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 Yii ) + (Yi j Yii )
= Yij Yi i Yi j + Yii
RBD
Source d.f
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_Yield
10
12
20
16
30
19
40
18
50
17
Dependent Variable: Y
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 distributed
the F-test is not robust to violations of independen
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
8.79886300
34.76645475
Mean Square
0.89543420
1.46647717
F V
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; (j=1 to t, t=5)
eij = the effect of random error; mean
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 sires (among groups); mean=0, var= a
2
eij are the rand
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, variance = b (j=1 to 4)
2
(ab)ij = interaction effect
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 = Y
z12 + z22 22
(Yi Y )2
~ n21
2
i =1
n
(n 1) 2
~ n21
2
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 type (j=1 to b, b=2)
(*)ij = the interaction between th
Completely Randomized Factorial (CRF) Designs
B1
B2
CRF - ab (Fixed Effects Model)
B1
B2
A1 11
12
1i
A1 Y11i
Y12 i
22
2 i
A2 Y21i Y22i
i 2
Yi1i
Yijk = + i + j + ( )ij + ijk
Yi 2 i
Source
iid
ijk N (0, e2 )
i = 1,., a
j = 1,., b
k = 1,., r
d.f.
Yiii )
Y
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
Measures the strength of the sample evidence against Ho
Def
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 Groups)
t
ni
i =1
ii
j =1
F
(n 1)
2
(W) N-t
(Y
Yi i )
2
Types of Studies
Experiment:
3 Principles of Experimental Design
researcher manipulates 1 or more independent variables (treatments)
1) Blocking (local control) to reduce experimental error
Experimental Unit (EU) One replication of the experiment
2) Rand