Graphical Aid in Correspondence Analysis Interpretation and Significance Testings.
CAinterprTools
vers 1.1.0
A number of interesting packages are available to perform Correspondence Analysis in R. At the best of my knowledge, however, they lack some tools to help users to eyeball some critical CA aspects (e.g., contribution of rows/cols categories to the principal axes, quality of the display,correlation of rows/cols categories with dimensions, etc). Besides providing those facilities, this package allows calculating the significance of the CA dimensions by means of the 'Average Rule', the Malinvaud test, and by permutation test. Further, it allows to also calculate the permuted significance of the CA total inertia.
The package comes with some datasets drawn from literature:
brand_coffee: after Kennedy R et al, Practical Applications of Correspondence Analysis to Categorical Data in Market Research, in Journal of Targeting Measurement and Analysis for Marketing, 1996
breakfast: after Bendixen M, A Practical Guide to the Use of Correspondence Analysis in Marketing Research, in Research on-line 1, 1996, 16-38 (table 5)
diseases: after Velleman P F, Hoaglin D C, Applications, Basics, and Computing of Exploratory Data Analysis, Wadsworth Pub Co 1984 (Exhibit 8-1)
fire_loss: after Li et al, Influences of Time, Location, and Cause Factors on the Probability of Fire Loss in China: A Correspondence Analysis, in Fire Technology 50(5), 2014, 1181-1200 (table 5)
greenacre_data: after Greenacre M, Correspondence Analysis in Practice, Boca Raton-London-New York, Chapman&Hall/CRC 2007 (exhibit 12.1)
List of implemented functions
aver.rule: average rule chart.caCluster: clustering row/column categories on the basis of Correspondence Analysis coordinates from a space of user-defined dimensionality.caCorr(): chart of correlation between rows and columns categories.caPercept(): perceptual map-like Correspondence Analysis scatterplot.caPlot(): intepretation-oriented Correspondence Analysis scatterplots, with informative and flexible (non-overlapping) labels.caPlus(): facility for interpretation-oriented CA scatterplot.caScatter(): basic scatterplot visualization facility.cols.cntr(): columns contribution chart.cols.cntr.scatter(): scatterplot for column categories contribution to dimensions.cols.qlt(): chart of columns quality of the display.groupBycoord(): define groups of categories on the basis of a selected partition into k groups employing the Jenks' natural break method on the selected dimension's coordinates.malinvaud(): Malinvaud's test for significance of the CA dimensions.rescale(): rescale row/column categories coordinates between a minimum and maximum value.rows.cntr(): rows contribution chart.rows.cntr.scatter(): scatterplot for row categories contribution to dimensions.rows.qlt(): chart of rows quality of the display.sig.dim.perm(): permuted significance of CA dimensions.sig.dim.perm.scree(): scree plot to test the significance of CA dimensions by means of a randomized procedure.sig.tot.inertia.perm(): permuted significance of the CA total inertia.table.collapse(): collapse rows and columns of a table on the basis of hierarchical clustering.
Description of implemented functions
aver.rule(): allows you to locate the number of dimensions which are important for CA interpretation, according to the so-called average rule. The reference line showing up in the returned histogram indicates the threshold of an optimal dimensionality of the solution according to the average rule.
caCluster(): plots the result of cluster analysis performed on the results of Correspondence Analysis, and plots a dendrogram, a silouette plot depicting the "quality" of the clustering solution, and a scatterplot with points coded according to the cluster membership. The function provides the facility to perform hierarchical cluster analysis of row and/or column categories on the basis of Correspondence Analysis result. The clustering is based on the row and/or colum categories' coordinates from:
- (1) a high-dimensional space corresponding to the whole dimensionality of the input contingency table;
- (2) a high-dimensional space of dimensionality smaller than the full dimensionality of the input dataset;
- (3) a bi-dimensional space defined by a pair of user-defined dimensions.
To obtain (1), the dim parameter must be left in its default value (NULL); to obtain (2), the dim parameter must be given an integer (needless to say, smaller than the full dimensionality of the input data); to obtain (3), the dim parameter must be given a vector (e.g., c(1,3)) specifying the dimensions the user is interested in.
The method by which the distance is calculated is specified using the dist.meth parameter, while the agglomerative method is speficied using the aggl.meth parameter. By default, they are set to euclidean and ward.D2 respectively.
The user may want to specify beforehand the desired number of clusters (i.e., the cluster solution). This is accomplished feeding an integer into the 'part' parameter. A dendrogram (with rectangles indicating the clustering solution), a silhouette plot (indicating the "quality" of the cluster solution), and a CA scatterplot (with points given colours on the basis of their cluster membership) are returned. Please note that, when a high-dimensional space is selected, the scatterplot will use the first 2 CA dimensions; the user must keep in mind that the clustering based on a higher-dimensional space may not be well reflected on the subspace defined by the first two dimensions only.
Also note:
if both row and column categories are subject to the clustering, the column categories will be flagged by an asterisk (*) in the dendrogram (and in the silhouette plot) just to make it easier to identify rows and columns;
the silhouette plot displays the average silhouette width as a dashed vertical line; the dimensionality of the CA space used is reported in the plot's title; if a pair of dimensions has been used, the individual dimensions are reported in the plot's title;
the silhouette plot's labels end with a number indicating the cluster to which each category is closer.
An optimal clustering solution can be obtained setting the opt.part parameter to TRUE. The optimal partition is selected by means of an iterative routine which locates at which cluster solution the highest average silhouette width is achieved. If the opt.part parameter is set to TRUE, an additional plot is returned along with the silhouette plot. It displays a scatterplot in which the cluster solution (x-axis) is plotted against the average silhouette width (y-axis). A vertical reference line indicate the cluster solution which maximize the silhouette width, corresponding to the suggested optimal partition.
The function returns a list storing information about the cluster membership (i.e., which categories belong to which cluster).
Further info and Disclaimer about the caCluster() function:
The silhouette plot is obtained from the silhouette() function out from the cluster package. For a detailed description of the silhouette plot, its rationale, and its interpretation, see:
- Rousseeuw P J. 1987. Silhouettes: A graphical aid to the interpretation and validation of cluster analysis, Journal of Computational and Applied Mathematics 20, 53-65
For the idea of clustering categories on the basis of the CA coordinates from a full high-dimensional space (or from a subset thereof), see:
- Ciampi et al. 2005. Correspondence analysis and two-way clustering, SORT 29 (1), 27-4
- Beh et al. 2011. A European perception of food using two methods of correspondence analysis, Food Quality and Preference 22(2), 226-231
Please note that the interpretation of the clustering when both row AND column categories are used must procede with caution due to the issue of inter-class points' distance interpretation. For a full description of the issue (also with further references), see:
- Greenacre M. 2007. Correspondence Analysis in Practice, Boca Raton-London-New York, Chapman&Hall/CRC, 267-268.
caCorr(): allows you to calculate the strenght of the correlation between rows and columns of the contingency table. A reference line indicates the threshold above which the correlation can be considered important.
caPercept(): plots a variant of the traditional Correspondence Analysis scatterplots that allows facilitating the interpretation of the results. It aims at producing what in marketing research is called perceptual map, a visual representation of the CA results that seeks to avoid the problem of interpreting inter-spatial distance. It represents only one type of points (say, column points), and "gives names to the axes" corresponding to the major row category contributors to the two selected dimensions.
caPlot(): plots different types of CA scatterplots, adding information that are relevant to the CA interpretation. Thanks to the ggrepel package, the labels tends to not overlap so producing a nicely readable chart. The function provides the facility to produce:
(1) a regular (symmetric) scatterplot, in which points' labels only report the categories' names;
(2) a scatterplot with advanced labels. If the user's interest lies (for instance) in interpreting the rows in the space defined by the column categories, by setting the parameter 'cntr' to "columns" the columns' labels will be coupled with two asterisks within round brackets; each asterisk (if present) will indicate if the category is a major contributor to the definition of the first selected dimension (if the first asterisk to the left is present) and/or if the same category is also a major contributor to the definition of the second selected dimension (if the asterisk to the right is present). The rows' labels will report the correlation (i.e., sqrt(COS2)) with the selected dimensions; the correlation values are reported between square brackets; the left-hand side value refers to the correlation with the first selected dimensions, while the right-hand side value refers to the correlation with the second selected dimension. If the parameter 'cntr' is set to "rows", the row categories' labels will indicate the contribution, and the column categories' labels will report the correlation values.
(3) a perceptual map, in which axes' poles are given names according to the categories (either rows or columns, as specified by the user) having a major contribution to the definition of the selected dimensions; rows' (or columns') labels will report the correlation with the selected dimensions.
The function returns a dataframe containing data about row and column points:
- (a) coordinates on the first selected dimension
- (b) coordinates on the second selected dimension
- (c) contribution to the first selected dimension
- (d) contribution to the second selected dimension
- (e) quality on the first selected dimension
- (f) quality on the second selected dimension
- (g) correlation with the first selected dimension
- (h) correlation with the second selected dimension
- (j) (k) asterisks indicating whether the corresponding category is a major contribution to the first and/or second selected dimension.
caPlus(): plots Correspondence Analysis scatterplots modified to help interpreting the analysis' results. In particular, the function aims at making easier to understand in the same visual context:
- (a) which (say, column) categories are actually contributing to the definition of given pairs of dimensions;
- (b) which (say, row) categories are more correlated to which dimension.
caScatter(): allows to get different types of CA scatterplots. It is just a wrapper for functions from the ca and FactoMineR packages.
cols.cntr(): column equivalent of rows.cntr() (see below).
cols.cntr.scatter(): column equivalent of rows.cntr.scatter() (see below).
cols.corr(): column equivalent of rows.corr() (see below).
cols.corr.scatter(): column equivalent of rows.corr.scatter() (see below).
cols.qlt(): column equivalent of rows.qlt() (see below).
groupBycoord(): allows to group the row/column categories into k user-defined partitions. K groups are created employing the Jenks' natural break method applied on the selected dimension's coordinates. A dotchart is returned representing the categories grouped into the selected partitions. At the bottom of the chart, the Goodness of Fit statistic is also reported. The function also returns a dataframe storing the categories' coordinates on the selected dimension and the group each category belongs to.
malinvaud(): performs the Malinvaud test, which assesses the significance of the CA dimensions. The function returns both a table and a plot. The former lists relevant information, among which the significance of each CA dimension. The dotchart graphically represents the p-value of each dimension; dimensions are grouped by level of significance; a red reference lines indicates the 0.05 threshold.
rescale(): allows to rescale the coordinates of a selected dimension to be constrained between a minimum and a maximum user-defined value. The rationale of the function is that users may wish to use the coordinates on a given dimension to devise a scale, along the lines of what is accomplished in: Greenacre M 2002, The Use of Correspondence Analysis in the Exploration of Health Survey Data, Documentos de Trabajo 5, Fundacion BBVA, pp. 7-39. The function returns a chart representing the row/column categories against the rescaled coordinates from the selected dimension. A dataframe is also returned containing the original values (i.e., the coordinates) and the corresponding rescaled values.
rows.cntr(): calculates the contribution of the row categories to a selected dimension. It displays the contribution of the categories as a dotplot. A reference line indicates the threshold above which a contribution can be considered important for the determination of the selected dimension. The parameter sort=TRUE sorts the categories in descending order of contribution to the inertia of the selected dimension. At the left-hand side of the plot, the categories' labels are given a symbol (+ or -) according to wheather each category is actually contributing to the definition of the positive or negative side of the dimension, respectively. The categories are grouped into two groups: 'major' and 'minor' contributors to the inertia of the selected dimension. At the right-hand side, a legend (which is enabled/disabled using the leg parameter) reports the correlation (sqrt(COS2)) of the column categories with the selected dimension. A symbol (+ or -) indicates with which side of the selected dimension each column category is correlated.
rows.cntr.scatter(): plots a scatterplot of the contribution of row categories to two selected dimensions. Two references lines (in RED) indicate the threshold above which the contribution can be considered important for the determination of the dimensions. A diagonal line (in BLACK) is a visual aid to eyeball whether a category is actually contributing more (in relative terms) to either of the two dimensions. The row categories' labels are coupled with + or - symbols within round brackets indicating to which side of the two selected dimensions the contribution values that can be read off from the chart are actually referring. The first symbol (i.e., the one to the left), either + or -, refers to the first of the selected dimensions (i.e., the one reported on the x-axis). The second symbol (i.e., the one to the right) refers to the second of the selected dimensions (i.e., the one reported on the y-axis).
rows.corr(): calculates and graphically displays the correlation (sqrt(COS2)) of the row categories with the selected dimension. The parameter sort=TRUE arranges the categories in decreasing order of correlation. In the returned chart, at the left-hand side, the categories' labels show a symbol (+ or -) according to which side of the selected dimension they are correlated, either positive or negative. The categories are grouped into two groups: categories correlated with the positive ('pole +') or negative ('pole -') pole of the selected dimension. At the right-hand side, a legend indicates the column categories' contribution (in permils) to the selected dimension (value enclosed within round brackets), and a symbol (+ or -) indicating whether they are actually contributing to the definition of the positive or negative side of the dimension, respectively. Further, an asterisk (*) flags the categories which can be considered major contributors to the definition of the dimension:
rows.corr.scatter(): plots a scatterplot of the correlation (sqrt(COS2)) of row categories with two selected dimensions. A diagonal line (in BLACK) is a visual aid to eyeball whether a category is actually more correlated (in relative terms) to either of the two dimensions. The row categories' labels are coupled with two + or - symbols within round brackets indicating to which side of the two selected dimensions the correlation values that can be read off from the chart are actually referring. The first symbol (i.e., the one to the left), either + or -, refers to the first of the selected dimensions (i.e., the one reported on the x-axis). The second symbol (i.e., the one to the right) refers to the second of the selected dimensions (i.e., the one reported on the y-axis).
rows.qlt(): plots the quality of row categories display on the sub-space determined by a pair of selected dimensions.
sig.dim.perm(): calculates the significance of a pair of selected dimensions via a permutation test, and displays the results as a scatterplot; a large RED dot indicates the observed inertia. Permuted p-values are reported in the axes' labels.
sig.dim.perm.scree(): tests the significance of the CA dimensions by means of permutation of the input contingency table. A scree-plot displays for each dimension the observed eigenvalue and the 95th percentile of the permuted distribution of the corresponding eigenvalue. Observed eigenvalues that are larger than the corresponding 95th percentile are significant at least at alpha 0.05. P-values are displayed into the chart.
sig.tot.inertia.perm(): calculates the significance of the CA total inertia via permutation test; a histogram of the permuted total inertia is displayed along with the observed total inertia and the 95th percentile of the permuted total inertia. The latter can be regarded as a 0.05 alpha threshold for the observed total inertia's significance.
table.collapse(): allows to collapse the rows and columns of the input contingency table on the basis of the results of a hierarchical clustering. The function returns a list containing the input table, the rows-collapsed table, the columns-collapsed table, and a table with both rows and columns collapsed. It optionally returns two dendrograms (one for the row profiles, one for the column profiles) representing the clusters. The hierarchical clustering is obtained using the FactoMineRs HCPC() function.
Rationale: clustering rows and/or columns of a table could interest the users who want to know where a significant association is concentrated by collecting together similar rows (or columns) in discrete groups (Greenacre M, Correspondence Analysis in Practice, Boca Raton-London-New York, Chapman&Hall/CRC 2007, pp. 116, 120). Rows and/or columns are progressively aggregated in a way in which every successive merging produces the smallest change in the table’s inertia. The underlying logic lies in the fact that rows (or columns) whose merging produces a small change in table’s inertia have similar profiles. This procedure can be thought of as maximizing the between-group inertia and minimizing the within-group inertia. A method essentially similar is that provided by the FactoMineR package (Husson F, Le S, Pages J, Exploratory Multivariate Analysis by Example Using R, Boca Raton-London-New York, CRC Press, pp. 177-185). The cluster solution is based on the following rationale: a division into Q (i.e., a given number of) clusters is suggested when the increase in between-group inertia attained when passing from a Q-1 to a Q partition is greater than that from a Q to a Q+1 clusters partition. In other words, during the process of rows (or columns) merging, if the following agggregation raises highly the within-group inertia, it means that at the further step very different profiles are being aggregated.
History
version 1.1.0:
minor changes to optimize the calculation of permuted p-values returned by the functions
sig.dim.perm(),sig.dim.perm.scree(), andsig.tot.inertia.perm().sig.dim.perm.scree()andsig.dim.perm()now return permuted p-values in a dataframe (besides reporting them in the output plots).minor improvements and typo fixes to the package's help documentation.
version 1.0.0: first release to CRAN.