---
title: "Hierarchical Bayesian Aldrich-McKelvey Scaling in R via Stan"
output: rmarkdown::html_vignette
bibliography: hbam_refs.bib 
csl: american-journal-of-political-science.csl
vignette: >
  %\VignetteIndexEntry{Hierarchical Bayesian Aldrich-McKelvey Scaling in R via Stan}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
resource_files:
  - p_stim.svg
  - p_resp.svg
  - p_trace_theta.svg
  - p_lambda.svg
  - p_eta.svg
  - p_chi.svg
  - p_alpha.svg
  - p_abs_beta.svg
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE)

load("data.rda")
library("dplyr")
```

```{=html}
<!-- avoid border around images -->
<style> img { border: 0; } </style>
```
The goal of the **hbamr** package is to enable users to efficiently perform Hierarchical Bayesian Aldrich-McKelvey (HBAM) scaling in R. Aldrich-McKelvey (AM) scaling is a method for estimating the ideological positions of survey respondents and political actors on a common scale using ideological survey data [@aldrich:1977]. The hierarchical versions of the AM model included in this package outperform other versions both in terms of yielding meaningful posterior distributions for all respondent positions and in terms of recovering true respondent positions in simulations. The original version of the default model is described in an [open access article](https://doi.org/10.1017/pan.2023.18) [@boelstad:2023].

The package mainly fits models via the NUTS algorithm in **rstan** -- a Markov chain Monte Carlo (MCMC) algorithm. However, it also offers a simplified model that can be fit using optimization to analyze large data sets quickly.

This vignette provides an overview of how to use the key functions in the **hbamr** package. The vignette walks through an applied example, showing how to prepare data, fit models, extract estimates, plot key results, and perform cross-validation.

## Example Data

For illustration, we will use data from the 1980 American National Election Study (ANES). This is the same data set that serves to illustrate the original AM model in the **basicspace** package. The data set is included in the **hbamr** package and can be loaded by running `data(LC1980)`.

The data set contains respondents' placements of themselves and six stimuli on 7-point Liberal-Conservative scales. The stimuli in question are: The Democratic and Republican parties, Democratic presidential candidate Jimmy Carter, Republican candidate Ronald Reagan, independent candidate (and former Republican) John B. Anderson, and Ted Kennedy (who challenged the incumbent Carter, but failed to win the Democratic nomination).

We load the data and re-code missing values as follows:

```{r load_data, echo = TRUE, eval = FALSE, include = TRUE}
library("hbamr")
data(LC1980)
LC1980[LC1980 == 0 | LC1980 == 8 | LC1980 == 9] <- NA 
self <- LC1980[, 1]
stimuli <- LC1980[, -1]
```

```{r show_data, eval = FALSE}
head(stimuli) 
```

```{r, echo = FALSE}
head_stimuli 
```

## Preparing the Data

The function `prep_data()` serves to prepare the data. This function can be run ahead of fitting the models, or it can be run implicitly as part of a single function call to fit the models (as shown below). The function takes a vector of $N$ ideological self-placements and an $N \times J$ matrix of stimulus placements. The self-placement vector is required to estimate respondent positions, but can be dropped if such data are not available or if the respondent positions are not of interest. 

`prep_data()` applies a set of inclusion criteria, performs any necessary data transformation, and returns a list of data suited for sampling in **rstan**. The stimuli data are stored in a vector as a long-form sparse matrix. If the stimuli data include column-names, these will be preserved for later use.

Any missing data must be set to `NA` before use. The `prep_data()` function allows the user to decide how many missing values should be permitted per respondent by specifying the argument `allow_miss`. (The default is `allow_miss = 2`. Alternatively, the argument `req_valid` specifies how many valid observations to require per respondent. The default is `req_valid = J - allow_miss`, but, if specified, `req_valid` takes precedence.) Similarly, the user may specify how may unique positions on the ideological scale each respondent is required to have used when placing the stimuli in order to be included in the analysis. The default is `req_unique = 2`, which means that respondents who place all stimuli in exactly the same place will not be included.

The data provided to `prep_data()` can be centered, but they do not have to be: The function will detect un-centered data and attempt to center these automatically, assuming that the highest and lowest observed values in the data mark the extremes of the scale.

To use the `prep_data()` function on the example data using the default settings, we would run:

```{r prep_data, echo = TRUE, eval = FALSE}
dat <- prep_data(self, stimuli)
```

Users who want to keep other covariates for subsequent analysis, may find it useful to run `prep_data()` separately from the call to fit the models. The list returned by this function includes the logical vector `keep`, which identifies the rows in the original data that have been kept. If we had a data set `x` containing covariates, and used `prep_data()` to produce the list `dat`, then we could use `x[dat$keep, ]` to get a subset of `x` corresponding to the data used in the analysis. (The order of the individuals/rows in the data remains unchanged by the functions in this package.)

## Models

This package provides several alternative models that can be selected using the names below. Users who are unsure which model to use are advised to use the default HBAM model. If speed or sampling diagnostics are an issue, HBAM_MINI may provide a useful alternative. (See also the section on cross-validation for further discussion on the issue of model selection.)

**HBAM** is the default model, which allows for scale flipping and employs hierarchical priors on the shift and stretch parameters. It also models heteroskedastic errors that vary by both individual and stimuli. Compared to the model in @boelstad:2023, this version has been slightly revised to provide faster sampling. A key difference from the original model is that the respondent positions are not treated as parameters, but rather calculated as a function of self-placements, individual-level parameters, and simulated errors. This makes the model considerably faster, while yielding very similar results. The model simulates errors in the self-placements of the same magnitude as that with which the respondent in question places the stimulus with the smallest errors. All models in the package use this approach.

**HBAM_MULTI** is a version that models differences between groups defined by the user. It requires a vector identifying the groups to be supplied as the argument `group_id`. The model gives each group separate hyperparameters for the locations of the prior distributions for the shift and stretch parameters. Rather than shrinking the estimates toward the mode for the whole dataset, this model shrinks the estimates toward the mode for the group. The vectors of hyperparameters are called `mu_alpha` and `mu_beta` and are constructed to have means of 0. The scales of the priors on these hyperparameters can be set by the user via the arguments `sigma_mu_alpha` and `sigma_mu_beta`. The default values are B / 10 and .2, respectively. (Here, B measures the length of the survey scale as the number of possible placements on one side of the center.) One potential use for this model is to supply self-placements as `group_id`, and thus give each self-placement group its own prior distribution for the shift and stretch parameters.

**HBAM_NF** (formerly HBAM_0) is a version of the HBAM model that does not allow for scale flipping. This may be useful if there are truly zero cases of scale flipping in the data. Such scenarios can be created artificially, but may also arise in real data. For example, expert surveys appear unlikely to contain many instances of scale flipping. For data that contain zero cases of flipping, models that allow for flipping contain superfluous parameters that lead to inefficient sampling. Models that do not allow for flipping will sample faster and typically yield slightly more accurate estimates. Such models are therefore usually preferable when no flipping is present.

**HBAM_MULTI_NF** is a version of the HBAM_MULTI model that does not allow for scale flipping.

**HBAM_MINI** is a version of the HBAM model that assumes the prediction errors in the stimuli placements to be homoskedastic. This model tends to sample faster faster than the standard HBAM model while yielding very similar point estimates. For large datasets, this model may provide a reasonable compromise between model complexity and estimation speed.

**FBAM_MINI** is a version of the HBAM_MINI model with fixed hyperparameters to allow fitting via optimization rather than MCMC -- which can be useful for large data sets. This model allows the user to specify the scales of the priors for the shift and (logged) stretch parameters via the arguments `sigma_alpha` and `sigma_beta`. The default values are B / 4 and .35, respectively. These defaults are intended to be realistic and weakly informative. Users who want to control the degree of shrinkage of the individual-level parameters may find it useful to fit this model -- or other FBAM models -- via either MCMC or optimization.

**FBAM_MULTI** is a version of the FBAM_MINI model that shares the group-modeling features of the HBAM_MULTI model. It allows the user to set the scales of the priors for the shift and stretch parameters via the arguments `sigma_alpha` and `sigma_beta`, and set the scales of the priors on `mu_alpha` and `mu_beta` via the arguments `sigma_mu_alpha` and `sigma_mu_beta`.

**FBAM_MULTI_NF** is a version of the FBAM_MULTI model that does not allow for scale flipping.

**HBAM_R_MINI** is a version of the HBAM_MINI model that incorporates the rationalization component of the ISR model in @boelstad:2020. This model requires additional data to be supplied as the argument `pref`: An N × J matrix of stimuli ratings from the respondents. The rationalization part of the model is simplified relative to the original ISR model: The direction in which respondents move disfavored stimuli is estimated as a common expectation for each possible self-placement on the scale.

**BAM** is an unpooled model with wide uniform priors on the shift and stretch parameters. It is similar to the JAGS version introduced by @hare:2015, although the version included here has been adjusted to yield stretch parameters with an average of approximately one (and thus produce a scale similar to those of the other models in the package). Like the other models, this version of the BAM model also simulates errors in the self-placements to yield a realistic level of uncertainty. While this model is simple and fast, it tends to overfit the data and produce invalid posterior distributions for some respondent positions [@boelstad:2023]. However, it could potentially be useful as a baseline for model comparisons in situations where respondent positions are not of interest.

**HBAM_2** has been replaced by the more general HBAM_MULTI model.

These models can also be used in situations where self-placements are not available and the only goal is to estimate stimulus positions or respondents' shift and stretch parameters. While the latent respondent positions will not be estimated, all other parameters are unaffected if the argument `self` is dropped when calling `prep_data()` or `hbam()`.

To see the Stan code for any of the models, use the `show_code()` function:

```{r, eval = FALSE}
show_code("HBAM_MULTI")
```

### Summary of Model Features

|                        | HBAM<br>  | HBAM<br>MULTI | HBAM<br>NF | HBAM<br>MULTI_NF | HBAM<br>MINI |
|-----------------------------|:------------:|:------------:|:------------:|:------------:|:------------:|
| Hierarchical priors    |     X     |       X       |     X      |        X         |      X       |
| Scale flipping         |     X     |       X       |            |                  |      X       |
| Group-differences      |           |       X       |            |        X         |              |
| Heteroskedastic errors |     X     |       X       |     X      |        X         |              |

: Recommended models for fitting via MCMC

|                            | FBAM<br>MINI | FBAM<br>MULTI | FBAM<br>MULTI_NF | HBAM<br>R_MINI | BAM<br>  |
|-----------------------------|:------------:|:------------:|:------------:|:------------:|:------------:|
| Hierarchical priors        |              |               |                  |       X        |          |
| User-defined priors\*      |      X       |       X       |        X         |                |          |
| Scale flipping             |      X       |       X       |                  |       X        |    X     |
| Group-differences          |              |       X       |        X         |                |          |
| Heteroskedastic errors     |              |               |                  |                |    X     |
| Rationalization            |              |               |                  |       X        |          |
| Fitting via optimization\* |      X       |       X       |        X         |                |          |

: Non-hierarchical and special purpose models ("\*" marks optional features)

## Fitting

The `hbam()` function can be used to fit all models in this package and obtain a `stanfit` object. The default model is HBAM, while other models can be specified via the argument `model`. 

Unless the user supplies pre-prepared data via the `data`-argument, `hbam()` will implicitly run `prep_data()`. It therefore takes the same arguments as the `prep_data()` function (i.e. `self`, `stimuli`, `group_id`, `allow_miss`, `req_valid`, and `req_unique`).

To fit the HBAM model using the default settings, we would run:

```{r fithbam1, eval = FALSE}
fit_hbam <- hbam(self, stimuli)
```

To fit the HBAM_MINI model while requiring complete data for all respondents, we would run:

```{r fitbam, eval = FALSE, cache = FALSE, message = FALSE, warning = FALSE}
fit_hbam_mini <- hbam(self, stimuli, model = "HBAM_MINI", allow_miss = 0)
```

If we wanted to run the `prep_data()` function separately before fitting the model, we would supply output from `prep_data()` as the argument `data`:

```{r fithbam2, cache = FALSE, eval = FALSE, message = FALSE, warning = FALSE}
dat <- prep_data(self, stimuli, allow_miss = 0) 
fit_hbam <- hbam(data = dat)
```

```{r prep_data3, include = FALSE, eval = FALSE}
dat <- prep_data(self, stimuli, allow_miss = 0) 
```

To fit the HBAM_MULTI or FBAM_MULTI model, we would need to supply a vector identifying the groups of interest. One option would be to supply respondents' self-placements as `group_id` and thus give each self-placement group their own prior distributions for the shift and stretch parameters. If we decided to fit the FBAM_MULTI model, we could also specify our own scales for the priors on key parameters:

```{r, eval = FALSE}
fit_fbam_multi <- hbam(self, stimuli, model = "FBAM_MULTI", group_id = self, 
                       sigma_alpha = .8, sigma_mu_alpha = .7, 
                       sigma_beta = .4, sigma_mu_beta = .25)
```

`hbam()` uses `rstan::sampling()`, and any additional arguments to `hbam()` will be passed on to the sampling function. By default, `hbam()` will run 4 chains and detect the number of available CPU cores. If possible, `hbam()` will use as many cores as there are chains. The other sampling defaults are: `warmup = 1000` and `iter = 2000`. These settings can all be overridden in the `hbam()` call.

### Fitting via Optimization

For very large data sets, optimization can be a useful and much faster alternative to MCMC. However, the optimization feature provided in this package only provides maximum a posteriori (MAP) point estimates.

The `fbam()` function fits FBAM models using `rstan::optimizing()`. The `fbam()` function works just like the `hbam()` function, except the arguments for `rstan::sampling()` do not apply. To fit the FBAM_MINI model using default settings, we would run:

```{r fitfbam, eval = FALSE}
fit_fbam <- fbam(self, stimuli)
```

### Execution Times

Models of the kind included in the **hbamr** package face an inevitable trade-off between nuance (i.e. model complexity) and execution times. For small data sets (like ANES 1980), all models provide reasonable running times, but for large data sets (like ANES 2012), more complex models tend to get slow. In these cases, the HBAM_MINI model may be a useful alternative. For very large data sets, fitting the FBAM_MINI model via optimization may be the best alternative.

|                                   | HBAM <br>iter=2000 | HBAM_NF <br>iter=2000 | HBAM_MINI <br>iter=2000 | FBAM_MINI<br>iter=2000 | FBAM_MINI <br>optimization |
|--------------|:----------:|:----------:|:----------:|:----------:|:----------:|
| **ANES 1980** <br>  (N=643, J=6)  |          5m          |         1m 20s          |          3m 30s           |          3m 45s          |             3s             |
| **ANES 2012** <br>  (N=4949, J=4) |          1h          |           15m           |            35m            |           40m            |            21s             |

: Execution times on an Intel Core i5 2.0-3.8 GHz CPU

## Plotting

The **hbamr** package contains several functions for creating presentable plots of the results. The package uses **ggplot2**, which means ggplot-themes can be added to the plots. 

### Stimuli Positions

The function `plot_stimuli()` plots the marginal posterior distributions of all stimuli in the data. By default, it will fill the distributions with shades from blue to red depending on the position on the scale. The argument `rev_color = TRUE` will reverse the order of the colors.

```{r, include = TRUE, eval = FALSE, fig.align = "center", fig.asp = .4, fig.width = 8, out.width = "100%"}
plot_stimuli(fit_hbam)
```

```{r, echo = FALSE, fig.align = "center", fig.asp = .4, fig.width = 8, out.width = "100%"}
knitr::include_graphics("p_stim.svg")
```

In this example, we see that John B. Anderson -- the former Republican who ran as an independent candidate -- gets a wider posterior distribution, suggesting that voters were more uncertain about where to place him relative to the others.

### Respondent Positions

The function `plot_respondents()` plots the distribution of estimated respondent positions. It illustrates the uncertainty of the estimates by calculating the population density for each of a set of posterior draws. The default is to use 15 draws for each respondent, but this can be altered by specifying the argument `n_draws`. The `plot_respondents()` function also plots the estimated stimulus positions by default, but this behavior can be turned off by adding the argument `inc_stimuli = FALSE`.

```{r, include = TRUE, eval = FALSE, fig.align = "center", fig.asp = .4, fig.width = 8, out.width = "100%"}
plot_respondents(fit_hbam, n_draws = 10)
```

```{r, echo = FALSE, fig.align = "center", fig.asp = .4, fig.width = 8, out.width = "100%"}
knitr::include_graphics("p_resp.svg")
```

Users who want to customize the plots further can obtain the underlying data by using the function `get_plot_data()`. This function accepts the same `n_draws`-argument as `plot_respondents()`. The output is a list of three tibbles: The first element contains the posterior mean stimulus positions, as well as the $x$- and $y$-values of the posterior modes (which can be useful for labeling the distributions). The second element contains the posterior draws for the stimulus positions (which can be used to calculate marginal posterior densities). The third element contains the selected number of posterior draws for each respondent (which form the key ingredient for the `plot_respondents()` function).

```{r get_plot_dat, include = TRUE, eval = FALSE, include= FALSE, eval = FALSE}
get_plot_data(fit_hbam)
```

### Individual Parameters over Self-Placements

The function `plot_over_self()` plots the distributions of key parameter estimates over the respondents' self-placements. The function will accept either a single `stanfit` object produced by `hbam()`, results from `fbam()`, or a list of such objects. 

The user specifies which parameter to show via the argument `par`. This can be either of the following: `"alpha"`, `"beta"`, `"abs_beta"`, `"lambda"`, or `"chi"`, where `"abs_beta"` calls for the absolute value of beta to be used. By default, the function uses posterior median estimates, but this can be changed by specifying `estimate = "mean"`.

#### Shifting

To compare the distributions of estimated shift parameters from the HBAM and HBAM_MINI models, we would run:

```{r plot_alpha, include = TRUE, eval = FALSE, echo = TRUE, fig.asp = .37, fig.width = 8.5, fig.align = "center", out.width = "100%"}
plot_over_self(list(fit_hbam, fit_hbam_mini), "alpha")
```

```{r, echo = FALSE, fig.align = "center", fig.asp = .37, fig.width = 8.5, fig.align = "center", out.width = "100%"}
knitr::include_graphics("p_alpha.svg")
```

#### Stretching

For models that allow for scale flipping, the draws for $\beta$ combine the separate parameters for each flipping-state. The absolute value of $\beta$ may therefore be better suited for examining the extent to which each individual stretches the ideological space. To inspect the distribution of these values across self-placements, we would run:

```{r plot_beta, include = TRUE, eval = FALSE, echo = TRUE, fig.asp = .37, fig.width = 8.5, fig.align = "center", out.width = "100%"}
plot_over_self(list(fit_hbam, fit_hbam_mini), "abs_beta")
```

```{r, echo = FALSE, fig.align = "center", fig.asp = .37, fig.width = 8.5, fig.align = "center", out.width = "100%"}
knitr::include_graphics("p_abs_beta.svg")
```

The pattern above, where respondents with more extreme self-placements have more extreme $\beta$ parameters, is exactly the kind of differential item functioning that the models in this package are intended to correct for: These respondents tend to place both stimuli and themselves further out on the scale than others do, thus appearing more extreme in comparison.

#### Flipping

To see whether the $\beta$ parameters are likely to be positive or negative, we can look at the expectations of the flipping parameters, $\lambda$. These parameters represent each respondent's probability of *not* flipping the scale:

```{r plot_lambda, include = TRUE, eval = FALSE, echo = TRUE, fig.asp = .37, fig.width = 8.5, fig.align = "center", out.width = "100%"}
plot_over_self(list(fit_hbam, fit_hbam_mini), "lambda")
```

```{r, echo = FALSE, fig.align = "center", fig.asp = .37, fig.width = 8.5, fig.align = "center", out.width = "100%"}
knitr::include_graphics("p_lambda.svg")
```

In this example, flipping is uncommon, but respondents who place themselves in the middle have a somewhat higher flipping-probability. This may suggest that some of these respondents are less informed about politics and provide less accurate answers.

#### Respondent Positions

It may also be useful to inspect the distribution of the scaled respondent positions over the self-placements. This illustrates the extent to which the model has transformed the original data. In this example, the impact of the models is generally modest, although a few respondents have been detected as having flipped the scale, and thus have had their self-placement flipped back.

```{r plot_chi, include = TRUE, eval = FALSE, echo = TRUE, fig.asp = .37, fig.width = 8.5, fig.align = "center", out.width = "100%"}
plot_over_self(list(fit_hbam, fit_hbam_mini), "chi")
```

```{r, echo = FALSE, fig.align = "center", fig.asp = .37, fig.width = 8.5, fig.align = "center", out.width = "100%"}
knitr::include_graphics("p_chi.svg")
```

#### Additional Parameters

Other individual-level parameters like `"eta"` can also be plotted if these have been passed to `hbam()` via the argument `extra_pars` when fitting the model. Parameters like $\eta$ are not stored in the `hbam()` results by default, as this would increase the post-processing time as well as the size of the model fits. (Note also that homoskedastic models have no `"eta"` parameters and "NF"-type models have no `"lambda"` or `"kappa"` parameters.) 

The estimated $\eta$ parameters yield information about the accuracy of respondents' answers. When the argument `par = "eta"` is specified, the plotting function will display $\sqrt{\eta_i} / J$, which equals the average error for each individual (the mean of $\sigma_{ij}$ for each $i$ across $j$). The point estimates will still be calculated using the posterior median, unless the argument `estimate = "mean"` is added.

```{r plot_eta, include = TRUE, eval = FALSE, echo = TRUE, fig.asp = .67, fig.width = 4.25, fig.align = "center", out.width = "50%"}
fit_hbam <- hbam(data = dat, extra_pars = "eta")
plot_over_self(fit_hbam, "eta")
```

```{r, echo = FALSE, fig.align = "center", fig.asp = .67, fig.width = 4.25, fig.align = "center", out.width = "50%"}
knitr::include_graphics("p_eta.svg")
```

## Posterior Summaries

The package also contains a wrapper for `rstan::summary()` called `get_est()`. This function takes the arguments `object` -- a `stanfit` object produced by `hbam()` or a list produced by `fbam()` -- and `par` -- the name of the parameter(s) to be summarized. The function returns a tibble, which by default contains the posterior mean, the 95% credible interval, the posterior median, the estimated number of effective draws, and the split R-hat. One can obtain other posterior quantiles by using the argument `probs`. To get a 50% credible interval (and no median), one would add the argument `probs = c(0.25, 0.75)`. To include the Monte Carlo standard error and the posterior standard deviation, use the argument `simplify = FALSE`. (When applied to outputs from `fbam()`, `get_est()` only returns point estimates.)

The posterior draws for the stimulus positions can be summarized as follows:

```{r get_est_t, include = TRUE, eval = FALSE}
get_est(fit_hbam, "theta")
```

```{r, echo = FALSE}
est_theta
```

To get summaries of the respondent positions, we would specify `par = "chi"`. If we want the results to match the rows in the original dataset by reporting rows of NAs for respondents who were not included in the analysis, we would add the argument `format_orig = TRUE`:

```{r get_est_v, include = TRUE, eval = FALSE}
get_est(fit_hbam, "chi", format_orig = TRUE)
```

```{r, echo = FALSE}
est_chi_orig
```

## Cross-Validation

A useful way to compare alternative models is to estimate their out-of-sample prediction accuracy. More specifically, we can estimate their expected log pointwise predictive density for a new data set (ELPD). 

The **rstan** and **loo** packages contain functions for estimating ELPDs by performing approximate leave-one-out (LOO) cross-validation using Pareto smoothed importance sampling (PSIS). These functions can be used on model fits from `hbam()` if the argument `extra_pars = "log_lik"` is specified when fitting the model:

```{r, eval = FALSE}
fit_hbam <- hbam(data = dat, extra_pars = "log_lik")
loo_hbam <- loo::loo(fit_hbam)
```

However, PSIS-LOO only works when all Pareto *k* values are sufficiently low, or when the number of high values is so low that moment matching can be used on the problematic cases. This is not always the case for the models in this package. 

The **hbamr** package therefore allows users to perform *K*-fold cross-validation. The function `hbam_cv()` is similar to `hbam()` and takes the same arguments, but will perform a *K*-fold cross-validation for the chosen model. In contrast to `hbam()`, the default for `hbam_cv()` is to not allow respondents to have any missing values (`allow_miss = 0`). The reason is that the cross-validation essentially creates missing values by excluding some data from each run. 

A key choice when performing cross-validation is to set *K* (the number of folds to use), and the default in this package is 10. To preserve memory, the function extracts the summaries of the log-likelihoods for the held-out data and drops the stanfit objects once this is done. The memory requirements of the function is therefore similar to running a single analysis with one chain per core. The function that splits the data into *K* folds uses a default seed to produce the same folds each time, unless a different seed is specified.

The `hbam_cv()` function is written to allow parallel computation via the `future` package to minimize execution time. The `future` package offers several computational strategies, of which "multisession" works on all operating systems. It most settings, it is advisable to use all physical CPU cores when performing cross-validation. To set up parallel computation using 4 cores via the `future` package, we could run:

```{r, eval = FALSE}
library(future)
plan(multisession, workers = 4)
```

To perform 10-fold cross-validation for a selection of models, we could run:

```{r, include = FALSE}
load("elpds.rda")
```

```{r run_cv_new, eval = FALSE}
kfold_bam <- hbam_cv(self, stimuli, model = "BAM")
kfold_hbam <- hbam_cv(self, stimuli, model = "HBAM")
kfold_hbam_nf <- hbam_cv(self, stimuli, model = "HBAM_NF")
kfold_hbam_multi <- hbam_cv(self, stimuli, group_id = self, 
                            model = "HBAM_MULTI")
```

The `hbam_cv()` function returns an object of classes `kfold` and `loo`, which can be further processed using the `loo` package. To compare the estimated ELPDs, we could run:

```{r, eval = FALSE}
print(loo::loo_compare(list(BAM = kfold_bam, 
                            HBAM = kfold_hbam, 
                            HBAM_NF = kfold_hbam_nf, 
                            HBAM_MULTI = kfold_hbam_multi)), simplify = FALSE)
```

```{r, echo = FALSE, message = FALSE, warning = FALSE}
library(loo)
print(loo1, simplify = FALSE)
```

We see that the unpooled BAM model is worse at predicting out-of-sample data than the other models, suggesting it overfits the data. The HBAM_NF model -- which does not allow for scale flipping -- also performs worse, suggesting it is too restrictive and underfits. The HBAM and HBAM_MULTI models outperform the other two models in this case. The HBAM_MULTI model -- with self-placements as `group_id` -- may have a slight edge, but the default HBAM model performs about equally well. 

We could also perform cross-validation for a couple of models that do not account for heteroskedastic errors:

```{r, eval = FALSE}
kfold_hbam_mini <- hbam_cv(self, stimuli, model = "HBAM_MINI")
kfold_fbam_mini <- hbam_cv(self, stimuli, model = "FBAM_MINI")
```

```{r, eval = FALSE}
print(loo::loo_compare(list(HBAM_MINI = kfold_hbam_mini, 
                            FBAM_MINI = kfold_fbam_mini)), simplify = FALSE)
```

```{r, echo = FALSE}
print(loo2, simplify = FALSE)
```

The ELPD for the FBAM_MINI model is lower than that of the HBAM_MINI model, which suggests that the default priors of the FBAM model are too wide to yield optimal results for these data. While the ELPD for the HBAM_MINI model is higher than that of the FBAM model, it is still notably lower than that of its heteroskedastic counterpart (HBAM). This suggest there is a considerable degree of heteroskedasticity in the data -- which is not surprising.

Modeling the heteroskedasticity increases prediction accuracy, but it should be noted that this does not necessarily translate into much more accurate estimates of key model outputs. As shown in section on plotting, the results for the HBAM and HBAM_MINI models are very similar. In fact, their estimated respondent positions correlate at .97 and .99, depending on whether we use the posterior means or medians. Their estimated stimulus positions also correlate at .99, but this masks the fact that the HBAM_MINI model places the less well known candidate John B. Anderson further to the left than the HBAM model does. There are some subtle differences that users should be aware of, even if these models tend to produce very similar results.

## Diagnostics

The `rstan::sampling()` function that `hbam()` uses automatically performs a number of key diagnostic checks after sampling and issues warnings when a potential issue is detected. The authors of **rstan** emphasize diagnostics and careful model development, and users of **rstan** will more frequently encounter warnings than users of **rjags**. One warning users of this package may encounter is that the Bulk or Tail Effective Sample Size (ESS) is too low (see <https://mc-stan.org/misc/warnings.html>). The most straightforward solution to this issue is to increase the number of posterior draws, using the `iter` argument. However, this increases the computational load, and users should consider carefully what level of accuracy they need.

Because the `hbam()` function returns a `stanfit` object, the model fit can be examined using the full range of diagnostic tools from the **rstan** package. Users should consult the **rstan** documentation for details on the various diagnostic tests and plots that are available. One example of the available tools is `traceplot()`:

```{r traceplot_code, eval = FALSE, include = TRUE}
rstan::traceplot(fit_hbam, pars = "theta")
```

```{r, echo = FALSE, fig.align = "center", fig.asp = .5, fig.width = 8, fig.align = "center", out.width = "100%", warning = FALSE, message = FALSE}
knitr::include_graphics("p_trace_theta.svg")
```

### Limits to Exact Replication

The functions in this package accept a seed argument for all operations that involve a random number generator. Supplying a seed is sufficient to get the exact same results in repeated runs on the same system, but it does not ensure exact replication across systems or software versions. The [Stan Reference Manual](https://mc-stan.org/docs/reference-manual/reproducibility.html) explains why:

> Floating point operations on modern computers are notoriously difficult to replicate because the fundamental arithmetic operations, right down to the IEEE 754 encoding level, are not fully specified. The primary problem is that the precision of operations varies across different hardware platforms and software implementations.
>
> Stan is designed to allow full reproducibility. However, this is only possible up to the external constraints imposed by floating point arithmetic.

In short, running the functions in this package on different systems will likely yield slightly different results. It should be noted, however, that the differences across systems will be minimal and substantively negligible as long as the user obtains a sufficient number of posterior draws.

## References
