How to detect outlier samples in gene expression studies? I have a matrix of  n observations where each observation has m variables. So I am faced in with a matrix of mxn. How may I determine which observations are outliers? 
Thank you in advance.
 A: The answer to your question depends generally on your objectives, and more specifically, on what you mean by 'outlier'.
I do not disagree with @DemetriPanthos' suggestion to
use Mahalanobis distance. However, to me this method seems best when all variables are from
the same family of distributions (e.g, normal).
[If you are unfamiliar with
Mahalanobis distance, I suggest you look at the relevant Wikipedia page.
Googling fetches a YouTube demo that might be
helpful.]
Also, Mahalanobis distance is a generalization of the idea of measuring the distance between the center of a univariate distribution and a point in terms of the number of standard deviations. 
In one dimension, if the SD $\sigma$ is estimated by the sample SD $S,$ then an outlier can inflate $S.$
So if the criterion for an outlier is expressed in terms of the number of SD's, the inflated $S$ may decrease
the apparent number of SD's, leading to failure to detect the presence of the outlier.
Depending on the nature of your data and the importance of outliers in your work, you may want to begin by
taking a quick look at boxplot outliers among the $n$
observations on each of your $m$ variables. If your
data are in an $n \times m$ matrix, then it is easy to
do this in R.
For example, suppose $n = 100$ with $m = 4$ and the relevant distributions are $\mathsf{Norm}(0,1),$ $\mathsf{Norm}(1, \sigma=.5),$ $\mathsf{Exp}(1),$ and $\mathsf{Unif}(0,1),$ respectively. Typically with $n = 100,$ we can expect a few boxplot
outliers among the normal observations, several among
the exponential observations, and none among the uniform ones.
set.seed(1225); n = 100
MAT = cbind(rnorm(n), rnorm(n,1,.5), rexp(n), runif(n))
boxplot(MAT, col="skyblue2", pch=20)

apply(MAT, 2, sd)
[1] 0.8875380 0.4192971 1.1709075 0.2949216


Notes: (1) If your criterion for outlier were to be more than $2.5\sigma$ from the mean, then some of the exponential outliers might go undetected because of the effect
of the extreme outliers on $\hat \sigma = S \approx 1.17.$
(2) For my fake data, the four variables are independent, so issues of correlation do not arise.
pairs(MAT, pch=20)


