5.9 THREE FACTOR SPLIT PLOT MODEL (IV) Y = C|S'(B|A) on a response of proportions Analysis of terms: C|B|A + S(B A) Data: S A B C Su Fa Fr 1 1 1 1 15 58 0.470462 2 1 1 1 10 76 0.347977 1 1 1 2 17 52 0.519405 2 1 1 2 19 13 0.879706 1 1 2 1 26 99 0.473574 2 1 2 1 21 41 0.621171 1 1 2 2 14 67 0.428756 2 1 2 2 12 9 0.857072 1 2 1 1 10 82 0.335975 2 2 1 1 12 96 0.339837 1 2 1 2 30 14 0.971482 2 2 1 2 32 1 1.395827 1 2 2 1 37 54 0.691440 2 2 2 1 30 12 1.006854 1 2 2 2 32 66 0.608246 2 2 2 2 37 7 1.160521 1 3 1 1 18 34 0.629015 2 3 1 1 18 55 0.519635 1 3 1 2 23 85 0.479662 2 3 1 2 21 34 0.666087 1 3 2 1 24 82 0.495909 2 3 2 1 27 65 0.572501 1 3 2 2 37 27 0.863845 2 3 2 2 37 60 0.665701 COMMENT: 'Su' is the frequency of successes, 'Fa' the frequency of failures, 'Fr' is the Arcsin-root transformed fraction: Su/(Su + Fa). Model 5.9(i) A, B and C are fixed factors, S is a random blocking factor: Method 1. Analysis of Fr assuming a normal distribution of residuals. Source DF SS MS F P 1 A 2 0.2650 0.1325 2.36 0.175 2 B 1 0.0330 0.0330 0.59 0.472 3 B*A 2 0.0043 0.0021 0.04 0.963 4 S(B*A) 6 0.3363 0.0561 - - 5 C 1 0.3730 0.3730 12.65 0.012 6 C*A 2 0.1158 0.0579 1.96 0.221 7 C*B 1 0.0997 0.0997 3.38 0.116 8 C*B*A 2 0.2747 0.1374 4.66 0.060 9 C*S(B*A) 6 0.1769 0.0295 - - 10 P(C*S(B*A) 0 - - Total 23 1.6787 Method 2. Analysis of Su and Fa frequencies assuming a binomial error structure. See the set of commands in R for obtaining deviances at: http://www.soton.ac.uk/~cpd/anovas/datasets/ANOVA%20in%20R.htm#model5_9binomial Because the residual deviance greatly exceeds the residual d.f. (28.123 > 6 for A|B and 38.790 > 6 for C and its interactions), significance tests need to compensate for overdispersion. This is done by treating the deviance for each term as a SS. Then MS = SS/d.f., and F = MS[term]/MS[Error], with MS[Error] obtained by dividing the residual deviance of the last-entered term in a step by its residual d.f. Deviance Source DF = SS MS F P 1 A 2 15.442 7.721 1.65 0.269 2 B 1 8.675 8.675 1.85 0.223 3 B*A 2 3.644 1.822 0.39 0.694 4 S(B*A) 6 28.123 4.687 - - 5 C 1 60.058 60.058 9.29 0.023 6 C*A 2 12.986 6.493 1.00 0.421 7 C*B 1 19.318 19.318 2.99 0.135 8 C*B*A 2 69.035 34.518 5.34 0.047 9 C*S(B*A) 6 38.790 6.465 - - 10 P(C*S(B*A) 0 - - Total 23 256.066 __________________________________________________________________ Doncaster, C. P. & Davey, A. J. H. (2007) Analysis of Variance and Covariance: How to Choose and Construct Models for the Life Sciences. Cambridge: Cambridge University Press. http://www.southampton.ac.uk/~cpd/anovas/datasets/