# PCA or MCA for binarized data

I am working with bioinformatics and I have data that looks like the following:

    H3K18Ac H3K27me3    H3K36me3    H3K4me1 H3K4me2 H3K4me3 H3K9Ac  H4K12Ac PolII
1   0   0   0   0   0   0   0   0   0
2   0   0   0   0   0   0   0   0   0
3   0   0   0   0   0   0   0   0   0
4   0   0   0   0   0   0   0   0   0
5   0   0   0   0   0   0   0   0   0
6   0   0   0   0   0   0   0   0   0
|
|
308792  0   1   0   0   1   0   0   0   0


So I have about 300,000 observations of 9 variables. They are binarized (ie the only value the variable could have is 0/1). I was told by my professor to run PCA on this. The goal is to determine the independence of the variables. After doing some research, it turns out that PCA is only for continuous variables and what I am looking for is MCA/MFA. This is because (and I am guessing here) that my data is categorical rather than continuous. Is this correct?

My question has two parts. Basically, I am trying to get an intuition on what PCA/MCA/MFA mean and their statistical interpretation.

EDIT: Based on the answer below, I ran PCA on my data. This is what I got back as a summary, can someone care to explain what it all means:

Call:
PCA(b)

Eigenvalues
Dim.1   Dim.2   Dim.3   Dim.4   Dim.5   Dim.6   Dim.7   Dim.8   Dim.9
Variance               3.604   1.115   1.045   0.938   0.713   0.539   0.456   0.300   0.290
% of var.             40.050  12.390  11.612  10.426   7.925   5.986   5.061   3.332   3.217
Cumulative % of var.  40.050  52.440  64.052  74.478  82.403  88.390  93.451  96.783 100.000

Individuals (the 10 first)
Dist    Dim.1    ctr   cos2    Dim.2    ctr   cos2    Dim.3    ctr   cos2
1        |  0.699 | -0.542  0.000  0.602 | -0.137  0.000  0.038 | -0.101  0.000  0.021 |
2        |  0.699 | -0.542  0.000  0.602 | -0.137  0.000  0.038 | -0.101  0.000  0.021 |
3        |  0.699 | -0.542  0.000  0.602 | -0.137  0.000  0.038 | -0.101  0.000  0.021 |
4        |  0.699 | -0.542  0.000  0.602 | -0.137  0.000  0.038 | -0.101  0.000  0.021 |
5        |  0.699 | -0.542  0.000  0.602 | -0.137  0.000  0.038 | -0.101  0.000  0.021 |
6        |  0.699 | -0.542  0.000  0.602 | -0.137  0.000  0.038 | -0.101  0.000  0.021 |
7        |  0.699 | -0.542  0.000  0.602 | -0.137  0.000  0.038 | -0.101  0.000  0.021 |
8        |  0.699 | -0.542  0.000  0.602 | -0.137  0.000  0.038 | -0.101  0.000  0.021 |
9        |  0.699 | -0.542  0.000  0.602 | -0.137  0.000  0.038 | -0.101  0.000  0.021 |
10       |  0.699 | -0.542  0.000  0.602 | -0.137  0.000  0.038 | -0.101  0.000  0.021 |

Variables
Dim.1    ctr   cos2    Dim.2    ctr   cos2    Dim.3    ctr   cos2
H3K18Ac  |  0.500  6.943  0.250 |  0.651 38.002  0.424 |  0.271  7.036  0.074 |
H3K27me3 | -0.084  0.195  0.007 | -0.372 12.421  0.139 |  0.584 32.635  0.341 |
H3K36me3 |  0.115  0.365  0.013 |  0.126  1.421  0.016 | -0.763 55.679  0.582 |
H3K4me1  |  0.555  8.543  0.308 |  0.593 31.535  0.352 |  0.175  2.933  0.031 |
H3K4me2  |  0.835 19.327  0.697 | -0.167  2.512  0.028 |  0.041  0.157  0.002 |
H3K4me3  |  0.796 17.592  0.634 | -0.253  5.760  0.064 | -0.046  0.207  0.002 |
H3K9Ac   |  0.845 19.828  0.715 | -0.190  3.249  0.036 |  0.001  0.000  0.000 |
H4K12Ac  |  0.800 17.736  0.639 | -0.085  0.643  0.007 | -0.035  0.116  0.001 |
PolII    |  0.584  9.472  0.341 | -0.223  4.457  0.050 | -0.114  1.236  0.013 |

> res$eig eigenvalue percentage of variance cumulative percentage of variance comp 1 3.6044591 40.049546 40.04955 comp 2 1.1151142 12.390158 52.43970 comp 3 1.0451003 11.612225 64.05193 comp 4 0.9383536 10.426151 74.47808 comp 5 0.7132522 7.925025 82.40311 comp 6 0.5387807 5.986452 88.38956 comp 7 0.4555308 5.061453 93.45101 comp 8 0.2998385 3.331539 96.78255 comp 9 0.2895705 3.217450 100.00000 > res$var
$coord Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 H3K18Ac 0.50024249 0.65096914 0.271178606 0.082130491 0.179898889 H3K27me3 -0.08393274 -0.37216436 0.584012215 0.716196755 -0.004918553 H3K36me3 0.11467656 0.12589380 -0.762825333 0.613452477 -0.066302036 H3K4me1 0.55491452 0.59300598 0.175082970 0.171327973 -0.090682110 H3K4me2 0.83463853 -0.16735597 0.040521897 -0.068620027 -0.192035432 H3K4me3 0.79629230 -0.25344621 -0.046489758 -0.054375461 -0.220167476 H3K9Ac 0.84540049 -0.19034355 0.001356118 -0.070513987 -0.087435840 H4K12Ac 0.79955158 -0.08468939 -0.034841216 0.003741303 -0.079640892 PolII 0.58429734 -0.22292558 -0.113651084 0.018482152 0.754258704$cor
Dim.1       Dim.2        Dim.3        Dim.4        Dim.5
H3K18Ac   0.50024249  0.65096914  0.271178606  0.082130491  0.179898889
H3K27me3 -0.08393274 -0.37216436  0.584012215  0.716196755 -0.004918553
H3K36me3  0.11467656  0.12589380 -0.762825333  0.613452477 -0.066302036
H3K4me1   0.55491452  0.59300598  0.175082970  0.171327973 -0.090682110
H3K4me2   0.83463853 -0.16735597  0.040521897 -0.068620027 -0.192035432
H3K4me3   0.79629230 -0.25344621 -0.046489758 -0.054375461 -0.220167476
H3K9Ac    0.84540049 -0.19034355  0.001356118 -0.070513987 -0.087435840
H4K12Ac   0.79955158 -0.08468939 -0.034841216  0.003741303 -0.079640892
PolII     0.58429734 -0.22292558 -0.113651084  0.018482152  0.754258704

$cos2 Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 H3K18Ac 0.250242546 0.423760821 7.353784e-02 6.745418e-03 3.236361e-02 H3K27me3 0.007044705 0.138506309 3.410703e-01 5.129378e-01 2.419216e-05 H3K36me3 0.013150713 0.015849249 5.819025e-01 3.763239e-01 4.395960e-03 H3K4me1 0.307930129 0.351656088 3.065405e-02 2.935327e-02 8.223245e-03 H3K4me2 0.696621476 0.028008020 1.642024e-03 4.708708e-03 3.687761e-02 H3K4me3 0.634081430 0.064234980 2.161298e-03 2.956691e-03 4.847372e-02 H3K9Ac 0.714701983 0.036230668 1.839057e-06 4.972222e-03 7.645026e-03 H4K12Ac 0.639282736 0.007172292 1.213910e-03 1.399735e-05 6.342672e-03 PolII 0.341403380 0.049695815 1.291657e-02 3.415899e-04 5.689062e-01$contrib
Dim.1     Dim.2        Dim.3        Dim.4       Dim.5
H3K18Ac   6.9425825 38.001561 7.036438e+00  0.718856660  4.53747063
H3K27me3  0.1954442 12.420818 3.263517e+01 54.663590856  0.00339181
H3K36me3  0.3648457  1.421312 5.567911e+01 40.104703429  0.61632615
H3K4me1   8.5430330 31.535431 2.933120e+00  3.128167613  1.15292246
H3K4me2  19.3266578  2.511673 1.571164e-01  0.501805286  5.17034591
H3K4me3  17.5915835  5.760395 2.068029e-01  0.315093446  6.79615372
H3K9Ac   19.8282728  3.249054 1.759695e-04  0.529887903  1.07185451
H4K12Ac  17.7358854  0.643189 1.161525e-01  0.001491692  0.88926070
PolII     9.4716952  4.456567 1.235917e+00  0.036403115 79.76227412

• Check out the related material at stats.stackexchange.com/search?q=pca+binary. The highest voted one at stats.stackexchange.com/q/5774 might be particularly helpful. Since you're new to PCA, make sure to look over the answers at stats.stackexchange.com/q/2691 too.
– whuber
Nov 5, 2014 at 23:51
• masfenix, Please de-cipher all the acronyms in your question. PCA, MCA, MFA... Otherwise, how can a reader know what they mean?! Nov 6, 2014 at 0:19
• PCA - Principle Component Analysis, MCA - Multiple Component Analysis, MFA - Multiple Factor Analysis Nov 6, 2014 at 20:36

What PCA does essentially is find orthogonal directions in an $N$ dimensional space (in your case $N=9$) which explain the most variance in your data.