实验设计与数据处理：Experimental Design

TSS
k i1
( y ni
j1 ij
y.. )2
dfTotal N 1
• Between Group (Sample) Variation
SST
t i 1
( y ni
j1 i.
y.. )2
t i1
ni
(
yi.
y..
)2
dfT t 1
• Within Group (Sample) Variation
Data Normal Design
Independent F-Test
Samples
1-Way
(CRD)
ANOVA
Paired Data F-Test
(RBD)
2-Way
ANOVA
Nonnormal
KruskalWallis Test
Friedman’s Test
Completely Randomized Design (CRD)
– As spread in (1,…,t) E(MST)/E(MSE)
– As (n1,…,nt) E(MST)/E(MSE) (when H0 false)
– As s2 E(MST)/E(MSE) (when H0 false)
0.09
0.09
0.08
0.08
0.07
0.07
0.06
j
ni 1
• Obtain the overall mean and sample size
N n1 ... nt
y..
n1 y1.
... nt N
yt.
y i j ij N
Analysis of Variance - Sums of Squares
• Total Variation
– True group effects (1,…,t)
– Group sample sizes (n1,…,nt)
– Within group variance (s2)
• Fobs = MST/MSE
• When H0 is true (1=…=t=0), E(MST)/E(MSE)=1
• Marginal Effects of each factor (all other factors fixed)
Experimental Design and the Analysis of Variance
Comparing t > 2 Groups - Numeric Responses
• Extension of Methods used to Compare 2 Groups • Independent Samples and Paired Data Designs • Normal and non-normal data distributions
s
2
i1 t
s2
1
ni
2 i
1
i 1
s 2 (t
1)
When
H0
:1
t
0
is
true,
E(MST ) E ( MSE )
1
otherwise
(Ha
is
true),
E(MST ) E ( MSE )
1
Expected Mean Squares
• 3 Factors effect magnitude of F-statistic (for fixed t)
• Statistical model yij is measurement from the jth subject from group i:
yij i ij i ij
where is the overall mean, i is the effect of treatment i , ij is a random error, and i is
the population mean for group i
1-Way ANOVA for Normal Data (CRD)
• For each group obtain the mean, standard deviation, and sample size:
yij
yi.
j
ni
si
( yij yi. )2
SSE
t i 1
( y ni
j1 ij
y i.Hale Waihona Puke Baidu)2
t i
1
(ni
1)si2
dfE N t
TSS SST SSE dfTotal dfT dfE
Analysis of Variance Table and F-Test
Source of Variation Treatments Error Total
0.06
0.05
0.05
0.04
0.03
0.02
0.01
0
0
20
40
60
80
100
120
140
160
180
200
A) =100, t1=-20, t2=0, t3=20, s = 20
E(MST ) E(MSE)
0.04
0.03
0.02
P val : P(F Fobs )
Expected Mean Squares
• Model: yij = +i + ij with ij ~ N(0,s2), Si = 0:
E(MSE) s 2
t
ni
2 i
E(MST ) s 2 i1
t 1
t
ni
2 i
t
E(MST ) E ( MSE )
Sum of Squares SST SSE TSS
Degrres of Freedom
t-1 N-t N-1
Mean Square MST=SST/(t-1) MSE=SSE/(N-t)
F F=MST/MSE
• Assumption: All distributions normal with common variance
•H0: No differences among Group Means (1 t =0) • HA: Group means are not all equal (Not all i are 0)
T.S. :
Fobs
MST MSE
R.R. : Fobs F ,t1,N t (Table 9)
• Controlled Experiments - Subjects assigned at random to one of the t treatments to be compared
• Observational Studies - Subjects are sampled from t existing groups