#
Individual Variance Detection #
Individual Variance Detection
ivd is an R package for random effects selection in the scale part of
Mixed Effects Location Scale Models (MELSM). ivd() fits a random
intercepts model with a spike-and-slab prior on the random effects of
the scale. The method is described in Carmo et al. (2026).
This package can be installed with
# install.packages("devtools")
# devtools::install_github("consistentlybetter/ivd")
library(ivd)
library(data.table)
#>
#> Attaching package: 'data.table'
#> The following object is masked from 'package:base':
#>
#> %notin%
The illustration uses openly accessible data from The Basic Education
Evaluation System (Saeb) conducted by Brazil’s National Institute for
Educational Studies and Research (Inep), available at
https://web.archive.org/web/20250202015037/https://www.gov.br/inep/pt-br/areas-de-atuacao/avaliacao-e-exames-educacionais/saeb/resultados.
It is also available as the saeb dataset in the ivd package.
Separate within- from between-school effects. That is, besides
student_ses, compute school_ses.
## Calculate school-level SES
school_ses <- saeb[, .(school_ses = mean(student_ses, na.rm = TRUE)), by = school_id]
## Join the school_ses back to the original dataset
saeb <- saeb[school_ses, on = "school_id"]
## Define student level SES as deviation from the school SES
saeb$student_ses <- saeb$student_ses - saeb$school_ses
## Grand mean center school ses
saeb$school_ses <- c(scale(saeb$school_ses, scale = FALSE))
Illustration of school level variability:
library(ggplot2)
plot0 <- ggplot( data = saeb, aes( x = school_id, y = math_proficiency) )
plot0 + geom_point(aes(color = school_id), show.legend = FALSE)
We will predict math_proficiency which is a standardized variable
capturing math proficiency at the end of grade 12.
Both, location (means) and scale (residual variances) are modeled as a
function of student and school SES. Note that the formula objects for
both location and scale follow lme4 notation.
out <- ivd(location_formula = math_proficiency ~ student_ses * school_ses + (1 | school_id),
scale_formula = ~ student_ses * school_ses + (1 | school_id),
data = saeb,
niter = 5000, nburnin = 6000,
progress = TRUE,
seed = 800056)
#> ivd: compiling and sampling 4 chains in parallel ...
The fitted object prints a concise overview of the model, the data, and a one-line convergence note:
out
#> Individual variance detection (ivd) model
#>
#> Location: math_proficiency ~ student_ses * school_ses + (1 | school_id)
#> Scale: ~student_ses * school_ses + (1 | school_id)
#> Data: 11386 observations in 160 clusters
#> Sampling: 4 chains, 5000 post-warmup draws each
#> Diagnostics: max split-Rhat = 1.013, min n_eff = 99
The summary shows the fixed and random effects and it returns all posterior inclusion probabilities (PIP) for each one of the 160 schools’ residual variance random effects. The PIP returns the probability of a school belonging to the slab, that is, the probability of the model having to include the random scale effect.
In other words, large PIP’s indicate schools that are substantially deviating from the fixed scale effects either because they are much more or much less variable compared to other schools in math proficiency.
One can readily convert those PIP’s to odds, indicating that a school with a PIP = .75 is three times as likely to belonging to the slab than belonging to the spike. With an .50 inclusion prior, these odds can be readily interpreted as Bayes Factors.
summary(out)
#> Summary statistics for ivd model:
#> Chains (workers): 4
#>
#> Mean SD Time-series SE 2.5% 50% 97.5%
#> R[scl_Intc, Intc] -0.684 0.175 0.006 -0.957 -0.706 -0.287
#> Intc 0.129 0.025 0.001 0.082 0.129 0.180
#> student_ses 0.082 0.010 0.000 0.063 0.082 0.100
#> school_ses 0.676 0.085 0.004 0.509 0.676 0.847
#> student_ses:school_ses -0.023 0.040 0.000 -0.100 -0.023 0.055
#> sd_Intc 0.268 0.019 0.001 0.232 0.267 0.308
#> sd_scl_Intc 0.080 0.015 0.000 0.051 0.079 0.112
#> pip[Intc, 1] 0.462 0.499 0.004 0.000 0.000 1.000
#> pip[Intc, 2] 0.488 0.500 0.004 0.000 0.000 1.000
#> pip[Intc, 3] 0.457 0.498 0.004 0.000 0.000 1.000
#> pip[Intc, 4] 0.481 0.500 0.005 0.000 0.000 1.000
#> pip[Intc, 5] 0.525 0.499 0.005 0.000 1.000 1.000
#> pip[Intc, 6] 0.423 0.494 0.004 0.000 0.000 1.000
#> pip[Intc, 7] 0.408 0.492 0.005 0.000 0.000 1.000
#> pip[Intc, 8] 0.464 0.499 0.004 0.000 0.000 1.000
#> pip[Intc, 9] 0.987 0.114 0.001 1.000 1.000 1.000
#> pip[Intc, 10] 0.442 0.497 0.004 0.000 0.000 1.000
#> pip[Intc, 11] 0.561 0.496 0.004 0.000 1.000 1.000
#> pip[Intc, 12] 0.360 0.480 0.004 0.000 0.000 1.000
#> pip[Intc, 13] 0.476 0.499 0.005 0.000 0.000 1.000
#> pip[Intc, 14] 0.600 0.490 0.006 0.000 1.000 1.000
#> pip[Intc, 15] 0.509 0.500 0.004 0.000 1.000 1.000
#> pip[Intc, 16] 0.480 0.500 0.004 0.000 0.000 1.000
#> pip[Intc, 17] 0.423 0.494 0.004 0.000 0.000 1.000
#> pip[Intc, 18] 0.531 0.499 0.004 0.000 1.000 1.000
#> pip[Intc, 19] 0.236 0.425 0.005 0.000 0.000 1.000
#> pip[Intc, 20] 0.519 0.500 0.004 0.000 1.000 1.000
#> pip[Intc, 21] 0.351 0.477 0.004 0.000 0.000 1.000
#> pip[Intc, 22] 0.565 0.496 0.005 0.000 1.000 1.000
#> pip[Intc, 23] 0.504 0.500 0.004 0.000 1.000 1.000
#> pip[Intc, 24] 0.484 0.500 0.004 0.000 0.000 1.000
#> pip[Intc, 25] 0.458 0.498 0.004 0.000 0.000 1.000
#> pip[Intc, 26] 0.516 0.500 0.004 0.000 1.000 1.000
#> pip[Intc, 27] 0.441 0.497 0.004 0.000 0.000 1.000
#> pip[Intc, 28] 0.395 0.489 0.004 0.000 0.000 1.000
#> pip[Intc, 29] 0.484 0.500 0.004 0.000 0.000 1.000
#> pip[Intc, 30] 0.449 0.497 0.005 0.000 0.000 1.000
#> pip[Intc, 31] 0.466 0.499 0.004 0.000 0.000 1.000
#> pip[Intc, 32] 0.477 0.499 0.005 0.000 0.000 1.000
#> pip[Intc, 33] 0.490 0.500 0.004 0.000 0.000 1.000
#> pip[Intc, 34] 0.543 0.498 0.004 0.000 1.000 1.000
#> pip[Intc, 35] 0.663 0.473 0.005 0.000 1.000 1.000
#> pip[Intc, 36] 0.360 0.480 0.006 0.000 0.000 1.000
#> pip[Intc, 37] 0.463 0.499 0.004 0.000 0.000 1.000
#> pip[Intc, 38] 0.452 0.498 0.004 0.000 0.000 1.000
#> pip[Intc, 39] 0.696 0.460 0.009 0.000 1.000 1.000
#> pip[Intc, 40] 0.456 0.498 0.004 0.000 0.000 1.000
#> pip[Intc, 41] 0.646 0.478 0.009 0.000 1.000 1.000
#> pip[Intc, 42] 0.465 0.499 0.004 0.000 0.000 1.000
#> pip[Intc, 43] 0.413 0.492 0.004 0.000 0.000 1.000
#> pip[Intc, 44] 0.447 0.497 0.004 0.000 0.000 1.000
#> pip[Intc, 45] 0.442 0.497 0.004 0.000 0.000 1.000
#> pip[Intc, 46] 0.999 0.026 0.000 1.000 1.000 1.000
#> pip[Intc, 47] 0.464 0.499 0.004 0.000 0.000 1.000
#> pip[Intc, 48] 0.612 0.487 0.004 0.000 1.000 1.000
#> pip[Intc, 49] 0.494 0.500 0.005 0.000 0.000 1.000
#> pip[Intc, 50] 0.519 0.500 0.004 0.000 1.000 1.000
#> pip[Intc, 51] 0.462 0.499 0.004 0.000 0.000 1.000
#> pip[Intc, 52] 0.543 0.498 0.004 0.000 1.000 1.000
#> pip[Intc, 53] 0.825 0.380 0.005 0.000 1.000 1.000
#> pip[Intc, 54] 0.607 0.488 0.004 0.000 1.000 1.000
#> pip[Intc, 55] 0.395 0.489 0.004 0.000 0.000 1.000
#> pip[Intc, 56] 0.470 0.499 0.004 0.000 0.000 1.000
#> pip[Intc, 57] 0.621 0.485 0.004 0.000 1.000 1.000
#> pip[Intc, 58] 0.406 0.491 0.004 0.000 0.000 1.000
#> pip[Intc, 59] 0.429 0.495 0.004 0.000 0.000 1.000
#> pip[Intc, 60] 0.523 0.499 0.004 0.000 1.000 1.000
#> pip[Intc, 61] 0.287 0.452 0.005 0.000 0.000 1.000
#> pip[Intc, 62] 0.417 0.493 0.004 0.000 0.000 1.000
#> pip[Intc, 63] 0.496 0.500 0.004 0.000 0.000 1.000
#> pip[Intc, 64] 0.732 0.443 0.004 0.000 1.000 1.000
#> pip[Intc, 65] 0.487 0.500 0.004 0.000 0.000 1.000
#> pip[Intc, 66] 0.631 0.483 0.004 0.000 1.000 1.000
#> pip[Intc, 67] 0.349 0.477 0.004 0.000 0.000 1.000
#> pip[Intc, 68] 0.374 0.484 0.005 0.000 0.000 1.000
#> pip[Intc, 69] 0.454 0.498 0.004 0.000 0.000 1.000
#> pip[Intc, 70] 0.394 0.489 0.005 0.000 0.000 1.000
#> pip[Intc, 71] 0.383 0.486 0.004 0.000 0.000 1.000
#> pip[Intc, 72] 0.379 0.485 0.004 0.000 0.000 1.000
#> pip[Intc, 73] 0.389 0.488 0.006 0.000 0.000 1.000
#> pip[Intc, 74] 0.500 0.500 0.004 0.000 1.000 1.000
#> pip[Intc, 75] 0.467 0.499 0.004 0.000 0.000 1.000
#> pip[Intc, 76] 0.466 0.499 0.004 0.000 0.000 1.000
#> pip[Intc, 77] 0.459 0.498 0.004 0.000 0.000 1.000
#> pip[Intc, 78] 0.437 0.496 0.004 0.000 0.000 1.000
#> pip[Intc, 79] 0.385 0.487 0.004 0.000 0.000 1.000
#> pip[Intc, 80] 0.507 0.500 0.004 0.000 1.000 1.000
#> pip[Intc, 81] 0.503 0.500 0.004 0.000 1.000 1.000
#> pip[Intc, 82] 0.521 0.500 0.004 0.000 1.000 1.000
#> pip[Intc, 83] 0.484 0.500 0.004 0.000 0.000 1.000
#> pip[Intc, 84] 0.512 0.500 0.004 0.000 1.000 1.000
#> pip[Intc, 85] 0.464 0.499 0.004 0.000 0.000 1.000
#> pip[Intc, 86] 0.533 0.499 0.004 0.000 1.000 1.000
#> pip[Intc, 87] 0.725 0.447 0.006 0.000 1.000 1.000
#> pip[Intc, 88] 0.480 0.500 0.004 0.000 0.000 1.000
#> pip[Intc, 89] 0.531 0.499 0.004 0.000 1.000 1.000
#> pip[Intc, 90] 0.495 0.500 0.004 0.000 0.000 1.000
#> pip[Intc, 91] 0.438 0.496 0.004 0.000 0.000 1.000
#> pip[Intc, 92] 0.719 0.450 0.005 0.000 1.000 1.000
#> pip[Intc, 93] 0.443 0.497 0.004 0.000 0.000 1.000
#> pip[Intc, 94] 0.476 0.499 0.004 0.000 0.000 1.000
#> pip[Intc, 95] 0.751 0.433 0.004 0.000 1.000 1.000
#> pip[Intc, 96] 0.449 0.497 0.004 0.000 0.000 1.000
#> pip[Intc, 97] 0.347 0.476 0.004 0.000 0.000 1.000
#> pip[Intc, 98] 0.464 0.499 0.004 0.000 0.000 1.000
#> pip[Intc, 99] 0.560 0.496 0.004 0.000 1.000 1.000
#> pip[Intc, 100] 0.451 0.498 0.004 0.000 0.000 1.000
#> pip[Intc, 101] 0.422 0.494 0.004 0.000 0.000 1.000
#> pip[Intc, 102] 0.503 0.500 0.005 0.000 1.000 1.000
#> pip[Intc, 103] 0.383 0.486 0.004 0.000 0.000 1.000
#> pip[Intc, 104] 0.456 0.498 0.004 0.000 0.000 1.000
#> pip[Intc, 105] 0.487 0.500 0.004 0.000 0.000 1.000
#> pip[Intc, 106] 0.457 0.498 0.005 0.000 0.000 1.000
#> pip[Intc, 107] 0.519 0.500 0.004 0.000 1.000 1.000
#> pip[Intc, 108] 0.540 0.498 0.006 0.000 1.000 1.000
#> pip[Intc, 109] 0.531 0.499 0.004 0.000 1.000 1.000
#> pip[Intc, 110] 0.396 0.489 0.004 0.000 0.000 1.000
#> pip[Intc, 111] 0.447 0.497 0.005 0.000 0.000 1.000
#> pip[Intc, 112] 0.462 0.499 0.005 0.000 0.000 1.000
#> pip[Intc, 113] 0.628 0.483 0.004 0.000 1.000 1.000
#> pip[Intc, 114] 0.892 0.310 0.003 0.000 1.000 1.000
#> pip[Intc, 115] 0.811 0.392 0.005 0.000 1.000 1.000
#> pip[Intc, 116] 0.464 0.499 0.005 0.000 0.000 1.000
#> pip[Intc, 117] 0.430 0.495 0.004 0.000 0.000 1.000
#> pip[Intc, 118] 0.386 0.487 0.004 0.000 0.000 1.000
#> pip[Intc, 119] 0.489 0.500 0.004 0.000 0.000 1.000
#> pip[Intc, 120] 0.561 0.496 0.004 0.000 1.000 1.000
#> pip[Intc, 121] 0.327 0.469 0.004 0.000 0.000 1.000
#> pip[Intc, 122] 0.540 0.498 0.008 0.000 1.000 1.000
#> pip[Intc, 123] 0.575 0.494 0.004 0.000 1.000 1.000
#> pip[Intc, 124] 0.752 0.432 0.004 0.000 1.000 1.000
#> pip[Intc, 125] 0.473 0.499 0.004 0.000 0.000 1.000
#> pip[Intc, 126] 0.504 0.500 0.004 0.000 1.000 1.000
#> pip[Intc, 127] 0.562 0.496 0.004 0.000 1.000 1.000
#> pip[Intc, 128] 0.514 0.500 0.004 0.000 1.000 1.000
#> pip[Intc, 129] 0.410 0.492 0.004 0.000 0.000 1.000
#> pip[Intc, 130] 0.461 0.499 0.004 0.000 0.000 1.000
#> pip[Intc, 131] 0.559 0.496 0.004 0.000 1.000 1.000
#> pip[Intc, 132] 0.400 0.490 0.004 0.000 0.000 1.000
#> pip[Intc, 133] 0.355 0.478 0.004 0.000 0.000 1.000
#> pip[Intc, 134] 0.510 0.500 0.004 0.000 1.000 1.000
#> pip[Intc, 135] 0.430 0.495 0.004 0.000 0.000 1.000
#> pip[Intc, 136] 0.390 0.488 0.004 0.000 0.000 1.000
#> pip[Intc, 137] 0.448 0.497 0.004 0.000 0.000 1.000
#> pip[Intc, 138] 0.439 0.496 0.004 0.000 0.000 1.000
#> pip[Intc, 139] 0.413 0.492 0.004 0.000 0.000 1.000
#> pip[Intc, 140] 0.568 0.495 0.004 0.000 1.000 1.000
#> pip[Intc, 141] 0.522 0.500 0.004 0.000 1.000 1.000
#> pip[Intc, 142] 0.473 0.499 0.004 0.000 0.000 1.000
#> pip[Intc, 143] 0.416 0.493 0.004 0.000 0.000 1.000
#> pip[Intc, 144] 0.481 0.500 0.004 0.000 0.000 1.000
#> pip[Intc, 145] 0.456 0.498 0.004 0.000 0.000 1.000
#> pip[Intc, 146] 0.454 0.498 0.004 0.000 0.000 1.000
#> pip[Intc, 147] 0.486 0.500 0.004 0.000 0.000 1.000
#> pip[Intc, 148] 0.635 0.481 0.004 0.000 1.000 1.000
#> pip[Intc, 149] 0.712 0.453 0.004 0.000 1.000 1.000
#> pip[Intc, 150] 0.381 0.486 0.005 0.000 0.000 1.000
#> pip[Intc, 151] 0.464 0.499 0.004 0.000 0.000 1.000
#> pip[Intc, 152] 0.461 0.498 0.004 0.000 0.000 1.000
#> pip[Intc, 153] 0.762 0.426 0.004 0.000 1.000 1.000
#> pip[Intc, 154] 0.341 0.474 0.005 0.000 0.000 1.000
#> pip[Intc, 155] 0.417 0.493 0.004 0.000 0.000 1.000
#> pip[Intc, 156] 0.581 0.493 0.004 0.000 1.000 1.000
#> pip[Intc, 157] 0.534 0.499 0.004 0.000 1.000 1.000
#> pip[Intc, 158] 0.462 0.499 0.004 0.000 0.000 1.000
#> pip[Intc, 159] 0.477 0.500 0.004 0.000 0.000 1.000
#> pip[Intc, 160] 0.516 0.500 0.004 0.000 1.000 1.000
#> scl_Intc -0.234 0.008 0.000 -0.251 -0.234 -0.217
#> scl_student_ses 0.031 0.009 0.000 0.014 0.031 0.048
#> scl_school_ses 0.119 0.035 0.001 0.051 0.119 0.187
#> scl_student_ses:school_ses 0.074 0.038 0.001 0.000 0.074 0.148
#> n_eff R-hat
#> R[scl_Intc, Intc] 374 1.007
#> Intc 129 1.008
#> student_ses 14605 1.000
#> school_ses 99 1.010
#> student_ses:school_ses 18448 1.000
#> sd_Intc 238 1.013
#> sd_scl_Intc 982 1.004
#> pip[Intc, 1] 20765 1.000
#> pip[Intc, 2] 18405 1.000
#> pip[Intc, 3] 15161 1.000
#> pip[Intc, 4] 12172 1.000
#> pip[Intc, 5] 10662 1.000
#> pip[Intc, 6] 17089 1.000
#> pip[Intc, 7] 10611 1.000
#> pip[Intc, 8] 19464 1.000
#> pip[Intc, 9] 6344 1.000
#> pip[Intc, 10] 17568 1.000
#> pip[Intc, 11] 8090 1.000
#> pip[Intc, 12] 12700 1.000
#> pip[Intc, 13] 11474 1.000
#> pip[Intc, 14] 6403 1.000
#> pip[Intc, 15] 20014 1.000
#> pip[Intc, 16] 18786 1.000
#> pip[Intc, 17] 15541 1.000
#> pip[Intc, 18] 15736 1.000
#> pip[Intc, 19] 8558 1.000
#> pip[Intc, 20] 16595 1.000
#> pip[Intc, 21] 11061 1.000
#> pip[Intc, 22] 11715 1.000
#> pip[Intc, 23] 13495 1.000
#> pip[Intc, 24] 19366 1.000
#> pip[Intc, 25] 20431 1.000
#> pip[Intc, 26] 18187 1.000
#> pip[Intc, 27] 18205 1.000
#> pip[Intc, 28] 13988 1.001
#> pip[Intc, 29] 10100 1.000
#> pip[Intc, 30] 10719 1.001
#> pip[Intc, 31] 15582 1.000
#> pip[Intc, 32] 10759 1.000
#> pip[Intc, 33] 14083 1.000
#> pip[Intc, 34] 15857 1.000
#> pip[Intc, 35] 5180 1.000
#> pip[Intc, 36] 2989 1.000
#> pip[Intc, 37] 19070 1.000
#> pip[Intc, 38] 18754 1.000
#> pip[Intc, 39] 1337 1.002
#> pip[Intc, 40] 19577 1.000
#> pip[Intc, 41] 1791 1.002
#> pip[Intc, 42] 13703 1.000
#> pip[Intc, 43] 13720 1.001
#> pip[Intc, 44] 16137 1.000
#> pip[Intc, 45] 17674 1.000
#> pip[Intc, 46] 20084 1.000
#> pip[Intc, 47] 15416 1.000
#> pip[Intc, 48] 10646 1.000
#> pip[Intc, 49] 6412 1.000
#> pip[Intc, 50] 20387 1.000
#> pip[Intc, 51] 16915 1.000
#> pip[Intc, 52] 14039 1.000
#> pip[Intc, 53] 5074 1.000
#> pip[Intc, 54] 13878 1.000
#> pip[Intc, 55] 16539 1.000
#> pip[Intc, 56] 15420 1.000
#> pip[Intc, 57] 14632 1.000
#> pip[Intc, 58] 12181 1.000
#> pip[Intc, 59] 14098 1.000
#> pip[Intc, 60] 12601 1.000
#> pip[Intc, 61] 6476 1.001
#> pip[Intc, 62] 14000 1.000
#> pip[Intc, 63] 16533 1.000
#> pip[Intc, 64] 8984 1.000
#> pip[Intc, 65] 16346 1.000
#> pip[Intc, 66] 12721 1.000
#> pip[Intc, 67] 11822 1.000
#> pip[Intc, 68] 10532 1.000
#> pip[Intc, 69] 18552 1.000
#> pip[Intc, 70] 6564 1.000
#> pip[Intc, 71] 14363 1.000
#> pip[Intc, 72] 12043 1.000
#> pip[Intc, 73] 2295 1.001
#> pip[Intc, 74] 17685 1.000
#> pip[Intc, 75] 21842 1.000
#> pip[Intc, 76] 19445 1.000
#> pip[Intc, 77] 16291 1.000
#> pip[Intc, 78] 18353 1.000
#> pip[Intc, 79] 14302 1.000
#> pip[Intc, 80] 17135 1.000
#> pip[Intc, 81] 13744 1.001
#> pip[Intc, 82] 12187 1.000
#> pip[Intc, 83] 18013 1.000
#> pip[Intc, 84] 18319 1.000
#> pip[Intc, 85] 17394 1.000
#> pip[Intc, 86] 11812 1.000
#> pip[Intc, 87] 4885 1.001
#> pip[Intc, 88] 13652 1.000
#> pip[Intc, 89] 17243 1.000
#> pip[Intc, 90] 21644 1.000
#> pip[Intc, 91] 18460 1.000
#> pip[Intc, 92] 5759 1.001
#> pip[Intc, 93] 18806 1.000
#> pip[Intc, 94] 18599 1.000
#> pip[Intc, 95] 8758 1.000
#> pip[Intc, 96] 16783 1.000
#> pip[Intc, 97] 13241 1.000
#> pip[Intc, 98] 15741 1.000
#> pip[Intc, 99] 13483 1.000
#> pip[Intc, 100] 15013 1.000
#> pip[Intc, 101] 13821 1.000
#> pip[Intc, 102] 8177 1.000
#> pip[Intc, 103] 12892 1.000
#> pip[Intc, 104] 18178 1.000
#> pip[Intc, 105] 18682 1.000
#> pip[Intc, 106] 7775 1.000
#> pip[Intc, 107] 16361 1.000
#> pip[Intc, 108] 4779 1.001
#> pip[Intc, 109] 18965 1.000
#> pip[Intc, 110] 13428 1.000
#> pip[Intc, 111] 11845 1.001
#> pip[Intc, 112] 10370 1.000
#> pip[Intc, 113] 12008 1.000
#> pip[Intc, 114] 6633 1.000
#> pip[Intc, 115] 4723 1.001
#> pip[Intc, 116] 9682 1.000
#> pip[Intc, 117] 16013 1.000
#> pip[Intc, 118] 14030 1.000
#> pip[Intc, 119] 17817 1.000
#> pip[Intc, 120] 17867 1.000
#> pip[Intc, 121] 10148 1.000
#> pip[Intc, 122] 2372 1.001
#> pip[Intc, 123] 18772 1.000
#> pip[Intc, 124] 8903 1.000
#> pip[Intc, 125] 13669 1.000
#> pip[Intc, 126] 15413 1.000
#> pip[Intc, 127] 17042 1.000
#> pip[Intc, 128] 15961 1.000
#> pip[Intc, 129] 14117 1.000
#> pip[Intc, 130] 18737 1.000
#> pip[Intc, 131] 17722 1.000
#> pip[Intc, 132] 16606 1.000
#> pip[Intc, 133] 13785 1.000
#> pip[Intc, 134] 15188 1.000
#> pip[Intc, 135] 13488 1.000
#> pip[Intc, 136] 13660 1.000
#> pip[Intc, 137] 13759 1.000
#> pip[Intc, 138] 15221 1.000
#> pip[Intc, 139] 15922 1.000
#> pip[Intc, 140] 17327 1.000
#> pip[Intc, 141] 18243 1.000
#> pip[Intc, 142] 16972 1.000
#> pip[Intc, 143] 13522 1.000
#> pip[Intc, 144] 19645 1.000
#> pip[Intc, 145] 7428 1.000
#> pip[Intc, 146] 19512 1.000
#> pip[Intc, 147] 18867 1.000
#> pip[Intc, 148] 12862 1.000
#> pip[Intc, 149] 14604 1.000
#> pip[Intc, 150] 11158 1.000
#> pip[Intc, 151] 20044 1.000
#> pip[Intc, 152] 17979 1.000
#> pip[Intc, 153] 9109 1.001
#> pip[Intc, 154] 9825 1.000
#> pip[Intc, 155] 14810 1.000
#> pip[Intc, 156] 13214 1.000
#> pip[Intc, 157] 17658 1.000
#> pip[Intc, 158] 16066 1.000
#> pip[Intc, 159] 14558 1.000
#> pip[Intc, 160] 18859 1.000
#> scl_Intc 777 1.003
#> scl_student_ses 3519 1.001
#> scl_school_ses 815 1.003
#> scl_student_ses:school_ses 3404 1.000
#>
#> WAIC: 27045.06
#> elppd: -13364.85
#> pWAIC: 157.6844
pip() returns the posterior inclusion probabilities as a plain data
frame, one row per school and scale random effect, ready for further
processing:
head(pip(out))
#> scale_var cluster_index cluster_id pip u_mean u_sd
#> 1 (Intercept) 1 1 0.46245 -0.018826194 0.04848226
#> 2 (Intercept) 2 2 0.48830 -0.008655035 0.03800933
#> 3 (Intercept) 3 3 0.45730 0.007382731 0.03178920
#> 4 (Intercept) 4 4 0.48055 0.011827067 0.03734019
#> 5 (Intercept) 5 5 0.52515 -0.019156138 0.04094284
#> 6 (Intercept) 6 6 0.42295 -0.012751469 0.03507899
Fixed and random effects are available through the usual accessors:
fixef(out)
#> (Intercept) student_ses
#> 0.12915529 0.08160147
#> school_ses student_ses:school_ses
#> 0.67579793 -0.02284366
#> scl_(Intercept) scl_student_ses
#> -0.23404500 0.03112850
#> scl_school_ses scl_student_ses:school_ses
#> 0.11903442 0.07434945
head(ranef(out))
#> (Intercept) scl_(Intercept)
#> 1 0.256085174 -0.018826194
#> 2 0.043759611 -0.008655035
#> 3 -0.007273998 0.007382731
#> 4 0.029942592 0.011827067
#> 5 0.014027889 -0.019156138
#> 6 0.219024550 -0.012751469
Grouping IDs do not need to form a consecutive integer index: character
or factor IDs and non-consecutive numeric codes (e.g. official school
codes) are recoded internally, and summary(), plot() and pip() can
report results in terms of the original IDs (labels = "original").
The PIPs depend on the prior inclusion probability (ss_prior_p, 0.5 by
default). pip_sensitivity() re-computes them analytically for a range
of priors – without refitting – by converting the posterior odds into
prior-independent Bayes factors:
sens <- pip_sensitivity(out)
plot(sens)
Schools whose classification at the 0.75 threshold flips within a plausible perturbation of the fitted prior (halving to doubling its odds, by default) are highlighted; robust schools are grey. Individual schools can be singled out:
pips <- pip(out)
top <- pips[order(pips$pip, decreasing = TRUE), ][1:3, ]
plot(sens, clusters = top$cluster_id)
plot(out, type = "pip")
plot(out, type = "funnel")
plot(out, type = "outcome")
codaplot(out, parameters = "Intc")
codaplot(out, parameters = "R[scl_Intc, Intc]")
This work was supported by the Tools Competition catalyst award for the project consistentlyBetter to PR. The content is solely the responsibility of the authors and does not necessarily represent the official views of the funding agency.
Carmo, M., Williams, D. R., & Rast, P. (2026). Beyond average scores: Identification of consistent and inconsistent academic achievement in grouping units. Journal of Educational and Behavioral Statistics. https://doi.org/10.3102/10769986261426004
Rodriguez, J. E., Williams, D. R., & Rast, P. (2024). Who is and is not" average’"? Random effects selection with spike-and-slab priors. Psychological Methods. https://doi.org/10.1037/met0000535
Williams, D. R., Martin, S. R., & Rast, P. (2022). Putting the individual into reliability: Bayesian testing of homogeneous within-person variance in hierarchical models. Behavior Research Methods, 54(3), 1272–1290. https://doi.org/10.3758/s13428-021-01646-x