# FWER, FDR and multiple comparisons for Beginners

Wikipedia:

"In statistics, family-wise error rate (FWER) is the probability of making one or more false discoveries, or type I errors, among all the hypotheses when performing multiple hypotheses tests."

"The false discovery rate (FDR) is one way of conceptualizing the rate of type I errors in null hypothesis testing when conducting multiple comparisons."

I don't understand the difference between these two concepts. How do they not mean the same?

Perhaps you can help me by further elaborating the following example:

Say the probability for an unbiased coin to substantially deviate from a 50/50 head/tail-distribution in a sequence of 1,000 tosses is 0.001.

If I want to find out if one coin is biased I throw it 1,000 times and if it shows heads ~500 times I can be quite sure it is not biased.

However if I throw a million coins 1,000 times and deem those biased who don't show a 50/50-distribution of heads and tails, I will categorize unbiased coins as biased, because the probability of an unbiased coin showing deviating from the 50/50-distribution is multiplied by the number of coins (1 million).

Thus from a set of one million unbiased coins, I have to expect about 1,000,000*0.001=1,000 coins to deviate substantially from the 50% tails, 50% heads-distribution.

As far as I understood this is multiple hypotheses testing (synonymous: multiple comparisons?) as I am testing the hypothesis "coin is unbiased" a million times, and the false discovery rate FDR is 1,000 in this example.

But what, then, is the FWER (family wise error rate)?

• Does this help? stats.stackexchange.com/questions/59681/… Jul 26 '16 at 12:50
• See the fdr section in stats.stackexchange.com/questions/166323/…
– user83346
Jul 26 '16 at 13:52
• @ChristophHanck what does $m_0$ (or $m$ for that matter) stand for? (I'm referring to your link) Jul 26 '16 at 14:13
• The number of true hypotheses. Jul 26 '16 at 14:15
• @ChristophHanck so $m$ is the number of all hypotheses? Jul 26 '16 at 14:18