生物统计习题及答案-4

生物统计与试验设计--作业四

彭继光3090100060 植物保护

Problem 2：In a field experiment, there are g varieties, d different plant densities and l different plant locations, the observation value can be expressed by following model: Yℎij=μ+Gℎ+D i+L j+GDℎi+GLℎj+DL ij+εℎij

Where, μ is the population mean；Gℎis the effect due to h’th variety，fixed effect；D i

is the effect due to i ’th density，D i~N(0，σD2)；L j is the effect due to j ’th location，

L j~N(0，σL2)；GDℎi is the random interaction effect between variety and plant density，GDℎi~ N(0，σGD2)，GLℎj is the random interaction effect between variety and location，GLℎj~ N(0，σGL2)，DL ij is the interaction effect between plant density and

location，random effect，DL ij~N(0，σDL2)，εℎij is the residual effect，εℎij~ N(0，σε2)。

(1)According to the design of experiment, write out the distribution of observation；

Yℎij~N(μ+μG，σD2+σL2+σGD2+σGL2+σDL2+σε2)

(2)Write out the formula of calculating sum of squares for each effect；

SSG=∑∑∑(Y̅ℎ∙∙−Y̅∙∙∙)2

j

i

ℎ=dl∑(Y̅ℎ∙∙−Y̅∙∙∙)2

ℎ

SSD=∑∑∑(Y̅∙i∙−Y̅∙∙∙)2

j

i

ℎ=gl∑(Y̅∙i∙−Y̅∙∙∙)2

i

SSL=∑∑∑(Y̅∙∙j−Y̅∙∙∙)2

j

i

ℎ=gd∑(Y̅∙∙j−Y̅∙∙∙)2

j

SSGD=∑∑∑(Y̅ℎi∙−Y̅ℎ∙∙−Y̅∙i∙+Y̅∙∙∙)2

j

i

ℎ=l∑∑(Y̅ℎi∙−Y̅ℎ∙∙−Y̅∙i∙+Y̅∙∙∙)2 i

ℎ

SSGL=∑∑∑(Y̅ℎ∙j−Y̅ℎ∙∙−Y̅∙∙j+Y̅∙∙∙)2

j

i

ℎ=d∑∑(Y̅ℎ∙j−Y̅ℎ∙∙−Y̅∙∙j+Y̅∙∙∙)2

j

ℎ

SSDL=∑∑∑(Y̅∙ij−Y̅∙i∙−Y̅∙∙j+Y̅∙∙∙)2

j

i

ℎ=g∑∑(Y̅∙ij−Y̅∙i∙−Y̅∙∙j+Y̅∙∙∙)2

i

ℎ

SSTO=SSG+SSD+SSL+SSGD+SSGL+SSDL+SSE=∑∑∑(Yℎij−Y̅∙∙∙)2

j

i

ℎ

(3) Write out the mean expectation of each sum squares for each effect ； E (MSG )=σε2+lσGD 2+dσGL 2+dl g −1∑G ℎ2ℎ

E (MSD )=σε2+glσD 2+gσDL 2

E (MSL )=σε2+gdσL 2+gσDL 2

E (MSGD )=σε2+lσGD

2 E (MSGL )=σε2+dσGL

2 E (MSDL )=σε2+gσDL 2

E (MSE )=σε2

(4) How to test the significance due to variety effect ？

H 0:∑G i 2=0i

H 1:∑G i 2>0i

当原假设成立时，F ∗=MSG MSGD+MSGL−MSE ~F(g −1,df)

df =(MSGD +MSGL −MSE)2

MSGD 2(g −1)(d −1)+MSGL 2(g −1)(l −1)+MSE 2(g −1)(d −1)(l −1)

(5) How to test the significance due to the interaction effect of variety and location ？

H 0:σGL 2=0

H 1:σGL 2>0

当原假设成立时，F ∗=MSGL MSE ~F((g −1)(l −1)，(g −1)(d −1)(l −1))