I'll explain the problem but what I am looking for is a few suggested methods to approach this problem.
You don't need to know what a microarray but if you are interested look here [link text](http://en.wikipedia.org/wiki/DNA_microarray) 
The info below is simplified, not addressed to a micro-biologist (I am not one)


Data:


There are about 250,000 probes on an array that each measure a part of a dna sample.
An array is specific to an individual(germ, person....) If the array is designed for my dna and we put yours on it some probes will match (we are both human I think) and some will not. If they match the measurement is high(the probe glows) as the dna at a probe matched less it glows less. There is a very high variance at each probe across the same dna. So it is very difficult to know if a individual sample matches (same dna) at the probe level.


If the data was nice we could just do a simple Hypothesis test to determine if the sample probe was the same as the dna the array was made for. Because of the high variance this will not work.


Question:


We are looking for a way to consider the measurements at other probes to improve the test at a probe. We kinda know this should work (based on the DNA) but we are not sure how to choose the probes that will best assist for each of the set of 250,000 probes or what statistical method would be best to use the information about the other probes to test an individual probe.


Different example, maybe easier to understand:


A person has 6 characteristics, lets say weight, height, age, income, date of birth (db), education, race. I measure characteristics for myself but my measurement tools are really bad so I make several measurements but I have a very large variance in my measurements. Now I measure another person and want to know if they have a different income than myself. But my measurement tools are so bad almost any other person’s measured income would be within my confidence interval. (I can only measure the other person once, It kills them ☺ ) But I know that other characteristics should be able to help me determine if the income is the same. I don’t know which ones (weight, height, age, db, education, race). I need a method to choose the most useful/helpful characteristics. Now lets say I know that height and age can help determine if our income are the same or different. What would be the statistical method used?


I have what might be called a training set. That is in know what is and is not the same.


@sheldon-cooper answer,
"Note: neither of this will give a "significance score"", A confidence value is good enough. I don't need a statistical "Test"


As an MD I hope that this is an acceptable question for MathOverFlow.net. and I hope you can understand the question as abridged as it is. If some would like to help me rewrite or has suggestion on how I can rewrite the question without all the bio background and as a more specific stats question, let me know.



MicroArray, tesing if a sample is the same with high variance data.  
In general, the approach of using additional measurements of other values (not the one you are interested in directly) was useful in many problems in the past, so it sounds like a good idea in your case as well. Here are a few things to look at:


Binary classification
---------------------


If the task is to determine whether a given probe is a 'match', you can just pose it as a binary classification problem. The input features to a classifier will be ALL the probes in the array. The classifier will determine automatically which of these are useful. If the variance at the target probe is actually small, it may decide to just look at that probe only. If other sites (probes) are helpful too, it may decide to look at them as well. Classifiers can automatically construct complex rules that take into account one or many probes, as needed.


This can be simple to try since there are packages readily available and there are not many parameters to tune. I'd try using a classifier called SVM, it works well in many cases, can deal with limited training data (looks like that's an issue for you). Again, there are SVM packages available that you can download and just run (libsvm is one example, but there are many others).


The disadvantage of this is that it may become resource-consuming if you want a classifier for each probe. Your input dimensionality is 250,000. A simple linear classifier needs roughly one number per dimension, so 250,000 numbers. If you need such a classifier for every probe, this will require 250,000^2 numbers total, which is a lot. There are ways to go around this, but that's one concern.


Collaborative filtering
-----------------------


Collaborative filtering basically allows you to predict the value of one item from the values of related items. Say you know that I liked two movies M1, M2, and disliked two others, M3 and M4. Using collaborative filtering, you can predict whether or not I will like a new movie M5. In your case, you are not doing a completely blind prediction for M5; you do have some (although very noisy) observation. So you would combine that observation with the prediction from other features. 


The advantage of this is that there is only one model (as opposed to one classifier per probe). So this is easier to scale. 


One model would work something like this. It would group probes that are either all "same" or all "different" most of the time. (These groups would be learned by the model from training data.) Then, given a new array, it would use these groups and data from other probes in each group to determine whether a group is, again, "same" or "different". [This](https://mathoverflow.net/questions/14412/matrix-factorization-model) question had a link to what looked like an easy intro to collaborative filtering.


The main disadvantage is, I'm not sure you can find an implementation of the model you need for microarrays online (or anywhere). You'd probably need to derive something suitable for your data, which is not difficult, but requires knowing what to do. 


Another disadvantage is, you'll probably need more training data for collaborative filtering to work.


Note: neither of this will give a "significance score" (as in t-test, for example). You'll get a confidence value, but you won't be able to say that the findings are statistically significant with a certain confidence.


