---
title: BCVA data comparison between Bayesian and frequentist MMRMs
date: "09/02/2024"
output:
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vignette: >
  %\VignetteIndexEntry{BCVA data comparison between Bayesian and frequentist MMRMs}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
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---



# 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)](#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",
+   "emmeans",
+   "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 `bcva_data` dataset contained in the `mmrm` package:


```r
> 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 800 rows and 7 variables. The primary endpoint for the analysis is `BCVA_CHG`.

* `USUBJID` (subject ID)
* `AVISITN` (visit number, numeric)
* `AVISIT` (visit number, factor)
* `ARMCD` (treatment, `TRT` or `CTL`)
* `RACE` (3-category race)
* `BCVA_BL` (BCVA at baseline)
* `BCVA_CHG` (BCVA change from baseline)

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
> bcva_data <- brm_data(
+   data = bcva_data,
+   outcome = "BCVA_CHG",
+   role = "change",
+   group = "ARMCD",
+   time = "AVISIT",
+   patient = "USUBJID",
+   baseline = "BCVA_BL",
+   reference_group = "CTL",
+   covariates = "RACE"
+ ) |>
+   mutate(ARMCD = factor(ARMCD), AVISIT = factor(AVISIT))
```

The following table shows the first rows of the dataset.


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

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<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <!--/html_preserve--><caption class='gt_caption'>Table 1. First rows of the pre-processed <code>bcva_data</code> dataset.</caption><!--html_preserve-->
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="BCVA_CHG">BCVA_CHG</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="BCVA_BL">BCVA_BL</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="ARMCD">ARMCD</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="AVISIT">AVISIT</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="USUBJID">USUBJID</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="RACE">RACE</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="BCVA_CHG" class="gt_row gt_right">5.058546</td>
<td headers="BCVA_BL" class="gt_row gt_right">71.70881</td>
<td headers="ARMCD" class="gt_row gt_center">CTL</td>
<td headers="AVISIT" class="gt_row gt_center">VIS01</td>
<td headers="USUBJID" class="gt_row gt_center">3</td>
<td headers="RACE" class="gt_row gt_left">Asian</td></tr>
    <tr><td headers="BCVA_CHG" class="gt_row gt_right">4.018582</td>
<td headers="BCVA_BL" class="gt_row gt_right">71.70881</td>
<td headers="ARMCD" class="gt_row gt_center">CTL</td>
<td headers="AVISIT" class="gt_row gt_center">VIS02</td>
<td headers="USUBJID" class="gt_row gt_center">3</td>
<td headers="RACE" class="gt_row gt_left">Asian</td></tr>
    <tr><td headers="BCVA_CHG" class="gt_row gt_right">3.572535</td>
<td headers="BCVA_BL" class="gt_row gt_right">71.70881</td>
<td headers="ARMCD" class="gt_row gt_center">CTL</td>
<td headers="AVISIT" class="gt_row gt_center">VIS03</td>
<td headers="USUBJID" class="gt_row gt_center">3</td>
<td headers="RACE" class="gt_row gt_left">Asian</td></tr>
    <tr><td headers="BCVA_CHG" class="gt_row gt_right">4.822669</td>
<td headers="BCVA_BL" class="gt_row gt_right">71.70881</td>
<td headers="ARMCD" class="gt_row gt_center">CTL</td>
<td headers="AVISIT" class="gt_row gt_center">VIS04</td>
<td headers="USUBJID" class="gt_row gt_center">3</td>
<td headers="RACE" class="gt_row gt_left">Asian</td></tr>
    <tr><td headers="BCVA_CHG" class="gt_row gt_right">7.348768</td>
<td headers="BCVA_BL" class="gt_row gt_right">71.70881</td>
<td headers="ARMCD" class="gt_row gt_center">CTL</td>
<td headers="AVISIT" class="gt_row gt_center">VIS05</td>
<td headers="USUBJID" class="gt_row gt_center">3</td>
<td headers="RACE" class="gt_row gt_left">Asian</td></tr>
    <tr><td headers="BCVA_CHG" class="gt_row gt_right">6.076615</td>
<td headers="BCVA_BL" class="gt_row gt_right">71.70881</td>
<td headers="ARMCD" class="gt_row gt_center">CTL</td>
<td headers="AVISIT" class="gt_row gt_center">VIS06</td>
<td headers="USUBJID" class="gt_row gt_center">3</td>
<td headers="RACE" class="gt_row gt_left">Asian</td></tr>
  </tbody>
  
  
</table>
</div><!--/html_preserve-->

## 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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  <!--/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;, 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>, 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;, 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>, N = 506<span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span></th>
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    <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>
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      <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>
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</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.")
```

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</style>
<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <!--/html_preserve--><caption class='gt_caption'>Table 3. Change from baseline.</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;, N = 494"><strong>CTL</strong>, N = 494</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;, N = 506"><strong>TRT</strong>, N = 506</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="label" class="gt_row gt_left">Visit  VIS01</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">    Mean (SD)</td>
<td headers="stat_1" class="gt_row gt_center">5.32 (1.23)</td>
<td headers="stat_2" class="gt_row gt_center">5.86 (1.33)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Median (IQR)</td>
<td headers="stat_1" class="gt_row gt_center">5.34 (4.51, 6.17)</td>
<td headers="stat_2" class="gt_row gt_center">5.86 (4.98, 6.81)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Range</td>
<td headers="stat_1" class="gt_row gt_center">1.83, 9.02</td>
<td headers="stat_2" class="gt_row gt_center">2.28, 10.30</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Unknown</td>
<td headers="stat_1" class="gt_row gt_center">12</td>
<td headers="stat_2" class="gt_row gt_center">5</td></tr>
    <tr><td headers="label" class="gt_row gt_left">Visit  VIS02</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">    Mean (SD)</td>
<td headers="stat_1" class="gt_row gt_center">5.59 (1.49)</td>
<td headers="stat_2" class="gt_row gt_center">6.33 (1.45)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Median (IQR)</td>
<td headers="stat_1" class="gt_row gt_center">5.53 (4.64, 6.47)</td>
<td headers="stat_2" class="gt_row gt_center">6.36 (5.34, 7.31)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Range</td>
<td headers="stat_1" class="gt_row gt_center">0.29, 10.15</td>
<td headers="stat_2" class="gt_row gt_center">2.35, 10.75</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Unknown</td>
<td headers="stat_1" class="gt_row gt_center">13</td>
<td headers="stat_2" class="gt_row gt_center">7</td></tr>
    <tr><td headers="label" class="gt_row gt_left">Visit  VIS03</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">    Mean (SD)</td>
<td headers="stat_1" class="gt_row gt_center">5.79 (1.61)</td>
<td headers="stat_2" class="gt_row gt_center">6.79 (1.71)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Median (IQR)</td>
<td headers="stat_1" class="gt_row gt_center">5.73 (4.64, 6.89)</td>
<td headers="stat_2" class="gt_row gt_center">6.82 (5.66, 7.93)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Range</td>
<td headers="stat_1" class="gt_row gt_center">1.53, 11.46</td>
<td headers="stat_2" class="gt_row gt_center">1.13, 11.49</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Unknown</td>
<td headers="stat_1" class="gt_row gt_center">23</td>
<td headers="stat_2" class="gt_row gt_center">17</td></tr>
    <tr><td headers="label" class="gt_row gt_left">Visit  VIS04</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">    Mean (SD)</td>
<td headers="stat_1" class="gt_row gt_center">6.18 (1.73)</td>
<td headers="stat_2" class="gt_row gt_center">7.29 (1.82)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Median (IQR)</td>
<td headers="stat_1" class="gt_row gt_center">6.14 (5.05, 7.40)</td>
<td headers="stat_2" class="gt_row gt_center">7.24 (6.06, 8.54)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Range</td>
<td headers="stat_1" class="gt_row gt_center">0.45, 11.49</td>
<td headers="stat_2" class="gt_row gt_center">2.07, 11.47</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Unknown</td>
<td headers="stat_1" class="gt_row gt_center">36</td>
<td headers="stat_2" class="gt_row gt_center">18</td></tr>
    <tr><td headers="label" class="gt_row gt_left">Visit  VIS05</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">    Mean (SD)</td>
<td headers="stat_1" class="gt_row gt_center">6.28 (1.97)</td>
<td headers="stat_2" class="gt_row gt_center">7.68 (1.94)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Median (IQR)</td>
<td headers="stat_1" class="gt_row gt_center">6.18 (4.96, 7.71)</td>
<td headers="stat_2" class="gt_row gt_center">7.75 (6.48, 8.93)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Range</td>
<td headers="stat_1" class="gt_row gt_center">0.87, 11.53</td>
<td headers="stat_2" class="gt_row gt_center">2.24, 14.10</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Unknown</td>
<td headers="stat_1" class="gt_row gt_center">40</td>
<td headers="stat_2" class="gt_row gt_center">35</td></tr>
    <tr><td headers="label" class="gt_row gt_left">Visit  VIS06</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">    Mean (SD)</td>
<td headers="stat_1" class="gt_row gt_center">6.69 (1.97)</td>
<td headers="stat_2" class="gt_row gt_center">8.31 (1.98)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Median (IQR)</td>
<td headers="stat_1" class="gt_row gt_center">6.64 (5.26, 8.14)</td>
<td headers="stat_2" class="gt_row gt_center">8.29 (6.92, 9.73)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Range</td>
<td headers="stat_1" class="gt_row gt_center">1.35, 12.95</td>
<td headers="stat_2" class="gt_row gt_center">1.93, 14.38</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Unknown</td>
<td headers="stat_1" class="gt_row gt_center">84</td>
<td headers="stat_2" class="gt_row gt_center">48</td></tr>
    <tr><td headers="label" class="gt_row gt_left">Visit  VIS07</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">    Mean (SD)</td>
<td headers="stat_1" class="gt_row gt_center">6.78 (2.09)</td>
<td headers="stat_2" class="gt_row gt_center">8.78 (2.11)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Median (IQR)</td>
<td headers="stat_1" class="gt_row gt_center">6.91 (5.46, 8.29)</td>
<td headers="stat_2" class="gt_row gt_center">8.67 (7.44, 10.25)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Range</td>
<td headers="stat_1" class="gt_row gt_center">-1.54, 11.99</td>
<td headers="stat_2" class="gt_row gt_center">3.21, 14.36</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Unknown</td>
<td headers="stat_1" class="gt_row gt_center">106</td>
<td headers="stat_2" class="gt_row gt_center">78</td></tr>
    <tr><td headers="label" class="gt_row gt_left">Visit  VIS08</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">    Mean (SD)</td>
<td headers="stat_1" class="gt_row gt_center">7.08 (2.25)</td>
<td headers="stat_2" class="gt_row gt_center">9.40 (2.26)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Median (IQR)</td>
<td headers="stat_1" class="gt_row gt_center">7.08 (5.57, 8.67)</td>
<td headers="stat_2" class="gt_row gt_center">9.35 (7.96, 10.86)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Range</td>
<td headers="stat_1" class="gt_row gt_center">0.97, 13.71</td>
<td headers="stat_2" class="gt_row gt_center">2.28, 15.95</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Unknown</td>
<td headers="stat_1" class="gt_row gt_center">123</td>
<td headers="stat_2" class="gt_row gt_center">86</td></tr>
    <tr><td headers="label" class="gt_row gt_left">Visit  VIS09</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">    Mean (SD)</td>
<td headers="stat_1" class="gt_row gt_center">7.39 (2.33)</td>
<td headers="stat_2" class="gt_row gt_center">10.01 (2.50)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Median (IQR)</td>
<td headers="stat_1" class="gt_row gt_center">7.47 (5.77, 9.05)</td>
<td headers="stat_2" class="gt_row gt_center">10.01 (8.19, 11.73)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Range</td>
<td headers="stat_1" class="gt_row gt_center">0.04, 14.61</td>
<td headers="stat_2" class="gt_row gt_center">4.22, 18.09</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Unknown</td>
<td headers="stat_1" class="gt_row gt_center">167</td>
<td headers="stat_2" class="gt_row gt_center">114</td></tr>
    <tr><td headers="label" class="gt_row gt_left">Visit  VIS10</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">    Mean (SD)</td>
<td headers="stat_1" class="gt_row gt_center">7.49 (2.58)</td>
<td headers="stat_2" class="gt_row gt_center">10.59 (2.36)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Median (IQR)</td>
<td headers="stat_1" class="gt_row gt_center">7.40 (5.73, 9.01)</td>
<td headers="stat_2" class="gt_row gt_center">10.71 (9.03, 12.24)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Range</td>
<td headers="stat_1" class="gt_row gt_center">-0.08, 15.75</td>
<td headers="stat_2" class="gt_row gt_center">3.24, 16.40</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Unknown</td>
<td headers="stat_1" class="gt_row gt_center">213</td>
<td headers="stat_2" class="gt_row gt_center">170</td></tr>
  </tbody>
  
  
</table>
</div><!--/html_preserve-->

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


```r
> 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()
```

<div class="figure" style="text-align: center">
<img src="bcva_figures/descr_3-1.png" alt="Figure 1. Change from baseline in BCVA over 4 visit time points." width="75%" />
<p class="caption">Figure 1. Change from baseline in BCVA over 4 visit time points.</p>
</div>

# 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,
+   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(VIS01,VIS02)     0.05      0.03    -0.01     0.11 1.00    64943    24580
#> cortime(VIS01,VIS03)     0.31      0.03     0.25     0.37 1.00    64795    24810
#> cortime(VIS02,VIS03)     0.05      0.03    -0.01     0.11 1.00    75095    23845
#> cortime(VIS01,VIS04)     0.21      0.03     0.15     0.27 1.00    48997    25597
#> cortime(VIS02,VIS04)     0.13      0.03     0.07     0.20 1.00    53822    27463
#> cortime(VIS03,VIS04)    -0.01      0.03    -0.07     0.05 1.00    49973    25432
#> cortime(VIS01,VIS05)     0.17      0.03     0.11     0.23 1.00    47899    25766
#> cortime(VIS02,VIS05)     0.12      0.03     0.05     0.18 1.00    55002    27003
#> cortime(VIS03,VIS05)    -0.01      0.03    -0.07     0.06 1.00    50643    26319
#> cortime(VIS04,VIS05)     0.38      0.03     0.32     0.43 1.00    55423    27026
#> cortime(VIS01,VIS06)     0.26      0.03     0.20     0.32 1.00    44737    28050
#> cortime(VIS02,VIS06)     0.20      0.03     0.14     0.27 1.00    54726    27792
#> cortime(VIS03,VIS06)     0.04      0.03    -0.02     0.11 1.00    52319    27395
#> cortime(VIS04,VIS06)     0.40      0.03     0.35     0.46 1.00    51509    26799
#> cortime(VIS05,VIS06)     0.39      0.03     0.34     0.45 1.00    55687    26874
#> cortime(VIS01,VIS07)     0.07      0.04    -0.00     0.13 1.00    67222    23189
#> cortime(VIS02,VIS07)     0.09      0.03     0.02     0.15 1.00    68013    23631
#> cortime(VIS03,VIS07)    -0.00      0.03    -0.07     0.07 1.00    67332    24340
#> cortime(VIS04,VIS07)     0.15      0.03     0.08     0.21 1.00    66288    24838
#> cortime(VIS05,VIS07)     0.19      0.03     0.13     0.26 1.00    70461    24208
#> cortime(VIS06,VIS07)     0.21      0.04     0.14     0.28 1.00    69109    24028
#> cortime(VIS01,VIS08)     0.05      0.04    -0.02     0.12 1.00    72277    22000
#> cortime(VIS02,VIS08)     0.10      0.04     0.03     0.17 1.00    73280    24125
#> cortime(VIS03,VIS08)    -0.03      0.04    -0.10     0.04 1.00    68920    24860
#> cortime(VIS04,VIS08)     0.17      0.03     0.10     0.24 1.00    67547    23658
#> cortime(VIS05,VIS08)     0.17      0.04     0.10     0.24 1.00    71694    24584
#> cortime(VIS06,VIS08)     0.16      0.04     0.09     0.23 1.00    69518    24126
#> cortime(VIS07,VIS08)     0.05      0.04    -0.02     0.13 1.00    62240    22292
#> cortime(VIS01,VIS09)     0.03      0.04    -0.04     0.10 1.00    68423    23721
#> cortime(VIS02,VIS09)    -0.01      0.04    -0.08     0.07 1.00    74240    21638
#> cortime(VIS03,VIS09)    -0.04      0.04    -0.12     0.03 1.00    70100    22754
#> cortime(VIS04,VIS09)     0.12      0.04     0.04     0.19 1.00    69789    22965
#> cortime(VIS05,VIS09)     0.09      0.04     0.02     0.16 1.00    72416    24164
#> cortime(VIS06,VIS09)     0.17      0.04     0.09     0.24 1.00    73834    24991
#> cortime(VIS07,VIS09)     0.02      0.04    -0.06     0.09 1.00    70843    23470
#> cortime(VIS08,VIS09)     0.06      0.04    -0.02     0.14 1.00    68629    22953
#> cortime(VIS01,VIS10)     0.02      0.04    -0.06     0.10 1.00    61683    25514
#> cortime(VIS02,VIS10)     0.13      0.04     0.05     0.20 1.00    62160    26376
#> cortime(VIS03,VIS10)     0.02      0.04    -0.06     0.10 1.00    62572    25325
#> cortime(VIS04,VIS10)     0.31      0.04     0.24     0.38 1.00    62792    23595
#> cortime(VIS05,VIS10)     0.24      0.04     0.16     0.31 1.00    64921    24888
#> cortime(VIS06,VIS10)     0.30      0.04     0.22     0.37 1.00    64007    23874
#> cortime(VIS07,VIS10)     0.06      0.05    -0.03     0.15 1.00    68579    23223
#> cortime(VIS08,VIS10)     0.09      0.04     0.01     0.18 1.00    67211    24274
#> cortime(VIS09,VIS10)     0.08      0.05    -0.01     0.17 1.00    68075    23303
#> 
#> Population-Level Effects: 
#>                      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept                4.29      0.17     3.95     4.62 1.00    54732    23560
#> BCVA_BL                 -0.00      0.00    -0.01     0.00 1.00    57530    22363
#> AVISITVIS02              0.28      0.07     0.14     0.42 1.00    31423    27345
#> AVISITVIS03              0.46      0.07     0.33     0.59 1.00    43974    27529
#> AVISITVIS04              0.86      0.08     0.71     1.01 1.00    28624    26174
#> AVISITVIS05              0.96      0.09     0.79     1.14 1.00    28737    25221
#> AVISITVIS06              1.33      0.09     1.17     1.50 1.00    28571    26120
#> AVISITVIS07              1.42      0.11     1.20     1.63 1.00    35204    26051
#> AVISITVIS08              1.71      0.12     1.48     1.94 1.00    34176    25768
#> AVISITVIS09              2.00      0.13     1.74     2.25 1.00    35133    25230
#> AVISITVIS10              2.10      0.14     1.82     2.38 1.00    32940    27567
#> RACEBlack                1.04      0.05     0.93     1.14 1.00    50156    25546
#> RACEWhite                2.01      0.05     1.90     2.11 1.00    53389    27446
#> AVISITVIS01:ARMCDTRT     0.54      0.06     0.42     0.67 1.00    34020    27641
#> AVISITVIS02:ARMCDTRT     0.72      0.08     0.57     0.88 1.00    50604    24914
#> AVISITVIS03:ARMCDTRT     1.01      0.09     0.83     1.19 1.00    47195    27254
#> AVISITVIS04:ARMCDTRT     1.10      0.10     0.91     1.30 1.00    37121    27502
#> AVISITVIS05:ARMCDTRT     1.38      0.12     1.15     1.61 1.00    37968    27516
#> AVISITVIS06:ARMCDTRT     1.63      0.12     1.40     1.86 1.00    34971    26561
#> AVISITVIS07:ARMCDTRT     2.02      0.14     1.75     2.29 1.00    45423    25443
#> AVISITVIS08:ARMCDTRT     2.35      0.15     2.05     2.64 1.00    42623    24864
#> AVISITVIS09:ARMCDTRT     2.66      0.17     2.33     2.99 1.00    42236    26456
#> AVISITVIS10:ARMCDTRT     3.07      0.18     2.72     3.43 1.00    38944    24776
#> sigma_AVISITVIS01       -0.01      0.02    -0.05     0.03 1.00    66786    24815
#> sigma_AVISITVIS02        0.23      0.02     0.18     0.27 1.00    73930    23895
#> sigma_AVISITVIS03        0.36      0.02     0.31     0.40 1.00    71987    25366
#> sigma_AVISITVIS04        0.44      0.02     0.40     0.49 1.00    59027    25866
#> sigma_AVISITVIS05        0.57      0.02     0.52     0.61 1.00    61245    24199
#> sigma_AVISITVIS06        0.58      0.02     0.53     0.63 1.00    56690    26883
#> sigma_AVISITVIS07        0.69      0.03     0.64     0.74 1.00    77367    22628
#> sigma_AVISITVIS08        0.74      0.03     0.69     0.79 1.00    71907    22165
#> sigma_AVISITVIS09        0.80      0.03     0.75     0.85 1.00    70162    24546
#> sigma_AVISITVIS10        0.84      0.03     0.79     0.90 1.00    64936    25928
#> 
#> 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 = bcva_data
+ )
```

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:        bcva_data (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  25.086  < 2e-16 ***
#> BCVA_BL              -9.933e-04  2.156e-03  9.906e+02  -0.461    0.645    
#> AVISITVIS02           2.810e-01  7.067e-02  9.995e+02   3.976 7.51e-05 ***
#> AVISITVIS03           4.573e-01  6.716e-02  9.747e+02   6.809 1.71e-11 ***
#> AVISITVIS04           8.570e-01  7.637e-02  9.795e+02  11.221  < 2e-16 ***
#> AVISITVIS05           9.638e-01  8.634e-02  9.629e+02  11.163  < 2e-16 ***
#> AVISITVIS06           1.334e+00  8.650e-02  9.450e+02  15.421  < 2e-16 ***
#> AVISITVIS07           1.417e+00  1.071e-01  8.698e+02  13.233  < 2e-16 ***
#> AVISITVIS08           1.711e+00  1.145e-01  8.467e+02  14.943  < 2e-16 ***
#> AVISITVIS09           1.996e+00  1.283e-01  7.784e+02  15.550  < 2e-16 ***
#> AVISITVIS10           2.101e+00  1.400e-01  7.025e+02  15.003  < 2e-16 ***
#> RACEBlack             1.038e+00  5.496e-02  1.011e+03  18.892  < 2e-16 ***
#> RACEWhite             2.005e+00  5.198e-02  9.769e+02  38.574  < 2e-16 ***
#> AVISITVIS01:ARMCDTRT  5.391e-01  6.281e-02  9.859e+02   8.583  < 2e-16 ***
#> AVISITVIS02:ARMCDTRT  7.248e-01  7.984e-02  9.803e+02   9.078  < 2e-16 ***
#> AVISITVIS03:ARMCDTRT  1.012e+00  9.163e-02  9.638e+02  11.039  < 2e-16 ***
#> AVISITVIS04:ARMCDTRT  1.104e+00  1.004e-01  9.653e+02  11.003  < 2e-16 ***
#> AVISITVIS05:ARMCDTRT  1.383e+00  1.147e-01  9.505e+02  12.065  < 2e-16 ***
#> AVISITVIS06:ARMCDTRT  1.630e+00  1.189e-01  9.157e+02  13.715  < 2e-16 ***
#> AVISITVIS07:ARMCDTRT  2.016e+00  1.382e-01  8.262e+02  14.592  < 2e-16 ***
#> AVISITVIS08:ARMCDTRT  2.347e+00  1.474e-01  8.041e+02  15.924  < 2e-16 ***
#> AVISITVIS09:ARMCDTRT  2.658e+00  1.644e-01  7.277e+02  16.173  < 2e-16 ***
#> AVISITVIS10:ARMCDTRT  3.072e+00  1.815e-01  6.621e+02  16.929  < 2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Covariance estimate:
#>        VIS01   VIS02   VIS03   VIS04   VIS05  VIS06   VIS07   VIS08   VIS09  VIS10
#> VIS01 0.9712  0.0630  0.4371  0.3314  0.3055 0.4686  0.1324  0.1019  0.0610 0.0585
#> VIS02 0.0630  1.5618  0.0871  0.2685  0.2635 0.4636  0.2180  0.2776 -0.0153 0.3762
#> VIS03 0.4371  0.0871  2.0221 -0.0216 -0.0189 0.1102 -0.0048 -0.0993 -0.1322 0.0719
#> VIS04 0.3314  0.2685 -0.0216  2.4114  1.0476 1.1409  0.4625  0.5660  0.4086 1.1481
#> VIS05 0.3055  0.2635 -0.0189  1.0476  3.0915 1.2592  0.6909  0.6307  0.3593 0.9999
#> VIS06 0.4686  0.4636  0.1102  1.1409  1.2592 3.1852  0.7539  0.6093  0.6821 1.2559
#> VIS07 0.1324  0.2180 -0.0048  0.4625  0.6909 0.7539  3.9273  0.2306  0.0723 0.3017
#> VIS08 0.1019  0.2776 -0.0993  0.5660  0.6307 0.6093  0.2306  4.3272  0.2682 0.4658
#> VIS09 0.0610 -0.0153 -0.1322  0.4086  0.3593 0.6821  0.0723  0.2682  4.8635 0.4138
#> VIS10 0.0585  0.3762  0.0719  1.1481  0.9999 1.2559  0.3017  0.4658  0.4138 5.3520
```

# 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("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 = "_")
+   )
```

## 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)) |>
+   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(),
+   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">4.2879</td>
<td headers="freq_estimate" class="gt_row gt_right">4.2881</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">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">0.0083</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis02</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.2816</td>
<td headers="freq_estimate" class="gt_row gt_right">0.2810</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0006</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0021</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis03</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.4579</td>
<td headers="freq_estimate" class="gt_row gt_right">0.4573</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0006</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0013</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis04</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.8570</td>
<td headers="freq_estimate" class="gt_row gt_right">0.8570</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">avisitvis05</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.9646</td>
<td headers="freq_estimate" class="gt_row gt_right">0.9638</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0008</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0008</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis06</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.3346</td>
<td headers="freq_estimate" class="gt_row gt_right">1.3339</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0007</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0005</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis07</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.4175</td>
<td headers="freq_estimate" class="gt_row gt_right">1.4167</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0008</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0005</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis08</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.7110</td>
<td headers="freq_estimate" class="gt_row gt_right">1.7107</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">avisitvis09</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.9959</td>
<td headers="freq_estimate" class="gt_row gt_right">1.9956</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">avisitvis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">2.1015</td>
<td headers="freq_estimate" class="gt_row gt_right">2.1005</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0010</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0005</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">raceblack</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.0386</td>
<td headers="freq_estimate" class="gt_row gt_right">1.0382</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0003</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0003</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">racewhite</td>
<td headers="bayes_estimate" class="gt_row gt_right">2.0056</td>
<td headers="freq_estimate" class="gt_row gt_right">2.0051</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0006</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0003</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis01:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.5393</td>
<td headers="freq_estimate" class="gt_row gt_right">0.5391</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0002</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0003</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis02:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.7243</td>
<td headers="freq_estimate" class="gt_row gt_right">0.7248</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">avisitvis03:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.0108</td>
<td headers="freq_estimate" class="gt_row gt_right">1.0115</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">avisitvis04:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.1047</td>
<td headers="freq_estimate" class="gt_row gt_right">1.1042</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">avisitvis05:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.3833</td>
<td headers="freq_estimate" class="gt_row gt_right">1.3834</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">avisitvis06:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.6293</td>
<td headers="freq_estimate" class="gt_row gt_right">1.6301</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0008</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0005</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis07:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">2.0156</td>
<td headers="freq_estimate" class="gt_row gt_right">2.0160</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">avisitvis08:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">2.3476</td>
<td headers="freq_estimate" class="gt_row gt_right">2.3469</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">avisitvis09:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">2.6579</td>
<td headers="freq_estimate" class="gt_row gt_right">2.6585</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0005</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0002</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis10:armcdtrt</td>
<td headers="bayes_estimate" class="gt_row gt_right">3.0717</td>
<td headers="freq_estimate" class="gt_row gt_right">3.0722</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0005</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0002</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">sigma_avisitvis01</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.9895</td>
<td headers="freq_estimate" class="gt_row gt_right">0.9855</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0040</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0040</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">sigma_avisitvis02</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.2555</td>
<td headers="freq_estimate" class="gt_row gt_right">1.2497</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0058</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0047</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">sigma_avisitvis03</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.4288</td>
<td headers="freq_estimate" class="gt_row gt_right">1.4220</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0068</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_avisitvis04</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.5563</td>
<td headers="freq_estimate" class="gt_row gt_right">1.5529</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0035</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0022</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">sigma_avisitvis05</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.7631</td>
<td headers="freq_estimate" class="gt_row gt_right">1.7583</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0048</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0027</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">sigma_avisitvis06</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.7882</td>
<td headers="freq_estimate" class="gt_row gt_right">1.7847</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0035</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0019</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">sigma_avisitvis07</td>
<td headers="bayes_estimate" class="gt_row gt_right">1.9928</td>
<td headers="freq_estimate" class="gt_row gt_right">1.9817</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0111</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0056</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">sigma_avisitvis08</td>
<td headers="bayes_estimate" class="gt_row gt_right">2.0926</td>
<td headers="freq_estimate" class="gt_row gt_right">2.0802</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0124</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0059</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">sigma_avisitvis09</td>
<td headers="bayes_estimate" class="gt_row gt_right">2.2204</td>
<td headers="freq_estimate" class="gt_row gt_right">2.2053</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0150</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0068</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">sigma_avisitvis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">2.3276</td>
<td headers="freq_estimate" class="gt_row gt_right">2.3134</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0142</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0061</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis01_vis02</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0490</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0511</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0021</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0420</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis01_vis03</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.3086</td>
<td headers="freq_estimate" class="gt_row gt_right">0.3119</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0033</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0106</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis02_vis03</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0483</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">−0.0154</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis01_vis04</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.2125</td>
<td headers="freq_estimate" class="gt_row gt_right">0.2165</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_vis02_vis04</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1349</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1383</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0034</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0248</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis03_vis04</td>
<td headers="bayes_estimate" class="gt_row gt_right">−0.0107</td>
<td headers="freq_estimate" class="gt_row gt_right">−0.0098</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0009</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.0910</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis01_vis05</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1723</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1763</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0041</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0230</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis02_vis05</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1165</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1199</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0034</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0283</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis03_vis05</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">0.0833</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis04_vis05</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.3764</td>
<td headers="freq_estimate" class="gt_row gt_right">0.3837</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0073</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0190</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis01_vis06</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">−0.0047</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_vis02_vis06</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">−0.0041</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0198</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis03_vis06</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0423</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0434</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0011</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0260</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis04_vis06</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.4041</td>
<td headers="freq_estimate" class="gt_row gt_right">0.4117</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0076</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_vis05_vis06</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.3938</td>
<td headers="freq_estimate" class="gt_row gt_right">0.4013</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0075</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0186</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis01_vis07</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0656</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0678</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0022</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0326</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis02_vis07</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0859</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0880</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0021</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_vis03_vis07</td>
<td headers="bayes_estimate" class="gt_row gt_right">−0.0021</td>
<td headers="freq_estimate" class="gt_row gt_right">−0.0017</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0003</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">0.2020</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis04_vis07</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1459</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1503</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0044</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0292</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis05_vis07</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1936</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1983</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0046</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_vis06_vis07</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.2080</td>
<td headers="freq_estimate" class="gt_row gt_right">0.2132</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0052</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0242</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis01_vis08</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0476</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0497</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0021</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0413</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis02_vis08</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1041</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1068</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0027</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0249</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis03_vis08</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">0.0003</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0096</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis04_vis08</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1707</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1752</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0045</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0255</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis05_vis08</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1682</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1724</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0043</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0248</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis06_vis08</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1595</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1641</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0046</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0282</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis07_vis08</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">−0.0385</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis01_vis09</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">−0.0426</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis02_vis09</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">0.1763</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis03_vis09</td>
<td headers="bayes_estimate" class="gt_row gt_right">−0.0418</td>
<td headers="freq_estimate" class="gt_row gt_right">−0.0421</td>
<td headers="diff_estimate" class="gt_row gt_right">0.0004</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0084</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis04_vis09</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">−0.0280</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis05_vis09</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0894</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0927</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0033</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0351</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis06_vis09</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">−0.0235</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis07_vis09</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">−0.0012</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0748</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis08_vis09</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0565</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0585</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0020</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0337</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis01_vis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0230</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0257</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0027</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.1053</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis02_vis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.1264</td>
<td headers="freq_estimate" class="gt_row gt_right">0.1301</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0037</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0287</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis03_vis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0215</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0219</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0003</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0142</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis04_vis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.3115</td>
<td headers="freq_estimate" class="gt_row gt_right">0.3196</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0081</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0254</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis05_vis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.2384</td>
<td headers="freq_estimate" class="gt_row gt_right">0.2458</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0075</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0303</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis06_vis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.2957</td>
<td headers="freq_estimate" class="gt_row gt_right">0.3042</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0085</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_vis07_vis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0627</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0658</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0031</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0478</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis08_vis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0931</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">−0.0380</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">correlation_vis09_vis10</td>
<td headers="bayes_estimate" class="gt_row gt_right">0.0780</td>
<td headers="freq_estimate" class="gt_row gt_right">0.0811</td>
<td headers="diff_estimate" class="gt_row gt_right">−0.0031</td>
<td headers="diff_relative_estimate" class="gt_row gt_right">−0.0380</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"
+   )
```

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<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <!--/html_preserve--><caption class='gt_caption'>Table 5. Comparison of parameter standard errors 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_se" class="gt_row gt_right">0.1710</td>
<td headers="freq_se" class="gt_row gt_right">0.1709</td>
<td headers="diff_se" class="gt_row gt_right">0.0001</td>
<td headers="diff_relative_se" class="gt_row gt_right">0.0005</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">bcva_bl</td>
<td headers="bayes_se" class="gt_row gt_right">0.0022</td>
<td headers="freq_se" class="gt_row gt_right">0.0022</td>
<td headers="diff_se" class="gt_row gt_right">0.0000</td>
<td headers="diff_relative_se" class="gt_row gt_right">0.0017</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis02</td>
<td headers="bayes_se" class="gt_row gt_right">0.0713</td>
<td headers="freq_se" class="gt_row gt_right">0.0707</td>
<td headers="diff_se" class="gt_row gt_right">0.0006</td>
<td headers="diff_relative_se" class="gt_row gt_right">0.0091</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis03</td>
<td headers="bayes_se" class="gt_row gt_right">0.0671</td>
<td headers="freq_se" class="gt_row gt_right">0.0672</td>
<td headers="diff_se" class="gt_row gt_right">−0.0001</td>
<td headers="diff_relative_se" class="gt_row gt_right">−0.0014</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis04</td>
<td headers="bayes_se" class="gt_row gt_right">0.0760</td>
<td headers="freq_se" class="gt_row gt_right">0.0764</td>
<td headers="diff_se" class="gt_row gt_right">−0.0003</td>
<td headers="diff_relative_se" class="gt_row gt_right">−0.0046</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis05</td>
<td headers="bayes_se" class="gt_row gt_right">0.0869</td>
<td headers="freq_se" class="gt_row gt_right">0.0863</td>
<td headers="diff_se" class="gt_row gt_right">0.0006</td>
<td headers="diff_relative_se" class="gt_row gt_right">0.0066</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis06</td>
<td headers="bayes_se" class="gt_row gt_right">0.0866</td>
<td headers="freq_se" class="gt_row gt_right">0.0865</td>
<td headers="diff_se" class="gt_row gt_right">0.0001</td>
<td headers="diff_relative_se" class="gt_row gt_right">0.0017</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis07</td>
<td headers="bayes_se" class="gt_row gt_right">0.1084</td>
<td headers="freq_se" class="gt_row gt_right">0.1071</td>
<td headers="diff_se" class="gt_row gt_right">0.0014</td>
<td headers="diff_relative_se" class="gt_row gt_right">0.0129</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis08</td>
<td headers="bayes_se" class="gt_row gt_right">0.1159</td>
<td headers="freq_se" class="gt_row gt_right">0.1145</td>
<td headers="diff_se" class="gt_row gt_right">0.0014</td>
<td headers="diff_relative_se" class="gt_row gt_right">0.0123</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis09</td>
<td headers="bayes_se" class="gt_row gt_right">0.1305</td>
<td headers="freq_se" class="gt_row gt_right">0.1283</td>
<td headers="diff_se" class="gt_row gt_right">0.0022</td>
<td headers="diff_relative_se" class="gt_row gt_right">0.0168</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis10</td>
<td headers="bayes_se" class="gt_row gt_right">0.1409</td>
<td headers="freq_se" class="gt_row gt_right">0.1400</td>
<td headers="diff_se" class="gt_row gt_right">0.0009</td>
<td headers="diff_relative_se" class="gt_row gt_right">0.0062</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">raceblack</td>
<td headers="bayes_se" class="gt_row gt_right">0.0546</td>
<td headers="freq_se" class="gt_row gt_right">0.0550</td>
<td headers="diff_se" class="gt_row gt_right">−0.0003</td>
<td headers="diff_relative_se" class="gt_row gt_right">−0.0058</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">racewhite</td>
<td headers="bayes_se" class="gt_row gt_right">0.0521</td>
<td headers="freq_se" class="gt_row gt_right">0.0520</td>
<td headers="diff_se" class="gt_row gt_right">0.0001</td>
<td headers="diff_relative_se" class="gt_row gt_right">0.0020</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis01:armcdtrt</td>
<td headers="bayes_se" class="gt_row gt_right">0.0634</td>
<td headers="freq_se" class="gt_row gt_right">0.0628</td>
<td headers="diff_se" class="gt_row gt_right">0.0006</td>
<td headers="diff_relative_se" class="gt_row gt_right">0.0093</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis02:armcdtrt</td>
<td headers="bayes_se" class="gt_row gt_right">0.0806</td>
<td headers="freq_se" class="gt_row gt_right">0.0798</td>
<td headers="diff_se" class="gt_row gt_right">0.0007</td>
<td headers="diff_relative_se" class="gt_row gt_right">0.0094</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis03:armcdtrt</td>
<td headers="bayes_se" class="gt_row gt_right">0.0922</td>
<td headers="freq_se" class="gt_row gt_right">0.0916</td>
<td headers="diff_se" class="gt_row gt_right">0.0005</td>
<td headers="diff_relative_se" class="gt_row gt_right">0.0060</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis04:armcdtrt</td>
<td headers="bayes_se" class="gt_row gt_right">0.0996</td>
<td headers="freq_se" class="gt_row gt_right">0.1004</td>
<td headers="diff_se" class="gt_row gt_right">−0.0008</td>
<td headers="diff_relative_se" class="gt_row gt_right">−0.0079</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis05:armcdtrt</td>
<td headers="bayes_se" class="gt_row gt_right">0.1159</td>
<td headers="freq_se" class="gt_row gt_right">0.1147</td>
<td headers="diff_se" class="gt_row gt_right">0.0013</td>
<td headers="diff_relative_se" class="gt_row gt_right">0.0110</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis06:armcdtrt</td>
<td headers="bayes_se" class="gt_row gt_right">0.1191</td>
<td headers="freq_se" class="gt_row gt_right">0.1189</td>
<td headers="diff_se" class="gt_row gt_right">0.0002</td>
<td headers="diff_relative_se" class="gt_row gt_right">0.0017</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis07:armcdtrt</td>
<td headers="bayes_se" class="gt_row gt_right">0.1390</td>
<td headers="freq_se" class="gt_row gt_right">0.1382</td>
<td headers="diff_se" class="gt_row gt_right">0.0008</td>
<td headers="diff_relative_se" class="gt_row gt_right">0.0061</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis08:armcdtrt</td>
<td headers="bayes_se" class="gt_row gt_right">0.1500</td>
<td headers="freq_se" class="gt_row gt_right">0.1474</td>
<td headers="diff_se" class="gt_row gt_right">0.0027</td>
<td headers="diff_relative_se" class="gt_row gt_right">0.0180</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis09:armcdtrt</td>
<td headers="bayes_se" class="gt_row gt_right">0.1677</td>
<td headers="freq_se" class="gt_row gt_right">0.1644</td>
<td headers="diff_se" class="gt_row gt_right">0.0033</td>
<td headers="diff_relative_se" class="gt_row gt_right">0.0200</td></tr>
    <tr><td headers="variable" class="gt_row gt_left">avisitvis10:armcdtrt</td>
<td headers="bayes_se" class="gt_row gt_right">0.1831</td>
<td headers="freq_se" class="gt_row gt_right">0.1815</td>
<td headers="diff_se" class="gt_row gt_right">0.0016</td>
<td headers="diff_relative_se" class="gt_row gt_right">0.0091</td></tr>
  </tbody>
  
  
</table>
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