Case Studies

Case Studies

open-source methods and models

The case studies on this page are intended to reflect best practices in Bayesian methodology and Stan programming.

Contributing Case Studies

To contribute a case study, please contact us through the users group. We require

  • a documented, reproducible example with narrative documentation (preferably coded using knitr or Jupyter) and

  • an open-source license (preferably BSD 3 clause); authors retain all rights, including copyright.

Stan Case Studies,   Volume 3   (2016)



Exact Sparse CAR Models in Stan

This document details sparse exact conditional autoregressive (CAR) models in Stan as an extension of previous work on approximate sparse CAR models in Stan. Sparse representations seem to give order of magnitude efficiency gains, scaling better for large spatial data sets.

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Author
Max Joseph
Keywords
conditional autorgressive (CAR), independent autoregressive (IAR), sparsity, spatial random effects, maps
Source Repository
mbjoseph/CARstan (GitHub)
R Package Dependencies
rstan, dplyr, ggmcmc, knitr, maptools, rgeos, spdep.
License
BSD (3 clause), CC-BY

A Primer on Bayesian Multilevel Modeling using PyStan

This case study replicates the analysis of home radon levels using hierarchical models of Lin, Gelman, Price, and Kurtz (1999). It illustrates how to generalize linear regressions to hierarchical models with group-level predictors and how to compare predictive inferences and evaluate model fits. Along the way it shows how to get data into Stan using pandas, how to sample using PyStan, and how to visualize the results using Seaborn.

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Author
Chris Fonnesbeck
Keywords
hierararchical/multilevel modeling, linear regression, model comparison, predictive inference, radon
Source Repository
fonnesbeck/stan_workshop_2016 (GitHub)
Python Package Dependencies
pystan, numpy, pandas, matplotlib, seaborn
License
Apache 2.0 (code), CC-BY 3 (text)

Reparameterization: MLE vs. Bayes

When changing variables, a Jacobian adjustment needs to be provided to account for the rate of change of the transform. Applying the adjustment preserves the probability distributions of quantities of interest, thus making Bayesian inference invariant to reparameterizations. In contrast, the maximum likelihood estimate (posterior mode) is changed when the distribution-preserving Jacobian adjustment is included for a parameter. In this note, we use Stan to code a repeated binary trial model parameterized by chance of success, along with its reparameterization in terms of log odds in order to demonstrate the effect of the Jacobian adjustment on the Bayesian posterior and maximum likelihood estimate. Along the way, we derive the logistic distribution by transforming a uniformly distributed variable.

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Author
Bob Carpenter
Keywords
MLE, Bayesian posterior, reparameterization, Jacobian, binomial
Source Repository
example-models/knitr/mle-params (GitHub)
R Package Dependencies
rstan
License
BSD (3 clause), CC-BY

Hierarchical Two-Parameter Logistic Item Response Model

This case study documents a Stan model for the two-parameter logistic model (2PL) with hierarchical priors. A brief simulation indicates that the Stan model successfully recovers the generating parameters. An example using a grade 12 science assessment is provided.

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Author
Daniel C. Furr
Keywords
education, item response theory, two-parameter logistic model, hierarchical priors
Source Repository
example-models/education/hierarchical_2pl (GitHub)
R Package Dependencies
rstan, ggplot2, mirt
License
BSD (3 clause), CC-BY

Generalized Rating Scale Model with Latent Regression

This case study documents a Stan model for the generalized rating scale model (GRSM) with latent regression. The latent regression portion of the model may be restricted to an intercept only, yielding a standard GRSM. A brief simulation indicates that the Stan model successfully recovers the generating parameters. An example using a survey of public perceptions of science and technology is provided.

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Authors
Daniel C. Furr
Keywords
education, item response theory, generalized rating scale model
Source Repository
example-models/education/grsm_latent_reg (GitHub)
R Package Dependencies
rstan, ggplot2, ltm
License
BSD (3 clause), CC-BY

Generalized Partial Credit Model with Latent Regression

This case study documents a Stan model for the generalized partial credit model (GPCM) with latent regression. The latent regression portion of the model may be restricted to an intercept only, yielding a standard GPCM. A brief simulation indicates that the Stan model successfully recovers the generating parameters. An example using the TIMSS 2011 mathematics assessment is provided

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Authors
Daniel C. Furr
Keywords
education, item response theory, generalized partial credit model
Source Repository
example-models/education/gpcm_latent_reg (GitHub)
R Package Dependencies
rstan, ggplot2, TAM
License
BSD (3 clause), CC-BY

Rating Scale Model with Latent Regression

This case study documents a Stan model for the rating scale model (RSM) with latent regression. The latent regression portion of the model may be restricted to an intercept only, yielding a standard RSM. A brief simulation indicates that the Stan model successfully recovers the generating parameters. An example using a survey of public perceptions of science and technology is provided.

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Authors
Daniel C. Furr
Keywords
education, item response theory, rating scale model
Source Repository
example-models/education/rsm_latent_reg (GitHub)
R Package Dependencies
rstan, ggplot2, ltm
License
BSD (3 clause), CC-BY

Partial Credit Model with Latent Regression

This case study documents a Stan model for the partial credit model (PCM) with latent regression. The latent regression portion of the model may be restricted to an intercept only, yielding a standard PCM. A brief simulation indicates that the Stan model successfully recovers the generating parameters. An example using the TIMSS 2011 mathematics assessment is provided.

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Authors
Daniel C. Furr
Keywords
education, item response theory, partial credit model
Source Repository
example-models/education/pcm_latent_reg (GitHub)
R Package Dependencies
rstan, ggplot2, TAM
License
BSD (3 clause), CC-BY

Rasch and Two-Parameter Logistic Item Response Models with Latent Regression

This case study documents Stan models for the Rasch and two-parameter logistic models with latent regression. The latent regression portion of the models may be restricted to an intercept only, yielding standard versions of the models. Simulations indicate that the two models successfully recover generating parameters. An example using a grade 12 science assessment is provided.

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Authors
Daniel C. Furr
Keywords
education, item response theory, rasch model, two-parameter logistic model
Source Repository
example-models/education/rasch_and_2pl.html (GitHub)
R Package Dependencies
edstan, rstan, ggplot2, TAM
License
BSD (3 clause), CC-BY

Two-Parameter Logistic Item Response Model

This tutorial introduces the R package edstan for estimating two-parameter logistic item response models using Stan without knowing the Stan language. Subsequently, the tutorial explains how the model can be expressed in the Stan language and fit using the rstan package. Specification of prior distributions and assessment of convergence are discussed. Using the Stan language directly has the advantage that it becomes quite easy to extend the model, and this is demonstrated by adding a latent regression and differential item functioning to the model. Posterior predictive model checking is also demonstrated.

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Author
Daniel C. Furr, Seung Yeon Lee, Joon-Ho Lee, and Sophia Rabe-Hesketh
Keywords
education, item response theory, two-parameter logistic model
Source Repository
example-models/education/tutorial_twopl (GitHub)
R Package Dependencies
rstan, reshape2, ggplot2, gridExtra, devtools, edstan
License
BSD (3 clause), CC-BY

Pooling with Hierarchical Models for Repeated Binary Trials

This note illustrates the effects on posterior inference of pooling data (aka sharing strength) across items for repeated binary trial data. It provides Stan models and R code to fit and check predictive models for three situations: (a) complete pooling, which assumes each item is the same, (b) no pooling, which assumes the items are unrelated, and (c) partial pooling, where the similarity among the items is estimated. We consider two hierarchical models to estimate the partial pooling, one with a beta prior on chance of success and another with a normal prior on the log odds of success. The note explains with working examples how to (i) fit models in RStan and plot the results in R using ggplot2, (ii) estimate event probabilities, (iii) evaluate posterior predictive densities to evaluate model predictions on held-out data, (iv) rank items by chance of success, (v) perform multiple comparisons in several settings, (vi) replicate new data for posterior p-values, and (vii) perform graphical posterior predictive checks.

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Author
Bob Carpenter
Keywords
binary trials, pooling, hierarchical models, baseball, epidemiology, prediction, posterior predictive checks
Source Repository
example-models/knitr/pool-binary-trials (GitHub)
R Package Dependencies
rstan, ggplot2, rmarkdown
License
BSD (3 clause), CC-BY

RStanARM version

There is also a version of this case study in which all models are fit using the RStanARM interface. Many of the visualizations are also created using RStanARM’s plotting functions.

View RStanARM version (HTML)

Author
Bob Carpenter, Jonah Gabry, Ben Goodrich

Stan Case Studies,   Volume 2   (2015)



Multiple Species-Site Occupancy Model

This case study replicates the analysis and output graphs of Dorazio et al. (2006) noisy-measurement occupancy model for multiple species abundance of butterflies. Going beyond the paper, the supercommunity assumptions are tested to show they are invariant to sizing, and posterior predictive checks are provided.

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Author
Bob Carpenter
Keywords
ecology, occupancy, species abundance, supercommunity, posterior predictive check
Source Repository
example-models/knitr/dorazio-royle-occupancy (GitHub)
License
BSD (3 clause), CC-BY
R Package Dependencies
rstan, ggplot2, rmarkdown

Stan Case Studies,   Volume 1   (2014)



Soil Carbon Modeling with RStan

This case study provides ordinary differential equation-based compartment models of soil carbon flux, with experimental data fitted with unknown initial compartment balance and noisy CO2 measurements. Results form Sierra and Müller’s (2014) soilR package are replicated.

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Author
Bob Carpenter
Keywords
biogeochemistry, compartment ODE, soil carbon respiration, incubation experiment
Source Repository
soil-metamodel/stan/soil-knit (GitHub)
License
BSD (3 clause), CC-BY
R Package Dependencies
rstan, ggplot2, rmarkdown