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I'm analyzing high through proteomic data and have several thousand 5x5 matrices, where in each matrix the rows and columns refer to different biological conditions, and each cell refers to gene expression in that pair of biological conditions.

I expect there to be three kinds of proteins in this dataset:

In most cases I expect there to be no large variation, but random fluctuations between cells:

  0  0 -1 -2  1
  2  1  0 -1 -1
  0  0  1  2  0  
  1 -2  1  3  0
 -3  1  2  0  0

In a few cases, some of the samples might be contaminated, and there might be some extreme outliers, e.g. in a case where (2,2) and (3,4) are outliers:

  1  0  0 -2  1
  0 84  1 -1 -1
 -1  0  0  2  0  
  0  1  99 3  0
  1 -3  2  0  0

However, the cases that I'm interested in, are those in which protein expression changes towards a certain optimal pair of biological conditions, e.g. where proteins change expression reaching an optimal point at the (3,4) cell:

  0  0  5 12  8
  1  5 20 36 21
  0 10 31 52 40  
  1 -2 17 23 30
 -3  1 12  5  8

Is there a statistical test to identify the optimal cell in the above example, that is also robust to handling outliers and random fluctuation as in the first two cases?

Thanks in advance.

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I'm just going to throw out some thoughts:

  1. There is no rigorous method to detect outliers for nonparametric data. If you can't say what the distribution really is, then you can't say what is an outlier. Therefore, you need to remove those by hand in a justifiable way.

  2. Mathematically, I think by "optimal" you mean "maximum." Assuming you have removed putative outliers, your problem is now how to determine a maximum value that you are confindent in with respect to noise. This is a signal processing problem, not a statistics problem. I would suggest taking a look at image processing algorithms which also deal with noisy 2D data sets.

Best I can offer.

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