# How to do hypothesis testing on Gaussian mixture model? [closed]

I am CS major, please be patient if my question is not well-stated. This is cross-posted from math as suggested by J.M.

The dataset is quantitative mass spectrometry (MS) data. By labeling proteins of two different samples A and B, we get the relative abundance of 100 to thousands of proteins in A/B. Alongside with this ratio, we can estimate its variance based on the signal intensities.

Wanted: A list of proteins significantly different from the set of all protein ratios.

Most proteins remain unchanged between A and B. The population of log-ratios distributes around 1. The histogram shows a bell shape with fat tails. Two-term Gaussian mixture model has been found to provide a good fit to experimental noise. I suppose it would work good for this data - think of experimental and biological noise.

How to test for significantly different ratios on such a two-term Gaussian mixture model?

 There is a lot of vagueness here. Are you saying (a) there is a set of proteins present in both samples and (b) each protein yields a ratio equal to its abundance in A divided by its abundance in B? If so, where do you obtain the value for "mean_population"? Is it a measurement? If so, how is it made? In fact, what do you really mean by "H0"? Are you looking for outliers in the log ratios or are you truly trying to compare the mean log ratio, averaged over all the proteins, to some known constant (the "mean_population")? – whuber♦ Nov 5 '10 at 16:27 There is some confusion here. Your hypothesis states 'means' whereas your text refers to 'ratios'. What do you mean by 'population of protein ratios'? Are these some sort of theoretical ratios that you expect to see? Can you clarify? – user28 Nov 5 '10 at 16:29 Thanks for your answers. whuber: I do say (a) and (b). I changed the definition of what I want to test to make it clearer. I want to get the significant outliers of the population of ratios at hand. – Florian Bw Nov 6 '10 at 9:06 Srikant Vadali: I tried to remove the disambiguation between text and H0. mean_protein: protein ratio; mean_population: the set of all protein ratios. – Florian Bw Nov 6 '10 at 9:11 @Florian I am still confused about your hypothesis. I think it will help a lot if you could identify the variables in your set-up and explicitly establish the hypothesis in those terms. For example, my current understanding is as follows: We have a set of ratios $r_i$ where $i$ indexes the proteins. You want to identify the set of proteins ${i,j}$ such that ${r_i \ne r_j}$. Is the above an accurate translation of what you want to do? – user28 Nov 8 '10 at 16:06