The q-value is an adjusted p-value, taking in to account the false discovery rate (FDR). Applying a FDR becomes necessary when we're measuring thousands of variables (e.g. gene expression levels) from a small sample set (e.g. a couple of individuals). A p-value of 0.05 implies that we are willing to accept that 5% of all tests will be false positives. An FDR-adjusted p-value (aka a q-value) of 0.05 implies that we are willing to accept that 5% of the tests found to be statistically significant (e.g. by p-value) will be false positives. Such an adjustment is necessary when we're making multiple tests on the same sample. See, for example, http://www.totallab..../pq-values.aspx.
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