# pass.lme.CLb.test: Calculate Power or Sample Size required for Contrasts of... In pass.lme: Power and Sample Size for Linear Mixed Effect Models

## Description

Interested parameters/linear combinations LB from more than one independent populations can be aggregrate togeter by appending mu vertically and Sigma/n diagonally

Consider Lb~N(MU,SIGMA) as the aggregrated estimates
Any comparison of interested parameters can be formulated by multiplying a contrast matrix C on LB and set

H0: C*LB=d for any vector of value d to be tested

We then have

C*Lb~N(C*MU,C*SIGMA*C')

and

(C*Lb-d)'*inv(C*SIGMA*C')*(C*Lb-d)~chisq(q,lambda)

where degree of freedom q=rank(C*SIGMA*C'),
non-centrality parameter lambda=(C*LB-d)'*inv(C*SIGMA*C')*(C*LB-d)

Power of the test H0 is given by 1-beta=P(chisq(q,lambda)>qchisq(1-alpha,lambda))
Required sample size for desired power can be obtained by bisection method.

## Usage

 ```1 2``` ```pass.lme.CLb.test(thetas, C = NULL, d = NULL, alpha = 0.05, power = NULL, n = NULL) ```

## Arguments

 `thetas` list of theta (LB and VLb), can be different for each group `C` Contrast of Matrix `d` Value vector to be tested for all contrast `alpha` significant level `power` desired power for sample size calculation `n` sample size for power calculation / or sample size ratio with power for sample size calculation (NULL for balanced design)

## Value

solved.power given sample size n, this gives the power for testing H0
solved.n given the desired power, this gives the sample size for H0

## Author(s)

Marco Chak Yan YU
Maintainer: Marco Chak Yan YU <[email protected]>

`lme.Lb.dist.theta`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45``` ```#Example 1 (test fixed effect coefficient 2=0) with power of 80% # for 1-level LME model, with covariates X, Z: (1,t), t=1,2,3 # for both fixed and random effects, with fixed effect coefficients B: (100,-0.5), # random effect variance D: (2 1;1 2), residual variance R: 0.2 B <- matrix(c(100,-0.5),2,1) D <- matrix(c(2,1,1,2),2,2) R <- 0.2 X <- cbind(rep(1,3),1:3) Z <- X theta <- lme.Lb.dist.theta(B,D,R,X,Z) pass.lme.CLb.test(list(theta),alpha=0.05,power=0.8) pass.lme.CLb.test(list(theta),alpha=0.05,n=66) #Example 2 (compare two fixed effect coefficient 2) with power of 80% # Consider above model as a control group model, # with an independent treatment group with model same as the control # except a different fixed effect coefficient 2 for treatment # = fixed effect coefficient 2 for control x 0.7 theta2 <- theta theta2\$mu <- theta\$mu *0.7 C <- matrix(c(1,-1),1,2) pass.lme.CLb.test(list(theta,theta2),C,alpha=0.05,power=0.8) pass.lme.CLb.test(list(theta,theta2),C,alpha=0.05,n=1468) #Example 3 (compare two fixed effect coefficient 2) with power of 80% # with sample size ratio, control:treatment = 1:2 pass.lme.CLb.test(list(theta,theta2),C,alpha=0.05,power=0.8,n=c(1,2)) pass.lme.CLb.test(list(theta,theta2),C,alpha=0.05,n=c(1101,2202)) #Example 4 (repeated-measures ANOVA for comparing 3 group means) with power of 80% # for 1-level LME model with mean for group 1, 2 and 3 are 100, 99, 102, respectively, # each subject to be measured 2 times, with within-subject variance = 15, residual variance = 10 B <- 100 D <- 15 R <- 10 X <- matrix(1,2,1) Z <- X theta <- lme.Lb.dist.theta(B,D,R,X,Z) theta2 <- theta theta3 <- theta theta2\$mu <- 99 theta3\$mu <- 102 C <- rbind(c(1,-1,0),c(1,0,-1)) pass.lme.CLb.test(list(theta,theta2,theta3),C,alpha=0.05,power=0.8) pass.lme.CLb.test(list(theta,theta2,theta3),C,alpha=0.05,n=41) ```