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As an alternative to conducting exploratory factor analysis on a set of data, with binary responses, I have been suggested to use Multiple Correspondence Analysis (MCA).

Following is a curtailed and slightly modified version of the output that I receive from Stata.

 . mca q*, method(burt) normalize(principal) compact

  Multiple/Joint correspondence analysis         
  Number of obs      =        61          Total inertia      =  .0440935
  Method: Burt/adjusted inertias          Number of axes     =         2

            |   principal               cumul 
  Dimension |    inertia     percent   percent
------------+----------------------------------
      dim 1 |    .0078404     17.78      17.78
      dim 2 |    .0066631     15.11      32.89
      dim 3 |     .005856     13.28      46.17
      dim 4 |    .0035926      8.15      54.32
      dim 5 |    .0026133      5.93      60.25
      dim 6 |    .0020177      4.58      64.82
      dim 7 |    .0016913      3.84      68.66
      dim 8 |    .0012484      2.83      71.49
      dim 9 |    .0008873      2.01      73.50
     dim 10 |     .000839      1.90      75.41
     dim 11 |    .0006795      1.54      76.95
     dim 12 |    .0004584      1.04      77.99
     dim 13 |    .0002605      0.59      78.58
     dim 14 |    .0002364      0.54      79.11
     dim 15 |    .0001877      0.43      79.54
     dim 16 |    .0000679      0.15      79.69
     dim 17 |     .000035      0.08      79.77
     dim 18 |    .0000192      0.04      79.82
     dim 19 |    1.73e-06      0.00      79.82
------------+----------------------------------
      Total |    .0440935    100.00

Statistics (x1000) for column categories in principal normalization

------------------- overall ---------- dimension 1 ------- dimension 2 ----
   Categories|  mass qualt %inert | coord sqcor contr | coord sqcor contr |
-------------+--------------------+-------------------+-------------------+
q1a          |                    |                   |                   |
           0 |     6   762    10  |    43    27     1 |    33    16     1 |
           1 |     9   762     7  |   -30    27     1 |   -23    16     1 |
-------------+--------------------+-------------------+-------------------+
q1c          |                    |                   |                   |
           0 |    13   771     2  |   -53   404     5 |   -17    43     1 |
           1 |     2   771    14  |   352   404    31 |   115    43     4 |
-------------+--------------------+-------------------+-------------------+

Can someone help me with interpreting this output and compare it to a factor analysis output.

I am assuming that for interpreting the MCA output, principal inertia should be interpreted similarly to eigenvalues. Does there exist a criteria like eigenvalue>1 for MCA as well for choosing the number of items/dimensions to retain?

Are the "contr" columns in the second table similar to factor loadings?

Thanks,

May

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I'm not Stata user and won't interpret the specific output you show, the so more that you gave only results, not the data to analyze it. Instead, I'll offer few lines about the relationship between the types of analyses, just to guide you.

Multiple Correspondence analysis (which is Correspondence analysis of 3+ dim. contingency table and with optional computation of individuals' "object scores"), MCA, is an optimal scaling dimensionality reduction / mapping technique for all the input variables being nominal.

It is actually a particular case of, and becomes equivalent to Categorical Principal Component analysis (CatPCA) when the latter uses multiple nominal quantification for all the input variables.

If all the variables are dichotomous then MCA is equivalent to CatPCA using any type of quantification - because a variable with just 2 categories can be quantified no otherwise than one way - linearly. And this way MCA/CatPCA becomes almost equivalent to usual linear PCA.

The equivalencies pertain to eigenvalues and object scores (= component scores). MCA does not or normally should not output component-variable loadings because MCA uses "multiple nominal quantification" which is the treating of every value (category) of a variable as a separate variable (a dummy). What will correspond in MCA to the variable "loadings" of PCA is the coordinates of centroids of the categories of the variables.

But in your case - because all the variables are dichotomous and so each variable is fully represented by one nonredundanrt category (one dummy) - you may use and interpret results of just PCA with its loadings, in place of MCA with its centroids coordinates. To repeate: since all variables binary, you actually don't need MCA, usual PCA suffice$^1$ $^2$.

By word inertia the scale is meant in literature: either squared (eigenvalues and their sum) or nonsquared (singular values).

To know more about MCA or simple (two-way) CA please read a good text.


$^1$ You may expect just minor differences, primarily due to MCA being iterative and PCA not. If the convergence is not complete, as often, these ignorable differences show up.

$^2$ When we are considering to use PCA with binary data it is worth thinking about doing or not doing variable centering before the analysis. Typically, PCA is done on centered or standardized data (= done based on covariances or correlations, respectively). Not centering the data can produce very different PCA results, but they can make more sense in some studies with binary categorical data. For example, in text analytics PCA is frequently done based on cosines which implies data normalization without centering.

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  • $\begingroup$ Thanks for the detailed help ttnphns. This is helping me a lot in understanding the similarities and differences between mca and (cat)pca. $\endgroup$ – May Ank Mar 28 '16 at 9:29
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@ttnphns's explanation is excellent. That said, Stata definitely decomposes the information in ways that aren't familiar to me, despite having used MCA for years. One thing that is consistent across all packages is the low dimensional nature of all CA output...two dimensions are always reported. The thing that's missing from this Stata output is the comparable and symmetric output for the rows. Since MCA works on a tabular information, that table is either row or column dominant -- the choice of which one to focus on is the analyst's.

Regarding the terminology, it's too complex for a quick summary (here) except to note that a key insight to understanding these metrics lies in the geometry behind them.

One of the architects of CA is Michael Greenacre who has been publishing seminal works on it since the 80s. His most recent book on this topic is Correspondence Analysis in Practice on Amazon here:

http://www.amazon.com/Correspondence-Analysis-Practice-Interdisciplinary-Statistics/dp/1584886161/ref=sr_1_1?ie=UTF8&qid=1459161494&sr=8-1&keywords=Correspondence+Analysis+in+Practice

You can access this book online. It contains, in the first chapters, thorough explanations of this terminology as well as of the geometry underlying the decompositions.

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  • $\begingroup$ To judge from works on a tabular information, that table is either row or column dominant I suspect you may be speaking about simple (2-way table) CA. But the OP's situation (although, unfortunately, the data have not been presented) seems different. It looks to me like there was 19 binary variables. That is, 2x2x2....x2 19-way frequency table, a task for Multiple CA. $\endgroup$ – ttnphns Mar 28 '16 at 11:22
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    $\begingroup$ @ttphns That is true but my point was different in noting that only two dimensions are reported. This is the standard output across CA software. $\endgroup$ – Mike Hunter Mar 28 '16 at 11:25
  • $\begingroup$ Thank you very much for the Greenacre reference, DJohnson. I am now going through the text. :-) $\endgroup$ – May Ank Mar 31 '16 at 20:22

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