Case Studies
opensource methods and models
The case studies on this page are intended to reflect best practices in Bayesian methodology and Stan programming. We aim to keep them current with the latest version of the Stan language, but there may be times when case studies need updating to reflect the latest Stan features and syntax.
Contributing Case Studies
To contribute a case study, please contact us through the Stan Forums. We require

a documented, reproducible example with narrative documentation (e.g., knitr or Jupyter with software/compiler versions noted and seeds fixed) and

an opensource code license (preferably BSD or GPL for code, Creative Commons for text); authors retain all copyright.
Stan Case Studies, Volume 10 (2023)
Instrumental Variables Analysis of Randomized Experiments with OneSided Noncompliance
In this document, we demonstrate how to implement Bayesian inference for causal effects in randomized experiments with onesided noncompliance using Stan. Specifically, we aim to replicate the analysis presented in Imbens and Rubin (1997). We present Stan models with and without the exclusion restriction assumption, showcasing a significant advantage of the Bayesian modelbased approach.
View (HTML)
 Authors
 JoonHo Lee, Avi Feller, Sophia RabeHesketh
 Keywords
 causal inference, instrumental variables analysis, onesided compliance, principal stratification
 Source Repository
 examplemodels/education/causal_iv_onesided (GitHub)
 Dependencies
 tidyverse, rstan, bayesplot, patchwork
 License
 BSD (3 clause), CCBY</small>
Bayesian Structural Equation Modeling using blavaan
In this case study, we fit Bayesian structural equation models (SEM) using Hamiltonian Monte Carlo sampling in Stanpowered R package blavaan and illustrate how to use confirmtory factor analysis and latent growth curve modeling as SEM’s special cases. We also compared the estimates from blavaan with its frequentist counterpart using lavaan.
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 Authors
 Feng Ji, Xingyao Xiao, Aybolek Amanmyradova, Sophia RabeHesketh
 Keywords
 Structural Equation Modeling (SEM), Lavant Variable Modeling, Latent Growth Curve Models, Confirmatory Factor Analysis (CFA), Growth Curve Modeling, Bayesian Model Evaluation
 Source Repository
 examplemodels/education/sem (GitHub)
 Dependencies
 blavaan, lavaan, rstan, MASS, mvtnorm, tidyverse, semPlot, magrittr, lavaan.survey
 License
 BSD (3 clause), CCBY
Stan Case Studies, Volume 9 (2022)
Multilevel regression modeling with CmdStanPy and plotnine
This notebook is a short introduction to multilevel regression modeling using the CmdStanPy interface and plotnine, a Python implementation of a grammar of graphics based on ggplot2.
View (HTML)
 Authors
 Mitzi Morris
 Keywords
 Python, CmdStanPy, plotnine, hierarchical/multilevel modeling, linear regression, posterior predictive checks, radon
 Source Repository
 examplemodels/jupyter/radon (GitHub)
 Dependencies
 cmdstanpy, numpy, pandas, matplotlib, plotnine, jupyter
 License
 BSD (3 clause), CCBY
HoloML in Stan: Lowphoton Image Reconstruction
In this case study, we perform image reconstruction in Stan by implementing the HoloML phase retrieval model and then solving the inverse problem with optimization. This case study requires Stan 2.30 or greater in order to use the Fourier transform functions added in that version.
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 Authors
 Brian Ward, Bob Carpenter, and David Barmherzig
 Keywords
 image reconstruction, phase retrieval, Fourier transforms, deconvolution
 Source Repository
 WardBrian/holomlinstan (GitHub)
 Dependencies
 cmdstanpy, numpy, matplotlib, scipy, jupyter
 License
 BSD (3 clause), CCBY
Stan Case Studies, Volume 8 (2021)
Bayesian Latent Class Models and Handling of Label Switching
In this case study, we fit the Bayesian latent class model using Hamiltonian Monte Carlo sampling and Variational Bayes in Stan and illustrate the issue of label switching and its treatment with simulated and empirical data.
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 Authors
 Feng Ji, Aybolek Amanmyradova, Sophia RabeHesketh
 Keywords
 latent class models, labelswitching, posthoc relabeling, variational Bayes
 Source Repository
 examplemodels/education/latent_class (GitHub)
 Dependencies
 label.switching, rstan, magrittr, knitr, poLCA
 License
 BSD (3 clause), CCBY
Stan Case Studies, Volume 7 (2020)
Bayesian model of planetary motion: exploring ideas for a modeling workflow when dealing with ordinary differential equations and multimodality
The Bayesian model of planetary motion is a simple but powerful example that illustrates important concepts, as well as gaps, in prescribed modeling workflows. Our focus is on Bayesian inference using Markov chains Monte Carlo for a model based on an ordinary differential equations (ODE). Our example presents unexpected multimodality, causing our inference to be unreliable and what is more, dramatically slowing down our ODE integrators. What do we do when our chains do not mix and do not forget their starting points? Reasoning about the computational statistics at hand and the physics of the modeled phenomenon, we diagnose how the modes arise and how to improve our inference. Our process for fitting the model is iterative, starting with a simplification and building the model back up, and makes extensive use of visualization.
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 Authors
 Charles Margossian and Andrew Gelman
 Keywords
 ordinary differential equations, multimodality, classical mechanics
 Source Repository
 examplemodels/knitr/planetary_motion (GitHub)
 Dependencies
 CmdStanR, posterior, ggplot2, dplyr, plyr, tidyr, boot, latex2exp
 License
 BSD (3 clause), CCBY
HMM Interface Example
Cmdstan 2.24 introduces a new interface for working with Hidden Markov Models (HMMs). This is an example of how to use that interface.
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 Authors
 Ben Bales
 Keywords
 Hidden Markov Models, HMMs, cmdstanr, Stan programming
 Source Repository
 examplemodels/knitr/hmmexample (GitHub)
 Dependencies
 CmdStanR, tidyverse, ggplot2, and posterior
 License
 BSD (3 clause), CCBY
Spatial models for plant neighborhood dynamics in Stan
In this case study, we demonstrate how Stan’s segment function can speed computation on sparse matrices of pairwise neighbors in plantplant interaction models. In addition, we present solutions to common problems of fitting neighborhood models with hierarchical effects, including a comparison of centered vs. noncentered parameterizations.
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 Authors
 Cristina Barber, Andrii Zaiats, Cara Applestein and T.Trevor Caughlin
 Keywords
 plants, neighbor interactions, sparse matrix, segment function
 Source Repository
 Cristinabarber/Neighbor_Interactions (GitHub)
 R Package Dependencies
 rstan
 License
 BSD (3 clause), CC BY NC
Predicting Engine Failure with Hierarchical Gaussian Process
This gaussian process case study is an extension of the StanCon talk, Failure prediction in hierarchical equipment system: spline fitting naval ship failure. Many comparison criteria exist, but in terms of prediction accuracy, the gaussian process model outperformed the spline model. However, this accuracy comes at a cost of a more detailed and iterative checking process. This case study shows how identification and underfitting problems diagnosed from pushforward and predictive checks are addressed through reparameterization and adding variables. Basically, our data is highly unbalanced per category with lots of missing data. Also, due to the hierarchical structure of the system, such as shared engine types, the hierarchical model is applicable. For a detailed explanation of the data and spline model, please refer to this notebook.
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 Author
 Hyunji Moon, Jungin Choi
 Keywords
 Hierarchical Gaussian process, Bayesian workflow
 Source Repository
 examplemodels/knitr/gaussianprocess (GitHub)
 Dependencies
 CmdStanR, Rstan
 License
 BSD (3 clause), CCBY
Upgrading to the new ODE interface
Cmdstan 2.24 introduces a new ODE interface intended to make it easier to specify the ODE system function. This document should serve as an overview of the interface changes as well as a tutorial for converting code written with the old ODE interface.
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 Authors
 Ben Bales, Sebastian Weber
 Keywords
 ordinary differential equations, cmdstanr, Stan programming
 Source Repository
 examplemodels/knitr/convertodes (GitHub)
 Dependencies
 CmdStanR
 License
 BSD (3 clause), CCBY
Bayesian Workflow for disease transmission modeling in Stan
This tutorial shows how to build, fit, and criticize disease transmission models in Stan, and should be useful to researchers interested in modeling the COVID19 outbreak and doing Bayesian inference. Bayesian modeling provides a principled way to quantify uncertainty and incorporate prior knowledge into the model. What is more, Stan’s main inference engine, Hamiltonian Monte Carlo sampling, is amiable to diagnostics, which means we can verify whether our inference is reliable. Stan is an expressive probabilistic programing language that abstracts the inference and allows users to focus on the modeling. The resulting code is readable and easily extensible, which makes the modeler’s work more transparent and flexible. In this tutorial, we demonstrate with a simple SusceptibleInfectedRecovered (SIR) model how to formulate, fit, and diagnose a compartmental model in Stan. We also introduce more advanced topics which can help practitioners fit sophisticated models; notably, how to use simulations to probe our model and our priors, and computational techniques to scale ODEbased models.
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 Authors
 Leo Grinsztajn, Elizaveta Semenova, Charles C. Margossian, and Julien Riou
 Keywords
 Disease transmission, Compartment models, Ordinary Differential Equations, Bayesian Workflow
 Source Repository
 charlesm93/disease_transmission_workflow (GitHub)
 Dependencies
 RStan
 License
 BSD (3 clause), CCBY
Reduce Sum Example: parallelization of a single chain across multiple cores
Stan 2.23 introduced reduce_sum
, a new way to parallelize the execution
of a single Stan chain across multiple cores. This introduction copies directly
from Richard McElreath’s
Multithreading and MapReduce in Stan 2.18.0: A Minimal Example
View (HTML)
 Author
 Ben Bales
 Keywords
 withinchain parallel computation, cmdstanr, Stan programming
 Source Repository
 examplemodels/knitr/reducesum (GitHub)
 Dependencies
 CmdStanR
 License
 BSD (3 clause), CCBY
Stan Notebooks in the Cloud
This report shows you how to author a Jupyter Notebook for your Stan model and data so that anyone with a modern web browser and a Google account can run your analysis with Google Colaboratory free cloud servers. It shows you how to quickly set up a Stan installation in the cloud and introduces two lightweight interfaces: CmdStanR and CmdStanPy.
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 Author
 Mitzi Morris
 Keywords
 Jupyter, Google Colab, teaching Stan, online classroom, cloud computing
 Source Repository
 examplemodels/knitr/cloudcompute2020 (GitHub)
 Dependencies
 internet connection, Google account
 License
 BSD (3 clause), CCBY
Stan Case Studies, Volume 6 (2019)
Modelbased Inference for Causal Effects in Completely Randomized Experiments
In this document, we discuss the implementation of Bayesian modelbased inference for causal effects in Stan. We start by providing an introduction to the Bayesian inferential framework by analyzing a simulated dataset generated under unconfounded treatment assignment. Then we analyze an example dataset obtained from a completely randomized experiment focusing on the specification of the joint distribution of the potential outcomes.
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 Author
 JoonHo Lee, Avi Feller and Sophia RabeHesketh
 Keywords
 causal inference, completely randomized experiments
 Source Repository
 examplemodels/education/causal_rct
 R Package Dependencies
 rstan, rstanarm, bayesplot, tidyverse, gridExtra, Matching
 License
 BSD (3 clause), CCBY
Tagging Basketball Events with HMM in Stan
This case study shows how we can apply Bayesian inference to Hidden Markov Models (HMMs) using Stan to extract useful information from basketball player tracking data. Specifically we show how to tag drive events and how to determine defensive assignment. Before diving into basketball data we show how to fit an HMM in Stan using a simple example. This should help build some intuition for those who are unfamiliar with HMMs and will also show how to specify an HMM using Stan.
View (HTML)
 Author
 Imad Ali
 Keywords
 hidden markov models, sports
 Source Repository
 imadmali/bballhmm (GitHub)
 R Package Dependencies
 rstan, bayesplot, dplyr
 License
 BSD (3 clause), CCBYNC
Model building and expansion for golf putting
In this case study, we use Stan to build a series of models to estimate the probability of a successful putt using data from professional golfers. We fit and check the fit of a series of models, demonstrating the benefits of modeling based on substantive (rather than purely statistical) principles. We successfully fit to a small dataset and then have to expand the model to fit a new, larger dataset. We use weakly informative priors and a modelmisfit error term to enable the fit.
View (HTML)
 Author
 Andrew Gelman
 Keywords
 nonlinear regression, sports
 Source Repository
 examplemodels/knitr/golf (GitHub)
 R Package Dependencies
 rstan
 License
 BSD (3 clause), CCBY
A Dyadic Item Response Theory Model: Stan Case Study
In this case study, we use Stan to fit the Dyadic Item Response Theory (dIRT) model proposed by (Gin et al. 2019) to measure interactions between pairs of individuals when the responses to items represent the actions/behaviors/perceptions of an individual (called the ‘actor’) made within the context of a dyad formed with another individual (called the ‘partner’). The dIRT model is fit using Stan (version 2.18.1) in R via the rstan package.
View (HTML)
 Author
 Nicholas Sim, Brian Gin, Anders Skrondal and Sophia RabeHesketh
 Keywords
 item response theory, social relations model, dyadic data
 Source Repository
 examplemodels/education/dyadic_irt_model (GitHub)
 R Package Dependencies
 rstan, tidyverse
 License
 BSD (3 clause), CCBY
Stan Case Studies, Volume 5 (2018)
Multilevel Linear Models using Rstanarm
In this tutorial, we illustrate how to fit a multilevel linear model within a full Bayesian framework using rstanarm. This tutorial is aimed primarily at educational researchers who have used lme4 in R to fit models to their data and who may be interested in learning how to fit Bayesian multilevel models. However, for readers who have not used lme4 before, we briefly review the use of the package for fitting multilevel models.
View (HTML)
 Author
 JoonHo Lee, Nicholas Sim, Feng Ji, and Sophia RabeHesketh
 Keywords
 education, rstanarm, multilevel models, linear mixed models, hierarchical linear models
 Source Repository
 examplemodels/education/tutorial_rstanarm (GitHub)
 R Package Dependencies
 rstanarm, mlmRev, ggplot2, lme4
 License
 BSD (3 clause), CCBY
PredatorPrey Population Dynamics: the LotkaVolterra model in Stan
Lotka (1925) and Volterra (1926) formulated parametric differential equations that characterize the oscillating populations of predators and prey. A statistical model to account for measurement error and unexplained variation uses the deterministic solutions to the LotkaVolterra equations as expected population sizes. Stan is used to encode the statistical model and perform full Bayesian inference to solve the inverse problem of inferring parameters from noisy data. The model is fit to Canadian lynx and snowshoe hare populations between 1900 and 1920, based on the number of pelts collected annually by the Hudson’s Bay Company. Posterior predictive checks for replicated data show the model fits this data well. Full Bayesian inference may be used to estimate future (or past) populations.
View (HTML)
 Author
 Bob Carpenter
 Keywords
 population dynamics, LotkaVolterra equations, differential equations, posterior predictive checks
 Source Repository
 standev/examplemodels/knitr/lotkavolterra (GitHub)
 R Package Dependencies
 rstan, ggplot2, gridExtra, knitr, reshape tufte
 License
 Code: BSD (3 clause), Text: CC BYNC 4.0
Nearest neighbor Gaussian process (NNGP) models in Stan
Nearest neighbor Gaussian process (NNGP) based models is a family of highly scalable Gaussian processes based models. In brief, NNGP extends the Vecchia’s approximation (Vecchia 1988) to a process using conditional independence given information from neighboring locations. This case study shows how to express and fit these models in Stan.
View (HTML)
 Author
 Lu Zhang
 Keywords
 Gaussian process, nearest neighbor Gaussian process, spatial models, latent process, regression
 Source Repository
 LuZhangstat/NNGP_STAN (GitHub)
 R Package Dependencies
 rstan
 License
 Code: BSD (3 clause), Text: CC BYNC 4.0
Stan Case Studies, Volume 4 (2017)
Extreme value analysis and user defined probability functions in Stan
This notebook demonstrates how to implement user defined probability functions in Stan language. As an example I use the generalized Pareto distribution (GPD) to model geomagnetic storm data from the World Data Center for Geomagnetism.
View (HTML)
 Author
 Aki Vehtari
 Keywords
 extreme value analysis, generalized Pareto distribution, user defined probability functions
 Source Repository
 avehtari/BDA_R_demos/demos_rstan/gpareto_functions (GitHub)
 R Package Dependencies
 rstan, bayesplot, loo, ggplot2, tidyr, dplyr, extraDistr, gridExtra
 License
 Code: BSD (3 clause), Text: CC BYNC 4.0
Modelling Loss Curves in Insurance with RStan
Loss curves are a standard actuarial technique for helping insurance companies assess the amount of reserve capital they need to keep on hand to cover claims from a line of business. Claims made and reported for a given accounting period are tracked separately over time. This enables the use of historical patterns of claim development to predict expected total claims for newer policies.
We model the growth of the losses in each accounting period as an increasing function of time, and use the model to estimate the parameters which determine the shape and form of this growth. We also use the sampler to estimate the values of the “ultimate loss ratio”, i.e. the ratio of the total claims on an accounting period to the total premium received to write those policies. We treat each accounting period as a cohort.
View (HTML)
 Author
 Mick Cooney
 Keywords
 actuarial science, loss curves, insurance, ultimate loss ratio, hierarchical model
 Source Repository
 kaybenleroll/stancasestudy_losscurves (GitHub)
 R Package Dependencies
 rstan, bayesplot, tidyverse, scales, cowplot
 License
 Code: BSD (3 clause), Text: CC BYNC 4.0
Splines in Stan
In this document, we discuss the implementation of splines in Stan. We start by providing a brief introduction to splines and then explain how they can be implemented in Stan. We also discuss a novel prior that alleviates some of the practical challenges of spline models.
View (HTML)
 Author
 Milad Kharratzadeh
 Keywords
 Bsplines, piecewise regression, knots, priors
 Source Repository
 milkha/Splines_in_Stan (GitHub)
 R Package Dependencies
 rstan, splines
 License
 Code: BSD (3 clause), Text: CC BYNC 4.0
Spatial Models in Stan: Intrinsic AutoRegressive Models for Areal Data
This case study shows how to efficiently encode and compute an Intrinsic Conditional AutoRegressive (ICAR) model in Stan. When data has a neighborhood structure, ICAR models provide spatial smoothing by averaging measurements of directly adjoining regions. The Besag, York, and Mollié (BYM) model is a Poisson GLM which includes both an ICAR component and an ordinary randomeffects component for nonspatial heterogeneity. We compare two variants of the BYM model and fit two datasets taken from epidemiological studies over 56 and 700 regions, respectively.
View (HTML)
 Author
 Mitzi Morris
 Keywords
 spatial modeling, CAR, ICAR, INLA, OpenBUGS, hierarchical models
 Source Repository
 standev/examplemodels (GitHub)
 R Package Dependencies
 cmdstanr, ggplot2, broom, reshape2, dplyr, maptools, spdep, RINLA, R2OpenBugs
 License
 Code: BSD (3 clause), Text: CC BYNC 4.0
The QR Decomposition for Regression Models
This case study reviews the QR decomposition, a technique for decorrelating covariates and, consequently, the resulting posterior distribution in regression models.
View (HTML)
 Author
 Michael Betancourt
 Keywords
 Markov chain Monte Carlo, regression, RStan
 Source Repository
 betanalpha/knitr_case_studies/qr_regression (GitHub)
 R Package Dependencies
 rstan, knitr.
 License
 Code: BSD (3 clause), Text: CC BYNC 4.0
Robust RStan Workflow
This case study demonstrates the recommended RStan workflow for ensuring robust inferences with the default dynamic Hamiltonian Monte Carlo algorithm.
View (HTML)
 Author
 Michael Betancourt
 Keywords
 Markov chain Monte Carlo, Hamiltonian Monte Carlo, divergences, RStan
 Source Repository
 betanalpha/knitr_case_studies/rstan_workflow (GitHub)
 R Package Dependencies
 rstan, knitr.
 License
 Code: BSD (3 clause), Text: CC BYNC 4.0
Robust PyStan Workflow
This case study demonstrates the recommended PyStan workflow for ensuring robust inferences with the default dynamic Hamiltonian Monte Carlo algorithm.
View (HTML)
 Author
 Michael Betancourt
 Keywords
 Markov chain Monte Carlo, Hamiltonian Monte Carlo, divergences, PyStan
 Source Repository
 betanalpha/jupyter_case_studies/pystan_workflow (GitHub)
 Python Package Dependencies
 rstan, pystan, pickle, numpy, md5.
 License
 Code: BSD (3 clause), Text: CC BYNC 4.0
Typical Sets and the Curse of Dimensionality
This case study illustrates the socalled “curse of dimensionality” using simple examples based on simulation to show that all points are far away in high dimensions and that the mode is an atypical draw from a multivariate normal. The informationtheoretic concept of typical set is illustrated with both discrete and continuous cases, which show that probability mass is a product of volume and density (or count and mass in the discrete case). It also illustrates Monte Carlo methods and relates distance to the log density of the normal distribution and the chisquared distribution.
View R version (HTML)
 Authors
 Bob Carpenter
 Keywords
 probability mass, typical sets, concentration of measure, Monte Carlo methods
 Source Repository (R)
 standev/examplemodels/knitr/cursedims (GitHub)
 R Package Dependencies
 ggplot2
 License
 Code: BSD (3 clause), Text: CC BYNC 4.0
View Python version (HTML)
 Author (Python translation)
 Aravind S (Python translation)
 Source Repository (Python)
 Aravindsds/StanCode/python notebooks/curse_dims (GitHub)
 Python Package Dependencies
 numpy, scipy, pandas, matplotlib, collections, sys
 License
 Code: BSD (3 clause), Text: CC BYNC 4.0
Diagnosing Biased Inference with Divergences
This case study discusses the subtleties of accurate Markov chain Monte Carlo estimation and how divergences can be used to identify biased estimation in practice.
View (HTML)
 Author
 Michael Betancourt
 Keywords
 Markov chain Monte Carlo, Hamiltonian Monte Carlo, divergences, RStan
 Source Repository
 betanalpha/knitr_case_studies/divergences_and_bias (GitHub)
 R Package Dependencies
 rstan, knitr.
 License
 Code: BSD (3 clause), Text: CC BYNC 4.0
Identifying Bayesian Mixture Models
This case study discusses the common pathologies of Bayesian mixture models as well as some strategies for identifying and overcoming them.
View (HTML)
 Author
 Michael Betancourt
 Keywords
 Markov chain Monte Carlo, Hamiltonian Monte Carlo, mixture models, multimodal models, RStan
 Source Repository
 betanalpha/knitr_case_studies/identifying_mixture_models (GitHub)
 R Package Dependencies
 rstan, knitr.
 License
 Code: BSD (3 clause), Text: CC BYNC 4.0
How the Shape of a Weakly Informative Prior Affects Inferences
This case study reviews the basics of weaklyinformative priors and how the choice of a specific shape of such a prior affects the resulting posterior distribution.
View (HTML)
 Author
 Michael Betancourt
 Keywords
 Markov chain Monte Carlo, Hamiltonian Monte Carlo, priors, weaklyinformative priors, RStan
 Source Repository
 betanalpha/knitr_case_studies/weakly_informative_shapes (GitHub)
 R Package Dependencies
 rstan, knitr.
 License
 Code: BSD (3 clause), Text: CC BYNC 4.0
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.
View (HTML)
 Author
 Max Joseph
 Keywords
 conditional autoregressive (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), CCBY
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 grouplevel 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.
View (HTML)
 Author
 Chris Fonnesbeck
 Keywords
 hierarchical/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), CCBY 3 (text)
The Impact of Reparameterization on Point Estimates
When changing variables, a Jacobian adjustment needs to be provided to account for the rate of change of the transform. Applying the adjustment ensures that inferences that are based on expectations over the posterior are invariant under reparameterizations. In contrast, the posterior mode changes as a result of the reparameterization. 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 the posterior mode. We contrast the posterior mode to the maximum likelihood estimate, which, like the Bayesian estimates, is invariant under reparameterization. Along the way, we derive the logistic distribution by transforming a uniformly distributed variable.
View (HTML)
 Author
 Bob Carpenter
 Keywords
 MLE, Bayesian posterior, reparameterization, Jacobian, binomial
 Source Repository
 examplemodels/knitr/mleparams (GitHub)
 R Package Dependencies
 rstan
 License
 BSD (3 clause), CCBY
Hierarchical TwoParameter Logistic Item Response Model
This case study documents a Stan model for the twoparameter 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.
View (HTML)
 Author
 Daniel C. Furr
 Keywords
 education, item response theory, twoparameter logistic model, hierarchical priors
 Source Repository
 examplemodels/education/hierarchical_2pl (GitHub)
 R Package Dependencies
 rstan, ggplot2, mirt
 License
 BSD (3 clause), CCBY
Rating Scale and Generalized Rating Scale Models with Latent Regression
This case study documents a Stan model for the rating scale model (RSM) and the generalized rating scale model (GRSM) with latent regression. The latent regression portion of the models may be restricted to an intercept only, yielding a standard RSM or GRSM. A brief simulation indicates that the Stan models successfully recover the generating parameters. An example using a survey of public perceptions of science and technology is provided.
View (HTML)
 Authors
 Daniel C. Furr
 Keywords
 education, item response theory, rating scale model, generalized rating scale model
 Source Repository
 examplemodels/education/rsm_and_grsm (GitHub)
 R Package Dependencies
 rstan, edstan, ggplot2, ltm
 License
 BSD (3 clause), CCBY
Partial Credit and Generalized Partial Credit Models with Latent Regression
This case study documents a Stan model for the partial credit model (PCM) and the generalized partial credit model (GPCM) with latent regression. The latent regression portion of the models may be restricted to an intercept only, yielding a standard PCM or GPCM. A brief simulation indicates that the Stan models successfully recover the generating parameters. An example using the TIMSS 2011 mathematics assessment is provided
View (HTML)
 Authors
 Daniel C. Furr
 Keywords
 education, item response theory, partial credit model, generalized partial credit model
 Source Repository
 examplemodels/education/pcm_and_gpcm (GitHub)
 R Package Dependencies
 rstan, edstan, ggplot2, TAM
 License
 BSD (3 clause), CCBY
Rasch and TwoParameter Logistic Item Response Models with Latent Regression
This case study documents Stan models for the Rasch and twoparameter 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.
View (HTML)
 Authors
 Daniel C. Furr
 Keywords
 education, item response theory, rasch model, twoparameter logistic model
 Source Repository
 examplemodels/education/rasch_and_2pl.html (GitHub)
 R Package Dependencies
 rstan, edstan, ggplot2, TAM
 License
 BSD (3 clause), CCBY
TwoParameter Logistic Item Response Model
This tutorial introduces the R package edstan for estimating twoparameter 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.
View (HTML)
 Author
 Daniel C. Furr, Seung Yeon Lee, JoonHo Lee, and Sophia RabeHesketh
 Keywords
 education, item response theory, twoparameter logistic model
 Source Repository
 examplemodels/education/tutorial_twopl (GitHub)
 R Package Dependencies
 rstan, reshape2, ggplot2, gridExtra, devtools, edstan
 License
 BSD (3 clause), CCBY
Cognitive Diagnosis Model: DINA model with independent attributes
This case study documents a Stan model for the DINA model with independent attributes. A Simulation indicates that the Stan model successfully recovers the generating parameters and predicts respondents’ attribute mastery. A Stan model with no structure of the attributes is also discussed and applied to the simulated data. An example using a subset of the fraction subtraction data is provided.
View (HTML)
 Author
 Seung Yeon Lee
 Keywords
 education, cognitive diagnosis model, diagnostic classification model, attribute mastery, DINA
 Source Repository
 examplemodels/education/dina_independent (GitHub)
 R Package Dependencies
 rstan, ggplot2, CDM
 License
 BSD (3 clause), CCBY
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 heldout data, (iv) rank items by chance of success, (v) perform multiple comparisons in several settings, (vi) replicate new data for posterior pvalues, and (vii) perform graphical posterior predictive checks.
View (HTML)
 Author
 Bob Carpenter
 Keywords
 binary trials, pooling, hierarchical models, baseball, epidemiology, prediction, posterior predictive checks
 Source Repository
 examplemodels/knitr/poolbinarytrials (GitHub)
 R Package Dependencies
 rstan, ggplot2, rmarkdown
 License
 BSD (3 clause), CCBY
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 SpeciesSite Occupancy Model
This case study replicates the analysis and output graphs of Dorazio et al. (2006) noisymeasurement 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.
View (HTML)
 Author
 Bob Carpenter
 Keywords
 ecology, occupancy, species abundance, supercommunity, posterior predictive check
 Source Repository
 examplemodels/knitr/dorazioroyleoccupancy (GitHub)
 License
 BSD (3 clause), CCBY
 R Package Dependencies
 rstan, ggplot2, rmarkdown
Stan Case Studies, Volume 1 (2014)
Soil Carbon Modeling with RStan
This case study provides ordinary differential equationbased compartment models of soil carbon flux, with experimental data fitted with unknown initial compartment balance and noisy CO_{2} 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
 soilmetamodel/stan/soilknit (GitHub)
 License
 BSD (3 clause), CCBY
 R Package Dependencies
 rstan, ggplot2, rmarkdown