# Identify points significantly outside a regression model

I would like to update my previous question "Methods to determine reliability of measurements using median and median absolute deviation".

I have data from biological experiments, that look like this:

v1 2 1.8 1.5 1.9 2.1 1.78 1.95 2.0 2.1
v2 2 100 -5.2
v3 1 -1.3 -2 2.3
v4 1 1.5 1.6 1.9 2.1 2.0 2.4 -1.1 2.3 1.5 1.6 1.9 1.8 1.6


These represent gene expressions.

Now, I would expect that all values of each variable (genes) are more or less similar, since the values are ripetute (?) measurement of the same gene. Having a variable with such huge difference, as v2, doesn't have sense. That means that it does not make sense to have a gene with such a huge variation in expression. Therefore, it has to come from a methodological error.

I was looking for a method (possible a statistical test) which could identify the "average variability" among my samples and report me which variables(genes) have a variability significantly greater. This means that for these gens my data are not good enough to estimate the expression, and I have to discard them.

** In the first place I would really appreciate any suggestion on test I could use for my purpose. **

I thought at something like that:

I consider for each gene the ratio between median absolute deviation (mad) and median (med). I would expect this ratio has a normal-like distribution, assuming that the variability comes from error in the measurement occurred by chance. Therefore, I could identify then all variables (gene) that have a mad/med ratio significantly higher that the average distribution, or if you want that are "positive outliers".

For me it sounds quite intuitive but I'm not sure how to perform that in an appropriate way. I would really appreciate your opinions about my method and also your suggestions on how I could perform the analysis.

• By appropriate way, do you mean you want to make assumptions about the distribution of "errors" and perform some kind of hyothesis testing? – Theja Jul 3 '14 at 12:57
• I want to create a sort of regression for the errors and remove the variables with error higher then the predicted regression. – efrem Jul 3 '14 at 13:36
• A few simulations do suggest that the ratio will at least in some situations have a somewhat normalish appearance; it may be plausible for your data, but with some cases that seem plausible to me it appears to be substantially right skew. – Glen_b Jul 3 '14 at 16:04
• Thank you for the clarifcation. I saw the previous question as well. I have a more basic question: have you looked at the tools and techniques related to anomaly detection and outliers? If so, maybe you can comment on why they are unsuitable in this context. – Theja Jul 3 '14 at 19:23
• Thanks. The variable with high dispersion are genes for which the quantification could not be achieved in a consistent way. For a gene in a specific time point there will be just 1 value that indicates its expression. If we measure more than once that gene we should always have similar values. In case the expression values we measure are too different, it means that the methods/conditions applied are not good enough to have an accurate measurement. Consequently, there is a big grade of uncertainty in the quantification of the gene expression. Therefore, we have to discard the gene. – efrem Jul 8 '14 at 7:21