<\body> We saw how to test the hypothesis that a point came from a certain normal distribution <\input| >> pnorm(5,mean=1,sd=2) <\output> [1] 0.9772499 <\input| >> \; > We also that to test if several points came from a certain normal, we use the t-test. Let us look deeper into that. Let us take 2 points that are taken from a normal, and look where they lie: <\input| >> x=rnorm(2000) <\input| >> y=rnorm(2000) <\input| >> plot(x,y,pch=".",xlim=c(-3,3),ylim=c(-3,3));v() <\output> |ps>||||||> <\input| >> grid(lwd=1,lty=1,col=2);v() <\output> |ps>||||||> <\input| >> \; > Let us count how many points are in each box <\input| >> x.cut=cut(x,-3:3) <\input| >> x.cut[1:20] <\output> \ [1] (0,1] \ \ (-1,0] \ (-2,-1] (-1,0] \ (-1,0] \ (0,1] \ \ (1,2] \ \ (-1,0] \ (0,1] \ [10] (1,2] \ \ (1,2] \ \ (-1,0] \ (-1,0] \ (0,1] \ \ (-1,0] \ (-1,0] \ (1,2] \ \ (-1,0]\ [19] (-1,0] \ (0,1] \ \; Levels: (-3,-2] (-2,-1] (-1,0] (0,1] (1,2] (2,3] <\input| >> \; > You see that each point was replaced by the interval that it is in. <\input| >> y.cut=cut(y,-3:3) <\input| >> tab.xy=table(x.cut,y.cut) <\input| >> tab.xy <\output> \ \ \ \ \ \ \ \ \ y.cut x.cut \ \ \ \ (-3,-2] (-2,-1] (-1,0] (0,1] (1,2] (2,3] \ \ (-3,-2] \ \ \ \ \ \ 2 \ \ \ \ \ \ 2 \ \ \ \ 15 \ \ \ 16 \ \ \ \ 5 \ \ \ \ 0 \ \ (-2,-1] \ \ \ \ \ \ 5 \ \ \ \ \ 46 \ \ \ \ 88 \ \ \ 97 \ \ \ 31 \ \ \ \ 8 \ \ (-1,0] \ \ \ \ \ \ 11 \ \ \ \ 121 \ \ \ 229 \ \ 233 \ \ \ 78 \ \ \ 10 \ \ (0,1] \ \ \ \ \ \ \ 16 \ \ \ \ \ 78 \ \ \ 228 \ \ 227 \ \ 103 \ \ \ 15 \ \ (1,2] \ \ \ \ \ \ \ \ 3 \ \ \ \ \ 40 \ \ \ \ 96 \ \ 105 \ \ \ 39 \ \ \ 10 \ \ (2,3] \ \ \ \ \ \ \ \ 1 \ \ \ \ \ \ 7 \ \ \ \ \ 9 \ \ \ 13 \ \ \ \ 7 \ \ \ \ 0 <\input| >> \; > Now we know how many are in each cell. Let us plot these: <\input| >> library(lattice) <\input| >> levelplot(tab.xy);v() <\output> |ps>||||||> <\input| >> \; > Now let us do the same, but with a finer grid, and more points: <\input| >> x=rnorm(1000000); y=rnorm(1000000) <\input| >> x.cut=cut(x, seq(-3,3, length=50) ); y.cut=cut(y, seq(-3,3,length=50) ) <\input| >> tab.xy=table(x.cut,y.cut) <\input| >> levelplot(tab.xy,colorkey=F,col.regions=rainbow(100));v() <\output> |ps>||||||> <\input| >> \; > Here we see why it makes sense to talk about sum of squares for normaly distributed data: All points for which +y=constant>, i.e. points on a circle, are equally likely. We can thus use +y> as a distance measure to see if x and y came from a normal. If we have more than 2 variables, we will also get balls with equi-likely radii. How likely are and to have come from a normal with mean 0 and variance 1? The chi-squared distribution does exactly that <\input| >> pchisq((1.2^2+3.4^2),2) <\output> [1] 0.9984966 <\input| >> x=rnorm(1000);y=rnorm(1000) <\input| >> p=pchisq(x^2+y^2,df=2) <\input| >> hist(p);v() <\output> |ps>||||||> <\input| >> \; > We need to tell pchisq how many dimensions the ball that we give the radius of has. That is the parameter . Which stands for degrees of freedom. When the variance is not 1, we also need to scale the ball. We simply divide by the variance: <\input| >> x=rnorm(1000,sd=2);y=rnorm(1000,sd=2) <\input| >> p=pchisq(x^2+y^2,df=2) <\input| >> hist(p);v() <\output> |ps>||||||> <\input| >> p=pchisq((x^2+y^2)/2^2,df=2) <\input| >> hist(p);v() <\output> |ps>||||||> <\input| >> \; > And, if the mean is not 0, we have to subtract the mean: <\input| >> x=rnorm(1000,mean=3,sd=2);y=rnorm(1000,mean=3,sd=2) <\input| >> p=pchisq((x^2+y^2)/2^2,df=2) <\input| >> hist(p);v() <\output> |ps>||||||> <\input| >> p=pchisq(( (x-3)^2+(y-3)^2)/2^2,df=2) <\input| >> hist(p);v() <\output> |ps>||||||> <\input| >> \; > That radii have constant probability is not true for other distributions. For example, the log-normal distribution: <\input| >> hist(exp(x),br=seq(0,1000,by=0.05),xlim=c(0,1.5));v() <\output> |ps>||||||> <\input| >> \; > Let us do a 2d plot in the same way here: <\input| >> x.cut=cut(exp(x), seq(0,1.5, length=50) ); y.cut=cut(exp(y), seq(0,1.5,length=50) ) <\input| >> tab.xy=table(x.cut,y.cut) <\input| >> levelplot(tab.xy,colorkey=F,col.regions=rainbow(100));v() <\output> |ps>||||||> <\input| >> \; > This is why for normally distributed data it makes sense to talk about sum square distance: because the radius is all that determines the likelyhood of a result. The same is not true for other distributions. <\input| >> setwd("~/R-course-2006/lecture7/") <\input| >> oneway=read.table("oneway.txt",header=T) <\input| >> dim(oneway) <\output> [1] 24 \ 2 <\input| >> oneway <\output> \ \ \ Growth Photoperiod 1 \ \ \ \ \ \ 2 \ \ \ \ \ \ Short 2 \ \ \ \ \ \ 3 \ \ \ \ \ \ Short 3 \ \ \ \ \ \ 1 \ \ \ \ \ \ Short 4 \ \ \ \ \ \ 1 \ \ \ \ \ \ Short 5 \ \ \ \ \ \ 2 \ \ \ \ \ \ Short 6 \ \ \ \ \ \ 1 \ \ \ \ \ \ Short 7 \ \ \ \ \ \ 3 \ \ \ \ \ \ Short 8 \ \ \ \ \ \ 4 \ \ \ \ \ \ Short 9 \ \ \ \ \ \ 2 \ \ \ \ \ \ Short 10 \ \ \ \ \ 1 \ \ \ \ \ \ Short 11 \ \ \ \ \ 2 \ \ \ \ \ \ Short 12 \ \ \ \ \ 1 \ \ \ \ \ \ Short 13 \ \ \ \ \ 3 \ \ \ \ \ \ \ Long 14 \ \ \ \ \ 5 \ \ \ \ \ \ \ Long 15 \ \ \ \ \ 1 \ \ \ \ \ \ \ Long 16 \ \ \ \ \ 2 \ \ \ \ \ \ \ Long 17 \ \ \ \ \ 2 \ \ \ \ \ \ \ Long 18 \ \ \ \ \ 2 \ \ \ \ \ \ \ Long 19 \ \ \ \ \ 4 \ \ \ \ \ \ \ Long 20 \ \ \ \ \ 6 \ \ \ \ \ \ \ Long 21 \ \ \ \ \ 2 \ \ \ \ \ \ \ Long 22 \ \ \ \ \ 2 \ \ \ \ \ \ \ Long 23 \ \ \ \ \ 2 \ \ \ \ \ \ \ Long 24 \ \ \ \ \ 3 \ \ \ \ \ \ \ Long <\input| >> names(oneway) <\output> [1] "Growth" \ \ \ \ \ "Photoperiod" <\input| >> \; > <\input| >> plot(1:24,oneway$Growth,ylab="growth",xlab="order") <\input| >> abline(mean(oneway$Growth),0);v() <\output> |ps>||||||> <\input| >> for(i in 1:24) lines(c(i,i),c(mean(oneway$Growth),oneway$Growth[i]));v() <\output> |ps>||||||> <\input| >> oneway <\output> \ \ \ Growth Photoperiod 1 \ \ \ \ \ \ 2 \ \ \ \ \ \ Short 2 \ \ \ \ \ \ 3 \ \ \ \ \ \ Short 3 \ \ \ \ \ \ 1 \ \ \ \ \ \ Short 4 \ \ \ \ \ \ 1 \ \ \ \ \ \ Short 5 \ \ \ \ \ \ 2 \ \ \ \ \ \ Short 6 \ \ \ \ \ \ 1 \ \ \ \ \ \ Short 7 \ \ \ \ \ \ 3 \ \ \ \ \ \ Short 8 \ \ \ \ \ \ 4 \ \ \ \ \ \ Short 9 \ \ \ \ \ \ 2 \ \ \ \ \ \ Short 10 \ \ \ \ \ 1 \ \ \ \ \ \ Short 11 \ \ \ \ \ 2 \ \ \ \ \ \ Short 12 \ \ \ \ \ 1 \ \ \ \ \ \ Short 13 \ \ \ \ \ 3 \ \ \ \ \ \ \ Long 14 \ \ \ \ \ 5 \ \ \ \ \ \ \ Long 15 \ \ \ \ \ 1 \ \ \ \ \ \ \ Long 16 \ \ \ \ \ 2 \ \ \ \ \ \ \ Long 17 \ \ \ \ \ 2 \ \ \ \ \ \ \ Long 18 \ \ \ \ \ 2 \ \ \ \ \ \ \ Long 19 \ \ \ \ \ 4 \ \ \ \ \ \ \ Long 20 \ \ \ \ \ 6 \ \ \ \ \ \ \ Long 21 \ \ \ \ \ 2 \ \ \ \ \ \ \ Long 22 \ \ \ \ \ 2 \ \ \ \ \ \ \ Long 23 \ \ \ \ \ 2 \ \ \ \ \ \ \ Long 24 \ \ \ \ \ 3 \ \ \ \ \ \ \ Long <\input| >> m.short=mean(oneway$Growth[ oneway$Photo=="Short" ] ) <\input| >> m.long=mean(oneway$Growth[ oneway$Photo=="Long" ] ) <\input| >> plot(1:24,oneway$Growth,ylab="growth",xlab="order",col=as.numeric(oneway$Photo));v() <\output> |ps>||||||> <\input| >> abline(m.short,0,col=2);abline(m.long,0,col=1);v() <\output> |ps>||||||> <\input| >> for(i in which( oneway$Photo=="Short" ) ) { \ \ lines(c(i,i),c(m.short,oneway$Growth[i]),col=2) } <\input| >> v() <\output> |ps>||||||> <\input| >> for(i in which( oneway$Photo=="Long" ) ) { \ \ lines(c(i,i),c(m.long,oneway$Growth[i]),col=1) } <\input| >> v() <\output> |ps>||||||> <\input| >> oneway4=oneway[c(1:2,19:20),] <\input| >> m.short=mean(oneway4$Growth[ oneway4$Photo=="Short" ] ) <\input| >> m.long=mean(oneway4$Growth[ oneway4$Photo=="Long" ] ) <\input| >> plot(1:4,oneway4$Growth,ylab="growth",xlab="order",col=as.numeric(oneway4$Photo)) <\input| >> abline(m.short,0,col=2);abline(m.long,0,col=1) <\input| >> for(i in which( oneway4$Photo=="Long" ) ) { \ \ lines(c(i,i),c(m.long,oneway4$Growth[i]),col=1) } <\input| >> for(i in which( oneway4$Photo=="Short" ) ) { \ \ lines(c(i,i),c(m.short,oneway4$Growth[i]),col=2) } <\input| >> \; <\input| >> v() <\output> |ps>||||||> <\input| >> plot(1:4,oneway4$Growth);abline(mean(oneway4$Growth),0) <\input| >> for(i in 1:4) lines(c(i,i),c(mean(oneway4$Growth),oneway4$Growth[i]));v() <\output> |ps>||||||> <\input| >> \; > \; \; <\initial> <\collection> <\references> <\collection> > > > > > |?>> |?>> |?>> |?>> |?>> <\auxiliary> <\collection> <\associate|toc> |math-font-series||Lecture 6 - Analysis of variance 1> |.>>>>|> |Introduction |.>>>>|> > |2 dimensional distributions |.>>>>|> > |Analysis of variance |.>>>>|> >