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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 i1 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 SYi 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