在对组均值进行推断时，可信区间是否对受试者内方差敏感，而置信区间则不敏感？

机器算法验证 r 置信区间 混合模式 锯齿 错误传播
2022-03-12 10:28:09

这是这个问题的一个衍生： 如何用 R 比较每个人的多个测量值的两组？

在那里的答案中（如果我理解正确的话）我了解到受试者内方差不会影响对组均值的推断，可以简单地取平均值来计算组均值，然后计算组内方差并使用它进行显着性检验。我想使用一种方法，在这种方法中，主题内的差异越大，我对小组的含义就越不确定，或者不明白为什么想要那样做是没有意义的。

这是原始数据的图以及一些使用相同主题均值的模拟数据，但使用这些均值和较小的主题内方差 (sd=.1) 从正态分布中对每个主题的单独测量值进行了采样。可以看出，组级置信区间（底行）不受此影响（至少我计算它们的方式）。

在此处输入图像描述

我还使用 rjags 以三种方式估计组均值。1) 使用原始原始数据 2) 仅使用主题均值 3) 使用主题内 sd 较小的模拟数据

结果如下。使用这种方法，我们看到在案例 #2 和 #3 中 95% 的可信区间更窄。这符合我在推断组均值时希望发生什么的直觉，但我不确定这是否只是我的模型的一些人工制品或可信区间的属性。

笔记。要使用 rjags，您需要先从这里安装 JAGS：http: //sourceforge.net/projects/mcmc-jags/files/

在此处输入图像描述

各种代码如下。

原始数据：

structure(c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 
3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 
6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 
10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 
12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 
15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 18, 
18, 18, 18, 18, 18, 2, 0, 16, 2, 16, 2, 8, 10, 8, 6, 4, 4, 8, 
22, 12, 24, 16, 8, 24, 22, 6, 10, 10, 14, 8, 18, 8, 14, 8, 20, 
6, 16, 6, 6, 16, 4, 2, 14, 12, 10, 4, 10, 10, 8, 4, 10, 16, 16, 
2, 8, 4, 0, 0, 2, 16, 10, 16, 12, 14, 12, 8, 10, 12, 8, 14, 8, 
12, 20, 8, 14, 2, 4, 8, 16, 10, 14, 8, 14, 12, 8, 14, 4, 8, 8, 
10, 4, 8, 20, 8, 12, 12, 22, 14, 12, 26, 32, 22, 10, 16, 26, 
20, 12, 16, 20, 18, 8, 10, 26), .Dim = c(108L, 3L), .Dimnames = list(
    NULL, c("Group", "Subject", "Value")))

获取主题均值并模拟具有较小主题内方差的数据：

#Get Subject Means
means<-aggregate(Value~Group+Subject, data=dat, FUN=mean)

#Initialize "dat2" dataframe
dat2<-dat

#Sample individual measurements for each subject
temp=NULL
for(i in 1:nrow(means)){
  temp<-c(temp,rnorm(6,means[i,3], .1))
}

#Set Simulated values
dat2[,3]<-temp

拟合 JAGS 模型的函数：

 require(rjags) 

#Jags fit function
jags.fit<-function(dat2){

  #Create JAGS model
  modelstring = "

  model{
  for(n in 1:Ndata){
  y[n]~dnorm(mu[subj[n]],tau[subj[n]]) T(0, )
  }

  for(s in 1:Nsubj){
  mu[s]~dnorm(muG,tauG) T(0, )
  tau[s] ~ dgamma(5,5)
  }


  muG~dnorm(10,.01) T(0, )
  tauG~dgamma(1,1)

  }
  "
  writeLines(modelstring,con="model.txt")

#############  

  #Format Data
  Ndata = nrow(dat2)
  subj = as.integer( factor( dat2$Subject ,
                             levels=unique(dat2$Subject ) ) )
  Nsubj = length(unique(subj))
  y = as.numeric(dat2$Value)

  dataList = list(
    Ndata = Ndata ,
    Nsubj = Nsubj ,
    subj = subj ,
    y = y
  )

  #Nodes to monitor
  parameters=c("muG","tauG","mu","tau")


  #MCMC Settings
  adaptSteps = 1000             
  burnInSteps = 1000            
  nChains = 1                   
  numSavedSteps= nChains*10000          
  thinSteps=20                      
  nPerChain = ceiling( ( numSavedSteps * thinSteps ) / nChains )            


  #Create Model
  jagsModel = jags.model( "model.txt" , data=dataList, 
                          n.chains=nChains , n.adapt=adaptSteps , quiet=FALSE )
  # Burn-in:
  cat( "Burning in the MCMC chain...\n" )
  update( jagsModel , n.iter=burnInSteps )

  # Getting DIC data:
  load.module("dic")


  # The saved MCMC chain:
  cat( "Sampling final MCMC chain...\n" )
  codaSamples = coda.samples( jagsModel , variable.names=parameters , 
                              n.iter=nPerChain , thin=thinSteps )  

  mcmcChain = as.matrix( codaSamples )

  result = list(codaSamples=codaSamples, mcmcChain=mcmcChain)

}

将模型拟合到每个数据集的每一组：

#Fit to raw data
groupA<-jags.fit(dat[which(dat[,1]==1),])
groupB<-jags.fit(dat[which(dat[,1]==2),])
groupC<-jags.fit(dat[which(dat[,1]==3),])

#Fit to subject mean data
groupA2<-jags.fit(means[which(means[,1]==1),])
groupB2<-jags.fit(means[which(means[,1]==2),])
groupC2<-jags.fit(means[which(means[,1]==3),])

#Fit to simulated raw data (within-subject sd=.1)
groupA3<-jags.fit(dat2[which(dat2[,1]==1),])
groupB3<-jags.fit(dat2[which(dat2[,1]==2),])
groupC3<-jags.fit(dat2[which(dat2[,1]==3),])

可信区间/最高密度区间函数：

#HDI Function
get.HDI<-function(sampleVec,credMass){ 
  sortedPts = sort( sampleVec )
  ciIdxInc = floor( credMass * length( sortedPts ) )
  nCIs = length( sortedPts ) - ciIdxInc
  ciWidth = rep( 0 , nCIs )
  for ( i in 1:nCIs ) {
    ciWidth[ i ] = sortedPts[ i + ciIdxInc ] - sortedPts[ i ]
  }
  HDImin = sortedPts[ which.min( ciWidth ) ]
  HDImax = sortedPts[ which.min( ciWidth ) + ciIdxInc ]
  HDIlim = c( HDImin , HDImax, credMass )
  return( HDIlim )
}

第一个情节：

layout(matrix(c(1,1,2,2,3,4),nrow=3,ncol=2, byrow=T))

boxplot(dat[,3]~dat[,2], 
xlab="Subject", ylab="Value", ylim=c(0, 1.2*max(dat[,3])),
col=c(rep("Red",length(which(dat[,1]==unique(dat[,1])[1]))/6),
rep("Green",length(which(dat[,1]==unique(dat[,1])[2]))/6),
rep("Blue",length(which(dat[,1]==unique(dat[,1])[3]))/6)
),
main="Original Data"
)
stripchart(dat[,3]~dat[,2], vert=T, add=T, pch=16)
legend("topleft", legend=c("Group A", "Group B", "Group C", "Individual Means +/- 95% CI"),
col=c("Red","Green","Blue", "Grey"), lwd=3, bty="n", pch=c(15),
pt.cex=c(rep(0.1,3),1),
ncol=3)

for(i in 1:length(unique(dat[,2]))){
  m<-mean(examp[which(dat[,2]==unique(dat[,2])[i]),3])
  ci<-t.test(dat[which(dat[,2]==unique(dat[,2])[i]),3])$conf.int[1:2]

  points(i-.3,m, pch=15,cex=1.5, col="Grey")
  segments(i-.3,
           ci[1],i-.3,
           ci[2], lwd=4, col="Grey"
  )
}



boxplot(dat2[,3]~dat2[,2], 
xlab="Subject", ylab="Value", ylim=c(0, 1.2*max(dat2[,3])),
col=c(rep("Red",length(which(dat2[,1]==unique(dat2[,1])[1]))/6),
rep("Green",length(which(dat2[,1]==unique(dat2[,1])[2]))/6),
rep("Blue",length(which(dat2[,1]==unique(dat2[,1])[3]))/6)
),
main=c("Simulated Data", "Same Subject Means but Within-Subject SD=.1")
)
stripchart(dat2[,3]~dat2[,2], vert=T, add=T, pch=16)
legend("topleft", legend=c("Group A", "Group B", "Group C", "Individual Means +/- 95% CI"),
col=c("Red","Green","Blue", "Grey"), lwd=3, bty="n", pch=c(15),
pt.cex=c(rep(0.1,3),1),
ncol=3)

for(i in 1:length(unique(dat2[,2]))){
  m<-mean(examp[which(dat2[,2]==unique(dat2[,2])[i]),3])
  ci<-t.test(dat2[which(dat2[,2]==unique(dat2[,2])[i]),3])$conf.int[1:2]

  points(i-.3,m, pch=15,cex=1.5, col="Grey")
  segments(i-.3,
           ci[1],i-.3,
           ci[2], lwd=4, col="Grey"
  )
}


means<-aggregate(Value~Group+Subject, data=dat, FUN=mean)

boxplot(means[,3]~means[,1], col=c("Red","Green","Blue"),
ylim=c(0,1.2*max(means[,3])), ylab="Value", xlab="Group",
main="Original Data"
)
stripchart(means[,3]~means[,1], pch=16, vert=T, add=T)

for(i in 1:length(unique(means[,1]))){
  m<-mean(means[which(means[,1]==unique(means[,1])[i]),3])
  ci<-t.test(means[which(means[,1]==unique(means[,1])[i]),3])$conf.int[1:2]

  points(i-.3,m, pch=15,cex=1.5, col="Grey")
  segments(i-.3,
           ci[1],i-.3,
           ci[2], lwd=4, col="Grey"
  )
}
legend("topleft", legend=c("Group Means +/- 95% CI"), bty="n", pch=15, lwd=3, col="Grey")


means2<-aggregate(Value~Group+Subject, data=dat2, FUN=mean)

boxplot(means2[,3]~means2[,1], col=c("Red","Green","Blue"),
ylim=c(0,1.2*max(means2[,3])), ylab="Value", xlab="Group",
main="Simulated Data Group Averages"
)
stripchart(means2[,3]~means2[,1], pch=16, vert=T, add=T)

for(i in 1:length(unique(means2[,1]))){
  m<-mean(means[which(means2[,1]==unique(means2[,1])[i]),3])
  ci<-t.test(means[which(means2[,1]==unique(means2[,1])[i]),3])$conf.int[1:2]

  points(i-.3,m, pch=15,cex=1.5, col="Grey")
  segments(i-.3,
           ci[1],i-.3,
           ci[2], lwd=4, col="Grey"
  )
}
legend("topleft", legend=c("Group Means +/- 95% CI"), bty="n", pch=15, lwd=3,   col="Grey")

第二个情节：

layout(matrix(c(1,2,3,4,4,4,5,5,5,6,6,6),nrow=4,ncol=3, byrow=T))

#Plot priors
plot(seq(0,10,by=.01),dgamma(seq(0,10,by=.01),5,5), type="l", lwd=4,
     xlab="Value", ylab="Density",
     main="Prior on Within-Subject Precision"
)
plot(seq(0,10,by=.01),dgamma(seq(0,10,by=.01),1,1), type="l", lwd=4,
     xlab="Value", ylab="Density",
     main="Prior on Within-Group Precision"
)
plot(seq(0,300,by=.01),dnorm(seq(0,300,by=.01),10,100), type="l", lwd=4,
     xlab="Value", ylab="Density",
     main="Prior on Group Means"
)


#Set overall xmax value
x.max<-1.1*max(groupA$mcmcChain[,"muG"],groupB$mcmcChain[,"muG"],groupC$mcmcChain[,"muG"],
               groupA2$mcmcChain[,"muG"],groupB2$mcmcChain[,"muG"],groupC2$mcmcChain[,"muG"],
               groupA3$mcmcChain[,"muG"],groupB3$mcmcChain[,"muG"],groupC3$mcmcChain[,"muG"]
)


#Plot result for raw data
#Set ymax
y.max<-1.1*max(density(groupA$mcmcChain[,"muG"])$y,density(groupB$mcmcChain[,"muG"])$y,density(groupC$mcmcChain[,"muG"])$y)

plot(density(groupA$mcmcChain[,"muG"]),xlim=c(0,x.max), 
     ylim=c(-.1*y.max,y.max), lwd=3, col="Red",
     main="Group Mean Estimates: Fit to Raw Data", xlab="Value"
)
lines(density(groupB$mcmcChain[,"muG"]), lwd=3, col="Green")
lines(density(groupC$mcmcChain[,"muG"]), lwd=3, col="Blue")

hdi<-get.HDI(groupA$mcmcChain[,"muG"], .95)
segments(hdi[1],-.033*y.max,hdi[2],-.033*y.max, lwd=3, col="Red")

hdi<-get.HDI(groupB$mcmcChain[,"muG"], .95)
segments(hdi[1],-.066*y.max,hdi[2],-.066*y.max, lwd=3, col="Green")

hdi<-get.HDI(groupC$mcmcChain[,"muG"], .95)
segments(hdi[1],-.099*y.max,hdi[2],-.099*y.max, lwd=3, col="Blue")

####

#Plot result for mean data

#x.max<-1.1*max(groupA2$mcmcChain[,"muG"],groupB2$mcmcChain[,"muG"],groupC2$mcmcChain[,"muG"])
y.max<-1.1*max(density(groupA2$mcmcChain[,"muG"])$y,density(groupB2$mcmcChain[,"muG"])$y,density(groupC2$mcmcChain[,"muG"])$y)

plot(density(groupA2$mcmcChain[,"muG"]),xlim=c(0,x.max), 
     ylim=c(-.1*y.max,y.max), lwd=3, col="Red",
     main="Group Mean Estimates: Fit to Subject Means", xlab="Value"
)
lines(density(groupB2$mcmcChain[,"muG"]), lwd=3, col="Green")
lines(density(groupC2$mcmcChain[,"muG"]), lwd=3, col="Blue")

hdi<-get.HDI(groupA2$mcmcChain[,"muG"], .95)
segments(hdi[1],-.033*y.max,hdi[2],-.033*y.max, lwd=3, col="Red")

hdi<-get.HDI(groupB2$mcmcChain[,"muG"], .95)
segments(hdi[1],-.066*y.max,hdi[2],-.066*y.max, lwd=3, col="Green")

hdi<-get.HDI(groupC2$mcmcChain[,"muG"], .95)
segments(hdi[1],-.099*y.max,hdi[2],-.099*y.max, lwd=3, col="Blue")




####
#Plot result for simulated data
#Set ymax
#x.max<-1.1*max(groupA3$mcmcChain[,"muG"],groupB3$mcmcChain[,"muG"],groupC3$mcmcChain[,"muG"])
y.max<-1.1*max(density(groupA3$mcmcChain[,"muG"])$y,density(groupB3$mcmcChain[,"muG"])$y,density(groupC3$mcmcChain[,"muG"])$y)

plot(density(groupA3$mcmcChain[,"muG"]),xlim=c(0,x.max), 
     ylim=c(-.1*y.max,y.max), lwd=3, col="Red",
     main=c("Group Mean Estimates: Fit to Simulated data", "(Within-Subject SD=0.1)"), xlab="Value"
)
lines(density(groupB3$mcmcChain[,"muG"]), lwd=3, col="Green")
lines(density(groupC3$mcmcChain[,"muG"]), lwd=3, col="Blue")

hdi<-get.HDI(groupA3$mcmcChain[,"muG"], .95)
segments(hdi[1],-.033*y.max,hdi[2],-.033*y.max, lwd=3, col="Red")

hdi<-get.HDI(groupB3$mcmcChain[,"muG"], .95)
segments(hdi[1],-.066*y.max,hdi[2],-.066*y.max, lwd=3, col="Green")

hdi<-get.HDI(groupC3$mcmcChain[,"muG"], .95)
segments(hdi[1],-.099*y.max,hdi[2],-.099*y.max, lwd=3, col="Blue")

用我个人版本的@StéphaneLaurent 的答案进行编辑

我使用他描述的模型从均值 = 0、主题方差 = 1 和主题误差/方差 = 0.1、1、10、100 之间的正态分布中采样。置信区间的一个子集显示在左侧面板中，而它们的宽度分布由相应的右侧面板显示。这让我确信他是 100% 正确的。但是，我仍然对上面的示例感到困惑，但将在此之后提出一个更集中的新问题。

在此处输入图像描述

上述模拟和图表的代码：

dev.new()
par(mfrow=c(4,2))


num.sims<-10000
sigmaWvals<-c(.1,1,10,100)
muG<-0  #Grand Mean
sigma.between<-1  #Between Experiment sd

for(sigma.w in sigmaWvals){

  sigma.within<-sigma.w #Within Experiment sd

  out=matrix(nrow=num.sims,ncol=2)
  for(i in 1:num.sims){

    #Sample the three experiment means (mui, i=1:3)
    mui<-rnorm(3,muG,sigma.between)

    #Sample the three obersvations for each experiment (muij, i=1:3, j=1:3)
    y1j<-rnorm(3,mui[1],sigma.within)
    y2j<-rnorm(3,mui[2],sigma.within)
    y3j<-rnorm(3,mui[3],sigma.within)


    #Put results in data frame
    d<-as.data.frame(cbind(
      c(rep(1,3),rep(2,3),rep(3,3)),
      c(y1j, y2j, y3j )
    ))
    d[,1]<-as.factor(d[,1])

    #Calculate means for each experiment
    dmean<-aggregate(d[,2]~d[,1], data=d, FUN=mean)

    #Add new confidence interval data to output
    out[i,]<-t.test(dmean[,2])$conf.int[1:2]

  }

  #Calculate % of intervals that contained muG
  cover<-matrix(nrow=nrow(out),ncol=1)
  for(i in 1:nrow(out)){
    cover[i]<-out[i,1]<muG & out[i,2]>muG
  }



  sub<-floor(seq(1,nrow(out),length=100))
  plot(out[sub,1], ylim=c(min(out[sub,1]),max(out[sub,2])),
       xlab="Simulation #", ylab="Value", xaxt="n",
       main=c(paste("# of Sims=",num.sims),
              paste("% CIs Including muG=",100*round(length(which(cover==T))/nrow(cover),3)))
  )
  axis(side=1, at=1:100, labels=sub)
  points(out[sub,2])

  cnt<-1
  for(i in sub){
    segments(cnt, out[i,1],cnt,out[i,2])
    cnt<-cnt+1
  }
  abline(h=0, col="Red", lwd=3)

  hist(out[,2]-out[,1], freq=F, xlab="Width of 95% CI",
       main=c(paste("muG=", muG), 
              paste("Sigma Between=",sigma.between), 
              paste("Sigma Within=",sigma.within))
  )

}
1个回答

在那里的答案中（如果我理解正确的话）我了解到受试者内方差不会影响对组均值的推断，可以简单地取平均值来计算组均值，然后计算组内方差并使用它进行显着性检验。

让我在这里发展这个想法。单个观察的模型是 ，其中：

yijk=μi+αij+ϵijk

  • yijk是组个测量值kji

  • αij∼iidN(0,σb2)是组的随机效应ji

  • ϵijk∼iidN(0,σw2)是误差内

在我对你的第一个问题的回答中，我建议你注意，一个人获得了一个经典的（固定效应）高斯线性模型，主题意味着。事实上，你可以很容易地检查与 假设次重复测量。这不过是具有固定因子的单向方差分析模型。y¯ij∙

y¯ij∙=μi+δij
δij=αij+1K∑kϵijk∼iidN(0,σ2)where σ2=σb2+σw2K,
K

然后我声称，为了得出关于的推断，您可以简单地考虑简单的经典线性模型，其观察结果是主题意味着。2014 年 12 月 4 日更新：这个想法的一些例子现在写在我的博客上：减少模型以获得置信区间。我的印象是，当我们在随机效应的水平上平均数据时，这总是有效的。μiy¯ij∙

在那里的答案中（如果我理解正确的话）我了解到受试者内方差不会影响对组均值的推断，可以简单地取平均值来计算组均值，然后计算组内方差并使用它进行显着性检验。我想使用一种方法，在这种方法中，主题内的差异越大，我对小组的含义就越不确定，或者不明白为什么想要那样做是没有意义的。

正如您从盒装公式中看到的那样，内方差在观察组均值的模型中发挥作用。σw2

其它你可能感兴趣的问题
上一篇为什么判别模型比生成模型更适合序列标记任务？ 下一篇来自 flexsurvreg 的标准错误