1. Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 18 Random Effects

2. Fixed vs. Random Factors Fixed Factors Levels are preselected, inferences limited to these specific levels Factors Shaft Sleeve Lubricant Manufacturer Speed Levels Steel, Aluminum Porous, Nonporous Lub 1, Lub 2, Lub 3, Lub 4 A, B High, Low

3. Fixed Factors (Effects) Fixed Factors Levels are preselected, inferences limited to these specific levels Fixed Levels Changes in the mean  i Changes in the mean  i One-Factor Model y ij =  +  i + e ij Main Effects  i -  i Parameters

4. Random Factors (Effects) Random Factor Levels are a random sample from a large population of possible levels. Inferences are desired on the population of levels. Factors Lawnmower Levels 1, 2, 3, 4, 5, 6 One-Factor Model y ij =  + a i + e ij Random Levels

5. Random Factors (Effects) One-Factor Model y ij =  + a i + e ij Main Effects Random a i Variability = Estimate Variance Components  a 2,  2

6. Skin Swelling Measurements Factors Laboratory animals (Random) Location of the measurement: Back, Ear (Fixed) Repeat measurements (2 / location)

7. Automatic Cutoff Times Factors Manufacturers: A, B (Fixed) Lawnmowers: 3 for each manufacturer (Random) Speeds: High, Low (Fixed) MGH Table 13.6 }

8. Random Factor Effects Assumption Factor levels are a random sample from a large population of possible levels Assumption Factor levels are a random sample from a large population of possible levels Subjects (people) in a medical study Laboratory animals Batches of raw materials Fields or farms in an agricultural study Blocks in a block design Inferences are desired on the population of levels, NOT just on the levels included in the design Inferences are desired on the population of levels, NOT just on the levels included in the design

9. Random Effects Model Assumptions (All Factors Random) Levels of each factor are a random sample of all possible levels of the factor Random factor effects and model error terms are distributed as mutually independent zero-mean normal variates; e.g., e i ~NID(0,  e 2 ), a i ~NID(0,  a 2 ), mutually independent Analysis of variance model contains random variables for each random factor and interaction Analysis of variance model contains random variables for each random factor and interaction Interactions of random factors are assumed random Interactions of random factors are assumed random

10. Skin Color Measurements Factors Participants -- representative of those from one ethnic group, in a well-defined geographic region of the U.S. Weeks -- No skin treatment, studying week-to-week variation (No Repeats -- must be able to assume no interaction) MGH Table 10.3

11. Two-Factor Random Effects Model: Main Effects Only Two-Factor Main Effects Model y ijk =  + a i + b j + e ijk i = 1,..., a j = 1,..., b Mutually Independent  0

12. Two-Factor Random Effects Model Two-Factor Model y ijk =  + a i + b j + (ab) ij + e ijk i = 1,..., a j = 1,..., b k = 1,..., r Mutually Independent  0

13. Two-Factor Model Differences  ij =  +  i +  j + (  ) ij Mean Variance Fixed Effects  ij =  Random Effects Variance Components Change the Mean

14. Expected Mean Squares Functions of model parameters Identify testable hypotheses Components set to zero under H 0 Identify appropriate F statistic ratios Under H 0, two E(MS) are identical

15. Properties of Quadratic Forms in Normally Distributed Random Variables

16. Expected Mean Squares One Factor, Fixed Effects y ij =  +  i + e ij i = 1,..., a ; j = 1,..., r e ij ~ NID(0,  e 2 )

17. Expected Mean Squares One Factor, Fixed Effects Sum of Squares

18. Expected Mean Squares One Factor, Fixed Effects Sum of Squares E{MS A )=  e 2   =   =... =  a

19. Expected Mean Squares Three-Factor Fixed Effects Model Source Mean SquareExpected Mean Square AMS A  e 2 + bcr Q  ABMS AB  e 2 + cr Q  ABCMS ABC  e + r Q  ErrorMS E  e 2 All effects tested against error Typical Main Effects and Interactions

20. Expected Mean Squares One Factor, Random Effects y ij =  + a i + e ij i = 1,..., a ; j = 1,..., r a i ~ NID(0,  a 2 ), e ij ~ NID(0,  e 2 ) Independent

21. Expected Mean Squares One Factor, Random Effects Sum of Squares

22. Expected Mean Squares One Factor, Random Effects Sum of Squares E{MS A )=  e 2  a 2 = 0

23. Skin Color Measurements Factors Participants -- representative of those from one ethnic group, in a well-defined geographic region of the U.S. Weeks -- No skin treatment, studying week-to-week variation (No Repeats -- Must be Able to Assume No Interaction)

24. Expected Mean Squares Three-Factor Random Effects Model Source Mean SquareExpected Mean Square AMS A  e 2 + r  abc  + cr  ab  + br  ac  + bcr  a  ABMS AB  e 2 + r  abc  + cr  ab  ABCMS ABC  e + r  abc  ErrorMS E  e 2 Effects not necessarily tested against error Test main effects even if interactions are significant May not be an exact test (three or more factors, random or mixed effects models; e.g. main effect for A) Effects not necessarily tested against error Test main effects even if interactions are significant May not be an exact test (three or more factors, random or mixed effects models; e.g. main effect for A)

25. Expected Mean Squares Balanced Random Effects Models Each E(MS) includes the error variance component Each E(MS) includes the variance component for the corresponding main effect or interaction Each E(MS) includes all higher-order interaction variance components that include the effect The multipliers on the variance components equal the number of data values in factor-level combination defined by the subscript(s) of the effect e.g., E(MS AB ) =  e 2 + r  abc  +cr  ab 

26. Expected Mean Squares Balanced Experimental Designs 1. Specify the ANOVA Model y ijk =  +  i +  j + (  ) ij + e ijk Two Factors, Fixed Effects MGH Appendix to Chapter 10

27. Expected Mean Squares Balanced Experimental Designs 2. Label a Two-Way Table a. One column for each model subscript b. Row for each effect in the model -- Ignore the constant term -- Express the error term as a nested effect

28. Two Factors, Fixed Effects y ijk =  +  i +  j + (  ) ij + e ijk

29. Expected Mean Squares Balanced Experimental Designs 3. Column Subscript Corresponds to a Fixed Effect. a. If the column subscript appears in the row effect & no other subscripts in the row effect are nested within the column subscript -- Enter 0 if the column effect is in a fixed row effect b. If the column subscript appears in the row effect & one or more other subscripts in the row effect are nested within the column subscript -- Enter 1 c. If the column subscript does not appear in the row effect -- Enter the number of levels of the factor

30. Two Factors, Fixed Effects Step 3a y ijk =  +  i +  j + (  ) ij + e ijk

31. Two Factors, Fixed Effects Step 3b y ijk =  +  i +  j + (  ) ij + e ijk

32. Two Factors, Fixed Effects Step 3c y ijk =  +  i +  j + (  ) ij + e ijk

33. Expected Mean Squares Balanced Experimental Designs 4. Column Subscript Corresponds to a Random Effect a. If the column subscript appears in the row effect -- Enter 1 b. If the column subscript does not appear in the row effect -- Enter the number of levels of the factor

34. Two Factors, Fixed Effects Step 4a y ijk =  +  i +  j + (  ) ij + e ijk

35. Two Factors, Fixed Effects Step 4b y ijk =  +  i +  j + (  ) ij + e ijk

36. Expected Mean Squares Balanced Experimental Designs 5. Notation a.  = Q factor(s) for fixed main effects and interactions b.  =  factor(s) 2 for random main effects and interactions List each  parameter in a column on the same line as its corresponding model term.

37. Two Factors, Fixed Effects Step 5 y ijk =  +  i +  j + (  ) ij + e ijk

38. Expected Mean Squares Balanced Experimental Designs 6. MS = Mean Square, C = Set of All Subscripts for the Corresponding Model Term a. Identify the  parameters whose model terms contain all the subscripts in C (Note: can have more than those in C) b. Multipliers for each  : -- Eliminate all columns having the subscripts in C -- Eliminate all rows not in 6a. -- Multiply remaining constants across rows for each  c. E(MS) is the linear combination of the coefficients from 6b and the corresponding  parameters; E(MS E ) =  e 2.

39. Two Factors, Fixed Effects Step 6a y ijk =  +  i +  j + (  ) ij + e ijk

40. Two Factors, Fixed Effects Step 6b: MS AB y ijk =  +  i +  j + (  ) ij + e ijk

41. Two Factors, Fixed Effects Step 6c y ijk =  +  i +  j + (  ) ij + e ijk

42. Two Factors, Fixed Effects Step 6b: MS B y ijk =  +  i +  j + (  ) ij + e ijk

43. Two Factors, Fixed Effects Step 6c y ijk =  +  i +  j + (  ) ij + e ijk

44. Two Factors, Fixed Effects Under appropriate null hypotheses, E(MS) for A, B, and AB same as E(MS E ) F = MS / MS E Under appropriate null hypotheses, E(MS) for A, B, and AB same as E(MS E ) F = MS / MS E