# FWER and FDR controlling procedures for multiple correlation tests The package compiles some multiple testing procedures which theoretically control asymptotically the FWER in the framework of correlation testing. Four tests statistics can be considered: the empirical correlation, the Student statistics, the Fisher's z-transform and the usual Gaussian statistics considering random variables $(X_i-mean(X_i))(X_j-mean(X_j))$. Four methods are implemented: Bonferroni (1935)'s, Šidák (1967)'s, Romano & Wolf (2005)'s bootstrap and (Drton & Perlman (2007)'s procedure based on the asymptotic distributions of the test statistics, called MaxTinfty. The package also includes some multiple testing procedures which are related to the control of the FDR : Cai & Liu (2016)'s procedures called LCT-N and LCT-B -which have been proven to control the FDR for correlation tests- and Benjamini & Hochberg (1995)'s -which has no theoretical results in correlation testing. package ## Details Consider $\lbrace \bold{X}_\ell = (X_{1\ell},\dots X_{p\ell}),\; \ell=1,...,n\rbrace$ a set of $n$ independent and identically distributed $R^p$-valued random variables. Denote `data` the array containing $\lbrace\mathbf{X_\ell},\; \ell=1,\dots,n\rbrace$, with observation indexes $l$ in row. The aim is to test simultaneously $$ (H_{0ij})~ Cor(X_i,X_j)=0 {~~against~~} (H_{1ij})~ Cor(X_i,X_j)\neq 0,~~ i,j=1,..., p,~ i ## References Benjamini, Y., & Hochberg, Y. (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the royal statistical society. Series B (Methodological), 289-300, [https://doi.org/10.1111/j.2517-6161.1995.tb02031.x](https://doi.org/10.1111/j.2517-6161.1995.tb02031.x). Bonferroni, C. E. (1935). Il calcolo delle assicurazioni su gruppi di teste. Studi in onore del professore salvatore ortu carboni, 13-60. Cai, T. T., & Liu, W. (2016). Large-scale multiple testing of correlations. Journal of the American Statistical Association, 111(513), 229-240, [https://doi.org/10.1080/01621459.2014.999157](https://doi.org/10.1080/01621459.2014.999157). Drton, M., & Perlman, M. D. (2007). Multiple testing and error control in Gaussian graphical model selection. Statistical Science, 22(3), 430-449, [https://doi.org/10.1214/088342307000000113](https://doi.org/10.1214/088342307000000113). Romano, J. P., & Wolf, M. (2005). Exact and approximate stepdown methods for multiple hypothesis testing. Journal of the American Statistical Association, 100(469), 94-108, [https://doi.org/10.1198/016214504000000539](https://doi.org/10.1198/016214504000000539). Roux, M. (2018). Graph inference by multiple testing with application to Neuroimaging, Ph.D., Université Grenoble Alpes, France, [https://tel.archives-ouvertes.fr/tel-01971574v1](https://tel.archives-ouvertes.fr/tel-01971574v1). Šidák, Z. (1967). Rectangular confidence regions for the means of multivariate normal distributions. Journal of the American Statistical Association, 62(318), 626-633. ## Examples ```r # Parameters for simulations Nsimu <- 100 # number of Monte-Carlo simulations seqn <- seq(100,400,100) # sample sizes p <- 10 # number of random variables considered rho <- 0.3 # value of non-zero correlations seed <- 156724 corr_theo <- diag(1,p) # the correlation matrix corr_theo[1,2:p] <- rho corr_theo[2:p,1] <- rho # Parameters for multiple testing procedure stat_test <- 'empirical' # test statistics for correlation tests method <- 'BootRW' # FWER controlling procedure SD <- FALSE # logical determining if stepdown is applied alpha <- 0.05 # FWER threshold Nboot <- 100 # number of bootstrap or simulated samples # Simulations and application of the chosen procedure res <- matrix(0,nrow=length(seqn),ncol=5) for(i in 1:length(seqn)){ temp <- SimuFwer(corr_theo,n=seqn[i],Nsimu=Nsimu,alpha=alpha,stat_test=stat_test, method='BootRW',Nboot=Nboot,stepdown=SD,seed=seed) res[i,] <- temp } rownames(res) <- seqn colnames(res) <- names(temp) # Display results par(mfrow=c(1,2)) plot(seqn,res[,'fwer'],type='b',ylim=c(0,max(alpha*1.1,max(res[,'fwer']))), main='FWER',ylab='fwer',xlab='number of observations') plot(seqn,res[,'sensitivity'],type='b',ylim=c(0,1.1), main='Power',ylab='sensitivity',xlab='number of observations') ```

TestCor-package