Is true discovery rate (TDR) the same as true positive rate (TPR)? This is a general question about the definition of true discovery rate (TDR) and true positive rate (TPR). It is more common to use TPR, defined as $\frac{TP}{TP+FN}$, and false discovery rate (FDR) defined as $\frac{FP}{TP+FP}$. I searched online for TDR, but it seems that people seldom use this term, and I didn't find its definition either. My question: is TDR and TPR the same thing? Thanks!
 A: Concluding form your definitions of FDR and TPR, TPR is not the same as TDR (that is, if TDR = 1 - FDR). In a way, the TDR is inverse to the TPR.
Consider this scheme of a confusion table and different characteristic proportions derived from it:

(From:  Beleites, C.; Salzer, R. & Sergo, V.: Validation of soft classification models using partial class memberships: An extended concept of sensitivity & co. applied to grading of astrocytoma tissues, Chemom Intell Lab Syst, 122, 12 - 22 (2013). DOI: 10.1016/j.chemolab.2012.12.003). The paper is about classification, but hypothesis testing can be summarized analogously.
So you can look at different proportions that summarize the confusion table. For statistical hypothesis testing, you'd get a 2x2 table having  e.g. true difference and truely no difference as reference and significant and non-significant as "prediction" of the hypothesis test*)
So, TPR and TDR answer different ("inverse") questions:


*

*TPR: In which fraction of tests of all hypotheses where truely a difference is ("true hypotheses") does the test yield "significant" outcome? 

*TDR: Which fraction of the hypotheses that are declared "significant" by the test really are true?


Note that the TDR is of more practical interest, but crucially depends on the proportion of true hypotheses among all hypotheses that were tested (the prior probability or prevalence of true hypotheses). 
* Note also that for the sake of a brief discussion, I speak of "true hypothesis" meaning the research idea and skip the further inversion of the statistical hypothesis testing logics that a Null hypothesis of "no effect" is formulated which should be rejected if there is an effect (i.e. the research idea was correct).
