CorAna

CorAna performs correspondence analysis

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Syntax

Description

example

out =CorAna(N) CorAna with all the default options.

example

out =CorAna(N, Name, Value) CorAna with name pairs.

Examples

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  • CorAna with all the default options.

    • load smoke
[N,~,~,labels] =crosstab(smoke{:,1},smoke{:,2});
[I,J]=size(N);
if verLessThan('matlab','8.2.0')==0
% Contingency table is supplied to CorAna in table format
Ntable=array2table(N,'RowNames',labels(1:I,1),'VariableNames',labels(1:J,2))
out=CorAna(Ntable);
else
out=CorAna(N);
end

  • CorAna with name pairs.

    • Input is the contingency table, labels for rows and columns are supplied. 

    % Data are read from the txt file
load('smoke.txt')
labels_rows= {'Senior-Managers' 'Junior-Managers' 'Senior-Employees' 'Junior-Employees' 'Secretaries'};
labels_columns= {'None' 'Light' 'Medium' 'Heavy'};
N=crosstab(smoke(:,1),smoke(:,2));
out=CorAna(N,'Lr',labels_rows,'Lc',labels_columns);
    Summary
             Singular_value     Inertia      Accounted_for    Cumulative
             ______________    __________    _____________    __________

    dim_1        0.27342         0.074759        0.87756       0.87756  
    dim_2        0.10009         0.010017        0.11759       0.99515  
    dim_3       0.020337       0.00041357      0.0048547             1  

ROW POINTS
Results for dimension: 1
                         Scores      CntrbPnt2In    CntrbDim2In
                        _________    ___________    ___________

    Senior_Managers     -0.065768     0.0032977      0.092232  
    Junior_Managers       0.25896      0.083659        0.5264  
    Senior_Employees     -0.38059       0.51201       0.99903  
    Junior_Employees      0.23295       0.33097       0.94193  
    Secretaries          -0.20109      0.070064       0.86535  

Results for dimension: 2
                         Scores     CntrbPnt2In    CntrbDim2In
                        ________    ___________    ___________

    Senior_Managers     -0.19374       0.21356        0.80034 
    Junior_Managers      -0.2433       0.55115        0.46468 
    Senior_Employees    -0.01066     0.0029976     0.00078372 
    Junior_Employees    0.057744       0.15177       0.057876 
    Secretaries         0.078911      0.080522        0.13326 

COLUMN POINTS
Results for dimension: 1
               Scores     CntrbPnt2In    CntrbDim2In
              ________    ___________    ___________

    None      -0.39331        0.654        0.99402  
    Light     0.099456      0.03085        0.32673  
    Medium     0.19632      0.16562        0.98185  
    Heavy      0.29378      0.14954         0.6844  

Results for dimension: 2
               Scores      CntrbPnt2In    CntrbDim2In
              _________    ___________    ___________

    None      -0.030492      0.029336      0.0059745 
    Light       0.14106       0.46317        0.65729 
    Medium    0.0073591     0.0017368      0.0013796 
    Heavy      -0.19777       0.50575        0.31015 

-----------------------------------------------------------
Overview ROW POINTS
                          Mass       Score_1     Score_2      Inertia     CntrbPnt2In_1    CntrbPnt2In_2    CntrbDim2In_1    CntrbDim2In_2
                        ________    _________    ________    _________    _____________    _____________    _____________    _____________

    Senior_Managers     0.056995    -0.065768    -0.19374    0.0026729      0.0032977          0.21356        0.092232           0.80034  
    Junior_Managers     0.093264      0.25896     -0.2433     0.011881       0.083659          0.55115          0.5264           0.46468  
    Senior_Employees     0.26425     -0.38059    -0.01066     0.038314        0.51201        0.0029976         0.99903        0.00078372  
    Junior_Employees     0.45596      0.23295    0.057744     0.026269        0.33097          0.15177         0.94193          0.057876  
    Secretaries          0.12953     -0.20109    0.078911     0.006053       0.070064         0.080522         0.86535           0.13326  

Overview COLUMN POINTS
               Mass      Score_1      Score_2      Inertia     CntrbPnt2In_1    CntrbPnt2In_2    CntrbDim2In_1    CntrbDim2In_2
              _______    ________    _________    _________    _____________    _____________    _____________    _____________

    None      0.31606    -0.39331    -0.030492     0.049186         0.654          0.029336         0.99402         0.0059745  
    Light     0.23316    0.099456      0.14106    0.0070588       0.03085           0.46317         0.32673           0.65729  
    Medium    0.32124     0.19632    0.0073591      0.01261       0.16562         0.0017368         0.98185         0.0013796  
    Heavy     0.12953     0.29378     -0.19777     0.016335       0.14954           0.50575          0.6844           0.31015  

-----------------------------------------------------------
Legend
Row scores in principal coordinates
Column scores in principal coordinates
CntrbPnt2In = relative contribution of points to explain total Inertia of the latent dimension
              The sum of the numbers in a column is equal to 1
CntrbDim2In = relative contribution of latent dimension to explain total Inertia of a point
              CntrbDim2In_1+CntrbDim2In_2+...+CntrbDim2In_K=1
    Click here for the graphical output of this example (link to Ro.S.A. website).

    Related Examples

    expand all

  • CorAna with original data matrix as input.

    • load smoke
out=CorAna(smoke,'datamatrix',true);

  • CorAna with supplementary rows and supplementary columns.

    • Children data Active rows = 1:15 Active columns = 1:5 

    N=[51	64	32	29	17	59	66	70;
53	90	78	75	22	115	117	86;
71	111	50	40	11	79	88	177;
1	7	5	5	4	9	8	5;
7	11	4	3	2	2	17	18;
7	13	12	11	11	18	19	17;
21	37	14	26	9	14	34	61;
12	35	19	6	7	21	30	28;
10	7	7	3	1	8	12	8;
4	7	7	6	2	7	6	13;
8	22	7	10	5	10	27	17;
25	45	38	38	13	48	59	52;
18	27	20	19	9	13	29	53;
35	61	29	14	12	30	63	58;
2	4	3	1	4	nan  nan	nan	  ;
2	8	2	5	2	nan  nan	nan;
1	5	4	6	3	nan  nan	nan;
3	3	1	3	4	nan  nan	nan];
% rowslab = cell containing row labels
rowslab={'money','future','unemployment','circumstances',...
'hard','economic','egoism','employment','finances',...
'war','housing','fear','health','work','comfort','disagreement',...
'world','to_live'};
% colslab = cell containing column labels
colslab={'unqualified','cep','bepc','high_school_diploma','university',...
'thirty','fifty','more_fifty'};
tableN=array2table(N,'VariableNames',colslab,'RowNames',rowslab);
% Extract just active rows and active columns
Nactive=tableN(1:14,1:5);
% Define tables containing supplementary rows and supplementary cols
Nsupr=tableN(15:18,1:5);
Nsupc=tableN(1:14,6:8);
Sup=struct;
Sup.r=Nsupr;
Sup.c=Nsupc;
out=CorAna(Nactive,'Sup',Sup);
    Summary
             Singular_value     Inertia     Accounted_for    Cumulative
             ______________    _________    _____________    __________

    dim_1        0.18815        0.035402       0.57043        0.57043  
    dim_2        0.11452        0.013115       0.21132        0.78175  
    dim_3       0.085447       0.0073011       0.11764        0.89939  
    dim_4       0.079018       0.0062439       0.10061              1  

ROW POINTS
Results for dimension: 1
                      Scores      CntrbPnt2In    CntrbDim2In
                     _________    ___________    ___________

    money             -0.11527      0.045499        0.42845 
    future             0.17645       0.17567        0.71562 
    unemployment      -0.21223       0.22616        0.87492 
    circumstances      0.40092      0.062745        0.58397 
    hard              -0.24998      0.029938        0.88369 
    economic           0.35396       0.12005        0.48362 
    egoism            0.059889     0.0068096       0.073339 
    employment        -0.13675      0.026215         0.1643 
    finances            -0.237      0.027904        0.27623 
    war                0.21682      0.021688        0.74907 
    housing          -0.006681    4.1183e-05     0.00072894 
    fear               0.20335       0.11666        0.90069 
    health             0.11165      0.020571        0.79911 
    work              -0.21168       0.12005        0.75402 

Results for dimension: 2
                       Scores      CntrbPnt2In    CntrbDim2In
                     __________    ___________    ___________

    money             -0.020046     0.0037146       0.012958 
    future             0.097863       0.14587        0.22013 
    unemployment       0.070718      0.067786       0.097145 
    circumstances      -0.33099       0.11544          0.398 
    hard               -0.06765     0.0059184       0.064717 
    economic           -0.32072       0.26604        0.39705 
    egoism             0.025667     0.0033763       0.013471 
    employment         -0.21539       0.17555         0.4076 
    finances            0.20598      0.056902        0.20867 
    war                0.074663     0.0069419       0.088821 
    housing            -0.12824       0.04096        0.26858 
    fear               0.058068      0.025678       0.073446 
    health           -0.0042912    8.2025e-05      0.0011804 
    work               -0.10888      0.085745        0.19951 

COLUMN POINTS
Results for dimension: 1
                            Scores     CntrbPnt2In    CntrbDim2In
                           ________    ___________    ___________

    unqualified            -0.20932       0.2511        0.67619  
    cep                    -0.13858      0.18297        0.64492  
    bepc                    0.10876     0.067579         0.3119  
    high_school_diploma     0.27404      0.37976        0.75817  
    university              0.23123      0.11859        0.31171  

Results for dimension: 2
                            Scores      CntrbPnt2In    CntrbDim2In
                           _________    ___________    ___________

    unqualified             0.080727      0.10082        0.10058  
    cep                    -0.056047     0.080794        0.10549  
    bepc                    0.028483     0.012512       0.021393  
    high_school_diploma      0.12134      0.20099        0.14865  
    university              -0.31786      0.60488          0.589  

-----------------------------------------------------------
Overview ROW POINTS
                       Mass       Score_1      Score_2       Inertia      CntrbPnt2In_1    CntrbPnt2In_2    CntrbDim2In_1    CntrbDim2In_2
                     ________    _________    __________    __________    _____________    _____________    _____________    _____________

    money             0.12123     -0.11527     -0.020046     0.0037595       0.045499        0.0037146          0.42845         0.012958  
    future            0.19975      0.17645      0.097863     0.0086904        0.17567          0.14587          0.71562          0.22013  
    unemployment      0.17776     -0.21223      0.070718     0.0091512        0.22616         0.067786          0.87492         0.097145  
    circumstances    0.013819      0.40092      -0.33099     0.0038038       0.062745          0.11544          0.58397            0.398  
    hard              0.01696     -0.24998      -0.06765     0.0011994       0.029938        0.0059184          0.88369         0.064717  
    economic          0.03392      0.35396      -0.32072     0.0087874        0.12005          0.26604          0.48362          0.39705  
    egoism           0.067211     0.059889      0.025667     0.0032871      0.0068096        0.0033763         0.073339         0.013471  
    employment       0.049623     -0.13675      -0.21539     0.0056484       0.026215          0.17555           0.1643           0.4076  
    finances         0.017588       -0.237       0.20598     0.0035763       0.027904         0.056902          0.27623          0.20867  
    war              0.016332      0.21682      0.074663      0.001025       0.021688        0.0069419          0.74907         0.088821  
    housing          0.032663    -0.006681      -0.12824     0.0020001     4.1183e-05          0.04096       0.00072894          0.26858  
    fear             0.099874      0.20335      0.058068     0.0045852        0.11666         0.025678          0.90069         0.073446  
    health           0.058417      0.11165    -0.0042912    0.00091131       0.020571       8.2025e-05          0.79911        0.0011804  
    work             0.094849     -0.21168      -0.10888     0.0056364        0.12005         0.085745          0.75402          0.19951  

Overview COLUMN POINTS
                             Mass      Score_1      Score_2      Inertia     CntrbPnt2In_1    CntrbPnt2In_2    CntrbDim2In_1    CntrbDim2In_2
                           ________    ________    _________    _________    _____________    _____________    _____________    _____________

    unqualified             0.20289    -0.20932     0.080727     0.013146        0.2511          0.10082          0.67619          0.10058   
    cep                     0.33731    -0.13858    -0.056047     0.010044       0.18297         0.080794          0.64492          0.10549   
    bepc                    0.20226     0.10876     0.028483    0.0076704      0.067579         0.012512           0.3119         0.021393   
    high_school_diploma     0.17902     0.27404      0.12134     0.017732       0.37976          0.20099          0.75817          0.14865   
    university             0.078518     0.23123     -0.31786     0.013468       0.11859          0.60488          0.31171            0.589   

-----------------------------------------------------------
Legend
Row scores in principal coordinates
Column scores in principal coordinates
CntrbPnt2In = relative contribution of points to explain total Inertia of the latent dimension
              The sum of the numbers in a column is equal to 1
CntrbDim2In = relative contribution of latent dimension to explain total Inertia of a point
              CntrbDim2In_1+CntrbDim2In_2+...+CntrbDim2In_K=1
    Click here for the graphical output of this example (link to Ro.S.A. website)

  • Example of interpretation of values close to the center.

    • N=[80	20	90	90	5	100	40
50	40	40	70	10	100	40
10	70	20	90	80	99	40
0	80	2	20	95	20	40
35	52	38	47	48	80	40];
rl=["Dog" "Cat" "Rat" "Cockroach" "Wallaby"];
cl=["Big" "Athletic" "Friendly"	"Trainable" "Resourceful" "Animal" "Lucky"];
Ntable=array2table(N,"RowNames",rl,"VariableNames",cl);
out=CorAna(Ntable);
% In the center of the map we have Wallaby and Lucky. Does this mean
% wallabies are lucky animals? No. Wallaby is pretty average on all the
% variables being measured. As it has nothing that differentiates it, the
% result is that it is in the middle of the map (i.e., near the origin).
% Similarly, Lucky does not differentiate, so it is also near the center.
% That they are both in the center tells us that they are both indistinct,
% and that is all that they have in common (in the data).
    Summary
             Singular_value     Inertia     Accounted_for    Cumulative
             ______________    _________    _____________    __________

    dim_1        0.50576          0.2558        0.89448       0.89448  
    dim_2        0.14914        0.022243       0.077779       0.97226  
    dim_3       0.081626       0.0066627       0.023299       0.99556  
    dim_4        0.03564       0.0012702      0.0044417             1  

ROW POINTS
Results for dimension: 1
                  Scores     CntrbPnt2In    CntrbDim2In
                 ________    ___________    ___________

    Dog          -0.59431        0.3295       0.94186  
    Cat           -0.3256      0.081449       0.81272  
    Rat           0.27706      0.068913       0.57861  
    Cockroach     0.95997       0.51987       0.96141  
    Wallaby      0.019153    0.00027378      0.033895  

Results for dimension: 2
                  Scores      CntrbPnt2In    CntrbDim2In
                 _________    ___________    ___________

    Dog           -0.12157      0.15856       0.039411  
    Cat           0.079165     0.055371       0.048044  
    Rat            0.22533      0.52422        0.38272  
    Cockroach     -0.18988      0.23391       0.037614  
    Wallaby      -0.057062     0.027946        0.30085  

COLUMN POINTS
Results for dimension: 1
                    Scores     CntrbPnt2In    CntrbDim2In
                   ________    ___________    ___________

    Big            -0.68224       0.17879       0.90397  
    Athletic        0.54545        0.1711       0.97126  
    Friendly       -0.60693       0.15363       0.86806  
    Trainable      -0.19488      0.026427       0.47685  
    Resourceful     0.89767       0.42097       0.98264  
    Animal          -0.2172      0.041317       0.59767  
    Lucky           0.13298     0.0077629       0.55129  

Results for dimension: 2
                    Scores      CntrbPnt2In    CntrbDim2In
                   _________    ___________    ___________

    Big             -0.21116      0.19698        0.086599 
    Athletic       -0.042213     0.011785       0.0058171 
    Friendly        -0.20525      0.20206        0.099279 
    Trainable        0.17768      0.25264          0.3964 
    Resourceful    -0.072334     0.031435       0.0063805 
    Animal           0.16308      0.26789         0.33696 
    Lucky           -0.08585      0.03721         0.22978 

-----------------------------------------------------------
Overview ROW POINTS
                  Mass      Score_1      Score_2      Inertia     CntrbPnt2In_1    CntrbPnt2In_2    CntrbDim2In_1    CntrbDim2In_2
                 _______    ________    _________    _________    _____________    _____________    _____________    _____________

    Dog          0.23863    -0.59431     -0.12157     0.089486         0.3295         0.15856          0.94186         0.039411   
    Cat          0.19652     -0.3256     0.079165     0.025635       0.081449        0.055371          0.81272         0.048044   
    Rat          0.22965     0.27706      0.22533     0.030465       0.068913         0.52422          0.57861          0.38272   
    Cockroach     0.1443     0.95997     -0.18988      0.13832        0.51987         0.23391          0.96141         0.037614   
    Wallaby       0.1909    0.019153    -0.057062    0.0020661     0.00027378        0.027946         0.033895          0.30085   

Overview COLUMN POINTS
                     Mass      Score_1      Score_2      Inertia     CntrbPnt2In_1    CntrbPnt2In_2    CntrbDim2In_1    CntrbDim2In_2
                   ________    ________    _________    _________    _____________    _____________    _____________    _____________

    Big            0.098259    -0.68224     -0.21116     0.050593        0.17879         0.19698          0.90397          0.086599  
    Athletic        0.14711     0.54545    -0.042213     0.045062         0.1711        0.011785          0.97126         0.0058171  
    Friendly        0.10668    -0.60693     -0.20525      0.04527        0.15363         0.20206          0.86806          0.099279  
    Trainable       0.17799    -0.19488      0.17768     0.014176       0.026427         0.25264          0.47685            0.3964  
    Resourceful     0.13363     0.89767    -0.072334      0.10958        0.42097        0.031435          0.98264         0.0063805  
    Animal          0.22403     -0.2172      0.16308     0.017683       0.041317         0.26789          0.59767           0.33696  
    Lucky            0.1123     0.13298     -0.08585    0.0036019      0.0077629         0.03721          0.55129           0.22978  

-----------------------------------------------------------
Legend
Row scores in principal coordinates
Column scores in principal coordinates
CntrbPnt2In = relative contribution of points to explain total Inertia of the latent dimension
              The sum of the numbers in a column is equal to 1
CntrbDim2In = relative contribution of latent dimension to explain total Inertia of a point
              CntrbDim2In_1+CntrbDim2In_2+...+CntrbDim2In_K=1
    Click here for the graphical output of this example (link to Ro.S.A. website)

    Input Arguments

    expand all

    N — Contingency table (default) or n-by-2 input dataset. 2D Array or Table.

    2D array or table or timetable which contains the input contingency table (say of size I-by-J) or the original data matrix X.

    In this last case N=crosstab(X(:,1),X(:,2)). As default procedure assumes that the input is a contingency table.

    Data Types: single| double

    Name-Value Pair Arguments

    Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

    Example: 'k',3 , 'Lr',{'a' 'b' 'c'} , 'Lc',{'c1' c2' 'c3' 'c4'} , 'Sup',Sup=struct; Sup.c={'c2' 'c4'} , 'datamatrix',true , 'plots',1 , 'dispresults',false , 'd1',2 , 'd2',3

    k —Number of dimensions to retain.scalar.

    Scalar which contains the number of dimensions to retain.

    The default value of k is 2.

    Example: 'k',3

    Data Types: double

    Lr —Vector of row labels.cell.

    Cell containing the labels of the rows of the input contingency matrix N. This option is unnecessary if N is a table, because in this case Lr=N.Properties.RowNames;

    Example: 'Lr',{'a' 'b' 'c'}

    Data Types: cell array of strings

    Lc —Vector of column labels.cell.

    Cell containing the labels of the columns of the input contingency matrix N. This option is unnecessary if N is a table, because in this case Lc=N.Properties.VariableNames;

    Example: 'Lc',{'c1' c2' 'c3' 'c4'}

    Data Types: cell array of strings

    Sup —Structure containing indexes or names of supplementary rows or columns.structure.

    Structure with the following fields.

    Value Description
    r

    vector containing row indexes or vector of cell array of strings or table or 2D numeric array, containing supplementary rows.

    If indexes or cell array of strings are supplied in a vector, we assume that supplementary rows belong to contingency table N. For example: - if Sup.r=[2 5] (that is Sup.r is a numeric vector which contains row indexes) we use rows 2 and 5 of the input contingency table as supplementary rows.

    - if Sup.r={'Junior-Managers' 'Senior-Employees'} (that is Sup.r is a cell array of strings) we use rows named 'Junior-Managers' and 'Senior-Employees' of the input contingency table as supplementary rows. Of course the length of Sup.r must be smaller than the number of rows of the contingency matrix divided by 2.

    - if Sup.r is a table, or a 2D array supplementary rows do not belong to N. Note that if Sup.r is a table, the labels of the rows are taken directly from the table. If on the other hand Sup.r is a matrix the names of the rows of the supplementary units can be given using Sup.Lr as a cell array of strings.
    Lr

    cell array of strings containing the labels of the supplementary units if Sup.r is a 2D numeric array.
    c

    vector containing column indexes or vector of cell array of strings or table or 2D numeric array use as supplementary columns, or table or 2D numeric array containing supplementary rows.

    If indexes or cell array of strings are supplied in a vector, we assume that supplementary columns belong to contingency table N. For example: - if Sup.c=[2 3] (that is Sup.c is a numeric vector which contains column indexes) we use columns 2 and 3 of the input contingency table as supplementary columns.

    - if Sup.c={'Smokers' 'NonSmokers'} (that is Sup.c is a cell array of strings) we use columns of the contingency table labeled 'Smokers' and 'NonSmokers' of the input contingency table N as supplementary columns.

    Of course the length of Sup.c must be smaller than the number of columns of the contingency matrix divided by 2.

    - If Sup.c is a table, or a 2D array supplementary columns do not belong to N. Note that if Sup.c is a table, the labels of the columns are taken directly from the table. If on the other hand Sup.c is a matrix the names of the columns of the supplementary units can be given using Sup.Lc as a cell array of strings.
    Lc

    cell array of strings containing the labels of the supplementary units if Sup.r is a 2D numeric array.

    REMARK: The default value of Sup is a missing value that is we assume that there are no supplementary rows or columns.

    Example: 'Sup',Sup=struct; Sup.c={'c2' 'c4'}

    Data Types: struct

    datamatrix —Data matrix or contingency table.boolean.

    If datamatrix is true the first input argument N is forced to be interpreted as a data matrix, else if the input argument is false N is treated as a contingency table. The default value of datamatrix is false, that is the procedure automatically considers N as a contingency table (in array or table format). If datamatrix is true, N can be an array or a table of size n-by-2. Note that if N has more than two columns correspondence analysis is based on the first two columns of N (and a warning is produced).

    Example: 'datamatrix',true

    Data Types: logical

    plots —Plot on the screen.scalar | structure.

    If plots = 1, a plot which shows the Principal coordinates of rows and columns is shown on the screen. If plots is a structure it may contain the following fields:

    Value Description
    alpha

    type of plot, scalar in the interval [0 1] or a string identifying the type of coordinates to use in the plot.

    If $plots.alpha='rowprincipal'$ the row points are in principal coordinates and the column coordinates are standard coordinates. Distances between row points are (approximated) chi-squared distances (row-metric-preserving). The position of the row points are at the weighted average of the column points.

    Note that 'rowprincipal' can also be specified setting plots.alpha=1.

    If $plots.alpha='colprincipal'$, the column coordinates are referred to as principal coordinates and the row coordinates as standard coordinates.

    Distances between column points are (approximated) chi-squared distances (column-metric-preserving). The position of the column points are at the weighted average of the row points.

    Note that 'colprincipal' can also be specified setting plots.alpha=0.

    If $plots.alpha='symbiplot'$, the row and column coordinates are scaled similarly. The sum of weighted squared coordinates for each dimension is equal to the corresponding singular values. These coordinates are often called symmetrical coordinates. This representation is particularly useful if one is primarily interested in the relationships between categories of row and column variables rather than in the distances among rows or among columns. 'symbiplot' can also be specified setting plots.alpha=0.5;

    If $plots.alpha='bothprincipal'$, both the rows and columns are depicted in principal coordinates. Such a plot is often referred to as a symmetrical plot or French symmetrical model. Note that such a symmetrical plot does not provide a feasible solution in the sense that it does not approximate matrix $D_r^{-0.5}(P-rc')D_c^{-0.5}$.
    FontSize

    scalar which specifies the font size of row (column) labels. The default value is 10.
    MarkerSize

    scalar which specifies the marker size of symbols associated with rows or columns. The default value is 10.

    Example: 'plots',1

    Data Types: scalar double | struct

    dispresults —Display results on the screen.boolean.

    If dispresults is true (default) it is possible to see on the screen all the summary results of the analysis.

    Example: 'dispresults',false

    Data Types: Boolean

    d1 —Dimension to show on the horizontal axis.positive integer.

    Positive integer in the range 1, 2, .., K which indicates the dimension to show on the x axis. The default value of d1 is 1.

    Example: 'd1',2

    Data Types: single | double

    d2 —Dimension to show on the vertical axis.positive integer.

    Positive integer in the range 1, 2, .., K which indicates the dimension to show on the y axis. The default value of d2 is 2.

    Example: 'd2',3

    Data Types: single | double

    Output Arguments

    expand all

    out — description Structure

    A structure containing the following fields

    Value Description
    Lr

    cell of length $I$ containing the labels of active rows (i.e. the rows which participated to the fit).
    Lc

    cell of length $J$ containing the labels of active columns (i.e. the columns which participated to the fit).
    N

    $I$-by-$J$-array containing contingency table referred to active rows and active columns (i.e. referred to the rows/columns which participated to the fit). The $(i,j)$-th element is equal to $n_{ij}$, $i=1, 2, \ldots, I$ and $j=1, 2, \ldots, J$. The sum of the elements of out.P is $n$ (the grand total).
    Ntable

    Same as out.N but in table format (with row and column names).

    This output is present just if your MATLAB version is not<2013b.
    I

    Number of active rows of contingency table.
    J

    Number of active columns of contingency table.
    n

    Grand total. out.n is equal to sum(sum(out.N)).

    This is the number of observations.
    Nhat

    $I$-by-$J$-array containing contingency table referred to active rows (i.e. referred to the rows which participated to the fit) under the independence hypothesis.

    The $(i,j)$-th element is equal to $n_{i.}n_{.j}/n$, $i=1, 2, \ldots, I$ and $j=1, 2, \ldots, J$. The sum of the elements of out.Nhat is $n$ (the grand total).
    Nhattable

    Same as out.Nhat but in table format (with row and column names).
    P

    $I$-by-$J$-array containing correspondence matrix (proportions). The $(i,j)$-th element is equal to $n_{ij}/n$, $i=1, 2, \ldots, I$ and $j=1, 2, \ldots, J$. The sum of the elements of out.P is 1.
    Ptable

    Same as out.P but in table format (with row and column names).

    This output is present just if your MATLAB version is not<2013b.
    r

    Vector of length $I$ containing row masses.

    \[ r=(f_{1.}, f_{2.}, \ldots, f_{I.})' \] $r$ is also the centroid of column profiles.
    Dr

    Square matrix of size $I$ containing on the diagonal the row masses. This is matrix $D_r$.

    \[ D_r=diag(r) \]
    c

    Vector of length $J$ containing column masses.

    \[ c=(f_{.1}, f_{.2}, \ldots, f_{.J})' \] $c$ is also the centroid of row profiles.
    Dc

    Square matrix of size $J$ containing on the diagonal the column masses. This is matrix $D_c$.

    \[ D_c=diag(c) \]
    ProfilesRows

    $I$-by-$J$-matrix containing row profiles.

    The $i,j$-th element of this matrix is given by $f_{ij}/f_{i.}=n_{ij}/n_{i.}$.

    Written in matrix form:

    \[ ProfilesRows = D_r^{-1} \times P \]
    ProfilesCols

    $I$-by-$J$-matrix containing column profiles.

    The $i,j$-th element of this matrix is given by $f_{ij}/f_{.j}=n_{ij}/n_{.j}$.

    Written in matrix form:

    \[ ProfilesCols = P \times D_c^{-1} \]
    K

    Scalar integer containing the maximum number of dimensions. $K = \min(I-1,J-1)$.
    k

    Scalar integer containing the number of retained dimensions.
    Residuals

    $I$-by-$J$-matrix containing standardized residuals.

    \[ Residuals = D_r^{1/2} (D_r^{-1} P - r c') D_c^{-1/2} = D_r^{-1/2} (P - r c') D_c^{-1/2} \] With the singular value decomposition (SVD) we obtain that: \[ Residuals = U \Gamma V' \]
    TotalInertia

    Scalar containing total inertia. Total inertia is equal (for example) to the sum of the squares of the elements of matrix out.Residuals.
    Chi2stat

    Scalar containing Chi-square statistic for the contingency table. $Chi2stat= TotalInertia \times n$.
    CramerV

    Scalar containing Cramer's $V$ index.

    \[ V=\sqrt{Chi2stat/(n (\min(I,J)-1))} \] Cramer's index goes between 0 and 1.
    InertiaExplained

    matrix with 4 columns.

    - First column contains the singular values (the sum of the squared singular values is the total inertia).

    - Second column contains the eigenvalues (the sum of the eigenvalues is the total inertia).

    - Third column contains the variance explained by each latent dimension.

    - Fourth column contains the cumulative variance explained by each dimension.
    RowsPri

    $I$-by-$K$ matrix containing principal coordinates of rows.

    \[ RowsPri = D_r^{-1/2} \times U \times \Gamma; \]
    ColsPri

    $J$-by-$K$ matrix containing Principal coordinates of columns.

    \[ ColsPri = D_c^{-1/2} \times V \times \Gamma; \]
    RowsSta

    $I$-by-$K$ matrix containing standard coordinates of rows.

    \[ RowsSta = RowsPri \times \Gamma^{-1} = D_r^{-1/2} U \Gamma \Gamma^{-1}= D_r^{-1/2} U \]
    ColsSta

    $J$-by-$K$ matrix containing standard coordinates of columns.

    \[ ColsSta = ColsPri \times \Gamma^{-1} = D_c^{-1/2} V \Gamma \Gamma^{-1}= D_c^{-1/2} V \]
    RowsSym

    $I$-by-$K$ matrix containing symmetrical coordinates of rows.

    \[ RowsSym = D_r^{-1/2} \times U \times \Gamma^{1/2} \]
    ColsSym

    $J$-by-$K$ matrix containing symmetrical coordinates of columns.

    \[ ColsSym = D_c^{-1/2} \times V \times \Gamma^{1/2} \]

    Symmetric plot represents the row and column profiles simultaneously in a common space (Bendixen, 2003). In this case, only the distance between row points or the distance between column points can be really interpreted.

    The distance between any row and column items is not meaningful! You can only make a general statements about the observed pattern. In order to interpret the distance between column and row points, the column profiles must be presented in row space or vice-versa. This type of map is called asymmetric biplot.
    InertiaRows

    $I$-by-$2$ matrix containing absolute and relative contribution of each row to TotalInertia.

    The inertia of a point is the squared distance of point $d_i^2$ to the centroid multiplied by its point mass (and is given in the first column).

    The sum of the inertia of the points is the total inertia. The relative contribution of each row is the absolute contribution of each row divided by the TotalInertia (and is given in the second column).

    1st column = absolute contribution of each row to TotalInertia. The sum of values of the first column is equal to TotalInertia;

    2nd column = relative contribution of each row to TotalInertia. The sum of the values of the second column is equal to 1.
    InertiaCols

    $J$-by-$2$ matrix containing absolute and relative contribution of each column to total inertia. The inertia of a point is the squared distance of point $d_i^2$ to the centroid multiplied by the mass (and is given in the first column).

    The sum of the inertia of the points is the total inertia. The relative contribution of each row is the absolute contribution of each row divided by the TotalInertia (and is given in the second column).

    1st column = absolute contribution of each column to TotalInertia. The sum of values of the first column is equal to TotalInertia;

    2nd column = relative contribution of each column to TotalInertia. The sum of values of the second column is equal to 1.
    Point2InertiaRows

    $I$-by-$K$ matrix containing relative contributions of rows to inertia of the dimension. The inertia of first latent dimension is given by $\lambda_1=\gamma_{11}^2$. The inertia of second latent dimension is given by $\lambda_2=\gamma_{22}^2$ .... The sum of each column of matrix Point2InertiaRows is equal to 1.

    Remark: the points with the larger value of Point2Inertia are those which contribute the most to the definition of the dimension. If the row contributions were uniform, the expected value would be 1/size(contingeny_table,1) For a given dimension, any row with a contribution larger than this threshold could be considered as important in contributing to that dimension.
    Point2InertiaCols

    $J$-by-$K$ matrix containing relative contributions of columns to inertia of the dimension. The sum of each column of matrix Point2InertiaCols is equal to 1.
    Dim2InertiaRows

    $I$-by-$K$ matrix containing relative contributions of latent dimensions to inertia of the row points. These numbers can be interpreted as squared correlations and measures the degree of association between row points and a particular axis. The sum of each row of matrix Dim2InertiaRows is equal to 1.
    Dim2InertiaCols

    $J$-by-$K$ matrix containing relative contributions of latent dimensions to inertia of the column points. These numbers can be interpreted as squared correlations and measure the degree of association between columns points and a particular axis. The sum of each row of matrix Dim2InertiaCols is equal to 1.
    cumsumDim2InertiaRows

    $I$-by-$K$ matrix containing cumulative sum of the contributions of latent dimensions to inertia of the row points. These cumulative sums are equivalent to the communalities in PCA.

    The last column of matrix cumsumDim2InertiaRows is equal to 1.
    cumsumDim2InertiaCols

    $J$-by-$K$ matrix containing cumulative sum of the contributions of latent dimensions to inertia of the column points. These cumulative sums are equivalent to the communalities in PCA.

    The last column of matrix cumsumDim2InertiaCols is equal to 1.
    sqrtDim2InertiaRows

    $I$-by-$K$ matrix containing correlation of rows points with latent dimension axes. Similar to component loadings in PCA
    sqrtDim2InertiaCols

    $I$-by-$K$ matrix containing correlation of column points with latent dimension axes. Similar to component loadings in PCA.
    Summary

    $K$-times-4 table containing summary results for correspondence analysis.

    First column contains the singular values (the sum of the squared singular values is the total inertia).

    Second column contains the eigenvalues (the sum of the eigenvalues is the total inertia).

    Third column contains the variance explained by each latent dimension. Fourth column contains the cumulative variance explained by each dimension.

    This output is present just if your MATLAB version is not<2013b.
    OverviewRows

    $I$-times-(k*3+2) table containing an overview of row points. More precisely, if we suppose that $k=2$, First column contains the row masses (vector $r$).

    Second column contains the scores of first dimension.

    Third column contains the scores of second dimension.

    Fourth column contains the inertia of each point, where inertia of point is the squared distance of point $d_i^2$ to the centroid.

    Fifth column contains the relative contribution of each point to the explanation of the inertia of the first dimension. The sum of the elements of this column is equal to 1.

    Sixth column contains the relative contribution of each point to the explanation of the inertia of the second dimension. The sum of the elements of this column is equal to 1.

    Seventh column contains the relative contribution of the first dimension to the explanation of the inertia of the point.

    Eight column contains the relative contribution of the second dimension to the explanation of the inertia of the point.
    OverviewCols

    $J$-times-(k*3+2) table containing an overview of row points. More precisely if we suppose that $k=2$ First column contains the column masses (vector $c$).

    Second column contains the scores of first dimension.

    Third column contains the scores of second dimension.

    Fourth column contains the inertia of each point, where inertia of point is the squared distance of point $d_i^2$ to the centroid.

    Fifth column contains the relative contribution of each point to the explanation of the inertia of the first dimension. The sum of the elements of this column is equal to 1.

    Sixth column contains the relative contribution of each point to the explanation of the inertia of the second dimension. The sum of the elements of this column is equal to 1.

    Seventh column contains the relative contribution of the first dimension to the explanation of the inertia of the point.

    Eight column contains the relative contribution of the second dimension to the explanation of the inertia of the point.
    LrSup

    cell containing the labels of the supplementary rows (i.e. the rows whicg did not participate to the fit).
    LcSup

    cell containing the labels of supplementary columns (i.e. the columns which did not participate to the fit).
    SupRowsN

    matrix of size length(LrSup)-by-c referred to supplementary rows. If there are no supplementary rows this field is not present.
    SupRowsNtable

    Same as out.SupRowsN but in table format (with row and column names). This is the contingency table referred to supplementary rows. If there are no supplementary rows this field is not present.

    This output is present just if your MATLAB version is not<2013b.
    SupColsN

    matlab of size r-by-length(LcSup) referred to supplementary columns.

    If there are no supplementary columns this field is not present.
    SupColsNtable

    Same as out.SupColsN but in table format (with row and column names). This is the contingency table referred to supplementary columns.

    If there are no supplementary columns this field is not present.

    This output is present just if your MATLAB version is not<2013b.
    RowsPriSup

    Principal coordinates of supplementary rows.

    If there are no supplementary rows this field is not present.
    RowsStaSup

    Standard coordinates of supplementary rows.

    If there are no supplementary rows this field is not present.
    RowsSymSup

    Symmetrical coordinates of supplementary rows.

    If there are no supplementary rows this field is not present.
    ColsPriSup

    Principal coordinates of supplementary columns.

    If there are no supplementary columns this field is not present.
    ColsStaSup

    Standard coordinates of of supplementary columns.

    If there are no supplementary columns this field is not present.
    ColsSymSup

    Symmetrical coordinates of supplementary columns.

    If there are no supplementary columns this field is not present.

    References

    Benzecri, J.-P. (1992), "Correspondence Analysis Handbook", New-York, Dekker.

    Benzecri, J.-P. (1980), "L'analyse des donnees tome 2: l'analyse des correspondances", Paris, Bordas.

    Greenacre, M.J. (1993), "Correspondence Analysis in Practice", London, Academic Press.

    Gabriel, K.R. and Odoroff, C. (1990), Biplots in biomedical research, "Statistics in Medicine", Vol. 9, pp. 469-485.

    Greenacre, M.J. (1993), Biplots in correspondence Analysis, "Journal of Applied Statistics", Vol. 20, pp. 251-269.

    Riani, M, Atkinson A.C., Torti, F., Corbellini A. (2023), Robust Correspondence Analysis, "Journal of the Royal Statistical Society Series C: Applied Statistics", Vol. 71, pp. 1381–1401, https://doi.org/10.1111/rssc.12580

    Acknowledgements

    This function has been inspired by the code developed by: Urbano Lorenzo-Seva (Rovira i Virgili University, Tarragona, Spain), Michel van de Velden (Erasmus University, Rotterdam, The Netherlands), and Henk A.L. Kiers (University of Groningen, Groningen, The Netherlands) (See References).

    See Also

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