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This vignette shows the results of a simulation-based calibration (SBC) study to validate the implementation of the models in brms.mmrm. SBC tests the ability of a Bayesian model to recapture the parameters used to simulate prior predictive data. For details on SBC, please read Talts et al. (2020) and the SBC R package (Kim et al. 2022). This particular SBC study uses the targets pipeline in the sbc subdirectory of the brms.mmrm package source code.

Simple scenario

In the simple scenario, we simulate datasets from the prior predictive distribution assuming 3 treatment groups, 4 time points, 100 patients per treatment group, no adjustment for covariates, and no missing responses. The model formula is:

#> response ~ 0 + group + group:time + time + unstr(time = time, gr = patient) 
#> sigma ~ 0 + time

The prior was randomly generated and used for both simulation and analysis:

fst::read_fst("sbc/prior_simple.fst")
#>                        prior    class         coef  dpar
#> 1  lkj_corr_cholesky(1.1759) Lcortime                   
#> 2     normal(-0.108, 1.9497)        b groupgroup_1      
#> 3     normal(0.0441, 1.3727)        b groupgroup_2      
#> 4    normal(-0.2303, 0.2806)        b groupgroup_3      
#> 5     normal(0.0631, 1.6016)        b   timetime_2      
#> 6       normal(0.16, 1.4724)        b   timetime_3      
#> 7     normal(0.0631, 1.7156)        b   timetime_4      
#> 8     normal(0.0389, 2.2844)        b   timetime_1 sigma
#> 9      normal(0.1793, 1.808)        b   timetime_2 sigma
#> 10    normal(0.1637, 1.4719)        b   timetime_3 sigma
#> 11    normal(0.0186, 0.7794)        b   timetime_4 sigma

The following histograms show the SBC rank statistics which compare the prior parameter draws draws to the posterior draws. If the data simulation code and modeling code are both correct and consistent, then the rank statistics should be uniformly distributed.

library(dplyr)
library(ggplot2)
library(tibble)
library(tidyr)

read_ranks <- function(path) {
  fst::read_fst(path) |>
    tibble::as_tibble() |>
    pivot_longer(
      cols = everything(),
      names_to = "parameter",
      values_to = "rank"
    )
}

plot_ranks <- function(ranks) {
  ggplot(ranks) +
    geom_histogram(
      aes(x = rank),
      breaks = seq(from = 0, to = max(ranks$rank), length.out = 10)
    ) +
    facet_wrap(~parameter)
}
simple_ranks <- read_ranks("sbc/simple.fst")

Fixed effect parameter ranks:

simple_ranks |>
  filter(grepl("^b_", parameter)) |>
  filter(!grepl("^b_sigma", parameter)) |>
  plot_ranks()

Log-scale standard deviation parameter ranks:

simple_ranks |>
  filter(grepl("b_sigma", parameter)) |>
  plot_ranks()

Correlation parameter ranks:

simple_ranks |>
  filter(grepl("cortime_", parameter)) |>
  plot_ranks()

Complex scenario

In the complex scenario, we simulate datasets from the prior predictive distribution assuming 2 treatment groups, 2 subgroup levels, 3 time points, 150 patients per treatment group, adjustment for two continuous and two categorical baseline covariates, 30% dropout, and an 8% rate of independent/sporadic missing values. The model formula is:

#> response ~ group + group:subgroup + group:subgroup:time + group:time + subgroup + subgroup:time + time + continuous1 + continuous2 + balanced + unbalanced + unstr(time = time, gr = patient) 
#> sigma ~ 0 + time

The prior was randomly generated and used for both simulation and analysis:

fst::read_fst("sbc/prior_complex.fst")
#>                        prior     class
#> 1  lkj_corr_cholesky(1.4621)  Lcortime
#> 2    normal(-0.1661, 2.3091) Intercept
#> 3    normal(-0.1831, 1.8732)         b
#> 4     normal(0.1578, 2.8225)         b
#> 5    normal(-0.1069, 1.7987)         b
#> 6    normal(-0.1071, 0.5407)         b
#> 7     normal(-0.1028, 0.464)         b
#> 8     normal(-0.007, 0.7129)         b
#> 9     normal(0.1007, 1.8179)         b
#> 10   normal(-0.1965, 2.9243)         b
#> 11   normal(-0.0911, 1.5937)         b
#> 12    normal(0.0536, 0.7669)         b
#> 13    normal(0.0973, 0.5859)         b
#> 14    normal(0.0741, 2.3458)         b
#> 15     normal(-0.17, 2.9882)         b
#> 16    normal(0.1211, 0.7527)         b
#> 17    normal(-0.234, 2.0401)         b
#> 18   normal(-0.0457, 2.2697)         b
#> 19    normal(-0.181, 2.8068)         b
#> 20    normal(0.1093, 2.3647)         b
#> 21    normal(-0.0878, 1.165)         b
#> 22   normal(-0.2374, 1.5208)         b
#>                                          coef  dpar
#> 1                                                  
#> 2                                                  
#> 3                              balancedlevel2      
#> 4                              balancedlevel3      
#> 5                                 continuous1      
#> 6                                 continuous2      
#> 7                                groupgroup_2      
#> 8             groupgroup_2:subgroupsubgroup_2      
#> 9  groupgroup_2:subgroupsubgroup_2:timetime_2      
#> 10 groupgroup_2:subgroupsubgroup_2:timetime_3      
#> 11                    groupgroup_2:timetime_2      
#> 12                    groupgroup_2:timetime_3      
#> 13                         subgroupsubgroup_2      
#> 14              subgroupsubgroup_2:timetime_2      
#> 15              subgroupsubgroup_2:timetime_3      
#> 16                                 timetime_2      
#> 17                                 timetime_3      
#> 18                           unbalancedlevel2      
#> 19                           unbalancedlevel3      
#> 20                                 timetime_1 sigma
#> 21                                 timetime_2 sigma
#> 22                                 timetime_3 sigma

The following histograms show the SBC rank statistics which compare the prior parameter draws draws to the posterior draws. If the data simulation code and modeling code are both correct and consistent, then the rank statistics should be uniformly distributed.

complex_ranks <- read_ranks("sbc/complex.fst")

Fixed effect parameter ranks:

complex_ranks |>
  filter(grepl("^b_", parameter)) |>
  filter(!grepl("^b_sigma", parameter)) |>
  plot_ranks()

Log-scale standard deviation parameter ranks:

complex_ranks |>
  filter(grepl("b_sigma", parameter)) |>
  plot_ranks()

Correlation parameter ranks:

complex_ranks |>
  filter(grepl("cortime_", parameter)) |>
  plot_ranks()

Conclusion

The SBC rank statistics look uniformly distributed. In other words, the posterior distribution from the brms/Stan MMRM modeling code matches the prior from which the datasets were simulated. This is evidence that both the subgroup and non-subgroup models in brms.mmrm are implemented correctly.

References

Kim, S., Moon, H., Modrák, M., and Säilynoja, T. (2022), SBC: Simulation based calibration for rstan/cmdstanr models.
Talts, S., Betancourt, M., Simpson, D., Vehtari, A., and Gelman, A. (2020), Validating bayesian inference algorithms with simulation-based calibration.”