Hierarchical Shrinkage Stan Models for Biomarker Selection
The hsstan package provides linear and logistic regression models penalized with hierarchical shrinkage priors for selection of biomarkers. Models are fitted with Stan, which allows to perform full Bayesian inference (Carpenter et al. (2017)).
It implements the horseshoe and regularized horseshoe priors (Piironen and Vehtari (2017)), and the projection predictive selection approach to recover a sparse set of predictive biomarkers (Piironen, Paasiniemi and Vehtari (2020)).
The approach is particularly suited to selection from high-dimensional panels of biomarkers, such as those that can be measured by MSMS or similar technologies.
Example
library(hsstan)
data(diabetes)
## if possible, allow using as many cores as cross-validation folds
options(mc.cores=10)
## baseline model with only clinical covariates
hs.base <- hsstan(diabetes, Y ~ age + sex)
## model with additional predictors
hs.biom <- hsstan(diabetes, Y ~ age + sex, penalized=colnames(diabetes)[3:10])
print(hs.biom)
# mean sd 2.5% 97.5% n_eff Rhat
# (Intercept) 0.00 0.03 -0.07 0.07 4483 1
# age 0.00 0.04 -0.07 0.08 4706 1
# sex -0.15 0.04 -0.22 -0.08 5148 1
# bmi 0.33 0.04 0.25 0.41 4228 1
# map 0.20 0.04 0.12 0.28 3571 1
# tc -0.45 0.25 -0.94 0.04 3713 1
# ldl 0.28 0.20 -0.12 0.68 3674 1
# hdl 0.01 0.12 -0.23 0.25 3761 1
# tch 0.07 0.08 -0.06 0.25 4358 1
# ltg 0.43 0.11 0.22 0.64 3690 1
# glu 0.02 0.03 -0.03 0.10 3034 1
## behaviour of the sampler
sampler.stats(hs.base)
# accept.stat stepsize divergences treedepth gradients warmup sample
# chain:1 0.9497 0.5723 0 3 6320 0.09 0.08
# chain:2 0.9357 0.6480 0 3 5938 0.09 0.08
# chain:3 0.9455 0.6014 0 3 6112 0.09 0.08
# chain:4 0.9488 0.5932 0 3 6238 0.09 0.08
# all 0.9449 0.6037 0 3 24608 0.36 0.32
sampler.stats(hs.biom)
# accept.stat stepsize divergences treedepth gradients warmup sample
# chain:1 0.9821 0.0191 0 8 233656 5.04 4.28
# chain:2 0.9891 0.0158 1 8 255994 5.88 4.72
# chain:3 0.9908 0.0143 0 9 274328 5.77 5.14
# chain:4 0.9933 0.0121 0 9 344984 5.98 6.70
# all 0.9888 0.0153 1 9 1108962 22.67 20.84
## approximate leave-one-out cross-validation with Pareto smoothed
## importance sampling
loo(hs.base)
# Computed from 4000 by 442 log-likelihood matrix
# Estimate SE
# elpd_loo -622.4 11.4
# p_loo 3.4 0.2
# looic 1244.9 22.7
# ------
# Monte Carlo SE of elpd_loo is 0.0.
#
# All Pareto k estimates are good (k < 0.5).
loo(hs.biom)
# Computed from 4000 by 442 log-likelihood matrix
# Estimate SE
# elpd_loo -476.5 13.7
# p_loo 9.8 0.7
# looic 953.0 27.5
# ------
# Monte Carlo SE of elpd_loo is 0.1.
#
# All Pareto k estimates are good (k < 0.5).
## run 10-folds cross-validation
set.seed(1)
folds <- caret::createFolds(diabetes$Y, k=10, list=FALSE)
cv.base <- kfold(hs.base, folds=folds)
cv.biom <- kfold(hs.biom, folds=folds)
## cross-validated performance
round(posterior_performance(cv.base), 2)
# mean sd 2.5% 97.5%
# r2 0.02 0.00 0.01 0.03
# llk -623.14 1.67 -626.61 -620.13
# attr(,"type")
# [1] "cross-validated"
round(posterior_performance(cv.biom), 2)
# mean sd 2.5% 97.5%
# r2 0.48 0.01 0.47 0.50
# llk -482.86 3.76 -490.45 -476.56
# attr(,"type")
# [1] "cross-validated"
## projection predictive selection
sel.biom <- projsel(hs.biom)
print(sel.biom, digits=4)
# var kl rel.kl.null rel.kl elpd delta.elpd
# 1 Intercept only 0.352283 0.00000 NA -627.3 -155.84260
# 2 Initial submodel 0.333156 0.05429 0.0000 -619.8 -148.39729
# 3 bmi 0.138629 0.60648 0.5839 -533.1 -61.69199
# 4 ltg 0.058441 0.83411 0.8246 -492.5 -21.09681
# 5 map 0.035970 0.89789 0.8920 -482.7 -11.25515
# 6 hdl 0.010304 0.97075 0.9691 -473.9 -2.41192
# 7 tc 0.005292 0.98498 0.9841 -472.2 -0.72490
# 8 ldl 0.002444 0.99306 0.9927 -471.8 -0.38292
# 9 tch 0.001105 0.99686 0.9967 -471.5 -0.07819
# 10 glu 0.000000 1.00000 1.0000 -471.4 0.00000References
M. Colombo, A. Asadi Shehni, I. Thoma et al., Quantitative levels of serum N-glycans in type 1 diabetes and their association with kidney disease, Glycobiology (2021) 31 (5): 613-623.
M. Colombo, S.J. McGurnaghan, L.A.K. Blackbourn et al., Comparison of serum and urinary biomarker panels with albumin creatinin ratio in the prediction of renal function decline in type 1 diabetes, Diabetologia (2020) 63 (4): 788-798.
M. Colombo, E. Valo, S.J. McGurnaghan et al., Biomarkers associated with progression of renal disease in type 1 diabetes, Diabetologia (2019) 62 (9): 1616-1627.