RS power

notes.Rmd

Shows the power of the Rao-Scott (Wald) test in a 2-d problem.  
Alternative is parametrised by displacement (fixed to give specified power for intrinsic test) and by angle theta.
Purple circle indicates power for intrinsic test (constant)
black points are corresponding power for RS-type test. 

Maybe facet this to get a three-d problem?

rspower.R

## find non-centrality with W0-power=85%

power<-function(nu){
	pchisq(qchisq(0.95,df=2),df=2,ncp=nu,lower.tail=FALSE)
	
}

nu_75<-uniroot(function(nu) power(nu)-0.75,interval=c(5,15))$root
lambda<-eigen(W1)$values
thetas<-seq(0,2*pi,length=100)
powers<-numeric(100)
for(i in 1:100){
	theta<-thetas[i]
	## evaluate W1-power by simulation
	beta<-MASS::mvrnorm(10000,mu=sqrt(nu_75)*c(cos(theta),sin(theta)),S=diag(2))
	Qstat<-rowSums((beta%*%W1)*beta)
	powers[i]<-mean(pchisqsum(Qstat,c(1,1),lambda,lower.tail=FALSE,method="int")<0.05)
	
}

plot(powers*cos(thetas),powers*sin(thetas),xlim=c(-1,1),ylim=c(-1,1))
points(0.75*cos(thetas),0.75*sin(thetas),col="purple")