The Importance
function considers variable importance (or
predictor importance) to be the effect that the variable has on
replicates y^rep (or
Y^rep) when the variable is removed from the
model by setting it equal to zero. Here, variable importance is
considered in terms of the comparison of posterior predictive
checks. This may be considered to be a form of sensitivity analysis,
and can be useful for model revision, variable selection, and model
interpretation.
Currently, this function only tests the variable importance of design matrix X.
1 2  Importance(object, Model, Data, Categorical=FALSE, Discrep, d=0, CPUs=1,
Type="PSOCK")

object 
An object of class 
Model 
The model specification function is required. 
Data 
A data set in a list is required. The dependent variable
is required to be named either 
Categorical 
Logical. If 
Discrep 
This optional argument allows a discrepancy statistic to
be included. For more information on discrepancy statistics, see

d 
This is an optional integer to be used with the

CPUs 
This argument accepts an integer that specifies the number
of central processing units (CPUs) of the multicore computer or
computer cluster. This argument defaults to 
Type 
This argument specifies the type of parallel processing to
perform, accepting either 
Variable importance is defined here as the impact of each variable (predictor, or column vector) in design matrix X on y^rep (or Y^rep), when the variable is removed.
First, the full model is predicted with the
predict.demonoid
, predict.iterquad
,
predict.laplace
, predict.pmc
, or
predict.vb
function, and summarized with the
summary.demonoid.ppc
,
summary.iterquad.ppc
, summary.laplace.ppc
,
summary.pmc.ppc
, or summary.vb.ppc
function, respectively. The results are stored in the first row of the
output. Each successive row in the output corresponds to the
application of predict
and summary
functions, but with
each variable in design matrix X being set to zero
and effectively removed. The results show the impact of sequentially
removing each predictor.
The criterion for variable importance may differ from model to model. As a default, BPIC is recommended. The Bayesian Predictive Information Criterion (BPIC) was introduced by Ando (2007). BPIC is a variation of the Deviance Information Criterion (DIC) that has been modified for predictive distributions. For more information on DIC (Spiegelhalter et al., 2002), see the accompanying vignette entitled "Bayesian Inference". BPIC = Dbar + 2pD.
With BPIC, variable importance has a positive relationship, such that larger values indicate a more important variable, because removing that variable resulted in a worse fit to the data. The best model has the lowest BPIC.
In a model in which the dependent variable is not categorical, it is
also recommended to consider the Lcriterion (Laud and Ibrahim, 1995),
provided that sample size is small enough that it does not result in
Inf
. For more information on the Lcriterion, see the
accompanying vignette entitled "Bayesian Inference".
With the Lcriterion, variable importance has a positive relationship,
such that larger values indicate a more important variable, because
removing that variable resulted in a worse fit to the data. Ibrahim
(1995) recommended considering the model with the lowest
Lcriterion, say as L[1], and the model with the closest
Lcriterion, say as L[2], and creating a comparison score
as phi = (L[2]L[1])/S[L], where
S.L
is from the L[1] model. If the comparison score,
phi is less than 2, then L[2] is within 2
standard deviations of L[1], and is the recommended cutoff
for model choice.
The Importance
function may suggest that a model fits the data
better with a variable removed. In which case, the user may
choose to leave the variable in the model (perhaps the model is
misspecified without the variable), investigate and possibly
respecify the relationship between the independent and dependent
variable(s), or remove the variable and update the model again.
In contrast to variable importance, the PosteriorChecks
function calculates parameter importance, which is the probability
that each parameter's marginal posterior distribution is greater than
zero, where an important parameter does not include zero in its
probability interval (see p.interval
). Parameter
importance and variable importance may disagree, and both should be
studied.
The Importance
function tends to indicate that a model fits the
data better when variables are removed that have parameters with
marginal posterior distributions that include 0 in the 95%
probability interval (variables associated with lower parameter
importance).
Often, in complicated models, it is difficult to assess variable importance by examining the marginal posterior distribution of the associated parameter(s). Consider polynomial regression, in which each variable may have multiple parameters.
The information provided by the Importance
function may be used
for model revision, or reporting the relative importance of variables.
The plot.importance
function is available to plot the
output of the Importance
function according to BPIC, predictive
concordance (Gelfand, 1996), the selected discrepancy statistic
(Gelman et al., 1996), or the Lcriterion.
Parallel processing may be performed when the user specifies
CPUs
to be greater than one, implying that the specified number
of CPUs exists and is available. Parallelization may be performed on a
multicore computer or a computer cluster. Either a Simple Network of
Workstations (SNOW) or Message Passing Interface is used (MPI). With
small data sets and few samples, parallel processing may be slower,
due to computer network communication. With larger data sets and more
samples, the user should experience a faster runtime.
Importance
returns an object of class importance
, which
is a matrix with a number of rows equal to the number of columns in
design matrix X + 1 (including the full model), and
4 columns, which are BPIC, Concordance (or Mean.Lift if categorical),
Discrep, and Lcriterion. Each row represents a model with a predictor
in X removed (except for the first row, which is the
full model), and the resulting posterior predictive checks. For
noncategorical dependent variables, an attribute is returned with the
object, and the attribute is a vector of S.L
, the calibration
number of the Lcriterion.
Statisticat, LLC. software@bayesianinference.com
Ando, T. (2007). "Bayesian Predictive Information Criterion for the Evaluation of Hierarchical Bayesian and Empirical Bayes Models". Biometrika, 94(2), p. 443–458.
Gelfand, A. (1996). "Model Determination Using Sampling Based Methods". In Gilks, W., Richardson, S., Spiegehalter, D., Chapter 9 in Markov Chain Monte Carlo in Practice. Chapman and Hall: Boca Raton, FL.
Laud, P.W. and Ibrahim, J.G. (1995). "Predictive Model Selection". Journal of the Royal Statistical Society, B 57, p. 247–262.
Spiegelhalter, D.J., Best, N.G., Carlin, B.P., and van der Linde, A. (2002). "Bayesian Measures of Model Complexity and Fit (with Discussion)". Journal of the Royal Statistical Society, B 64, p. 583–639.
is.importance
,
IterativeQuadrature
,
LaplaceApproximation
,
LaplacesDemon
,
PMC
,
plot.importance
,
PosteriorChecks
,
p.interval
,
predict.demonoid
,
predict.iterquad
,
predict.laplace
,
predict.pmc
,
predict.vb
,
summary.demonoid.ppc
,
summary.iterquad.ppc
,
summary.laplace.ppc
,
summary.pmc.ppc
,
summary.vb.ppc
, and
VariationalBayes
.
1 2 3 4  #First, update the model with the LaplacesDemon function, such as
#the example with linear regression, creating an object called Fit.
#Then
#Importance(Fit, Model, MyData, Discrep="ChiSquare", CPUs=1)

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