PCA on count-based fractions, taking uncertainties into account I'm looking to do a PCA analysis on count based data itself rather than averages. I'm hoping this will help for variable observation depths; for example, 3/4 reads is not really equivalent to 15/20. There is more confidence in the 15/20 being near 75% than for the 3/4. 
Any ideas how how I could do this? 
Here is some example data, each site is the number of positive reads and the number of total reads. 
Individual, Site1, Site 2, ...
Indiv1, 7/9, 4/5, ...
Indiv2, 5/11, 7/22, ...
Indiv3, 14/29, 3/5, ...

 A: My pick of your problem
You ask how to incorporate the statistical uncertainties on the data in your table to do PCA. Since 4 out of 5 has a larger uncertainty than 40 out of 50.
The solution
Put the uncertainty into your data. I'll try to explain below.
First an assumption
You have to make an assumption, though, which is that your measurements (the fractions) follow a certain distribution, which should reflect the statistical uncertainties we wish to incorporate into the data.
I'd recommend the beta distribution.
The procedure
Try the following procedure:


*

*Consider each data point $p_{i}$ in your data table and determine the numbers $i_{i}$ and $n_{i}$ such that $p_{i}\equiv\frac{i_{i}}{n_{i}}$. For example, if $p_{i}=\frac{3}{4}$, then $i_{i}=3$ and $n_{i}=4$.

*Generate extra tables $j$, for which you generate each data point $p_{i,j}$ according to a beta distribution $\texttt{B}\left(\alpha, \beta\right)$. Where $\alpha=i_{i}$ and $\beta=n_{i}-i_{i}+1$. (See here for why this is so.) Leave all the other data points as they were, only manipulate the $p_{i}$.

*Do your PCA analysis treating all your tables as "real" data.


Keep adding new "fake" data until the outcomes don't change significantly any more.
Let us know if that worked for you.
Edit (2016 Sep 13)
I improved my answer to accommodate for the situation amoeba sketches in the comments below.
