sim_markov("violet") simulates daily 4-state trajectories resembling the
VIOLET trial of early high-dose
vitamin D3 versus placebo in critically ill adults with vitamin D deficiency.
The transition model is the partial proportional-odds Markov model that
Frank Harrell fitted to the VIOLET data to simulate his large simlongord
trial (Harrell, 2022), and patients are drawn from the same 1,351 VIOLET
patients. Its daily state proportions agree with simlongord within
simulation error. The states are:
at home
in hospital or another health care facility
on a ventilator or with acute respiratory distress syndrome (ARDS)
dead
The output has columns id, time, y, yprev, age, sofa, and tx,
28 rows per patient by default, and dead patients stay in state 4.
Details
Patients. Each simulated patient's starting state (yprev), age, and
SOFA (Sequential Organ Failure Assessment) score are drawn together, with
replacement, from the 1,351 VIOLET patients summarized in
violet_baseline. About 65% of patients start in hospital (state 2) and
35% on a ventilator or with ARDS (state 3). Treatment (tx = 1) is
assigned independently of these characteristics with probability
treatment_prob; control patients have tx = 0. A baseline_data that
you supply must contain the columns id, yprev, age, sofa, and tx.
Transition model. Write \(Y_t\) for the state on day \(t\) and
\(k\) for the state on the previous day. The log odds of being in state
\(j\) or worse on day \(t\) is
$$\mathrm{logit}\,\Pr(Y_t \ge j \mid Y_{t-1} = k) = \alpha_j +
\gamma_k + \delta_j t + \delta'_j \max(t - 2, 0) +
0.0105\,\mathrm{age} + 0.046\,\mathrm{SOFA} +
\beta(t)\,\mathrm{tx}$$
for the cutoffs \(j = 2, 3, 4\). Here \(\beta(t)\) is the treatment
effect on day \(t\) described under treatment_effect_decay in
sim_markov(). Positive values make worse states more likely: for example,
each additional SOFA point multiplies the odds of being in a worse state by
\(\exp(0.046) = 1.05\).
The thresholds \(\alpha_j\) and time slopes \(\delta_j\) and \(\delta'_j\) differ between the cutoffs, so the mix of states can change over follow-up (a partial proportional odds model for time):
| Cutoff \(j\) | States counted | \(\alpha_j\) | \(\delta_j\) | \(\delta'_j\) |
| 4 | dead | -4.713 | 0.412 | -0.410 |
| 3 | ventilator/ARDS or dead | 1.741 | -0.481 | 0.525 |
| 2 | hospital or worse | 10.034 | -1.315 | 1.384 |
Time enters as a straight line whose slope changes after day 2, because very few patients went home on day 1. The previous state shifts all three cutoffs equally: \(\gamma_k\) is -15.797 after a day at home (\(k = 1\)), -6.364 after a day in hospital (\(k = 2\)), and 0 after a day on a ventilator or with ARDS (\(k = 3\)). The large negative values mean that patients usually stay in, or improve from, their previous state.
Treatment effect. The default treatment_effect of log(0.75) is the
odds ratio of 0.75 that Harrell built into simlongord, the same for every
previous state. In VIOLET itself, vitamin D had no detectable effect
(fitted log odds ratio -0.028, standard error 0.050).
Relation to Harrell's model. Harrell numbered the states from dead
(lowest) to home (highest), modeled the probability of a better state, and
published the coefficients rounded to three decimals. This preset numbers
the states from best to worst, so each intercept and coefficient above is
his printed value with the opposite sign; the transition probabilities are
the same. The one exception is age. Harrell printed 0.011 (with his sign,
-0.011), but simlongord was evidently simulated with a value close to
0.0105: refitting his model to all 250,000 simulated patients gives 0.0105
(standard error 0.00015), while every other coefficient agrees with its
printed value. A difference of 0.0005 per year of age moves a typical
patient's log odds by about 0.03 every day, so with 0.011 about 1 percentage
point fewer patients are at home by day 28 than in simlongord. With 0.0105
the daily state proportions agree with simlongord within simulation
error.
He fitted the model to days 1 to 27, because ventilator use on day 28
looked anomalous, and simulated 28 days, which is the default here. Later
days extend the fitted time trends. Harrell assigned treatment once to each
of the 1,351 VIOLET patients before resampling them, so in simlongord the
arms differ slightly in their starting states and covariates (for example,
33.9% versus 36.0% start on a ventilator). This preset assigns treatment
independently to every simulated patient, so the arms differ only by chance.
References
National Heart, Lung, and Blood Institute PETAL Clinical Trials Network (2019). "Early High-Dose Vitamin D3 for Critically Ill, Vitamin D-Deficient Patients." New England Journal of Medicine, 381(26), 2529-2540. doi:10.1056/NEJMoa1911124
Harrell FE Jr (2022). "Simulating a Large Clinical Trial From VIOLET." https://hbiostat.org/data/repo/simlongord
See also
sim_markov() to simulate, and violet_baseline for the VIOLET
patients.
Examples
violet <- sim_markov("violet", n_patients = 20, seed = 123)
head(violet)
#> id time y yprev age sofa tx
#> 1 1 1 2 2 68 6 0
#> 2 1 2 2 2 68 6 0
#> 3 1 3 2 2 68 6 0
#> 4 1 4 2 2 68 6 0
#> 5 1 5 3 2 68 6 0
#> 6 1 6 3 3 68 6 0
# A trial in which treatment has no effect
null_trial <- sim_markov(
"violet",
n_patients = 20,
treatment_effect = 0,
seed = 123
)