FDR calculation for each p value

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? 1 Answer 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. • 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 Mar 28 '19 at 21:51 • @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! :) Mar 28 '19 at 22:34 • can you at least explain what is T, what is m? Apr 3 '19 at 2:41 •$m$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. Apr 3 '19 at 14:39
• 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 ? Apr 3 '19 at 15:14