Statistical Intervals and Maximum Likelihood Estimation
This informative content delves into statistical intervals such as confidence intervals, credible intervals, and high-density intervals. It also explains the concept of maximum likelihood estimation using models like the binomial distribution. The material further explores Clopper-Pearson confidence intervals and Bayesian equivalents of confidence intervals, providing insights into posterior probability density functions. Additionally, it discusses the difference between the two intervals and presents a Bayesian analysis scenario.
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Confidence interval, credible interval and high density interval Xuhua Xia xxia@uottawa.ca http://dambe.bio.uottawa.ca
Maximum likelihood illustration The likelihood approach always needs a model. As a fish is either a male or a female, we use the model of binomial distribution, and the likelihood function is ! ( | , ) !( M N M N = = M N M (1 ) . L p M N p p p )! The maximum likelihood method finds the value of p that maximizes the likelihood value. This maximization process is simplified by maximizing the natural logarithm of L instead: ln ln( ) ( L A M p N = + + )ln(1 ) M p ln L M N M = = 0 1 p p p M = . p N The likelihood estimate of the variance of p is the negative reciprocal of the second derivative, 1 1 ( ) ln( ) (1 p p p (1 ) p p = = = . Var p M N M 2 L N 2 2 ) 2 Slide 2 Xuhua Xia
Clopper-Pearson confidence interval 1 1 (1 ) p p ? = = = ( ) . Var p M N M 2 ln( ) L N 2 2 (1 ) p p 2 p Most frequently used confidence interval for proportions ? ?? = ? + ? ? + 1 ?? 2 ,?1,?2;?1 = 2 ? ? 1 ;?2 = 2? ? + 1 ?? 2 ,?1 ? ? + (? + 1)?? 2 ,?1 ,?2 = ?2+ 2;?2 = ?1 2 ?? = ;?1 ,?2 3.5 6 Likelihood 3 5 Posterior 2.5 4 Likelihood 2 f(p) 3 1.5 Prior 2 1 1 0.5 0 0 0.45 0.01 0.05 0.09 0.13 0.17 0.21 0.25 0.29 0.33 0.37 0.41 0.49 0.53 0.57 0.61 0.65 0.69 0.73 0.77 0.81 0.85 0.89 0.93 0.97 p
R code for CP confidence interval CP95 <- function(X,n) { if(X==0) LL = 0 else { v1<-2*(n-X+1); v2<-2*X LL<-X/(X+(n-X+1)*qf(0.025,v1,v2,lower.tail=FALSE)) } if (X==n) UL = 1 else { u1=v2+2; u2=v1-2 Finv<-qf(0.025,u1,u2,lower.tail=FALSE) UL <-(X+1)*Finv / (n -X+(X+1)*Finv) } return(c(LL, UL)) } CP95(5,30) # will output 0.0564217 and 0.3472117 Slide 4
Bayesian equivalents of C.I. Posterior probability density function known Credible interval (exact) High density interval (exact) Posterior probability density function unknown Credible interval (approximate) High density interval (approximate) Slide 5
Difference between the two intervals 0.4822 0.9398 3 2.5 2 density 1.5 1 0.959 0.5 0.517 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p
The posterior ( | f y p f p ) ( ) = ( | ) f p y ( | f y p f p dp ) ( ) 8 2 30 (1 ) A B p p 8 2 = = = ( | ) f p y 495 (1 ) p p 2/33 3.5 6 Likelihood 3 5 Posterior 2.5 4 Likelihood 2 f(p) 3 1.5 Prior 2 1 1 0.5 0 0 0.45 0.01 0.05 0.09 0.13 0.17 0.21 0.25 0.29 0.33 0.37 0.41 0.49 0.53 0.57 0.61 0.65 0.69 0.73 0.77 0.81 0.85 0.89 0.93 0.97 Slide 7 Xuhua Xia p
MCMC: Metropolis N <- 50000 z <- sample(1:10000,N,replace=T)/20000 meanz <- mean(z) rnd <- sample(1:10000,N,replace=T)/10000 p <- rep(0,N) p[1] <- 0.1 # or just any number between 0 and 1 Add=TRUE for (i in seq(1:(N-1))) { p[i+1] <- p[i] + (if(Add) z[i] else -z[i]) if(p[i+1]>1) { p[i+1] <- p[i]-z[i] } else if(p[i+1]<0) { p[i+1] <- p[i]+z[i] } fp0 <- dbeta(p[i],3,3) fp1 <- dbeta(p[i+1],3,3) L0 <- p[i]^6 L1 <- p[i+1]^6 numer <- (fp1*L1) denom <- (fp0*L0) if(numer>denom) { if(p[i+1]>p[i]) Add=TRUE else Add=FALSE } else { if(p[i+1]>p[i]) Add=FALSE else Add=TRUE Alpha <- numer/denom # Alpha is (0,1) if(rnd[i] > Alpha) { p[i+1] <- p[i] p[i] <- 0 } } } postp <- p[(N-9999):N] postp <- postp[postp>0] freq <- hist(postp) mean(postp) sd(postp) Run with = 3 and = 3 Run again with = 1, = 1 Slide 8 Xuhua Xia
Posterior PDF known # Credible interval qbeta(c(0.025,0.975),9,3) # generates 0.4822441, 0.9397823 # High density interval p<-seq(0,1,by=0.001) # reduce 'by' value to increase precision db<-dbeta(p,9,3) # get corresponding prob. density sb<-sample(db,1e5,replace=TRUE,prob=db) # sample prob. density critVal <- quantile(sb, 0.05) # a value to horizontally cut the peak phigh <-p[db >= critVal] lo<-phigh[1] hi<-phigh[length(phigh)] lo;hi # display lo = 0.517 and hi = 0.959. Slide 9
Approximate posterior PDF 2.5% 0.4745 97.5% 0.9455 3.5 3 2.5 Density 2 1.5 1 0.5 0 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 p dx Edx
Posterior PDF unknown # Quick and dirty credible interval quantile(postp,c(0.025,0.975)) # generates credible interval # More refined method res<-hist(postp,100) x<-res$mids; dx<-res$density fit25<-loess(dx~x,span=0.25) # higher span leads to more smooth fit p<-seq(min(x),max(x),by=0.001) # reduce 'by' value to increase precision dp<-pmax(0,predict(fit25,data.frame(x=p))) # pmax changes all negative values to 0. samp<-sample(p,1e5,replace=TRUE,prob=dp) quantile(samp,c(0.025,0.975)) # produce credible interval samp <- sample(dp, 1e5, replace = TRUE, prob = dp) # sample probability densities crit <- quantile(samp, 0.05) # a horizontal line to cut the peak phigh <-p[dp >= crit] lo<-phigh[1]; hi<-phigh[length(phigh)] lo; hi # produce high density interval Slide 11
