knitr::opts_chunk$set( collapse = TRUE, warning = FALSE, fig.height = 6, fig.width = 7, fig.path = "fig/tut04-", dev = "png", comment = "##" ) # save some typing knitr::set_alias(w = "fig.width", h = "fig.height", cap = "fig.cap") # Load packages set.seed(1071) library(vcd) library(vcdExtra) library(ggplot2) library(seriation) data(HairEyeColor) data(PreSex) data(Arthritis, package="vcd") art <- xtabs(~Treatment + Improved, data = Arthritis) if(!file.exists("fig")) dir.create("fig")
Mosaic plots provide an ideal method both for visualizing contingency tables and for visualizing the fit--- or more importantly--- lack of fit of a loglinear model.
For a two-way table, mosaic()
, by default, fits a model of independence, $[A][B]$
or ~A + B
as an R formula. The vcdExtra
package extends this to models fit
using glm(..., family=poisson)
, which can include specialized models for
ordered factors, or square tables that are intermediate between the saturated model,
$[A B]$ = A * B
, and the independence model $[A][B]$.
For $n$-way tables, vcd::mosaic()
can fit any loglinear model, and can also be
used to plot a model fit with MASS:loglm()
. The vcdExtra
package extends this
to models fit using stats::glm()
and, by extension, to non-linear models fit
using the gnm package.
See @vcd:Friendly:1994, @vcd:Friendly:1999 for the statistical ideas behind these uses of mosaic displays in connection with loglinear models. Our book @FriendlyMeyer:2016:DDAR gives a detailed discussion of mosaic plots and many more examples.
The essential ideas are to:
recursively sub-divide a unit square into rectangular "tiles" for the cells of the table, such that the area of each tile is proportional to the cell frequency. Tiles are split in a sequential order:
First according to the marginal proportions of a first variable, V1
...
For a given loglinear model, the tiles can then be shaded in various ways to reflect the residuals (lack of fit) for a given model.
The pattern of residuals can then be used to suggest a better model or understand where a given model fits or does not fit.
mosaic()
provides a wide range of options for the directions of splitting,
the specification of shading, labeling, spacing, legend and many other details.
It is actually implemented as a special case of a more general
class of displays for $n$-way tables called strucplot
, including
sieve diagrams, association plots, double-decker plots as well as mosaic
plots.
For details, see help(strucplot)
and the "See also" links therein,
and also @vcd:Meyer+Zeileis+Hornik:2006b, which is available as
an R vignette via vignette("strucplot", package="vcd")
.
Example:
A mosaic plot for the Arthritis treatment data fits the model of independence,
~ Treatment + Improved
and displays the association in the pattern of
residual shading. The goal is to visualize the difference in the proportions
of Improved
for the two levels of Treatment
: "Placebo" and "Treated".
The plot below is produced with the following call to mosaic()
.
With the first split by Treatment
and the shading used, it is easy to see
that more people given the placebo experienced no improvement, while more
people given the active treatment reported marked improvement.
#| Arthritis1, #| fig.height = 6, #| fig.width = 7, #| fig.cap = "Mosaic plot for the `Arthritis` data, using `shading_max`" data(Arthritis, package="vcd") art <- xtabs(~Treatment + Improved, data = Arthritis) mosaic(art, gp = shading_max, split_vertical = TRUE, main="Arthritis: [Treatment] [Improved]")
gp = shading_max
specifies that color in the plot signals a significant residual at a 90% or 99% significance level,
with the more intense shade for 99%.
Note that the residuals for the independence model were not large
(as shown in the legend),
yet the association between Treatment
and Improved
is highly significant.
summary(art)
In contrast, one of the other shading schemes, from @vcd:Friendly:1994
(use: gp = shading_Friendly
),
uses fixed cutoffs of $\pm 2, \pm 4$,
to shade cells which are individually significant
at approximately $\alpha = 0.05$ and $\alpha = 0.001$ levels, respectively.
The plot below uses gp = shading_Friendly
.
#| Arthritis2, #| fig.height = 6, #| fig.width = 7, #| fig.cap = "Mosaic plot for the `Arthritis` data, using `shading_Friendly`" mosaic(art, gp = shading_Friendly, split_vertical = TRUE, main="Arthritis: gp = shading_Friendly")
Mosaic plots using tables or frequency data frames as input typically take the levels of the table variables in the order presented in the dataset. For character variables, this is often alphabetical order. That might be helpful for looking up a value, but is unhelpful for seeing and understanding the pattern of association.
It is usually much better to order the levels of the row and column variables to help reveal the nature of their association. This is an example of effect ordering for data display [@FriendlyKwan:02:effect].
Example:
Data from @Glass:54 gave this 5 x 5 table on the occupations of 3500 British fathers and their sons, where the occupational categories are listed in alphabetic order.
data(Glass, package="vcdExtra") (glass.tab <- xtabs(Freq ~ father + son, data=Glass))
The mosaic display shows very strong association, but aside from the
diagonal cells, the pattern is unclear. Note the use of
set_varnames
to give more descriptive labels for the variables
and abbreviate the occupational category labels.
and interpolate
to set the shading levels for the mosaic.
largs <- list(set_varnames=list(father="Father's Occupation", son="Son's Occupation"), abbreviate=10) gargs <- list(interpolate=c(1,2,4,8)) mosaic(glass.tab, shade=TRUE, labeling_args=largs, gp_args=gargs, main="Alphabetic order", legend=FALSE, rot_labels=c(20,90,0,70))
The occupational categories differ in status, and can be reordered correctly as follows,
from Professional
down to Unskilled
.
# reorder by status ord <- c(2, 1, 4, 3, 5) row.names(glass.tab)[ord]
The revised mosaic plot can be produced by indexing the rows and columns of
the table using ord
.
mosaic(glass.tab[ord, ord], shade=TRUE, labeling_args=largs, gp_args=gargs, main="Effect order", legend=FALSE, rot_labels=c(20,90,0,70))
From this, and for the examples in the next section, it is useful to re-define
father
and son
as ordered factors in the original Glass
frequency data.frame.
Glass.ord <- Glass Glass.ord$father <- ordered(Glass.ord$father, levels=levels(Glass$father)[ord]) Glass.ord$son <- ordered(Glass.ord$son, levels=levels(Glass$son)[ord]) str(Glass.ord)
For mobility tables such as this, where the rows and columns refer to the same occupational categories it comes as no surprise that there is a strong association in the diagonal cells: most often, sons remain in the same occupational categories as their fathers.
However, the re-ordered mosaic display also reveals something subtler: when a son differs in occupation from the father, it is more likely that he will appear in a category one-step removed than more steps removed. The residuals seem to decrease with the number of steps from the diagonal.
For such tables, specialized loglinear models provide interesting cases
intermediate between the independence model, [A] [B],
and the saturated model, [A B]. These can be fit using glm()
, with the data
in frequency form,
glm(Freq ~ A + B + assoc, data = ..., family = poisson)
where assoc
is a special term to handle a restricted form of association, different from
A:B
which specifies the saturated model in this notation.
Quasi-independence: Asserts independence, but ignores the diagonal cells by fitting them exactly. The loglinear model is: $\log m_{ij} = \mu + \lambda^A_i + \lambda^B_j + \delta_i I(i = j)$, where $I()$ is the indicator function.
Symmetry: This model asserts that the joint distribution of the row and column variables is symmetric, that is $\pi_{ij} = \pi_{ji}$: A son is equally likely to move from their father's occupational category $i$ to another category, $j$, as the reverse, moving from $j$ to $i$. Symmetry is quite strong, because it also implies marginal homogeneity, that the marginal probabilities of the row and column variables are equal, $\pi{i+} = \sum_j \pi_{ij} = \sum_j \pi_{ji} = \pi_{+i}$ for all $i$.
Quasi-symmetry: This model uses the standard main-effect terms in the loglinear model, but asserts that the association parameters are symmetric, $\log m_{ij} = \mu + \lambda^A_i + \lambda^B_j + \lambda^{AB}{ij}$, where $\lambda^{AB}{ij} = \lambda^{AB}_{ji}$.
The gnm package provides a variety of these functions:
gnm::Diag()
, gnm::Symm()
and gnm::Topo()
for an interaction factor as specified by an array of levels, which may be arbitrarily structured.
For example, the following generates a term for a diagonal factor in a $4 \times 4$ table. The diagonal values reflect parameters fitted for each diagonal cell. Off-diagonal values, "." are ignored.
rowfac <- gl(4, 4, 16) colfac <- gl(4, 1, 16) diag4by4 <- Diag(rowfac, colfac) matrix(Diag(rowfac, colfac, binary = FALSE), 4, 4)
Symm()
constructs parameters for symmetric cells. The particular values don't matter.
All that does matter is that the same value, e.g., 1:2
appears in both the (1,2) and
(2,1) cells.
symm4by4 <- Symm(rowfac, colfac) matrix(symm4by4, 4, 4)
Example:
To illustrate, we fit the four models below, starting with the independence model
Freq ~ father + son
and then adding terms to reflect the restricted
forms of association, e.g., Diag(father, son)
for diagonal terms and
Symm(father, son)
for symmetry.
library(gnm) glass.indep <- glm(Freq ~ father + son, data = Glass.ord, family=poisson) glass.quasi <- glm(Freq ~ father + son + Diag(father, son), data = Glass.ord, family=poisson) glass.symm <- glm(Freq ~ Symm(father, son), data = Glass.ord, family=poisson) glass.qsymm <- glm(Freq ~ father + son + Symm(father, son), data = Glass.ord, family=poisson)
We can visualize these using the vcdExtra::mosaic.glm()
method, which extends
mosaic displays to handle fitted glm
objects. Technical note: for
models fitted using glm()
, standardized residuals, residuals_type="rstandard"
have better statistical properties than the default Pearson residuals in mosaic
plots and analysis.
mosaic(glass.quasi, residuals_type="rstandard", shade=TRUE, labeling_args=largs, gp_args=gargs, main="Quasi-Independence", legend=FALSE, rot_labels=c(20,90,0,70) )
Mosaic plots for the other models would give further visual assessment of these models,
however we can also test differences among them. For nested models, anova()
gives tests
of how much better a more complex model is compared to the previous one.
# model comparisons: for *nested* models anova(glass.indep, glass.quasi, glass.qsymm, test="Chisq")
Alternatively, vcdExtra::LRstats()
gives model summaries for a collection of
models, not necessarily nested, with AIC and BIC statistics reflecting
model parsimony.
models <- glmlist(glass.indep, glass.quasi, glass.symm, glass.qsymm) LRstats(models)
By all criteria, the model of quasi symmetry fits best. The residual deviance $G^2
is not significant. The mosaic is largely unshaded, indicating a good fit, but there
are a few shaded cells that indicate the remaining positive and negative residuals.
For comparative mosaic displays, it is sometimes useful to show the $G^2$ statistic
in the main title, using vcdExtra::modFit()
for this purpose.
mosaic(glass.qsymm, residuals_type="rstandard", shade=TRUE, labeling_args=largs, gp_args=gargs, main = paste("Quasi-Symmetry", modFit(glass.qsymm)), legend=FALSE, rot_labels=c(20,90,0,70) )
When natural orders for row and column levels are not given a priori, we can find orderings that make more sense using correspondence analysis.
The general ideas are that:
Correspondence analysis assigns scores to the row and column variables to best account for the association in 1, 2, ... dimensions
The first CA dimension accounts for largest proportion of the Pearson $\chi^2$
Therefore, permuting the levels of the row and column variables by the CA Dim1 scores gives a more coherent mosaic plot, more clearly showing the nature of the association.
The seriation package now has a method to order variables in frequency tables using CA.
Example:
As an example, consider the HouseTasks
dataset, a 13 x 4 table of frequencies of household tasks performed by couples, either by the Husband
, Wife
, Alternating
or Jointly
.
You can see from the table that some tasks (Repairs) are done largely by the husband; some (laundry, main meal)
are largely done by the wife, while others are done jointly or alternating between husband and
wife. But the Task
and Who
levels are both in alphabetical order.
data("HouseTasks", package = "vcdExtra") HouseTasks
The naive mosaic plot for this dataset is shown below, splitting first by Task
and then by Who
. Due to the length of the factor labels, some features of labeling
were used to make the display more readable.
require(vcd) mosaic(HouseTasks, shade = TRUE, labeling = labeling_border(rot_labels = c(45,0, 0, 0), offset_label =c(.5,5,0, 0), varnames = c(FALSE, TRUE), just_labels=c("center","right"), tl_varnames = FALSE), legend = FALSE)
Correspondence analysis, using the ca package, shows that nearly 89% of the $\chi^2$ can be accounted for in two dimensions.
require(ca) HT.ca <- ca(HouseTasks) summary(HT.ca, rows=FALSE, columns=FALSE)
The CA plot has a fairly simple interpretation: Dim1 is largely the distinction between tasks primarily done by the wife vs. the husband. Dim2 distinguishes tasks that are done singly vs. those that are done jointly.
plot(HT.ca, lines = TRUE)
So, we can use the CA
method of seriation::seriate()
to find the order of permutations of
Task
and Who
along the CA dimensions.
require(seriation) order <- seriate(HouseTasks, method = "CA") # the permuted row and column labels rownames(HouseTasks)[order[[1]]] colnames(HouseTasks)[order[[2]]]
Now, use seriation::permute()
to use order
for the permutations of Task
and Who
,
and plot the resulting mosaic:
# do the permutation HT_perm <- permute(HouseTasks, order, margin=1) mosaic(HT_perm, shade = TRUE, labeling = labeling_border(rot_labels = c(45,0, 0, 0), offset_label =c(.5,5,0, 0), varnames = c(FALSE, TRUE), just_labels=c("center","right"), tl_varnames = FALSE), legend = FALSE)
It is now easy to see the cluster of tasks (laundry and cooking) done largely by the wife at the top, and those (repairs, driving) done largely by the husband at the bottom.
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