BCVA data comparison between Bayesian and frequentist MMRMs
In brms.mmrm: Bayesian MMRMs using 'brms'

About {.unnumbered}

This vignette uses the bcva_data dataset from the mmrm package to compare 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 bcva_data dataset contained in the mmrm package:

> data(bcva_data, package = "mmrm")

According to https://openpharma.github.io/mmrm/latest-tag/articles/mmrm_review_methods.html:

The BCVA dataset contains data from a randomized longitudinal ophthalmology trial evaluating the change in baseline corrected visual acuity (BCVA) over the course of 10 visits. BCVA corresponds to the number of letters read from a visual acuity chart.

The dataset is a tibble with 8605 rows and the following notable variables.

USUBJID (subject ID)
AVISIT (visit number, factor)
VISITN (visit number, numeric)
ARMCD (treatment, TRT or CTL)
RACE (3-category race)
BCVA_BL (BCVA at baseline)
BCVA_CHG (BCVA change from baseline, primary endpoint for the analysis)

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().

> bcva_data <- bcva_data |>
+   mutate(AVISIT = gsub("VIS0*", "VIS", as.character(AVISIT))) |>
+   brm_data(
+     outcome = "BCVA_CHG",
+     group = "ARMCD",
+     time = "AVISIT",
+     patient = "USUBJID",
+     baseline = "BCVA_BL",
+     reference_group = "CTL",
+     covariates = "RACE"
+   ) |>
+   brm_data_chronologize(order = "VISITN")

The following table shows the first rows of the dataset.

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

Table 1. First rows of the pre-processed `bcva_data` dataset.
USUBJID	AVISIT	VISITN	ARMCD	RACE	BCVA_BL	BCVA_CHG
3	VIS1	1	CTL	Asian	71.70881	5.058546
3	VIS10	10	CTL	Asian	71.70881	10.152565
3	VIS2	2	CTL	Asian	71.70881	4.018582
3	VIS3	3	CTL	Asian	71.70881	3.572535
3	VIS4	4	CTL	Asian	71.70881	4.822669
3	VIS5	5	CTL	Asian	71.70881	7.348768

Descriptive statistics

Table of baseline characteristics:

```{.r .fold-hide}

bcva_data |> + select(ARMCD, USUBJID, RACE, BCVA_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.")

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<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <!--/html_preserve--><caption class='gt_caption'>Table 2. Baseline characteristics.</caption><!--html_preserve-->
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="&lt;strong&gt;Characteristic&lt;/strong&gt;"><strong>Characteristic</strong></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="&lt;strong&gt;CTL&lt;/strong&gt;&lt;br /&gt;&#10;N = 494&lt;span class=&quot;gt_footnote_marks&quot; style=&quot;white-space:nowrap;font-style:italic;font-weight:normal;&quot;&gt;&lt;sup&gt;1&lt;/sup&gt;&lt;/span&gt;"><strong>CTL</strong><br />
N = 494<span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="&lt;strong&gt;TRT&lt;/strong&gt;&lt;br /&gt;&#10;N = 506&lt;span class=&quot;gt_footnote_marks&quot; style=&quot;white-space:nowrap;font-style:italic;font-weight:normal;&quot;&gt;&lt;sup&gt;1&lt;/sup&gt;&lt;/span&gt;"><strong>TRT</strong><br />
N = 506<span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span></th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="label" class="gt_row gt_left">RACE</td>
<td headers="stat_1" class="gt_row gt_center"><br /></td>
<td headers="stat_2" class="gt_row gt_center"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Asian</td>
<td headers="stat_1" class="gt_row gt_center">151 / 494 (31%)</td>
<td headers="stat_2" class="gt_row gt_center">146 / 506 (29%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Black</td>
<td headers="stat_1" class="gt_row gt_center">149 / 494 (30%)</td>
<td headers="stat_2" class="gt_row gt_center">168 / 506 (33%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    White</td>
<td headers="stat_1" class="gt_row gt_center">194 / 494 (39%)</td>
<td headers="stat_2" class="gt_row gt_center">192 / 506 (38%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">BCVA_BL</td>
<td headers="stat_1" class="gt_row gt_center">75 (10)</td>
<td headers="stat_2" class="gt_row gt_center">75 (10)</td></tr>
  </tbody>

  <tfoot class="gt_footnotes">
    <tr>
      <td class="gt_footnote" colspan="3"><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span> n / N (%); Mean (SD)</td>
    </tr>
  </tfoot>
</table>
</div><!--/html_preserve-->

Table of change from baseline in BCVA over 52 weeks:


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

Table 3. Change from baseline.
Characteristic	CTL N = 494	TRT N = 506
Visit VIS1
Mean (SD)	5.32 (1.23)	5.86 (1.33)
Median (Q1, Q3)	5.34 (4.51, 6.17)	5.86 (4.98, 6.81)
Min, Max	1.83, 9.02	2.28, 10.30
Unknown	12	5
Visit VIS2
Mean (SD)	5.59 (1.49)	6.33 (1.45)
Median (Q1, Q3)	5.53 (4.64, 6.47)	6.36 (5.34, 7.31)
Min, Max	0.29, 10.15	2.35, 10.75
Unknown	13	7
Visit VIS3
Mean (SD)	5.79 (1.61)	6.79 (1.71)
Median (Q1, Q3)	5.73 (4.64, 6.91)	6.82 (5.66, 7.93)
Min, Max	1.53, 11.46	1.13, 11.49
Unknown	23	17
Visit VIS4
Mean (SD)	6.18 (1.73)	7.29 (1.82)
Median (Q1, Q3)	6.14 (5.05, 7.41)	7.24 (6.05, 8.54)
Min, Max	0.45, 11.49	2.07, 11.47
Unknown	36	18
Visit VIS5
Mean (SD)	6.28 (1.97)	7.68 (1.94)
Median (Q1, Q3)	6.18 (4.96, 7.71)	7.75 (6.48, 8.95)
Min, Max	0.87, 11.53	2.24, 14.10
Unknown	40	35
Visit VIS6
Mean (SD)	6.69 (1.97)	8.31 (1.98)
Median (Q1, Q3)	6.64 (5.26, 8.14)	8.29 (6.92, 9.74)
Min, Max	1.35, 12.95	1.93, 14.38
Unknown	84	48
Visit VIS7
Mean (SD)	6.78 (2.09)	8.78 (2.11)
Median (Q1, Q3)	6.91 (5.46, 8.29)	8.67 (7.44, 10.26)
Min, Max	-1.54, 11.99	3.21, 14.36
Unknown	106	78
Visit VIS8
Mean (SD)	7.08 (2.25)	9.40 (2.26)
Median (Q1, Q3)	7.08 (5.55, 8.68)	9.35 (7.96, 10.86)
Min, Max	0.97, 13.71	2.28, 15.95
Unknown	123	86
Visit VIS9
Mean (SD)	7.39 (2.33)	10.01 (2.50)
Median (Q1, Q3)	7.47 (5.76, 9.05)	10.01 (8.19, 11.74)
Min, Max	0.04, 14.61	4.22, 18.09
Unknown	167	114
Visit VIS10
Mean (SD)	7.49 (2.58)	10.59 (2.36)
Median (Q1, Q3)	7.40 (5.73, 9.01)	10.71 (9.03, 12.25)
Min, Max	-0.08, 15.75	3.24, 16.40
Unknown	213	170

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

> bcva_data |>
+   group_by(ARMCD) |>
+   ggplot(aes(x = AVISIT, y = BCVA_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 BCVA",
+     fill = "Treatment"
+   ) +
+   scale_fill_manual(values = c("darkgoldenrod2", "coral2")) +
+   theme_bw()

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

Fitting MMRMs

Bayesian model

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

```{.r .fold-hide}

b_mmrm_formula <- brm_formula( + data = bcva_data, + intercept = TRUE, + baseline = TRUE, + group = FALSE, + time = TRUE, + baseline_time = FALSE, + group_time = TRUE, + correlation = "unstructured" + ) print(b_mmrm_formula)

> BCVA_CHG ~ BCVA_BL + ARMCD:AVISIT + AVISIT + RACE + unstr(time = AVISIT, gr = USUBJID)

> sigma ~ 0 + AVISIT

We fit the model using `brms.mmrm::brm_model()`. The computation takes several minutes because of the size of the dataset. 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(bcva_data, !is.na(BCVA_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: BCVA_CHG ~ BCVA_BL + ARMCD:AVISIT + AVISIT + RACE + unstr(time = AVISIT, gr = USUBJID)

> sigma ~ 0 + AVISIT

> Data: data[!is.na(data[[attr(data, "brm_outcome")]]), ] (Number of observations: 8605)

> 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) 0.05 0.03 -0.01 0.11 1.00 63561 23159

> cortime(VIS1,VIS3) 0.31 0.03 0.25 0.36 1.00 70330 25831

> cortime(VIS2,VIS3) 0.05 0.03 -0.02 0.11 1.00 67715 22226

> cortime(VIS1,VIS4) 0.21 0.03 0.15 0.27 1.00 46375 28108

> cortime(VIS2,VIS4) 0.14 0.03 0.07 0.20 1.00 50232 27277

> cortime(VIS3,VIS4) -0.01 0.03 -0.07 0.05 1.00 50449 26940

> cortime(VIS1,VIS5) 0.17 0.03 0.11 0.23 1.00 49366 27023

> cortime(VIS2,VIS5) 0.12 0.03 0.05 0.18 1.00 53327 28297

> cortime(VIS3,VIS5) -0.01 0.03 -0.07 0.06 1.00 52752 26884

> cortime(VIS4,VIS5) 0.38 0.03 0.32 0.43 1.00 49514 26959

> cortime(VIS1,VIS6) 0.26 0.03 0.20 0.32 1.00 45483 26765

> cortime(VIS2,VIS6) 0.20 0.03 0.14 0.27 1.00 48236 27168

> cortime(VIS3,VIS6) 0.04 0.03 -0.02 0.11 1.00 51506 27189

> cortime(VIS4,VIS6) 0.40 0.03 0.35 0.46 1.00 48730 25696

> cortime(VIS5,VIS6) 0.39 0.03 0.34 0.45 1.00 55438 25998

> cortime(VIS1,VIS7) 0.07 0.04 -0.00 0.13 1.00 66961 24586

> cortime(VIS2,VIS7) 0.09 0.03 0.02 0.15 1.00 66564 23212

> cortime(VIS3,VIS7) -0.00 0.03 -0.07 0.07 1.00 62299 24284

> cortime(VIS4,VIS7) 0.15 0.03 0.08 0.22 1.00 70101 23346

> cortime(VIS5,VIS7) 0.19 0.03 0.13 0.26 1.00 71412 24243

> cortime(VIS6,VIS7) 0.21 0.04 0.14 0.28 1.00 69307 23697

> cortime(VIS1,VIS8) 0.05 0.04 -0.02 0.12 1.00 70424 22845

> cortime(VIS2,VIS8) 0.10 0.04 0.03 0.17 1.00 71230 23497

> cortime(VIS3,VIS8) -0.03 0.04 -0.10 0.04 1.00 65689 22667

> cortime(VIS4,VIS8) 0.17 0.03 0.10 0.24 1.00 68079 23681

> cortime(VIS5,VIS8) 0.17 0.04 0.10 0.24 1.00 73436 24011

> cortime(VIS6,VIS8) 0.16 0.04 0.09 0.23 1.00 68602 23567

> cortime(VIS7,VIS8) 0.05 0.04 -0.02 0.13 1.00 68688 23661

> cortime(VIS1,VIS9) 0.03 0.04 -0.04 0.10 1.00 70389 23613

> cortime(VIS2,VIS9) -0.01 0.04 -0.08 0.07 1.00 72988 22674

> cortime(VIS3,VIS9) -0.04 0.04 -0.12 0.03 1.00 73818 23450

> cortime(VIS4,VIS9) 0.12 0.04 0.04 0.19 1.00 73299 24366

> cortime(VIS5,VIS9) 0.09 0.04 0.02 0.16 1.00 72264 22069

> cortime(VIS6,VIS9) 0.17 0.04 0.10 0.24 1.00 74018 24561

> cortime(VIS7,VIS9) 0.02 0.04 -0.06 0.09 1.00 70521 22326

> cortime(VIS8,VIS9) 0.06 0.04 -0.02 0.14 1.00 71301 22488

> cortime(VIS1,VIS10) 0.02 0.04 -0.06 0.10 1.00 62930 25421

> cortime(VIS2,VIS10) 0.13 0.04 0.05 0.20 1.00 58101 25684

> cortime(VIS3,VIS10) 0.02 0.04 -0.06 0.10 1.00 60757 24802

> cortime(VIS4,VIS10) 0.31 0.04 0.24 0.38 1.00 62762 26583

> cortime(VIS5,VIS10) 0.24 0.04 0.16 0.31 1.00 66606 25076

> cortime(VIS6,VIS10) 0.30 0.04 0.22 0.37 1.00 67998 23891

> cortime(VIS7,VIS10) 0.06 0.04 -0.03 0.15 1.00 68944 23170

> cortime(VIS8,VIS10) 0.09 0.04 0.01 0.18 1.00 71353 23530

> cortime(VIS9,VIS10) 0.08 0.05 -0.01 0.17 1.00 65710 22799

>

> Regression Coefficients:

> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS

> Intercept 4.29 0.17 3.96 4.62 1.00 56813

> BCVA_BL -0.00 0.00 -0.01 0.00 1.00 59119

> AVISIT2 0.28 0.07 0.14 0.42 1.00 29890

> AVISIT3 0.46 0.07 0.33 0.59 1.00 44348

> AVISIT4 0.86 0.08 0.70 1.01 1.00 27610

> AVISIT5 0.96 0.09 0.79 1.13 1.00 29630

> AVISIT6 1.33 0.09 1.16 1.50 1.00 28672

> AVISIT7 1.42 0.11 1.21 1.63 1.00 34514

> AVISIT8 1.71 0.11 1.49 1.94 1.00 34167

> AVISIT9 2.00 0.13 1.75 2.25 1.00 35177

> AVISIT10 2.10 0.14 1.82 2.38 1.00 33084

> RACEBlack 1.04 0.05 0.93 1.15 1.00 53517

> RACEWhite 2.01 0.05 1.90 2.11 1.00 54553

> AVISITVIS1:ARMCDTRT 0.54 0.06 0.41 0.66 1.00 34057

> AVISITVIS2:ARMCDTRT 0.72 0.08 0.57 0.88 1.00 50542

> AVISITVIS3:ARMCDTRT 1.01 0.09 0.83 1.19 1.00 48732

> AVISITVIS4:ARMCDTRT 1.10 0.10 0.91 1.31 1.00 36650

> AVISITVIS5:ARMCDTRT 1.38 0.12 1.16 1.61 1.00 38946

> AVISITVIS6:ARMCDTRT 1.63 0.12 1.40 1.86 1.00 36052

> AVISITVIS7:ARMCDTRT 2.02 0.14 1.74 2.29 1.00 45530

> AVISITVIS8:ARMCDTRT 2.35 0.15 2.06 2.64 1.00 44496

> AVISITVIS9:ARMCDTRT 2.66 0.16 2.33 2.98 1.00 44251

> AVISITVIS10:ARMCDTRT 3.07 0.18 2.71 3.43 1.00 41207

> sigma_AVISITVIS1 -0.01 0.02 -0.05 0.03 1.00 63843

> sigma_AVISITVIS2 0.23 0.02 0.18 0.27 1.00 77180

> sigma_AVISITVIS3 0.36 0.02 0.31 0.40 1.00 68147

> sigma_AVISITVIS4 0.44 0.02 0.40 0.49 1.00 54719

> sigma_AVISITVIS5 0.57 0.02 0.52 0.61 1.00 60122

> sigma_AVISITVIS6 0.58 0.02 0.54 0.63 1.00 54741

> sigma_AVISITVIS7 0.69 0.02 0.64 0.74 1.00 67848

> sigma_AVISITVIS8 0.74 0.03 0.69 0.79 1.00 73959

> sigma_AVISITVIS9 0.80 0.03 0.75 0.85 1.00 73387

> sigma_AVISITVIS10 0.84 0.03 0.79 0.90 1.00 69664

> Tail_ESS

> Intercept 25046

> BCVA_BL 22844

> AVISIT2 25900

> AVISIT3 26347

> AVISIT4 26145

> AVISIT5 25959

> AVISIT6 25061

> AVISIT7 27504

> AVISIT8 26821

> AVISIT9 25947

> AVISIT10 25296

> RACEBlack 25805

> RACEWhite 27113

> AVISITVIS1:ARMCDTRT 27968

> AVISITVIS2:ARMCDTRT 25650

> AVISITVIS3:ARMCDTRT 27016

> AVISITVIS4:ARMCDTRT 26502

> AVISITVIS5:ARMCDTRT 25407

> AVISITVIS6:ARMCDTRT 26418

> AVISITVIS7:ARMCDTRT 26547

> AVISITVIS8:ARMCDTRT 26731

> AVISITVIS9:ARMCDTRT 26034

> AVISITVIS10:ARMCDTRT 25859

> sigma_AVISITVIS1 24881

> sigma_AVISITVIS2 24252

> sigma_AVISITVIS3 23768

> sigma_AVISITVIS4 25358

> sigma_AVISITVIS5 25761

> sigma_AVISITVIS6 27071

> sigma_AVISITVIS7 24330

> sigma_AVISITVIS8 22567

> sigma_AVISITVIS9 22205

> sigma_AVISITVIS10 25249

>

> 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 = BCVA_CHG ~ BCVA_BL + ARMCD:AVISIT + AVISIT + RACE +
+     us(AVISIT | USUBJID),
+   data = mutate(
+     bcva_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: BCVA_CHG ~ BCVA_BL + ARMCD:AVISIT + AVISIT + RACE + us(AVISIT |

> USUBJID)

> Data:

> mutate(bcva_data, AVISIT = factor(as.character(AVISIT), ordered = FALSE)) (used

> 8605 observations from 1000 subjects with maximum 10 timepoints)

> Covariance: unstructured (55 variance parameters)

> Method: Satterthwaite

> Vcov Method: Asymptotic

> Inference: REML

>

> Model selection criteria:

> AIC BIC logLik deviance

> 32181.0 32451.0 -16035.5 32071.0

>

> Coefficients:

> Estimate Std. Error df t value Pr(>|t|)

> (Intercept) 4.288e+00 1.709e-01 1.065e+03 25.085 < 2e-16 ***

> BCVA_BL -9.935e-04 2.156e-03 9.905e+02 -0.461 0.645

> AVISITVIS10 2.101e+00 1.400e-01 7.025e+02 15.003 < 2e-16 ***

> AVISITVIS2 2.810e-01 7.067e-02 9.995e+02 3.976 7.51e-05 ***

> AVISITVIS3 4.573e-01 6.716e-02 9.747e+02 6.809 1.71e-11 ***

> AVISITVIS4 8.570e-01 7.636e-02 9.796e+02 11.222 < 2e-16 ***

> AVISITVIS5 9.638e-01 8.634e-02 9.630e+02 11.163 < 2e-16 ***

> AVISITVIS6 1.334e+00 8.650e-02 9.451e+02 15.421 < 2e-16 ***

> AVISITVIS7 1.417e+00 1.071e-01 8.698e+02 13.233 < 2e-16 ***

> AVISITVIS8 1.711e+00 1.145e-01 8.467e+02 14.944 < 2e-16 ***

> AVISITVIS9 1.996e+00 1.283e-01 7.784e+02 15.549 < 2e-16 ***

> RACEBlack 1.038e+00 5.496e-02 1.011e+03 18.891 < 2e-16 ***

> RACEWhite 2.005e+00 5.198e-02 9.768e+02 38.573 < 2e-16 ***

> AVISITVIS1:ARMCDTRT 5.391e-01 6.282e-02 9.859e+02 8.582 < 2e-16 ***

> AVISITVIS10:ARMCDTRT 3.072e+00 1.815e-01 6.620e+02 16.929 < 2e-16 ***

> AVISITVIS2:ARMCDTRT 7.248e-01 7.984e-02 9.803e+02 9.078 < 2e-16 ***

> AVISITVIS3:ARMCDTRT 1.012e+00 9.163e-02 9.638e+02 11.039 < 2e-16 ***

> AVISITVIS4:ARMCDTRT 1.104e+00 1.004e-01 9.653e+02 11.003 < 2e-16 ***

> AVISITVIS5:ARMCDTRT 1.383e+00 1.147e-01 9.505e+02 12.065 < 2e-16 ***

> AVISITVIS6:ARMCDTRT 1.630e+00 1.189e-01 9.157e+02 13.715 < 2e-16 ***

> AVISITVIS7:ARMCDTRT 2.016e+00 1.382e-01 8.262e+02 14.592 < 2e-16 ***

> AVISITVIS8:ARMCDTRT 2.347e+00 1.474e-01 8.041e+02 15.924 < 2e-16 ***

> AVISITVIS9:ARMCDTRT 2.658e+00 1.644e-01 7.277e+02 16.172 < 2e-16 ***

> ---

> Signif. codes: 0 '' 0.001 '' 0.01 '' 0.05 '.' 0.1 ' ' 1

>

> Covariance estimate:

> VIS1 VIS10 VIS2 VIS3 VIS4 VIS5 VIS6 VIS7 VIS8

> VIS1 0.9713 0.0587 0.0630 0.4371 0.3315 0.3056 0.4688 0.1325 0.1020

> VIS10 0.0587 5.3519 0.3761 0.0719 1.1478 0.9997 1.2558 0.3021 0.4658

> VIS2 0.0630 0.3761 1.5618 0.0871 0.2684 0.2635 0.4636 0.2180 0.2776

> VIS3 0.4371 0.0719 0.0871 2.0221 -0.0216 -0.0189 0.1102 -0.0048 -0.0993

> VIS4 0.3315 1.1478 0.2684 -0.0216 2.4113 1.0475 1.1409 0.4625 0.5659

> VIS5 0.3056 0.9997 0.2635 -0.0189 1.0475 3.0916 1.2593 0.6911 0.6308

> VIS6 0.4688 1.2558 0.4636 0.1102 1.1409 1.2593 3.1853 0.7540 0.6094

> VIS7 0.1325 0.3021 0.2180 -0.0048 0.4625 0.6911 0.7540 3.9272 0.2306

> VIS8 0.1020 0.4658 0.2776 -0.0993 0.5659 0.6308 0.6094 0.2306 4.3272

> VIS9 0.0611 0.4141 -0.0153 -0.1321 0.4085 0.3594 0.6823 0.0723 0.2683

> VIS9

> VIS1 0.0611

> VIS10 0.4141

> VIS2 -0.0153

> VIS3 -0.1321

> VIS4 0.4085

> VIS5 0.3594

> VIS6 0.6823

> VIS7 0.0723

> VIS8 0.2683

> VIS9 4.8635

# 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(bcva_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" + )

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<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <!--/html_preserve--><caption class='gt_caption'>Table 4. Comparison of parameter estimates between Bayesian and frequentist MMRMs.</caption><!--html_preserve-->
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="Variable">Variable</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="Bayesian">Bayesian</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="Frequentist">Frequentist</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="Difference">Difference</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="Relative">Relative</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="variable" class="gt_row gt_left">intercept</td>
<td headers="bayes_estimate" class="gt_row gt_right">4.2889</td>
<td headers="freq_estimate" class="gt_row gt_right">4.2881</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0009</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0002</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">bcva_bl</td>
<td headers="bayes_estimate" class="gt_row gt_right">−0.0010</td>
<td headers="freq_estimate" class="gt_row gt_right">−0.0010</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0000</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0143</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisit2</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.2806</td>
<td headers="freq_estimate" class="gt_row gt_right">0.2810</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0004</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0014</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisit3</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.4577</td>
<td headers="freq_estimate" class="gt_row gt_right">0.4573</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0005</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0010</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisit4</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.8564</td>
<td headers="freq_estimate" class="gt_row gt_right">0.8570</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0005</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0006</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisit5</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.9631</td>
<td headers="freq_estimate" class="gt_row gt_right">0.9638</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0007</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0007</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisit6</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.3333</td>
<td headers="freq_estimate" class="gt_row gt_right">1.3339</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0006</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0005</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisit7</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.4161</td>
<td headers="freq_estimate" class="gt_row gt_right">1.4167</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0006</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0005</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisit8</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.7106</td>
<td headers="freq_estimate" class="gt_row gt_right">1.7107</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0001</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0001</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisit9</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.9955</td>
<td headers="freq_estimate" class="gt_row gt_right">1.9956</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0001</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0000</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisit10</td>
<td headers="bayes_estimate" class="gt_row gt_right">2.0997</td>
<td headers="freq_estimate" class="gt_row gt_right">2.1005</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0008</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0004</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">raceblack</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.0385</td>
<td headers="freq_estimate" class="gt_row gt_right">1.0382</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0002</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0002</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">racewhite</td>
<td headers="bayes_estimate" class="gt_row gt_right">2.0054</td>
<td headers="freq_estimate" class="gt_row gt_right">2.0051</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0003</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0002</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisit1:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.5391</td>
<td headers="freq_estimate" class="gt_row gt_right">0.5391</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0000</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0001</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisit2:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.7249</td>
<td headers="freq_estimate" class="gt_row gt_right">0.7248</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0001</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0001</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisit3:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.0110</td>
<td headers="freq_estimate" class="gt_row gt_right">1.0115</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0005</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0005</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisit4:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.1049</td>
<td headers="freq_estimate" class="gt_row gt_right">1.1042</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0007</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0007</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisit5:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.3843</td>
<td headers="freq_estimate" class="gt_row gt_right">1.3834</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0009</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0007</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisit6:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.6304</td>
<td headers="freq_estimate" class="gt_row gt_right">1.6301</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0003</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0002</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisit7:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">2.0168</td>
<td headers="freq_estimate" class="gt_row gt_right">2.0160</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0009</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0004</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisit8:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">2.3471</td>
<td headers="freq_estimate" class="gt_row gt_right">2.3469</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0002</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0001</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisit9:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">2.6592</td>
<td headers="freq_estimate" class="gt_row gt_right">2.6585</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0007</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0003</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisit10:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">3.0742</td>
<td headers="freq_estimate" class="gt_row gt_right">3.0723</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0019</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0006</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">sigma_avisit1</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.9893</td>
<td headers="freq_estimate" class="gt_row gt_right">0.9855</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0037</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0038</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">sigma_avisit2</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.2557</td>
<td headers="freq_estimate" class="gt_row gt_right">1.2497</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0060</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0048</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">sigma_avisit3</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.4289</td>
<td headers="freq_estimate" class="gt_row gt_right">1.4220</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0069</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0048</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">sigma_avisit4</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.5568</td>
<td headers="freq_estimate" class="gt_row gt_right">1.5528</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0040</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0026</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">sigma_avisit5</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.7633</td>
<td headers="freq_estimate" class="gt_row gt_right">1.7583</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0050</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0028</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">sigma_avisit6</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.7888</td>
<td headers="freq_estimate" class="gt_row gt_right">1.7847</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0041</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0023</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">sigma_avisit7</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.9931</td>
<td headers="freq_estimate" class="gt_row gt_right">1.9817</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0113</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0057</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">sigma_avisit8</td>
<td headers="bayes_estimate" class="gt_row gt_right">2.0922</td>
<td headers="freq_estimate" class="gt_row gt_right">2.0802</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0120</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0058</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">sigma_avisit9</td>
<td headers="bayes_estimate" class="gt_row gt_right">2.2208</td>
<td headers="freq_estimate" class="gt_row gt_right">2.2053</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0155</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0070</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">sigma_avisit10</td>
<td headers="bayes_estimate" class="gt_row gt_right">2.3279</td>
<td headers="freq_estimate" class="gt_row gt_right">2.3134</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0145</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0063</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis1_vis2</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0489</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0512</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0023</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0441</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis1_vis3</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.3084</td>
<td headers="freq_estimate" class="gt_row gt_right">0.3119</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0036</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0114</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis2_vis3</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0482</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0490</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0008</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0164</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis1_vis4</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.2126</td>
<td headers="freq_estimate" class="gt_row gt_right">0.2166</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0040</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0184</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis2_vis4</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1351</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1383</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0033</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0237</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis3_vis4</td>
<td headers="bayes_estimate" class="gt_row gt_right">−0.0106</td>
<td headers="freq_estimate" class="gt_row gt_right">−0.0098</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0008</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0869</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis1_vis5</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1722</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1764</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0041</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0234</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis2_vis5</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1167</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1199</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0032</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0265</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis3_vis5</td>
<td headers="bayes_estimate" class="gt_row gt_right">−0.0082</td>
<td headers="freq_estimate" class="gt_row gt_right">−0.0076</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0006</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0849</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis4_vis5</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.3770</td>
<td headers="freq_estimate" class="gt_row gt_right">0.3836</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0066</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0173</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis1_vis6</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.2617</td>
<td headers="freq_estimate" class="gt_row gt_right">0.2665</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0048</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0181</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis2_vis6</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.2038</td>
<td headers="freq_estimate" class="gt_row gt_right">0.2079</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0040</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0194</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis3_vis6</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0422</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0434</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0012</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0279</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis4_vis6</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.4044</td>
<td headers="freq_estimate" class="gt_row gt_right">0.4117</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0073</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0177</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis5_vis6</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.3941</td>
<td headers="freq_estimate" class="gt_row gt_right">0.4013</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0072</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0179</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis1_vis7</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0654</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0679</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0024</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0360</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis2_vis7</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0857</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0880</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0023</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0266</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis3_vis7</td>
<td headers="bayes_estimate" class="gt_row gt_right">−0.0019</td>
<td headers="freq_estimate" class="gt_row gt_right">−0.0017</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0002</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.1039</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis4_vis7</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1464</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1503</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0040</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0263</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis5_vis7</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1941</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1983</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0042</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0214</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis6_vis7</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.2083</td>
<td headers="freq_estimate" class="gt_row gt_right">0.2132</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0048</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0227</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis1_vis8</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0478</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0497</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0019</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0382</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis2_vis8</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1044</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1068</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0024</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0225</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis3_vis8</td>
<td headers="bayes_estimate" class="gt_row gt_right">−0.0332</td>
<td headers="freq_estimate" class="gt_row gt_right">−0.0336</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0004</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0112</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis4_vis8</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1712</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1752</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0040</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0229</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis5_vis8</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1683</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1725</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0041</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0240</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis6_vis8</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1597</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1641</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0045</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0273</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis7_vis8</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0538</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0559</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0022</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0392</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis1_vis9</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0269</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0281</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0012</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0432</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis2_vis9</td>
<td headers="bayes_estimate" class="gt_row gt_right">−0.0065</td>
<td headers="freq_estimate" class="gt_row gt_right">−0.0056</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0010</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.1708</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis3_vis9</td>
<td headers="bayes_estimate" class="gt_row gt_right">−0.0416</td>
<td headers="freq_estimate" class="gt_row gt_right">−0.0421</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0005</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0124</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis4_vis9</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1160</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1193</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0033</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0273</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis5_vis9</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0898</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0927</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0029</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0313</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis6_vis9</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1692</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1733</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0041</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0238</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis7_vis9</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0153</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0165</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0013</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0761</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis8_vis9</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0569</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0585</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0016</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0267</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis1_vis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0229</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0257</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0029</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.1112</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis2_vis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1266</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1301</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0035</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0267</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis3_vis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0217</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0219</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0002</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0070</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis4_vis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.3115</td>
<td headers="freq_estimate" class="gt_row gt_right">0.3195</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0080</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0251</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis5_vis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.2385</td>
<td headers="freq_estimate" class="gt_row gt_right">0.2458</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0073</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0298</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis6_vis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.2959</td>
<td headers="freq_estimate" class="gt_row gt_right">0.3041</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0082</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0271</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis7_vis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0631</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0659</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0028</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0422</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis8_vis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0932</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0968</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0037</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0377</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis9_vis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0781</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0812</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0031</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0383</td></tr>
  </tbody>


</table>
</div><!--/html_preserve-->


```{.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	0.1695	0.1709	−0.0015	−0.0086
bcva_bl	0.0021	0.0022	0.0000	−0.0100
avisit2	0.0709	0.0707	0.0003	0.0038
avisit3	0.0675	0.0672	0.0003	0.0052
avisit4	0.0771	0.0764	0.0007	0.0094
avisit5	0.0868	0.0863	0.0005	0.0055
avisit6	0.0869	0.0865	0.0004	0.0042
avisit7	0.1081	0.1071	0.0011	0.0102
avisit8	0.1147	0.1145	0.0002	0.0017
avisit9	0.1276	0.1283	−0.0007	−0.0057
avisit10	0.1418	0.1400	0.0018	0.0130
raceblack	0.0548	0.0550	−0.0001	−0.0024
racewhite	0.0518	0.0520	−0.0001	−0.0029
avisit1:armcdtrt	0.0632	0.0628	0.0003	0.0054
avisit2:armcdtrt	0.0806	0.0798	0.0007	0.0093
avisit3:armcdtrt	0.0925	0.0916	0.0008	0.0092
avisit4:armcdtrt	0.1017	0.1004	0.0014	0.0136
avisit5:armcdtrt	0.1157	0.1147	0.0010	0.0088
avisit6:armcdtrt	0.1189	0.1189	0.0000	0.0003
avisit7:armcdtrt	0.1390	0.1382	0.0008	0.0060
avisit8:armcdtrt	0.1484	0.1474	0.0010	0.0066
avisit9:armcdtrt	0.1643	0.1644	−0.0001	−0.0004
avisit10:armcdtrt	0.1837	0.1815	0.0022	0.0122

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brms.mmrm Bayesian MMRMs using 'brms'

BCVA data comparison between Bayesian and frequentist MMRMs In brms.mmrm: Bayesian MMRMs using 'brms'

About {.unnumbered}

Prerequisites

Data

Pre-processing

Descriptive statistics

Fitting MMRMs

Bayesian model

> BCVA_CHG ~ BCVA_BL + ARMCD:AVISIT + AVISIT + RACE + unstr(time = AVISIT, gr = USUBJID)

> sigma ~ 0 + AVISIT

> Family: gaussian

> Links: mu = identity; sigma = log

> Formula: BCVA_CHG ~ BCVA_BL + ARMCD:AVISIT + AVISIT + RACE + unstr(time = AVISIT, gr = USUBJID)

> sigma ~ 0 + AVISIT

> Data: data[!is.na(data[[attr(data, "brm_outcome")]]), ] (Number of observations: 8605)

> 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) 0.05 0.03 -0.01 0.11 1.00 63561 23159

> cortime(VIS1,VIS3) 0.31 0.03 0.25 0.36 1.00 70330 25831

> cortime(VIS2,VIS3) 0.05 0.03 -0.02 0.11 1.00 67715 22226

> cortime(VIS1,VIS4) 0.21 0.03 0.15 0.27 1.00 46375 28108

> cortime(VIS2,VIS4) 0.14 0.03 0.07 0.20 1.00 50232 27277

> cortime(VIS3,VIS4) -0.01 0.03 -0.07 0.05 1.00 50449 26940

> cortime(VIS1,VIS5) 0.17 0.03 0.11 0.23 1.00 49366 27023

> cortime(VIS2,VIS5) 0.12 0.03 0.05 0.18 1.00 53327 28297

> cortime(VIS3,VIS5) -0.01 0.03 -0.07 0.06 1.00 52752 26884

> cortime(VIS4,VIS5) 0.38 0.03 0.32 0.43 1.00 49514 26959

> cortime(VIS1,VIS6) 0.26 0.03 0.20 0.32 1.00 45483 26765

> cortime(VIS2,VIS6) 0.20 0.03 0.14 0.27 1.00 48236 27168

> cortime(VIS3,VIS6) 0.04 0.03 -0.02 0.11 1.00 51506 27189

> cortime(VIS4,VIS6) 0.40 0.03 0.35 0.46 1.00 48730 25696

> cortime(VIS5,VIS6) 0.39 0.03 0.34 0.45 1.00 55438 25998

> cortime(VIS1,VIS7) 0.07 0.04 -0.00 0.13 1.00 66961 24586

> cortime(VIS2,VIS7) 0.09 0.03 0.02 0.15 1.00 66564 23212

> cortime(VIS3,VIS7) -0.00 0.03 -0.07 0.07 1.00 62299 24284

> cortime(VIS4,VIS7) 0.15 0.03 0.08 0.22 1.00 70101 23346

> cortime(VIS5,VIS7) 0.19 0.03 0.13 0.26 1.00 71412 24243

> cortime(VIS6,VIS7) 0.21 0.04 0.14 0.28 1.00 69307 23697

> cortime(VIS1,VIS8) 0.05 0.04 -0.02 0.12 1.00 70424 22845

> cortime(VIS2,VIS8) 0.10 0.04 0.03 0.17 1.00 71230 23497

> cortime(VIS3,VIS8) -0.03 0.04 -0.10 0.04 1.00 65689 22667

> cortime(VIS4,VIS8) 0.17 0.03 0.10 0.24 1.00 68079 23681

> cortime(VIS5,VIS8) 0.17 0.04 0.10 0.24 1.00 73436 24011

> cortime(VIS6,VIS8) 0.16 0.04 0.09 0.23 1.00 68602 23567

> cortime(VIS7,VIS8) 0.05 0.04 -0.02 0.13 1.00 68688 23661

> cortime(VIS1,VIS9) 0.03 0.04 -0.04 0.10 1.00 70389 23613

> cortime(VIS2,VIS9) -0.01 0.04 -0.08 0.07 1.00 72988 22674

> cortime(VIS3,VIS9) -0.04 0.04 -0.12 0.03 1.00 73818 23450

> cortime(VIS4,VIS9) 0.12 0.04 0.04 0.19 1.00 73299 24366

> cortime(VIS5,VIS9) 0.09 0.04 0.02 0.16 1.00 72264 22069

> cortime(VIS6,VIS9) 0.17 0.04 0.10 0.24 1.00 74018 24561

> cortime(VIS7,VIS9) 0.02 0.04 -0.06 0.09 1.00 70521 22326

> cortime(VIS8,VIS9) 0.06 0.04 -0.02 0.14 1.00 71301 22488

> cortime(VIS1,VIS10) 0.02 0.04 -0.06 0.10 1.00 62930 25421

> cortime(VIS2,VIS10) 0.13 0.04 0.05 0.20 1.00 58101 25684

> cortime(VIS3,VIS10) 0.02 0.04 -0.06 0.10 1.00 60757 24802

> cortime(VIS4,VIS10) 0.31 0.04 0.24 0.38 1.00 62762 26583

> cortime(VIS5,VIS10) 0.24 0.04 0.16 0.31 1.00 66606 25076

> cortime(VIS6,VIS10) 0.30 0.04 0.22 0.37 1.00 67998 23891

> cortime(VIS7,VIS10) 0.06 0.04 -0.03 0.15 1.00 68944 23170

> cortime(VIS8,VIS10) 0.09 0.04 0.01 0.18 1.00 71353 23530

> cortime(VIS9,VIS10) 0.08 0.05 -0.01 0.17 1.00 65710 22799

>

> Regression Coefficients:

> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS

> Intercept 4.29 0.17 3.96 4.62 1.00 56813

> BCVA_BL -0.00 0.00 -0.01 0.00 1.00 59119

> AVISIT2 0.28 0.07 0.14 0.42 1.00 29890

> AVISIT3 0.46 0.07 0.33 0.59 1.00 44348

> AVISIT4 0.86 0.08 0.70 1.01 1.00 27610

> AVISIT5 0.96 0.09 0.79 1.13 1.00 29630

> AVISIT6 1.33 0.09 1.16 1.50 1.00 28672

> AVISIT7 1.42 0.11 1.21 1.63 1.00 34514

> AVISIT8 1.71 0.11 1.49 1.94 1.00 34167

> AVISIT9 2.00 0.13 1.75 2.25 1.00 35177

> AVISIT10 2.10 0.14 1.82 2.38 1.00 33084

brms.mmrm
Bayesian MMRMs using 'brms'

BCVA data comparison between Bayesian and frequentist MMRMs
In brms.mmrm: Bayesian MMRMs using 'brms'