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ANOVA TABLE

ANOVA TABLE. Factorial Experiment Completely Randomized Design. Anova table for the 3 factor Experiment. Sum of squares entries. Similar expressions for SS B , and SS C. Similar expressions for SS BC , and SS AC. Sum of squares entries. Finally.

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ANOVA TABLE

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  1. ANOVA TABLE Factorial Experiment Completely Randomized Design

  2. Anova table for the 3 factor Experiment

  3. Sum of squares entries Similar expressions for SSB , and SSC. Similar expressions for SSBC , and SSAC.

  4. Sum of squares entries Finally

  5. The statistical model for the 3 factor Experiment

  6. Anova table for the 3 factor Experiment

  7. The testing in factorial experiments • Test first the higher order interactions. • If an interaction is present there is no need to test lower order interactions or main effects involving those factors. All factors in the interaction affect the response and they interact • The testing continues with lower order interactions and main effects for factors which have not yet been determined to affect the response.

  8. Random Effects and Fixed Effects Factors

  9. So far the factors that we have considered are fixed effects factors • This is the case if the levels of the factor are a fixed set of levels and the conclusions of any analysis is in relationship to these levels. • If the levels have been selected at random from a population of levels the factor is called a random effects factor • The conclusions of the analysis will be directed at the population of levels and not only the levels selected for the experiment

  10. Example - Fixed Effects Source of Protein, Level of Protein, Weight Gain Dependent • Weight Gain Independent • Source of Protein, • Beef • Cereal • Pork • Level of Protein, • High • Low

  11. Example - Random Effects In this Example a Taxi company is interested in comparing the effects of three brands of tires (A, B and C) on mileage (mpg). Mileage will also be effected by driver. The company selects b = 4 drivers at random from its collection of drivers. Each driver has n = 3 opportunities to use each brand of tire in which mileage is measured. Dependent • Mileage Independent • Tire brand (A, B, C), • Fixed Effect Factor • Driver (1, 2, 3, 4), • Random Effects factor

  12. The Model for the fixed effects experiment where m, a1, a2, a3, b1, b2, (ab)11 , (ab)21 , (ab)31 , (ab)12 , (ab)22 , (ab)32 , are fixed unknown constants And eijk is random, normally distributed with mean 0 and variance s2. Note:

  13. The Model for the case when factor B is a random effects factor where m, a1, a2, a3, are fixed unknown constants And eijk is random, normally distributed with mean 0 and variance s2. bj is normal with mean 0 and variance and (ab)ij is normal with mean 0 and variance Note: This model is called a variance components model

  14. The Anova table for the two factor model

  15. The Anova table for the two factor model (A, B – fixed) EMS = Expected Mean Square

  16. The Anova table for the two factor model (A – fixed, B - random) Note: The divisor for testing the main effects of A is no longer MSError but MSAB.

  17. Rules for determining Expected Mean Squares (EMS) in an Anova Table Both fixed and random effects Formulated by Schultz[1] Schultz E. F., Jr. “Rules of Thumb for Determining Expectations of Mean Squares in Analysis of Variance,”Biometrics, Vol 11, 1955, 123-48.

  18. The EMS for Error is s2. • The EMS for each ANOVA term contains two or more terms the first of which is s2. • All other terms in each EMS contain both coefficients and subscripts (the total number of letters being one more than the number of factors) (if number of factors is k = 3, then the number of letters is 4) • The subscript of s2 in the last term of each EMS is the same as the treatment designation.

  19. The subscripts of all s2 other than the first contain the treatment designation. These are written with the combination involving the most letters written first and ending with the treatment designation. • When a capital letter is omitted from a subscript , the corresponding small letter appears in the coefficient. • For each EMS in the table ignore the letter or letters that designate the effect. If any of the remaining letters designate a fixed effect, delete that term from the EMS.

  20. Replace s2 whose subscripts are composed entirely of fixed effects by the appropriate sum.

  21. Example: 3 factors A, B, C – all are random effects

  22. Example: 3 factors A fixed, B, C random

  23. Example: 3 factors A , B fixed, C random

  24. Example: 3 factors A , B and C fixed

  25. Example - Random Effects In this Example a Taxi company is interested in comparing the effects of three brands of tires (A, B and C) on mileage (mpg). Mileage will also be effected by driver. The company selects at random b = 4 drivers at random from its collection of drivers. Each driver has n = 3 opportunities to use each brand of tire in which mileage is measured. Dependent • Mileage Independent • Tire brand (A, B, C), • Fixed Effect Factor • Driver (1, 2, 3, 4), • Random Effects factor

  26. The Data

  27. Asking SPSS to perform Univariate ANOVA

  28. Select the dependent variable, fixed factors, random factors

  29. The Output The divisor for both the fixed and the random main effect is MSAB This is contrary to the advice of some texts

  30. The Anova table for the two factor model (A – fixed, B - random) Note: The divisor for testing the main effects of A is no longer MSError but MSAB. References Guenther, W. C. “Analysis of Variance” Prentice Hall, 1964

  31. The Anova table for the two factor model (A – fixed, B - random) Note: In this case the divisor for testing the main effects of A is MSAB .This is the approach used by SPSS. References Searle “Linear Models” John Wiley, 1964

  32. Crossed and Nested Factors

  33. The factors A, B are called crossed if every level of A appears with every level of B in the treatment combinations. Levels of B Levels of A

  34. Factor B is said to be nested within factor A if the levels of B differ for each level of A. Levels of A Levels of B

  35. Example: A company has a = 4 plants for producing paper. Each plant has 6 machines for producing the paper. The company is interested in how paper strength (Y) differs from plant to plant and from machine to machine within plant Plants Machines

  36. Machines (B) are nested within plants (A) The model for a two factor experiment with B nested within A.

  37. The ANOVA table Note: SSB(A )= SSB + SSAB and a(b – 1) = (b – 1) + (a - 1)(b – 1)

  38. Example: A company has a = 4 plants for producing paper. Each plant has 6 machines for producing the paper. The company is interested in how paper strength (Y) differs from plant to plant and from machine to machine within plant. Also we have n = 5 measurements of paper strength for each of the 24 machines

  39. The Data

  40. Anova Table Treating Factors (Plant, Machine) as crossed

  41. Anova Table: Two factor experiment B(machine) nested in A (plant)

  42. Graph Paper Strength

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