knitr::opts_chunk$set(collapse = TRUE, comment = "", message = FALSE, warning = FALSE) old_plot_new <- getHook("plot.new") old_before_plot_new <- getHook("before.plot.new") setHook("plot.new", list(), "replace") setHook("before.plot.new", rev(old_before_plot_new)[1], "replace") old_par <- par(no.readonly = TRUE) old_options <- options(show.signif.stars = FALSE) library(ggplot2) old_theme <- theme_get()
This extension package to the classical MASS
package (Venables &
Ripley, of ancient lineage), whose origins go back to nearly 30 years,
comes about for a number of reasons.
Firstly, in my teaching I found I was using some of the old functions in the package with consistently different argument settings to the defaults. I was also interested in supplying various convenience extensions that simplified teaching and including various tweaks to improve the interface. Examples follow below.
Secondly, I wanted to provide a few functions that were mainly useful
as programming examples. For example, the function zs
and its
allies zu
, zq
and zr
are mainly alternatives to base::scale
,
but they can be used to show how to write functions that can be used
in fitting models in such a way that they work as they should when the
fitted model object is used for prediction with new data.
select
from other packagesFinally, there is the perennial select
problem. When MASS
is used
with other packages, such as dplyr
the select
function can easily
be masked, causing confusion for users. MASS::select
is rarely
used, but dplyr::select
is fundamental. There are standard ways of
managing this kind of masking, but what we have done in MASSExtra
is
to export the more common functions used from MASS
along with the
extensions, in such a way that users will not need to have MASS
attached to the search path at all, and hence masking is unlikely.
The remainder of this document will do a walk-through of some of the new functions provided by the package. We begin by setting the computational context:
suppressPackageStartupMessages({ library(visreg) library(knitr) library(dplyr) library(ggplot2) library(patchwork) library(MASSExtra) }) options(knitr.kable.NA = "") theme_set(theme_bw() + theme(plot.title = element_text(hjust = 0.5)))
We now consider some of the extensions that the package offers to the originals. Most of the extensions will have a name that includes an underscore of two somewhere to distinguish it from the V&R original. Note that the original version is also exported so that scripts that use it may do so without change, via the new package.
box_cox
extensionsThis original version, boxcox
has a fairly rigid display for the
plotted output which has been changed to give a more easily
appreciated result. The $y-$axis has been changed to give the
likelihood-ratio statistic rather than the log-likelihood, and for the
$x-$axis some attempt has been made to focus on the crucial region for
the transformation parameter, $\lambda$,
The following example shows the old and new plot versions for a simple example.
par(mfrow = c(1, 2)) mod0 <- lm(MPG.city ~ Weight, Cars93) boxcox(mod0) ## MASS box_cox(mod0) ## MASSExtra tweak
In addition, there are functions bc
to evaluate the transformation
for a given exponent, and a function lambda
which finds the optimum
exponent (not that a precise exponent will usually be needed).
It is interesting to see how in this instance the transformation can both straighten the relationship and provide a scale in which the variance is more homogeneous. See Figure 2.
p0 <- ggplot(Cars93) + aes(x = Weight) + geom_point(colour = "#2297E6") + xlab("Weight (lbs)") + geom_smooth(se = FALSE, method = "loess", formula = y ~ x, size=0.7, colour = "black") p1 <- p0 + aes(y = MPG.city) + ylab("Miles per gallon (MPG)") + ggtitle("Untransformed response") p2 <- p0 + aes(y = bc(MPG.city, lambda(mod0))) + ggtitle("Transformed response") + ylab(bquote(bc(MPG, .(round(lambda(mod0), 2))))) p1 + p2
A more natural scale to use, consistent with the Box-Cox suggestion, would be the reciprocal. For example we could use $\mbox{GPM} = 100/\mbox{MPG}$ the "gallons per 100 miles" scale, which would have the added benefit of being more-or-less what the rest of the world uses to gauge fuel efficiency outside the USA. Readers should try this for themselves.
The primary MASS
functions for refining linear models and their
allies are dropterm
and stepAIC
. The package provides a few
extensions to these, but mainly a change of defaults in the argument
settings.
drop_term
is a front-end to MASS::dropterm
with a few tweaks.
By default the result is arranged in sorted order, i.e. with
sorted = TRUE
, and also by default with test = TRUE
(somewhat
in defiance of much advice to the contrary given by experienced
practitioners: caveat emptor!).The user may specify the test to use in the normal way, but the
default test is decided by an ancillary generic function,
default_test
, which guesses the appropriate test from the object
itself. This is an S3 generic and further methods can be supplied
for new fitted model objects.
There is also a function add_term
which provides similar
enhancements to those provided by drop_term
. In this case, of
course, the consequences of adding individual terms to the model
are displayed, rather than of dropping them. It follows that
using add_term
you will always need to provide a scope
specification, that is, some specification
of what extra terms are possible additions.
In addition drop_term
and add_term
return an object which
retains information on the criterion used, AIC
, BIC
, GIC
(see
below) or some specific penalty value k
. The object also has a
class "drop_term"
for which a plot
method is provided. Both
the plot
and print
methods display the criterion. See the
example below for how this is done.
step_AIC
is a front-end to MASS::stepAIC
with the default
argument trace = FALSE
set. This may of course be over-ruled,
but it seems the most frequent choice by users, anyway. In
addition the actual criterion used, by dafault k = 2
, i.e. AIC,
is retained with the result and passed on to methods in much the
same say as for drop_term
above.Since the (default) criterion name is encoded in the function name,
two further versions are supplied, namely step_BIC
and step_GIC
(again, see below), which use a different, and obvious, default
criterion.
In any of step_AIC
, step_BIC
or step_GIC
a different value of
k
may be specified in which case that value of k
is retained
with the object and displayed as appropriate in further methods.
Finally in any of these functions k
may be specified either as a
numeric penalty, such as k = 4
for example, or by character
string k = "AIC"
or k = "BIC"
with an obvious meaning in either
case.
k = 2
and the Bayesian Information Criterion,
BIC, corresponds to k = log(n)
where n
is the sample size. In
addition to these two the present functions offer an intermediate
default penalty k = (2 + log(n))/2
which is "not too strong and
not too weak", making it the Goldilocks Information Criterion,
GIC. There is also a standalone function GIC
to evaluate this
k
if need be.This suggestion appears to be original, but no particular claim is made for it other than with intermediate to largish data sets it has proved useful for exploratory purposes in our experience.
Our strong advice is that these tools should only be used for exploratory purposes in any case, and should never be used in isolation. They have a well-deserved very negative reputation when misused, as they commonly are.
We consider the well-known (and much maligned) Boston house price
data. See ?Boston
. We begin by fitting a model that has more terms
in it than the usual model, as it contains a few extra quadratic
terms, including some key linear by linear interactions.
big_model <- lm(medv ~ . + (rm + tax + lstat + dis)^2 + poly(dis, 2) + poly(rm, 2) + poly(tax, 2) + poly(lstat, 2), Boston) big_model %>% drop_term(k = "GIC") %>% plot() %>% kable(booktabs=TRUE, digits=3)
Unlike MASS::dropterm
, the table shows the terms beginning with the
most important ones, that is those which, if dropped, would increase
the criterion and ending with those of least looking importance, that
is those whose removal would most decrease the criterion. And also
note that here we are using the GIC
, which is displayed in the
output.
Note particularly that rather than give the value of the criterion by default the table and plot show change in the criterion which would result if the term is removed from the model at that point. This is a more meaningful quantity, and invariant with respect to the way in which the log-likelihood is defined.
The plot
method gives a graphical view of the same key bits of
information, in the same vertical order as given in the table. Terms
whose removal would (at this point) improve the model are shown in
red and those which would not, and hence should (again, at this
point) be retained are shown in blue.
With all stepwise methods it is critically important to notice that the whole picture can change once any change is made to the current model. This terms which appear "promising" at this stage may not seem so once any variable is removed from the model or some other variable brought into it. This is a notoriously tricky area for the inexperienced.
Notice that the plot
method returns the original object, which can
then be passed on via a pipe to more operations. (kable
does not,
so this pipe sequence cannot be changed.)
We now consider a refinement of this model by stepwise means, but rather than use the large model as the starting point, we begin with a more modest one which has no quadratic terms.
base_model <- lm(medv ~ ., Boston) gic_model <- step_GIC(base_model, scope = list(lower = ~1, upper = formula(big_model))) drop_term(gic_model) %>% plot() %>% kable(booktabs = TRUE, digits = 3)
The model is likely to be over-fitted. To follow up on this we could look at profiles of the fitted terms as an informal way of model 'criticism'.
capture.output(suppressWarnings({ g1 <- visreg(gic_model, "dis", plot = FALSE) g2 <- visreg(gic_model, "lstat", plot = FALSE) plot(g1, gg = TRUE) + plot(g2, gg = TRUE) })) -> void
The case for curvature appears to be fairly weak, in each case with
departure from a straight line dependence depending on a relatively
few observations with high values for the predictor. (Notice how hard
you have to work to prevent visreg
from generating unwanted output.)
As an example of add_term
, consider going from what we have called
the base_model
to the big_model
, or at least what might be the
initial step:
add_term(base_model, scope = formula(big_model), k = "gic") %>% plot() %>% kable(booktabs = TRUE, digits = 3)
So in this case, your best first step would be to add the term which most decreases the criterion, that is, the one nearest the bottom of the table (or display).
For a non-gaussian model consider the Quine data (?quine
) example
discussed in the MASS book. We begin by fitting a full negative
binomial model and refine it using a stepwise algorithm.
quine_full <- glm.nb(Days ~ Age*Eth*Sex*Lrn, data = quine) drop_term(quine_full) %>% kable(booktabs = TRUE, digits = 4) quine_gic <- step_GIC(quine_full) drop_term(quine_gic) %>% kable(booktabs = TRUE, digits = 4)
So GIC refinement has led to the same model as in the MASS book,
which in more understandable form would be written Days ~ Sex/(Age +
Eth*Lrn)
. Note also that the default test is in this case
the likelihood ratio test.
For a different example, consider an alternative way to model the MPG data, rather than transforming to an inverse scale, using a generalized linear model with an inverse link and a gamma response.
mpg0 <- glm(MPG.city ~ Weight + Cylinders + EngineSize + Origin, family = Gamma(link = "inverse"), data = Cars93) drop_term(mpg0) %>% kable(booktabs = TRUE, digits = 3) mpg_gic <- step_down(mpg0, k = "gic") ## simple backward elimination, mainly used for GLMMs drop_term(mpg_gic) %>% kable(booktabs = TRUE, digits = 3)
We can see something of how well the final model is performing by
looking at a slightly larger model fitted on the fly in ggplot
:
ggplot(Cars93) + aes(x = Weight, y = MPG.city, colour = Cylinders) + geom_point() + geom_smooth(method = "glm", method.args = list(family = Gamma), formula = y ~ x) + ylab("Miles per gallon (city driving)") + scale_colour_brewer(palette = "Dark2")
The function step_down
is a simple implementation of backward
elimination with the sole virtue that it works for (Generalised)
Linear Mixed Models, or at least for the fixed effect component of
them, whereas other stepwise methods do not (yet). As fitting GLMMs
can be very slow, going through a full stepwise process could be very
time consuming in any case.
When a function such as base::scale
, stats:poly
or splines:ns
(also exported from MASSExtra
) is used in modelling it is important
that the fitted model object has enough information so that when it is
used in prediction for new data, the same transformation can be put in
place with the new predictor variable values. Setting up functions in
such a way to enable this is a slightly tricky exercise. It involves
writing a method function for the S3 generic function
stats::makepredictcall
. To illustrate this we have supplied four
simple functions that
employ the technique.
zs
("z-score") is essentially the same as base::scale
with the default argument settings,zu
allows re-scaling to a fixed range of [0, 1], often used in neural network models,zq
allows a quantile scaling where the location is the lower quartile and the scale is the inter-quartile range.
In other words, the scaling is to [0,1] within the box of a boxplot. (Go figure.)zr
allows a "robust" scaling where the location is the median and the scale uses stats::mad
The only real interest in these very minor convenience functions lies in how they are programmed. See the code itself for more details.
Release 1.1.0 contains two functions for kernel density estimation: one- and two-dimensional.
kde_1d
offers a similar functionality to stats::density
, though with two additional
features that may be useful in some situations, namelydemoKde::kernelBiweight
or demoKde::kernalGaussian
,
using the same argument list with the same intrinsic meanings for the arguments themselves.The kernel density estimate may be "folded" to emulate the effect of fitting a density with a known finite range for the underlying distribution. This amounts to fitting the kde initially with unrestricted range and "folding back" the parts beyond the known range, adding them on to the mirror image components inside the range. This strategy appears to give a credible result, though no particular claim is made for it on theoretical grounds. See the examples.
kde_2d
uses much the same computational ideas as in MASS::kde2d
(due to Prof. Brian Ripley),
but uses an approximation that allows the algorithm to scale much better for both large data sets
and large resolution in the result. Indeed the approximation improves as the resolution increases,
so the default size is now $512\times512$ rather than $25\times25$ as it is for MASS::kde2d
. This
function also allows the kernel function(s) to be either specified or user-defined, as for
kde_1d
above. Folding is not implemented, however.
Both functions produce objects with a class agreeing with the name of the calling function, and
suitable plot
and print
methods are provided.
Two examples follow. The first shows (mainly) the surprising capacity for a log-transformation to amplify what is essentially a trivial effect into something that appears impressive!
Criminality <- with(Boston, log(crim)) kcrim <- kde_1d(Criminality, n = 1024, kernel = demoKde::kernelBiweight) kcrim plot(kcrim)
We now take this further into a two-dimensional example
Spaciousness <- with(Boston, sqrt(rm)) kcrimrm <- kde_2d(Criminality, Spaciousness, n = 512, kernel = "opt") kcrimrm plot(kcrimrm, ## col = hcl.colors(25, rev = TRUE), xlab = expression(italic(Criminality)), ylab = expression(italic(Spaciousness))) contour(kcrimrm, col = "dark green", add = TRUE)
An even more deceptive plot uses persp
:
with(kcrimrm, persp(x, 10*y, 3*z, border="transparent", col = "powder blue", theta = 30, phi = 15, r = 100, scale = FALSE, shade = TRUE, xlab = "Criminality", ylab = "Spaciousness", zlab = "kde"))
In this final section we mainly give a list of functions provided by the package, and their origins.
We begin by giving a list of functions in the MASS
package which are
not re-exported from the MASSExtra
package. If you need any of
these you will need either to attach the MASS
package itself, or use
the qualified form MASS::<name>
.
setdiff(getNamespaceExports("MASS"), c("lmwork", getNamespaceExports("MASSExtra"))) %>% sort() %>% noquote()
The following objects are re-exported from the MASSExtra
package,
and hence may be used directly, if needed.
intersect(getNamespaceExports("MASS"), getNamespaceExports("MASSExtra")) %>% sort() %>% noquote()
The following functions are new to the MASSExtra
package, some of
which are obviously refinements of their MASS
workhorse counterparts.
setdiff(getNamespaceExports("MASSExtra"), getNamespaceExports("MASS")) %>% setdiff(getNamespaceExports("splines")) %>% grep("^[.]__", ., invert = TRUE, value = TRUE) %>% ## exclude S4 class objects sort() %>% noquote()
Finally the following objects are re-exported from splines
:
intersect(getNamespaceExports("MASSExtra"), getNamespaceExports("splines")) %>% sort() %>% noquote()
Only four of the MASS
data sets are included in MASSExtra
, namely
Cars93
, Boston
, quine
and whiteside
. Other data sets from
MASS
itself will need to be accessed directly, e.g. MASS::immer
.
setHook("plot.new", old_plot_new, "replace") setHook("before.plot.new", old_before_plot_new, "replace") par(old_par) options(old_options) theme_set(old_theme)
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