Transcript Completely Randomized Design

The Statistical Analysis
 Partitions the total variation in the data into components
associated with sources of variation
– For a Completely Randomized Design (CRD)
• Treatments --- Error
– For a Randomized Complete Block Design (RBD)
• Treatments --- Blocks --- Error
 Provides an estimate of experimental error (s2)
n
s2 
2
(Y

Y)
 i
i1
n 1
– Used to construct interval estimates and significance tests
 Provides a way to test the significance of variance sources
Analysis of Variance (ANOVA)
Assumptions
 The error terms are…
randomly, independently, and normally distributed,
with a mean of zero and a common variance.
 The main effects are additive
Linear additive model for a Completely Randomized Design (CRD)
mean
Yij =  + i + ij
observation
random error
treatment effect
The CRD Analysis
We can:
 Estimate the treatment means
 Estimate the standard error of a treatment mean
 Test the significance of differences among the
treatment means
SiSj Yij=Y..
What?
 i represents the treatment number (varies from 1 to t=3)
 j represents the replication number (varies from 1 to r=4)
 S is the symbol for summation
Treatment (i)
1
1
1
1
2
2
2
2
3
3
3
3
Replication (j)
1
2
3
4
1
2
3
4
1
2
3
4
Observation (Yij)
47.9
50.6
43.5
42.6
62.8
50.9
61.8
49.1
66.4
60.6
64.0
64.0
C
P
K
47.9
62.5
66.4
50.6
50.9
60.6
43.5
61.8
64.0
42.6
49.1
64.0
The CRD Analysis - How To:
 Set up a table of observations and compute the
treatment means and deviations
 Yij
Y  Y.. 
, where N   ri
N
Yi .  j Yij
Yi 

ri
ri
Ti  (Yi  Y)
grand mean
mean of the i-th treatment
deviation of the i-th treatment
mean from the grand mean
The CRD Analysis, cont’d.
 Separate sources of variation
– Variation between treatments
– Variation within treatments (error)
 Compute degrees of freedom (df)
– 1 less than the number of observations
– total df = N-1
– treatment df = t-1
– error df = N-t or t(r-1) if each treatment has the same r
Skeleton ANOVA for CRD
Source
Total
df
N-1
Treatments
t-1
Within treatments
(Error)
N-t
SS
MS
F
P >F
The CRD Analysis, cont’d.
 Compute Sums of Squares
– Total
– Treatment
– Error SSE = SSTot - SST

SST   r  Y  Y 

SSTot   i  j Yij  Y
2
2
i i
i
SSE  i  j  Yij  Yi 
 Compute mean squares
– Treatment
MST = SST / (t-1)
– Error
MSE = SSE / (N-t)
 Calculate F statistic for treatments
– FT = MST/MSE
2
Using the ANOVA
 Use FT to judge whether treatment means differ significantly
– If FT is greater than F in the table, then differences are significant
 MSE = s2 or the sample estimate of the experimental error
– Used to compute standard errors and interval estimates
– Standard Error of a treatment mean
MSE
SY 
r
– Standard Error of the difference between two means
1 1 
SYi Yi   MSE  
 ri ri 
Numerical Example
 A set of on-farm demonstration plots were located
throughout an agricultural district. A single plot was
located within a lentil field on each of 20 farms in the
district.
 Each plot was fertilized and treated to control weevils
and weeds.
 A portion of each plot was harvested for yield and the
farms were classified by soil type.
 A CRD analysis was used to see if there were yield
differences due to soil type.
Table of observations, means, and deviations
1
2
3
4
5
42.2
28.4
18.8
41.5
33.0
34.9
28.0
19.5
36.3
26.0
29.7
22.8
13.1
31.7
30.6
18.5
10.1
31.0
19.4
Mean
35.60
ri
3
Dev
8.42
23.42
5
28.2
15.38
4
-3.77 -11.81
33.74
Mean
29.87
5
3
6.55
2.68
27.18
20
ANOVA Table
Source
df
Total
19
1,439.2055
4
1,077.6313
269.4078
15
361.5742
24.1049
Soil Type
Error
SS
MS
Fcritical(α=0.05; 4,15 df) = 3.06
** Significant at the 1% level
F
11.18**
Formulae and Computations
 MSE 
 24.1049 
Coefficient of Variation CV  
100  
100  18.1%
 Y 
 27.18 
Standard Error of a Mean
s Y  MSE r i  24.1049 3  2.83
Confidence Interval Estimate of a Mean (soil type 4)
L   i   Y i  t  MSE r i  33.74  2.131 24.1049 5  33.74  4.69
Standard Error of the Difference between Two Means (soils 1 and 2)
1 1
1 1
s Y Y   MSE     24.1049     3.58
1
2
3 5
 r1 r2 
Test statistic with N-t df
12.18
Y
1  Y2
t

 3.40
MSE(1/ r1  1/ r2 ) 3.58
Mean Yields and Standard Errors
Soil Type
Mean Yield
1
35.60
2
23.42
3
15.38
4
33.74
5
29.87
Replications
3
5
4
5
3
Standard error
2.83
2.20
2.45
2.20
2.83
CV = 18.1%
95% interval estimate of soil type 4 = 33.74 + 4.69
Standard error of difference between 1 and 2 = 3.58
Report of Analysis
 Analysis of yield data indicates highly significant
differences in yield among the five soil types
 Soil type 1 produces the highest yield of lentil seed,
though not significantly different from type 4
 Soil type 3 is clearly inferior to the others
1
4
5
2
3