-plot of chunk trace13a
-
-plot of chunk trace13a
-
When the sampler has converged the chains show one horizontal band, as in the above figure.
@@ -81,10 +78,7 @@ traceplot(mod13a, ncol = 3, use_ggplot = TRUE) +
scale_color_brewer(palette = 'Dark2')
```
-
-plot of chunk ggtrace15a
-
### Density plot
@@ -103,10 +97,7 @@ densplot(mod13a, ncol = 3, col = c("darkred", "darkblue", "darkgreen"),
col = grey(0.8))))
```
-
-plot of chunk dens13a-2
-
or marking the posterior mean and 2.5% and 97.5% quantiles:
``` r
@@ -122,10 +113,7 @@ densplot(mod13a, ncol = 3,
)
```
-
-plot of chunk densplot15a
-
Like with `traceplot()` it is possible to use the **ggplot2** version of `densplot()`
@@ -179,10 +167,7 @@ p13a +
labels = c('JointAI', 'compl.case'))
```
-
-plot of chunk ggdens15a
-
## Model Summary
@@ -319,10 +304,7 @@ estimated posterior distribution.
The figure visualizes three examples of posterior distributions
and the corresponding minimum of $Pr(\theta > 0)$ and $Pr(\theta < 0)$
(shaded area):
-
-plot of chunk tailprob
-
## Evaluation criteria
@@ -411,10 +393,7 @@ plot(MC_error(mod13a)) # left panel: all iterations 101:600
plot(MC_error(mod13a, end = 250)) # right panel: iterations 101:250
```
-
-plot of chunk MCE15a
-
## Subset of output {#sec:subset}
@@ -515,10 +494,7 @@ densplot(mod13c, subset = list(analysis_main = FALSE,
other = c('beta[4]', 'beta[5]')), nrow = 1)
```
-
-plot of chunk densplot13c
-
This also works when a subset of the imputed values should be displayed:
@@ -537,10 +513,7 @@ sub3
traceplot(mod13d, subset = list(analysis_main = FALSE, other = sub3), ncol = 2)
```
-
-plot of chunk trace15d
-
When the number of imputed values is large or in order to check convergence
of random effects, it may not be feasible to plot and inspect all trace plots.
@@ -561,10 +534,7 @@ traceplot(mod13e, subset = list(analysis_main = FALSE,
other = sample(ri, size = 12)), ncol = 4)
```
-
-plot of chunk ri
-
### Subset of MCMC samples
@@ -586,47 +556,32 @@ mod14 <- lm_imp(SBP ~ gender + WC + alc + creat, data = NHANES, n.iter = 100,
densplot(mod14, ncol = 3)
```
-
-plot of chunk mod14
-
``` r
densplot(mod14, exclude_chains = c(2,4), ncol = 3)
```
-
-plot of chunk mod14
-
``` r
traceplot(mod14, thin = 10, ncol = 3)
```
-
-plot of chunk trace14
-
``` r
traceplot(mod14, start = 150, ncol = 3)
```
-
-plot of chunk trace14
-
``` r
traceplot(mod14, end = 120, ncol = 3)
```
-
-plot of chunk trace14
-
@@ -697,10 +652,7 @@ matplot(pred$newdata$age, pred$fitted,
xlab = 'age in months', ylab = 'predicted value')
```
-
-plot of chunk predplot
-
It is possible to have multiple variables vary and to set values for these
variables:
@@ -734,10 +686,7 @@ ggplot(pred$newdata, aes(x = age, y = fit, color = factor(HEIGHT_M),
scale_y_continuous(name = 'Expected BMI', breaks = seq(15, 18, 0.5))
```
-
-plot of chunk predplotgg
-
@@ -785,8 +734,5 @@ of the observed and imputed values.
plot_imp_distr(impDF, nrow = 1)
```
-
-plot of chunk plotimpdistr
-
diff --git a/vignettes/AfterFitting.Rmd.orig b/vignettes/AfterFitting.Rmd.orig
index dd256d5e..f7295bd2 100644
--- a/vignettes/AfterFitting.Rmd.orig
+++ b/vignettes/AfterFitting.Rmd.orig
@@ -19,6 +19,8 @@ knitr::opts_chunk$set(
fig.width = 7,
out.width = '100%',
dev = "svglite",
+ fig.cap = "",
+ fig.scap = "",
fig.align = 'center',
fig.path = "figures_AfterFitting/"
)
@@ -79,7 +81,7 @@ rows and columns for the layout of the grid of plots.
With the argument `use_ggplot` it is possible to get a [**ggplot2**](https://CRAN.R-project.org/package=ggplot2)
version of the traceplot. It can be extended using standard **ggplot2** syntax.
-```{r ggtrace15a, fig.width = 7, fig.height = 3.5, out.width = '100%'}
+```{r ggtrace15a, fig.width = 7, fig.height = 3.5, out.width = '100%', message = FALSE}
library(ggplot2)
traceplot(mod13a, ncol = 3, use_ggplot = TRUE) +
theme(legend.position = 'bottom') +
diff --git a/vignettes/MinimalExample.Rmd b/vignettes/MinimalExample.Rmd
index 78134533..485c2a4f 100644
--- a/vignettes/MinimalExample.Rmd
+++ b/vignettes/MinimalExample.Rmd
@@ -2,7 +2,7 @@
title: "Minimal Example"
description: "A minimal example to demonstrate the usage of the R package JointAI."
author: "Nicole Erler"
-date: January 03, 2026
+date: February 22, 2026
resource_files:
- man/figures/JointAI_square.png
opengraph:
@@ -71,10 +71,7 @@ Convergence can be evaluated visually with a trace plot.
traceplot(lm1)
```
-
-plot of chunk results_lm1
-
The function [`traceplot()`](https://nerler.github.io/JointAI/reference/traceplot.html)
produces a plot of the sampled values across
@@ -105,22 +102,22 @@ summary(lm1)
#>
#> Posterior summary:
#> Mean SD 2.5% 97.5% tail-prob. GR-crit MCE/SD
-#> (Intercept) 59.667 22.5323 13.8296 104.999 0.00800 1.00 0.0263
-#> genderfemale -3.088 2.2801 -7.3085 1.497 0.18533 1.00 0.0258
-#> age 0.367 0.0730 0.2256 0.512 0.00000 1.01 0.0262
-#> raceOther Hispanic 0.857 5.1357 -8.8799 10.758 0.88933 1.00 0.0258
-#> raceNon-Hispanic White -1.286 3.0295 -6.9351 4.658 0.66267 1.00 0.0258
-#> raceNon-Hispanic Black 9.061 3.4765 2.3074 15.679 0.01067 1.01 0.0258
-#> raceother 3.919 3.4728 -2.6699 11.094 0.25467 1.00 0.0258
-#> WC 0.243 0.0811 0.0835 0.401 0.00267 1.00 0.0258
-#> alc>=1 7.249 2.2744 2.5975 11.650 0.00000 1.00 0.0325
-#> educhigh -3.392 2.2355 -8.0464 0.830 0.10400 1.01 0.0267
-#> albu 5.390 4.1032 -2.9169 13.239 0.18133 1.01 0.0260
-#> bili -5.456 4.8697 -14.3350 4.293 0.25467 1.01 0.0285
+#> (Intercept) 60.879 22.8704 19.1247 106.227 0.0040 1.00 0.0271
+#> genderfemale -3.177 2.2549 -7.6423 1.099 0.1640 1.00 0.0258
+#> age 0.364 0.0708 0.2196 0.505 0.0000 1.01 0.0258
+#> raceOther Hispanic 0.518 5.0580 -9.2403 10.675 0.9373 1.00 0.0258
+#> raceNon-Hispanic White -1.451 3.0937 -7.2049 4.686 0.6227 1.00 0.0276
+#> raceNon-Hispanic Black 9.047 3.6389 2.0441 16.042 0.0147 1.00 0.0267
+#> raceother 3.686 3.4550 -3.0852 10.725 0.2653 1.00 0.0268
+#> WC 0.236 0.0822 0.0754 0.392 0.0040 1.00 0.0258
+#> alc>=1 7.272 2.3870 2.5418 11.763 0.0040 1.03 0.0284
+#> educhigh -3.397 2.1457 -7.3896 0.869 0.1120 1.00 0.0258
+#> albu 5.320 4.0590 -2.7810 12.922 0.1987 1.00 0.0291
+#> bili -5.516 4.9060 -15.2486 4.592 0.2413 1.01 0.0274
#>
#> Posterior summary of residual std. deviation:
#> Mean SD 2.5% 97.5% GR-crit MCE/SD
-#> sigma_SBP 13.2 0.716 11.9 14.7 1 0.0281
+#> sigma_SBP 13.2 0.738 11.9 14.7 1.02 0.0282
#>
#>
#> MCMC settings:
@@ -155,10 +152,7 @@ $$2\times\min\left\{Pr(\theta > 0), Pr(\theta < 0)\right\}$$
In the following graphics, the shaded areas represent the minimum of
$Pr(\theta > 0)$ and $Pr(\theta < 0)$:
-
-plot of chunk tailprob
-
#### Gelman-Rubin criterion
The Gelman-Rubin^[Gelman, A and Rubin, DB (1992) Inference from iterative
@@ -206,10 +200,7 @@ The posterior distributions can be visualized using the function `densplot()`:
densplot(lm1)
```
-
-plot of chunk densplot
-
By default, `densplot()` plots the empirical distribution of each of the chains
separately. When `joined = TRUE` the distributions of the combined chains
diff --git a/vignettes/MinimalExample.Rmd.orig b/vignettes/MinimalExample.Rmd.orig
index 543edf2a..d74985e4 100644
--- a/vignettes/MinimalExample.Rmd.orig
+++ b/vignettes/MinimalExample.Rmd.orig
@@ -29,6 +29,8 @@ knitr::opts_chunk$set(
fig.width = 7,
fig.align = 'center',
out.width = '100%',
+ fig.cap = "",
+ fig.scap = "",
dev = "svglite",
fig.path = "figures_MinimalExample/"
)
diff --git a/vignettes/ModelSpecification.Rmd b/vignettes/ModelSpecification.Rmd
index 438b543a..f93902c0 100644
--- a/vignettes/ModelSpecification.Rmd
+++ b/vignettes/ModelSpecification.Rmd
@@ -1,6 +1,6 @@
---
title: "Model Specification"
-date: January 03, 2026
+date: February 22, 2026
output:
rmarkdown::html_vignette:
toc: true
@@ -685,14 +685,16 @@ distributions they are ordered so that longitudinal (level-1) variables may have
baseline (level-2) variables in their linear predictors but not vice versa.
For example:
-$$\begin{align}
+
+\begin{align}
p(y, x, b, \theta) = & p(y \mid x_1, ..., x_4, b_y, \theta_y) && \text{analysis model}\\
& p(x_1\mid \theta_{x1}) && \text{model for a complete baseline covariate}\\
& p(x_2\mid x_1, \theta_{x2}) && \text{model for an incomplete baseline covariate}\\
& p(x_3\mid x_1, x_2, b_{x3}, \theta_{x3}) && \text{model for a complete longitudinal covariate}\\
& p(x_4\mid x_1, x_2, x_3, b_{x4}, \theta_{x4}) && \text{model for an incomplete longitudinal covariate}\\
& p(b_y|\theta_b) p(b_{x3}|\theta_b) p(b_{x4}|\theta_b) && \text{models for the random effects}\\
-& p(\theta_y) p(\theta_{x1}) \ldots p(\theta_{x4}) p(\theta_b) && \text{prior distributions}\end{align}$$
+& p(\theta_y) p(\theta_{x1}) \ldots p(\theta_{x4}) p(\theta_b) && \text{prior distributions}
+\end{align}
Since the parameter vectors $\theta_{x1}$, $\theta_{x2}$, ... are assumed to be
a priori independent, and furthermore $x_1$ is completely observed and modelled
@@ -1125,7 +1127,7 @@ refs_mod10 <- set_refcat(NHANES, formula = formula(mod10b))
#> How do you want to specify the reference categories?
#>
#> 1: Use the first category for each variable.
-#> 2: Use the last category for each variable.
+#> 2: Use the last category for each variabe.
#> 3: Use the largest category for each variable.
#> 4: Specify the reference categories individually.
```
diff --git a/vignettes/ModelSpecification.Rmd.orig b/vignettes/ModelSpecification.Rmd.orig
index 8b40382c..47bb46ef 100644
--- a/vignettes/ModelSpecification.Rmd.orig
+++ b/vignettes/ModelSpecification.Rmd.orig
@@ -17,6 +17,8 @@ knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>",
fig.width = 7,
+ fig.cap = "",
+ fig.scap = "",
fig.align = 'center'
)
library(JointAI)
@@ -536,14 +538,16 @@ distributions they are ordered so that longitudinal (level-1) variables may have
baseline (level-2) variables in their linear predictors but not vice versa.
For example:
-$$\begin{align}
+
+\begin{align}
p(y, x, b, \theta) = & p(y \mid x_1, ..., x_4, b_y, \theta_y) && \text{analysis model}\\
& p(x_1\mid \theta_{x1}) && \text{model for a complete baseline covariate}\\
& p(x_2\mid x_1, \theta_{x2}) && \text{model for an incomplete baseline covariate}\\
& p(x_3\mid x_1, x_2, b_{x3}, \theta_{x3}) && \text{model for a complete longitudinal covariate}\\
& p(x_4\mid x_1, x_2, x_3, b_{x4}, \theta_{x4}) && \text{model for an incomplete longitudinal covariate}\\
& p(b_y|\theta_b) p(b_{x3}|\theta_b) p(b_{x4}|\theta_b) && \text{models for the random effects}\\
-& p(\theta_y) p(\theta_{x1}) \ldots p(\theta_{x4}) p(\theta_b) && \text{prior distributions}\end{align}$$
+& p(\theta_y) p(\theta_{x1}) \ldots p(\theta_{x4}) p(\theta_b) && \text{prior distributions}
+\end{align}
Since the parameter vectors $\theta_{x1}$, $\theta_{x2}$, ... are assumed to be
a priori independent, and furthermore $x_1$ is completely observed and modelled
diff --git a/vignettes/SelectingParameters.Rmd b/vignettes/SelectingParameters.Rmd
index e0cda946..0ef36239 100644
--- a/vignettes/SelectingParameters.Rmd
+++ b/vignettes/SelectingParameters.Rmd
@@ -1,7 +1,7 @@
---
title: "Parameter Selection"
subtitle: "How to specify which parameters to monitor and display in the results"
-date: January 03, 2026
+date: February 22, 2026
output:
rmarkdown::html_vignette:
toc: true
@@ -217,20 +217,20 @@ terms and transformations of variables.
``` r
head(lm2$data_list$M_lvlone)
-#> SBP alc occup smoke WC (Intercept) age alc>=1 smokeformer smokecurrent
-#> 10 108.0000 NA 1 1 99.0 1 35 NA NA NA
-#> 14 105.3333 0 1 1 82.7 1 38 NA NA NA
-#> 41 110.0000 0 3 1 94.9 1 78 NA NA NA
-#> 77 106.0000 1 2 1 82.4 1 23 NA NA NA
-#> 91 114.6667 0 3 1 93.1 1 40 NA NA NA
-#> 105 139.3333 1 NA 3 105.4 1 54 NA NA NA
-#> occuplooking for work occupnot working
-#> 10 NA NA
-#> 14 NA NA
-#> 41 NA NA
-#> 77 NA NA
-#> 91 NA NA
-#> 105 NA NA
+#> SBP alc occup smoke WC (Intercept) age alc>=1 smokeformer
+#> 10 108.0000 NA 1 1 99.0 1 35 NA NA
+#> 14 105.3333 0 1 1 82.7 1 38 NA NA
+#> 41 110.0000 0 3 1 94.9 1 78 NA NA
+#> 77 106.0000 1 2 1 82.4 1 23 NA NA
+#> 91 114.6667 0 3 1 93.1 1 40 NA NA
+#> 105 139.3333 1 NA 3 105.4 1 54 NA NA
+#> smokecurrent occuplooking for work occupnot working
+#> 10 NA NA NA
+#> 14 NA NA NA
+#> 41 NA NA NA
+#> 77 NA NA NA
+#> 91 NA NA NA
+#> 105 NA NA NA
```
@@ -284,8 +284,8 @@ list_models(lm2)
#> family: gaussian
#> link: identity
#> * Predictor variables:
-#> (Intercept), age, WC, alc>=1, smokeformer, smokecurrent, occuplooking for
-#> work, occupnot working
+#> (Intercept), age, WC, alc>=1, smokeformer, smokecurrent,
+#> occuplooking for work, occupnot working
#> * Regression coefficients:
#> beta[1:8] (normal prior(s) with mean 0 and precision 1e-04)
#> * Precision of "SBP" :
@@ -297,8 +297,8 @@ list_models(lm2)
#> link: logit
#> * Reference category: "<1"
#> * Predictor variables:
-#> (Intercept), age, WC, smokeformer, smokecurrent, occuplooking for work,
-#> occupnot working
+#> (Intercept), age, WC, smokeformer, smokecurrent,
+#> occuplooking for work, occupnot working
#> * Regression coefficients:
#> alpha[1:7] (normal prior(s) with mean 0 and precision 1e-04)
#>
@@ -446,9 +446,11 @@ lme3b <- lme_imp(bmi ~ age + EDUC, random = ~age | ID, data = simLong, n.adapt =
RinvD_main = FALSE,
ranef_main = FALSE))
#> Warning:
-#> It is currently not possible to use "contr.poly" for incomplete categorical
-#> covariates. I will use "contr.treatment" instead. You can specify (globally)
-#> which types of contrasts are used by changing "options('contrasts')".
+#> It is currently not possible to use "contr.poly" for
+#> incomplete categorical covariates. I will use
+#> "contr.treatment" instead. You can specify (globally) which
+#> types of contrasts are used by changing
+#> "options('contrasts')".
parameters(lme3b)
#> outcome outcat varname coef
@@ -525,16 +527,16 @@ summary(lm5)
#>
#>
#> Posterior summary:
-#> Mean SD 2.5% 97.5% tail-prob. GR-crit MCE/SD
-#> (Intercept) 81.096 9.6988 60.971 97.166 0.00000 1.01 0.0577
-#> genderfemale 0.676 2.5491 -3.776 5.439 0.81333 1.00 0.0577
-#> WC 0.300 0.0699 0.164 0.425 0.00000 1.02 0.0577
-#> alc>=1 6.645 2.2974 2.500 11.345 0.00667 1.02 0.0689
-#> creat 8.206 8.1859 -7.115 24.301 0.34000 1.07 0.0577
+#> Mean SD 2.5% 97.5% tail-prob. GR-crit MCE/SD
+#> (Intercept) 81.7399 9.6925 63.439 100.547 0.000 1.04 0.0577
+#> genderfemale 0.0802 2.6383 -4.441 5.527 0.933 1.00 0.0577
+#> WC 0.3074 0.0767 0.168 0.451 0.000 1.01 0.0577
+#> alc>=1 6.3855 2.5770 1.542 11.284 0.020 1.01 0.0670
+#> creat 6.8877 7.4473 -5.536 21.447 0.387 1.13 0.0586
#>
#> Posterior summary of residual std. deviation:
#> Mean SD 2.5% 97.5% GR-crit MCE/SD
-#> sigma_SBP 14.4 0.806 13 16.3 1.02 0.0577
+#> sigma_SBP 14.4 0.704 13.1 15.9 1 0.0577
#>
#>
#> MCMC settings:
@@ -549,10 +551,7 @@ summary(lm5)
traceplot(lm5)
```
-
-plot of chunk unnamed-chunk-9
-
``` r
@@ -560,10 +559,7 @@ traceplot(lm5)
densplot(lm5)
```
-
-plot of chunk unnamed-chunk-9
-
``` r
@@ -572,26 +568,26 @@ GR_crit(lm5)
#> Potential scale reduction factors:
#>
#> Point est. Upper C.I.
-#> (Intercept) 1.00 1.01
-#> genderfemale 1.00 1.00
-#> WC 1.00 1.02
-#> alc>=1 1.00 1.02
-#> creat 1.03 1.07
-#> sigma_SBP 1.00 1.02
+#> (Intercept) 1.009 1.04
+#> genderfemale 0.999 1.00
+#> WC 1.003 1.01
+#> alc>=1 1.003 1.01
+#> creat 1.035 1.13
+#> sigma_SBP 0.997 1.00
#>
#> Multivariate psrf
#>
-#> 1.03
+#> 1.04
# Monte Carlo Error of the MCMC sample
MC_error(lm5)
-#> est MCSE SD MCSE/SD
-#> (Intercept) 81.10 0.560 9.70 0.058
-#> genderfemale 0.68 0.147 2.55 0.058
-#> WC 0.30 0.004 0.07 0.058
-#> alc>=1 6.64 0.158 2.30 0.069
-#> creat 8.21 0.473 8.19 0.058
-#> sigma_SBP 14.42 0.047 0.81 0.058
+#> est MCSE SD MCSE/SD
+#> (Intercept) 81.74 0.5596 9.692 0.058
+#> genderfemale 0.08 0.1523 2.638 0.058
+#> WC 0.31 0.0044 0.077 0.058
+#> alc>=1 6.39 0.1727 2.577 0.067
+#> creat 6.89 0.4361 7.447 0.059
+#> sigma_SBP 14.38 0.0407 0.704 0.058
```
When `analysis_main` was not switched on the default behaviour is that all
@@ -607,10 +603,7 @@ lm4 <- lm_imp(SBP ~ gender + WC + alc + creat,
traceplot(lm4, ncol = 4)
```
-
-plot of chunk unnamed-chunk-10
-
### Select a subset of the variables to display
To display other parts of the MCMC sample, `subset` needs to be specified:
@@ -621,21 +614,21 @@ GR_crit(lm5, subset = c(analysis_main = FALSE, other_models = TRUE))
#> Potential scale reduction factors:
#>
#> Point est. Upper C.I.
-#> alc: (Intercept) 1.081 1.268
-#> alc: genderfemale 1.104 1.327
-#> alc: WC 1.015 1.061
-#> alc: creat 1.061 1.210
-#> creat: (Intercept) 1.015 1.056
-#> creat: genderfemale 0.996 0.998
-#> creat: WC 1.018 1.060
-#> WC: (Intercept) 1.003 1.021
-#> WC: genderfemale 1.001 1.013
-#> sigma_creat 1.006 1.022
-#> sigma_WC 1.007 1.010
+#> alc: (Intercept) 0.997 1.001
+#> alc: genderfemale 1.005 1.025
+#> alc: WC 1.030 1.086
+#> alc: creat 1.007 1.034
+#> creat: (Intercept) 0.996 0.997
+#> creat: genderfemale 0.999 1.000
+#> creat: WC 0.995 0.996
+#> WC: (Intercept) 1.011 1.039
+#> WC: genderfemale 1.003 1.024
+#> sigma_creat 1.010 1.040
+#> sigma_WC 1.007 1.032
#>
#> Multivariate psrf
#>
-#> 1.1
+#> 1.05
```
To select only some of the parameters, they can be specified directly by
@@ -652,16 +645,16 @@ summary(lm5, subset = list(other = c('creat', 'alc>=1')))
#>
#>
#> Posterior summary:
-#> Mean SD 2.5% 97.5% tail-prob. GR-crit MCE/SD
-#> (Intercept) 81.096 9.6988 60.971 97.166 0.00000 1.01 0.0577
-#> genderfemale 0.676 2.5491 -3.776 5.439 0.81333 1.00 0.0577
-#> WC 0.300 0.0699 0.164 0.425 0.00000 1.02 0.0577
-#> alc>=1 6.645 2.2974 2.500 11.345 0.00667 1.02 0.0689
-#> creat 8.206 8.1859 -7.115 24.301 0.34000 1.07 0.0577
+#> Mean SD 2.5% 97.5% tail-prob. GR-crit MCE/SD
+#> (Intercept) 81.7399 9.6925 63.439 100.547 0.000 1.04 0.0577
+#> genderfemale 0.0802 2.6383 -4.441 5.527 0.933 1.00 0.0577
+#> WC 0.3074 0.0767 0.168 0.451 0.000 1.01 0.0577
+#> alc>=1 6.3855 2.5770 1.542 11.284 0.020 1.01 0.0670
+#> creat 6.8877 7.4473 -5.536 21.447 0.387 1.13 0.0586
#>
#> Posterior summary of residual std. deviation:
#> Mean SD 2.5% 97.5% GR-crit MCE/SD
-#> sigma_SBP 14.4 0.806 13 16.3 1.02 0.0577
+#> sigma_SBP 14.4 0.704 13.1 15.9 1 0.0577
#>
#>
#> MCMC settings:
@@ -693,10 +686,7 @@ sub3
traceplot(lm2, subset = list(other = sub3), ncol = 2)
```
-
-plot of chunk lm2_2
-
### Random subset of subject-specific values
@@ -719,10 +709,7 @@ rs <- grep('^b_bmi_ID\\[[[:digit:]]+,2\\]$', colnames(lme4$MCMC[[1]]), value = T
traceplot(lme4, subset = list(other = sample(ri, size = 8)), ncol = 4)
```
-
-plot of chunk unnamed-chunk-11
-
``` r
@@ -730,8 +717,5 @@ traceplot(lme4, subset = list(other = sample(ri, size = 8)), ncol = 4)
traceplot(lme4, subset = list(other = sample(rs, size = 8)), ncol = 4)
```
-
-plot of chunk unnamed-chunk-11
-
diff --git a/vignettes/SelectingParameters.Rmd.orig b/vignettes/SelectingParameters.Rmd.orig
index 40ae870f..32cc9e2d 100644
--- a/vignettes/SelectingParameters.Rmd.orig
+++ b/vignettes/SelectingParameters.Rmd.orig
@@ -20,6 +20,8 @@ knitr::opts_chunk$set(
comment = "#>",
dev = "svglite",
out.width = "100%",
+ fig.cap = "",
+ fig.scap = "",
fig.width = 7,
fig.align = 'center',
fig.path = "figures_SelectingParameters/"
@@ -339,7 +341,7 @@ Note that the model summary will contain separate parts per sub-model when
regression coefficients from different models are monitored.
This also works when a subset of the imputed values should be displayed:
-```{r lm2_2, fig.height = 2, fig.width = 5, out.width = "50%", warning = FALSE}
+```{r lm2_2, fig.height = 2, fig.width = 5, out.width = "60%", warning = FALSE}
# Re-run the model from above, now creating MCMC samples
lm2 <- lm_imp(SBP ~ age + WC + alc + smoke + occup,
data = NHANES, n.iter = 100, progress.bar = 'none',
diff --git a/vignettes/TheoreticalBackground.Rmd b/vignettes/TheoreticalBackground.Rmd
index 18e9ed9f..92a72b69 100644
--- a/vignettes/TheoreticalBackground.Rmd
+++ b/vignettes/TheoreticalBackground.Rmd
@@ -1,6 +1,6 @@
---
title: "Theoretical Background"
-date: "2020-06-20"
+date: "2026-02-22"
output:
bookdown::html_document2:
toc: true
@@ -8,6 +8,8 @@ output:
number_sections: false
pkgdown:
as_is: true
+template:
+ math-rendering: katex
vignette: >
%\VignetteIndexEntry{Theoretical Background}
%\VignetteEncoding{UTF-8}
@@ -221,16 +223,19 @@ implemented as a link in
### Cumulative logit (mixed) models
Cumulative logit mixed models are of the form
-\begin{eqnarray*}
-y_{ij} &\sim& \text{Mult}(\pi_{ij,1}, \ldots, \pi_{ij,K}),\\[2ex]
-\pi_{ij,1} &=& P(y_{ij} \leq 1),\\
-\pi_{ij,k} &=& P(y_{ij} \leq k) - P(y_{ij} \leq k-1), \quad k \in 2, \ldots, K-1,\\
-\pi_{ij,K} &=& 1 - \sum_{k = 1}^{K-1}\pi_{ij,k},\\[2ex]
-\text{logit}(P(y_{ij} \leq k)) &=& \gamma_k + \eta_{ij}, \quad k \in 1,\ldots,K,\\
-\eta_{ij} &=& \mathbf x_{ij}^\top\boldsymbol\beta + \mathbf z_{ij}^\top\mathbf b_i,\\[2ex]
-\gamma_1,\delta_1,\ldots,\delta_{K-1} &\overset{iid}{\sim}& N(\mu_\gamma, \sigma_\gamma^2),\\
-\gamma_k &\sim& \gamma_{k-1} + \exp(\delta_{k-1}),\quad k = 2,\ldots,K,
-\end{eqnarray*}
+
+
+\begin{align*}
+y_{ij} &\sim \text{Mult}(\pi_{ij,1}, \ldots, \pi_{ij,K}),\\[2ex]
+\pi_{ij,1} &= P(y_{ij} \leq 1),\\
+\pi_{ij,k} &= P(y_{ij} \leq k) - P(y_{ij} \leq k-1), \quad k \in 2, \ldots, K-1,\\
+\pi_{ij,K} &= 1 - \sum_{k = 1}^{K-1}\pi_{ij,k},\\[2ex]
+\text{logit}(P(y_{ij} \leq k)) &= \gamma_k + \eta_{ij}, \quad k \in 1,\ldots,K,\\
+\eta_{ij} &= \mathbf x_{ij}^\top\boldsymbol\beta + \mathbf z_{ij}^\top\mathbf b_i,\\[2ex]
+\gamma_1,\delta_1,\ldots,\delta_{K-1} &\overset{iid}{\sim} N(\mu_\gamma, \sigma_\gamma^2),\\
+\gamma_k &\sim \gamma_{k-1} + \exp(\delta_{k-1}),\quad k = 2,\ldots,K,
+\end{align*}
+
where $\pi_{ij,k} = P(y_{ij} = k)$ and $\text{logit}(x) = \log\left(\frac{x}{1-x}\right)$.
A cumulative logit regression model for a univariate outcome $y_i$ can be
obtained by dropping the index \(j\) and omitting \(\mathbf z_{ij}^\top\mathbf b_i\).
@@ -251,13 +256,13 @@ a parametric model which assumes a Weibull distribution for the true
Cox proportional hazards model.
The parametric survival model is implemented as
-\begin{eqnarray*}
-T_i^* &\sim& \text{Weibull}(1, r_i, s),\\
-D_i &\sim& \unicode{x1D7D9}(T_i^* \geq C_i),\\
-\log(r_j) &=& - \mathbf x_i^\top\boldsymbol\beta,\\
-s &\sim& \text{Exp}(0.01),
-\end{eqnarray*}
-where $\unicode{x1D7D9}$ is the indicator function which is one if $T_i^*\geq C_i$,
+\begin{align*}
+T_i^* &\sim \text{Weibull}(1, r_i, s),\\
+D_i &\sim \text{𝟙}(T_i^* \geq C_i),\\
+\log(r_j) &= - \mathbf x_i^\top\boldsymbol\beta,\\
+s &\sim \text{Exp}(0.01),
+\end{align*}
+where 𝟙 is the indicator function which is one if $T_i^*\geq C_i$,
and zero otherwise.
The Cox proportional hazards model can be written as
@@ -281,12 +286,12 @@ A convenient way to specify the joint distribution of the incomplete covariates
\(\mathbf X_{mis} = (\mathbf x_{mis_1}, \ldots, \mathbf x_{mis_q})\) is to use
a sequence of conditional univariate distributions [@Ibrahim2002; @Erler2016]
-\begin{eqnarray}
+\begin{align}
p(\mathbf x_{mis_1}, \ldots, \mathbf x_{mis_q} \mid \mathbf X_{obs}, \boldsymbol\theta_{x})
-& = & p(\mathbf x_{mis_1} \mid \mathbf X_{obs}, \boldsymbol\theta_{x_1})\\
-& & \prod_{\ell=2}^q p(\mathbf x_{mis_{\ell}} \mid \mathbf X_{obs}, \mathbf x_{mis_1},
+& = p(\mathbf x_{mis_1} \mid \mathbf X_{obs}, \boldsymbol\theta_{x_1})\nonumber\\
+& \phantom{=} \prod_{\ell=2}^q p(\mathbf x_{mis_{\ell}} \mid \mathbf X_{obs}, \mathbf x_{mis_1},
\ldots, \mathbf x_{mis_{\ell-1}}, \boldsymbol\theta_{x_\ell}),(\#eq:factorization)
-\end{eqnarray}
+\end{align}
with $\boldsymbol\theta_{x} = (\boldsymbol\theta_{x_1}^\top, \ldots,
\boldsymbol\theta_{x_q}^\top)^\top$.
@@ -318,26 +323,26 @@ and covariates.
When, for instance, the analysis model is a GLM, the full conditional
distribution of an incomplete covariate $x_{i, mis_{\ell}}$ can be written as
-\begin{eqnarray} \nonumber
+\begin{align} \nonumber
p(x_{i, mis_{\ell}} \mid \mathbf y_i, \mathbf x_{i,obs},
\mathbf x_{i,mis_{-\ell}}, \boldsymbol\theta)
-&\propto& p \left(y_i \mid \mathbf x_{i, obs}, \mathbf x_{i, mis},
+\propto& p \left(y_i \mid \mathbf x_{i, obs}, \mathbf x_{i, mis},
\boldsymbol\theta_{y\mid x}
\right)
p(\mathbf x_{i, mis}\mid \mathbf x_{i, obs}, \boldsymbol\theta_{x})\,
p(\boldsymbol\theta_{y\mid x})\, p(\boldsymbol\theta_{x})\\\nonumber
-&\propto& p \left(y_i \mid \mathbf x_{i, obs}, \mathbf x_{i, mis},
+\propto& p \left(y_i \mid \mathbf x_{i, obs}, \mathbf x_{i, mis},
\boldsymbol\theta_{y\mid x}
\right)\\\nonumber
-& & p(x_{i, mis_\ell} \mid \mathbf x_{i, obs}, \mathbf x_{i, mis_{<\ell}}, \boldsymbol\theta_{x_\ell})\\\nonumber
-& & \left\{
+ & p(x_{i, mis_\ell} \mid \mathbf x_{i, obs}, \mathbf x_{i, mis_{<\ell}}, \boldsymbol\theta_{x_\ell})\\\nonumber
+ & \left\{
\prod_{k=\ell+1}^q p(x_{i,mis_k}\mid \mathbf x_{i, obs},
\mathbf x_{i, mis_{