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Informative prior archetypes
In brms.mmrm: Bayesian MMRMs using 'brms'
Informative prior archetypes allow users to conveniently set informative priors in brms.mmrm in a robust way, guarding against common pitfalls such as reference level issues, interpretation problems, and rank deficiency.
Constructing an archetype
We begin with the FEV dataset from the mmrm package, an artificial (simulated) dataset of a clinical trial investigating the effect of an active treatment on FEV1 (forced expired volume in one second), compared to placebo. FEV1 is a measure of how quickly the lungs can be emptied and low levels may indicate chronic obstructive pulmonary disease (COPD).
The dataset is a tibble with 800 rows and 7 variables:
USUBJID (subject ID),AVISIT (visit number),ARMCD (treatment, TRT or PBO),RACE (3-category race),SEX (sex),FEV1_BL (FEV1 at baseline, %),FEV1 (FEV1 at study visits),WEIGHT (weighting variable).
We will derive FEV1_CHG = FEV1 - FEV1_BL and analyze FEV1_CHG as the outcome variable.
library(brms.mmrm)
data(fev_data, package = "mmrm")
data <- fev_data |>
brm_data(
outcome = "FEV1",
group = "ARMCD",
time = "AVISIT",
patient = "USUBJID",
reference_time = "VIS1",
reference_group = "PBO",
covariates = c("WEIGHT", "SEX")
) |>
brm_data_chronologize(order = "VISITN")
data
#> # A tibble: 800 × 10
#> USUBJID AVISIT ARMCD RACE SEX FEV1_BL FEV1 WEIGHT VISITN VISITN2
#> <fct> <ord> <fct> <fct> <fct> <dbl> <dbl> <dbl> <int> <dbl>
#> 1 PT2 VIS1 PBO Asian Male 45.0 NA 0.465 1 0.330
#> 2 PT2 VIS2 PBO Asian Male 45.0 31.5 0.233 2 -0.820
#> 3 PT2 VIS3 PBO Asian Male 45.0 36.9 0.360 3 0.487
#> 4 PT2 VIS4 PBO Asian Male 45.0 48.8 0.507 4 0.738
#> 5 PT3 VIS1 PBO Black or African A… Fema… 43.5 NA 0.682 1 0.576
#> 6 PT3 VIS2 PBO Black or African A… Fema… 43.5 36.0 0.892 2 -0.305
#> 7 PT3 VIS3 PBO Black or African A… Fema… 43.5 NA 0.128 3 1.51
#> 8 PT3 VIS4 PBO Black or African A… Fema… 43.5 37.2 0.222 4 0.390
#> 9 PT5 VIS1 PBO Black or African A… Male 43.6 32.3 0.411 1 -0.0162
#> 10 PT5 VIS2 PBO Black or African A… Male 43.6 NA 0.422 2 0.944
#> # ℹ 790 more rows
The functions listed at https://openpharma.github.io/brms.mmrm/reference/index.html#informative-prior-archetypes can create different kinds of informative prior archetypes from a dataset like the one above. For example, suppose we want to place informative priors on the successive differences between adjacent time points. This approach is appropriate and desirable in many situations because the structure naturally captures the prior correlations among adjacent visits of a clinical trial. To do this, we create an instance of the "successive cells" archetype.
archetype <- brm_archetype_successive_cells(data, baseline = FALSE)
The instance of the archetype is an ordinary tibble, but it adds new columns with prefixes "x_" and "nuisance_". These new columns constitute a custom model matrix to describe the desired parameterization.
archetype
#> # A tibble: 800 × 20
#> x_PBO_VIS1 x_PBO_VIS2 x_PBO_VIS3 x_PBO_VIS4 x_TRT_VIS1 x_TRT_VIS2 x_TRT_VIS3
#> * <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 0 0 0 0 0 0
#> 2 1 1 0 0 0 0 0
#> 3 1 1 1 0 0 0 0
#> 4 1 1 1 1 0 0 0
#> 5 1 0 0 0 0 0 0
#> 6 1 1 0 0 0 0 0
#> 7 1 1 1 0 0 0 0
#> 8 1 1 1 1 0 0 0
#> 9 1 0 0 0 0 0 0
#> 10 1 1 0 0 0 0 0
#> # ℹ 790 more rows
#> # ℹ 13 more variables: x_TRT_VIS4 <dbl>, nuisance_WEIGHT <dbl>,
#> # nuisance_SEX_Male <dbl>, USUBJID <fct>, AVISIT <ord>, ARMCD <fct>, RACE <fct>,
#> # SEX <fct>, FEV1_BL <dbl>, FEV1 <dbl>, WEIGHT <dbl>, VISITN <int>, VISITN2 <dbl>
We have effects of interest to express successive differences:
attr(archetype, "brm_archetype_interest")
#> [1] "x_PBO_VIS1" "x_PBO_VIS2" "x_PBO_VIS3" "x_PBO_VIS4" "x_TRT_VIS1" "x_TRT_VIS2"
#> [7] "x_TRT_VIS3" "x_TRT_VIS4"
We also have nuisance variables. Some nuisance variables are continuous covariates, while others are levels of one-hot-encoded concomitant factors or interactions of those concomitant factors with baseline and/or subgroup. All nuisance variables are centered at their means so the reference level of the model is at the "center" of the data and not implicitly conditional on a subset of the data.^[brm_recenter_nuisance() can retroactively recenter a nuisance column to a fixed value other than its mean.] In addition, some nuisance variables are automatically dropped in order to ensure the model matrix is full-rank, and automatic centering in brms is disabled^[brm_formula() assigns center = FALSE in brmsformula() for all informative prior archetypes.]. This is critically important to preserve the interpretation of the columns of interest and make sure the informative priors behave as expected.
attr(archetype, "brm_archetype_nuisance")
#> [1] "nuisance_WEIGHT" "nuisance_SEX_Male"
The factors of interest linearly map to marginal means. To see the mapping, call summary() on the archetype. The printed output helps build intuition on how the archetype is parameterized and what those parameters are doing.^[summary() also invisibly returns a simple character vector with the equations below.]
summary(archetype)
#> # This is the "successive cells" informative prior archetype in brms.mmrm.
#> # The following equations show the relationships between the
#> # marginal means (left-hand side) and fixed effect parameters
#> # (right-hand side).
#> #
#> # PBO:VIS1 = x_PBO_VIS1
#> # PBO:VIS2 = x_PBO_VIS1 + x_PBO_VIS2
#> # PBO:VIS3 = x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3
#> # PBO:VIS4 = x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3 + x_PBO_VIS4
#> # TRT:VIS1 = x_TRT_VIS1
#> # TRT:VIS2 = x_TRT_VIS1 + x_TRT_VIS2
#> # TRT:VIS3 = x_TRT_VIS1 + x_TRT_VIS2 + x_TRT_VIS3
#> # TRT:VIS4 = x_TRT_VIS1 + x_TRT_VIS2 + x_TRT_VIS3 + x_TRT_VIS4
Above, x_PBO_VIS1 serves as the intercept, and x_TRT_VIS1 is defined relative to x_TRT_VIS1. The rest of the parameters keep their original interpretations.
Informative priors
Let's assume you want to assign informative priors to the fixed effect parameters of interest declared in the archetype, such as x_group_1_time_2 and x_group_2_time_3. Your priors may come from expert elicitation, historical data, or some other method, and you might consider distributional families recommended by the Stan team. However you construct these priors, brms.mmrm helps you assign them to the model without having to guess at the automatically-generated names of model coefficients in R.
In the printed output from summary(archetype), parameters of interest such as x_group_1_time_2 and x_group_2_time_3 are always labeled using treatment groups and time points in the data (and subgroup levels, if applicable). This labeling mechanism is the same regardless of which archetype you choose, and it the way brms.mmrm helps you assign priors.
brm_prior_label() is one way to create a labeling scheme. Each call to brm_prior_label() below assigns a univariate prior to a fixed effect parameter. Each univariate prior is a Stan code string. Possible choices are documented in the Stan function reference at https://mc-stan.org/docs/functions-reference/unbounded_continuous_distributions.html.
label <- NULL |>
brm_prior_label(code = "student_t(4, -7.57, 4.96)", group = "PBO", time = "VIS1") |>
brm_prior_label(code = "student_t(4, 3.14, 7.86)", group = "PBO", time = "VIS2") |>
brm_prior_label(code = "student_t(4, 8.78, 8.18)", group = "PBO", time = "VIS3") |>
brm_prior_label(code = "student_t(4, 3.36, 8.10)", group = "PBO", time = "VIS4") |>
brm_prior_label(code = "student_t(4, -2.96, 4.78)", group = "TRT", time = "VIS1") |>
brm_prior_label(code = "student_t(4, 3.13, 7.64)", group = "TRT", time = "VIS2") |>
brm_prior_label(code = "student_t(4, 7.65, 8.24)", group = "TRT", time = "VIS3") |>
brm_prior_label(code = "student_t(4, 4.64, 8.21)", group = "TRT", time = "VIS4")
label
#> # A tibble: 8 × 3
#> code group time
#> <chr> <chr> <chr>
#> 1 student_t(4, -7.57, 4.96) PBO VIS1
#> 2 student_t(4, 3.14, 7.86) PBO VIS2
#> 3 student_t(4, 8.78, 8.18) PBO VIS3
#> 4 student_t(4, 3.36, 8.10) PBO VIS4
#> 5 student_t(4, -2.96, 4.78) TRT VIS1
#> 6 student_t(4, 3.13, 7.64) TRT VIS2
#> 7 student_t(4, 7.65, 8.24) TRT VIS3
#> 8 student_t(4, 4.64, 8.21) TRT VIS4
As an alternative to brm_prior_label(), you can start with a template and manually fill in the Stan code.
template <- brm_prior_template(archetype)
template
#> # A tibble: 8 × 3
#> code group time
#> <chr> <chr> <chr>
#> 1 <NA> PBO VIS1
#> 2 <NA> PBO VIS2
#> 3 <NA> PBO VIS3
#> 4 <NA> PBO VIS4
#> 5 <NA> TRT VIS1
#> 6 <NA> TRT VIS2
#> 7 <NA> TRT VIS3
#> 8 <NA> TRT VIS4
label <- template |>
mutate(
code = c(
"student_t(4, -7.57, 4.96)",
"student_t(4, 3.14, 7.86)",
"student_t(4, 8.78, 8.18)",
"student_t(4, 3.36, 8.10)",
"student_t(4, -2.96, 4.78)",
"student_t(4, 3.13, 7.64)",
"student_t(4, 7.65, 8.24)",
"student_t(4, 4.64, 8.21)"
)
)
label
#> # A tibble: 8 × 3
#> code group time
#> <chr> <chr> <chr>
#> 1 student_t(4, -7.57, 4.96) PBO VIS1
#> 2 student_t(4, 3.14, 7.86) PBO VIS2
#> 3 student_t(4, 8.78, 8.18) PBO VIS3
#> 4 student_t(4, 3.36, 8.10) PBO VIS4
#> 5 student_t(4, -2.96, 4.78) TRT VIS1
#> 6 student_t(4, 3.13, 7.64) TRT VIS2
#> 7 student_t(4, 7.65, 8.24) TRT VIS3
#> 8 student_t(4, 4.64, 8.21) TRT VIS4
After you have a labeling scheme, brm_prior_archetype() can create a brms prior for the important fixed effects.^[brms priors are documented in https://paulbuerkner.com/brms/reference/set_prior.html.]
prior <- brm_prior_archetype(label = label, archetype = archetype)
prior
#> prior class coef group resp dpar nlpar lb ub source
#> student_t(4, -7.57, 4.96) b x_PBO_VIS1 <NA> <NA> user
#> student_t(4, 3.14, 7.86) b x_PBO_VIS2 <NA> <NA> user
#> student_t(4, 8.78, 8.18) b x_PBO_VIS3 <NA> <NA> user
#> student_t(4, 3.36, 8.10) b x_PBO_VIS4 <NA> <NA> user
#> student_t(4, -2.96, 4.78) b x_TRT_VIS1 <NA> <NA> user
#> student_t(4, 3.13, 7.64) b x_TRT_VIS2 <NA> <NA> user
#> student_t(4, 7.65, 8.24) b x_TRT_VIS3 <NA> <NA> user
#> student_t(4, 4.64, 8.21) b x_TRT_VIS4 <NA> <NA> user
In less common situations, you may wish to assign priors to nuisance parameters. For example, our model accounts for interactions between baseline and discrete time, and it may be reasonable to assign priors to these slopes based on high-quality historical data. This requires a thorough understanding of the fixed effect structure of the model, but it can be done directly through brms. First, check the formula for the included nuisance parameters. brm_formula() automatically understands archetypes.
brm_formula(archetype)
#> FEV1 ~ 0 + x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3 + x_PBO_VIS4 + x_TRT_VIS1 + x_TRT_VIS2 + x_TRT_VIS3 + x_TRT_VIS4 + nuisance_WEIGHT + nuisance_SEX_Male + unstr(time = AVISIT, gr = USUBJID)
#> sigma ~ 0 + AVISIT
The "nuisance_*" terms are the nuisance variables, and the ones involving baseline are nuisance_FEV1_BL.AVISITVIS1, nuisance_FEV1_BL.AVISITVIS2, nuisance_FEV1_BL.AVISITVIS3, and nuisance_FEV1_BL.AVISITVIS4. Because there is no overall slope for baseline, we can interpret each term as the linear rate of change in the outcome variable per unit increase in baseline for a given discrete time point. Suppose we use this interpretation to construct informative priors student_t(4, -0.83, 1), student_t(4, -0.78, 1), student_t(4, -0.86, 1), and student_t(4, -0.82, 1), respectively. Use brms::set_prior() and c() to append these priors to our existing prior object:
The model still has many parameters where we did not set priors, and brms sets automatic defaults. You can see these defaults with brms::get_prior().
https://paulbuerkner.com/brms/reference/set_prior.html documents many of the default priors set by brms. In particular, "(flat)" denotes an improper uniform prior over all the real numbers.
Modeling and analysis
The downstream methods in brms.mmrm automatically understand how to work with informative prior archetypes. Notably, the formula uses custom interest and nuisance variables instead of the original variables in the data.
formula <- brm_formula(archetype)
formula
#> FEV1 ~ 0 + x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3 + x_PBO_VIS4 + x_TRT_VIS1 + x_TRT_VIS2 + x_TRT_VIS3 + x_TRT_VIS4 + nuisance_WEIGHT + nuisance_SEX_Male + unstr(time = AVISIT, gr = USUBJID)
#> sigma ~ 0 + AVISIT
The model can accept the archetype, formula, and prior. Usage is the same as in non-archetype workflows.
model <- brm_model(
data = archetype,
formula = formula,
prior = prior,
refresh = 0
)
#> Compiling Stan program...
#> Start sampling
brms::prior_summary(model)
#> prior class coef group resp dpar nlpar lb ub
#> (flat) b
#> (flat) b nuisance_SEX_Male
#> (flat) b nuisance_WEIGHT
#> student_t(4, -7.57, 4.96) b x_PBO_VIS1
#> student_t(4, 3.14, 7.86) b x_PBO_VIS2
#> student_t(4, 8.78, 8.18) b x_PBO_VIS3
#> student_t(4, 3.36, 8.10) b x_PBO_VIS4
#> student_t(4, -2.96, 4.78) b x_TRT_VIS1
#> student_t(4, 3.13, 7.64) b x_TRT_VIS2
#> student_t(4, 7.65, 8.24) b x_TRT_VIS3
#> student_t(4, 4.64, 8.21) b x_TRT_VIS4
#> (flat) b sigma
#> (flat) b AVISITVIS1 sigma
#> (flat) b AVISITVIS2 sigma
#> (flat) b AVISITVIS3 sigma
#> (flat) b AVISITVIS4 sigma
#> lkj_corr_cholesky(1) Lcortime
#> source
#> default
#> (vectorized)
#> (vectorized)
#> user
#> user
#> user
#> user
#> user
#> user
#> user
#> user
#> default
#> (vectorized)
#> (vectorized)
#> (vectorized)
#> (vectorized)
#> default
Marginal mean estimation, post-processing, and visualization automatically understand the archetype without any user intervention.
draws <- brm_marginal_draws(
data = archetype,
formula = formula,
model = model
)
summaries_model <- brm_marginal_summaries(draws)
summaries_data <- brm_marginal_data(archetype)
brm_plot_compare(model = summaries_model, data = summaries_data)
plot of chunk archetype_compare_data
All archetypes
brms.mmrm supports a variety of informative prior archetypes with different kinds of fixed effects. For example, brms.mmrm supports simple cell mean and treatment effect parameterizations.
summary(brm_archetype_cells(data, intercept = FALSE))
#> # This is the "cells" informative prior archetype in brms.mmrm.
#> # The following equations show the relationships between the
#> # marginal means (left-hand side) and fixed effect parameters
#> # (right-hand side).
#> #
#> # PBO:VIS1 = x_PBO_VIS1
#> # PBO:VIS2 = x_PBO_VIS2
#> # PBO:VIS3 = x_PBO_VIS3
#> # PBO:VIS4 = x_PBO_VIS4
#> # TRT:VIS1 = x_TRT_VIS1
#> # TRT:VIS2 = x_TRT_VIS2
#> # TRT:VIS3 = x_TRT_VIS3
#> # TRT:VIS4 = x_TRT_VIS4
summary(brm_archetype_effects(data, intercept = FALSE))
#> # This is the "effects" informative prior archetype in brms.mmrm.
#> # The following equations show the relationships between the
#> # marginal means (left-hand side) and fixed effect parameters
#> # (right-hand side).
#> #
#> # PBO:VIS1 = x_PBO_VIS1
#> # PBO:VIS2 = x_PBO_VIS2
#> # PBO:VIS3 = x_PBO_VIS3
#> # PBO:VIS4 = x_PBO_VIS4
#> # TRT:VIS1 = x_PBO_VIS1 + x_TRT_VIS1
#> # TRT:VIS2 = x_PBO_VIS2 + x_TRT_VIS2
#> # TRT:VIS3 = x_PBO_VIS3 + x_TRT_VIS3
#> # TRT:VIS4 = x_PBO_VIS4 + x_TRT_VIS4
There are archetypes to parameterize the average across all time points in the data. Below, x_group_1_time_2 is the average across time points for group 1 because it is the algebraic result of simplifying (group_1:time_2 + group_1:time_3 + group_1:time_3) / 3.
summary(brm_archetype_average_cells(data, intercept = FALSE))
#> # This is the "average cells" informative prior archetype in brms.mmrm.
#> # The following equations show the relationships between the
#> # marginal means (left-hand side) and fixed effect parameters
#> # (right-hand side).
#> #
#> # PBO:VIS1 = 4*x_PBO_VIS1 - x_PBO_VIS2 - x_PBO_VIS3 - x_PBO_VIS4
#> # PBO:VIS2 = x_PBO_VIS2
#> # PBO:VIS3 = x_PBO_VIS3
#> # PBO:VIS4 = x_PBO_VIS4
#> # TRT:VIS1 = 4*x_TRT_VIS1 - x_TRT_VIS2 - x_TRT_VIS3 - x_TRT_VIS4
#> # TRT:VIS2 = x_TRT_VIS2
#> # TRT:VIS3 = x_TRT_VIS3
#> # TRT:VIS4 = x_TRT_VIS4
There is also a treatment effect version where x_group_2_time_2 becomes the time-averaged treatment effect of group 2 relative to group 1.
summary(brm_archetype_average_effects(data, intercept = FALSE))
#> # This is the "average effects" informative prior archetype in brms.mmrm.
#> # The following equations show the relationships between the
#> # marginal means (left-hand side) and fixed effect parameters
#> # (right-hand side).
#> #
#> # PBO:VIS1 = 4*x_PBO_VIS1 - x_PBO_VIS2 - x_PBO_VIS3 - x_PBO_VIS4
#> # PBO:VIS2 = x_PBO_VIS2
#> # PBO:VIS3 = x_PBO_VIS3
#> # PBO:VIS4 = x_PBO_VIS4
#> # TRT:VIS1 = 4*x_PBO_VIS1 - x_PBO_VIS2 - x_PBO_VIS3 - x_PBO_VIS4 + 4*x_TRT_VIS1 - x_TRT_VIS2 - x_TRT_VIS3 - x_TRT_VIS4
#> # TRT:VIS2 = x_PBO_VIS2 + x_TRT_VIS2
#> # TRT:VIS3 = x_PBO_VIS3 + x_TRT_VIS3
#> # TRT:VIS4 = x_PBO_VIS4 + x_TRT_VIS4
The example in this vignette uses the "successive cells" archetype, where fixed effects represent successive differences between adjacent time points.
summary(brm_archetype_successive_cells(data, intercept = FALSE))
#> # This is the "successive cells" informative prior archetype in brms.mmrm.
#> # The following equations show the relationships between the
#> # marginal means (left-hand side) and fixed effect parameters
#> # (right-hand side).
#> #
#> # PBO:VIS1 = x_PBO_VIS1
#> # PBO:VIS2 = x_PBO_VIS1 + x_PBO_VIS2
#> # PBO:VIS3 = x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3
#> # PBO:VIS4 = x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3 + x_PBO_VIS4
#> # TRT:VIS1 = x_TRT_VIS1
#> # TRT:VIS2 = x_TRT_VIS1 + x_TRT_VIS2
#> # TRT:VIS3 = x_TRT_VIS1 + x_TRT_VIS2 + x_TRT_VIS3
#> # TRT:VIS4 = x_TRT_VIS1 + x_TRT_VIS2 + x_TRT_VIS3 + x_TRT_VIS4
There is also a treatment effect version of the successive differences archetype:
summary(brm_archetype_successive_effects(data, intercept = FALSE))
#> # This is the "successive effects" informative prior archetype in brms.mmrm.
#> # The following equations show the relationships between the
#> # marginal means (left-hand side) and fixed effect parameters
#> # (right-hand side).
#> #
#> # PBO:VIS1 = x_PBO_VIS1
#> # PBO:VIS2 = x_PBO_VIS1 + x_PBO_VIS2
#> # PBO:VIS3 = x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3
#> # PBO:VIS4 = x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3 + x_PBO_VIS4
#> # TRT:VIS1 = x_PBO_VIS1 + x_TRT_VIS1
#> # TRT:VIS2 = x_PBO_VIS1 + x_PBO_VIS2 + x_TRT_VIS1 + x_TRT_VIS2
#> # TRT:VIS3 = x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3 + x_TRT_VIS1 + x_TRT_VIS2 + x_TRT_VIS3
#> # TRT:VIS4 = x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3 + x_PBO_VIS4 + x_TRT_VIS1 + x_TRT_VIS2 + x_TRT_VIS3 + x_TRT_VIS4
Variations on archetypes
Archetypes can be customized. As an example, consider the simple cell means archetype.
summary(brm_archetype_cells(data))
#> # This is the "cells" informative prior archetype in brms.mmrm.
#> # The following equations show the relationships between the
#> # marginal means (left-hand side) and fixed effect parameters
#> # (right-hand side).
#> #
#> # PBO:VIS1 = x_PBO_VIS1
#> # PBO:VIS2 = x_PBO_VIS2
#> # PBO:VIS3 = x_PBO_VIS3
#> # PBO:VIS4 = x_PBO_VIS4
#> # TRT:VIS1 = x_TRT_VIS1
#> # TRT:VIS2 = x_TRT_VIS2
#> # TRT:VIS3 = x_TRT_VIS3
#> # TRT:VIS4 = x_TRT_VIS4
To include an intercept term which all the marginal means share, set intercept = TRUE.
summary(brm_archetype_cells(data, intercept = TRUE))
#> # This is the "cells" informative prior archetype in brms.mmrm.
#> # The following equations show the relationships between the
#> # marginal means (left-hand side) and fixed effect parameters
#> # (right-hand side).
#> #
#> # PBO:VIS1 = x_PBO_VIS1
#> # PBO:VIS2 = x_PBO_VIS1 + x_PBO_VIS2
#> # PBO:VIS3 = x_PBO_VIS1 + x_PBO_VIS3
#> # PBO:VIS4 = x_PBO_VIS1 + x_PBO_VIS4
#> # TRT:VIS1 = x_PBO_VIS1 + x_TRT_VIS1
#> # TRT:VIS2 = x_PBO_VIS1 + x_TRT_VIS2
#> # TRT:VIS3 = x_PBO_VIS1 + x_TRT_VIS3
#> # TRT:VIS4 = x_PBO_VIS1 + x_TRT_VIS4
To set up constrained longitudinal data analysis (cLDA), set clda = TRUE. This constraint pools all treatment groups at baseline, and it can help model clinical trials where a baseline measurement is observed before randomization. Some archetypes cannot support cLDA (e.g. brm_archetype_average_cells() and brm_archetype_average_effects()).
summary(brm_archetype_cells(data, clda = TRUE))
#> # This is the "cells" informative prior archetype in brms.mmrm.
#> # The following equations show the relationships between the
#> # marginal means (left-hand side) and fixed effect parameters
#> # (right-hand side).
#> #
#> # PBO:VIS1 = x_PBO_VIS1
#> # PBO:VIS2 = x_PBO_VIS2
#> # PBO:VIS3 = x_PBO_VIS3
#> # PBO:VIS4 = x_PBO_VIS4
#> # TRT:VIS1 = x_PBO_VIS1
#> # TRT:VIS2 = x_TRT_VIS2
#> # TRT:VIS3 = x_TRT_VIS3
#> # TRT:VIS4 = x_TRT_VIS4
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Informative prior archetypes allow users to conveniently set informative priors in brms.mmrm in a robust way, guarding against common pitfalls such as reference level issues, interpretation problems, and rank deficiency.
Constructing an archetype
We begin with the FEV dataset from the mmrm package, an artificial (simulated) dataset of a clinical trial investigating the effect of an active treatment on FEV1 (forced expired volume in one second), compared to placebo. FEV1 is a measure of how quickly the lungs can be emptied and low levels may indicate chronic obstructive pulmonary disease (COPD).
The dataset is a tibble with 800 rows and 7 variables:
USUBJID(subject ID),AVISIT(visit number),ARMCD(treatment, TRT or PBO),RACE(3-category race),SEX(sex),FEV1_BL(FEV1 at baseline, %),FEV1(FEV1 at study visits),WEIGHT(weighting variable).
We will derive FEV1_CHG = FEV1 - FEV1_BL and analyze FEV1_CHG as the outcome variable.
library(brms.mmrm) data(fev_data, package = "mmrm") data <- fev_data |> brm_data( outcome = "FEV1", group = "ARMCD", time = "AVISIT", patient = "USUBJID", reference_time = "VIS1", reference_group = "PBO", covariates = c("WEIGHT", "SEX") ) |> brm_data_chronologize(order = "VISITN") data #> # A tibble: 800 × 10 #> USUBJID AVISIT ARMCD RACE SEX FEV1_BL FEV1 WEIGHT VISITN VISITN2 #> <fct> <ord> <fct> <fct> <fct> <dbl> <dbl> <dbl> <int> <dbl> #> 1 PT2 VIS1 PBO Asian Male 45.0 NA 0.465 1 0.330 #> 2 PT2 VIS2 PBO Asian Male 45.0 31.5 0.233 2 -0.820 #> 3 PT2 VIS3 PBO Asian Male 45.0 36.9 0.360 3 0.487 #> 4 PT2 VIS4 PBO Asian Male 45.0 48.8 0.507 4 0.738 #> 5 PT3 VIS1 PBO Black or African A… Fema… 43.5 NA 0.682 1 0.576 #> 6 PT3 VIS2 PBO Black or African A… Fema… 43.5 36.0 0.892 2 -0.305 #> 7 PT3 VIS3 PBO Black or African A… Fema… 43.5 NA 0.128 3 1.51 #> 8 PT3 VIS4 PBO Black or African A… Fema… 43.5 37.2 0.222 4 0.390 #> 9 PT5 VIS1 PBO Black or African A… Male 43.6 32.3 0.411 1 -0.0162 #> 10 PT5 VIS2 PBO Black or African A… Male 43.6 NA 0.422 2 0.944 #> # ℹ 790 more rows
The functions listed at https://openpharma.github.io/brms.mmrm/reference/index.html#informative-prior-archetypes can create different kinds of informative prior archetypes from a dataset like the one above. For example, suppose we want to place informative priors on the successive differences between adjacent time points. This approach is appropriate and desirable in many situations because the structure naturally captures the prior correlations among adjacent visits of a clinical trial. To do this, we create an instance of the "successive cells" archetype.
archetype <- brm_archetype_successive_cells(data, baseline = FALSE)
The instance of the archetype is an ordinary tibble, but it adds new columns with prefixes "x_" and "nuisance_". These new columns constitute a custom model matrix to describe the desired parameterization.
archetype #> # A tibble: 800 × 20 #> x_PBO_VIS1 x_PBO_VIS2 x_PBO_VIS3 x_PBO_VIS4 x_TRT_VIS1 x_TRT_VIS2 x_TRT_VIS3 #> * <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> #> 1 1 0 0 0 0 0 0 #> 2 1 1 0 0 0 0 0 #> 3 1 1 1 0 0 0 0 #> 4 1 1 1 1 0 0 0 #> 5 1 0 0 0 0 0 0 #> 6 1 1 0 0 0 0 0 #> 7 1 1 1 0 0 0 0 #> 8 1 1 1 1 0 0 0 #> 9 1 0 0 0 0 0 0 #> 10 1 1 0 0 0 0 0 #> # ℹ 790 more rows #> # ℹ 13 more variables: x_TRT_VIS4 <dbl>, nuisance_WEIGHT <dbl>, #> # nuisance_SEX_Male <dbl>, USUBJID <fct>, AVISIT <ord>, ARMCD <fct>, RACE <fct>, #> # SEX <fct>, FEV1_BL <dbl>, FEV1 <dbl>, WEIGHT <dbl>, VISITN <int>, VISITN2 <dbl>
We have effects of interest to express successive differences:
attr(archetype, "brm_archetype_interest") #> [1] "x_PBO_VIS1" "x_PBO_VIS2" "x_PBO_VIS3" "x_PBO_VIS4" "x_TRT_VIS1" "x_TRT_VIS2" #> [7] "x_TRT_VIS3" "x_TRT_VIS4"
We also have nuisance variables. Some nuisance variables are continuous covariates, while others are levels of one-hot-encoded concomitant factors or interactions of those concomitant factors with baseline and/or subgroup. All nuisance variables are centered at their means so the reference level of the model is at the "center" of the data and not implicitly conditional on a subset of the data.^[brm_recenter_nuisance() can retroactively recenter a nuisance column to a fixed value other than its mean.] In addition, some nuisance variables are automatically dropped in order to ensure the model matrix is full-rank, and automatic centering in brms is disabled^[brm_formula() assigns center = FALSE in brmsformula() for all informative prior archetypes.]. This is critically important to preserve the interpretation of the columns of interest and make sure the informative priors behave as expected.
attr(archetype, "brm_archetype_nuisance") #> [1] "nuisance_WEIGHT" "nuisance_SEX_Male"
The factors of interest linearly map to marginal means. To see the mapping, call summary() on the archetype. The printed output helps build intuition on how the archetype is parameterized and what those parameters are doing.^[summary() also invisibly returns a simple character vector with the equations below.]
summary(archetype) #> # This is the "successive cells" informative prior archetype in brms.mmrm. #> # The following equations show the relationships between the #> # marginal means (left-hand side) and fixed effect parameters #> # (right-hand side). #> # #> # PBO:VIS1 = x_PBO_VIS1 #> # PBO:VIS2 = x_PBO_VIS1 + x_PBO_VIS2 #> # PBO:VIS3 = x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3 #> # PBO:VIS4 = x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3 + x_PBO_VIS4 #> # TRT:VIS1 = x_TRT_VIS1 #> # TRT:VIS2 = x_TRT_VIS1 + x_TRT_VIS2 #> # TRT:VIS3 = x_TRT_VIS1 + x_TRT_VIS2 + x_TRT_VIS3 #> # TRT:VIS4 = x_TRT_VIS1 + x_TRT_VIS2 + x_TRT_VIS3 + x_TRT_VIS4
Above, x_PBO_VIS1 serves as the intercept, and x_TRT_VIS1 is defined relative to x_TRT_VIS1. The rest of the parameters keep their original interpretations.
Informative priors
Let's assume you want to assign informative priors to the fixed effect parameters of interest declared in the archetype, such as x_group_1_time_2 and x_group_2_time_3. Your priors may come from expert elicitation, historical data, or some other method, and you might consider distributional families recommended by the Stan team. However you construct these priors, brms.mmrm helps you assign them to the model without having to guess at the automatically-generated names of model coefficients in R.
In the printed output from summary(archetype), parameters of interest such as x_group_1_time_2 and x_group_2_time_3 are always labeled using treatment groups and time points in the data (and subgroup levels, if applicable). This labeling mechanism is the same regardless of which archetype you choose, and it the way brms.mmrm helps you assign priors.
brm_prior_label() is one way to create a labeling scheme. Each call to brm_prior_label() below assigns a univariate prior to a fixed effect parameter. Each univariate prior is a Stan code string. Possible choices are documented in the Stan function reference at https://mc-stan.org/docs/functions-reference/unbounded_continuous_distributions.html.
label <- NULL |> brm_prior_label(code = "student_t(4, -7.57, 4.96)", group = "PBO", time = "VIS1") |> brm_prior_label(code = "student_t(4, 3.14, 7.86)", group = "PBO", time = "VIS2") |> brm_prior_label(code = "student_t(4, 8.78, 8.18)", group = "PBO", time = "VIS3") |> brm_prior_label(code = "student_t(4, 3.36, 8.10)", group = "PBO", time = "VIS4") |> brm_prior_label(code = "student_t(4, -2.96, 4.78)", group = "TRT", time = "VIS1") |> brm_prior_label(code = "student_t(4, 3.13, 7.64)", group = "TRT", time = "VIS2") |> brm_prior_label(code = "student_t(4, 7.65, 8.24)", group = "TRT", time = "VIS3") |> brm_prior_label(code = "student_t(4, 4.64, 8.21)", group = "TRT", time = "VIS4") label #> # A tibble: 8 × 3 #> code group time #> <chr> <chr> <chr> #> 1 student_t(4, -7.57, 4.96) PBO VIS1 #> 2 student_t(4, 3.14, 7.86) PBO VIS2 #> 3 student_t(4, 8.78, 8.18) PBO VIS3 #> 4 student_t(4, 3.36, 8.10) PBO VIS4 #> 5 student_t(4, -2.96, 4.78) TRT VIS1 #> 6 student_t(4, 3.13, 7.64) TRT VIS2 #> 7 student_t(4, 7.65, 8.24) TRT VIS3 #> 8 student_t(4, 4.64, 8.21) TRT VIS4
As an alternative to brm_prior_label(), you can start with a template and manually fill in the Stan code.
template <- brm_prior_template(archetype) template #> # A tibble: 8 × 3 #> code group time #> <chr> <chr> <chr> #> 1 <NA> PBO VIS1 #> 2 <NA> PBO VIS2 #> 3 <NA> PBO VIS3 #> 4 <NA> PBO VIS4 #> 5 <NA> TRT VIS1 #> 6 <NA> TRT VIS2 #> 7 <NA> TRT VIS3 #> 8 <NA> TRT VIS4
label <- template |> mutate( code = c( "student_t(4, -7.57, 4.96)", "student_t(4, 3.14, 7.86)", "student_t(4, 8.78, 8.18)", "student_t(4, 3.36, 8.10)", "student_t(4, -2.96, 4.78)", "student_t(4, 3.13, 7.64)", "student_t(4, 7.65, 8.24)", "student_t(4, 4.64, 8.21)" ) ) label #> # A tibble: 8 × 3 #> code group time #> <chr> <chr> <chr> #> 1 student_t(4, -7.57, 4.96) PBO VIS1 #> 2 student_t(4, 3.14, 7.86) PBO VIS2 #> 3 student_t(4, 8.78, 8.18) PBO VIS3 #> 4 student_t(4, 3.36, 8.10) PBO VIS4 #> 5 student_t(4, -2.96, 4.78) TRT VIS1 #> 6 student_t(4, 3.13, 7.64) TRT VIS2 #> 7 student_t(4, 7.65, 8.24) TRT VIS3 #> 8 student_t(4, 4.64, 8.21) TRT VIS4
After you have a labeling scheme, brm_prior_archetype() can create a brms prior for the important fixed effects.^[brms priors are documented in https://paulbuerkner.com/brms/reference/set_prior.html.]
prior <- brm_prior_archetype(label = label, archetype = archetype) prior #> prior class coef group resp dpar nlpar lb ub source #> student_t(4, -7.57, 4.96) b x_PBO_VIS1 <NA> <NA> user #> student_t(4, 3.14, 7.86) b x_PBO_VIS2 <NA> <NA> user #> student_t(4, 8.78, 8.18) b x_PBO_VIS3 <NA> <NA> user #> student_t(4, 3.36, 8.10) b x_PBO_VIS4 <NA> <NA> user #> student_t(4, -2.96, 4.78) b x_TRT_VIS1 <NA> <NA> user #> student_t(4, 3.13, 7.64) b x_TRT_VIS2 <NA> <NA> user #> student_t(4, 7.65, 8.24) b x_TRT_VIS3 <NA> <NA> user #> student_t(4, 4.64, 8.21) b x_TRT_VIS4 <NA> <NA> user
In less common situations, you may wish to assign priors to nuisance parameters. For example, our model accounts for interactions between baseline and discrete time, and it may be reasonable to assign priors to these slopes based on high-quality historical data. This requires a thorough understanding of the fixed effect structure of the model, but it can be done directly through brms. First, check the formula for the included nuisance parameters. brm_formula() automatically understands archetypes.
brm_formula(archetype) #> FEV1 ~ 0 + x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3 + x_PBO_VIS4 + x_TRT_VIS1 + x_TRT_VIS2 + x_TRT_VIS3 + x_TRT_VIS4 + nuisance_WEIGHT + nuisance_SEX_Male + unstr(time = AVISIT, gr = USUBJID) #> sigma ~ 0 + AVISIT
The "nuisance_*" terms are the nuisance variables, and the ones involving baseline are nuisance_FEV1_BL.AVISITVIS1, nuisance_FEV1_BL.AVISITVIS2, nuisance_FEV1_BL.AVISITVIS3, and nuisance_FEV1_BL.AVISITVIS4. Because there is no overall slope for baseline, we can interpret each term as the linear rate of change in the outcome variable per unit increase in baseline for a given discrete time point. Suppose we use this interpretation to construct informative priors student_t(4, -0.83, 1), student_t(4, -0.78, 1), student_t(4, -0.86, 1), and student_t(4, -0.82, 1), respectively. Use brms::set_prior() and c() to append these priors to our existing prior object:
The model still has many parameters where we did not set priors, and brms sets automatic defaults. You can see these defaults with brms::get_prior().
https://paulbuerkner.com/brms/reference/set_prior.html documents many of the default priors set by brms. In particular, "(flat)" denotes an improper uniform prior over all the real numbers.
Modeling and analysis
The downstream methods in brms.mmrm automatically understand how to work with informative prior archetypes. Notably, the formula uses custom interest and nuisance variables instead of the original variables in the data.
formula <- brm_formula(archetype) formula #> FEV1 ~ 0 + x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3 + x_PBO_VIS4 + x_TRT_VIS1 + x_TRT_VIS2 + x_TRT_VIS3 + x_TRT_VIS4 + nuisance_WEIGHT + nuisance_SEX_Male + unstr(time = AVISIT, gr = USUBJID) #> sigma ~ 0 + AVISIT
The model can accept the archetype, formula, and prior. Usage is the same as in non-archetype workflows.
model <- brm_model( data = archetype, formula = formula, prior = prior, refresh = 0 ) #> Compiling Stan program... #> Start sampling brms::prior_summary(model) #> prior class coef group resp dpar nlpar lb ub #> (flat) b #> (flat) b nuisance_SEX_Male #> (flat) b nuisance_WEIGHT #> student_t(4, -7.57, 4.96) b x_PBO_VIS1 #> student_t(4, 3.14, 7.86) b x_PBO_VIS2 #> student_t(4, 8.78, 8.18) b x_PBO_VIS3 #> student_t(4, 3.36, 8.10) b x_PBO_VIS4 #> student_t(4, -2.96, 4.78) b x_TRT_VIS1 #> student_t(4, 3.13, 7.64) b x_TRT_VIS2 #> student_t(4, 7.65, 8.24) b x_TRT_VIS3 #> student_t(4, 4.64, 8.21) b x_TRT_VIS4 #> (flat) b sigma #> (flat) b AVISITVIS1 sigma #> (flat) b AVISITVIS2 sigma #> (flat) b AVISITVIS3 sigma #> (flat) b AVISITVIS4 sigma #> lkj_corr_cholesky(1) Lcortime #> source #> default #> (vectorized) #> (vectorized) #> user #> user #> user #> user #> user #> user #> user #> user #> default #> (vectorized) #> (vectorized) #> (vectorized) #> (vectorized) #> default
Marginal mean estimation, post-processing, and visualization automatically understand the archetype without any user intervention.
draws <- brm_marginal_draws( data = archetype, formula = formula, model = model ) summaries_model <- brm_marginal_summaries(draws) summaries_data <- brm_marginal_data(archetype) brm_plot_compare(model = summaries_model, data = summaries_data)
plot of chunk archetype_compare_data
All archetypes
brms.mmrm supports a variety of informative prior archetypes with different kinds of fixed effects. For example, brms.mmrm supports simple cell mean and treatment effect parameterizations.
summary(brm_archetype_cells(data, intercept = FALSE)) #> # This is the "cells" informative prior archetype in brms.mmrm. #> # The following equations show the relationships between the #> # marginal means (left-hand side) and fixed effect parameters #> # (right-hand side). #> # #> # PBO:VIS1 = x_PBO_VIS1 #> # PBO:VIS2 = x_PBO_VIS2 #> # PBO:VIS3 = x_PBO_VIS3 #> # PBO:VIS4 = x_PBO_VIS4 #> # TRT:VIS1 = x_TRT_VIS1 #> # TRT:VIS2 = x_TRT_VIS2 #> # TRT:VIS3 = x_TRT_VIS3 #> # TRT:VIS4 = x_TRT_VIS4
summary(brm_archetype_effects(data, intercept = FALSE)) #> # This is the "effects" informative prior archetype in brms.mmrm. #> # The following equations show the relationships between the #> # marginal means (left-hand side) and fixed effect parameters #> # (right-hand side). #> # #> # PBO:VIS1 = x_PBO_VIS1 #> # PBO:VIS2 = x_PBO_VIS2 #> # PBO:VIS3 = x_PBO_VIS3 #> # PBO:VIS4 = x_PBO_VIS4 #> # TRT:VIS1 = x_PBO_VIS1 + x_TRT_VIS1 #> # TRT:VIS2 = x_PBO_VIS2 + x_TRT_VIS2 #> # TRT:VIS3 = x_PBO_VIS3 + x_TRT_VIS3 #> # TRT:VIS4 = x_PBO_VIS4 + x_TRT_VIS4
There are archetypes to parameterize the average across all time points in the data. Below, x_group_1_time_2 is the average across time points for group 1 because it is the algebraic result of simplifying (group_1:time_2 + group_1:time_3 + group_1:time_3) / 3.
summary(brm_archetype_average_cells(data, intercept = FALSE)) #> # This is the "average cells" informative prior archetype in brms.mmrm. #> # The following equations show the relationships between the #> # marginal means (left-hand side) and fixed effect parameters #> # (right-hand side). #> # #> # PBO:VIS1 = 4*x_PBO_VIS1 - x_PBO_VIS2 - x_PBO_VIS3 - x_PBO_VIS4 #> # PBO:VIS2 = x_PBO_VIS2 #> # PBO:VIS3 = x_PBO_VIS3 #> # PBO:VIS4 = x_PBO_VIS4 #> # TRT:VIS1 = 4*x_TRT_VIS1 - x_TRT_VIS2 - x_TRT_VIS3 - x_TRT_VIS4 #> # TRT:VIS2 = x_TRT_VIS2 #> # TRT:VIS3 = x_TRT_VIS3 #> # TRT:VIS4 = x_TRT_VIS4
There is also a treatment effect version where x_group_2_time_2 becomes the time-averaged treatment effect of group 2 relative to group 1.
summary(brm_archetype_average_effects(data, intercept = FALSE)) #> # This is the "average effects" informative prior archetype in brms.mmrm. #> # The following equations show the relationships between the #> # marginal means (left-hand side) and fixed effect parameters #> # (right-hand side). #> # #> # PBO:VIS1 = 4*x_PBO_VIS1 - x_PBO_VIS2 - x_PBO_VIS3 - x_PBO_VIS4 #> # PBO:VIS2 = x_PBO_VIS2 #> # PBO:VIS3 = x_PBO_VIS3 #> # PBO:VIS4 = x_PBO_VIS4 #> # TRT:VIS1 = 4*x_PBO_VIS1 - x_PBO_VIS2 - x_PBO_VIS3 - x_PBO_VIS4 + 4*x_TRT_VIS1 - x_TRT_VIS2 - x_TRT_VIS3 - x_TRT_VIS4 #> # TRT:VIS2 = x_PBO_VIS2 + x_TRT_VIS2 #> # TRT:VIS3 = x_PBO_VIS3 + x_TRT_VIS3 #> # TRT:VIS4 = x_PBO_VIS4 + x_TRT_VIS4
The example in this vignette uses the "successive cells" archetype, where fixed effects represent successive differences between adjacent time points.
summary(brm_archetype_successive_cells(data, intercept = FALSE)) #> # This is the "successive cells" informative prior archetype in brms.mmrm. #> # The following equations show the relationships between the #> # marginal means (left-hand side) and fixed effect parameters #> # (right-hand side). #> # #> # PBO:VIS1 = x_PBO_VIS1 #> # PBO:VIS2 = x_PBO_VIS1 + x_PBO_VIS2 #> # PBO:VIS3 = x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3 #> # PBO:VIS4 = x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3 + x_PBO_VIS4 #> # TRT:VIS1 = x_TRT_VIS1 #> # TRT:VIS2 = x_TRT_VIS1 + x_TRT_VIS2 #> # TRT:VIS3 = x_TRT_VIS1 + x_TRT_VIS2 + x_TRT_VIS3 #> # TRT:VIS4 = x_TRT_VIS1 + x_TRT_VIS2 + x_TRT_VIS3 + x_TRT_VIS4
There is also a treatment effect version of the successive differences archetype:
summary(brm_archetype_successive_effects(data, intercept = FALSE)) #> # This is the "successive effects" informative prior archetype in brms.mmrm. #> # The following equations show the relationships between the #> # marginal means (left-hand side) and fixed effect parameters #> # (right-hand side). #> # #> # PBO:VIS1 = x_PBO_VIS1 #> # PBO:VIS2 = x_PBO_VIS1 + x_PBO_VIS2 #> # PBO:VIS3 = x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3 #> # PBO:VIS4 = x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3 + x_PBO_VIS4 #> # TRT:VIS1 = x_PBO_VIS1 + x_TRT_VIS1 #> # TRT:VIS2 = x_PBO_VIS1 + x_PBO_VIS2 + x_TRT_VIS1 + x_TRT_VIS2 #> # TRT:VIS3 = x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3 + x_TRT_VIS1 + x_TRT_VIS2 + x_TRT_VIS3 #> # TRT:VIS4 = x_PBO_VIS1 + x_PBO_VIS2 + x_PBO_VIS3 + x_PBO_VIS4 + x_TRT_VIS1 + x_TRT_VIS2 + x_TRT_VIS3 + x_TRT_VIS4
Variations on archetypes
Archetypes can be customized. As an example, consider the simple cell means archetype.
summary(brm_archetype_cells(data)) #> # This is the "cells" informative prior archetype in brms.mmrm. #> # The following equations show the relationships between the #> # marginal means (left-hand side) and fixed effect parameters #> # (right-hand side). #> # #> # PBO:VIS1 = x_PBO_VIS1 #> # PBO:VIS2 = x_PBO_VIS2 #> # PBO:VIS3 = x_PBO_VIS3 #> # PBO:VIS4 = x_PBO_VIS4 #> # TRT:VIS1 = x_TRT_VIS1 #> # TRT:VIS2 = x_TRT_VIS2 #> # TRT:VIS3 = x_TRT_VIS3 #> # TRT:VIS4 = x_TRT_VIS4
To include an intercept term which all the marginal means share, set intercept = TRUE.
summary(brm_archetype_cells(data, intercept = TRUE)) #> # This is the "cells" informative prior archetype in brms.mmrm. #> # The following equations show the relationships between the #> # marginal means (left-hand side) and fixed effect parameters #> # (right-hand side). #> # #> # PBO:VIS1 = x_PBO_VIS1 #> # PBO:VIS2 = x_PBO_VIS1 + x_PBO_VIS2 #> # PBO:VIS3 = x_PBO_VIS1 + x_PBO_VIS3 #> # PBO:VIS4 = x_PBO_VIS1 + x_PBO_VIS4 #> # TRT:VIS1 = x_PBO_VIS1 + x_TRT_VIS1 #> # TRT:VIS2 = x_PBO_VIS1 + x_TRT_VIS2 #> # TRT:VIS3 = x_PBO_VIS1 + x_TRT_VIS3 #> # TRT:VIS4 = x_PBO_VIS1 + x_TRT_VIS4
To set up constrained longitudinal data analysis (cLDA), set clda = TRUE. This constraint pools all treatment groups at baseline, and it can help model clinical trials where a baseline measurement is observed before randomization. Some archetypes cannot support cLDA (e.g. brm_archetype_average_cells() and brm_archetype_average_effects()).
summary(brm_archetype_cells(data, clda = TRUE)) #> # This is the "cells" informative prior archetype in brms.mmrm. #> # The following equations show the relationships between the #> # marginal means (left-hand side) and fixed effect parameters #> # (right-hand side). #> # #> # PBO:VIS1 = x_PBO_VIS1 #> # PBO:VIS2 = x_PBO_VIS2 #> # PBO:VIS3 = x_PBO_VIS3 #> # PBO:VIS4 = x_PBO_VIS4 #> # TRT:VIS1 = x_PBO_VIS1 #> # TRT:VIS2 = x_TRT_VIS2 #> # TRT:VIS3 = x_TRT_VIS3 #> # TRT:VIS4 = x_TRT_VIS4
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