ST 524
NCSU - Fall 2008
RCBD and Power
Types of Error in Hypothesis testing
To compare two treatment means, we carry out a test of
We make a Type I error if
H o : 1  2 against H1 : 1  2 .
H o is true  1  2  but we conclude H1  1  2  .
H o when in fact 1  2 .
We make a Type II error if we fail to reject
Power = P(Type I Error) = probability of a Type II error,
Power = probability that the test will reject H o , given that
H o is false.
True Situation
Decision Taken
1  2
1  2
Do Not Reject Ho
Reject Ho
Correct Decision
Type I Error
Type II Error
Correct Decision
 =P  Reject Ho/1  2   P  Type I Error 
 =P  Do not Reject Ho/1  2   P  Type II Error 
Power  P  Reject Ho/1  2   1  
Power
For testing
H o : 1  2 against H1 : 1  2 , power depends on:
a) True value of 1  2 ,
b)  , measure of variability.
c) r, number of repetitions per treatment, and on
d)  , significance level of test.
2

Power increases as
r  1  2 
2
2
increases.
Sample size
Sample Size Determination
For a Randomized Complete Block Design
Yij     i  b j  eij
iidN  0,  2 
eij
i
is the effect of ith treatment
 i  i '
difference between means for treatments i and i’.
Number of repetitions per treatment is equal to the number of blocks in a balanced design.
Number of replicates required depends on
 Hypothesis is being tested
 Whether test is one- or two-tailed (what are the possible alternative hypotheses?)
 Significance level,  , to be used, P(Type I Error).
   2  1

Size of the difference

What assurance is desired to detect the difference ( Power  1   )?

An estimate of the variability of the data.  2 .
to be detected.
Sample size required to detect a mean difference at least equal to
Tuesday September 4, 2008
  1  2 o is given by
1
ST 524
NCSU - Fall 2008
RCBD and Power
r

2 2  z 2  z 
2
Express the desired difference to detect,  , as a multiple of the true standard deviation 
r  2  z 2  z 

2
2
 
 
 
2
Correction when residual variance is used instead of true variance
 Error df  3 
rnew  r 

 Error df  1 
Example 1
Table 9.2 ST&D (p. 207)
Oil Content of Redwing Flaxseed inoculated at different stages of growth
with S. linicola, Winnipeg, 1947 (in percentage)
Analysis of Variance
Source of Variation
Blocks
Treatments
Error
Total
df
r-1 = 3
t–1=5
(r-1)(t-1) = 15
n – 1= 23
SS
MS
F
3.14
31.65
19.72
54.51
1.05
6.33
1.31
4.83
Calculate the number of repetitions per treatment to detect a difference effect between treatments of at
least 2.5% oil, regardless of direction, at a significance level of 0.05 with a 90% assurance of detecting a
true difference of 2.5%
2 1.311.96  1.28 
 4.4 ,
2.52
2
r
We need r = 5 blocks to attain desired power in detecting a difference effect of 2.5%
Example 2. Calculate sample size for detecting difference D between clones 2 and 5, MSE = 11793
D
n
55
82
75
45
95
28
115
19
135
14
165
10
185
8
205
6
225
5
255
4
80
n=2*(11793)*(1.96+1.28)^2/(c(55,75,95,115,135,165,185,205,225,255)^2)
n
[1] 81.850047 44.017137 27.434503 18.721845 13.585536 9.094450 7.234372
[8] 5.891645 4.890793 3.807711
n
20
40
60
power=0.90
var=11793
50
100
150
200
250
D
Tuesday September 4, 2008
2
ST 524
NCSU - Fall 2008
RCBD and Power
Randomized Block Design
treat
block_1
block_2
block_3
block_4
Early_Bloom
33.3
31.9
34.9
37.1
Full_Bloom
34.4
34.0
34.5
33.1
Full_Bloom_P
36.8
36.6
37.0
36.4
Ripening
36.3
34.9
35.9
37.1
Seedling
34.4
35.9
36.0
34.1
Uninoculated
36.4
37.3
37.7
36.7
Field Layout
block
plot_1
plot_2
plot_3
plot_4
plot_5
plot_6
1
6
2
3
5
4
1
2
5
4
3
2
1
6
3
3
1
6
5
4
2
4
4
2
5
6
1
3
Linear Model
Treatments and Block as fixed effect factors
Yij     i   j  eij
 33.3 1
31.9  1

 
34.9  1

 
 37.1 1
34.4  1

 
34.0  1
 34.5 1

 
 33.1 1
 36.8 1

 
36.6  1

 
37.0  1
36.4  1


 36.3 1
34.9  1

 
35.9  1
 37.1 1

 
34.4  1

 
35.9  1
36.0  1

 
 34.1 1
36.4  1

 
 37.3 1
37.7  1

 
36.7  1
1 0 0 0 0 0
1 0 0 0
 e11 

e 
1 0 0
 12 
 e13 
0 0 1 0

 
0 0 0 1
 e14 

 e21 
1 0 0 0

 
0 1 0 0
e22 
e 
0 0 1 0
     23 
0 0 0 1    e24 
1
1 0 0 0     e31 
 2   
0 1 0 0     e32 
 3
 
0 0 1 0     e33 
 4 

0 0 0 1    e34 
  5    
1 0 0 0     e41 
6
0 1 0 0    e42 
  1   
0 0 1 0     e43 
2
0 0 0 1    e44 
  3   
1 0 0 0     e51 
 4   
0 1 0 0
 e52 
 e53 
0 0 1 0

 
0 0 0 1
 e54 
 e61 
1 0 0 0

 
0 1 0 0
 e62 
e 
0 0 1 0

 63 
0 0 0 1 
 e64 
1 0 0 0 0 0 0
1 0 0 0 0 0
1 0 0 0 0 0
0
0
1 0 0 0 0
1 0 0 0 0
0
1 0 0 0 0
0
1 0 0 0 0
0 0
0 0
1 0 0 0
1 0 0 0
0 0
1 0 0 0
0 0
1 0 0 0
0 0 0
1 0 0
0 0 0
0 0 0
1 0 0
1 0 0
0 0 0
1 0 0
0 0 0 0
1 0
0 0 0 0
0 0 0 0
1 0
1 0
0 0 0 0
1 0
0 0 0 0 0
1
0 0 0 0 0
1
0 0 0 0 0
0 0 0 0 0
1
1
Y  Xβ  e

, eij ~ iidN 0,  e2
Tuesday September 4, 2008
title "RCBD Block and Treat fixed
effects";
proc mixed data=redwing;
class block treat;
model y= block treat ;
run;

3
ST 524
NCSU - Fall 2008
RCBD and Power
Model Information
Data Set
Dependent Variable
Covariance Structure
Estimation Method
Residual Variance Method
Fixed Effects SE Method
Degrees of Freedom Method
WORK.REDWING
y
Diagonal
REML
Profile
Model-Based
Residual
Class Level Information
Class
Levels
block
treat
4
6
Values
1 2 3 4
Early_Bloom Full_Bloom
Full_Bloom_P Ripening Seedling
uninoculated
Dimensions
Covariance Parameters
Columns in X
Columns in Z
Subjects
Max Obs Per Subject
1
11
0
1
24
Number of Observations
Number of Observations Read
Number of Observations Used
Number of Observations Not Used
24
24
0
The Mixed Procedure
Covariance Parameter
Estimates
Cov Parm
Estimate
Residual
1.3144
ˆ e2  1.3144
Fit Statistics
-2 Res Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
59.0
61.0
61.3
61.7
Type 3 Tests of Fixed Effects
Effect
block
treat
Num
DF
Den
DF
F Value
Pr > F
3
5
15
15
0.80
4.82
0.5147
0.0080
Block and Treat are fixed-effect factors
Expected Mean Squares
ANALYSIS of VARIANCE TABLE
Type 3 Analysis of Variance
Source
DF
Sum of
Squares
block
treat
Residual
3
5
15
3.141250
31.652083
19.716250
Mean Square
1.047083
6.330417
1.314417
Tuesday September 4, 2008
Expected Mean Square
Error Term
Var(Residual) + Q(block)
Var(Residual) + Q(treat)
Var(Residual)
MS(Residual)
MS(Residual)
.
Error
DF
F Value
Pr > F
15
15
.
0.80
4.82
.
0.5147
0.0080
.
4
ST 524
NCSU - Fall 2008
RCBD and Power
r   i  
t
Q Treat  
i 1
Var(Residual)
 t  1
is


t  j  
r
2
Q  Block  
,
j 1

2
 r  1
2
e
Test of Hypothesis
a. Block H o : 1  2  3  4  0
H1 : at least one  j  0, j  1 4
b. Treatments
H o : 1   2   3   4   5   6  0
H1 : at least one  i  0,
p value= 0.0080, Reject Ho at 0.05 significance level.
i  1 6
Treatments as fixed-effect factor and Block as random-effect factor
Yij     i  b j  eij
b j ~ iidN  0,  b2  , eij ~ iidN  0,  e2 
 33.3 1
31.9  1

 
34.9  1

 
 37.1 1
34.4  1

 
34.0  1
 34.5 1

 
 33.1 1
 36.8 1

 
36.6  1

 
37.0  1
36.4  1


 36.3 1
34.9  1

 
35.9  1
 37.1 1

 
34.4  1

 
35.9  1
36.0  1

 
 34.1 1
36.4  1

 
 37.3 1
37.7  1

 
36.7  1
1 0 0 0 0 0
1


1 0 0 0 0 0
0
0
1 0 0 0 0 0


1 0 0 0 0 0    0
 
0 1 0 0 0 0    1


0 1 0 0 0 0    0
  

0 1 0 0 0 0  1  0


0 1 0 0 0 0    0
  

0 0 1 0 0 0    1
0 0 1 0 0 0   2  0


0 0 1 0 0 0    0
 
0 0 1 0 0 0    0


0 0 0 1 0 0   3   1
  

0 0 0 1 0 0   0


0 0 0 1 0 0    0
  

0 0 0 1 0 0  4  0


0 0 0 0 1 0    1




0 0 0 0 1 0    0
 
0 0 0 0 1 0   5  0


0 0 0 0 1 0    0
  

0 0 0 0 0 1  6  1


0 0 0 0 0 1
0

0
0 0 0 0 0 1


0 0 0 0 0 1 
0
Y  Xβ  Zb  e
0 0 0
 e11 

e 
1 0 0
 12 
 e13 
0 1 0

 
0 0 1
 e14 
 e21 
0 0 0

 
1 0 0
e22 

e 
0 1 0

 23 
0 0 1
e24 
e 
0 0 0 
 31 
 e32 
1 0 0

 
0 1 0   b1   e33 
0 0 1  b2   e34 
 

0 0 0  b3   e41 
   

1 0 0 b4  e42

 
0 1 0
 e43 

e 
0 0 1

 44 
0 0 0
 e51 

 
1 0 0
 e52 
 e53 
0 1 0

 
0 0 1
 e56 

 e61 
0 0 0

 
1 0 0
 e62 

e 
0 1 0

 63 
0 0 1 
 e64 
title "RCBD Block and Treat fixed
effects";
proc mixed data=redwing;
class block treat;
model y= treat ;
random intercept / subject=block;
run;
Model Information
Data Set
Dependent Variable
Covariance Structure
Subject Effect
Estimation Method
Residual Variance Method
Fixed Effects SE Method
Degrees of Freedom Method
WORK.REDWING
y
Variance Components
block
REML
Profile
Model-Based
Satterthwaite
Class Level Information
Tuesday September 4, 2008
5
ST 524
NCSU - Fall 2008
RCBD and Power
Class
block
treat
Levels
4
6
Values
1 2 3 4
Early_Bloom Full_Bloom
Full_Bloom_P Ripening Seedling
uninoculated
Dimensions
Covariance Parameters
Columns in X
Columns in Z Per Subject
Subjects
Max Obs Per Subject
2
7
1
4
6
Number of Observations
Number of Observations Read
Number of Observations Used
Number of Observations Not Used
24
24
0
Covariance Parameter
Estimates
Cov Parm
Estimate
block
Residual
0
1.2699
> 3.141250+19.716250
[1] 22.8575
> 22.8575/18
[1] 1.269861
Fit Statistics
-2 Res Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
63.7
65.7
65.9
65.1
Type 3 Tests of Fixed Effects
Effect
treat
Num
DF
Den
DF
F Value
Pr > F
5
15
4.99
0.0069
> 6.330417/1.2699
[1] 4.984973
How many degrees of freedom for
Error? 15+3 = 18
Block is random-effect factor and Treat is fixed-effect factor
Type 3 Analysis of Variance
Source
DF
Sum of
Squares
treat
block
Residual
5
3
15
31.652083
3.141250
19.716250
Var(block) is
 b2
Var(Residual)
is
Mean Square
6.330417
1.047083
1.314417
Error Term
Var(Residual) + Q(treat)
Var(Residual) + 6 Var(block)
Var(Residual)
MS(Residual)
MS(Residual)
.
Error
DF
F Value
Pr > F
15
15
.
4.82
0.80
.
0.0080
0.5147
.
 e2
Method of Moments to estimate
ˆ b2 
Expected Mean Square
 b2
BlockMS  ErrorMS 1.047083  1.314417

 -0.04456
t
6
Since estimated value for  b2 is negative we can assume that variance for block effects is 0.
Tuesday September 4, 2008
6
ST 524
NCSU - Fall 2008
RCBD and Power
Need to correct the number of degrees of freedom in Type 3 test of hypothesis for fixed effects
RCBD Block random effects and Treat fixed effects
Satterthwaite correction for degrees of freedom
Covariance Parameter
Estimates
title "RCBD Block and Treat fixed
effects";
proc mixed data=redwing;
class block treat;
model y= treat/ddfm=satter ;
random intercept / subject=block;
run;
Cov Parm
Estimate
block
Residual
0
1.2699
ˆ b2  0
ˆ e2  1.2699
Fit Statistics
-2 Res Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
63.7
65.7
65.9
65.1
Type 3 Tests of Fixed Effects
Effect
Num
DF
Den
DF
F Value
Pr > F
5
18
4.99
0.0049
treat
Test of Hypothesis
2
b.
Block H o :  b  0
c.
H1 :  b2  0
Treatments H o :  1   2   3   4   5   6  0
H1 : at least one  i  0, i  1 6
p value= 0.0049, Reject Ho at 0.05 significance level.
Power Study
Assume a RCBD, Fixed Effects, with t treatments and r blocks. Assume that the true treatment means
are 1 , 2 ,
*
, 6* , and that the error variance is  e2 .
*
The power of the test to test H o : 1  2 
 t
H1 : at least one i is different from others, i  1 t
depends on the noncentrality
t
parameter
 , where  
1
 e2
r 
t
*
i

 and 
* 2
i
*


i 1
*
i
t
. As the noncentrality parameter increases,
power increases.
To compute power for a given number of reps ® and a given alternative H1, that specifies values for
1* ,
, t*
we may use SAS:
b.
c.
1* , , t* under H1 and compute  .
State significance level  and critical value for testing null hypothesis
State
 e2
and values for
a.
H o : 1  2  5  t ,
From ANOVA table for RCBD, the test statistic is F  TreatmentMS , with (t-1) and (t-1)(r-1)
ErrorMS
degrees of freedom.
b.
Critical value of F distribution is
c.
Calculate power as
Fcrit  Ft 1,t 1 r 1, = F  Hdf , Edf 
Power  P  Reject H o | H1 
 P  Fcalc  Fcrit |  , Hdf , Edf 
 1  P  Fcalc  Fcrit |  , Hdf , Edf 
Tuesday September 4, 2008
7
ST 524
NCSU - Fall 2008
RCBD and Power
Example, based in results for example 9.2 (STD)
Error MS = 1.3144
Treatment means: 34.3, 34, 36.7, 36.05, 35.1, 37.025 and Overall mean = 35.5292, = 0.05, Fcrit = F(5,15,0.05) = 2.9013
Obs
grndmn
dfnum
mse
t
r
trt
mu
1
2
3
4
5
6
35.5292
35.5292
35.5292
35.5292
35.5292
35.5292
5
5
5
5
5
5
1.31
1.31
1.31
1.31
1.31
1.31
6
6
6
6
6
6
4
4
4
4
4
4
A
B
C
D
E
F
34.300
34.000
36.700
36.050
35.100
37.025
Noncentrality parameter ==
24.0810
Power = 1- P(F<2.9013| 24.0810,5,15) =
diff_mn2
1.51085
2.33835
1.37085
0.27127
0.18418
2.23752
term
4.61328
7.14000
4.18580
0.82830
0.56239
6.83211
0.90594
Power vs sample size - example STD-9.2
How does power change as number of blocks (repetitions) varies?
Power vs lambda (noncentrality parameter) - example STD-9.2
How does power change asnoncentrality parameter varies?
Tuesday September 4, 2008
8