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I am trying to understand how FDR is calculated and what to understand from it. I selected R as language so the example is in R

Lets look at an example of 10 samples

df<- structure(list(Names = structure(c(1L, 3L, 4L, 5L, 6L, 7L, 8L, 
9L, 10L, 2L), .Label = c("Sample1", "Sample10", "Sample2", "Sample3", 
"Sample4", "Sample5", "Sample6", "Sample7", "Sample8", "Sample9"
), class = "factor"), p_value_1 = c(0.01, 0.02, 0.03, 0.05, 0.013, 
0.014, 0.019, 0.35, 0.5, 0.63)), class = "data.frame", row.names = c(NA, 
-10L))

To calculate FDR values , one should use adjusted-p_value which I think is influenced by the size of the sample , how ? I don't know. One can calculate it based on different method for example

FDR <- p.adjust(df$p_value_1, method="fdr")

I came across a study which I cannot understand. is the following formula FDR?

enter image description here

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How to do the math
To adjust for multiple comparisons using the Benjamini-Hochberg procedure for controlling the False Discovery Rate (FDR) of $\alpha$ on $m$ different tests ($m$ is not sample size):

  1. Compute the p-value for each test.
  2. Order the p-values from largest to smallest.
  3. In the first (ordered) test ($i=1$), compare the -value to $\frac{\alpha\times(m+1–1)}{m}$ (if you are using, say, $p=P(|T|>|t|)$ for two-tailed tests, if using $p=P(T>|t|)$, then replace $\alpha$ with $\alpha/2$).
  4. In the second test ($i=2$), compare the -value to $\frac{\alpha\times(m+1–2)}{m}$.
  5. In the third test ($i=3$), compare the -value to $\frac{\alpha\times(m+1–3)}{m}$.
  6. In the ith test, compare the -value to $\frac{\alpha\times(m+1–i)}{m}$.
  7. Using Benjamini & Hochberg’s method, we reject all tests including and following the first test for which we reject the null hypothesis.

Adjusted p-values simply invert the above math to transform p, instead of alter the rejection threshold. So reject if $p \le \frac{\alpha (m+1–i)}{m}$ becomes reject if $\frac{mp}{(m+1–i)} \le \alpha$

Important things to note about adjusted p values

  • Adjusted p-values can be computed as $>1$ which is incoherent, since p-values are supposed to be probabilities. These are typically reported simply as $1$.
  • One cannot determine whether any given test is significant or not simply by looking at the adjusted p-values alone, since these decisions also depend on the ordering of the unadjusted p-values.
  • p.adjust() for some reason has a arithmetic error relative to Benjamini & Hochberg: the first two adjusted p values it reports are always given the same value in my experience.
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  • $\begingroup$ I want to accept and like your answer, however, I have some questions regarding your answer. 1- can you give an example how to calculate it? second what about the definition of null hypothesis? can you please explain that ? Please look at this manuscript jstor.org/stable/2346101?seq=1#page_scan_tab_contents $\endgroup$
    – Learner
    Mar 28 '19 at 21:51
  • $\begingroup$ @Learner Hi Learner... I will be happy to add a worked example, but it may be a few hours. I am indeed familiar with the article you cite, and have read it a few times. I am not sure I follow you about null hypotheses? If you have a question about what they are you should ask a new question! :) $\endgroup$
    – Alexis
    Mar 28 '19 at 22:34
  • $\begingroup$ can you at least explain what is T, what is m? $\endgroup$
    – Learner
    Apr 3 '19 at 2:41
  • $\begingroup$ $m$ is the number of tests. $P(T>|t|)$ literally "the probability of observing some value of a t test statistic greater than or equal to the absolute value of $t$." $P(|T|>|t|)$ literally "the probability of observing some value of a t test statistic greater than or equal to the absolute value of $t$ or less than or equal to negative one time the absolute value of $t$." But t test statistics are just an example. FDR works for any group of hypothesis tests. $\endgroup$
    – Alexis
    Apr 3 '19 at 14:39
  • $\begingroup$ how can I calculate the probabilities with hand? you make me more confused than I should be!!!!!! Can you give an example of 5 p values ? lets say I have a group (control) and a group of treated of 5 patients, I obtain 5 p values. Now how can I calculate the FDR with hand ? $\endgroup$
    – Learner
    Apr 3 '19 at 15:14

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