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Fixed and Random Effects in Analysis of Variance

This theory explores how to determine appropriate F-tests for expected mean squares in ANOVA, including coefficients and interactions to consider, with detailed examples. It covers fixed and random effects, pooling sums of squares, and multiple comparisons.

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Fixed and Random Effects in Analysis of Variance

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  1. Fixed and Random Effects

  2. Theory of Analysis of Variance [e2 + kt2]/e2 = 1, if kt2 = 0

  3. Expected Mean Squares • Dependant on whether factor effects are Fixed or Random. • Necessary to determine which F-tests are appropriate and which are not.

  4. Setting Expected Mean Squares • The expected mean square for a source of variation (say X) contains. • the error term. • a term in 2x. • a variance term for other selected interactions involving X.

  5. Coefficients for EMS Coefficient for error mean square is always 1 Coefficient of other expected mean squares is # reps times the product of factors levels that do not appear in the factor name.

  6. Expected Mean Squares • Which interactions to include in an EMS? • All the factors appear in the interaction. • All the other factors in the interaction are Random Effects.

  7. A and B Fixed Effects

  8. A and B Fixed Effects

  9. A and B Fixed Effects

  10. A and B Random Effects

  11. A and B Random Effects

  12. A and B Random Effects

  13. A Fixed and B Random

  14. A Fixed and B Random

  15. A Fixed and B Random

  16. A, B, and C are Fixed

  17. A, B, and C are Random

  18. A, B, and C are Random

  19. A, B, and C are Random

  20. A, B, and C are Random

  21. A Fixed, B and C are Random

  22. A Fixed, B and C are Random

  23. A Fixed, B and C are Random

  24. A Fixed, B and C are Random

  25. Pooling Sums of Squares

  26. Example • Ten yellow mustard lines. • Five different nitrogen levels (50, 75, 100, 125, 150 units of N). • Three replicates. • Raw data presented in Table 7 (Page 110 & 111)

  27. Example

  28. Example

  29. Example ~ Genotype Main-plot

  30. Example ~ Genotype Main-plot

  31. Example ~ Genotypes Main-plot

  32. Example ~ Nitrogen Main-plot

  33. Example ~ Strip-plot

  34. Another Example • Four species of Brassica. • Ten lines within each species. • Three insecticide treatments (Thiodan, Furidan, none). • Three replicates. • Raw data in Table 8.

  35. Example

  36. Example

  37. Example

  38. Cross-Classified

  39. Cross-Classified

  40. Nested

  41. Example ~ Nesting

  42. Example ~ Nested Split-plot

  43. Example ~ Nest Split-plot

  44. Multiple Comparisons

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