R：正态性检验

（1）QQ概率图

功能和原理：检验样本的概率分布是否服从某种理论分布。PP概率图的原理是检验实际累积概率分布与理论累积概率分布是否吻合，若吻合，则散点应围绕在一条直线周围，或者实际概率与理论概率之差分布在对称于以0为水平轴的带内。QQ概率图的原理是检验实际分位数与理论分位数之差分布是否吻合，若吻合，则散点应围绕在一条直线周围，或者实际分位数与理论分位数之差分布在对称于以0为水平轴的带内。QQ概率图以样本的分位数为横轴，以指定理论分布的分位数为纵轴绘制散点图。

> library(DAAG)
> data(possum)
> attach(possum)
The following object(s) are masked from 'possum (position 12)':

    age, belly, case, chest, earconch, eye, footlgth, hdlngth, Pop,
    sex, site, skullw, taill, totlngth
> fpossum <- possum[possum$sex=="f",]
> mean = mean(totlngth)
> sd = sd(totlngth)
> x <- sort(totlngth)
> n <- length(x)
> y <- (1:n)/n
>
> plot(x,y,
+       type = 's',
+       main = "Empirical CDF of ")
> curve(pnorm(x, mean, sd),
+       col = 'red',
+       lwd = 2,
+       add = T)

图形表示，数据与正态性略有差异，特别是中部区域。

（2）与正态密度函数直接比较

> library(DAAG)
> data(possum)
> attach(possum)
The following object(s) are masked from 'possum (position 13)':

    age, belly, case, chest, earconch, eye, footlgth, hdlngth, Pop,
    sex, site, skullw, taill, totlngth
> fpossum <- possum[possum$sex=="f",]
> dens <- density(totlngth)
> xlim <- range(dens$x)
> ylim <- range(dens$y)
> mean = mean(totlngth)
> sd = sd(totlngth)
> par(mfrow=c(1,2))
>
> hist(totlngth,
+       breaks=72.5+(0:5)*5,
+       xlim = xlim ,
+       ylim = ylim ,
+       probability = T ,
+       xlab = "total length",
+       main = "A:Breaks at 72.5...")
> lines(dens,
+       col = par('fg'),
+       lty = 2)
> curve( dnorm(x, mean, sd),
+         col = 'red',
+         add = T)
>        
> hist(totlngth,
+       breaks = 75 + (0:5) * 5 ,
+       xlim = xlim,
+       ylim = ylim,
+       probability = T,
+       xlab="total length",
+       main = "B:Breaks at 75")
> lines(dens,
+       col = par('fg'),
+       lty = 2)
> curve(dnorm(x,mean,sd),
+       col = 'red',
+       add = T)

看图直接看和正态密度函数的差异度。

（3）使用经验分布函数，直接比较数据的经验分布函数和正态分布的分布函数对比。

> library(DAAG)
> data(possum)
> attach(possum)
The following object(s) are masked from 'possum (position 14)':

    age, belly, case, chest, earconch, eye, footlgth, hdlngth, Pop,
    sex, site, skullw, taill, totlngth
> fpossum <- possum[possum$sex=="f",]
> mean = mean(totlngth)
> sd = sd(totlngth)
> x <- sort(totlngth)
> n <- length(x)
> y <- (1:n)/n
>
> plot(x,y,
+       type = 's',
+       main = "Empirical CDF of ")
> curve(pnorm(x, mean, sd),
+       col = 'red',
+       lwd = 2,
+       add = T)

总体来说，数据并不完全服从正态分布，需要做进一步检验，看和正态分布的差距多大，是否在接受范围之内？？