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pdftitle={Homogeneity Analysis of Durant Bend Sherds},
pdfauthor={Jan de Leeuw and C. Roger Nance},
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\title{Homogeneity Analysis of Durant Bend Sherds}
\pretitle{\vspace{\droptitle}\centering\huge}
\posttitle{\par}
\author{Jan de Leeuw and C. Roger Nance}
\preauthor{\centering\large\emph}
\postauthor{\par}
\predate{\centering\large\emph}
\postdate{\par}
\date{Version February 24, 2018}
\begin{document}
\maketitle
\begin{abstract}
This paper is a non-technical and mostly graphical introduction to
\emph{Homogeneity Analysis}, also known as \emph{Multiple Correspondence
Analysis}. It is meant as an explanation and justification of a
non-standard application of \emph{Correspondence Analysis} to an example
from archeology.
\end{abstract}
{
\setcounter{tocdepth}{3}
\tableofcontents
}
\section{Data}\label{data}
The data for this paper are 5888 pot sherds excavated in 1971 and 1975
from several sites in the vicinity of Durant Bend, Dalla County,
Alabama. Each sherd was labeled by site and by the depth level in the
excavation. In addition each sherd was classified usng three binary
variables: design (check stamped vs plain), paste color (dark vs light),
and thickness (thick vs thin). For the details we refer to Nance (1976)
and Nance and De Leeuw (2018). In the following table the sherds are
aggregated over sites/depths.
\begin{verbatim}
## CS Plain Dark Light Thin Thick
## Ds73S1 118 555 425 248 157 516
## Ds73S2 171 302 304 169 126 347
## Ds73S3 83 156 163 76 67 172
## Ds73S4 36 51 59 28 24 63
## Ds73S5+ 25 50 46 29 19 56
## Ds73NAM 203 964 656 511 183 984
## Ds73NUM 196 389 342 243 100 485
## Ds73NLM 164 199 229 134 64 299
## Ds73NBM 74 85 102 57 27 132
## Ds791 11 292 170 133 125 178
## Ds792 17 247 163 101 116 148
## Ds793 24 163 100 87 62 125
## Ds794+ 3 39 25 17 18 24
## Au1131 7 34 30 11 30 11
## Au1132 14 40 38 16 32 22
## Au1133+ 15 18 25 8 22 11
## Ds98PZ 20 219 166 73 183 56
## DBUCC 11 110 36 85 85 36
## DBLCC 9 45 32 22 47 7
## DBBCC1 13 20 19 14 29 4
## DBBCC2 28 34 45 17 49 13
## DBBCC3 90 79 132 37 147 22
## DBBCC4 83 67 106 44 111 39
## DBBCC5+ 66 45 81 30 81 30
## DS971 1 75 60 16 49 27
## DS972 7 67 47 27 51 23
## DS973+ 9 45 43 11 38 16
\end{verbatim}
\section{Homogeneity Analysis}\label{homogeneity-analysis}
The technique we will use to analyse the Durent Bend data is Homogeneity
Analysis (Gifi (1990)), which is more widely known as \emph{Multiple
Correspondence Analysis} (Greenacre (1984), Greenacre and Blasius
(2006)). We give a graphical introduction to Homogeneity Analysis,
without using formulas.
Suppose we have \(m\) categorical \emph{variables}, and that variable
\(j\) has \(k_j\) \emph{categories} \((j=1,\cdots,m)\). The \(m\)
variables \emph{categorize} or \emph{measure} \(n\) \emph{objects}.
Variable \(j\) partitions the set of \(n\) objects into \(k_j\) subsets,
one subset for each category. Before we get to the analysis of the
Durant Bend sherds, we will illustrate the main concepts of our approach
with a small example in which three variables partition ten objects. The
first two variables have three categories, the last one has two
categories.
\begin{verbatim}
## first second third
## 01 a p u
## 02 b q v
## 03 a r v
## 04 a p u
## 05 b p v
## 06 c p v
## 07 a p u
## 08 a p v
## 09 c p v
## 10 a p v
\end{verbatim}
In Homogeneity Analysis we aim to make a \emph{joint plot} of the
objects and the categories of the variables. Joint plots are also known
as \emph{biplots} (Gower and Hand (1996)). In a joint plot both objects
and categories are represented as points in a low-dimensional space,
usually the plane, in such a way that the relations in the data are
represented as precisely as possible in the plot. We will specify what
we mean by ``as precisely as possible'' below. Homogeneity Analysis is
defined by defining a measure of the loss of information in a certain
way, and then choosing the representation that minimizes that loss.
The \(n\) points in the plan, or more generally in \(p\)-dimensional
space, representing the objects (sherds) are collected in a matrix of
\emph{object scores}. The \(k_j\) points representing the categories of
variable \(j\) are in a matrix of \emph{category quantifications} for
variabe \(j\). For each variable we can make a \emph{graph plot} in
which each of the \(n\) object scores is connected by a straight line to
the quantification of the category that this object falls in. Thus there
is one line departing from each object point, while the number of lines
arriving at a category point is equal to the number of objects in the
category. One graph plot has \(n\) lines, all graph plots together have
\(n\times m\) lines.
If all these lines have length zero, then all objects coincide with
``their'' categories for that variable, and thus we have reproduced the
data exactly. If there is more than one variable, however, we cannot
expect to have such a perfect representation, because objects which are
together in a category for one variable may not be together for another
variable.
Homogeneity Analysis is defined as the technique that produces a joint
plot of objects and category quantification in such a way that the total
length of all \(n\times m\) lines in the \(m\) graph plots is a small as
possible. Some qualifications are needed, however. We actually minimizes
the sum of the squared length of the lines, for the same reason that we
use the squares of the residuals in a regression analysis. It simplifies
the mathematics and the computation to use squared distances. Secondly,
we could trivially gain our objective of minimizing line length by
collapsing all object scores and category quantifications in a single
point, which makes our loss function equal to zero, but is useless in
representing or reproducing the data. Thus we need some form of
\emph{normalization} to prevent this trivial solution from happening. In
Homogeneity Analysis we require the columns of the object score matrix
add up to zero, have sum of squares equal to one, and are uncorrelated.
Let's illustrate this with our small example. We start with a completely
aribitrary \emph{initial configuration}. The ten objects are placed at
equal distances on a circle, and the categories for each of the
variables are on the horizontal axis. This leads to the first three
graph plots, which we have superimposed to get the fourth plot with
\(n\times m = 30\) lines.
\includegraphics{sherds_files/figure-latex/points-1.pdf}
Figure 1: Graph Plots Initial Configuration, Small Example
For this arbitrary initial configuration the loss, i.e.~the sum of
squares of the line lengths, or the sum of the squred distances between
objects and the categories they fall in, is equal to 8.1081098.
\section{Reciprocal Averaging}\label{reciprocal-averaging}
In Homogeneity Analysis we minimize loss by what is known as
\emph{reciprocal averaging} or \emph{alternating least squares}. We
alternate two substeps. The first substep improves the category
quantifications for a given set of object scores, the second substep
improves and normalizes the object scores for a given set of category
quantifications, namely those we have just computed in the first
substep. Taken together these two substeps are an \emph{iteration}. So
each iteration starts with object scores and category quantifications
and uses its two substeps to improve both. Each of the two substeps
decreases the loss functions, i.e.~the total squared length of the lines
in the graph plots.
The two substeps are both very simple. Let's look at the first one. We
compute optimal category quantifications for given object scores by
taking the averages (or \emph{centroids}) of the objects scores in each
of the categories. The corresponding graph plots are
\includegraphics{sherds_files/figure-latex/updateY-1.pdf}
Figure 2: Graph Plots, Iteration 1, substep 1, Small Example
and the loss has decreased to 5.2016654. Note that we have not improved
the object scores yet, so they are still their initial configuration,
equally spaced on a circle. Also note, in variable 2 for instance, that
category quantifications coincide with object scores, and thus
contribute zero to the loss, if the object is the only observation in
the category. In addition, because category quantifications are averages
of objects points, they are in the convex hull of the object points,
which means in this figure that they are within the circle. Averaging
objects points makes the category quantifications move closer to the
origin.
The second substep improves the object scores, while keeping the
category quantifications in the locations we have just computed in the
first substep. The second substaep has itself two substeps, say \(2A\)
and \(2B\). In the substep \(2A\) the score of an object for given
category quantifications is computed as the average or centroid of the
\(m\) category quantifications the object is in.
\includegraphics{sherds_files/figure-latex/updateXa-1.pdf}
Figure 3: Graph Plots, Iteration 1, substep 2A, Small Example
The loss function is down all the way to 0.4852424. This is not a proper
loss value, however, because the object scores are no longer centered,
standardized, and uncorrelated, and that was a Homogeneity Analysis
requirement. Substep 1 shrinks the object scores towards the origin by
averaging, substep 2A takes the resulting category quantifications and
shrinks them more by even more averaging. Thus in substep \(2B\) we have
to renormalize the object scores such that they are centered,
standardized, and uncorrelated. This gives
\includegraphics{sherds_files/figure-latex/updateXb-1.pdf}
Figure 4: Graph Plots, Iteration 1, substep 2B, Small Example
Loss, which is now the proper loss for a normalized configuration, has
decreased to 4.5538169.
Now that we have new category quantifications and new suitably
normalized object scores we can start the next iteration, and again
improve both in two substeps. Ultimately, after a certain number of
iterations, there is no change any more from one iteration to another,
and we have reached the optimal solution. In other words, there is
convergence, and our Homogeneity Analysis is finished. Note that the
renormalization in step 2B is necessary, because without it both object
scores and category quantifications would become smaller and smaller,
and converge to the origin. Of course the origin does have loss zero,
but it is never a proper description of the data.
The optimal graph plots, after the iterations have converged, are
\includegraphics{sherds_files/figure-latex/optimum-1.pdf}
Figure 5: Graph Plots, Optimum Solution, Small Example
The minimum loss for these data is 2.8377218.
Because the object scores are in deviations from the mean, and the
category quantifications are weighted means of object scores it follows
that category quantifications are in deviations from the weighted mean,
with weights equal to the marginals of the variable. Thus category
quantifications for each variable are distributed around the origin.
In Homogeneity Analysis the graph plots for individual variables are
often called \emph{star plots}, because the optimal category
quantification is the centroid of the scores of the objects in the
category. Thus it is somewhere in the middle of a bunch of objects,
which are connected to it by lines. Thus the subset of the graph for
each category is a star graph, and the corresponding plot for the
variable with these stars is a star plot. The top three plots in figure
5 are examples of such star plots. One could formulate the objective of
Homeogeneity Analysis as finding normalized object scores in such a way
that the stars (over all categories of all variables) are as small as
possible. Or, in yet another formulation, we want to maximize the
between-category variation and minimize the within-category variation.
Variables with a small star, which is necessarily close to the origin,
have poor discrimination power. The average object score for each
category is about the same. In general, categories with a large number
of observations will have an average close to the average of all
observations, and thus they will be close to the origin. And,
conversely, categories with a small number of observations will tend to
be relatively far from the origin.
\section{Specifics}\label{specifics}
\subsection{Passive, Supplementary, and Constraining
Variables}\label{passive-supplementary-and-constraining-variables}
Now, going back to the Durent Bend data, we do not have the values of
all 5888 sherds on the three variables. The sherds are aggregated over
various site/depth combinations and the original data cannot be
recovered from the aggregated table. But the framework of Homogeneity
Analysis can still be applied by using \emph{equality restrictions} (Van
Buuren and De Leeuw (1992)). The only thing added is that we require
that sherds in the same site/depth get the same object score. Or,
geometrically, all sherds in the same site/depth are mapped into the
same point in the joint plot.
The Homogeneity Analysis loss function is still minimized in two steps.
The first step, updating the category quantifications, is still the same
as in Homogeneity Analysis without equality restrictions. The second
step, which updates the object scores, now has three substeps instead of
two. In substep \(2A\) we compute the average of the category
quantifications of the categories the sherd is in. In substep \(2B\) we
replace these tentative object scores for the sherds by the site/depth
averages, and in substep \(2C\) we normalize the object scores, making
them centered, standardized, and uncorrelated.
The graph plots on unconstrained Homogeneity Analysis must now be
replaced by plots of \emph{valued graphs}. For any variable each
site/depth point is now connected to all category points, and the edge
connecting the object and category point has a value equal to the number
of sherds in the category.
As an example we use the GALO data, which has been used innumerable
times before as a Homogeneity Analysis example, and is in the
\texttt{Gifi} package in R (P. Mair and De Leeuw (2017)). The GALO data
can be used to show the difference between working with aggregated data
(over sherds in the same site or students in the same school) and the
raw data, which are actually unavailable for Durant Bend. Here is the
description of the GALO data in the help file of the package.
\begin{verbatim}
galo Gifi R Documentation
GALO dataset
Description
The objects (individuals) are 1290 school children in the sixth grade of elementary school in the city of Groningen (Netherlands) in 1959.
Usage
galo
Format
Data frame with the five variables Gender, IQ, Advice, SES and School. IQ (original range 60 to 144) has been categorized into 9 ordered categories and the schools are enumerated from 1 to 37.
SES:
LoWC = Lower white collar; MidWC = Middle white collar; Prof = Professional, Managers; Shop = Shopkeepers; Skil = Schooled labor; Unsk = Unskilled labor.
Advice:
Agr = Agricultural; Ext = Extended primary education; Gen = General; Grls = Secondary school for girls; Man = Manual, including housekeeping; None = No further education; Uni = Pre-University.
References
Peschar, J.L. (1975). School, Milieu, Beroep. Groningen: Tjeek Willink.
\end{verbatim}
Note that IQ is measured by the GIT (Groningen Intelligence Test) and
that Advice refers to the sixth grade teachers advice abut the most
appropriate form of secondary educations for the students.
We first ignore the school variable, and only analyze the four variables
Gender, IQ, Advice, and SES. Separate joint plots for the four
variables, with both object scores and category quatifications, are in
figure 6. We do not make star plots (by drawing the lines from the
object points to the category points they are in) in this case, because
1290 lines in a plot just create a big black blob.
The joint plots show a curved one-dimensional solution with good
students on the left and poor students on the right. Such curved
solutions, sometimes called \emph{horseshoes}, are a familiar outcome of
Homogeneity Analysis when there is a dominant single dimension
explaining the results (in this case student achievement). Both IQ and
Advice differentiate students well (mainly because teachers rely on IQ
scores in their advice), which means they will have the smallest stars.
Girls tend to be better students than boys, and SES mainly contrasts the
two extremes categories PROF and UNSK.
\begin{center}\includegraphics{sherds_files/figure-latex/galo_sex-1} \end{center}
Figure 6: Joint Plots, GALO Example without School
Our first analysis does not use the School variable at all. In the
terminology of Gifi School is a \emph{passive variable}. There are a
number of different ways in which we can incorporate School. The first
is the obvious one: just repeat the Homogeneity Analysis over all five
variables and include School. The joint plots of the first four
variables are in figure 7.
\begin{center}\includegraphics{sherds_files/figure-latex/galo_with_school-1} \end{center}
Figure 7: Joint Plots, GALO Example with School
The horseshoe pattern is still there, but it is less pronounced, mainly
because of several outlying students at the bottom of the plot more or
less defining the vertical dimension. This is due to including the
School variable. The joint plot for the School variable is in figure 8.
We see the outliers are in school 25, a small school with 11 low-IQ
students, possibly some type of special education school. The figure
also shows the star for school 25. The actual data for the eleven
students are
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{galo[galo[,}\DecValTok{5}\NormalTok{]}\OperatorTok{==}\StringTok{"25"}\NormalTok{,]}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
## gender IQ advice SES School
## 898 F 3 None Skil 25
## 899 F 4 Agr Skil 25
## 900 F 5 Man Skil 25
## 901 F 5 Man Skil 25
## 902 F 5 Man Skil 25
## 903 F 5 Agr LoWC 25
## 904 F 4 None Skil 25
## 905 M 4 None Prof 25
## 906 F 4 Man LoWC 25
## 907 F 2 None Skil 25
## 908 F 3 None Skil 25
\end{verbatim}
\begin{center}\includegraphics{sherds_files/figure-latex/galo_school_sep-1} \end{center}
Figure 8: Joint Plot, GALO Example with School, School Variable
There is another, and perhaps more interesting, way to incorporate
School in our analysis, by using it as what is commonly known as a
\emph{supplementary variable}. Such a supplementary variable does not
actively enter into the Homogeneity Analysis, but after the analysis of
the remaining variables we can compute category quantifications of the
supplementary variable as centroids of object scores in the categories.
Thus we can make star plots for the passive variables that have not been
used in the analysis. This is done in figure 9. The horseshoe of object
scores does not change from the one in figure 6, but by not including
School in the analysis we do not give school 25 the opportunity to
dominate the second dimension. It is still true that the same schools
(5, 24, 25, 37) perform poorly, and the same schools (4, 17, 31, 32)
perform well, but generaly the category quantifications are more
well-behaved.
\begin{center}\includegraphics{sherds_files/figure-latex/supple-1} \end{center}
Figure 9: Object Score Plot, School Supplementary, GALO Example
We now repeat the analysis, requiring that students in the same school
get the same object score. We treat school as a \emph{constraining
variable}, performing a Homogeneity Analysis with equality restrictions
on the object scores (Van Buuren and De Leeuw (1992)). In terms of the
joint plot we require the stars for the school variable to collapse into
single points. Computationally this is easiest to do using the the R
package \texttt{anacor} (De Leeuw and Mair (2009)). In this constrained
Homogeneity Analysis each school gets an object score, and these scores
are plotted in figure 10. Not surprisingly school 25 is now even more of
an outlier, but otherwise schools are dispersed pretty much in the same
way as before. For the GALO example this constrained analysis throws
away useful information and gives a result which is inferior to the
supplementary variable approach. For the archeological data we do not
ignore within-site information, because the data are aggregated over
sherds in the same site to begin with.
\begin{center}\includegraphics{sherds_files/figure-latex/aggregate-1} \end{center}
Figure 10: Object Score Plot, School Constraints, GALO Example
\subsection{Binary Variables}\label{binary-variables}
Besides aggregation, another proerty of the Durant Bend data is that the
three variables describing the sherds are binary (CS/Plain, Dark/Light,
Thin/Thick). This implies some special properties of the Homogeneity
Analysis.
We have seen that category quantifications are in deviations from the
weighted mean, with the weights equal to the marginal frequencies of the
variable. If a variable has only two categories, and our Homogeneity
Analysis has two dimensions, that means that the two category
quantifications for a variable are on a line through the origin. The
direction of the line is determined by the marginals of the variables.
What Homogeneity Analysis gives us is how far away from the origin the
category quantifications are placed on the line to get the smallest
stars.
We have said very little so far about the number of dimensions we choose
for our Homogeneity Analysis. The default is to choose two, because
two-dimensional joint and graph plots are the easiest to look at. The
maximum number of dimensions in Homogeneity Analysis, i.e.~the number of
dimensions that are needed to represent all variation in the data, is
equal to the total number of categories minus the number of variables.
In the GALO example (without School) that is \(2+9+6+7-4=20\) but in the
Durant Bend example it is \(2+2+2-3=3\). Only three dimensions will
capture all variation.
\section{Analysis Durant Bend Data}\label{analysis-durant-bend-data}
The Durant Bend analysis is an aggregated Homogeneity Analysis of three
binary variables, requiring equal object scores for all sherds in the
same site/depth. The joint plot is in figure 11. For a discussion of
these results we refer to the companion paper by Nance and De Leeuw
(2018). There is not much variation in the category quantifications of
the three variables (the lines are rather short). In particular, the
averages for light sherds and for dark sherds are very close, indicating
not much discriminatiry power for that variable.
\includegraphics{sherds_files/figure-latex/anacor-1.pdf}
Figure 11: Joint Plot, Durant Bend, Two Dimensions
We have to realize, of course, that there are only three dimensions
available to describe our data. This makes it interesting to look at the
three-dimensional solution, specifically at light vs dark sherds. We
more or less expect each variable to define a dimension, indicating
relatively low correlations between the three variables, and
consequently not much difference between sites. The summary of the three
dimensional Homogeneity Analysis from \texttt{anacor} is
\begin{verbatim}
##
## CA fit:
##
## Total chi-square value: 2230.319
## Sum of eigenvalues (total inertia): 0.126
## Eigenvalues (principal inertias):
## 0.079 0.042 0.005
##
## Benzecri RMSE columns: 2.928824e-20
##
## Chi-square decomposition:
## Chisq Proportion Cumulative Proportion
## Dimension 1 1391.515 0.624 0.624
## Dimension 2 744.667 0.334 0.958
## Dimension 3 94.136 0.042 1.000
## Dimension 4 0.000 0.000 1.000
## Dimension 5 0.000 0.000 1.000
\end{verbatim}
The two three-dimensional scatterplots, one for categoiry
quantifications and one for sites, are in figures 14 and 15. We see that
the third dimension indeed separates the light from the dark.
\begin{center}\includegraphics{sherds_files/figure-latex/3dc-1} \end{center}
Figure 12: Category Quantifiations, Durant Bend, Three Dimensions
\begin{center}\includegraphics{sherds_files/figure-latex/3dr-1} \end{center}
Figure 13: Object Scores, Durant Bend, Three Dimensions
\section*{References}\label{references}
\addcontentsline{toc}{section}{References}
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\href{\%7Bhttps://R-Forge.R-project.org/projects/psychor/\%7D}{\{https://R-Forge.R-project.org/projects/psychor/\}}.
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\end{document}