--- title: FEV1 data comparison between Bayesian and frequentist MMRMs date: "2024-09-07" output: rmarkdown::html_document: theme: spacelab highlight: default toc: yes toc_float: yes number_sections: true vignette: > %\VignetteIndexEntry{FEV1 data comparison between Bayesian and frequentist MMRMs} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} editor_options: chunk_output_type: console markdown: wrap: 72 --- # About {.unnumbered} 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)](#Summary) at the end. # Prerequisites This comparison workflow requires the following packages. ```r > 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. ```r > set.seed(123L) ``` # Data ## Pre-processing This analysis exercise uses the `fev_dat` dataset contained in the `mmrm`-package: ```r > 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`. ```r > 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()`. ```r > 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. ```r > 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: ```{.r .fold-hide} > 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: ```{.r .fold-hide} > 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. ```r > 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.

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. ```{.r .fold-hide} > 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. ```{.r .fold-hide} > 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. ```{.r .fold-hide} > 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. ```{.r .fold-hide} > 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. ```{.r .fold-hide} > 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. ```{.r .fold-hide} > b_mmrm_draws <- b_mmrm_fit |> + as_draws_df() > visit_levels <- sort(unique(as.character(fev_data$AVISIT))) > for (level in visit_levels) { + name <- paste0("b_sigma_AVISIT", level) + b_mmrm_draws[[name]] <- exp(b_mmrm_draws[[name]]) + } > b_mmrm_summary <- b_mmrm_draws |> + summarize_draws() |> + select(variable, mean, sd) |> + filter(!(variable %in% c("Intercept", "lprior", "lp__"))) |> + rename(bayes_estimate = mean, bayes_se = sd) |> + mutate( + variable = variable |> + tolower() |> + gsub(pattern = "b_", replacement = "") |> + gsub(pattern = "b_sigma_AVISIT", replacement = "sigma_") |> + gsub(pattern = "cortime", replacement = "correlation") |> + gsub(pattern = "__", replacement = "_") |> + gsub(pattern = "avisitvis", replacement = "avisit") + ) ``` ## Extract estimates from frequentist model We extract and standardize the frequentist estimates. ```{.r .fold-hide} > f_mmrm_fixed <- summary(f_mmrm_fit)$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) ``` ```{.r .fold-hide} > f_mmrm_variance <- tibble( + variable = paste0("sigma_AVISIT", visit_levels) |> + tolower() |> + gsub(pattern = "avisitvis", replacement = "avisit"), + freq_estimate = sqrt(diag(f_mmrm_fit$cov)) + ) ``` ```{.r .fold-hide} > f_diagonal_factor <- diag(1 / sqrt(diag(f_mmrm_fit$cov))) > f_corr_matrix <- f_diagonal_factor %*% f_mmrm_fit$cov %*% f_diagonal_factor > colnames(f_corr_matrix) <- visit_levels ``` ```{.r .fold-hide} > 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) ``` ```{.r .fold-hide} > f_mmrm_summary <- bind_rows( + f_mmrm_fixed, + f_mmrm_variance, + f_mmrm_correlation + ) |> + mutate(variable = gsub("\\s+", "", variable) |> tolower()) ``` ## Summary {#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`. ```{.r .fold-hide} > 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")) ``` ```{.r .fold-hide} > 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
```{.r .fold-hide} > 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 {#session} ```r > 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 ```