Simulations of beta regression as compared to other techniques designed for working on response variables on a unit (0,1) scale.

sims3.r

######
#script to generate simulations comparing the power and type I error rate
#of a variety of different methods of analyzing response variables on a unit (0-1) scale
#
# Inspired by "The Arcsine is Asinine" in a 2011 issue of Ecology
#
# byrnes@nceas.ucsb.edu
# last updated 5/19/11
#
# Changelog: Added visualization code
# 5/19/11 Fixed species factor error
# 5/19/11 Added censored regression
######

library(plyr)
library(betareg)
library(lmtest)
library(car) #for logit transform
library(multicore)
#source("./betareg.fit2.R") #I whipped up something to use nlminb instead of optim.  It was faster, but let's go with optim for now

#the function to generate power and error for different methods of analysis
#the basic ideas is that, say you have % of something eaten by different species.  
#we're assuming a basic linear model here, but, you can't have negative amounts eaten or >100% eaten - that's just 0 or 100.
sim3	<-	function(n.sim=100, sp.slope=c(10,20,30,40), n=10, p.crit=0.05, dens.range=0:10, stdDev=30, mc.cores=1){
  dataset<-data.frame(expand.grid(rep(dens.range, n), 1:length(sp.slope)))
  names(dataset)<-c("density", "species")
  
  #get a p value from either an F or Chisq LR test
  getp<-function(mod1, mod2, type="F"){
  	if(type=="F") return(anova(mod1, mod2)$"Pr(>F)"[2])
  	if(type=="ch") return(lrtest(mod1, mod2)$"Pr(>Chisq)"[2])
  	return(NA)
  }
  
  #create a list with all of the p value tests
 # cat("goin' in!\n") #debug
 
   #use mclapply to run the simulations, this leveraging multi-core computers
  eval_list<-mclapply(1:n.sim, function(x) {
 # 	cat(paste(x, n, "\n", sep="  ")) #so we know things are going...
 
 	#simulate a response variable for which there is an interaction
	dataset$y1<-rnorm(n*length(sp.slope)*length(dens.range), dataset$density*sp.slope[dataset$species], stdDev)
	dataset$y1[which(dataset$y1>100)]<-99.9 #correction to fit in scale
	dataset$y1[which(dataset$y1<0)]<-0.001 #correction to fit in scale
	
	#note, there are other options for correcting the 0 and 100 values, but, I'm using a simple one here
	
	#simulate a response variable for which there isn't even a species effect
	dataset$y2<-rnorm(n*length(sp.slope)*length(dens.range), dataset$density*mean(sp.slope), stdDev)
	dataset$y2[which(dataset$y2>100)]<-99.9
	dataset$y2[which(dataset$y2<0)]<-0.001

	dataset$species<-as.factor(dataset$species)


	
	
	return(c( "untrans_power" = getp(lm(y1 ~ density*species, data=dataset), lm(y1 ~ density+species, data=dataset)),
			  "untrans_error" = getp(lm(y2 ~ density*species, data=dataset), lm(y2 ~ density+species, data=dataset)),
			"arcsin_power" = getp(lm(asin(sqrt(y1/100)) ~ density*species, data=dataset), lm(asin(sqrt(y1/100)) ~ density+species, data=dataset)),
			  "arcsin_error" = getp(lm(asin(sqrt(y2/100)) ~ density*species, data=dataset), lm(asin(sqrt(y2/100)) ~ density+species, data=dataset)),
			  
			  "logit_power" = getp(lm(logit(y1) ~ density*species, data=dataset), lm(logit(y1) ~ density+species, data=dataset)),
			  "logit_error" = getp(lm(logit(y2) ~ density*species, data=dataset), lm(logit(y2) ~ density+species, data=dataset)),
		
			"censReg_power" = getp(censReg(y1~ density*species, data=dataset, left=0.001, right=99.9), censReg(y1 ~ density+species, data=dataset, left=0.001, right=99.9), type="ch"),
			"censReg_error" = getp(censReg(y2 ~ density*species, data=dataset, left=0.001, right=99.9), censReg(y2 ~ density+species, data=dataset, left=0.001, right=99.9), type="ch"),

	  
			#"glm_power" = getp(glm(y1/100 ~ density*species, data=dataset, family=binomial), glm(y1/100 ~ density+species, data=dataset, family=binomial), type="ch"),
			#"glm_error" = getp(glm(y2/100 ~ density*species, data=dataset, family=binomial), glm(y2/100 ~ density+species, data=dataset, family=binomial), type="ch"),
			  
			"betareg_power" = getp(betareg(I(y1/100)~ density*species, data=dataset), betareg(I(y1/100) ~ density+species, data=dataset), type="ch"),
			"betareg_error" = getp(betareg(I(y2/100) ~ density*species, data=dataset), betareg(I(y2/100) ~ density+species, data=dataset), type="ch")
		))
  },  mc.cores= mc.cores)
  
  eval_df<-data.frame(t(eval_list[[1]]))
  for(i in 2:length(eval_list)) {eval_df<-rbind(eval_df, eval_list[[i]])}
  
  ret<-apply(eval_df, 2, function(x) sum((x<=p.crit), na.rm=T)/(n.sim-length(which(is.na(x)))))
  #colSums(eval_df<=p.crit, na.rm=T)/n.sim
  
  return(ret)
 }

#note, this is setup for EOS @ nceas, so...yeah, 10 cores.
simDf<-ddply(data.frame(n=c(3,6,9,12,14,16,18,21,24,27)), "n", .progress="text", function(x) sim3(n.sim=1000, n=x$n, mc.cores=10, stdDev=40))

#write.csv(simDf[1:10,], "beta_pow.csv", row.names=F) #plyr kept giving me repeated rows, so, I only need the first 10

####graphing results
#simDf<-read.csv("./beta_pow.csv")

errDf<-melt(simDf, id.vars="n", measure.vars=c(3,5,7,11))

errDf$variable<-gsub("_error", "", errDf$variable)

qplot(n, value, data=errDf, colour=variable, geom=c("point", "line"), ylab="Type I Error Rate\n", xlab="\nSample Size")+theme_bw(base_size=18)+scale_colour_hue(name="Analysis")




powDf<-melt(simDf, id.vars="n", measure.vars=c(2,4,6,10))

powDf$variable<-gsub("_power", "", powDf$variable)

qplot(n, value, data=powDf, colour=variable, geom=c("point", "line"), ylab="Power\n", xlab="\nSample Size")+theme_bw()+theme_bw(base_size=18)+scale_colour_hue(name="Analysis")