Completely Randomized Design - Crop and Soil Science

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Completely Randomized Design - Crop and Soil Science Powered By Docstoc
					 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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posted:5/3/2014
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