This vignette illustrates 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 (approximately) valid post-selection inference.
The projection predictive variable selection is based on the ideas of @goutis_model_1998 and @dupuis_variable_2003.
The methods implemented in projpred are described in detail in @piironen_projective_2020, @catalina_projection_2022, @weber_projection_2023, and @catalina_latent_2021.
A comparison to many other methods may also be found in @piironen_comparison_2017.
For details on how to cite projpred, see the projpred citation info on CRAN^[The citation information can be accessed offline by typing print(citation("projpred"), bibtex = TRUE)
within R.].
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") dat_gauss <- data.frame(y = df_gaussian$y, df_gaussian$x)
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_projective_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^[We will cover custom reference models more deeply in a future vignette.].
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
).
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_sparsity_2017] with the hyperprior for its global shrinkage parameter following @piironen_hyperprior_2017 and @piironen_sparsity_2017. In R code, these are the preparation steps for the regularized horseshoe prior:
# Number of regression coefficients: ( D <- sum(grepl("^X", names(dat_gauss))) ) # 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^[More generally, the number of chains is split up as evenly as possible among the number of CPU cores.].
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(507801) 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 )
Usually, we would now have to check the convergence diagnostics (see, e.g., ?posterior::diagnostics
and ?posterior::default_convergence_measures
; the bulk-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 bulk-ESS warning).
Now, projpred comes into play.
library(projpred)
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_practical_2017; @vehtari_pareto_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
(which is currently only available for cv_method = "LOO"
), 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^[Currently, neither varsel()
nor cv_varsel()
(not even cv_varsel()
with validate_search = TRUE
) guard against overfitting in the selection of the submodel size. This is why we added "approximately" to "valid post-selection inference" in section "Introduction". Typically, however, the overfitting induced by the size selection should be comparatively small [@piironen_comparison_2017].], 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).
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 nclusters_pred
to a comparatively low value of 20
.
# Preliminary cv_varsel() run: cvvs_fast <- cv_varsel( refm_fit, validate_search = FALSE, ### Only for the sake of speed (not recommended in general): method = "L1", nclusters_pred = 20, ### nterms_max = 20, ### In interactive use, we recommend not to deactivate the verbose mode: verbose = FALSE ### )
In this case, we ignore the Pareto $\hat{k}$ warnings due to the reduced values for chains
and iter
in the reference model fit.
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).
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 (see ?plot.vsel
for a description) 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: nterms_max = 9
) to ensure that we see the MLPD leveling off.
For this final cv_varsel()
run, 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"
(with validate_search = TRUE
) or a larger value for K
if this is possible in terms of computation time.
We also illustrate how projpred's CV 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).
# For the CV parallelization (cv_varsel()'s argument `parallel`): doParallel::registerDoParallel(ncores) # Final cv_varsel() run: cvvs <- cv_varsel( refm_fit, cv_method = "kfold", ### Only for the sake of speed (not recommended in general): K = 2, 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()
Again, we ignore the bulk-ESS warnings due to the reduced values for chains
and iter
in the reference model fit.
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)
Based on that 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 general, only cv_varsel()
with validate_search = TRUE
(which we have here) allows to judge whether the submodels perform better than the reference model or not. Such a judgment is not possible with varsel()
or cv_varsel()
with validate_search = FALSE
(in general).]).
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:
size_decided <- 6
If the focus of our predictive model had been accuracy (not sparsity), size 8 would have been a natural choice because at size 8, the submodel MLPD "levels off" (in fact, size 8 here even comes with the maximum submodel MLPD among all plotted sizes, but in general, this does not need to be the case). Further below, 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")
With this heuristic, we would get the same final submodel size (6
) as by our manual (sparsity-based) decision.
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()
^[print.vsel()
is the method that is called when simply printing an object resulting from varsel()
or cv_varsel()
.].
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)
As highlighted by the message above, the predictor ranking from column solution_terms
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()
:
rk <- ranking(cvvs)
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) )
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 set of the two most relevant predictor terms (X1
and X14
) as well as on their order.
Since the column names of the matrix returned by cv_proportions()
follow the full-data predictor ranking, we can infer that X1
and X14
are also the two most relevant predictor terms in the full-data predictor ranking.
To see this more explicitly, we can access element fulldata
of the ranking()
output:
rk[["fulldata"]]
This is the same as column solution_terms
in the summary.vsel()
output above.
(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()
:
plot(pr_rk)
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) )
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 confirms (from a slightly different perspective) 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 two most relevant predictors (X1
and X14
): When looking at <=2
on the x-axis, all tiles above and including the second main diagonal element are at 100 %.
Similarly, the two CV folds agree on the sets of the six and nine 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 ninth 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.
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^[During the search, the reference model is projected onto all candidate models (this is where arguments ndraws
and nclusters
of varsel()
and cv_varsel()
come into play; note that in case of an L1 search, the projection is L1-penalized). For the evaluation of the submodels along the predictor ranking returned by the search, the reference model is projected onto these submodels again (this is where arguments ndraws_pred
and nclusters_pred
of varsel()
and cv_varsel()
come into play; note that this only holds if argument refit_prj
of varsel()
and cv_varsel()
is set to TRUE
, as by default). Within project()
, refit_prj = FALSE
allows to re-use the submodel fits (that is, the projections) from the full-data search of a vsel
object, but usually, the search relies on a rather coarse clustering or thinning of the reference model's posterior draws (by default, varsel()
and cv_varsel()
use nclusters = 20
---or nclusters = 1
in case of L1 search), which would then imply the same coarseness for a project()
call where refit_prj
is set to FALSE
. In general, we want the final projection (that post-selection inference is based on) to be as accurate as possible, so here we call project()
with the defaults of refit_prj = TRUE
and ndraws = 400
. For more accurate results, we could increase argument ndraws
of project()
(up to the number of posterior draws in the reference model). However, this would increase the runtime, which we don't want in this vignette.]:
prj <- project( refm_fit, solution_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^[In general, this implies that projected regression coefficients do not reflect isolated effects of the predictors. For example, especially in case of highly correlated predictors, it is possible that projected regression coefficients "absorb" effects from predictors that have been excluded in the projection.].
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^[proj_predict()
also has an argument return_draws_matrix
, but it simply converts the return value type. In proj_predict()
, different weights of the projected draws are taken into account via argument nresample_clusters
.]).
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)
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)
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) + 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 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 ) )
Then proj_linpred()
can calculate the linear predictors^[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).] 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$pred))
If dat_gauss_new
also contained response values (i.e., y
values in this example), then proj_linpred()
would also evaluate the log predictive density at these (conditional on each of the projected parameter draws if integrated = FALSE
and integrated over the projected parameter draws---before taking the logarithm---if integrated = TRUE
).
With proj_predict()
, we can obtain draws from predictive distributions based on the final submodel.
In contrast to proj_linpred(<...>, integrated = FALSE)
, this encompasses not only the uncertainty arising from parameter estimation, but also the uncertainty arising from the observation (or "sampling") model for the response^[In case of the Gaussian family we are using here, the uncertainty arising from the observation model is the uncertainty due to the residual standard deviation.].
This is useful for what is usually termed a posterior predictive check (PPC), but would have to be termed something like a posterior-projection predictive check (PPPC) here:
prj_predict <- proj_predict(prj) # Using the 'bayesplot' package: ppc_dens_overlay(y = dat_gauss$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.
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()
^[Via brms::get_refmodel.brmsfit()
, the brms::bernoulli()
family is supported as well.] and the poisson()
family.
The families supported by projpred's augmented-data projection [@weber_projection_2023] are binomial()
^[Currently, the augmented-data support for the binomial()
family does not include binomial distributions with more than one trial.
In such a case, a workaround is to de-aggregate the Bernoulli trials which belong to the same (aggregated) observation, i.e., to use a "long" dataset.] ^[Like the traditional projection, the augmented-data projection also supports the brms::bernoulli()
family via brms::get_refmodel.brmsfit()
.], brms::cumulative()
, rstanarm::stan_polr()
fits, and brms::categorical()
^[For the augmented-data projection based on a "typical" brms reference model object, brms version 2.17.0 or later is needed.] ^[More brms families might be supported in the future.].
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_latent_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()
^[Currently, the latent-projection support for the binomial()
family does not include binomial distributions with more than one trial. In such a case, a workaround is to de-aggregate the Bernoulli trials which belong to the same (aggregated) observation, i.e., to use a "long" dataset.] ^[Like the traditional projection, the latent projection also supports the brms::bernoulli()
family via brms::get_refmodel.brmsfit()
.], poisson()
, brms::cumulative()
, and rstanarm::stan_polr()
fits^[For the latent projection based on a "typical" brms reference model object, brms version 2.19.0 or later is needed.].
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^[Multilevel models are also known as hierarchical models or models with partially pooled, group-level, or---in frequentist terms---random effects.], and---as an experimental feature---additive^[Additive terms are also known as smooth terms.] 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`
).
Sometimes, the predictor ranking makes sense, but for an increasing submodel size, the predictive performance of the submodels does not approach the reference model's predictive performance so that the submodels exhibit a predictive performance that stays worse than the reference model's. There are different reasons that can explain this behavior (the following list might not be exhaustive, though):
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.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
.
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.
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.
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
(which requires cv_method = "LOO"
) instead of validate_search = TRUE
.
This approach (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 (with validate_search = TRUE
).
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_projective_2020 and @pavone_using_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 may be undesired [@piironen_projective_2020, section 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 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).
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