About

This vignette provides an example comparison of a Bayesian MMRM fit, obtained by brms.mmrm::brm_model(), and a frequentist MMRM fit, obtained by mmrm::mmrm(). An overview of parameter estimates and differences by type of MMRM is given in the summary (Tables 4 and 5) at the end.

Prerequisites

This comparison workflow requires the following packages.

> packages <- c(
+   "dplyr",
+   "tidyr",
+   "ggplot2",
+   "gt",
+   "gtsummary",
+   "purrr",
+   "parallel",
+   "brms.mmrm",
+   "mmrm",
+   "posterior"
+ )
> invisible(lapply(packages, library, character.only = TRUE))

We set a seed for the random number generator to ensure statistical reproducibility.

> set.seed(123L)

Data

Pre-processing

This analysis exercise uses the fev_dat dataset contained in the mmrm-package:

> data(fev_data, package = "mmrm")

It is an artificial (simulated) dataset of a clinical trial investigating the effect of an active treatment on FEV1 (forced expired volume in one second), compared to placebo. FEV1 is a measure of how quickly the lungs can be emptied and low levels may indicate chronic obstructive pulmonary disease (COPD).

The dataset is a tibble with 800 rows and the following notable variables:

USUBJID (subject ID)
AVISIT (visit number, factor)
VISITN (visit number, numeric)
ARMCD (treatment, TRT or PBO)
RACE (3-category race)
SEX (female or male)
FEV1_BL (FEV1 at baseline, %)
FEV1 (FEV1 at study visits)
WEIGHT (weighting variable)

The primary endpoint for the analysis is change from baseline in FEV1, which we derive below and denote FEV1_CHG.

> fev_data <- fev_data |>
+   mutate("FEV1_CHG" = FEV1 - FEV1_BL)

The rest of the pre-processing steps create factors for the study arm and visit and apply the usual checking and standardization steps of brms.mmrm::brm_data().

> fev_data <- brm_data(
+   data = fev_data,
+   outcome = "FEV1_CHG",
+   group = "ARMCD",
+   time = "AVISIT",
+   patient = "USUBJID",
+   baseline = "FEV1_BL",
+   reference_group = "PBO",
+   covariates = c("RACE", "SEX")
+ ) |>
+   brm_data_chronologize(order = "VISITN")

The following table shows the first rows of the dataset.

> head(fev_data) |>
+   gt() |>
+   tab_caption(caption = md("Table 1. First rows of the pre-processed `fev_dat` dataset."))

Table 1. First rows of the pre-processed `fev_dat` dataset.
USUBJID	AVISIT	ARMCD	RACE	SEX	FEV1_BL	FEV1	WEIGHT	VISITN	VISITN2	FEV1_CHG
PT2	VIS1	PBO	Asian	Male	45.02477	NA	0.4651848	1	0.3295078	NA
PT2	VIS2	PBO	Asian	Male	45.02477	31.45522	0.2330974	2	-0.8204684	-13.569552
PT2	VIS3	PBO	Asian	Male	45.02477	36.87889	0.3600763	3	0.4874291	-8.145878
PT2	VIS4	PBO	Asian	Male	45.02477	48.80809	0.5073795	4	0.7383247	3.783324
PT3	VIS1	PBO	Black or African American	Female	43.50070	NA	0.6821642	1	0.5757814	NA
PT3	VIS2	PBO	Black or African American	Female	43.50070	35.98699	0.8917896	2	-0.3053884	-7.513705

Descriptive statistics

Table of baseline characteristics:

> fev_data |>
+   select(ARMCD, USUBJID, SEX, RACE, FEV1_BL) |>
+   distinct() |>
+   select(-USUBJID) |>
+   tbl_summary(
+     by = c(ARMCD),
+     statistic = list(
+       all_continuous() ~ "{mean} ({sd})",
+       all_categorical() ~ "{n} / {N} ({p}%)"
+     )
+   ) |>
+   modify_caption("Table 2. Baseline characteristics.")

Table 2. Baseline characteristics.
Characteristic	PBO N = 105¹	TRT N = 95¹
SEX
Male	50 / 105 (48%)	44 / 95 (46%)
Female	55 / 105 (52%)	51 / 95 (54%)
RACE
Asian	38 / 105 (36%)	32 / 95 (34%)
Black or African American	46 / 105 (44%)	29 / 95 (31%)
White	21 / 105 (20%)	34 / 95 (36%)
FEV1_BL	40 (9)	40 (9)
¹ n / N (%); Mean (SD)

Table of change from baseline in FEV1 over 52 weeks:

> fev_data |>
+   pull(AVISIT) |>
+   unique() |>
+   sort() |>
+   purrr::map(
+     .f = ~ fev_data |>
+       filter(AVISIT %in% .x) |>
+       tbl_summary(
+         by = ARMCD,
+         include = FEV1_CHG,
+         type = FEV1_CHG ~ "continuous2",
+         statistic = FEV1_CHG ~ c(
+           "{mean} ({sd})",
+           "{median} ({p25}, {p75})",
+           "{min}, {max}"
+         ),
+         label = list(FEV1_CHG = paste("Visit ", .x))
+       )
+   ) |>
+   tbl_stack(quiet = TRUE) |>
+   modify_caption("Table 3. Change from baseline.")

Table 3. Change from baseline.
Characteristic	PBO N = 105	TRT N = 95
Visit VIS1
Mean (SD)	-8 (9)	-2 (10)
Median (Q1, Q3)	-9 (-16, 0)	-4 (-9, 7)
Min, Max	-26, 12	-24, 20
Unknown	37	29
Visit VIS2
Mean (SD)	-3 (8)	2 (9)
Median (Q1, Q3)	-3 (-10, 1)	2 (-4, 8)
Min, Max	-20, 15	-22, 23
Unknown	36	24
Visit VIS3
Mean (SD)	2 (8)	5 (9)
Median (Q1, Q3)	2 (-2, 8)	6 (0, 11)
Min, Max	-15, 20	-19, 30
Unknown	34	37
Visit VIS4
Mean (SD)	8 (12)	13 (13)
Median (Q1, Q3)	6 (1, 20)	12 (5, 23)
Min, Max	-20, 39	-14, 47
Unknown	38	28

The following figure shows the primary endpoint over the four study visits in the data.

> fev_data |>
+   group_by(ARMCD) |>
+   ggplot(aes(x = AVISIT, y = FEV1_CHG, fill = factor(ARMCD))) +
+   geom_hline(yintercept = 0, col = "grey", linewidth = 1.2) +
+   geom_boxplot(na.rm = TRUE) +
+   labs(
+     x = "Visit",
+     y = "Change from baseline in FEV1",
+     fill = "Treatment"
+   ) +
+   scale_fill_manual(values = c("darkgoldenrod2", "coral2")) +
+   theme_bw()

Figure 1. Change from baseline in FEV1 over 4 visit time points.

Fitting MMRMs

Bayesian model

The formula for the Bayesian model includes additive effects for baseline, study visit, race, sex, and study-arm-by-visit interaction.

> b_mmrm_formula <- brm_formula(
+   data = fev_data,
+   intercept = TRUE,
+   baseline = TRUE,
+   group = FALSE,
+   time = TRUE,
+   baseline_time = FALSE,
+   group_time = TRUE,
+   correlation = "unstructured"
+ )
> print(b_mmrm_formula)
#> FEV1_CHG ~ FEV1_BL + ARMCD:AVISIT + AVISIT + RACE + SEX + unstr(time = AVISIT, gr = USUBJID) 
#> sigma ~ 0 + AVISIT

We fit the model using brms.mmrm::brm_model(). To ensure a good basis of comparison with the frequentist model, we put an extremely diffuse prior on the intercept. The parameters already have diffuse flexible priors by default.

> b_mmrm_fit <- brm_model(
+   data = filter(fev_data, !is.na(FEV1_CHG)),
+   formula = b_mmrm_formula,
+   prior = brms::prior(class = "Intercept", prior = "student_t(3, 0, 1000)"),
+   iter = 10000,
+   warmup = 2000,
+   chains = 4,
+   cores = 4,
+   seed = 1,
+   refresh = 0
+ )

Here is a posterior summary of model parameters, including fixed effects and pairwise correlation among visits within patients.

> summary(b_mmrm_fit)
#>  Family: gaussian 
#>   Links: mu = identity; sigma = log 
#> Formula: FEV1_CHG ~ FEV1_BL + ARMCD:AVISIT + AVISIT + RACE + SEX + unstr(time = AVISIT, gr = USUBJID) 
#>          sigma ~ 0 + AVISIT
#>    Data: data[!is.na(data[[attr(data, "brm_outcome")]]), ] (Number of observations: 537) 
#>   Draws: 4 chains, each with iter = 10000; warmup = 2000; thin = 1;
#>          total post-warmup draws = 32000
#> 
#> Correlation Structures:
#>                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> cortime(VIS1,VIS2)     0.36      0.08     0.18     0.52 1.00    48758    26086
#> cortime(VIS1,VIS3)     0.14      0.10    -0.05     0.33 1.00    49018    26172
#> cortime(VIS2,VIS3)     0.04      0.10    -0.16     0.23 1.00    49178    25472
#> cortime(VIS1,VIS4)     0.17      0.11    -0.06     0.38 1.00    49528    25555
#> cortime(VIS2,VIS4)     0.11      0.09    -0.07     0.28 1.00    49509    24007
#> cortime(VIS3,VIS4)     0.01      0.10    -0.18     0.21 1.00    45294    24353
#> 
#> Regression Coefficients:
#>                            Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
#> Intercept                     24.34      1.41    21.60    27.09 1.00    43678
#> FEV1_BL                       -0.84      0.03    -0.89    -0.78 1.00    56286
#> AVISIT2                        4.80      0.82     3.20     6.40 1.00    31792
#> AVISIT3                       10.37      0.83     8.73    12.01 1.00    29808
#> AVISIT4                       15.20      1.33    12.60    17.83 1.00    35293
#> RACEBlackorAfricanAmerican     1.41      0.59     0.27     2.55 1.00    46945
#> RACEWhite                      5.46      0.63     4.23     6.69 1.00    47801
#> SEXFemale                      0.35      0.51    -0.64     1.36 1.00    49193
#> AVISITVIS1:ARMCDTRT            3.98      1.07     1.89     6.06 1.00    31646
#> AVISITVIS2:ARMCDTRT            3.93      0.83     2.31     5.55 1.00    46665
#> AVISITVIS3:ARMCDTRT            2.98      0.68     1.65     4.31 1.00    52457
#> AVISITVIS4:ARMCDTRT            4.41      1.68     1.06     7.71 1.00    45307
#> sigma_AVISITVIS1               1.83      0.06     1.71     1.95 1.00    50344
#> sigma_AVISITVIS2               1.59      0.06     1.47     1.71 1.00    48248
#> sigma_AVISITVIS3               1.33      0.06     1.21     1.46 1.00    48058
#> sigma_AVISITVIS4               2.28      0.06     2.16     2.41 1.00    51078
#>                            Tail_ESS
#> Intercept                     25394
#> FEV1_BL                       24494
#> AVISIT2                       24396
#> AVISIT3                       24188
#> AVISIT4                       24810
#> RACEBlackorAfricanAmerican    25405
#> RACEWhite                     23816
#> SEXFemale                     24919
#> AVISITVIS1:ARMCDTRT           26255
#> AVISITVIS2:ARMCDTRT           23809
#> AVISITVIS3:ARMCDTRT           24705
#> AVISITVIS4:ARMCDTRT           25026
#> sigma_AVISITVIS1              26156
#> sigma_AVISITVIS2              24526
#> sigma_AVISITVIS3              24328
#> sigma_AVISITVIS4              23975
#> 
#> Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
#> and Tail_ESS are effective sample size measures, and Rhat is the potential
#> scale reduction factor on split chains (at convergence, Rhat = 1).

Frequentist model

The formula for the frequentist model is the same, except for the different syntax for specifying the covariance structure of the MMRM. We fit the model below.

> f_mmrm_fit <- mmrm::mmrm(
+   formula = FEV1_CHG ~ FEV1_BL + ARMCD:AVISIT + AVISIT + RACE + SEX +
+     us(AVISIT | USUBJID),
+   data = mutate(
+     fev_data,
+     AVISIT = factor(as.character(AVISIT), ordered = FALSE)
+   )
+ )

The parameter summaries of the frequentist model are below.

> summary(f_mmrm_fit)
#> mmrm fit
#> 
#> Formula:     
#> FEV1_CHG ~ FEV1_BL + ARMCD:AVISIT + AVISIT + RACE + SEX + us(AVISIT |  
#>     USUBJID)
#> Data:        
#> mutate(fev_data, AVISIT = factor(as.character(AVISIT), ordered = FALSE)) (used 
#> 537 observations from 197 subjects with maximum 4 timepoints)
#> Covariance:  unstructured (10 variance parameters)
#> Method:      Satterthwaite
#> Vcov Method: Asymptotic
#> Inference:   REML
#> 
#> Model selection criteria:
#>      AIC      BIC   logLik deviance 
#>   3381.4   3414.2  -1680.7   3361.4 
#> 
#> Coefficients: 
#>                                Estimate Std. Error        df t value Pr(>|t|)
#> (Intercept)                    24.35372    1.40754 257.97000  17.302  < 2e-16
#> FEV1_BL                        -0.84022    0.02777 190.27000 -30.251  < 2e-16
#> AVISITVIS2                      4.79036    0.79848 144.82000   5.999 1.51e-08
#> AVISITVIS3                     10.36601    0.81318 157.08000  12.748  < 2e-16
#> AVISITVIS4                     15.19231    1.30857 139.25000  11.610  < 2e-16
#> RACEBlack or African American   1.41921    0.57874 169.56000   2.452 0.015211
#> RACEWhite                       5.45679    0.61626 157.54000   8.855 1.65e-15
#> SEXFemale                       0.33812    0.49273 166.43000   0.686 0.493529
#> AVISITVIS1:ARMCDTRT             3.98329    1.04540 142.32000   3.810 0.000206
#> AVISITVIS2:ARMCDTRT             3.93076    0.81351 142.26000   4.832 3.46e-06
#> AVISITVIS3:ARMCDTRT             2.98372    0.66567 129.61000   4.482 1.61e-05
#> AVISITVIS4:ARMCDTRT             4.40400    1.66049 132.88000   2.652 0.008970
#>                                  
#> (Intercept)                   ***
#> FEV1_BL                       ***
#> AVISITVIS2                    ***
#> AVISITVIS3                    ***
#> AVISITVIS4                    ***
#> RACEBlack or African American *  
#> RACEWhite                     ***
#> SEXFemale                        
#> AVISITVIS1:ARMCDTRT           ***
#> AVISITVIS2:ARMCDTRT           ***
#> AVISITVIS3:ARMCDTRT           ***
#> AVISITVIS4:ARMCDTRT           ** 
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Covariance estimate:
#>         VIS1    VIS2    VIS3    VIS4
#> VIS1 37.8301 11.3255  3.4796 10.6844
#> VIS2 11.3255 23.5476  0.7760  5.5103
#> VIS3  3.4796  0.7760 13.8037  0.5683
#> VIS4 10.6844  5.5103  0.5683 92.9625

Comparison

This section compares the Bayesian posterior parameter estimates from brms.mmrm to the frequentist parameter estimates of the mmrm package.

Extract estimates from Bayesian model

We extract and standardize the Bayesian estimates.

> b_mmrm_draws <- b_mmrm_fit |>
+   as_draws_df()
> visit_levels <- sort(unique(as.character(fev_data$AVISIT)))
> for (level in visit_levels) {
+   name <- paste0("b_sigma_AVISIT", level)
+   b_mmrm_draws[[name]] <- exp(b_mmrm_draws[[name]])
+ }
> b_mmrm_summary <- b_mmrm_draws |>
+   summarize_draws() |>
+   select(variable, mean, sd) |>
+   filter(!(variable %in% c("Intercept", "lprior", "lp__"))) |>
+   rename(bayes_estimate = mean, bayes_se = sd) |>
+   mutate(
+     variable = variable |>
+       tolower() |>
+       gsub(pattern = "b_", replacement = "") |>
+       gsub(pattern = "b_sigma_AVISIT", replacement = "sigma_") |>
+       gsub(pattern = "cortime", replacement = "correlation") |>
+       gsub(pattern = "__", replacement = "_") |>
+       gsub(pattern = "avisitvis", replacement = "avisit")
+   )

Extract estimates from frequentist model

We extract and standardize the frequentist estimates.

> f_mmrm_fixed <- summary(f_mmrm_fit)$coefficients |>
+   as_tibble(rownames = "variable") |>
+   mutate(variable = tolower(variable)) |>
+   mutate(variable = gsub("(", "", variable, fixed = TRUE)) |>
+   mutate(variable = gsub(")", "", variable, fixed = TRUE)) |>
+   mutate(variable = gsub("avisitvis", "avisit", variable)) |>
+   rename(freq_estimate = Estimate, freq_se = `Std. Error`) |>
+   select(variable, freq_estimate, freq_se)

> f_mmrm_variance <- tibble(
+   variable = paste0("sigma_AVISIT", visit_levels) |>
+     tolower() |>
+     gsub(pattern = "avisitvis", replacement = "avisit"),
+   freq_estimate = sqrt(diag(f_mmrm_fit$cov))
+ )

> f_diagonal_factor <- diag(1 / sqrt(diag(f_mmrm_fit$cov)))
> f_corr_matrix <- f_diagonal_factor %*% f_mmrm_fit$cov %*% f_diagonal_factor
> colnames(f_corr_matrix) <- visit_levels

> f_mmrm_correlation <- f_corr_matrix |>
+   as.data.frame() |>
+   as_tibble() |>
+   mutate(x1 = visit_levels) |>
+   pivot_longer(
+     cols = -any_of("x1"),
+     names_to = "x2",
+     values_to = "freq_estimate"
+   ) |>
+   filter(
+     as.numeric(gsub("[^0-9]", "", x1)) < as.numeric(gsub("[^0-9]", "", x2))
+   ) |>
+   mutate(variable = sprintf("correlation_%s_%s", x1, x2)) |>
+   select(variable, freq_estimate)

> f_mmrm_summary <- bind_rows(
+   f_mmrm_fixed,
+   f_mmrm_variance,
+   f_mmrm_correlation
+ ) |>
+   mutate(variable = gsub("\\s+", "", variable) |> tolower())

Summary

The first table below summarizes the parameter estimates from each model and the differences between estimates (Bayesian minus frequentist). The second table shows the standard errors of these estimates and differences between standard errors. In each table, the “Relative” column shows the relative difference (the difference divided by the frequentist quantity).

Because of the different statistical paradigms and estimation procedures, especially regarding the covariance parameters, it would not be realistic to expect the Bayesian and frequentist approaches to yield virtually identical results. Nevertheless, the absolute and relative differences in the table below show strong agreement between brms.mmrm and mmrm.

> b_f_comparison <- full_join(
+   x = b_mmrm_summary,
+   y = f_mmrm_summary,
+   by = "variable"
+ ) |>
+   mutate(
+     diff_estimate = bayes_estimate - freq_estimate,
+     diff_relative_estimate = diff_estimate / freq_estimate,
+     diff_se = bayes_se - freq_se,
+     diff_relative_se = diff_se / freq_se
+   ) |>
+   select(variable, ends_with("estimate"), ends_with("se"))

> table_estimates <- b_f_comparison |>
+   select(variable, ends_with("estimate"))
> gt(table_estimates) |>
+   fmt_number(decimals = 4) |>
+   tab_caption(
+     caption = md(
+       paste(
+         "Table 4. Comparison of parameter estimates between",
+         "Bayesian and frequentist MMRMs."
+       )
+     )
+   ) |>
+   cols_label(
+     variable = "Variable",
+     bayes_estimate = "Bayesian",
+     freq_estimate = "Frequentist",
+     diff_estimate = "Difference",
+     diff_relative_estimate = "Relative"
+   )

Table 4. Comparison of parameter estimates between Bayesian and frequentist MMRMs.
Variable	Bayesian	Frequentist	Difference	Relative
intercept	24.3351	24.3537	−0.0186	−0.0008
fev1_bl	−0.8399	−0.8402	0.0003	−0.0004
avisit2	4.7996	4.7904	0.0093	0.0019
avisit3	10.3730	10.3660	0.0070	0.0007
avisit4	15.1989	15.1923	0.0066	0.0004
raceblackorafricanamerican	1.4131	1.4192	−0.0061	−0.0043
racewhite	5.4556	5.4568	−0.0012	−0.0002
sexfemale	0.3456	0.3381	0.0075	0.0221
avisit1:armcdtrt	3.9842	3.9833	0.0009	0.0002
avisit2:armcdtrt	3.9330	3.9308	0.0023	0.0006
avisit3:armcdtrt	2.9795	2.9837	−0.0042	−0.0014
avisit4:armcdtrt	4.4066	4.4040	0.0026	0.0006
sigma_avisit1	6.2317	6.1506	0.0811	0.0132
sigma_avisit2	4.9146	4.8526	0.0620	0.0128
sigma_avisit3	3.7775	3.7153	0.0622	0.0167
sigma_avisit4	9.7953	9.6417	0.1536	0.0159
correlation_vis1_vis2	0.3607	0.3795	−0.0187	−0.0493
correlation_vis1_vis3	0.1419	0.1523	−0.0104	−0.0683
correlation_vis2_vis3	0.0396	0.0430	−0.0034	−0.0791
correlation_vis1_vis4	0.1680	0.1802	−0.0121	−0.0674
correlation_vis2_vis4	0.1110	0.1178	−0.0067	−0.0571
correlation_vis3_vis4	0.0144	0.0159	−0.0015	−0.0929

> table_se <- b_f_comparison |>
+   select(variable, ends_with("se")) |>
+   filter(!is.na(freq_se))
> gt(table_se) |>
+   fmt_number(decimals = 4) |>
+   tab_caption(
+     caption = md(
+       paste(
+         "Table 5. Comparison of parameter standard errors between",
+         "Bayesian and frequentist MMRMs."
+       )
+     )
+   ) |>
+   cols_label(
+     variable = "Variable",
+     bayes_se = "Bayesian",
+     freq_se = "Frequentist",
+     diff_se = "Difference",
+     diff_relative_se = "Relative"
+   )

Table 5. Comparison of parameter standard errors between Bayesian and frequentist MMRMs.
Variable	Bayesian	Frequentist	Difference	Relative
intercept	1.4125	1.4075	0.0050	0.0035
fev1_bl	0.0279	0.0278	0.0001	0.0046
avisit2	0.8176	0.7985	0.0191	0.0239
avisit3	0.8335	0.8132	0.0203	0.0250
avisit4	1.3283	1.3086	0.0198	0.0151
raceblackorafricanamerican	0.5857	0.5787	0.0070	0.0120
racewhite	0.6252	0.6163	0.0089	0.0144
sexfemale	0.5119	0.4927	0.0192	0.0390
avisit1:armcdtrt	1.0651	1.0454	0.0197	0.0188
avisit2:armcdtrt	0.8260	0.8135	0.0125	0.0154
avisit3:armcdtrt	0.6771	0.6657	0.0114	0.0171
avisit4:armcdtrt	1.6805	1.6605	0.0200	0.0120

Session info

> sessionInfo()
#> R version 4.4.0 (2024-04-24)
#> Platform: aarch64-apple-darwin20
#> Running under: macOS Sonoma 14.5
#> 
#> Matrix products: default
#> BLAS:   /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRblas.0.dylib 
#> LAPACK: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.0
#> 
#> locale:
#> [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
#> 
#> time zone: America/Indiana/Indianapolis
#> tzcode source: internal
#> 
#> attached base packages:
#> [1] parallel  stats     graphics  grDevices utils     datasets  methods  
#> [8] base     
#> 
#> other attached packages:
#>  [1] posterior_1.5.0      mmrm_0.3.11          brms.mmrm_1.0.1.9005
#>  [4] purrr_1.0.2          gtsummary_1.9.9.9003 gt_0.10.1           
#>  [7] ggplot2_3.5.1        tidyr_1.3.1          dplyr_1.1.4         
#> [10] knitr_1.46          
#> 
#> loaded via a namespace (and not attached):
#>  [1] tidyselect_1.2.1     svUnit_1.0.6         farver_2.1.2        
#>  [4] loo_2.7.0            tidybayes_3.0.6      fastmap_1.2.0       
#>  [7] TH.data_1.1-2        tensorA_0.36.2.1     digest_0.6.35       
#> [10] estimability_1.5     lifecycle_1.0.4      StanHeaders_2.32.8  
#> [13] processx_3.8.4       survival_3.5-8       magrittr_2.0.3      
#> [16] compiler_4.4.0       rlang_1.1.4          sass_0.4.9          
#> [19] tools_4.4.0          utf8_1.2.4           labeling_0.4.3      
#> [22] bridgesampling_1.1-2 pkgbuild_1.4.4       curl_5.2.1          
#> [25] plyr_1.8.9           xml2_1.3.6           abind_1.4-5         
#> [28] multcomp_1.4-25      withr_3.0.0          grid_4.4.0          
#> [31] stats4_4.4.0         fansi_1.0.6          xtable_1.8-4        
#> [34] colorspace_2.1-0     inline_0.3.19        emmeans_1.10.1      
#> [37] scales_1.3.0         gtools_3.9.5         MASS_7.3-60.2       
#> [40] ggridges_0.5.6       cli_3.6.2            mvtnorm_1.2-4       
#> [43] generics_0.1.3       RcppParallel_5.1.7   binom_1.1-1.1       
#> [46] reshape2_1.4.4       commonmark_1.9.1     rstan_2.32.6        
#> [49] stringr_1.5.1        splines_4.4.0        bayesplot_1.11.1    
#> [52] matrixStats_1.3.0    brms_2.21.0          vctrs_0.6.5         
#> [55] V8_4.4.2             Matrix_1.7-0         sandwich_3.1-0      
#> [58] jsonlite_1.8.8       callr_3.7.6          arrayhelpers_1.1-0  
#> [61] ggdist_3.3.2         glue_1.7.0           ps_1.7.6            
#> [64] codetools_0.2-20     distributional_0.4.0 stringi_1.8.4       
#> [67] gtable_0.3.5         QuickJSR_1.1.3       munsell_0.5.1       
#> [70] tibble_3.2.1         pillar_1.9.0         htmltools_0.5.8.1   
#> [73] Brobdingnag_1.2-9    TMB_1.9.11           R6_2.5.1            
#> [76] Rdpack_2.6           evaluate_0.23        lattice_0.22-6      
#> [79] highr_0.10           markdown_1.12        cards_0.1.0.9046    
#> [82] rbibutils_2.2.16     backports_1.4.1      trialr_0.1.6        
#> [85] rstantools_2.4.0     Rcpp_1.0.12          coda_0.19-4.1       
#> [88] gridExtra_2.3        nlme_3.1-164         checkmate_2.3.1     
#> [91] xfun_0.43            zoo_1.8-12           pkgconfig_2.0.3

FEV1 data comparison between Bayesian and frequentist MMRMs