Question: Hi everybody ! I'm trying to use PCA over Rna-seq data just to understand PCA. I saw pcaExplorer and I did not quite understand what it actually does. I have a count matrix of 90 samples (healthy and cancer). I have then thousands of genes. I want to obtain the genes that differ my samples the most for further classification ( I know there are other techniques but I just wanted to exercise a bit, with PCA ). Thats what I have done :

Let z be the transpose of a count matrix, normalized with method TMM = samples as rows and genes as columns

z <- t(counts.tmm)
pca <- prcomp(z, center = TRUE,scale. = TRUE)
summary(pca)
pca.class <- sample.cond$class # here sample.cond$class will be HC or Pancreas (Tumor)
ggbiplot(pca,ellipse=TRUE,groups = pca.class,choices= c(1,2),var.axes=FALSE)
dev.off()

The result is the following :

Importance of components:
PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9 PC10 PC11 PC12
Standard deviation 22.6979 8.01951 6.89380 5.85445 5.39476 5.30267 5.2139 5.00639 4.99198 4.96105 4.85406 4.7934
Proportion of Variance 0.2937 0.03667 0.02709 0.01954 0.01659 0.01603 0.0155 0.01429 0.01421 0.01403 0.01343 0.0131
Cumulative Proportion 0.2937 0.33039 0.35749 0.37703 0.39362 0.40965 0.4251 0.43944 0.45365 0.46768 0.48111 0.4942
PC13 PC14 PC15 PC16 PC17 PC18 PC19 PC20 PC21 PC22 PC23 PC24
Standard deviation 4.74638 4.69116 4.66121 4.62983 4.56994 4.55543 4.50351 4.46740 4.44247 4.36019 4.32935 4.3111
Proportion of Variance 0.01284 0.01255 0.01239 0.01222 0.01191 0.01183 0.01156 0.01138 0.01125 0.01084 0.01069 0.0106
Cumulative Proportion 0.50706 0.51960 0.53199 0.54421 0.55612 0.56795 0.57951 0.59089 0.60214 0.61298 0.62367 0.6343
PC25 PC26 PC27 PC28 PC29 PC30 PC31 PC32 PC33 PC34 PC35 PC36
Standard deviation 4.26250 4.21480 4.15619 4.14941 4.04881 4.01388 3.94854 3.93874 3.91634 3.86539 3.81318 3.79119
Proportion of Variance 0.01036 0.01013 0.00985 0.00982 0.00935 0.00919 0.00889 0.00884 0.00874 0.00852 0.00829 0.00819
Cumulative Proportion 0.64462 0.65475 0.66460 0.67441 0.68376 0.69295 0.70183 0.71068 0.71942 0.72794 0.73623 0.74443
PC37 PC38 PC39 PC40 PC41 PC42 PC43 PC44 PC45 PC46 PC47 PC48
Standard deviation 3.77294 3.73427 3.72646 3.62194 3.59297 3.56278 3.51904 3.42127 3.38418 3.37216 3.32199 3.30519
Proportion of Variance 0.00812 0.00795 0.00792 0.00748 0.00736 0.00724 0.00706 0.00667 0.00653 0.00648 0.00629 0.00623
Cumulative Proportion 0.75254 0.76049 0.76841 0.77589 0.78325 0.79049 0.79755 0.80422 0.81075 0.81723 0.82352 0.82975
PC49 PC50 PC51 PC52 PC53 PC54 PC55 PC56 PC57 PC58 PC59 PC60
Standard deviation 3.27842 3.2430 3.21268 3.18443 3.17152 3.09140 3.05199 3.04186 3.03361 2.99736 2.98039 2.94492
Proportion of Variance 0.00613 0.0060 0.00588 0.00578 0.00573 0.00545 0.00531 0.00528 0.00525 0.00512 0.00506 0.00494
Cumulative Proportion 0.83588 0.8419 0.84776 0.85354 0.85928 0.86472 0.87003 0.87531 0.88056 0.88568 0.89074 0.89569
PC61 PC62 PC63 PC64 PC65 PC66 PC67 PC68 PC69 PC70 PC71 PC72
Standard deviation 2.92100 2.90968 2.86032 2.8395 2.82278 2.79180 2.74172 2.73852 2.70586 2.68556 2.64009 2.60152
Proportion of Variance 0.00486 0.00483 0.00466 0.0046 0.00454 0.00444 0.00429 0.00428 0.00417 0.00411 0.00397 0.00386
Cumulative Proportion 0.90055 0.90538 0.91004 0.9146 0.91918 0.92363 0.92791 0.93219 0.93636 0.94047 0.94445 0.94831
PC73 PC74 PC75 PC76 PC77 PC78 PC79 PC80 PC81 PC82 PC83 PC84
Standard deviation 2.58597 2.57208 2.52419 2.49683 2.45577 2.43039 2.4041 2.36481 2.31208 2.28884 2.24954 2.20218
Proportion of Variance 0.00381 0.00377 0.00363 0.00355 0.00344 0.00337 0.0033 0.00319 0.00305 0.00299 0.00289 0.00276
Cumulative Proportion 0.95212 0.95589 0.95952 0.96308 0.96652 0.96988 0.9732 0.97637 0.97941 0.98240 0.98529 0.98805
PC85 PC86 PC87 PC88 PC89 PC90
Standard deviation 2.18536 2.1373 2.06398 1.94600 1.88876 7.458e-15
Proportion of Variance 0.00272 0.0026 0.00243 0.00216 0.00203 0.000e+00
Cumulative Proportion 0.99077 0.9934 0.99581 0.99797 1.00000 1.000e+00

And the image looks like this :

![enter image description here][1]

My Questions are :

1) Why the PC are the samples? Shouldn't they be the genes? If I do `var.axes=TRUE` it actually shows me the arrows that are the genes so is correct but I did not get this point.

2) How can I get the genes that make my samples differ by condition the best from the PCA that i have computed?

If I do not transpose the matrix :

z <- counts.tmm # samples are in the columns and genes as rows
pca <- prcomp(z, center = TRUE,scale. = FALSE)
summary(pca)
ggbiplot(pca,ellipse=TRUE,choices= c(1,2),var.axes=FALSE)
dev.off()

thats what I get :

Importance of components:
PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9 PC10 PC11 PC12
Standard deviation 13.0772 4.07904 2.49366 2.04542 1.82674 1.50391 1.46282 1.41308 1.39098 1.33582 1.32702 1.30732
Proportion of Variance 0.5983 0.05821 0.02175 0.01464 0.01167 0.00791 0.00749 0.00699 0.00677 0.00624 0.00616 0.00598
Cumulative Proportion 0.5983 0.65649 0.67825 0.69288 0.70456 0.71247 0.71996 0.72694 0.73371 0.73996 0.74612 0.75210
PC13 PC14 PC15 PC16 PC17 PC18 PC19 PC20 PC21 PC22 PC23 PC24
Standard deviation 1.29400 1.28199 1.26799 1.25235 1.24683 1.21349 1.19880 1.18708 1.18165 1.17765 1.16516 1.15687
Proportion of Variance 0.00586 0.00575 0.00562 0.00549 0.00544 0.00515 0.00503 0.00493 0.00488 0.00485 0.00475 0.00468
Cumulative Proportion 0.75795 0.76370 0.76933 0.77482 0.78025 0.78541 0.79043 0.79536 0.80025 0.80510 0.80985 0.81453
PC25 PC26 PC27 PC28 PC29 PC30 PC31 PC32 PC33 PC34 PC35 PC36
Standard deviation 1.15105 1.13993 1.12768 1.1218 1.11720 1.10388 1.09192 1.08421 1.07808 1.07271 1.06507 1.05828
Proportion of Variance 0.00464 0.00455 0.00445 0.0044 0.00437 0.00426 0.00417 0.00411 0.00407 0.00403 0.00397 0.00392
Cumulative Proportion 0.81917 0.82371 0.82816 0.8326 0.83693 0.84119 0.84537 0.84948 0.85354 0.85757 0.86154 0.86546
PC37 PC38 PC39 PC40 PC41 PC42 PC43 PC44 PC45 PC46 PC47 PC48 PC49
Standard deviation 1.05420 1.0415 1.03585 1.02542 1.0145 1.00792 0.99140 0.97666 0.97385 0.96267 0.9568 0.95232 0.94773
Proportion of Variance 0.00389 0.0038 0.00375 0.00368 0.0036 0.00355 0.00344 0.00334 0.00332 0.00324 0.0032 0.00317 0.00314
Cumulative Proportion 0.86935 0.8731 0.87689 0.88057 0.8842 0.88773 0.89117 0.89450 0.89782 0.90106 0.9043 0.90744 0.91058
PC50 PC51 PC52 PC53 PC54 PC55 PC56 PC57 PC58 PC59 PC60 PC61
Standard deviation 0.9416 0.93214 0.92504 0.91419 0.91151 0.89366 0.88537 0.88206 0.87722 0.86586 0.86438 0.85371
Proportion of Variance 0.0031 0.00304 0.00299 0.00292 0.00291 0.00279 0.00274 0.00272 0.00269 0.00262 0.00261 0.00255
Cumulative Proportion 0.9137 0.91672 0.91972 0.92264 0.92555 0.92834 0.93108 0.93381 0.93650 0.93912 0.94173 0.94428
PC62 PC63 PC64 PC65 PC66 PC67 PC68 PC69 PC70 PC71 PC72 PC73
Standard deviation 0.8449 0.84420 0.83856 0.83176 0.82325 0.80923 0.7924 0.78655 0.78025 0.77028 0.76573 0.75772
Proportion of Variance 0.0025 0.00249 0.00246 0.00242 0.00237 0.00229 0.0022 0.00216 0.00213 0.00208 0.00205 0.00201
Cumulative Proportion 0.9468 0.94928 0.95174 0.95416 0.95653 0.95882 0.9610 0.96318 0.96531 0.96738 0.96944 0.97144
PC74 PC75 PC76 PC77 PC78 PC79 PC80 PC81 PC82 PC83 PC84 PC85
Standard deviation 0.75455 0.74868 0.74169 0.73250 0.73053 0.72453 0.71426 0.70595 0.6973 0.68962 0.6761 0.66666
Proportion of Variance 0.00199 0.00196 0.00192 0.00188 0.00187 0.00184 0.00178 0.00174 0.0017 0.00166 0.0016 0.00155
Cumulative Proportion 0.97344 0.97540 0.97732 0.97920 0.98107 0.98290 0.98469 0.98643 0.9881 0.98980 0.9914 0.99295
PC86 PC87 PC88 PC89 PC90
Standard deviation 0.66129 0.6555 0.64137 0.61824 0.59554
Proportion of Variance 0.00153 0.0015 0.00144 0.00134 0.00124
Cumulative Proportion 0.99448 0.9960 0.99742 0.99876 1.00000

![enter image description here][2]

But the list of PC is always 90. Now here it makes sense that they are 90 (samples) but before I don't get why is 90 when the genes are thousands.

4) Last question , for my purpose (obtain genes that shows more difference between my samples), is it more suitable the first (PC as genes) or the second (PC as samples)?

From pcaExplorer I have seen that they relate this last PCA (samples as columns and genes as rows) with the study of the Heatmap so I assume that this last one is the one to use for gene view? But I don't get why.

[1]: /media/images/cccd8f68-e7e2-4db2-9754-9c030b54
[2]: /media/images/a174713a-facf-4890-9899-3cd2f6aa

Answer: Not an expert by any means, but I have a couple ideas for you.

About your data and the genes-vs-samples thing, I think you're right that you want counts per gene being the components, not counts per sample. But I think you already have that when your samples are on rows-- it's just hard to see in the output, especially because the number of components you see there is throwing you off. You can look at `pca$x` to see the actual transformed data.

Here's a stupid example with ten samples and three "measurements" where there's really only one dimension.

```
> x <- matrix(1:30, ncol=3)
> x
[,1] [,2] [,3]
[1,] 1 11 21
[2,] 2 12 22
[3,] 3 13 23
[4,] 4 14 24
[5,] 5 15 25
[6,] 6 16 26
[7,] 7 17 27
[8,] 8 18 28
[9,] 9 19 29
[10,] 10 20 30
> pca <- prcomp(x)
> summary(pca)
Importance of components:
PC1 PC2 PC3
Standard deviation 5.244 3.55e-16 6.418e-32
Proportion of Variance 1.000 0.00e+00 0.000e+00
Cumulative Proportion 1.000 1.00e+00 1.000e+00
```

It rotated it so PC1 has all the variation (and the others essentially none, since this only requires one dimension to fully explain). If you print the object you see that rotation:

```
> pca
Standard deviations (1, .., p=3):
[1] 5.244044e+00 3.549628e-16 6.417917e-32

Rotation (n x k) = (3 x 3):
PC1 PC2 PC3
[1,] 0.5773503 -0.8164966 0.0000000
[2,] 0.5773503 0.4082483 -0.7071068
[3,] 0.5773503 0.4082483 0.7071068
```

And then `pca$x` contains the same ten points, with the rotation applied, where they all lie along that line:

```
> pca$x
PC1 PC2 PC3
[1,] -7.7942286 2.220446e-15 -4.440892e-16
[2,] -6.0621778 1.776357e-15 -4.440892e-16
[3,] -4.3301270 1.110223e-15 -2.220446e-16
[4,] -2.5980762 7.771561e-16 -2.220446e-16
[5,] -0.8660254 2.498002e-16 -5.551115e-17
[6,] 0.8660254 -2.498002e-16 5.551115e-17
[7,] 2.5980762 -7.771561e-16 2.220446e-16
[8,] 4.3301270 -1.110223e-15 2.220446e-16
[9,] 6.0621778 -1.776357e-15 4.440892e-16
[10,] 7.7942286 -2.220446e-15 4.440892e-16
```
If you give input that's wider than it is long (more genes than samples), you should see that the number of rows of the output is still what you expect, and it's just your columns that are fewer than you might expect.

```
> x2 <- matrix(1:50, ncol=10)
> x2
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 1 6 11 16 21 26 31 36 41 46
[2,] 2 7 12 17 22 27 32 37 42 47
[3,] 3 8 13 18 23 28 33 38 43 48
[4,] 4 9 14 19 24 29 34 39 44 49
[5,] 5 10 15 20 25 30 35 40 45 50
> pca2 <- prcomp(x2)
> pca2$x
PC1 PC2 PC3 PC4 PC5
[1,] -6.324555 -2.498002e-15 4.440892e-16 4.440892e-16 4.440892e-16
[2,] -3.162278 -1.249001e-15 2.220446e-16 2.220446e-16 2.220446e-16
[3,] 0.000000 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
[4,] 3.162278 1.249001e-15 -2.220446e-16 -2.220446e-16 -2.220446e-16
[5,] 6.324555 2.498002e-15 -4.440892e-16 -4.440892e-16 -4.440892e-16
```

Thinking it through with a very small number of samples and dimensions to make my brain hurt less, this makes sense. If you had just two points, you would never need more than two axes (actually, one, right?) to fully define the difference between them. So you'll never need more components in the output than you have samples going in.

**So long story short, give your samples on rows and genes on columns.**

P.S. Be careful juggling different matrices and data frames and such. `ggbiplot` would have complained about the transposed case, except that happens that the length of the `groups` happened to match up. (I usually aim for putting more into data frames wherever possible and then subsetting/slicing them as needed, just because of how often that sort of thing comes up in R.)