# Statistics Software Mac Correspondence Analysis

Here, we describe the simple correspondence analysis, which is used to analyze frequencies formed by two categorical data, a data table known as contengency table. It provides factor scores (coordinates) for both row and column points of contingency table. These coordinates are used to visualize graphically the association between row and column elements in the contingency table.

## Statistics Software Mac Correspondence Analysis

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The result of the analysis shows that, the contingency table has been successfully represented in low dimension space using correspondence analysis. The two dimensions 1 and 2 are sufficient to retain 88.6% of the total inertia (variation) contained in the data.

As mentioned above, the standard plot of correspondence analysis is a symmetric biplot in which both rows (blue points) and columns (red triangles) are represented in the same space using the principal coordinates. These coordinates represent the row and column profiles. In this case, only the distance between row points or the distance between column points can be really interpreted.

Only some of the rows and columns will be used to perform the correspondence analysis (CA). The coordinates of the remaining (supplementary) rows/columns on the factor map will be predicted after the CA.

In conclusion, we described how to perform and interpret correspondence analysis (CA). We computed CA using the CA() function [FactoMineR package]. Next, we used the factoextra R package to produce ggplot2-based visualization of the CA results.

This tutorial will help you set up and interpret a Multiple Correspondence Analysis in Excel using the XLSTAT software.Not sure if this is the right multivariate data analysis tool you need? Refer to this guide.

The next table shows the eight non-null eigenvalues and the corresponding % of inertia. However, unlike with CA (correspondence analysis performed on only 2 variables), the % of inertia are here pessimistic estimates of the quality of the representation, the latter being for the user "how close is the representation to the reality".Then, a table displays the coordinates of the categories in the factors space. The results that correspond to the supplementary variable are displayed in blue color.The coordinates of the observations are displayed further down.The contributions, the test values and the squared cosines help in the interpretation of the results. Before interpreting that two categories are close on the map, one should check that their contribution to the axes of the map, or that their squared cosines are high.The three following charts respectively correspond to the map of categories, the map of observations and the biplot containing both coordinates of observations and categories on the first two axes.From these charts, we may suggest that a customer will come back only if he is satisfied with the intervention, the welcome and the price. We also notice that there seems to be a link between the fact that the repair was not satisfactory and the fact that the welcome was bad. This should be investigated further: has the customer described the problem not precisely enough because he had been badly welcome or has the person called back to mention that the problem was still there and has been badly welcome by the representative?

Hi Jimmy,The standard Excel ToolPak supports this test, and so I have not added it to the Real Statistics data analysis tools.You can see how to use the Excel ToolPak to perform this test at -statistics.com/sampling-distributions/comparing-two-means-with-known-variancesi/You can see a complete description of how to access the ToolPak at -statistics.com/data-analysis-tools/CharlesCharles

Randy,It is not a data analysis tool. You can access it via the SlopesTest function. See -statistics.com/regression/hypothesis-testing-significance-regression-line-slope/comparing-slopes-two-independent-samples/Charles

The purpose of statistics hasbeen described as to summarise, simplify and eventually explain(Greenacre, 1984). Since codon usage by its very nature ismultivariate, it is necessary to analyse this datawith multivariate statistical techniques (i.e Correspondenceanalysis). If one examines the set of conventional statisticaltechniques in use today, it is clear that the statistician canrarely proceed without introducing a certain degree ofsubjectivity into the analysis (Greenacre, 1984). It is thereforeadvantageous to use a method that can examine data without apriori assumptions and there are several multivariateanalysis techniques that satisfy this condition (Greenacre,1984).

Multivariate analyses (MVA) areused to simplify rectangular matrices in which (for our purposes)the columns represent some measurement of codon usage or aminoacid usage and the rows represent individual genes. Examples ofMVA techniques that have been successfully applied to theanalysis of codon usage are cluster analysis and correspondenceanalysis. Cluster analysis partitions data into discrete groupsbased on the trends within the data, but has the disadvantagethat it sometimes forces arbitrary divisions of a dataset evenwhen presented with continuous variation. Correspondence analysisis an ordination technique that identifies the major trends inthe variation of the data and distributes genes along continuousaxes in accordance with these trends. Correspondence analysis hasthe advantage that it does not assume that the data falls intodiscrete clusters and therefore can therefore represent continuous variationaccurately.

There is an alternative tocreating these Fop/CAI input files by hand. During acorrespondence analysis (of codon usage) the files"fop.coa", "cbi.coa", and "cai.coa" are generatedautomatically (fop.coa is also the appropriate file for the CBIindices). These files can then be used to calculate theirrespective indices. However, in order to make these files CodonW makesimportant assumptions about the dataset, which must be verifiedby the researcher. These are:

Correspondence analysis can be adversely affected by genesthat have atypical codon or amino acid usage. These genes (i.e.plasmid genes, transposons, etc.) are normally excluded from theanalysis. However, it is often interesting to view where thesegenes lie relative to the genes used to identify the trends. Thisis possible by identifying the trends using a dataset of"good" genes (the main input file), then using these totransform the codon usage of the additional genes intoco-ordinates. This is possible if the "add additional genes after COA" isselected under the "advanced correspondence analysis"menu. After the initial COA, the second file (containg the additional genes)will be prompted for.

The most complex analysis thatCodonW performs is correspondence analysis of either amino acidor codon usage. In essence, correspondence analysis creates aseries of orthogonal axes to identify trends that explain thedata variation, with each subsequent axis explaining a decreasingamount of the variation (Benzecri 1992). Correspondence analysisassigns ordination for each gene and codon (or amino acid) onthese axes, and a very useful property is that the ordination ofthe genes and codons are superimposable. When used to analysecodon usage data, we routinely include the 59 synonymous sensecodons, which generate up to 58 axes, while the alternative analysis ofthe 20 standard amino acid usage produces 19 axes.

Correspondence analysisgenerates a large volume of data and CodonW writes the core datanecessary to interpret the correspondence analysis to the file"summary.coa". To aid analysis most of thisinformation is duplicated and compartmentalised into separatefiles. To reduce the memory requirements, additional data fromintermediate stages of the analysis are also stored as temporaryfiles. Depending on the precise options selected more than 20permanent and temporary files may be created, however, to minimise amountof user interaction required for a correspondence analysis thesefiles are created, overwritten and deleted automatically.Permanent correspondence analysis files created by CodonW havethe extension ".coa". For a description of their formatand contents see the Table below.

To identify the putativeoptimal codons, the genes are sorted according to their positionon the principal axis, and the sorted codon usage of these genes iswritten to the file "cusort.coa". Then a number ofgenes, decided by the advanced correspondence analysis menuoption "number of genes used to identify optimalcodons", are read from the start and end of this file (i.e.equivalent to the extremes of the principle axis), and the codon usageof each set of genes is totalled. A prerequisite for theidentification of optimal codons is the identification of whichgroup of genes are "more highly" expressed. As theorientation of the principal axis is arbitrary, highly expressedgenes can be located at either the positive or negative extremeof the axis. Fortuitously, in those species where codon usage is afunction of gene expression, selection for optimal codons in morehighly expressed genes has the effect of increasing their codonbias. The codon bias of the totalled codon usage fromboth sets of genes is estimated using Nc(an index independent of the choice of optimal codons) and the setof genes with the lower Nc (more highly biased) is putativelyidentified as the more highly expressed.

CodonW calculates two indicesthat measure the degree to which the codon usage of a gene hasadapted towards the usage of optimal codons, namely thefrequency of optimal codons (Fop) and the codonadaptation index (CAI). Before either of these indices can becalculated information about codon usage in the species beinganalysed is required. The index Fop requires theidentification of the optimal codons for th

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