Reference model
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 methods get_refmodel.stanreg() or
brms::get_refmodel.brmsfit() (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 submodels 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
submodels 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.
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).
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:
( D <- sum(grepl("^X", names(dat_gauss))) )
[1] 20
# Prior guess for the number of relevant (i.e., non-zero) regression
# coefficients:
p0 <- 5
# Number of observations:
N <- nrow(dat_gauss)
# Hyperprior scale for tau, the global shrinkage parameter (note that for the
# Gaussian family, 'rstanarm' will automatically scale this by the residual
# standard deviation):
tau0 <- p0 / (D - p0) * 1 / sqrt(N)
We now fit the reference model to the data. To make this vignette
build faster, we use only 2 MCMC chains and 1000 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. 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:
ncores <- parallel::detectCores(logical = FALSE)
### Only for technical reasons in this vignette (you can omit this when running
### the code yourself):
ncores <- min(ncores, 2L)
###
options(mc.cores = ncores)
set.seed(5078022)
refm_fit <- stan_glm(
y ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10 + X11 + X12 + X13 + X14 +
X15 + X16 + X17 + X18 + X19 + X20,
family = gaussian(),
data = dat_gauss,
prior = hs(global_scale = tau0),
### Only for the sake of speed (not recommended in general):
chains = 2, iter = 1000,
###
QR = TRUE, refresh = 0
)
Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
https://mc-stan.org/misc/warnings.html#tail-ess
Usually, we would now have to check the convergence diagnostics (see,
e.g., ?posterior::diagnostics and
?posterior::default_convergence_measures; the tail-ESS
warning already indicates a problem). However, due to the technical
reasons for which we reduced chains and iter,
we skip this step here (and hence ignore the tail-ESS warning).
Variable selection
Now, projpred comes into play.
From the reference model fit (called refm_fit here), we
create a reference model object (i.e., an object of class
refmodel) since this avoids redundant calculations in the
remainder of this vignette:
refm_obj <- get_refmodel(refm_fit)
In projpred, the projection predictive variable
selection relies on a so-called search part and a so-called
evaluation part. The search part determines the predictor
ranking (also known as solution path), i.e., the best submodel for each
submodel size (the size is given by the number of predictor terms). The
evaluation part determines the predictive performance of the
increasingly complex submodels along the predictor ranking.
There are two functions for running the combination of search and
evaluation: varsel() and cv_varsel(). In
contrast to varsel(), cv_varsel() performs a
cross-validation (CV). With cv_method = "LOO" (the
default), cv_varsel() runs a Pareto-smoothed importance
sampling leave-one-out CV (PSIS-LOO CV, see Vehtari, Gelman, and Gabry 2017;
Vehtari et al. 2022). With
cv_method = "kfold", cv_varsel() runs a \(K\)-fold CV. The extent of the CV depends
on cv_varsel()’s argument validate_search: If
validate_search = TRUE (the default), the search part is
run with the training data of each CV fold separately and the evaluation
part is run with the corresponding test data of each CV fold. If
validate_search = FALSE, the search is excluded from the CV
so that only a single full-data search is run. Because of its most
thorough protection against overfitting,
cv_varsel() with validate_search = TRUE is
recommended over varsel() and cv_varsel() with
validate_search = FALSE. Nonetheless, a preliminary and
comparatively fast run of varsel() or
cv_varsel() with validate_search = FALSE can
give a rough idea of the performance of the submodels and can be used
for finding a suitable value for argument nterms_max in
subsequent runs (argument nterms_max imposes a limit on the
submodel size up to which the search is continued and is thus able to
reduce the runtime considerably).
Preliminary cv_varsel() run
To illustrate a preliminary cv_varsel() run with
validate_search = FALSE, we set nterms_max to
the number of predictor terms in the full model, i.e.,
nterms_max = 20. To speed up the building of the vignette
(this is not recommended in general), we choose the "L1"
search method
and set refit_prj to FALSE.
# Preliminary cv_varsel() run:
cvvs_fast <- cv_varsel(
refm_obj,
validate_search = FALSE,
### Only for the sake of speed (not recommended in general):
method = "L1",
refit_prj = FALSE,
###
nterms_max = 20,
### In interactive use, we recommend not to deactivate the verbose mode:
verbose = FALSE
###
)
Warning: Some Pareto k diagnostic values are slightly high. See help('pareto-k-diagnostic') for details.
Warning in warn_pareto(n07 = sum(pareto_k > 0.7), n05 = sum(0.7 >= pareto_k & :
In the calculation of the reference model's PSIS-LOO CV weights, 11 (out of
100) Pareto k-values are in the interval (0.5, 0.7]. Moment matching (see the
loo package), mixture importance sampling (see the loo package), and
`reloo`-ing (see the brms package) are not supported by projpred. If these
techniques (run outside of projpred, i.e., for the reference model only; note
that `reloo`-ing may be computationally costly) result in a markedly different
reference model ELPD estimate than ordinary PSIS-LOO CV does, we recommend to
use K-fold CV within projpred.
In this case, we ignore the Pareto \(\hat{k}\) warnings due to the reduced
values for chains and iter in the reference
model fit.
To find a suitable value for nterms_max in subsequent
cv_varsel() runs, we take a look at a plot of at least one
predictive performance statistic in dependence of the submodel size.
Here, we choose the mean log predictive density (MLPD; see the
documentation for argument stats of
summary.vsel() for details) as the only performance
statistic. Since we will be using the following plot only to determine
nterms_max for subsequent cv_varsel() runs, we
can omit the predictor ranking from the plot by setting
ranking_nterms_max to NA:
plot(cvvs_fast, stats = "mlpd", ranking_nterms_max = NA)
This plot suggests that the submodel MLPD levels off for the first time
from submodel size 9 to 10 and that it increases only slightly
afterwards. However, we used L1 search with
refit_prj = FALSE, which means that the projections
employed for the predictive performance evaluation are L1-penalized,
which is usually undesired (Piironen, Paasiniemi, and Vehtari
2020, sec. 4). Thus, to investigate the impact of
refit_prj = FALSE, we re-run cv_varsel(), but
this time with the default of refit_prj = TRUE and re-using
the search results (as well as the CV-related arguments such as
validate_search; see section “Usage” in
?cv_varsel.vsel) from the former cv_varsel()
call (this is done by applying cv_varsel() to
cvvs_fast instead of refm_obj so that the
cv_varsel() generic dispatches to the
cv_varsel.vsel() method that was introduced in
projpred 2.8.0). To save time, we also set
nclusters_pred to a comparatively low value of
20:
# Preliminary cv_varsel() run with `refit_prj = TRUE`:
cvvs_fast_refit <- cv_varsel(
cvvs_fast,
### Only for the sake of speed (not recommended in general):
nclusters_pred = 20,
###
### In interactive use, we recommend not to deactivate the verbose mode:
verbose = FALSE
###
)
Warning: Some Pareto k diagnostic values are slightly high. See help('pareto-k-diagnostic') for details.
Warning in warn_pareto(n07 = sum(pareto_k > 0.7), n05 = sum(0.7 >= pareto_k & :
In the calculation of the reference model's PSIS-LOO CV weights, 11 (out of
100) Pareto k-values are in the interval (0.5, 0.7]. Moment matching (see the
loo package), mixture importance sampling (see the loo package), and
`reloo`-ing (see the brms package) are not supported by projpred. If these
techniques (run outside of projpred, i.e., for the reference model only; note
that `reloo`-ing may be computationally costly) result in a markedly different
reference model ELPD estimate than ordinary PSIS-LOO CV does, we recommend to
use K-fold CV within projpred.
Warning in loo_varsel(refmodel = refmodel, method = method, nterms_max =
nterms_max, : The projected draws used for the performance evaluation have
different (i.e., nonconstant) weights, so using standard importance sampling
(SIS) instead of Pareto-smoothed importance sampling (PSIS). In general, PSIS
is recommended over SIS.
As explained above, we ignore the Pareto \(\hat{k}\) warnings
here. We
also ignore the warning that SIS is used instead of PSIS (this is due to
nclusters_pred = 20 which we used only to speed up the
building of the vignette).
With the refit_prj = TRUE results, the predictive
performance plot from above now looks as follows:
plot(cvvs_fast_refit, stats = "mlpd", ranking_nterms_max = NA)
This refined plot shows that the submodel MLPD does not change much
after submodel size 8, so in our final cv_varsel() run, we
set nterms_max to a value slightly higher than 8 (here: 9)
to ensure that we see the MLPD leveling off. The search results from the
initial cvvs_fast object could now be re-used again (via
cv_varsel.vsel()) for investigating the sensitivity of the
results to changes in nclusters_pred (or
ndraws_pred). Here, we skip this for the sake of brevity
and instead head over to the final cv_varsel() run.
Final cv_varsel() run
For this final cv_varsel() run (with
validate_search = TRUE, as recommended), we use a \(K\)-fold CV with a small number of folds
(K = 2) to make this vignette build faster. In practice, we
recommend using either the default of cv_method = "LOO" or
a larger value for K if this is possible in terms of
computation time.
Here, we also perform the \(K\)
reference model refits outside of cv_varsel(). Although not
strictly necessary here, this is helpful in practice because often,
cv_varsel() needs to be re-run multiple times in order to
try out different argument settings. We also illustrate how
projpred’s CV (i.e., the CV comprising search and
performance evaluation, after refitting the reference model \(K\) times) can be parallelized, even though
this is of little use here (we have only K = 2 folds and
the fold-wise searches and performance evaluations are quite fast, so
the parallelization overhead eats up any runtime improvements). Note
that for this final cv_varsel() run, we cannot make use of
cv_varsel.vsel() (by applying cv_varsel() to
cvvs_fast or cvvs_fast_refit instead of
refm_obj) because of the change in nterms_max
(here: 9, for cvvs_fast and cvvs_fast_refit:
20).
# Refit the reference model K times:
cv_fits <- run_cvfun(
refm_obj,
### Only for the sake of speed (not recommended in general):
K = 2
###
)
# For running projpred's CV in parallel (see cv_varsel()'s argument `parallel`):
doParallel::registerDoParallel(ncores)
# Final cv_varsel() run:
cvvs <- cv_varsel(
refm_obj,
cv_method = "kfold",
cvfits = cv_fits,
### Only for the sake of speed (not recommended in general):
method = "L1",
nclusters_pred = 20,
###
nterms_max = 9,
parallel = TRUE,
### In interactive use, we recommend not to deactivate the verbose mode:
verbose = FALSE
###
)
# Tear down the CV parallelization setup:
doParallel::stopImplicitCluster()
foreach::registerDoSEQ()
We can now select a final submodel size by looking at a predictive
performance plot similar to the one created for the preliminary
cv_varsel() run above. By default, the performance
statistics are plotted on their original scale, but with
deltas = TRUE, they are plotted as differences from a
baseline model (which is the reference model by default, at least in the
most common cases). Since the differences and the (frequentist)
uncertainty in their estimation are usually of more interest than the
original-scale performance statistics (at least with regard to the
decision for a final submodel size), we directly plot with
deltas = TRUE here:
plot(cvvs, stats = "mlpd", deltas = TRUE)
Decision for final submodel size
Based on that final predictive performance plot, we decide for a
submodel size. Usually, the aim is to find the smallest submodel size
where the predictive performance of the submodels levels off and is
close enough to the reference model’s predictive performance (the dashed
red horizontal line). Sometimes (as here), the plot may be ambiguous
because after reaching the reference model’s performance, the submodels’
performance may keep increasing (and hence become even better than the
reference model’s performance). In that case, one has to find a suitable
trade-off between predictive performance (accuracy) and model size
(sparsity) in the context of subject-matter knowledge. Here, we assume
that the focus of our predictive model is sparsity (not accuracy).
Hence, based on the plot, we decide for a submodel size of 6 because
this is the smallest size where the submodel MLPD is close enough to the
reference model MLPD:
If the focus of our predictive model had been accuracy (not
sparsity), size 7 (or even 8) would have been a natural choice because
at size 7 (at the latest at size 8), the submodel MLPD can be considered
to “level off”. Further down, the predictor ranking and the (CV) ranking
proportions that are shown in the plot (below the submodel sizes on the
x-axis) are explained in detail—and also how they could have been
incorporated into our decision for a submodel size.
The suggest_size() function offered by
projpred 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, stat = "mlpd")
[1] 6
With this heuristic, we would get the same final submodel size
(6) as by our manual (sparsity-based) decision.
summary() method for vsel objects
A tabular representation of the plot created by
plot.vsel() can be achieved via
summary.vsel(). For the output of
summary.vsel(), there is a sophisticated
print() method (print.vselsummary()) which is
also called by the shortcut method print.vsel().
Specifically, to create the summary table matching the predictive
performance plot above as closely as possible (and to also adjust the
minimum number of printed significant digits), we may call
summary.vsel() and print.vselsummary() as
follows:
smmry <- summary(cvvs, stats = "mlpd", type = c("mean", "lower", "upper"),
deltas = TRUE)
print(smmry, digits = 1)
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
Projection method: traditional
CV method: K-fold CV with K = 2 and search included (i.e., fold-wise searches)
Search method: L1
Maximum submodel size for the search: 9
Number of projected draws in the search: 1 (from clustered projection)
Number of projected draws in the performance evaluation: 20 (from clustered projection)
Argument `refit_prj`: TRUE
Submodel performance evaluation summary with `deltas = TRUE` and `cumulate = FALSE`:
size ranking_fulldata cv_proportions_diag mlpd mlpd.lower mlpd.upper
0 (Intercept) NA -0.90 -0.980 -0.82
1 X1 1.0 -0.70 -0.787 -0.62
2 X14 0.5 -0.59 -0.669 -0.50
3 X5 0.5 -0.43 -0.521 -0.35
4 X20 0.5 -0.34 -0.418 -0.26
5 X6 0.0 -0.11 -0.175 -0.05
6 X3 0.5 0.05 0.003 0.11
7 X8 1.0 0.13 0.082 0.18
8 X11 0.5 0.14 0.094 0.18
9 X10 0.0 0.12 0.086 0.16
Reference model performance evaluation summary with `deltas = TRUE`:
mlpd mlpd.lower mlpd.upper
0 0 0
Predictor ranking(s) and identification of the selected
submodel
As indicated by its column name, the predictor ranking from column
ranking_fulldata of the summary.vsel() output
is based on the full-data search. In case of cv_varsel()
with validate_search = TRUE, there is not only the
full-data search, but also fold-wise searches, implying that there are
also fold-wise predictor rankings. All of these predictor rankings (the
full-data one and—if available—the fold-wise ones) can be retrieved via
ranking():
In addition to inspecting the full-data predictor ranking, it usually
makes sense to investigate the ranking proportions derived from the
fold-wise predictor rankings (only available in case of
cv_varsel() with validate_search = TRUE, which
we have here) in order to get a sense for the variability in the ranking
of the predictors. For a given predictor \(x\) and a given submodel size \(j\), the ranking proportion is the
proportion of CV folds which have predictor \(x\) at position \(j\) of their predictor ranking. To compute
these ranking proportions, we use cv_proportions():
( pr_rk <- cv_proportions(rk) )
predictor
size X1 X14 X5 X20 X6 X3 X8 X11 X10
1 1 0.0 0.0 0.0 0.0 0.0 0 0.0 0
2 0 0.5 0.0 0.0 0.0 0.5 0 0.0 0
3 0 0.5 0.5 0.0 0.0 0.0 0 0.0 0
4 0 0.0 0.0 0.5 0.5 0.0 0 0.0 0
5 0 0.0 0.5 0.5 0.0 0.0 0 0.0 0
6 0 0.0 0.0 0.0 0.5 0.5 0 0.0 0
7 0 0.0 0.0 0.0 0.0 0.0 1 0.0 0
8 0 0.0 0.0 0.0 0.0 0.0 0 0.5 0
9 0 0.0 0.0 0.0 0.0 0.0 0 0.5 0
attr(,"class")
[1] "cv_proportions"
Here, the ranking proportions are of little use as we have used
K = 2 (in the final cv_varsel() call above)
for the sake of speed. Nevertheless, we can see that the two CV folds
agree on the most relevant predictor term (X1). Since the
column names of the matrix returned by cv_proportions()
follow the full-data predictor ranking, we can infer that
X1 is also the most relevant predictor term in the
full-data predictor ranking. To see this more explicitly, we can access
element fulldata of the ranking() output:
[1] "X1" "X14" "X5" "X20" "X6" "X3" "X8" "X11" "X10"
This is the same as column ranking_fulldata in the
summary.vsel() output above (apart from the intercept). (As
also stated by the message thrown by print.vselsummary(),
column cv_proportions_diag of the
summary.vsel() output contains the main diagonal of the
matrix that we stored manually as pr_rk.)
Note that we have cut off the search at nterms_max = 9
(which is smaller than the number of predictor terms in the full model,
20 here), so the ranking proportions in the pr_rk matrix do
not need to sum to 100 % (neither column-wise nor row-wise).
The transposed matrix of ranking proportions can be
visualized via plot.cv_proportions():
Apart from visualizing the variability in the ranking of the predictors
(here, this is of little use because of K = 2), this plot
will be helpful later.
To retrieve the predictor terms of the final submodel (except for the
intercept which is always included in the submodels), we combine the
chosen submodel size of 6 with the full-data predictor ranking:
( predictors_final <- head(rk[["fulldata"]], size_decided) )
[1] "X1" "X14" "X5" "X20" "X6" "X3"
At this place, it is again helpful to take the ranking proportions
into account, but now in a cumulated fashion:
plot(cv_proportions(rk, cumulate = TRUE))
This plot shows that the two fold-wise searches (as well as the
full-data search, whose predictor ranking determines the order of the
predictors on the y-axis) agree on the set of the 6 most
relevant predictors: When looking at <=6 on the x-axis,
all tiles above and including the 6th main diagonal element
are at 100 %. Similarly, the two CV folds agree on the set of
the 7 most relevant predictors (and also on the set of the most relevant
predictor, which is a singleton).
Although not demonstrated here, the cumulated ranking proportions
could also have guided the decision for a submodel size (if we had not
been willing to follow a strict rule based on accuracy or sparsity):
From their plot, we can see that size 9 might have been an unfortunate
choice because X10 (which—by cutting off the full-data
predictor ranking at size 9—would then have been selected as the
9th predictor in the final submodel) is not included among
the first 9 terms by any CV fold. However, since K = 2 is
too small for reliable statements regarding the variability of the
predictor ranking, we did not take the cumulated ranking proportions
into account when we made our decision for a submodel size above.
In a real-world application, we might also be able to incorporate the
full-data predictor ranking into our decision for a submodel size
(usually, this requires to also take into account the variability of the
predictor ranking, as reflected by the—possibly cumulated—ranking
proportions). For example, the predictors might be associated with
different measurement costs, so that we might want to select a costly
predictor only if the submodel size at which it would be selected
(according to the full-data predictor ranking, but taking into account
that there might be variability in the ranking of the predictors) comes
with a considerable increase in predictive performance.
Post-selection inference
The project() function returns an object of class
projection which forms the basis for convenient
post-selection inference. By the following project() call,
we project the reference model onto the final submodel once again:
prj <- project(
refm_obj,
predictor_terms = predictors_final,
### In interactive use, we recommend not to deactivate the verbose mode:
verbose = FALSE
###
)
Next, we create a matrix containing the projected posterior draws
stored in the depths of project()’s output:
prj_mat <- as.matrix(prj)
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. Beware that in case
of clustered projection (i.e., a non-NULL argument
nclusters in the project() call), the
projected draws have different (i.e., nonconstant) weights, which needs
to be taken into account when performing post-selection (or, more
generally, post-projection) inference, see
as_draws_matrix.projection() (proj_linpred()
and proj_predict() offer similar functionality via
arguments return_draws_matrix and
nresample_clusters, respectively).
Marginals of the projected posterior
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:
prj_drws <- as_draws_matrix(prj_mat)
prj_smmry <- summarize_draws(
prj_drws,
"median", "mad", function(x) quantile(x, probs = c(0.025, 0.975))
)
# Coerce to a `data.frame` because pkgdown versions > 1.6.1 don't print the
# tibble correctly:
prj_smmry <- as.data.frame(prj_smmry)
print(prj_smmry, digits = 1)
variable median mad 2.5% 97.5%
1 (Intercept) 0.04 0.10 -0.2 0.2
2 X1 1.38 0.09 1.2 1.6
3 X14 -1.12 0.09 -1.3 -0.9
4 X5 -0.92 0.11 -1.2 -0.7
5 X20 -1.13 0.12 -1.3 -0.9
6 X6 0.54 0.09 0.3 0.7
7 X3 0.79 0.10 0.6 1.0
8 sigma 1.13 0.09 1.0 1.3
A visualization of the projected posterior can be achieved with the
bayesplot
package, for example using its mcmc_intervals()
function:
bayesplot_theme_set(ggplot2::theme_bw())
mcmc_intervals(prj_mat) +
ggplot2::coord_cartesian(xlim = c(-1.5, 1.6))
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:
refm_mat <- as.matrix(refm_fit)
mcmc_intervals(refm_mat, pars = colnames(prj_mat)) +
ggplot2::coord_cartesian(xlim = c(-1.5, 1.6))
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
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:
( dat_gauss_new <- setNames(
as.data.frame(replicate(length(predictors_final), c(-1, 0, 1))),
predictors_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 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 (beware that the following code refers to 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 achieve such expected values):
prj_linpred <- proj_linpred(prj, newdata = dat_gauss_new, integrated = TRUE)
cbind(dat_gauss_new, linpred = as.vector(prj_linpred$pred))
X1 X14 X5 X20 X6 X3 linpred
1 -1 -1 -1 -1 -1 -1 0.48904558
2 0 0 0 0 0 0 0.03647046
3 1 1 1 1 1 1 -0.41610466
If dat_gauss_new also contained response values (i.e.,
y values in this example), then proj_linpred()
would also evaluate the log predictive density at these (conditional on
each of the projected parameter draws if integrated = FALSE
and integrated over the projected parameter draws—before taking the
logarithm—if integrated = TRUE).
With proj_predict(), we can obtain draws from predictive
distributions based on the final submodel. In contrast to
proj_linpred(<...>, integrated = FALSE), this
encompasses not only the uncertainty arising from parameter estimation,
but also the uncertainty arising from the observation (or “sampling”)
model for the response. This is useful for what is usually
termed a posterior predictive check (PPC), but would have to be termed
something like a posterior-projection predictive check (PPPC) here:
prj_predict <- proj_predict(prj)
# Using the 'bayesplot' package:
ppc_dens_overlay(y = dat_gauss$y, yrep = prj_predict)
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.
Supported types of models
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.
In the following and throughout projpred’s
documentation, the term “traditional projection” is used whenever the
projection type is neither “augmented-data” nor “latent” (see below for
a description of these).
Apart from the gaussian() response family used in this
vignette, projpred’s traditional projection also
supports the binomial() and the poisson()
family.
The families supported by projpred’s augmented-data
projection (Weber, Glass, and Vehtari
2023) are binomial() ,
brms::cumulative(), rstanarm::stan_polr()
fits, and brms::categorical() . See
?extend_family (which is called by
init_refmodel()) for an explanation how to apply the
augmented-data projection to “custom” reference model objects. For
“typical” reference model objects (i.e., those created by
get_refmodel.stanreg() or
brms::get_refmodel.brmsfit()), the augmented-data
projection is applied automatically if the family is supported by the
augmented-data projection and neither binomial() nor
brms::bernoulli(). For applying the augmented-data
projection to the binomial() (or
brms::bernoulli()) family, see ?extend_family
as well as ?augdat_link_binom and
?augdat_ilink_binom. Finally, we note that there are some
restrictions with respect to the augmented-data projection;
projpred will throw an informative error if a requested
feature is currently not supported for the augmented-data
projection.
The latent projection (Catalina, Bürkner, and Vehtari
2021) is a quite general principle for extending
projpred’s traditional projection to more response
families. The latent projection is applied when setting argument
latent of extend_family() (which is called by
init_refmodel()) to TRUE. The families for
which full latent-projection functionality (in particular,
resp_oscale = TRUE, i.e., post-processing on the original
response scale) is currently available are binomial() ,
poisson(), brms::cumulative(), and
rstanarm::stan_polr() fits. For all other
families, you can try to use the latent projection (by setting
latent = TRUE) and projpred should tell
you if any features are not available and how to make them available.
More details concerning the latent projection are given in the
corresponding latent-projection
vignette. Note that there are some restrictions with respect to the
latent projection; projpred will throw an informative
error if a requested feature is currently not supported for the latent
projection.
On the side of the predictors, projpred not only
supports linear main effects as shown in this vignette, but also
interactions, multilevel, and—as an experimental
feature—additive terms.
Transferring this vignette to such more complex problems is
straightforward (also because this vignette employs a “typical”
reference model object): 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")
refm_fit <- stan_glmer(
r2 ~ btype + situ + mode + (btype + situ + mode | id),
family = binomial(),
data = VerbAgg,
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):
lasrosas.corn$year <- as.factor(lasrosas.corn$year)
refm_fit <- stan_gamm4(
yield ~ year + topo + t2(nitro, bv),
family = gaussian(),
data = lasrosas.corn,
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")
refm_fit <- stan_gamm4(
disease ~ field + leaf + s(water),
random = ~ (1 | row) + (1 | quadrat),
family = binomial(),
data = gumpertz.pepper,
QR = TRUE, refresh = 0
)
In case of multilevel models (currently only non-additive ones),
projpred has two global options that may be relevant
for users: projpred.mlvl_pred_new and
projpred.mlvl_proj_ref_new. These are explained in detail
in the general package documentation (available online
or by typing ?`projpred-package`).
Troubleshooting
Overfitting
If you are using varsel(), then the lack of 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).
Similarly, cv_varsel() with
validate_search = FALSE is more prone to overfitting than
cv_varsel() with validate_search = TRUE.
Issues with the traditional projection
For multilevel binomial models, the traditional projection may not
work properly and give suboptimal results, see #353 on GitHub
(the underlying issue is described in lme4 issue #682). With
suboptimality of the results, we mean that in such cases, the relevance
of the group-level terms can be underestimated. According to the
simulation-based case study from #353, the
latent projection should be considered as a currently available
remedy.
For multilevel Poisson models, the traditional projection may take
very long, see #353. According
to the simulation-based case study from #353, the
latent projection should be considered as a currently available
remedy.
Finally, as illustrated in the Poisson example of the
latent-projection vignette, the latent projection can be beneficial for
non-multilevel models with a (non-Gaussian) family that is also
supported by the traditional projection, at least in case of the Poisson
family and L1 search.
Issues with the augmented-data projection
For multilevel models, the augmented-data projection seems to suffer
from the same issue as the traditional projection for the binomial
family (see above), i.e., it may not work properly and give suboptimal
results, see #353 (the
underlying issue is probably similar to the one described in
lme4 issue #682). With
suboptimality of the results, we mean that in such cases, the relevance
of the group-level terms can be underestimated. According to the
simulation-based case study from #353, the
latent projection should be considered as a remedy in such cases.
Speed
There are many ways to speed up projpred, but in
general, such speed-ups lead to results that are less accurate and hence
should only be considered as preliminary results. Some speed-up
possibilities are:
Using cv_varsel() with
validate_search = FALSE instead of
validate_search = TRUE. In case of
cv_method = "LOO", cv_varsel() with
validate_search = FALSE has comparable runtime to
varsel(), but accounts for some overfitting, namely that
induced by varsel()’s in-sample predictions during the
predictive performance evaluation. However, as explained in section “Variable selection” (see also section “Overfitting”), cv_varsel() with
validate_search = FALSE is more prone to overfitting than
cv_varsel() with
validate_search = TRUE.
Using cv_varsel() with \(K\)-fold CV instead of PSIS-LOO CV. Whether
this provides a speed improvement mainly depends on the number of
observations and the complexity of the reference model. Note that
PSIS-LOO CV is often more accurate than \(K\)-fold CV if argument K is
(much) smaller than the number of observations.
Using a “custom” reference model object with a dimension
reduction technique for the predictor data (e.g., by computing principal
components from the original predictors, using these principal
components as predictors when fitting the reference model, and then
performing the variable selection in terms of the original
predictor terms). Examples are given in Piironen,
Paasiniemi, and Vehtari (2020) and Pavone et al. (2022). This approach makes
sense if there is a large number of predictor variables, in which case
this aims at improving the runtime required for fitting the reference
model and hence improving the runtime of \(K\)-fold CV.
Using varsel() with its argument d_test
for evaluating predictive performance on a hold-out dataset instead of
doing this with cv_varsel()’s CV approach. Typically, the
hold-out approach requires a large amount of data.
Reducing nterms_max in varsel() or
cv_varsel(). The resulting predictive performance plot(s)
should be inspected to check that the search is not terminated too early
(i.e., before the submodel performance levels off), which would indicate
that nterms_max has been reduced too much.
Reducing argument nclusters (of
varsel() or cv_varsel()) below 20
and/or setting nclusters_pred to some non-NULL
(and smaller than 400, the default for
ndraws_pred) value. If setting nclusters_pred
as low as nclusters (and using forward search),
refit_prj can instead be set to FALSE, see
below.
Using L1 search (see argument method of
varsel() or cv_varsel()) instead of forward
search. Note that L1 search implies nclusters = 1 and is
not always supported.
In general, forward search is more accurate than L1 search and hence
more desirable (see section “Details” in ?varsel or
?cv_varsel for a more detailed comparison of the two). The
issue demonstrated in the Poisson example from the latent-projection
vignette is related to this.
Setting argument refit_prj (of varsel()
or cv_varsel()) to FALSE, which basically
means to set ndraws_pred = ndraws and
nclusters_pred = nclusters, but in a more efficient (i.e.,
faster) way. In case of L1 search, this means that the L1-penalized
projections of the regression coefficients are used for the predictive
performance evaluation, which is usually undesired (Piironen,
Paasiniemi, and Vehtari 2020, sec. 4). In case of forward
search, this issue does not exist.
Parallelizing costly parts of the CV implied by
cv_varsel() (this was demonstrated in the example above;
see argument parallel of cv_varsel()). When
using project(), parallelizing the projection might also
help (see the general package documentation available online
or by typing ?`projpred-package`).
Using varsel.vsel() or cv_varsel.vsel()
to re-use previous search results for new performance evaluation(s).
This is helpful if the performance evaluation part is run multiple times
based on the same search results (e.g., when only arguments
ndraws_pred or nclusters_pred of
varsel() or cv_varsel() are changed). In the
example above, this was illustrated when cv_varsel() was
applied to cvvs_fast instead of refm_obj to
yield cvvs_fast_refit. In that example, search and
performance evaluation were effectively run separately by the
cv_varsel() calls yielding cvvs_fast and
cvvs_fast_refit, respectively. This is because
cv_varsel() with refit_prj = FALSE (used for
cvvs_fast) has almost only the computational cost of the
search (the performance evaluation with refit_prj = FALSE
has almost no computational cost) and cv_varsel.vsel()
(used for cvvs_fast_refit) has almost only the
computational cost of the performance evaluation (the search in
cv_varsel.vsel() has no computational cost at
all).
Using run_cvfun() in case of repeated \(K\)-fold CV with the same \(K\) reference model refits. The output of
run_cvfun() is typically used as input for argument
cvfits of cv_varsel.refmodel() (so in order to
have a speed improvement, the output of run_cvfun() needs
to be assigned to an object which is then re-used in multiple
cv_varsel() calls).