For models fit using MCMC, compute approximate leave-one-out cross-validation (LOO, LOOIC) or, less preferably, the Widely Applicable Information Criterion (WAIC) using the loo package. (For $$K$$-fold cross-validation see kfold.stanreg.) Functions for model comparison, and model weighting/averaging are also provided.

Note: these functions are not guaranteed to work properly unless the data argument was specified when the model was fit. Also, as of loo version 2.0.0 the default number of cores is now only 1, but we recommend using as many (or close to as many) cores as possible by setting the cores argument or using options(mc.cores = VALUE) to set it for an entire session.

# S3 method for stanreg
loo(
x,
...,
cores = getOption("mc.cores", 1),
save_psis = FALSE,
k_threshold = NULL
)

# S3 method for stanreg
waic(x, ...)

# S3 method for stanreg
loo_compare(x, ..., criterion = c("loo", "kfold", "waic"), detail = FALSE)

# S3 method for stanreg_list
loo_compare(x, ..., criterion = c("loo", "kfold", "waic"), detail = FALSE)

# S3 method for stanreg_list
loo_model_weights(x, ..., cores = getOption("mc.cores", 1), k_threshold = NULL)

compare_models(..., loos = list(), detail = FALSE)

## Arguments

x For loo and waic, a fitted model object returned by one of the rstanarm modeling functions. See stanreg-objects. For the loo_model_weights method, x should be a "stanreg_list" object, which is a list of fitted model objects created by stanreg_list. loo_compare also allows x to be a single stanreg object, with the remaining objects passed via ..., or a single stanreg_list object. For loo_compare.stanreg, ... can contain objects returned by the loo, kfold, or waic method (see the Examples section, below). For loo_model_weights, ... should contain arguments (e.g. method) to pass to the default loo_model_weights method from the loo package. Passed to loo. Threshold for flagging estimates of the Pareto shape parameters $$k$$ estimated by loo. See the How to proceed when loo gives warnings section, below, for details. For loo_compare.stanreg and loo_compare.stanreg_list, should the comparison be based on LOO-CV (criterion="loo"), K-fold-CV (criterion="kfold"), or WAIC (criterion="waic"). The default is LOO-CV. See the Comparing models and Examples sections below. For loo_compare.stanreg and loo_compare.stanreg_list, if TRUE then extra information about each model (currently just the model formulas) will be printed with the output. a list of objects produced by the loo function

## Value

The structure of the objects returned by loo and waic methods are documented in detail in the Value section in loo and waic (from the loo package).

loo_compare returns a matrix with class 'compare.loo'. See the Comparing models section below for more details.

## Approximate LOO CV

The loo method for stanreg objects provides an interface to the loo package for approximate leave-one-out cross-validation (LOO). The LOO Information Criterion (LOOIC) has the same purpose as the Akaike Information Criterion (AIC) that is used by frequentists. Both are intended to estimate the expected log predictive density (ELPD) for a new dataset. However, the AIC ignores priors and assumes that the posterior distribution is multivariate normal, whereas the functions from the loo package do not make this distributional assumption and integrate over uncertainty in the parameters. This only assumes that any one observation can be omitted without having a major effect on the posterior distribution, which can be judged using the diagnostic plot provided by the plot.loo method and the warnings provided by the print.loo method (see the How to Use the rstanarm Package vignette for an example of this process).

### How to proceed when loo gives warnings (k_threshold)

The k_threshold argument to the loo method for rstanarm models is provided as a possible remedy when the diagnostics reveal problems stemming from the posterior's sensitivity to particular observations. Warnings about Pareto $$k$$ estimates indicate observations for which the approximation to LOO is problematic (this is described in detail in Vehtari, Gelman, and Gabry (2017) and the loo package documentation). The k_threshold argument can be used to set the $$k$$ value above which an observation is flagged. If k_threshold is not NULL and there are $$J$$ observations with $$k$$ estimates above k_threshold then when loo is called it will refit the original model $$J$$ times, each time leaving out one of the $$J$$ problematic observations. The pointwise contributions of these observations to the total ELPD are then computed directly and substituted for the previous estimates from these $$J$$ observations that are stored in the object created by loo. Another option to consider is K-fold cross-validation, which is documented on a separate page (see kfold).

Note: in the warning messages issued by loo about large Pareto $$k$$ estimates we recommend setting k_threshold to at least $$0.7$$. There is a theoretical reason, explained in Vehtari, Gelman, and Gabry (2017), for setting the threshold to the stricter value of $$0.5$$, but in practice they find that errors in the LOO approximation start to increase non-negligibly when $$k > 0.7$$.

## Comparing models

"loo" (or "waic" or "kfold") objects can be passed to the loo_compare function in the loo package to perform model comparison. rstanarm also provides a loo_compare.stanreg method that can be used if the "loo" (or "waic" or "kfold") object has been added to the fitted model object (see the Examples section below for how to do this). This second method allows rstanarm to perform some extra checks that can't be done by the loo package itself (e.g., verifying that all models to be compared were fit using the same outcome variable).

loo_compare will return a matrix with one row per model and columns containing the ELPD difference and the standard error of the difference. In the first row of the matrix will be the model with the largest ELPD (smallest LOOIC) and will contain zeros (there is no difference between this model and itself). For each of the remaining models the ELPD difference and SE are reported relative to the model with the best ELPD (the first row). See the Details section at the loo_compare page in the loo package for more information.

## Model weights

The loo_model_weights method can be used to compute model weights for a "stanreg_list" object, which is a list of fitted model objects made with stanreg_list. The end of the Examples section has a demonstration. For details see the loo_model_weights documentation in the loo package.

## References

Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. 27(5), 1413--1432. doi:10.1007/s11222-016-9696-4. arXiv preprint: http://arxiv.org/abs/1507.04544/

Yao, Y., Vehtari, A., Simpson, D., and Gelman, A. (2018) Using stacking to average Bayesian predictive distributions. Bayesian Analysis, advance publication, doi:10.1214/17-BA1091. (online).

Gabry, J. , Simpson, D. , Vehtari, A. , Betancourt, M. and Gelman, A. (2019), Visualization in Bayesian workflow. J. R. Stat. Soc. A, 182: 389-402. doi:10.1111/rssa.12378, (journal version, arXiv preprint, code on GitHub)

• The new loo package vignettes and various rstanarm vignettes for more examples using loo and related functions with rstanarm models.

• pareto-k-diagnostic in the loo package for more on Pareto $$k$$ diagnostics.

• log_lik.stanreg to directly access the pointwise log-likelihood matrix.

## Examples

# \donttest{
fit1 <- stan_glm(mpg ~ wt, data = mtcars, refresh = 0)
fit2 <- stan_glm(mpg ~ wt + cyl, data = mtcars, refresh = 0)

# (for bigger models use as many cores as possible)
loo1 <- loo(fit1, cores = 2)
print(loo1)#>
#> Computed from 4000 by 32 log-likelihood matrix
#>
#>          Estimate  SE
#> elpd_loo    -83.6 4.3
#> p_loo         3.3 1.2
#> looic       167.1 8.6
#> ------
#> Monte Carlo SE of elpd_loo is 0.1.
#>
#> All Pareto k estimates are good (k < 0.5).
#> See help('pareto-k-diagnostic') for details.loo2 <- loo(fit2, cores = 2)
print(loo2)#>
#> Computed from 4000 by 32 log-likelihood matrix
#>
#>          Estimate  SE
#> elpd_loo    -78.6 4.6
#> p_loo         4.1 1.3
#> looic       157.1 9.2
#> ------
#> Monte Carlo SE of elpd_loo is 0.1.
#>
#> Pareto k diagnostic values:
#>                          Count Pct.    Min. n_eff
#> (-Inf, 0.5]   (good)     31    96.9%   711
#>  (0.5, 0.7]   (ok)        1     3.1%   456
#>    (0.7, 1]   (bad)       0     0.0%   <NA>
#>    (1, Inf)   (very bad)  0     0.0%   <NA>
#>
#> All Pareto k estimates are ok (k < 0.7).
#> See help('pareto-k-diagnostic') for details.
# when comparing models the loo objects can be passed to loo_compare
# as individual arguments or as a list of loo objects
loo_compare(loo1, loo2)#>      elpd_diff se_diff
#> fit2  0.0       0.0
#> fit1 -5.0       2.8   loo_compare(list(loo1, loo2))#>      elpd_diff se_diff
#> fit2  0.0       0.0
#> fit1 -5.0       2.8
# if the fitted model objects contain a loo object in the component "loo"
# then the model objects can be passed directly or as a stanreg_list
fit1$loo <- loo1 fit2$loo <- loo2
loo_compare(fit1, fit2)#> Model comparison based on LOO-CV:
#>      elpd_diff se_diff
#> fit2  0.0       0.0
#> fit1 -5.0       2.8
# if the fitted model objects contain a loo object _and_ a waic or kfold
# object, then the criterion argument determines which of them the comparison
# is based on
fit1$waic <- waic(fit1)#> Warning: #> 3 (9.4%) p_waic estimates greater than 0.4. We recommend trying loo instead.fit2$waic <- waic(fit2)#> Warning:
#> 3 (9.4%) p_waic estimates greater than 0.4. We recommend trying loo instead.loo_compare(fit1, fit2, criterion = "waic")#> Model comparison based on WAIC:
#>      elpd_diff se_diff
#> fit2  0.0       0.0
#> fit1 -5.0       2.8
# the models can also be combined into a stanreg_list object, and more
# informative model names can be provided to use when printing
model_list <- stanreg_list(fit1, fit2, model_names = c("Fewer predictors", "More predictors"))
loo_compare(model_list)#> Model comparison based on LOO-CV:
#>                  elpd_diff se_diff
#> More predictors   0.0       0.0
#> Fewer predictors -5.0       2.8
fit3 <- stan_glm(mpg ~ disp * as.factor(cyl), data = mtcars, refresh = 0)
loo3 <- loo(fit3, cores = 2, k_threshold = 0.7)#> All pareto_k estimates below user-specified threshold of 0.7.
#> Returning loo object.loo_compare(loo1, loo2, loo3)#>      elpd_diff se_diff
#> fit3  0.0       0.0
#> fit2 -1.0       4.2
#> fit1 -6.0       5.1
# setting detail=TRUE will also print model formulas if used with
# loo_compare.stanreg or loo_compare.stanreg_list
fit3\$loo <- loo3
model_list <- stanreg_list(fit1, fit2, fit3)
loo_compare(model_list, detail=TRUE)#> Model formulas:
#>  fit1:  mpg ~ wt
#>  fit2:  mpg ~ wt + cyl
#>  fit3:  mpg ~ disp * as.factor(cyl)
#>
#> Model comparison based on LOO-CV:
#>      elpd_diff se_diff
#> fit3  0.0       0.0
#> fit2 -1.0       4.2
#> fit1 -6.0       5.1
# Computing model weights
#
# if the objects in model_list already have 'loo' components then those
# will be used. otherwise loo will be computed for each model internally
# (in which case the 'cores' argument may also be used and is passed to loo())
loo_model_weights(model_list)  # defaults to method="stacking"#> Method: stacking
#> ------
#>      weight
#> fit1 0.000
#> fit2 0.494
#> fit3 0.506 loo_model_weights(model_list,  method = "pseudobma")#> Method: pseudo-BMA+ with Bayesian bootstrap
#> ------
#>      weight
#> fit1 0.034
#> fit2 0.396
#> fit3 0.570 loo_model_weights(model_list,  method = "pseudobma", BB = FALSE)#> Method: pseudo-BMA
#> ------
#>      weight
#> fit1 0.002
#> fit2 0.273
#> fit3 0.725
# you can also pass precomputed loo objects directly to loo_model_weights
loo_list <- list(A = loo1, B = loo2, C = loo3) # names optional (affects printing)
loo_model_weights(loo_list)#> Method: stacking
#> ------
#>   weight
#> A 0.000
#> B 0.494
#> C 0.506 # }