1.317  0.18774    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Names of linear predictors: logitlink(P[Y>=2]), logitlink(P[Y>=3]), 
logitlink(P[Y>=4])

Residual deviance: 4509.04 on 38962 degrees of freedom

Log-likelihood: -2254.52 on 38962 degrees of freedom

Number of Fisher scoring iterations: 9 

Warning: Hauck-Donner effect detected in the following estimate(s):
'(Intercept):3'


Exponentiated coefficients:
yprevIn Hospital/Facility            yprevVent/ARDS          lsp(day, 2)day:1 
             6.458278e+03              3.754021e+06              2.389609e-01 
         lsp(day, 2)day:2          lsp(day, 2)day:3         lsp(day, 2)day':1 
             4.929322e-01              1.147340e+00              4.412427e+00 
        lsp(day, 2)day':2         lsp(day, 2)day':3                       age 
             2.108581e+00              8.871915e-01              1.011371e+00 
                     sofa                        tx 
             1.066182e+00              8.945744e-01

Code

g <- vglm(ordered(y) ~ yprev + lsp(day, c(2, 4, 8, 15)) +
            lsp(age, c(35, 60, 75)) + lsp(sofa, c(2, 6, 10)) + tx,
          cumulative(reverse=TRUE, parallel=FALSE ~ lsp(day, c(2, 4, 8, 15))),
          data=d)
summary(g)

Call:
vglm(formula = ordered(y) ~ yprev + lsp(day, c(2, 4, 8, 15)) + 
    lsp(age, c(35, 60, 75)) + lsp(sofa, c(2, 6, 10)) + tx, family = cumulative(reverse = TRUE, 
    parallel = FALSE ~ lsp(day, c(2, 4, 8, 15))), data = d)

Coefficients: 
                                    Estimate Std. Error z value Pr(>|z|)    
(Intercept):1                      -4.540695   0.886608  -5.121 3.03e-07 ***
(Intercept):2                     -12.871450   0.856614 -15.026  < 2e-16 ***
(Intercept):3                     -18.852185   1.181311 -15.959  < 2e-16 ***
yprevIn Hospital/Facility           8.768798   0.268604  32.646  < 2e-16 ***
yprevVent/ARDS                     15.119643   0.325089  46.509  < 2e-16 ***
lsp(day, c(2, 4, 8, 15))day:1      -1.553745   0.305276  -5.090 3.59e-07 ***
lsp(day, c(2, 4, 8, 15))day:2      -0.645513   0.300866  -2.146   0.0319 *  
lsp(day, c(2, 4, 8, 15))day:3      -0.218295   0.602222  -0.362   0.7170    
lsp(day, c(2, 4, 8, 15))day':1      1.722567   0.353662   4.871 1.11e-06 ***
lsp(day, c(2, 4, 8, 15))day':2      0.675690   0.396113   1.706   0.0880 .  
lsp(day, c(2, 4, 8, 15))day':3      0.587743   0.812592   0.723   0.4695    
lsp(day, c(2, 4, 8, 15))day'':1    -0.142458   0.140217  -1.016   0.3096    
lsp(day, c(2, 4, 8, 15))day'':2    -0.011895   0.207304  -0.057   0.9542    
lsp(day, c(2, 4, 8, 15))day'':3    -0.434170   0.385428  -1.126   0.2600    
lsp(day, c(2, 4, 8, 15))day''':1    0.024947   0.076121   0.328   0.7431    
lsp(day, c(2, 4, 8, 15))day''':2   -0.020957   0.121498  -0.172   0.8631    
lsp(day, c(2, 4, 8, 15))day''':3   -0.029000   0.228337  -0.127   0.8989    
lsp(day, c(2, 4, 8, 15))day'''':1   0.001783   0.052002   0.034   0.9726    
lsp(day, c(2, 4, 8, 15))day'''':2   0.092686   0.080835   1.147   0.2515    
lsp(day, c(2, 4, 8, 15))day'''':3   0.211000   0.168854   1.250   0.2114    
lsp(age, c(35, 60, 75))age         -0.003840   0.019177  -0.200   0.8413    
lsp(age, c(35, 60, 75))age'         0.019171   0.023483   0.816   0.4143    
lsp(age, c(35, 60, 75))age''       -0.015406   0.016060  -0.959   0.3374    
lsp(age, c(35, 60, 75))age'''       0.035960   0.026792   1.342   0.1795    
lsp(sofa, c(2, 6, 10))sofa          0.170889   0.095970   1.781   0.0750 .  
lsp(sofa, c(2, 6, 10))sofa'        -0.136896   0.122389  -1.119   0.2633    
lsp(sofa, c(2, 6, 10))sofa''        0.018298   0.069084   0.265   0.7911    
lsp(sofa, c(2, 6, 10))sofa'''       0.074134   0.089369   0.830   0.4068    
tx                                 -0.122940   0.085320  -1.441   0.1496    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Names of linear predictors: logitlink(P[Y>=2]), logitlink(P[Y>=3]), 
logitlink(P[Y>=4])

Residual deviance: 4497.307 on 38947 degrees of freedom

Log-likelihood: -2248.653 on 38947 degrees of freedom

Number of Fisher scoring iterations: 9 

Warning: Hauck-Donner effect detected in the following estimate(s):
'(Intercept):3'


Exponentiated coefficients:
        yprevIn Hospital/Facility                    yprevVent/ARDS 
                     6.430436e+03                      3.684490e+06 
    lsp(day, c(2, 4, 8, 15))day:1     lsp(day, c(2, 4, 8, 15))day:2 
                     2.114547e-01                      5.243932e-01 
    lsp(day, c(2, 4, 8, 15))day:3    lsp(day, c(2, 4, 8, 15))day':1 
                     8.038880e-01                      5.598882e+00 
   lsp(day, c(2, 4, 8, 15))day':2    lsp(day, c(2, 4, 8, 15))day':3 
                     1.965389e+00                      1.799922e+00 
  lsp(day, c(2, 4, 8, 15))day'':1   lsp(day, c(2, 4, 8, 15))day'':2 
                     8.672238e-01                      9.881755e-01 
  lsp(day, c(2, 4, 8, 15))day'':3  lsp(day, c(2, 4, 8, 15))day''':1 
                     6.478024e-01                      1.025260e+00 
 lsp(day, c(2, 4, 8, 15))day''':2  lsp(day, c(2, 4, 8, 15))day''':3 
                     9.792613e-01                      9.714168e-01 
lsp(day, c(2, 4, 8, 15))day'''':1 lsp(day, c(2, 4, 8, 15))day'''':2 
                     1.001785e+00                      1.097118e+00 
lsp(day, c(2, 4, 8, 15))day'''':3        lsp(age, c(35, 60, 75))age 
                     1.234913e+00                      9.961678e-01 
      lsp(age, c(35, 60, 75))age'      lsp(age, c(35, 60, 75))age'' 
                     1.019356e+00                      9.847118e-01 
    lsp(age, c(35, 60, 75))age'''        lsp(sofa, c(2, 6, 10))sofa 
                     1.036615e+00                      1.186359e+00 
      lsp(sofa, c(2, 6, 10))sofa'      lsp(sofa, c(2, 6, 10))sofa'' 
                     8.720611e-01                      1.018467e+00 
    lsp(sofa, c(2, 6, 10))sofa'''                                tx 
                     1.076951e+00                      8.843170e-01

Code

lrtest(g, f)

Likelihood ratio test

Model 1: ordered(y) ~ yprev + lsp(day, c(2, 4, 8, 15)) + lsp(age, c(35, 
    60, 75)) + lsp(sofa, c(2, 6, 10)) + tx
Model 2: ordered(y) ~ yprev + lsp(day, 2) + age + sofa + tx
    #Df  LogLik Df  Chisq Pr(>Chisq)
1 38947 -2248.7                     
2 38962 -2254.5 15 11.733     0.6991

Code

AIC(f); AIC(g)

[1] 4537.04

[1] 4555.307

We will use the simpler model, which has the better (smaller) AIC. Check the PO assumption on time by comparing the simpler model’s AIC to the AIC from a fully PO model.

Code

h <- vglm(ordered(y) ~ yprev + lsp(day, 2) + age + sofa + tx,
          cumulative(reverse=TRUE, parallel=TRUE), data=d)
lrtest(f, h)

Likelihood ratio test

Model 1: ordered(y) ~ yprev + lsp(day, 2) + age + sofa + tx
Model 2: ordered(y) ~ yprev + lsp(day, 2) + age + sofa + tx
    #Df  LogLik Df Chisq Pr(>Chisq)  
1 38962 -2254.5                      
2 38966 -2259.8  4 10.67    0.03054 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Code

AIC(f); AIC(h)

[1] 4537.04

[1] 4539.71

The model allowing for non-PO in time is better. Now show Wald tests on the parameters.

Code

wald <- function(f) {
  se <- sqrt(diag(vcov(f)))
  s <- round(cbind(beta=coef(f), SE=se, Z=coef(f) / se), 3)
  a <- c('>= in hospital/facility', '>= vent/ARDS', 'dead',
         'previous state in hospital/facility',
         'previous state vent/ARDS',
         'initial slope for day, >= hospital/facility',
         'initial slope for day, >= vent/ARDS',
         'initial slope for day, dead',
         'slope increment, >= hospital/facilty',
         'slope increment, >= vent/ARDS',
         'slope increament, dead',
         'baseline age linear effect',
         'baseline SOFA score linear effect',
         'treatment log OR')
  rownames(s) <- a
  s
}
wald(f)

                                               beta    SE       Z
>= in hospital/facility                      -5.030 0.629  -7.996
>= vent/ARDS                                -13.193 0.578 -22.827
dead                                        -19.580 0.936 -20.914
previous state in hospital/facility           8.773 0.266  33.008
previous state vent/ARDS                     15.138 0.322  47.022
initial slope for day, >= hospital/facility  -1.431 0.284  -5.036
initial slope for day, >= vent/ARDS          -0.707 0.257  -2.752
initial slope for day, dead                   0.137 0.469   0.293
slope increment, >= hospital/facilty          1.484 0.286   5.191
slope increment, >= vent/ARDS                 0.746 0.261   2.857
slope increament, dead                       -0.120 0.476  -0.252
baseline age linear effect                    0.011 0.003   4.054
baseline SOFA score linear effect             0.064 0.013   5.088
treatment log OR                             -0.111 0.085  -1.317

We see evidence for a benefit of treatment. Compute the treatment transition OR and approximate 0.95 confidence interval.

Code

lor <- coef(f)['tx']
se  <- sqrt(vcov(f)['tx', 'tx'])
b   <- exp(lor + qnorm(0.975) * se * c(0, -1, 1))
names(b) <- c('OR', 'Lower', 'Upper')
round(b, 3)

   OR Lower Upper 
0.895 0.758 1.056

The maximum likelihood estimate of the OR is somewhat at odds with the true OR of 0.75 on which the simulations were based.

22.3.3 Covariate Effects

Most interesting covariate effect is effect of time since randomization

Code

w <- d[day == 1]
dat <- expand.grid(yprev = 'In Hospital/Facility', age=median(w$age),
                   sofa=median(w$sofa), tx=0, day=1:28)
ltrans <- function(fit, mod) {
  p <- predict(fit, dat)
  u <- data.frame(day=as.vector(row(p)), y=as.vector(col(p)), logit=as.vector(p))
  u$y   <- factor(u$y, 1:3, paste('>=', levels(d$y)[-1]))
  u$mod <- mod
  u
}
u <- rbind(ltrans(f, 'model with few knots'),
           ltrans(g, 'model with more knots'))
ggplot(u, aes(x=day, y=logit, color=y)) + geom_line() +
  facet_wrap(~ mod, ncol=2) +
  xlab('Day') + ylab('Log Odds') +
  labs(caption='Relative log odds of transitioning from in hospital/facility to indicated status')

Figure 22.3: Estimated time trends in relative log odds of transitions. Variables not shown are set to median/mode and `tx=0`.

22.3.4 Correlation Structure

The data were simulated under a first-order Markov process so it doesn’t make sense to check correlation pattern assumptions for our model
When the simulated data were created, the within-patient correlation pattern was checked against the pattern from the fitted model by simulating a large trial from the model fit and comparing correlations in the simulated data to those in the real data
It showed excellent agreement
Let’s compute the Spearman \(\rho\) correlation matrix on the 500 patient dataset and show the matrix from the real data next to it
Delete day 28 from the new correlation matrix to conform with correlation matrix computed on real data
Also show correlation matrix from 10,000 patient sample
Heights of bars are proportional to Spearman \(\rho\)

Code

# Tall and thin -> short and wide data table
w <- dcast(d[, .(id, day, y=as.numeric(y))],
           id ~ day, value.var='y')
r <- cor(as.matrix(w[, -1]), use='pairwise.complete.obs',
         method='spearman')[-28, -28]
p <- plotCorrM(r, xangle=90)
p[[1]] + theme(legend.position='none') +
  labs(caption='Spearman correlation matrix from 500 patient dataset')

Code

vcorr <- readRDS('markov-vcorr.rds')
ra    <- vcorr$r.actual
plotCorrM(ra, xangle=90)[[1]] + theme(legend.position='none') +
  labs(caption='Spearman correlation matrix from actual data')

Code

rc    <- vcorr$r.simulated
plotCorrM(rc, xangle=90)[[1]] + theme(legend.position='none') +
  labs(caption='Spearman correlation matrix from 10,000 simulated patients')

Estimating the whole correlation matrix from 500 patients is noisy
Compute the mean absolute difference between two correlation matrices (the first on 10,000 simulated patients assuming a first-order Markov process and the second from the real data)
Compute means of mean absolute differences stratified by the day involved

Code

ad <- abs(rc[-28, -28] - ra)
round(apply(ad, 1, mean), 2)

   1    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16 
0.09 0.05 0.05 0.05 0.04 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.03 0.02 0.02 
  17   18   19   20   21   22   23   24   25   26   27 
0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02

Actual and simulated with-patient correlations agree well except when day 1 is involved.

Code

p[[2]]

Figure 22.4: Variogram-like graph for checking intra-patient correlation structure. \(x\)-axis shows the number of days between two measurements.

Usual serial correlation declining pattern; outcome status values become less correlated within patient as time gap increases
Also see non-isotropic pattern: correlations depend also on absolute time, not just gap

Formal Goodness of Fit Assessments for Correlation Structure

Data simulation model \(\rightarrow\) we already know that the first order Markov process has to fit
Do two formal assessments to demonstrate how this can be done in general. Both make the correlation structure more versatile.
- Add patient-specific intercepts to see if a compound symmetry structure adds anything to the first-order Markov structure
- Add a dependency on state before last in addition to our model’s dependency on the last state to see if a second-order Markov process fits better

Add Random Effects

Bayesian models handle random effects more naturally than frequentist models \(\rightarrow\) use a Bayesian partial PO first-order Markov model (R rmsb package)
Use cmdstan software in place of the default of rstan

Code

require(rmsb)
cmdstanr::set_cmdstan_path(cmdstan.loc)   # cmdstan.loc is defined in ~/.Rprofile
options(prType='html')
# Use all but one core
options(mc.cores = parallel::detectCores() - 1, rmsb.backend='cmdstan')
seed <- 2   # The following took 15m using 4 cores
b <- blrm(y ~ yprev + lsp(day, 2) + age + sofa + tx + cluster(id),
          ~ lsp(day, 2), data=d, file='markov-bppo.rds')
stanDx(b)

Iterations: 2000 on each of 4 chains, with 4000 posterior distribution samples saved

For each parameter, n_eff is a crude measure of effective sample size
and Rhat is the potential scale reduction factor on split chains
(at convergence, Rhat=1)

Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
6 of 4000 (0.15%) transitions ended with a divergence.
These divergent transitions indicate that HMC is not fully able to explore the posterior distribution.
Try increasing adapt delta closer to 1.
If this doesn't remove all divergences, try to reparameterize the model.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete.
Divergent samples: 0 0 3 3 

EBFMI: 0.452 0.415 0.49 0.35 

   Parameter  Rhat ESS bulk ESS tail
1   alpha[1] 1.002     2707     2238
2   alpha[2] 1.003     1596     1793
3   alpha[3] 1.005     1242     1893
4    beta[1] 1.003     2257     2264
5    beta[2] 1.001     2583     2543
6    beta[3] 1.013      469      895
7    beta[4] 1.000     3453     3040
8    beta[5] 1.001     3199     2700
9    beta[6] 1.005      750     1432
10   beta[7] 1.002     3410     1888
11 sigmag[1] 1.029      174      443
12  tau[1,1] 1.002     4497     3196
13  tau[2,1] 1.000     3716     3278
14  tau[1,2] 1.003     3880     2937
15  tau[2,2] 1.002     4488     3212

Code

Bayesian Partial Proportional Odds Ordinal Logistic Model

Dirichlet Priors With Concentration Parameter 0.455 for Intercepts

blrm(formula = y ~ yprev + lsp(day, 2) + age + sofa + tx + cluster(id), 
    ppo = ~lsp(day, 2), data = d, file = "markov-bppo.rds")

Frequencies of Responses

                Home In Hospital/Facility            Vent/ARDS 
                8141                 3887                  913 
                Dead 
                  51

	Mixed Calibration/ Discrimination Indexes	Discrimination Indexes	Rank Discrim. Indexes
Obs 12992	B 0.028 [0.028, 0.029]	g 5.349 [5.12, 5.621]	C 0.982 [0.982, 0.982]
Draws 4000		g_p 0.453 [0.45, 0.456]	D_xy 0.964 [0.964, 0.965]
Chains 4		EV 0.877 [0.864, 0.887]
Time 211.4s		v 26.552 [24.625, 29.382]
p 7		vp 0.205 [0.201, 0.208]
Cluster on `id`
Clusters 500
σ_γ 0.3913 [0.0071, 0.6328]

	Mean β	Median β	S.E.	Lower	Upper	Pr(β>0)	Symmetry
y≥In Hospital/Facility	-4.8317	-4.8527	0.6449	-6.1173	-3.6028	0.0000	1.07
y≥Vent/ARDS	-13.1432	-13.1439	0.5720	-14.2189	-12.0002	0.0000	0.99
y≥Dead	-19.8810	-19.8297	0.9875	-21.8128	-18.0560	0.0000	0.89
yprev=In Hospital/Facility	8.6424	8.6395	0.2791	8.0763	9.1543	1.0000	1.05
yprev=Vent/ARDS	15.0620	15.0566	0.3321	14.4023	15.7044	1.0000	1.05
day	-1.4992	-1.4899	0.2876	-2.0645	-0.9417	0.0000	0.92
day'	1.5420	1.5325	0.2886	0.9892	2.1160	1.0000	1.09
age	0.0125	0.0124	0.0032	0.0064	0.0192	1.0000	1.02
sofa	0.0764	0.0759	0.0164	0.0473	0.1122	1.0000	1.12
tx	-0.1156	-0.1126	0.0976	-0.2888	0.0875	0.1192	0.99
day:y≥Vent/ARDS	0.7327	0.7301	0.3784	-0.0151	1.4705	0.9748	1.02
day':y≥Vent/ARDS	-0.7466	-0.7425	0.3828	-1.5146	-0.0157	0.0240	0.97
day:y≥Dead	1.6845	1.6666	0.5684	0.6111	2.7775	0.9998	1.13
day':y≥Dead	-1.7186	-1.6939	0.5754	-2.8060	-0.6102	0.0005	0.89

Note that blrm parameterizes the partial PO parameters differently than vglm.
Posterior median of the standard deviation of the random effects \(\sigma_\gamma\) is 0.39
This is fairly small on the logit scale in which most of the action takes place in \([-4, 4]\)
Random intercepts add an inconsequential improvement in the fit, justifying the Markov process’ conditional (on prior state) independence assumption
Another useful analysis would entail comparing the SD of the posterior distributions for the main parameters with and without inclusion of the random effects

Second-order Markov Process

On follow-up days 2-28
Frequentist partial PO model

Code

# Derive time-before-last states (lag-1 `yprev`)
h <- d[, yprev2 := shift(yprev), by=id]
h <- h[day > 1, ]
# Fit first-order model ignoring day 1 so can compare to second-order
# We have to make time linear since no day 1 data
f1 <- vglm(ordered(y) ~ yprev + day + age + sofa + tx,
           cumulative(reverse=TRUE, parallel=FALSE ~ day), data=h)
f2 <- vglm(ordered(y) ~ yprev + yprev2 + day + age + sofa + tx,
           cumulative(reverse=TRUE, parallel=FALSE ~ day), data=h)
lrtest(f2, f1)

Likelihood ratio test

Model 1: ordered(y) ~ yprev + yprev2 + day + age + sofa + tx
Model 2: ordered(y) ~ yprev + day + age + sofa + tx
    #Df  LogLik Df  Chisq Pr(>Chisq)
1 37463 -2094.5                     
2 37465 -2095.0  2 1.1236     0.5702

Code

AIC(f1); AIC(f2)

[1] 4212.051

[1] 4214.928

First-order model has better fit “for the money” by AIC
Formal chunk test of second-order terms not impressive

22.3.5 Computing Derived Quantities

From the fitted Markov state transition model, compute for one covariate setting and two treatments:

state occupancy probabilities
mean time in state
differences between treatments in mean time in state

To specify covariate setting:

most common initial state is In Hospital/Facility so use that
within that category look at relationship between the two covariates
they have no correlation so use the individual medians

Code

istate <- 'In Hospital/Facility'
w <- d[day == 1 & yprev == istate, ]
w[, cor(age, sofa, method='spearman')]

[1] 0.01651803

Code

x <- w[, lapply(.SD, median), .SDcols=Cs(age, sofa)]
adjto <- x[, paste0('age=', x[, age], '  sofa=', x[, sofa],
                    '  initial state=', istate)]
# Expand to cover both treatments and initial state
x <- cbind(tx=0:1, yprev=istate, x)
x

      tx                yprev   age  sofa
   <int>               <char> <num> <num>
1:     0 In Hospital/Facility    56     5
2:     1 In Hospital/Facility    56     5

Compute all SOPs for each treatment. soprobMarkovOrdm is in Hmisc.

Code

S <- z <- NULL
for(Tx in 0:1) {
  s <- soprobMarkovOrdm(f, x[tx == Tx, ], times=1:28, ylevels=levels(d$y),
                        absorb='Dead', tvarname='day')
  S <- rbind(S, cbind(tx=Tx, s))
  u <- data.frame(day=as.vector(row(s)), y=as.vector(col(s)), p=as.vector(s))
  u$tx <- Tx
  z <- rbind(z, u)
}
z$y <- factor(z$y, 1:4, levels(d$y))
revo <- function(z) {
  z <- as.factor(z)
  factor(z, levels=rev(levels(as.factor(z))))
}
ggplot(z, aes(x=factor(day), y=p, fill=revo(y))) +
    facet_wrap(~ paste('Treatment', tx), nrow=1) + geom_col() +
    xlab('Day') + ylab('Probability') +
    guides(fill=guide_legend(title='Status')) +
    labs(caption=paste0('Estimated state occupancy probabilities for\n',
                        adjto)) +
    theme(legend.position='bottom',
          axis.text.x=element_text(angle=90, hjust=1))

Figure 22.5: State occupancy probabilities for each treatment

Compute by treatment the mean time unwell (expected number of days not at home). Expected days in state is simply the sum over days of daily probabilities of being in that state.

Code

mtu  <- tapply(1. - S[, 'Home'], S[, 'tx'], sum)
dmtu <- diff(mtu)
w    <- c(mtu, dmtu)
names(w) <- c('tx=0', 'tx=1', 'Days Difference')
w

           tx=0            tx=1 Days Difference 
      9.5270484       8.5866986      -0.9403498

We estimate that patients on treatment 1 have 3 less days unwell than those on treatment 0 for the given covariate settings.

Do a similar calculation for the expected number of days alive out of 28 days (similar to restricted mean survival time.

Code

mta <- tapply(1. - S[, 'Dead'], S[, 'tx'], sum)
w <- c(mta, diff(mta))
names(w) <- c('tx=0', 'tx=1', 'Days Difference')
w

           tx=0            tx=1 Days Difference 
    27.65094449     27.74380920      0.09286472

22.3.6 Bootstrap Confidence Interval for Difference in Mean Time Unwell

Need to sample with replacement from patients, not records
- code taken from rms package’s bootcov function
- sampling patients entails including some patients multiple times and omitting others
- save all the record numbers, group them by patient ID, sample from these IDs, then use all the original records whose record numbers correspond to the sampled IDs
Use the basic bootstrap to get 0.95 confidence intervals
Speed up the model fit by having each bootstrap fit use as starting parameter estimates the values from the original data fit

Code

B         <- 500    # number of bootstrap resamples
recno     <- split(1 : nrow(d), d$id)
npt       <- length(recno)   # 500
startbeta <- coef(f)
seed      <- 3
if(file.exists('markov-boot.rds')) {
  z        <- readRDS('markov-boot.rds')
  betas    <- z$betas
  diffmean <- z$diffmean
} else {
  betas <- diffmean <- numeric(B)
  ylev  <- levels(d$y)
  for(i in 1 : B) {
    j <- unlist(recno[sample(npt, npt, replace=TRUE)])
    g <- vglm(ordered(y) ~ yprev + lsp(day, 2) + age + sofa + tx,
              cumulative(reverse=TRUE, parallel=FALSE ~ lsp(day, 2)),
              coefstart=startbeta, data=d[j, ])
    betas[i] <- coef(g)['tx']
    s0 <- soprobMarkovOrdm(g, x[tx == 0, ], times=1:28, ylevels=ylev,
                           absorb='Dead', tvarname='day')
    s1 <- soprobMarkovOrdm(g, x[tx == 1, ], times=1:28, ylevels=ylev,
                           absorb='Dead', tvarname='day')
    # P(not at home) =  1 - P(home); sum these probs to get E[days]
    mtud <- sum(1. - s1[, 'Home']) - sum(1. - s0[, 'Home'])
    diffmean[i] <- mtud
  }
  saveRDS(list(betas=betas, diffmean=diffmean), 'markov-boot.rds', compress='xz')
}

See how bootstrap treatment log ORs relate to differences in days unwell.

Code

ggfreqScatter(betas, diffmean,
              xlab='Log OR', ylab='Difference in Mean Days Unwell')

Figure 22.6: Relationship between bootstrap log ORs and differences in mean days unwell

Compute basic bootstrap 0.95 confidence interval for OR and differences in mean time

Code

# bootBCa is in the rms package and uses the boot package
clb <- exp(bootBCa(coef(f)['tx'], betas, seed=seed, n=npt, type='basic'))
clm <- bootBCa(dmtu, diffmean, seed=seed, n=npt, type='basic')
a   <- round(c(clb, clm), 3)[c(1,3,2,4)]
data.frame(Quantity=c('OR', 'Difference in mean days unwell'),
            Lower=a[1:2], Upper=a[3:4])

                        Quantity  Lower Upper
1                             OR  0.743 1.063
2 Difference in mean days unwell -2.522 0.496

22.3.7 Notes on Inference

Differences between treatments in mean time in state(s) is zero if and only if the treatment OR=1
- note agreement in bootstrap estimates
- will not necessarily be true if PO is relaxed for treatment
- inference about any treatment effect is the same for all covariate settings that do not interact with treatment
- \(\rightarrow p\)-values are the same for the two metrics, and Bayesian posterior probabilities are also identical
Bayesian posterior probabilities for mean time in state \(> \epsilon\), for \(\epsilon > 0\), will vary with covariate settings (sicker patients at baseline have more room to move)