# Individual Variance Detection
# Individual Variance Detection
ivd is an R package for random effects selection in the
scale part of Mixed Effects Location Scale Modlels (MELSM).
ivd() fits a random intercepts model with a spike-and-slab
prior on the random effects of the scale.
This package can be installed with
# install.packages("devtools")
# devtools::install_github("consistentlybetter/ivd")library(ivd)
library(data.table)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 = 1000, nburnin = 6000)
#> ===== Monitors =====
#> thin = 1: beta, sigma_rand, ss, Ustar, z, zeta
#> ===== Samplers =====
#> RW_lkj_corr_cholesky sampler (1)
#> - Ustar[1:2, 1:2]
#> RW sampler (326)
#> - z[] (320 elements)
#> - zeta[] (4 elements)
#> - sigma_rand[] (2 elements)
#> conjugate sampler (4)
#> - beta[] (4 elements)
#> binary sampler (320)
#> - ss[] (320 elements)
#> thin = 1: beta, mu, R, sigma_rand, ss, tau, u, zeta
#> |-------------|-------------|-------------|-------------|
#> |-------------------------------------------------------|
#> [Warning] There are 3 individual pWAIC values that are greater than 0.4. This may indicate that the WAIC estimate is unstable (Vehtari et al., 2017), at least in cases without grouping of data nodes or multivariate data nodes.
#> Defining model
#> Building model
#> Setting data and initial values
#> [Note] 'Z' is provided in 'data' but is not a variable in the model and is being ignored.
#> [Note] 'Z_scale' is provided in 'data' but is not a variable in the model and is being ignored.
#> Running calculate on model
#> [Note] Any error reports that follow may simply reflect missing values in model variables.
#> Checking model sizes and dimensions
#> [Note] This model is not fully initialized. This is not an error.
#> To see which variables are not initialized, use model$initializeInfo().
#> For more information on model initialization, see help(modelInitialization).
#> Compiling
#> [Note] This may take a minute.
#> [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details.
#> Compiling
#> [Note] This may take a minute.
#> [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details.
#> [Note] 'Z' is provided in 'data' but is not a variable in the model and is being ignored.
#> [Note] 'Z_scale' is provided in 'data' but is not a variable in the model and is being ignored.
#> running chain 1...
#> ===== Monitors =====
#> thin = 1: beta, sigma_rand, ss, Ustar, z, zeta
#> ===== Samplers =====
#> RW_lkj_corr_cholesky sampler (1)
#> - Ustar[1:2, 1:2]
#> RW sampler (326)
#> - z[] (320 elements)
#> - zeta[] (4 elements)
#> - sigma_rand[] (2 elements)
#> conjugate sampler (4)
#> - beta[] (4 elements)
#> binary sampler (320)
#> - ss[] (320 elements)
#> thin = 1: beta, mu, R, sigma_rand, ss, tau, u, zeta
#> |-------------|-------------|-------------|-------------|
#> |-------------------------------------------------------|
#> [Warning] There are 3 individual pWAIC values that are greater than 0.4. This may indicate that the WAIC estimate is unstable (Vehtari et al., 2017), at least in cases without grouping of data nodes or multivariate data nodes.
#> Defining model
#> Building model
#> Setting data and initial values
#> [Note] 'Z' is provided in 'data' but is not a variable in the model and is being ignored.
#> [Note] 'Z_scale' is provided in 'data' but is not a variable in the model and is being ignored.
#> Running calculate on model
#> [Note] Any error reports that follow may simply reflect missing values in model variables.
#> Checking model sizes and dimensions
#> [Note] This model is not fully initialized. This is not an error.
#> To see which variables are not initialized, use model$initializeInfo().
#> For more information on model initialization, see help(modelInitialization).
#> Compiling
#> [Note] This may take a minute.
#> [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details.
#> Compiling
#> [Note] This may take a minute.
#> [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details.
#> [Note] 'Z' is provided in 'data' but is not a variable in the model and is being ignored.
#> [Note] 'Z_scale' is provided in 'data' but is not a variable in the model and is being ignored.
#> running chain 1...
#> ===== Monitors =====
#> thin = 1: beta, sigma_rand, ss, Ustar, z, zeta
#> ===== Samplers =====
#> RW_lkj_corr_cholesky sampler (1)
#> - Ustar[1:2, 1:2]
#> RW sampler (326)
#> - z[] (320 elements)
#> - zeta[] (4 elements)
#> - sigma_rand[] (2 elements)
#> conjugate sampler (4)
#> - beta[] (4 elements)
#> binary sampler (320)
#> - ss[] (320 elements)
#> thin = 1: beta, mu, R, sigma_rand, ss, tau, u, zeta
#> |-------------|-------------|-------------|-------------|
#> |-------------------------------------------------------|
#> [Warning] There are 3 individual pWAIC values that are greater than 0.4. This may indicate that the WAIC estimate is unstable (Vehtari et al., 2017), at least in cases without grouping of data nodes or multivariate data nodes.
#> Defining model
#> Building model
#> Setting data and initial values
#> [Note] 'Z' is provided in 'data' but is not a variable in the model and is being ignored.
#> [Note] 'Z_scale' is provided in 'data' but is not a variable in the model and is being ignored.
#> Running calculate on model
#> [Note] Any error reports that follow may simply reflect missing values in model variables.
#> Checking model sizes and dimensions
#> [Note] This model is not fully initialized. This is not an error.
#> To see which variables are not initialized, use model$initializeInfo().
#> For more information on model initialization, see help(modelInitialization).
#> Compiling
#> [Note] This may take a minute.
#> [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details.
#> Compiling
#> [Note] This may take a minute.
#> [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details.
#> [Note] 'Z' is provided in 'data' but is not a variable in the model and is being ignored.
#> [Note] 'Z_scale' is provided in 'data' but is not a variable in the model and is being ignored.
#> running chain 1...
#> ===== Monitors =====
#> thin = 1: beta, sigma_rand, ss, Ustar, z, zeta
#> ===== Samplers =====
#> RW_lkj_corr_cholesky sampler (1)
#> - Ustar[1:2, 1:2]
#> RW sampler (326)
#> - z[] (320 elements)
#> - zeta[] (4 elements)
#> - sigma_rand[] (2 elements)
#> conjugate sampler (4)
#> - beta[] (4 elements)
#> binary sampler (320)
#> - ss[] (320 elements)
#> thin = 1: beta, mu, R, sigma_rand, ss, tau, u, zeta
#> |-------------|-------------|-------------|-------------|
#> |-------------------------------------------------------|
#> [Warning] There are 4 individual pWAIC values that are greater than 0.4. This may indicate that the WAIC estimate is unstable (Vehtari et al., 2017), at least in cases without grouping of data nodes or multivariate data nodes.
#> Defining model
#> Building model
#> Setting data and initial values
#> [Note] 'Z' is provided in 'data' but is not a variable in the model and is being ignored.
#> [Note] 'Z_scale' is provided in 'data' but is not a variable in the model and is being ignored.
#> Running calculate on model
#> [Note] Any error reports that follow may simply reflect missing values in model variables.
#> Checking model sizes and dimensions
#> [Note] This model is not fully initialized. This is not an error.
#> To see which variables are not initialized, use model$initializeInfo().
#> For more information on model initialization, see help(modelInitialization).
#> Compiling
#> [Note] This may take a minute.
#> [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details.
#> Compiling
#> [Note] This may take a minute.
#> [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details.
#> [Note] 'Z' is provided in 'data' but is not a variable in the model and is being ignored.
#> [Note] 'Z_scale' is provided in 'data' but is not a variable in the model and is being ignored.
#> running chain 1...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.
s_out <- summary(out)
#> Summary statistics for ivd model:
#> Chains (workers): 4
#>
#> Mean SD Time-series SE 2.5% 50% 97.5% n_eff R-hat
#> R[scl_Intc, Intc] -0.659 0.172 0.012 -0.946 -0.677 -0.264 135 1.017
#> Intc 0.129 0.024 0.003 0.082 0.130 0.176 49 1.038
#> student_ses 0.082 0.010 0.000 0.063 0.082 0.101 2466 1.001
#> school_ses 0.753 0.089 0.009 0.580 0.751 0.928 65 1.096
#> student_ses:school_ses -0.036 0.038 0.001 -0.112 -0.036 0.038 1193 1.002
#> sd_Intc 0.269 0.020 0.002 0.236 0.268 0.312 82 1.056
#> sd_scl_Intc 0.077 0.014 0.001 0.049 0.076 0.106 143 1.012
#> pip[Intc, 1] 0.468 0.499 0.008 0.000 0.000 1.000 2864 1.000
#> pip[Intc, 2] 0.482 0.500 0.008 0.000 0.000 1.000 4304 0.999
#> pip[Intc, 3] 0.459 0.498 0.009 0.000 0.000 1.000 3505 1.000
#> pip[Intc, 4] 0.495 0.500 0.009 0.000 0.000 1.000 2441 1.001
#> pip[Intc, 5] 0.519 0.500 0.009 0.000 1.000 1.000 2177 1.000
#> pip[Intc, 6] 0.443 0.497 0.008 0.000 0.000 1.000 3639 1.000
#> pip[Intc, 7] 0.402 0.490 0.010 0.000 0.000 1.000 1394 1.000
#> pip[Intc, 8] 0.483 0.500 0.008 0.000 0.000 1.000 3410 1.000
#> pip[Intc, 9] 0.980 0.140 0.005 1.000 1.000 1.000 612 1.006
#> pip[Intc, 10] 0.444 0.497 0.008 0.000 0.000 1.000 3977 0.999
#> pip[Intc, 11] 0.576 0.494 0.010 0.000 1.000 1.000 2601 1.000
#> pip[Intc, 12] 0.370 0.483 0.008 0.000 0.000 1.000 2616 1.000
#> pip[Intc, 13] 0.470 0.499 0.010 0.000 0.000 1.000 2436 1.000
#> pip[Intc, 14] 0.615 0.487 0.013 0.000 1.000 1.000 1298 1.003
#> pip[Intc, 15] 0.509 0.500 0.008 0.000 1.000 1.000 4422 1.001
#> pip[Intc, 16] 0.478 0.500 0.008 0.000 0.000 1.000 5244 1.000
#> pip[Intc, 17] 0.429 0.495 0.009 0.000 0.000 1.000 2220 1.000
#> pip[Intc, 18] 0.541 0.498 0.008 0.000 1.000 1.000 2661 1.000
#> pip[Intc, 19] 0.260 0.439 0.010 0.000 0.000 1.000 1760 1.001
#> pip[Intc, 20] 0.531 0.499 0.008 0.000 1.000 1.000 3615 1.001
#> pip[Intc, 21] 0.371 0.483 0.010 0.000 0.000 1.000 1313 1.000
#> pip[Intc, 22] 0.565 0.496 0.010 0.000 1.000 1.000 2713 1.002
#> pip[Intc, 23] 0.495 0.500 0.009 0.000 0.000 1.000 2919 1.000
#> pip[Intc, 24] 0.489 0.500 0.008 0.000 0.000 1.000 3719 0.999
#> pip[Intc, 25] 0.462 0.499 0.008 0.000 0.000 1.000 4512 1.000
#> pip[Intc, 26] 0.506 0.500 0.008 0.000 1.000 1.000 3151 1.000
#> pip[Intc, 27] 0.460 0.498 0.009 0.000 0.000 1.000 3611 1.000
#> pip[Intc, 28] 0.398 0.490 0.009 0.000 0.000 1.000 3267 1.001
#> pip[Intc, 29] 0.487 0.500 0.009 0.000 0.000 1.000 3601 1.000
#> pip[Intc, 30] 0.440 0.496 0.010 0.000 0.000 1.000 2689 1.000
#> pip[Intc, 31] 0.476 0.499 0.008 0.000 0.000 1.000 3012 1.000
#> pip[Intc, 32] 0.488 0.500 0.010 0.000 0.000 1.000 2311 1.000
#> pip[Intc, 33] 0.491 0.500 0.009 0.000 0.000 1.000 1722 1.000
#> pip[Intc, 34] 0.587 0.492 0.008 0.000 1.000 1.000 3476 0.999
#> pip[Intc, 35] 0.644 0.479 0.012 0.000 1.000 1.000 643 1.003
#> pip[Intc, 36] 0.364 0.481 0.012 0.000 0.000 1.000 1428 1.001
#> pip[Intc, 37] 0.476 0.499 0.008 0.000 0.000 1.000 3883 1.001
#> pip[Intc, 38] 0.432 0.495 0.008 0.000 0.000 1.000 3967 1.000
#> pip[Intc, 39] 0.739 0.439 0.017 0.000 1.000 1.000 231 1.008
#> pip[Intc, 40] 0.466 0.499 0.008 0.000 0.000 1.000 2991 1.000
#> pip[Intc, 41] 0.662 0.473 0.018 0.000 1.000 1.000 628 1.006
#> pip[Intc, 42] 0.467 0.499 0.009 0.000 0.000 1.000 3458 1.000
#> pip[Intc, 43] 0.410 0.492 0.009 0.000 0.000 1.000 3101 0.999
#> pip[Intc, 44] 0.440 0.496 0.008 0.000 0.000 1.000 3499 1.000
#> pip[Intc, 45] 0.426 0.494 0.008 0.000 0.000 1.000 3502 1.001
#> pip[Intc, 46] 0.998 0.042 0.001 1.000 1.000 1.000 2620 1.000
#> pip[Intc, 47] 0.479 0.500 0.008 0.000 0.000 1.000 3566 1.000
#> pip[Intc, 48] 0.623 0.485 0.009 0.000 1.000 1.000 3266 1.000
#> pip[Intc, 49] 0.524 0.499 0.013 0.000 1.000 1.000 1210 1.005
#> pip[Intc, 50] 0.519 0.500 0.008 0.000 1.000 1.000 3912 1.000
#> pip[Intc, 51] 0.461 0.499 0.008 0.000 0.000 1.000 3811 1.000
#> pip[Intc, 52] 0.543 0.498 0.008 0.000 1.000 1.000 3180 0.999
#> pip[Intc, 53] 0.829 0.376 0.010 0.000 1.000 1.000 1567 1.001
#> pip[Intc, 54] 0.597 0.491 0.009 0.000 1.000 1.000 2746 1.001
#> pip[Intc, 55] 0.408 0.492 0.008 0.000 0.000 1.000 2827 1.000
#> pip[Intc, 56] 0.477 0.500 0.009 0.000 0.000 1.000 3273 1.000
#> pip[Intc, 57] 0.611 0.488 0.008 0.000 1.000 1.000 3604 1.000
#> pip[Intc, 58] 0.416 0.493 0.009 0.000 0.000 1.000 1093 1.001
#> pip[Intc, 59] 0.432 0.495 0.008 0.000 0.000 1.000 4394 1.001
#> pip[Intc, 60] 0.512 0.500 0.009 0.000 1.000 1.000 3043 1.000
#> pip[Intc, 61] 0.301 0.459 0.010 0.000 0.000 1.000 1613 1.000
#> pip[Intc, 62] 0.428 0.495 0.009 0.000 0.000 1.000 3201 1.000
#> pip[Intc, 63] 0.501 0.500 0.008 0.000 1.000 1.000 3662 0.999
#> pip[Intc, 64] 0.673 0.469 0.009 0.000 1.000 1.000 2949 1.001
#> pip[Intc, 65] 0.510 0.500 0.008 0.000 1.000 1.000 4009 1.000
#> pip[Intc, 66] 0.628 0.483 0.010 0.000 1.000 1.000 1622 1.000
#> pip[Intc, 67] 0.369 0.483 0.009 0.000 0.000 1.000 2463 1.000
#> pip[Intc, 68] 0.368 0.482 0.009 0.000 0.000 1.000 2315 1.001
#> pip[Intc, 69] 0.447 0.497 0.008 0.000 0.000 1.000 3411 1.000
#> pip[Intc, 70] 0.388 0.487 0.011 0.000 0.000 1.000 1515 1.002
#> pip[Intc, 71] 0.377 0.485 0.009 0.000 0.000 1.000 3173 0.999
#> pip[Intc, 72] 0.403 0.491 0.009 0.000 0.000 1.000 2217 1.001
#> pip[Intc, 73] 0.389 0.488 0.014 0.000 0.000 1.000 1081 1.000
#> pip[Intc, 74] 0.477 0.500 0.008 0.000 0.000 1.000 3920 1.000
#> pip[Intc, 75] 0.469 0.499 0.008 0.000 0.000 1.000 4021 1.001
#> pip[Intc, 76] 0.464 0.499 0.008 0.000 0.000 1.000 3986 1.000
#> pip[Intc, 77] 0.456 0.498 0.011 0.000 0.000 1.000 1825 1.000
#> pip[Intc, 78] 0.448 0.497 0.008 0.000 0.000 1.000 3851 1.001
#> pip[Intc, 79] 0.402 0.490 0.008 0.000 0.000 1.000 3029 1.000
#> pip[Intc, 80] 0.499 0.500 0.008 0.000 0.000 1.000 5631 1.000
#> pip[Intc, 81] 0.522 0.500 0.009 0.000 1.000 1.000 3309 1.000
#> pip[Intc, 82] 0.528 0.499 0.010 0.000 1.000 1.000 2846 1.001
#> pip[Intc, 83] 0.482 0.500 0.008 0.000 0.000 1.000 4271 1.000
#> pip[Intc, 84] 0.497 0.500 0.008 0.000 0.000 1.000 3408 1.000
#> pip[Intc, 85] 0.466 0.499 0.008 0.000 0.000 1.000 3625 1.000
#> pip[Intc, 86] 0.487 0.500 0.010 0.000 0.000 1.000 2085 0.999
#> pip[Intc, 87] 0.713 0.452 0.012 0.000 1.000 1.000 1309 1.004
#> pip[Intc, 88] 0.476 0.499 0.009 0.000 0.000 1.000 2538 1.000
#> pip[Intc, 89] 0.535 0.499 0.008 0.000 1.000 1.000 4741 1.001
#> pip[Intc, 90] 0.500 0.500 0.008 0.000 0.000 1.000 3440 1.000
#> pip[Intc, 91] 0.430 0.495 0.008 0.000 0.000 1.000 2614 1.001
#> pip[Intc, 92] 0.703 0.457 0.011 0.000 1.000 1.000 1493 1.000
#> pip[Intc, 93] 0.439 0.496 0.008 0.000 0.000 1.000 3724 1.001
#> pip[Intc, 94] 0.485 0.500 0.008 0.000 0.000 1.000 3591 1.000
#> pip[Intc, 95] 0.729 0.445 0.009 0.000 1.000 1.000 1997 1.001
#> pip[Intc, 96] 0.453 0.498 0.008 0.000 0.000 1.000 3484 1.000
#> pip[Intc, 97] 0.360 0.480 0.009 0.000 0.000 1.000 3129 1.002
#> pip[Intc, 98] 0.467 0.499 0.010 0.000 0.000 1.000 2464 1.001
#> pip[Intc, 99] 0.550 0.498 0.008 0.000 1.000 1.000 3663 1.002
#> pip[Intc, 100] 0.448 0.497 0.008 0.000 0.000 1.000 3470 0.999
#> pip[Intc, 101] 0.442 0.497 0.009 0.000 0.000 1.000 2902 1.001
#> pip[Intc, 102] 0.515 0.500 0.012 0.000 1.000 1.000 1405 1.001
#> pip[Intc, 103] 0.368 0.482 0.009 0.000 0.000 1.000 2961 1.001
#> pip[Intc, 104] 0.450 0.498 0.008 0.000 0.000 1.000 4608 1.000
#> pip[Intc, 105] 0.488 0.500 0.008 0.000 0.000 1.000 4520 1.000
#> pip[Intc, 106] 0.420 0.494 0.010 0.000 0.000 1.000 2130 1.000
#> pip[Intc, 107] 0.544 0.498 0.008 0.000 1.000 1.000 3017 1.000
#> pip[Intc, 108] 0.548 0.498 0.015 0.000 1.000 1.000 1039 1.002
#> pip[Intc, 109] 0.552 0.497 0.008 0.000 1.000 1.000 4418 1.001
#> pip[Intc, 110] 0.388 0.487 0.009 0.000 0.000 1.000 2888 1.000
#> pip[Intc, 111] 0.452 0.498 0.010 0.000 0.000 1.000 2069 1.003
#> pip[Intc, 112] 0.464 0.499 0.011 0.000 0.000 1.000 1876 1.000
#> pip[Intc, 113] 0.629 0.483 0.009 0.000 1.000 1.000 2737 1.000
#> pip[Intc, 114] 0.896 0.306 0.007 0.000 1.000 1.000 1783 1.000
#> pip[Intc, 115] 0.832 0.374 0.010 0.000 1.000 1.000 1129 1.002
#> pip[Intc, 116] 0.472 0.499 0.010 0.000 0.000 1.000 2389 1.000
#> pip[Intc, 117] 0.437 0.496 0.008 0.000 0.000 1.000 4329 1.000
#> pip[Intc, 118] 0.391 0.488 0.009 0.000 0.000 1.000 2556 1.001
#> pip[Intc, 119] 0.495 0.500 0.009 0.000 0.000 1.000 2635 1.000
#> pip[Intc, 120] 0.553 0.497 0.008 0.000 1.000 1.000 3419 1.000
#> pip[Intc, 121] 0.336 0.472 0.009 0.000 0.000 1.000 2606 1.000
#> pip[Intc, 122] 0.561 0.496 0.015 0.000 1.000 1.000 962 1.004
#> pip[Intc, 123] 0.558 0.497 0.008 0.000 1.000 1.000 4447 1.000
#> pip[Intc, 124] 0.747 0.435 0.010 0.000 1.000 1.000 1999 1.004
#> pip[Intc, 125] 0.461 0.499 0.008 0.000 0.000 1.000 3404 1.001
#> pip[Intc, 126] 0.507 0.500 0.009 0.000 1.000 1.000 3278 1.001
#> pip[Intc, 127] 0.566 0.496 0.009 0.000 1.000 1.000 2377 1.001
#> pip[Intc, 128] 0.507 0.500 0.009 0.000 1.000 1.000 2997 0.999
#> pip[Intc, 129] 0.421 0.494 0.008 0.000 0.000 1.000 3656 1.001
#> pip[Intc, 130] 0.468 0.499 0.008 0.000 0.000 1.000 2749 1.001
#> pip[Intc, 131] 0.553 0.497 0.008 0.000 1.000 1.000 3658 1.000
#> pip[Intc, 132] 0.403 0.491 0.008 0.000 0.000 1.000 4398 1.000
#> pip[Intc, 133] 0.360 0.480 0.009 0.000 0.000 1.000 2017 1.000
#> pip[Intc, 134] 0.525 0.499 0.009 0.000 1.000 1.000 2387 1.000
#> pip[Intc, 135] 0.426 0.495 0.008 0.000 0.000 1.000 2417 1.001
#> pip[Intc, 136] 0.393 0.488 0.008 0.000 0.000 1.000 3930 0.999
#> pip[Intc, 137] 0.424 0.494 0.009 0.000 0.000 1.000 3196 1.001
#> pip[Intc, 138] 0.467 0.499 0.008 0.000 0.000 1.000 3386 1.000
#> pip[Intc, 139] 0.422 0.494 0.008 0.000 0.000 1.000 3666 1.001
#> pip[Intc, 140] 0.559 0.497 0.008 0.000 1.000 1.000 3592 1.001
#> pip[Intc, 141] 0.533 0.499 0.008 0.000 1.000 1.000 4050 1.000
#> pip[Intc, 142] 0.470 0.499 0.008 0.000 0.000 1.000 3139 1.000
#> pip[Intc, 143] 0.422 0.494 0.008 0.000 0.000 1.000 3135 1.000
#> pip[Intc, 144] 0.462 0.499 0.008 0.000 0.000 1.000 4450 1.000
#> pip[Intc, 145] 0.493 0.500 0.011 0.000 0.000 1.000 1338 1.001
#> pip[Intc, 146] 0.466 0.499 0.008 0.000 0.000 1.000 4173 1.000
#> pip[Intc, 147] 0.492 0.500 0.008 0.000 0.000 1.000 4603 1.000
#> pip[Intc, 148] 0.629 0.483 0.008 0.000 1.000 1.000 2250 1.000
#> pip[Intc, 149] 0.723 0.448 0.008 0.000 1.000 1.000 3194 1.002
#> pip[Intc, 150] 0.387 0.487 0.011 0.000 0.000 1.000 1886 1.001
#> pip[Intc, 151] 0.464 0.499 0.008 0.000 0.000 1.000 3980 1.000
#> pip[Intc, 152] 0.472 0.499 0.008 0.000 0.000 1.000 4126 1.000
#> pip[Intc, 153] 0.756 0.429 0.009 0.000 1.000 1.000 2225 1.000
#> pip[Intc, 154] 0.371 0.483 0.010 0.000 0.000 1.000 1394 1.001
#> pip[Intc, 155] 0.423 0.494 0.009 0.000 0.000 1.000 2756 0.999
#> pip[Intc, 156] 0.529 0.499 0.008 0.000 1.000 1.000 3433 1.002
#> pip[Intc, 157] 0.544 0.498 0.008 0.000 1.000 1.000 3874 1.000
#> pip[Intc, 158] 0.494 0.500 0.008 0.000 0.000 1.000 4450 0.999
#> pip[Intc, 159] 0.480 0.500 0.008 0.000 0.000 1.000 3816 1.000
#> pip[Intc, 160] 0.526 0.499 0.008 0.000 1.000 1.000 3942 1.000
#> scl_Intc -0.231 0.008 0.000 -0.248 -0.231 -0.215 202 1.007
#> scl_student_ses 0.032 0.009 0.000 0.015 0.032 0.049 452 1.005
#> scl_school_ses 0.144 0.034 0.001 0.076 0.144 0.208 301 1.008
#> scl_student_ses:school_ses 0.065 0.032 0.001 0.003 0.066 0.129 453 1.003
#>
#> WAIC: 27043.51
#> elppd: -13365
#> pWAIC: 156.7514plot(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.
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