Complete Block Designs
Yij = + i + j + ij
iid
ij N (0, e2 ) i = 1,., t j = 1,., r
CRF
Residuals
Source d.f.
ij = Yij ( + i
A
a-1
2 + b a2
B
b-1
2 + a b2
Error
(a-1)(b-1)
2
+ j
)
= Yij Yii + (Yi i Y
C = k11 + k 2 2 +
t
Y1i
k1
K = Y =
k
Y
t
ti
+ kt t
k =0
i =1 i
C = k1Y1i + k 2Y2 i +
Y1i
kt ) =
Y
ti
Class
DENSITY
C3 = 2 3
C4 =
1 + 2
3
2
Values
5
10 20 30 40 50
Density Mean_Yi
iid
Model : Yij = + i + ij , ij N(0, e2 )
iid
Yij = + i + ij , ij N (0, e2 ) ij = eij = Yij Yij = Yij Yi i
Assessing ANOVA Assumptions
ANOVA Assumptions
SRS:
the t populations are normally distribut
Stat 231
Chapter 5
8.
a) See the answer in the book for the model.
The ANOVA table an partial list of E(MS) from SAS are:
Source
Model
Error
Corrected Total
DF
29
6
35
Sum of
Squares
25.96759175
8.798
Stat231
Chapter 8
Chapter 8, Problem 2
a) The linear model for this experiment is: yij i j eij
= overall mean
i = the fixed effect of the block; (i=1 to r, r=5)
j = the fixed effect of the treatment;
Stat231
Chapter 5
5.1
a) The random effects model for the data is:
yij = + ai + eij , in which i=1 to 5, j=1 to 8
yij are the individual observations
is the overall mean
ai are the random effects in
Stat 231
Chapter 7
Chapter 7, #1
a) yijk = + ai + b j + (ab )ij + eijk
= overall mean
2
ai = random effect of the patient; mean = 0, variance = a (i=1 to 5)
2
bj = random effect of the run; mean = 0,
iid
=Y
Yi N (, 2 )
n
(Yi Y )2
n 1
are unbiased
2
n
E i =1 (Yi Y ) = (n 1) 2
2 unbiased for 2
zi =
2 = i =1
&
Yi iid
N (0,1)
2
Y
zi2 = i 12
z 2 n2
i =1 i
n
(Yi )2
~ n2
i =1
2
n
Estimating with =
Stat231
Chapter 6
6. 1
a) The linear model for this data is: yijk = + i + j + ( )ij + ijk
= the overall mean
i = the fixed effect of the alcohol type (i=1 to a, a=3)
j = the fixed effect of the base
Hypothesis Testing
Null Hypothesis (Ho ) vs. Alternative Hypothesis (HA )
= P(Type I Error)
the level of significance (l.o.s.)
= P(Type II Error)
power = 1
Meta-experiment
experiment
p-value
Measu
2
n
E j =1 (Yij Yi i ) = (n 1) 2
iid
Yij N (i , 2 )
Recall:
Recall:
Y
j=1 i
2
n
z
2
j =1 i
2
n
n
n2
2
Y Y
j=1 i n21
n
Source
d.f. SS
MS
(Between Groups) Treatment (A) t-1
(Y
Yii )
(Within Gr