1

As part of a larger experiment, Dale (1992) looked at six samples of a wetland soil undergoing a simulated snowmelt. Three were randomly se- lected for treatment with a neutral pH snowmelt; the other three got a reduced pH snowmelt. The observed response was the number of Copepoda removed from each microcosm during the first 14 days of snowmelt.

Reduced pH Neutral pH
256 159 149 54 123 248

h6(Using randomization methods, test the null hypothesis that the two treatments have equal average numbers of Copepoda versus a two-sided alternative.

ph <- c(256, 159, 149, 54, 123, 248)
s <- (sum(ph[1:3])-sum(ph[4:6]))/3
r <- NULL
for(i in 1:10000){
  ph1 <- sample(ph ,6,F)
  exp <-(sum(ph1[1:3])-sum(ph1[4:6]) )/3
  r <- c(r,exp)
}
ggplot(as.data.frame(r),aes(x = r))+
  geom_histogram(bins = 20)+
  geom_vline(xintercept = c(-46.333333,46.333333), color = "blue")+
  labs(title = "模擬結果",
         x = "r", y = "count")+
  theme(text  = element_text(size = 10 ,family = "STHeitiTC-Light"))

p_value <- (sum(r<=(-s))+sum(r>=s))/10000
p_value
## [1] 0.4082

2

Derive
\(\sum_{i=1}^{g}\sum_{j=1}^{n_i}(y_{ij}-\bar y{..})^2 = \sum_{i=1}^{g}n_i(\bar y_{i.}-\bar y_{..})^2+\sum_{i=1}^{g}\sum_{j=1}^{n_i}(\bar y_{ij}-\bar y{i.})^2\)


\(\sum_{i=1}^{g}\sum_{j=1}^{n_i}(y_{ij}-\bar y{..})^2 = \sum_{i=1}^{g}\sum_{j=1}^{n_i}(y_{i.}-\bar y_{ij}+\bar y_{ij}-\bar y{..})^2\\=\sum_{i=1}^{g}\sum_{j=1}^{n_i}(y_{ij}-\bar y{i.})^2 +2 \sum_{i=1}^{g}\sum_{j=1}^{n_i}(y_{ij}-\bar y{i.})(\bar y_{i.}-\bar y{..})+\sum_{i=1}^{g}\sum_{j=1}^{n_i}(\bar y_{i.}-\bar y{..})^2\)

其中

\(\sum_{i=1}^{g}\sum_{j=1}^{n_i}(y_{ij}-\bar y{i.})(\bar y_{i.}-\bar y{..})=\sum_{i=1}^{g}(\bar y_{i.}-\bar y _{..})\sum_{j=1}^{n_i}(y_{ij}-\bar y_{i.})\\\sum_{j=1}^{n_i}(y_{ij}-\bar y_{i.}) = \sum_{i=1}^{ni}y_{ij}-n_i\bar y_i=0\\\therefore\sum_{i=1}^{g}\sum_{j=1}^{n_i}(y_{ij}-\bar y{i.})(\bar y_{i.}-\bar y{..})\)

=>
\(\sum_{i=1}^{g}\sum_{j=1}^{n_i}(y_{ij}-\bar y{..})^2=\sum_{i=1}^{g}\sum_{j=1}^{n_i}(y_{ij}-\bar y{i.})^2+\sum_{i=1}^{g}\sum_{j=1}^{n_i}(\bar y_{i.}-\bar y{..})^2=\\\sum_{i=1}^{g}\sum_{j=1}^{n_i}(y_{ij}-\bar y{i.})^2+\sum_{i=1}^{g}n_i(\bar y_{i.}-\bar y{..})^2\)

又
\(\bar y_{i.}-\bar y_{..}=\alpha_i\)
\(\therefore \sum_{i=1}^{g}\sum_{j=1}^{n_i}(y_{ij}-\bar y{..})^2=\sum_{i=1}^gn_i\alpha_i+\sum_{i=1}^{g}\sum_{j=1}^{n_i}(y_{ij}-\bar y{i.})^2\)

3

\(E(MSE)=E(\frac{SSE}{N-g})=\frac{1}{N-g}[(\sum_{i=1}^g(Y_{ij}-\bar Y_{i.})^2]\\=\frac{1}{N-g}\sum_{i=1}^{g}\sum_{j=1}^{n_i}Y_{ij}^2-2\sum_{i=1}^{g}\sum_{j=1}^{n_i}Y\bar Y_{i.}Y_{ij}+\sum_{i=1}^{g}\sum_{j=1}^{n_i}\bar Y_{i.}\\=\frac{1}{N-g}(\sum_{i=1}^{g}\sum_{j=1}^{n_i}E(Y_{ij})-\sum_{i=1}^{g}n_i\bar Y_{i.})\\=\frac{1}{N-g}\sum_{i=1}^{g}\sum_{j=1}^{n_i}(\sigma^2+\mu_i^2)\sum_{i=1}^{g}n_i(\frac{\sigma^2}{n_i}+\mu_i^2)\\=\frac{1}{N-g}(N\sigma^2+\sum_{i=1}^gn_i\mu_i^2-g\sigma^2-\sum_{i=1}^gn_i\mu_i^2)\\=\frac{1}{N-g}(N-g)\sigma^2=\sigma^2\)