1. 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 (s 2 ) –Used to construct interval estimates and significance tests Provides a way to test the significance of variance sources

2. 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 observation Y ij =  +  i +  ij treatment effect random error

3. 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

4.  i  j Y ij =Y.. What? i represents the treatment number (varies from 1 to t=3) j represents the replication number (varies from 1 to r=4)  is the symbol for summation Treatment (i)Replication (j)Observation (Y ij ) 1147.9 1250.6 1343.5 1442.6 2162.8 2250.9 2361.8 2449.1 3166.4 3260.6 3364.0 3464.0 CPK 47.962.566.4 50.650.960.6 43.561.864.0 42.649.164.0

5. The CRD Analysis - How To: 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

6. 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 The CRD Analysis, cont’d.

7. Skeleton ANOVA for CRD SourcedfSSMSFP >F TotalN-1 Treatmentst-1 Within treatments (Error) N-t

8. Compute Sums of Squares –Total –Treatment –Error SSE = SSTot - SST Compute mean squares –TreatmentMST = SST / (t-1) –ErrorMSE = SSE / (N-t) Calculate F statistic for treatments –F T = MST/MSE The CRD Analysis, cont’d.

9. Using the ANOVA Use F T to judge whether treatment means differ significantly –If F T is greater than F in the table, then differences are significant MSE = s 2 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

10. 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.

11. Table of observations, means, and deviations 12345 12345 42.228.418.841.533.0 34.928.019.536.326.0 29.722.813.131.730.6 18.510.131.0 19.428.2 Mean Mean35.6023.4215.3833.7429.8727.18 r i 3545320 Dev8.42-3.77-11.816.552.68

12. ANOVA Table SourcedfSSMSF Total191,439.2055 Soil Type41,077.6313269.407811.18** Error15361.574224.1049 F critical(α=0.05; 4,15 df) = 3.06 ** Significant at the 1% level

13. 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

14. Mean Yields and Standard Errors Soil Type12345 Mean Yield35.6023.4215.3833.7429.87 Replications35453 Standard error2.832.202.452.202.83 CV = 18.1% 95% interval estimate of soil type 4 = 33.74 + 4.69 Standard error of difference between 1 and 2 = 3.58

15. 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 45 2 3