# Sample size determination in the context of multiple continuous endpoints with a control of the q-gFWER, for a given value of r-power (generalized disjunctive power). This function computes the sample size for an analysis of `m` multiple tests with a control of the q-gFWER. ```r indiv.rm.ssc(method, asympt = FALSE, r, m, p = m, nCovernE = 1, muC = NULL, muE = NULL, d = NULL, delta = NULL, SigmaC = NULL, SigmaE = NULL, power = 0.8, alpha = 0.05, interval = c(2, 2000), q = 1, maxpts = 25000, abseps = 0.001, releps = 0, nbcores = 1, LB = FALSE, orig.Hochberg = FALSE) ``` ## Arguments - `method`: "Bonferroni", "Hochberg" or "Holm". When `method = "Hochberg"`, we use critical values involving the D1 term in formula (11) of Romano et al. in order to control strongly the $q$-FWER. If you want to use the original Hochberg's procedure, set `orig.Hochberg` to `TRUE`. Even for $q=1$, this is a bad idea except when the p-values can be assumed independent. - `asympt`: logical. `TRUE` for the use of the asymptotic approximation by a multivariate normal distribution or `FALSE` for the multivariate Student distribution. - `r`: integer, r = 1, ..., m. Desired number of endpoints to be declared significant. - `m`: integer. Number of endpoints. - `p`: integer, p = 1, ..., m. Indicates the number of false null hypotheses. - `nCovernE`: ratio of `nC` over `nE`. - `muC`: `NULL` or a vector of length `m` of the true means of the control group for all endpoints under the alternative hypothesis. If `muC`, `muE` and `d` are `NULL`, then `delta` should be provided instead. - `muE`: `NULL` or a vector of length `m` of the true means of the experimental (test) group for all endpoints under the alternative hypothesis. - `d`: `NULL` or a a vector of length `m` indicating the true value of the differences in means under the null hypothesis. - `delta`: should be `NULL` if `muC`, `muE` and `d` are provided. If not, it is equal to `muE - muC - d` and these parameters should be set to `NULL`. - `SigmaC`: matrix giving the covariances between the `m` primary endpoints in the control group. - `SigmaE`: matrix giving the covariances between the `m` primary endpoints in the experimental (test) group. - `power`: a value which correponds to the chosen r-power. - `alpha`: a value which corresponds to the chosen q-gFWER type-I control bound. - `interval`: an interval of values in which to search for the sample size. Left endpoint should be greater than or equal to 2. - `q`: integer. Value of 'q' (q=1,...,m) in the q-gFWER of Romano et al., which is the probability to make at least `q` false rejections. The default value `q=1` corresponds to the classical FWER control. - `maxpts`: convergence parameter used in the `GenzBretz` function. A good choice is `min(25000 * 10 ^ true.complexity, .Machine$integer.max)` where `true.complexity` is computed with the `complexity` function. But note that this might considerably increase the computation time! - `abseps`: convergence parameter used in the `GenzBretz` function. A good choice is `max(0.001 / true.complexity, 1e-08)` where `true.complexity` is computed with the `complexity` function. But note that this might considerably increase the computation time! - `releps`: relative error tolerance as double used in the `GenzBretz` function. - `nbcores`: integer. Number of cores to use for parallel computations. - `LB`: logical. Should we use a load balancing parallel computation. - `orig.Hochberg`: logical. To use the standard Hochberg's procedure. ## Note Results can differ from one time to another because the results of the function `pmvt` are random. If this is the case, you should consider increasing `maxpts` and decreasing `abseps`. In any case, you should always double check using one of the functions `Psirms`, `Psirmu` or `Psirmd` if the sample size you obtained gives you the intended power, with an acceptable error (or at least compute the power a few times with various seeds to see if results are stable). ## Returns The required sample size. ## References Delorme P., Lafaye de Micheaux P., Liquet B., Riou, J. (2015). Type-II Generalized Family-Wise Error Rate Formulas with Application to Sample Size Determination. Submitted to **Statistics in Medicine.** Romano J. and Shaikh A. (2006) Stepup Procedures For Control of Generalizations of the Familywise Error Rate. **The Annals of Statistics**, 34(4), 1850--1873. ## Author(s) P. Lafaye de Micheaux, B. Liquet and J. Riou ## See Also `indiv.analysis`, ## Examples ```r ## Not run: # Pneumovacs example (takes 37 mn to compute on 1 core) # Treatment effect delta <- c(0.55, 0.34, 0.38, 0.20, 0.70, 0.38, 0.86) # Variances of the m endpoints var <- c(0.3520, 0.6219, 0.5427, 0.6075, 0.6277, 0.5527, 0.8066) ^ 2 # Covariance matrix cov <- matrix(1, ncol = 7, nrow = 7) cov[1, 2:7] <- cov[2:7, 1] <- c(0.1341692, 0.1373891, 0.07480123, 0.1401267, 0.1280336, 0.1614103) cov[2, 3:7] <- cov[3:7, 2] <- c(0.2874531, 0.18451960, 0.3156895, 0.2954996, 0.3963837) cov[3, 4:7] <- cov[4:7, 3] <- c(0.19903400, 0.2736123, 0.2369907, 0.3423579) cov[4, 5:7] <- cov[5:7, 4] <- c(0.1915028, 0.1558958, 0.2376056) cov[5, 6:7] <- cov[6:7, 5] <- c(0.2642217, 0.3969920) cov[6, 7] <- cov[7, 6] <- 0.3352029 diag(cov) <- var indiv.rm.ssc(method = "Hochberg", asympt = FALSE, r = 3, m = 7, p = 7, nCovernE = 1, muC = NULL, muE = NULL, d = NULL, delta = delta, SigmaC = cov, SigmaE = cov, power = 0.8, alpha = 0.05, interval = c(10, 2000), q = 1) # Pre-RELAX-AHF example from the paper by Teerlink et al. (2009), # Relaxin for the treatment of patients with acute heart failure # (Pre-RELAX-AHF): a multicentre, randomised, # placebo-controlled, parallel-group, dose-finding phase IIb # study, Lancet, 373: 1429--39 # Table 2 page 1432: # ------------------ # Proportion with moderately or markedly better dyspnoea at 6 h, 12 h, and 24 h (Likert): 23% 40% # Dyspnoea AUC change from baseline to day 5 (VAS [mmxh]): 1679 (2556) 2567 (2898) # Worsening heart failure through day 5: 21% 12% # Length of stay (days): 12.0 (7.3) 10.2 (6.1) # Days alive out of hospital: 44.2 (14.2) 47.9 (10.1) # KM cardiovascular death or readmission (HR, 95% CI): 17.2% 2.6% (0.13, 0.02--1.03); p=0.053 # KM cardiovascular death (HR, 95% CI): 14.3% 0.0% (0.00, 0.00--0.98); p=0.046 # Table 4 page 1436: # ------------------ # >=25% increase at day 5: 8 (13%) 9 (21%) # >=26 micro-mol/L increase at days 5 and 14: 4 (7%) 3 (7%) muC <- c(23 / 100, 1679, 1 - 21 / 100, -12.0, 44.2, 1 - 17.2 / 100, 1 - 14.3 / 100, 13 / 100, 7 / 100) muE <- c(40 / 100, 2567, 1 - 12 / 100, -10.2, 47.9, 1 - 2.60 / 100, 1, 21 / 100, 7 / 100) sdC <- c(sqrt(0.23 * (1 - 0.23)), 2556, sqrt(0.79 * (1 - 0.79)), 7.3, 14.2, sqrt(0.828 * (1 - 0.828)), sqrt(0.857 * (1 - 0.857)), sqrt(0.13 * (1 - 0.13)), sqrt(0.07 * (1 - 0.07))) sdE <- c(sqrt(0.4 * (1 - 0.4)) , 2898, sqrt(0.88 * (1 - 0.88)), 6.1, 10.1, sqrt(0.974 * (1 - 0.974)), 1e-12 , sqrt(0.21 * (1 - 0.21)), sqrt(0.07 * (1 - 0.07))) m <- 9 rho <- 0.1 cor <- matrix(rho, nrow = m, ncol = m) diag(cor) <- 1 sd.pooled <- sqrt(0.5 * sdE + 0.5 * sdC) SigmaE <- diag(sdE) %*% cor %*% diag(sdE) SigmaC <-diag(sdC) %*% cor %*% diag(sdC) indiv.rm.ssc(method = "Bonferroni", asympt = FALSE, r = 6, m = 9, p = 9, nCovernE = 1, muC = NULL, muE = NULL, d = rep(0.0, m), delta = (muE - muC) / sd.pooled, SigmaC = cor, SigmaE = cor, power = 0.8, alpha = 0.1, interval = c(2, 500), q = 1, maxpts = 25000, abseps = 0.01, nbcores = 1, LB = TRUE) ## End(Not run) ```

indiv.rm.ssc function