statistical principles in experimental design(winer)
#statistical principles in experimental design
#p.31
#H1:mu_a=mu_b
na=7;nb=10
Xa=c(3,5,2,4,6,2,7)
Xb=c(6,5,7,8,9,4,7,8,9,7)
La=na*sum(Xa^2)-sum(Xa)^2
Lb=nb*sum(Xb^2)-sum(Xb)^2
t2=(na+nb-2)*(nb*sum(Xa)-na*sum(Xb))^2/((na+nb)*(nb*La+na*Lb))
alpha=0.05
qt(1-alpha/2,df=(na+nb-2))^2
#test for homogeneity of variance
sa2=15.58;sb2=28.2
na=10;nb=8
F_obs=sb2/sa2
qf(1-0.05,nb-1,na-1)
#2.6 testing hypotheses about the difference between two means
#small sample sizes
#H1:mu_a=mu_b
na=16;nb=8
mean_Xa=30;mean_Xb=21
sa2=32;sb2=80
t_obs=(mean_Xa-mean_Xb)/sqrt(sa2/na+sb2/nb)
alpha=0.05
c=(sa2/na)/(sa2/na+sb2/nb)
#degree of freedom
V=sa2/na;W=sb2/nb;U=V+W
fa=na-1;fb=nb-1
#(2)
f1=(U^2)/((V^2/(na+1))+(W^2/(nb+1)))-2
#(3)
f2=(fa*fb)/(fb*c^2+fa*(1-c)^2)
f=min(f1,f2)
qt(1-0.05/2,df=f)
#2.7 testing hypotheses about the difference between two means-correlated observations
#H:mu_a-mu_b=0
data=data.frame(person_number=c(1:7),before_treatment=c(3,8,4,6,9,2,12),after_treatment=c(6,14,8,4,16,7,19))
data=data%>%mutate(difference=after_treatment-before_treatment)
L_d=nrow(data)*sum(data$difference^2)-(sum(data$difference)^2)
sd2=L_d/(nrow(data)*(nrow(data)-1))
t_obs=(mean(data$difference))/sqrt(sd2/nrow(data))
qt(1-0.05/2,df=nrow(data)-1)
#Before treatment
B=data$before_treatment
L_b=nrow(data)*sum(B^2)-sum(B)^2
sb2=L_b/(nrow(data)*(nrow(data)-1))
#after treatment
A=data$after_treatment
L_a=nrow(data)*sum(A^2)-sum(A)^2
sa2=L_a/(nrow(data)*(nrow(data)-1))
#Product treatment
L_ab=nrow(data)*sum(A*B)-sum(A)*sum(B)
rab=L_ab/sqrt(L_a*L_b)
#table 2.8-1
t_obs=c(0.87,0.54,1.1,1.5,1.3)
probability=1-pt(t_obs,df=14)
kai2=-log(probability)
sum(kai2)
qchisq(1-0.05/2,df=length(t_obs)*2)
#3.2 Definitions and numerical examples
method1=c(3,5,2,4,8,4,3,9)
method2=c(4,4,3,8,7,4,2,5)
method3=c(6,7,8,6,7,9,10,9)
t1=sum(method1)
t2=sum(method2)
t3=sum(method3)
SS1=sum(method1^2)-t1^2/length(method1)
SS2=sum(method2^2)-t2^2/length(method2)
SS3=sum(method3^2)-t3^2/length(method3)
G=t1+t2+t3
#with-in variance
SS_w=sum(method1^2)+sum(method2^2)+sum(method3^2)-(sum(method1)^2+sum(method2)^2+sum(method3)^2)/length(method1)
G_bar=G/(3*length(method1))
SS_method=length(method1)*sum((c(t1,t2,t3)/length(method1)-G_bar)^2)
SS_total=sum((cbind(method1,method2,method3)-G_bar)^2)
SS_error=SS_total-SS_method
MS_method=SS_method/(3-1)
MS_error=SS_error/(3*(length(method1)-1))
MS_error=sum(var(method1)+var(method2)+var(method3))/(3)
MS_treat=length(method1)*sum((c(t1,t2,t3)/length(method1)-G_bar)^2)/(3-1)
#H:population means area equal
f=MS_method/MS_error
qf(1-0.05,3-1,3*(length(method1)-1))
#H:treatment effects are equal
f=MS_treat/MS_error
qf(1-0.05,3-1,3*(length(method1)-1))
#comparisons among treatment means p.65
#H:sum(C*t)=0
t=c(t1,t2,t3)
#C=c(1,-1,0)
C=c(1,-1,0)
f=sum(C*t)^2/(length(method1)*sum(C^2)*MS_error)
k=length(C[abs(C)>0])
qf(1-0.05,k-1,3*(length(method1)-1))
#p.71
#There are six displays(k=6) and each sample is observed(n=10) under only one of the displays.
#The hypothesis that the mean reaction times for the displays are equal
#For example,six different Web page sagements(complexity)
k=6;n=10
t=c(100,110,120,180,190,210)
X2=c(1180,1210,1600,3500,3810,4610)
G=sum(t)
SS_displays=sum(t^2)/n-G^2/(k*n)
SS_error=sum(X2)-sum(t^2)/n
SS_total=sum(X2)-G^2/(k*n)
f=(SS_displays/(k-1))/(SS_error/((k)*(n-1)))
qf(1-0.05,k-1,(k-1)*(n-1))
#linear(1次),Quadratic(2次),Cubic(3次) trend
#linear
coef=c(-5,-3,-1,1,3,5)
D=sum(coef^2)*n
C_lin=sum(coef*t)
C2D=C_lin^2/D;SS_lin=C2D
f_lin=C2D/(SS_error/((k)*(n-1)))
qf(1-0.05,k-1,k*(n-1))
#quadratic
coef=c(5,-1,-4,-4,-1,5)
D=sum(coef^2)*n
C_lin=sum(coef*t)
C2D=C_lin^2/D
f_quad=C2D/(SS_error/((k)*(n-1)))
qf(1-0.05,k-1,k*(n-1))
#cubic
coef=c(-5,7,4,-4,-7,5)
D=sum(coef^2)*n
C_lin=sum(coef*t)
C2D=C_lin^2/D
f_cub=C2D/(SS_error/((k)*(n-1)))
qf(1-0.05,k-1,k*(n-1))
#non-linear
SS_nonlinear=SS_displays-SS_lin
f_nonlinear=(SS_nonlinear/(k-2))/MS_error
qf(1-0.05,k-2,k*(n-1))
#linear equation
SSK=n*(k^3-k)/12
b=sqrt(SS_lin/SSK)
K_bar=(k+1)/2
X_bar=G/(k*n)
a=X_bar-b*K_bar
r=sqrt(SS_lin/SS_total)
#p.77 Use of the Studentized Range Statics
#H:treatment effects are equal
method1=c(3,5,2,4,8,4,3,9)
method2=c(4,4,3,8,7,4,2,5)
method3=c(6,7,8,6,7,9,10,9)
t1=sum(method1)
t2=sum(method2)
t3=sum(method3)
t=c(t1,t2,t3)
MS_error=sum(var(method1)+var(method2)+var(method3))/(3)
#Newman-Kleus test
#biostatics analysis p.215
q=(max(t)-min(t))/sqrt(MS_error/length(method1))
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