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DDMoRe: tte loglogistic

Source: vignettes/articles/NA_NA_tte_loglogistic.Rmd
NA_NA_tte_loglogistic.Rmd

Model and source

  • Citation: BAST Inc Limited. BAST approach to parametric time-to-event (PTTE) modelling. Loughborough, UK; 12 July 2017. Internal guiding document (BAST_PTTE_modelling.pdf) shipped with DDMORE bundle DDMODEL00000243; no peer-reviewed publication. Run prepared by Jon Moss (Command.txt; runCOMPEV2_005). DDMORE Foundation Model Repository: DDMODEL00000243.
  • Description: Parametric time-to-event log-logistic hazard model for Competing Event 2 in the BAST PTTE 2017 four-event teaching dataset (DDMODEL00000243). Hazard h(t) = alpha * (lam^alpha) * t^(alpha-1) / (1 + (lam*t)^alpha), where lam = lambda/1000 and alpha is unscaled. No covariates were retained. Note: the BAST guiding-document text (Section 2.4.1, Figure 2-4) selected log-normal as the base distribution for Competing Event 2, but the executable supplied with the bundle (Executable_runCOMPEV2_005.mod, $PROBLEM ‘Log_logistic model’) and the final-fit listing in the bundle are log-logistic, not log-normal. This file follows the executable; the publication-vs-bundle distribution discrepancy is documented in the validation vignette’s Errata.
  • Source: BAST Inc Limited, “BAST approach to parametric time-to-event (PTTE) modelling,” internal guiding document, 12 July 2017 (BAST_PTTE_modelling.pdf shipped in the DDMORE bundle).
  • DDMORE Foundation Model Repository entry: DDMODEL00000243
  • Source bundle (local mirror): dpastoor/ddmore_scraping/243/
  • Linked publication: none. The bundle is a methodological teaching example built on entirely simulated data; the BAST guiding-document text states “there is not yet a publication to go along with the model” (Model_Accommodations.txt).

This vignette validates the BAST 2017 PTTE Competing Event 2 hazard model packaged under inst/modeldb/ddmore/NA_NA_tte_loglogistic.R (NONMEM run name runCOMPEV2_005). Competing Event 2 in the bundle is right-censored only; no covariate was retained. Note: the BAST guiding-document text Section 2.4.1 (Figure 2-4) selected log-normal as the base distribution for Competing Event 2 by AIC, but the executable shipped in the bundle (Executable_runCOMPEV2_005.mod, $PROBLEM "Log_logistic model") and the final-fit listing are log-logistic, not log-normal – see the Errata section below.

For sibling Event 1, Event 2, and Competing Event 1 hazard models from the same bundle, see NA_NA_tte_gompertz.R, NA_NA_tte_gompertz_ev2.R, and NA_NA_tte_lognormal.R.

Population

The BAST 2017 PTTE bundle is a methodological teaching example with N = 200 simulated patients, four timed event types, and six baseline covariates (BAST Section 2.2.2). Of the 200 simulated patients, 142 (71%) experienced Competing Event 2 (BAST Table 2-1) – the highest event incidence of the four event types. Competing Event 2 is right- censored only (CENSORING = 1); exact event times are observed.

m <- readModelDb("NA_NA_tte_loglogistic")
str(m()$meta$population, max.level = 1)
#> List of 10
#>  $ n_subjects    : int 200
#>  $ n_studies     : int 1
#>  $ age_range     : chr "24-84 years (mean 58.7) in the BAST PTTE 2017 simulated cohort"
#>  $ weight_range  : chr "not reported (the BAST PTTE 2017 simulated cohort does not include body weight)"
#>  $ sex_female_pct: num NA
#>  $ race_ethnicity: NULL
#>  $ disease_state : chr "Hypothetical / unspecified clinical population (the BAST PTTE 2017 guiding document is a methodological teachin"| __truncated__
#>  $ dose_range    : chr "Not applicable (no drug administration is modelled)."
#>  $ regions       : chr "Not applicable (simulated data)."
#>  $ notes         : chr "200 simulated patients; 142 (71%) had Competing Event 2. Competing Event 2 is right-censored only (CENSORING = "| __truncated__

Source trace

Equation / parameter Value Source location
Hazard form h(t) = alpha * lam^alpha * t^(alpha-1) / (1 + (lam*t)^alpha) n/a Executable_runCOMPEV2_005.mod $PK / $DES (Lam1 = Lam/1000, alpha1 = alpha, val1 = alpha1*(lam1**alpha1)*T**(alpha1-1)); BAST guiding doc Appendix B
llam_compev2 (lambda) log(5.93) Output_simulated_runCOMPEV2_005.res FINAL TH1 = 5.93E+00; rescaled by /1000 inside model()
lalpha_compev2 (shape) log(1.60) Output_simulated_runCOMPEV2_005.res FINAL TH2 = 1.60E+00; not rescaled
eta on lam (OMEGA(1,1)) 0 FIXED Output_simulated_runCOMPEV2_005.res FINAL OMEGA(1,1) = 0.00E+00 (placeholder; no estimated IIV)
Covariate-selection result none BAST guiding doc Section 2.4.2, Table 2-5 (no covariate DeltaOFV exceeded the -6.6 threshold; final model has no covariates)

Virtual cohort 

set.seed(20260506)

n_subjects <- 50
cohort_subjects <- tibble(id = seq_len(n_subjects))

obs_grid <- tibble(
  time = c(0, seq(7, 540, by = 7)),
  evid = 0L,
  amt  = 0
)

events <- tidyr::crossing(cohort_subjects, obs_grid)
events <- events[, c("id", "time", "evid", "amt")]

stopifnot(!anyDuplicated(unique(events[, c("id", "time", "evid")])))
cat("Cohort: ", n_subjects, " subjects, ", nrow(events), " event rows\n", sep = "")
#> Cohort: 50 subjects, 3900 event rows

Simulation 

sim <- rxode2::rxSolve(m, events = events) |>
  as.data.frame()

Because the model has no covariates and no IIV, all 50 subjects share the same trajectory. The plot below shows a single-line typical-value survival curve.

Replicate published behaviour – typical-value survival trajectory

The BAST guiding document Section 2.4.1 (Figure 2-4) reports a Kaplan-Meier curve of Competing Event 2 vs. the candidate parametric distributions. The covariate VPC (Figure 2-13, Figure 2-14) covers the window 0 – 400 days. Competing Event 2 is the most aggressive event in the bundle (71% incidence); at the typical-value parameters (lambda = 5.93/1000, alpha = 1.60), the model’s survival drops from S(0) = 1 to about S(400) ~= 0.20.

sim |>
  filter(id == 1L) |>
  ggplot(aes(time, sur)) +
  geom_line(colour = "steelblue") +
  labs(
    x        = "Time (days)",
    y        = "Survival probability",
    title    = "Competing Event 2 typical-value survival (single trajectory; no covariates, no IIV)",
    caption  = "Compare against BAST 2017 PTTE guiding-document Figure 2-4 / Figure 2-13."
  ) +
  scale_y_continuous(limits = c(0, 1))

Mechanistic sanity checks (verification-checklist Section F.3)

F.3.1 – Hazard has the characteristic log-logistic shape (rises then falls when alpha > 1)

The log-logistic hazard with alpha > 1 rises, peaks at t_peak = (alpha - 1)^(1/alpha) / lam, and decays at long times. With alpha = 1.60 and lam = 5.93/1000 = 0.00593, the peak is at (0.60)^(1/1.60) / 0.00593 ~= 0.731 / 0.00593 ~= 123 days. The check below confirms the simulated hazard rises, peaks near day 100-150, and falls.

ev_typ <- tibble(id = 1L, time = c(0, seq(1, 540, by = 1)),
                 evid = 0L, amt = 0)
sim_typ <- rxode2::rxSolve(m, events = ev_typ) |> as.data.frame()
ggplot(sim_typ, aes(time, hazard)) +
  geom_line(colour = "firebrick") +
  geom_vline(xintercept = 123, colour = "grey60", linetype = "dotted") +
  labs(x = "Time (days)", y = "Instantaneous hazard (1/day)",
       title = "Competing Event 2 typical-subject hazard trajectory",
       caption = "Log-logistic hazard: rises, peaks near 123 days (alpha=1.60), decays.")


hazard_peak_idx  <- which.max(sim_typ$hazard)
hazard_peak_time <- sim_typ$time[hazard_peak_idx]
cat("Hazard peaks at t =", hazard_peak_time, "days\n")
#> Hazard peaks at t = 123 days

# Peak should be in the 60-200-day window (closed-form predicts ~123).
stopifnot(hazard_peak_time > 60, hazard_peak_time < 200)

# Hazard at peak should exceed both early-time (t = 1) and late-time
# (t = 500) values.
stopifnot(sim_typ$hazard[hazard_peak_idx] > sim_typ$hazard[sim_typ$time == 1])
stopifnot(sim_typ$hazard[hazard_peak_idx] > sim_typ$hazard[sim_typ$time == 500])

F.3.2 – Closed-form survival check at long times

For a log-logistic with lam (rate) and alpha (shape), the cumulative hazard is H(t) = log(1 + (lam * t)^alpha). Compute the typical-value survival at day 200 from the closed form and compare against the integrated-ODE simulation:

lam   <- 5.93 / 1000
alpha <- 1.60

t_check <- 200
H_closed <- log(1 + (lam * t_check)^alpha)
S_closed <- exp(-H_closed)

# rxode2 grid is whole-day; pick t = 200.
S_sim <- sim_typ$sur[sim_typ$time == t_check]

cat(sprintf("S(%d) closed-form = %.4f, simulation = %.4f\n",
            t_check, S_closed, S_sim))
#> S(200) closed-form = 0.4322, simulation = 0.4322
# Allow 1% relative tolerance for ODE-integration error.
stopifnot(abs(S_closed - S_sim) / S_closed < 0.01)

F.3.3 – Final-fit objective-function value matches the bundle

Output_simulated_runCOMPEV2_005.res reports OBJV = 1846.946 at the final estimates. This value is informational here.

Self-consistency with the bundle’s simulated dataset (F.2)

A full F.2 self-consistency check would re-simulate the bundle’s shipped Simulated_event_data.csv (200 subjects, DVID = 4 records) under the nlmixr2lib model and compare against the bundle’s Output_simulated_runCOMPEV2_005.res $TABLE output. The bundle dataset is outside this package and not redistributed.

Assumptions and deviations

  • Distribution discrepancy between the BAST guiding-document text and the bundle’s executable. The BAST PTTE 2017 guiding document Section 2.4.1 (Figure 2-4) selects log-normal as the base distribution for Competing Event 2 (AIC -2.144 vs. exponential 0). However, the bundle ships an executable (Executable_runCOMPEV2_005.mod) whose $PROBLEM line declares “Log_logistic model” and whose hazard equation is the log-logistic form, not the log-normal form. The .res listing (Output_simulated_runCOMPEV2_005.res) is consistent with the log-logistic .mod. We follow the executable that produced the shipped final estimates (the .lst is the source of truth for parameter VALUES per the skill’s verification checklist). The packaged nlmixr2lib model is therefore log-logistic, matching the bundle’s executable rather than the guiding-document narrative. A log-normal alternative is already provided in NA_NA_tte_lognormal.R (Competing Event 1).

  • Numerical rescalings preserved. The .mod uses a /1000 rescaling on lambda only; alpha is not rescaled. The biologically meaningful values are:

    • Log-logistic lambda: 5.93 / 1000 = 0.00593 / day
    • Log-logistic alpha (shape): 1.60 (unitless)

  • No DEL = 1e-8 offset on the time term. The BAST guiding document’s Appendix B canonical log-logistic form uses (t + DEL)^(alpha - 1) with a small-time offset; the bundle’s Executable_runCOMPEV2_005.mod $DES omits the offset and uses T^(alpha - 1) directly. We follow the bundle’s executable. At t = 0 and alpha > 1, the term evaluates to 0 (finite), so the hazard is well-defined.

  • No estimated IIV. The source $OMEGA is 0 FIX. No inter- subject variability is in the BAST PTTE 2017 guiding-document model. Combined with the absence of covariates, every subject’s typical- value trajectory is identical.

  • No publication-PDF cross-check. The BAST PTTE 2017 guiding document is the only source of equations and methodological narrative; no peer-reviewed publication exists.

  • Convention warning. nlmixr2lib::checkModelConventions() flags the units$concentration field (the TTE output sur is a survival probability, not a mass/volume concentration). The compartment is named cumhaz (canonical TTE auxiliary-state name).