..-
help for ^mca^                                                (Philippe Van Kerm)
..-

Multiple Correspondence Analysis
--------------------------------

      ^mca^ varlist  [^if^ exp] [^in^ range] ^[^weight^],^[^d(^#^) q(^#^) n^otrans]


^aweight^s and ^fweight^s are allowed; see help @weights@.

To reset problem-size limits, see help @matsize@.


Description
-----------

The command ^mca^ produces numerical results as well as graphical 
representations for multiple correspondence analyses (MCA).

^mca^ actually conducts an adjusted simple correspondence analysis on the Burt
matrix constructed with varlist (i.e. matrix of frequency counts resulting 
from all two-way cross-tabulations of the variables in varlist including
the cross-tabulations of each variable with itself). It can be shown that
the total inertia of the Burt matrix is high due to the fitting of the diagonal
sub-matrices. Consequently, a simple correspondence analysis applied to this
matrix usually results in maps of apparently poor quality. As a remedy, if not
otherwise specified (see Options), ^mca^ adjusts the obtained principal
inertias (eigenvalues) following a method suggested by Benzecri and presented
in Greenacre (1984).


The computation algorithm draws most largely on Blasius and Greenacre [2]
and Greenacre [5].  


Options
-------

^d(^#^)^ specifies the number of dimensions to be considered (for both numerical
    and graphical displays). If ^d(0)^ is specified, then ^mca^ provides no 
    graphical display and returns the numerical output for all non-trivial 
    dimensions. For maps to be readable, # must be set larger than 1.
    Furthermore, consistent maps can only be obtained by specifying # lower
    than or equal to the number of underlying non-trivial dimensions. Default
    # is 0. 
    

^q(^#^)^ specifies a quality of representation threshold (0<#<=1). It restricts
    the mappings to points satisfying the condition that their quality of
    representation (sum of contributions of principal axes) in the ^d(^#^)^ first
    dimensions is higher than or equal to #. Rejected points are still
    mapped but symbolized by a dot. 
    
^n^otrans requests that no adjustments to eigenvalues are to be made. 


Remarks and Restrictions
------------------------

- All variables in varlist must be numeric categorical variables. No string 
variables are allowed. 

- ^mca^ disregards all observations with missing values in any of the variables
in varlist.
 
- Internally using the @tabulate@ command, the variables in varlist can take on
a maximum of 20 values.

- The points on the outputs are represented by the first five characters of the
variable names followed by _X (where X is the numeric value of each category
composing the considered variable). All variables in varlist must therefore
be distinguishable with the their first 5 characters only. Note that the exact
coordinates of the points are located right in the middle of the label name.

- Beware of the possible aspect ratio distortion of the maps.



Example
-------

.. use "C:\Stata\auto.dta", clear
(1978 Automobile Data)

.. mca foreign hdroom rep78 ,d(3) 
------------------------------------------------------------------------------

                   MULTIPLE CORRESPONDENCE ANALYSIS

------------------------------------------------------------------------------


 Total Inertia :      0.431

 Principal Inertia Components :

      Inertia    Share    Cumul
Dim1    0.308    0.713    0.713
Dim2    0.060    0.138    0.851
Dim3    0.039    0.090    0.941

 Coordinates : 

             Mass  Inertia     Dim1     Dim2     Dim3
 forei_0    0.232    0.035   -0.387    0.007   -0.016
 forei_1    0.101    0.079    0.884   -0.017    0.036
hdroo_1.    0.014    0.025    0.129    1.252   -0.275
 hdroo_2    0.063    0.009   -0.156    0.061    0.274
hdroo_2.    0.068    0.036    0.721   -0.077   -0.040
 hdroo_3    0.053    0.018    0.568   -0.137    0.012
hdroo_3.    0.063    0.023   -0.498   -0.212   -0.256
 hdroo_4    0.048    0.017   -0.422    0.165   -0.050
hdroo_4.    0.019    0.012   -0.745   -0.252    0.041
 hdroo_5    0.005    0.015   -1.026    0.149    1.367
 rep78_1    0.010    0.024   -0.361    1.360   -0.041
 rep78_2    0.039    0.035   -0.752    0.066    0.554
 rep78_3    0.145    0.026   -0.336   -0.196   -0.143
 rep78_4    0.087    0.022    0.339    0.237   -0.145
 rep78_5    0.053    0.055    0.973   -0.150    0.230

 Explained inertia of axes  :

            Dim1    Dim2    Dim3
 forei_0  0.1126  0.0002  0.0014
 forei_1  0.2574  0.0005  0.0033
hdroo_1.  0.0008  0.3815  0.0282
 hdroo_2  0.0050  0.0040  0.1214
hdroo_2.  0.1143  0.0068  0.0028
 hdroo_3  0.0558  0.0167  0.0002
hdroo_3.  0.0507  0.0474  0.1059
 hdroo_4  0.0279  0.0220  0.0031
hdroo_4.  0.0348  0.0206  0.0008
 hdroo_5  0.0165  0.0018  0.2321
 rep78_1  0.0041  0.3003  0.0004
 rep78_2  0.0710  0.0028  0.3056
 rep78_3  0.0531  0.0931  0.0758
 rep78_4  0.0326  0.0823  0.0467
 rep78_5  0.1634  0.0201  0.0721

 Contributions of principal axes :

            Dim1    Dim2    Dim3
 forei_0  0.9979  0.0003  0.0016
 forei_1  0.9979  0.0003  0.0016
hdroo_1.  0.0097  0.9203  0.0445
 hdroo_2  0.1622  0.0250  0.4991
hdroo_2.  0.9759  0.0112  0.0031
 hdroo_3  0.9357  0.0541  0.0004
hdroo_3.  0.6844  0.1240  0.1807
 hdroo_4  0.5021  0.0765  0.0071
hdroo_4.  0.8794  0.1008  0.0026
 hdroo_5  0.3352  0.0071  0.5942
 rep78_1  0.0532  0.7558  0.0007
 rep78_2  0.6304  0.0048  0.3426
 rep78_3  0.6317  0.2144  0.1140
 rep78_4  0.4584  0.2240  0.0831
 rep78_5  0.9072  0.0216  0.0506


Author
------

	Philippe VAN KERM <philippe.vankerm@@fundp.ac.be>
	University of Namur, Department of Economics
	Rempart de la Vierge 8
	B-5000 Namur, Belgium.


References
----------

[1] Benzecri J.-P. and F. Benzecri (1980) , Analyse des correspondances: 
    expose elementaire, Dunod, Paris.

[2] Blasius J. and M. Greenacre (1994), 'Computation of Correspondence 
    Analysis' in Greenacre M. and J. Blasius (Eds.), Correspondence Analysis
    in the Social Sciences - Recent Developments and Applications, Academic
    Press, London.

[3] Greenacre Michael J. (1984), Theory and Applications of Correspondence 
    Analysis, Academic Press, London.
 
[4] Greenacre Michael J. (1993), Correspondence Analysis in Practice, Academic 
    Press, London.
 
[5] Greenacre Michael J. (1994), 'Multiple and Joint Correspondence Analysis'
    in Greenacre M. and J. Blasius (Eds.), Correspondence Analysis in the
    Social Sciences - Recent Developments and Applications, Academic Press,
    London.

[6] Greenacre M. and J. Blasius (Eds.) (1994), Correspondence Analysis in the
    Social Sciences - Recent Developments and Applications, Academic Press,
    London.

[7] Volle Michel (1985), L'analyse des donnees, 3e ed., Economica.    
  
  
Also see
--------

@coranal@, @factor@, @pca@, @canon@, @tabulate@, @matrix@




