# Completely Randomized Design - Crop and Soil Science by malj

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```									 The Statistical Analysis
ü 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
ü Provides an estimate of experimental error (s2)

– Used to construct interval estimates and significance tests
ü Provides a way to test the significance of variance sources
Analysis of Variance (ANOVA)
Assumptions
ü The error terms are…
randomly, independently, and normally distributed,
with a mean of zero and a common variance.
ü The main effects are additive

Linear additive model for a Completely Randomized Design (CRD)

mean
Yij = m + ti + eij
observation                random error
treatment effect
The CRD Analysis

We can:
ü Estimate the treatment means
ü Estimate the standard error of a treatment mean
ü Test the significance of differences among the
treatment means
SiSj Yij=Y..             What?
ü i represents the treatment number (varies from 1 to t=3)
ü j represents the replication number (varies from 1 to r=4)
ü S is the symbol for summation

Treatment (i)   Replication (j)   Observation (Yij)
1               1                 47.9
1               2                 50.6                C      P      K
1               3                 43.5
1               4                 42.6                47.9   62.5   66.4
2               1                 62.8
2               2                 50.9
50.6   50.9   60.6
2               3                 61.8
2               4                 49.1
3               1                 66.4                43.5   61.8   64.0
3               2                 60.6
3               3                 64.0                42.6   49.1   64.0
3               4                 64.0
The CRD Analysis - How To:
ü Set up a table of observations and compute the
treatment means and deviations

grand mean

mean of the i-th treatment

deviation of the i-th treatment
mean from the grand mean
The CRD Analysis, cont’d.
ü Separate sources of variation
– Variation between treatments
– Variation within treatments (error)
ü 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              df    SS   MS   F   P >F
Total               N-1

Treatments          t-1

Within treatments   N-t
(Error)
The CRD Analysis, cont’d.
ü Compute Sums of Squares
– Total

– Treatment

– Error SSE = SSTot - SST

ü Compute mean squares
– Treatment MST = SST / (t-1)
– Error     MSE = SSE / (N-t)
ü Calculate F statistic for treatments
– FT = MST/MSE
Using the ANOVA
ü Use FT to judge whether treatment means differ significantly
– If FT is greater than F in the table, then differences are significant

ü MSE = s2 or the sample estimate of the experimental error
– Used to compute standard errors and interval estimates

– Standard Error of a treatment mean

– Standard Error of the difference between two means
Numerical Example
ü 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.
ü Each plot was fertilized and treated to control weevils
and weeds.
ü A portion of each plot was harvested for yield and the
farms were classified by soil type.
ü 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            28.2            Mean
Mean     35.60   23.42   15.38   33.74   29.87   27.18
ri        3       5       4       5       3      20
Dev       8.42   -3.77 -11.81     6.55    2.68
ANOVA Table

Source          df           SS                 MS        F
Total          19              1,439.2055
Soil Type        4             1,077.6313269.4078 11.18**
Error          15           361.5742            24.1049

Fcritical(α=0.05; 4,15 df) = 3.06
** Significant at the 1% level
Formulae and Computations

Coefficient of Variation

Standard Error of a Mean

Confidence Interval Estimate of a Mean (soil type 4)

Standard Error of the Difference between Two Means (soils 1 and 2)

Test statistic with N-t df
Mean Yields and Standard Errors

Soil Type             1       2       3       4        5
Mean Yield           35.60   23.42   15.38   33.74    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
ü Analysis of yield data indicates highly significant
differences in yield among the five soil types
ü Soil type 1 produces the highest yield of lentil seed,
though not significantly different from type 4
ü Soil type 3 is clearly inferior to the others

1              4           5            2     3

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