Masking select from other packages

Finally, 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(tidyverse)
  library(patchwork)
  library(MASSExtra)
})
options(knitr.kable.NA = "")
theme_set(theme_bw() + theme(plot.title = element_text(hjust = 0.5)))

The box_cox extensions

This 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

Box-cox, old and new displays

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

The Box-Cox transformation effect

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.

Stepwise model building extensions

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.

1. 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.

2. 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.

3. Criteria. The Akaike Information Criterion, AIC, corresponds to a penalty 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.

Examples

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

Two component visualisations for the Boston house price model

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).

Non-gaussian models

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)

	Df	delta_AIC	LRT	Pr(Chi)
		0.0000
Age:Eth:Sex:Lrn	2	-2.5962	1.4038	0.4956

quine_gic <- step_GIC(quine_full)
drop_term(quine_gic) %>% kable(booktabs = TRUE, digits = 4)

	Df	delta_GIC	LRT	Pr(Chi)
Eth:Sex:Lrn	1	8.9298	12.4216	4e-04
Age:Sex	3	8.1439	18.6193	3e-04
		0.0000

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)

	Df	Deviance	delta_AIC	F value	Pr(F)
Weight	1	1.032	34.400	38.499	0.000
Cylinders	5	0.867	7.919	3.790	0.004
		0.708	0.000
Origin	1	0.721	-0.566	1.516	0.222
EngineSize	1	0.712	-1.560	0.465	0.497

mpg_gic <- step_down(mpg0, k = "gic")      ## simple backward elimination, mainly used for GLMMs
drop_term(mpg_gic) %>% kable(booktabs = TRUE, digits = 3)

	Df	Deviance	delta_GIC	F value	Pr(F)
Weight	1	1.490	79.688	89.042	0.000
Cylinders	5	0.901	2.097	3.956	0.003
		0.732	0.000

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")

MPG model using a Gamma family with inverse link

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.

Scaling functions

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.

Kernel density estimation

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, namely
  • The kernel function may be specified as an R function, as well as as a character string to select from a preset list. Users wishing to write a special kernel function should do so in line with, for example demoKde::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
A Kernel Density Estimate of class  kde_1d 

Response   : Criminality
Sample size: 506
Range      : -6.744 < x < 6.169
Bandwidth  : 0.5601
Resolution : 1024

Methods exist for generics: plot, print 
plot(kcrim)

A one-dimensional KDE of a predictor variable from the Boston data set

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
A Two-dimensional Kernel Density Estimate

Responses  : x, y
Sample size: 506
Ranges     : -5.724 < x < 5.148, 1.853 < y < 2.997
Bandwidths : 0.6597, 0.03361
Resolution : 512, 512

Methods exist for generics: hr_levels, plot, print 
plot(kcrimrm, ## col = hcl.colors(25, rev = TRUE),
     xlab = expression(italic(Criminality)),
     ylab = expression(italic(Spaciousness)))
contour(kcrimrm, col = "dark green", add = TRUE)

A two-dimensional KDE using two predictors from the Boston data set

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"))

A two-dimensional KDE presented as a perspective plot