# Power calculation in joint modeling of longitudinal and survival data - k-th Order Trajectories and Unknown Sigma Compute the power in joint modeling of longitudinal and survival data when the variance-covariance matrix Sigma_Theta is unknown and the trajectories are order k. The function computes power for a one-sided test, either [REMOVE_ME]$$ H_0: \beta = 0 \quad \mbox{ and } \quad H_{1A}: \beta > 0%H_0: \beta = 0 and H_1A: \beta > 0 [REMOVE_ME_2]$$ or [REMOVE_ME]$$ H_0: \beta = 0 \quad \mbox{ and } \quad H_{1B}: \beta < 0%H_0: \beta = 0 and H_1B: \beta < 0 [REMOVE_ME_2]$$ with Type I error $\alpha$. The choice of the alternative is determined by the sign of $\beta$. Negative values for $\beta$ indicate that the alternative hypothesis is $H_{1B}$, while $\beta \ge 0$ indicates that it is $H_{1A}$. It creates a `powerLongSurv` object. ```r powerLongSurv( N, nevents, tmedian, meantf, p, t, SigmaTheta, sigmae_2, ordtraj, beta = 0, alpha = 0.05, tol = 1.5e-08 ) ``` ## Arguments - `N`: numeric specifying the total sample size; minimum 20. - `nevents`: numeric specifying the number of events; at least 20 and at most N. - `tmedian`: numeric specifying the median survival time; positive - `meantf`: numeric specifying the mean follow-up time; positive and no greater than max(t). - `p`: numeric vector of estimated subject proportions with 2,3,... measurements, respectively, zero proportions allowed. - `t`: numeric vector of measurement times, distinct positive components; same length as `p`. - `SigmaTheta`: numeric matrix specifying the covariance matrix Sigma_Theta - `sigmae_2`: numeric specifying the measurement error; positive. - `ordtraj`: integer specifying the order of trajectories, must be less the order of Sigma_Theta - `beta`: numeric specifying the effect of the trajectory; default value 0. - `alpha`: numeric, strictly between 0.0 and 1.0, specifying the Type-I Error (2-sided), default value 0.05. - `tol`: numeric, For floating point objects x and y, if |x-y| <= tol, x==y. Passed to R function all.equal. ## Returns An object of S4 class `powerLongSurv`, which has the following 12 components - **title**: character string - **subtitle**: character string - **t**: numeric vector - **p**: numeric vector - **N**: integer - **nevents**: integer - **censr**: numeric - **tmedian**: numeric - **meantf**: numeric - **SigmaTheta**: numeric matrix - **ordtraj**: integer - **BSigma**: numeric matrix - **beta**: numeric - **alpha**: numeric - **power**: numeric ## Description Compute the power in joint modeling of longitudinal and survival data when the variance-covariance matrix Sigma_Theta is unknown and the trajectories are order k. The function computes power for a one-sided test, either $$ H_0: \beta = 0 \quad \mbox{ and } \quad H_{1A}: \beta > 0%H_0: \beta = 0 and H_1A: \beta > 0 $$ or $$ H_0: \beta = 0 \quad \mbox{ and } \quad H_{1B}: \beta < 0%H_0: \beta = 0 and H_1B: \beta < 0 $$ with Type I error $\alpha$. The choice of the alternative is determined by the sign of $\beta$. Negative values for $\beta$ indicate that the alternative hypothesis is $H_{1B}$, while $\beta \ge 0$ indicates that it is $H_{1A}$. It creates a `powerLongSurv` object. ## Details The function `powerLongSurv` is used to calculate the power in joint modeling of longitudinal and survival data. ## Examples ```r ## Example 1. ## ********** ## Input elements of Sigma_theta in forumula 4.6; SigmaTheta <- matrix(c(1.2,0.0,0.0,0.0,0.7,0.0,0.0,0.0,0.8),nrow=3,ncol=3) N <- 200; # Total sample size; nevents <- 140; # Number of events; tmedian <- 0.7; # median survival; meantf <- 1.4; # mean follow-up time; beta <- 0.2; # Effect of the trajectory; alpha <- 0.05;# Type-I Error (2-sided); sigmae_2 <- 0.09; # measurement error; ## schedule of measurement; t <- c(0.4, 0.8, 1.2, 1.6, 2) ; # maximum 2 year follow-up; ## Input estimated proportion subjects with 2,3,4,5,6 measurements; ## This is \xi in formula 4.6; ## The data is obtained from the simulated data for the calculation in table 2; p <- c(0.3, 0.4, 0.15, 0.1, 0.05); ## Input the order of trajectories ordtraj <- 1 ## linear trajectories ## Call function ## Linear Trajectories pLSl <- powerLongSurv(N, nevents, tmedian, meantf, p, t, SigmaTheta, sigmae_2, ordtraj, beta, alpha=0.05) pLSl show(pLSl) unclass(pLSl) ## Constant Trajectories powerLongSurv(N, nevents, tmedian, meantf, p, t, SigmaTheta, sigmae_2, ordtraj=0, beta, alpha=0.05) ## Quadratic Trajectories powerLongSurv(N, nevents, tmedian, meantf, p, t, SigmaTheta, sigmae_2, ordtraj=2, beta, alpha=0.05) ## *************************************************************************** ## Example 2. ## ********** ## Input elements of Sigma_theta in forumula 4.6; SigmaTheta <- matrix(c(1.2,0.0,0.0,0.0,0.7,0.0,0.0,0.0,0.8),nrow=3,ncol=3) N <- 200; # Total sample size; nevents <- 140; # Number of events; tmedian <- 0.7; # median survival; meantf <- 1.4; # mean follow-up time; beta <- 0.2; # Effect of the trajectory; alpha <- 0.05;# Type-I Error (2-sided); sigmae_2 <- 0.09; # measurement error; ## schedule of measurement; t <- c(0.4, 0.8, 1.2, 1.6); ## Input estimated proportion subjects with 2,3,4,5,6 measurements; ## This is \xi in formula 4.6; ## The data is obtained from the simulated data for the calculation in table 2; p <- c(0.3, 0.4, 0.2, 0.1); ## Input the order of trajectories ordtraj <- 2 ## quadratic trajectories ## Call function ## Quadratic Trajectories pLSq <- powerLongSurv(N,nevents,tmedian,meantf,p,t,SigmaTheta,sigmae_2,ordtraj,beta, alpha = 0.05) pLSq show(pLSq) unclass(pLSq) ## Constant Trajectories powerLongSurv(N, nevents, tmedian, meantf, p, t, SigmaTheta, sigmae_2, ordtraj=0, beta, alpha=0.05) ## Linear Trajectories powerLongSurv(N, nevents, tmedian, meantf, p, t, SigmaTheta, sigmae_2, ordtraj=1, beta, alpha=0.05) ``` ## References L. M. Chen, J. G. Ibrahim, and H. Chu. Sample size and power determination in joint modeling of longitudinal and survival data. Statist. Med. 2011, 30 2295-2309 ## See Also `powerLongSurv-class`, `show-methods` ## Author(s) Emil A. Cornea, Liddy M. Chen, Bahjat F. Qaqish, Haitao Chu, and Joseph G. Ibrahim

powerLongSurv function