This vignette shows the main functionalities of the
**projpred** package, which implements the projection
predictive variable selection for various regression models (see section
“Supported types of models” below for more
details on supported model types). What is special about the projection
predictive variable selection is that it not only performs a variable
selection, but also allows for valid post-selection inference.

The projection predictive variable selection is based on the ideas of
Goutis and Robert (1998) and
Dupuis and Robert (2003).
The methods implemented in **projpred** are described in
detail in Piironen, Paasiniemi, and Vehtari (2020) and Catalina, Bürkner, and Vehtari (2022). They are evaluated in comparison
to many other methods in Piironen and Vehtari (2017a). For details on how to cite
**projpred**, see the projpred
citation info on CRAN.^{1}

For this vignette, we use **projpred**’s
`df_gaussian`

data. It contains 100 observations of 20
continuous predictor variables `X1`

, …, `X20`

(originally stored in a sub-matrix; we turn them into separate columns
below) and one continuous response variable `y`

.

```
data("df_gaussian", package = "projpred")
<- data.frame(y = df_gaussian$y, df_gaussian$x) dat_gauss
```

First, we have to construct a reference model for the projection predictive variable selection. This model is considered as the best (“reference”) solution to the prediction task. The aim of the projection predictive variable selection is to find a subset of a set of candidate predictors which is as small as possible but achieves a predictive performance as close as possible to that of the reference model.

Usually (and this is also the case in this vignette), the reference
model will be an **rstanarm** or **brms**
fit. To our knowledge, **rstanarm** and
**brms** are currently the only packages for which a
`get_refmodel()`

method (which establishes the compatibility
with **projpred**) exists. Creating a reference model
object via one of these `get_refmodel.stanreg()`

or
`brms::get_refmodel.brmsfit()`

methods (either implicitly by
a call to a top-level function such as `project()`

,
`varsel()`

, and `cv_varsel()`

, as done below, or
explicitly by a call to `get_refmodel()`

) leads to a
“typical” reference model object. In that case, all candidate models are
actual *sub*models of the reference model. In general, however,
this assumption is not necessary for a projection predictive variable
selection (see, e.g., Piironen,
Paasiniemi, and Vehtari 2020). This is why “custom” (i.e.,
non-“typical”) reference model objects allow to avoid this assumption
(although the candidate models of a “custom” reference model object will
still be actual *sub*models of the full `formula`

used
by the search procedure—which does not have to be the same as the
reference model’s `formula`

, if the reference model possesses
a `formula`

at all). Such “custom” reference model objects
can be constructed via `init_refmodel()`

(or
`get_refmodel.default()`

), as shown in section “Examples” of
the `?init_refmodel`

help.^{2}

Here, we use the **rstanarm** package to fit the
reference model. If you want to use the **brms** package,
simply replace the **rstanarm** fit (of class
`stanreg`

) in all the code below by your
**brms** fit (of class `brmsfit`

). Only note
that in case of a **brms** fit, we recommend to specify
argument `brms_seed`

of
`brms::get_refmodel.brmsfit()`

.

`library(rstanarm)`

For our **rstanarm** reference model, we use the
Gaussian distribution as the `family`

for our response. With
respect to the predictors, we only include the linear main effects of
all 20 predictor variables. Compared to the more complex types of
reference models supported by **projpred** (see section “Supported types of models” below), this is a quite
simple reference model which is sufficient, however, to demonstrate the
interplay of **projpred**’s functions.

We use **rstanarm**’s default priors in our reference
model, except for the regression coefficients for which we use a
regularized horseshoe prior (Piironen and
Vehtari 2017c) with the hyperprior for its global shrinkage
parameter following Piironen and Vehtari (2017b) and Piironen and Vehtari (2017c). In R code, these are the
preparation steps for the regularized horseshoe prior:

```
# Number of regression coefficients:
<- sum(grepl("^X", names(dat_gauss))) ) ( D
```

`[1] 20`

```
# Prior guess for the number of relevant (i.e., non-zero) regression
# coefficients:
<- 5
p0 # Number of observations:
<- nrow(dat_gauss)
N # Hyperprior scale for tau, the global shrinkage parameter (note that for the
# Gaussian family, 'rstanarm' will automatically scale this by the residual
# standard deviation):
<- p0 / (D - p0) * 1 / sqrt(N) tau0
```

We now fit the reference model to the data. To make this vignette
build faster, we use only 2 MCMC chains and 500 iterations per chain
(with half of them being discarded as warmup draws). In practice, 4
chains and 2000 iterations per chain are reasonable defaults.
Furthermore, we make use of **rstan**’s parallelization,
which means to run each chain on a separate CPU core.^{3} If you run the
following code yourself, you can either rely on an automatic mechanism
to detect the number of CPU cores (like the
`parallel::detectCores()`

function shown below) or adapt
`ncores`

manually to your system.

```
# Set this manually if desired:
<- parallel::detectCores(logical = FALSE)
ncores ### Only for technical reasons in this vignette (you can omit this when running
### the code yourself):
<- min(ncores, 2L)
ncores ###
options(mc.cores = ncores)
<- stan_glm(
refm_fit ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10 + X11 + X12 + X13 + X14 +
y + X16 + X17 + X18 + X19 + X20,
X15 family = gaussian(),
data = dat_gauss,
prior = hs(global_scale = tau0),
### Only for the sake of speed (not recommended in general):
chains = 2, iter = 500,
###
seed = 2052109, QR = TRUE, refresh = 0
)
```

Usually, we would now have to check the convergence diagnostics (see,
e.g., `?posterior::diagnostics`

and
`?posterior::default_convergence_measures`

). However, due to
the technical reasons for which we reduced `chains`

and
`iter`

, we skip this step here.

Now, **projpred** comes into play.

`library(projpred)`

In **projpred**, the projection predictive variable
selection consists of a *search* part and an *evaluation*
part. The search part determines the solution path, i.e., the best
submodel for each submodel size (number of predictor terms). The
evaluation part determines the predictive performance of the submodels
along the solution path.

There are two functions for performing the variable selection:
`varsel()`

and `cv_varsel()`

. In contrast to
`varsel()`

, `cv_varsel()`

performs a
cross-validation (CV) by running the search part with the training data
of each CV fold separately (an exception is
`validate_search = FALSE`

, see `?cv_varsel`

and
below) and running the evaluation part on the corresponding test set of
each CV fold. Because of this CV, `cv_varsel()`

is
recommended over `varsel()`

. Thus, we use
`cv_varsel()`

here. Nonetheless, running
`varsel()`

first can offer a rough idea of the performance of
the submodels (after projecting the reference model onto them). A more
principled **projpred** workflow is work under
progress.

Here, we use only some of the available arguments; see the
documentation of `cv_varsel()`

for the full list of
arguments. By default, `cv_varsel()`

runs a leave-one-out
(LOO) CV (see argument `cv_method`

) which also
cross-validates the search (see argument `validate_search`

).
Here, we set argument `validate_search`

to `FALSE`

to obtain rough preliminary results and make this vignette build faster.
If possible (in terms of computation time), we recommend using the
default of `validate_search = TRUE`

to avoid overfitting in
the selection of the submodel size. Here, we also set
`nclusters_pred`

to a low value of `20`

only to
speed up the building of the vignette. By modifying argument
`nterms_max`

, we impose a limit on the submodel size until
which the search is continued. Typically, one has to run the variable
selection with a large `nterms_max`

first (the default value
may not even be large enough) and only after inspecting the results from
this first run, one is able to set a reasonable `nterms_max`

in subsequent runs. The value we are using here (`9`

) is
based on such a first run (which is not shown here, though).

```
<- cv_varsel(
cvvs
refm_fit,### Only for the sake of speed (not recommended in general):
validate_search = FALSE,
nclusters_pred = 20,
###
nterms_max = 9,
seed = 411183
)
```

The first step after running the variable selection should be the
decision for a final submodel size. This should be the first step (in
particular, before inspecting the solution path) in order to avoid a
user-induced selection bias (which could occur if the user made the
submodel size decision dependent on the solution path). To decide for a
submodel size, there are several performance statistics we can plot as a
function of the submodel size. Here, we use the expected log (pointwise)
predictive density (for a new dataset) (ELPD; empirically, this is the
sum of the log predictive densities of the observations in the
evaluation—or “test”—set) and the root mean squared error (RMSE). By
default, the performance statistics are plotted on their original scale,
but with `deltas = TRUE`

, they are calculated as differences
from a baseline model (which is the reference model by default, at least
in the most common cases). Since the differences are usually of more
interest (with regard to the submodel size decision), we directly plot
with `deltas = TRUE`

here (note that as
`validate_search = FALSE`

, this result is slightly
optimistic, and the plot looks different when
`validate_search = TRUE`

is used):

`plot(cvvs, stats = c("elpd", "rmse"), deltas = TRUE, seed = 54548)`

Based on that plot (see `?plot.vsel`

for a description),
we would decide for a submodel size of 6 because that’s the point where
the performance measures level off and are close enough to the reference
model’s performance (note that since the plot is affected by
`validate_search = FALSE`

, this manual decision based on the
plot is affected, too):

`<- 6 modsize_decided `

Note that **projpred** offers the
`suggest_size()`

function which may help in the decision for
a submodel size, but this is a rather heuristic method and needs to be
interpreted with caution (see `?suggest_size`

):

`suggest_size(cvvs)`

`[1] 6`

Here, we would get the same final submodel size (`6`

) as
by our manual decision (`suggest_size()`

is also affected by
`validate_search = FALSE`

). Note that by default,
`suggest_size()`

uses the ELPD as performance statistic.

Only now, after we have made a decision for the submodel size, we
inspect further results from the variable selection and, in particular,
the solution path. For example, we can simply `print()`

the
resulting object:

` cvvs`

```
Family: gaussian
Link function: identity
Formula: y ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10 + X11 +
X12 + X13 + X14 + X15 + X16 + X17 + X18 + X19 + X20
Observations: 100
CV method: LOO search not included
Search method: l1, maximum number of terms 9
Number of clusters used for selection: 1
Number of clusters used for prediction: 20
Suggested Projection Size: 6
Selection Summary:
size solution_terms elpd.loo se diff diff.se
0 <NA> -249.2 5.6 -102.1 7.7
1 X1 -230.6 6.1 -83.5 8.2
2 X14 -205.0 6.2 -58.0 7.5
3 X5 -196.5 7.2 -49.4 7.6
4 X20 -177.7 5.9 -30.6 6.3
5 X6 -168.8 5.0 -21.7 6.0
6 X3 -146.6 5.1 0.5 4.6
7 X8 -138.2 4.5 8.9 3.3
8 X11 -135.4 4.5 11.7 2.3
9 X10 -135.0 4.4 12.1 2.2
```

```
### Alternative modifying the number of printed decimal places:
# print(cvvs, digits = 2)
###
```

The solution path can be seen in the `print()`

output
(column `solution_terms`

), but it is also accessible through
the `solution_terms()`

function:

`<- solution_terms(cvvs) ) ( soltrms `

`[1] "X1" "X14" "X5" "X20" "X6" "X3" "X8" "X11" "X10"`

Combining the decided submodel size of 6 with the solution path leads to the following terms (as well as the intercept) as the predictor terms of the final submodel:

`<- head(soltrms, modsize_decided) ) ( soltrms_final `

`[1] "X1" "X14" "X5" "X20" "X6" "X3" `

The `project()`

function returns an object of class
`projection`

which forms the basis for convenient
post-selection inference. By the following code, `project()`

will project the reference model onto the final submodel once again^{4}:

`<- project(refm_fit, solution_terms = soltrms_final) prj `

For more accurate results, we could have increased argument
`ndraws`

of `project()`

(up to the number of
posterior draws in the reference model). This increases the runtime,
which we don’t want in this vignette.

Next, we create a matrix containing the projected posterior draws
stored in the depths of `project()`

’s output:

`<- as.matrix(prj) prj_mat `

This matrix is all we need for post-selection inference. It can be used like any matrix of draws from MCMC procedures, except that it doesn’t reflect a typical posterior distribution, but rather a projected posterior distribution, i.e., the distribution arising from the deterministic projection of the reference model’s posterior distribution onto the parameter space of the final submodel.

The **posterior** package provides a general way to deal
with posterior distributions, so it can also be applied to our projected
posterior. For example, to calculate summary statistics for the
marginals of the projected posterior:

```
library(posterior)
<- as_draws_matrix(prj_mat)
prj_drws # In the following call, as.data.frame() is used only because pkgdown
# versions > 1.6.1 don't print the tibble correctly.
as.data.frame(summarize_draws(
prj_drws,"median", "mad", function(x) quantile(x, probs = c(0.025, 0.975))
))
```

```
variable median mad 2.5% 97.5%
1 (Intercept) 0.0458985 0.10183583 -0.1505310 0.2205816
2 X1 1.3834312 0.09130185 1.1956560 1.5532896
3 X14 -1.1157218 0.09405313 -1.2834051 -0.9318850
4 X5 -0.9136915 0.11058452 -1.1096153 -0.7263205
5 X20 -1.1185014 0.10942521 -1.3460276 -0.9092710
6 X6 0.5395952 0.09545216 0.3489350 0.7149177
7 X3 0.7767160 0.09659702 0.5948437 0.9772419
8 sigma 1.1304147 0.08527177 0.9919125 1.3107667
```

A visualization of the projected posterior can be achieved with the
**bayesplot** package, for example using its
`mcmc_intervals()`

function:

```
library(bayesplot)
bayesplot_theme_set(ggplot2::theme_bw())
mcmc_intervals(prj_mat) +
::coord_cartesian(xlim = c(-1.5, 1.6)) ggplot2
```

Note that we only visualize the *1-dimensional* marginals of
the projected posterior here. To gain a more complete picture, we would
have to visualize at least some *2-dimensional* marginals of the
projected posterior (i.e., marginals for pairs of parameters).

For comparison, consider the marginal posteriors of the corresponding parameters in the reference model:

```
<- as.matrix(refm_fit)
refm_mat mcmc_intervals(refm_mat, pars = colnames(prj_mat)) +
::coord_cartesian(xlim = c(-1.5, 1.6)) ggplot2
```

Here, the reference model’s marginal posteriors differ only slightly from the marginals of the projected posterior. This does not necessarily have to be the case.

Predictions from the final submodel can be made by
`proj_linpred()`

and `proj_predict()`

.

We start with `proj_linpred()`

. For example, suppose we
have the following new observations:

```
<- setNames(
( dat_gauss_new as.data.frame(replicate(length(soltrms_final), c(-1, 0, 1))),
soltrms_final ) )
```

```
X1 X14 X5 X20 X6 X3
1 -1 -1 -1 -1 -1 -1
2 0 0 0 0 0 0
3 1 1 1 1 1 1
```

Then `proj_linpred()`

can calculate the linear
predictors^{5} for all new observations from
`dat_gauss_new`

. Depending on argument
`integrated`

, these linear predictors can be averaged across
the projected draws (within each new observation). For instance, the
following computes the expected values of the new observations’
predictive distributions:^{6}

```
<- proj_linpred(prj, newdata = dat_gauss_new, integrated = TRUE)
prj_linpred cbind(dat_gauss_new, linpred = as.vector(prj_linpred$pred))
```

```
X1 X14 X5 X20 X6 X3 linpred
1 -1 -1 -1 -1 -1 -1 0.50053930
2 0 0 0 0 0 0 0.04160798
3 1 1 1 1 1 1 -0.41732334
```

If `dat_gauss_new`

also contained response values (i.e.,
`y`

values in this example), then `proj_linpred()`

would also evaluate the log predictive density at these.

With `proj_predict()`

, we can obtain draws from predictive
distributions based on the final submodel. In contrast to
`proj_linpred(<...>, integrated = FALSE)`

, this
encompasses not only the uncertainty arising from parameter estimation,
but also the uncertainty arising from the observational (or “sampling”)
model for the response.^{7} This is useful for what is usually termed a
posterior predictive check (PPC), but would have to be termed something
like a posterior-projection predictive check (PPPC) here:

```
<- proj_predict(prj, .seed = 762805)
prj_predict # Using the 'bayesplot' package:
ppc_dens_overlay(y = dat_gauss$y, yrep = prj_predict, alpha = 0.9, bw = "SJ")
```

This PPPC shows that our final projection is able to generate predictions similar to the observed response values, which indicates that this model is reasonable, at least in this regard.

In principle, the projection predictive variable selection requires only little information about the form of the reference model. Although many aspects of the reference model coincide with those from the submodels if a “typical” reference model object is used, this does not need to be the case if a “custom” reference model object is used (see section “Reference model” above for the definition of “typical” and “custom” reference model objects). This explains why in general, the following remarks refer to the submodels and not to the reference model.

Apart from the `gaussian()`

response family used in this
vignette, **projpred** also supports the
`binomial()`

^{8} and the `poisson()`

family. On
the side of the predictors, **projpred** not only supports
linear main effects as shown in this vignette, but also interactions,
multilevel^{9}, and—as an experimental feature—additive^{10}
terms.

Transferring this vignette (which employs a “typical” reference
model) to such more complex problems is straightforward: Basically, only
the code for fitting the reference model via **rstanarm**
or **brms** needs to be adapted. The
**projpred** code stays almost the same. Only note that in
case of multilevel or additive reference models,
some **projpred** functions then have slightly different
options for a few arguments. See the documentation for details.

For example, to apply **projpred** to the
`VerbAgg`

dataset from the **lme4** package, a
corresponding multilevel reference model for the binary response
`r2`

could be created by the following code:

```
data("VerbAgg", package = "lme4")
<- stan_glmer(
refm_fit ~ btype + situ + mode + (btype + situ + mode | id),
r2 family = binomial(),
data = VerbAgg,
seed = 82616169, QR = TRUE, refresh = 0
)
```

As an example for an additive (non-multilevel) reference model,
consider the `lasrosas.corn`

dataset from the
**agridat** package. A corresponding reference model for
the continuous response `yield`

could be created by the
following code (note that `pp_check(refm_fit)`

gives a bad
PPC in this case, so there’s still room for improvement):

```
data("lasrosas.corn", package = "agridat")
# Convert `year` to a `factor` (this could also be solved by using
# `factor(year)` in the formula, but we avoid that here to put more emphasis on
# the demonstration of the smooth term):
$year <- as.factor(lasrosas.corn$year)
lasrosas.corn<- stan_gamm4(
refm_fit ~ year + topo + t2(nitro, bv),
yield family = gaussian(),
data = lasrosas.corn,
seed = 4919670, QR = TRUE, refresh = 0
)
```

As an example for an additive multilevel reference model, consider
the `gumpertz.pepper`

dataset from the
**agridat** package. A corresponding reference model for
the binary response `disease`

could be created by the
following code:

```
data("gumpertz.pepper", package = "agridat")
<- stan_gamm4(
refm_fit ~ field + leaf + s(water),
disease random = ~ (1 | row) + (1 | quadrat),
family = binomial(),
data = gumpertz.pepper,
seed = 14209013, QR = TRUE, refresh = 0
)
```

Sometimes, the ordering of the predictor terms in the solution path makes sense, but for increasing submodel size, the performance measures of the submodels do not approach that of the reference model. There are different reasons that can explain this behavior (the following list might not be exhaustive, though):

- The reference model’s posterior may be so wide that the default
`ndraws_pred`

could be too small. Usually, this comes in combination with a difference in predictive performance which is comparatively small. Increasing`ndraws_pred`

should help, but it also increases the computational cost. Re-fitting the reference model and thereby ensuring a narrower posterior (usually by employing a stronger sparsifying prior) should have a similar effect. - For non-Gaussian models, the discrepancy may be due to the fact that the penalized iteratively reweighted least squares (PIRLS) algorithm might have convergence issues (Catalina, Bürkner, and Vehtari 2021). In this case, the latent-space approach by Catalina, Bürkner, and Vehtari (2021) might help.
- If you are using
`varsel()`

, then the lack of the CV in`varsel()`

may lead to overconfident and overfitted results. In this case, try running`cv_varsel()`

instead of`varsel()`

(which you should in any case for your final results).

Catalina, Alejandro, Paul-Christian Bürkner, and Aki Vehtari. 2022.
“Projection Predictive Inference for Generalized Linear and
Additive Multilevel Models.” In *Proceedings of
The 25th International Conference on
Artificial Intelligence and Statistics*,
edited by Gustau Camps-Valls, Francisco J. R. Ruiz, and Isabel Valera,
151:4446–61. Proceedings of Machine Learning Research.
PMLR. https://proceedings.mlr.press/v151/catalina22a.html.

Catalina, Alejandro, Paul Bürkner, and Aki Vehtari. 2021. “Latent
Space Projection Predictive Inference.” arXiv. https://doi.org/10.48550/arXiv.2109.04702.

Dupuis, Jérome A., and Christian P. Robert. 2003. “Variable
Selection in Qualitative Models via an Entropic Explanatory
Power.” *Journal of Statistical Planning and Inference*
111 (1–2): 77–94. https://doi.org/10.1016/S0378-3758(02)00286-0.

Goutis, Constantinos, and Christian P. Robert. 1998. “Model Choice
in Generalised Linear Models: A Bayesian Approach via
Kullback-Leibler Projections.” *Biometrika*
85 (1): 29–37.

Piironen, Juho, Markus Paasiniemi, and Aki Vehtari. 2020.
“Projective Inference in High-Dimensional Problems:
Prediction and Feature Selection.” *Electronic
Journal of Statistics* 14 (1): 2155–97. https://doi.org/10.1214/20-EJS1711.

Piironen, Juho, and Aki Vehtari. 2017a. “Comparison of
Bayesian Predictive Methods for Model Selection.”
*Statistics and Computing* 27 (3): 711–35. https://doi.org/10.1007/s11222-016-9649-y.

———. 2017b. “On the Hyperprior Choice for the Global Shrinkage
Parameter in the Horseshoe Prior.” In *Proceedings of the 20th
International Conference on Artificial
Intelligence and Statistics*, edited by Aarti
Singh and Jerry Zhu, 54:905–13. Proceedings of Machine Learning
Research. PMLR. https://proceedings.mlr.press/v54/piironen17a.html.

———. 2017c. “Sparsity Information and Regularization in the
Horseshoe and Other Shrinkage Priors.” *Electronic Journal of
Statistics* 11 (2): 5018–51. https://doi.org/10.1214/17-EJS1337SI.

The citation information can be accessed offline by typing

`print(citation("projpred"), bibtex = TRUE)`

within R.↩︎We will cover custom reference models more deeply in a future vignette.↩︎

More generally, the number of chains is split up as evenly as possible among the number of CPU cores.↩︎

During the forward search, the reference model has already been projected onto all candidate models (this was where arguments

`ndraws`

and`nclusters`

of`cv_varsel()`

came into play). During the evaluation of the submodels along the solution path, the reference model has already been projected onto those submodels (this was where arguments`ndraws_pred`

and`nclusters_pred`

of`cv_varsel()`

came into play). In principle, one could use the results from the evaluation part for post-selection inference, but due to a bug in the current implementation (see GitHub issue #168), we currently have to project once again.↩︎`proj_linpred()`

can also transform the linear predictor to response scale, but here, this is the same as the linear predictor scale (because of the identity link function).↩︎Beware that this statement is correct here because of the Gaussian family with the identity link function. For other families (which usually come in combination with a different link function), one would typically have to use

`transform = TRUE`

in order to make this statement correct.↩︎In case of the Gaussian family we are using here, the uncertainty arising from the observational model is the uncertainty due to the residual standard deviation.↩︎

Via

`brms::get_refmodel.brmsfit()`

, the`brms::bernoulli()`

family is supported as well.↩︎Multilevel models are also known as

*hierarchical*models or models with*partially pooled*,*group-level*, or—in frequentist terms—*random*effects.↩︎Additive terms are also known as

*smooth*terms.↩︎