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Statistical Analysis

Statistical Analysis. Professor Lynne Stokes Department of Statistical Science Lecture 18 Random Effects. Fixed vs. Random Factors. Fixed Factors Levels are preselected , inferences limited to these specific levels. Factors Shaft Sleeve Lubricant Manufacturer Speed.

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Statistical Analysis

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  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 One-Factor Model yij = m + ai + eij Main Effects mi - m = aiParameters Changes in the mean mi Fixed Levels

  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 yij = m + ai + eij Random Levels

  5. Random Factors (Effects) One-Factor Model yij = m + ai + eij Main Effects Random ai Variability = Estimate Variance Components sa2 , s2

  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 • 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

  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., ei~NID(0,se2) , ai~NID(0,sa2), mutually independent Analysis of variance model contains random variables for each random factor and interaction 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. Mutually Independent s 0 Two-Factor Random Effects Model: Main Effects Only Two-Factor Main Effects Model yijk = m + ai + bj + eijk i = 1, ..., a j = 1, ..., b

  12. Mutually Independent s 0 Two-Factor Random Effects Model Two-Factor Model yijk = m + ai + bj + (ab)ij + eijk i = 1, ..., a j = 1, ..., b k = 1, ..., r

  13. Random Effects mij = m Variance Components Two-Factor Model Differences Fixed Effects Mean mij = m + ai + bj + (ab)ij Change the Mean Variance

  14. Expected Mean Squares • Functions of model parameters • Identify testable hypotheses • Components set to zero under H0 • Identify appropriate F statistic ratios • Under H0, two E(MS) are identical

  15. Properties of Quadratic Forms in Normally Distributed Random Variables

  16. Expected Mean Squares One Factor, Fixed Effects yij = m + ai + eij i = 1, ... , a ; j = 1, ... , r eij ~ NID(0,se2)

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

  18. Expected Mean Squares One Factor, Fixed Effects Sum of Squares E{MSA)=se2a1 = a2 = ... = aa

  19. Expected Mean Squares Three-Factor Fixed Effects Model Source Mean Square Expected Mean Square A MSAse2 + bcr Qa AB MSABse2 + cr Qab ABC MSABCse + r Qabg Error MSEse2 Typical Main Effects and Interactions • All effects tested against error

  20. Expected Mean Squares One Factor, Random Effects yij = m + ai + eij i = 1, ... , a ; j = 1, ... , r ai ~ NID(0,sa2) , eij ~ NID(0,se2) Independent

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

  22. Expected Mean Squares One Factor, Random Effects Sum of Squares E{MSA)=se2sa2 = 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 Square Expected Mean Square A MSAse2 + rsabc2 + crsab2 + brsac2 + bcrsa2 AB MSABse2 + rsabc2 + crsab2 ABC MSABCse + rsabc2 Error MSEse2 • 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 SquaresBalanced 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(MSAB) = se2 + rsabc2 +crsab2

  26. Expected Mean SquaresBalanced Experimental Designs 1. Specify the ANOVA Model yijk = m + ai + bj + (ab)ij + eijk Two Factors, Fixed Effects MGH Appendix to Chapter 10

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

  28. Two Factors, Fixed Effects yijk = m + ai + bj + (ab)ij + eijk

  29. Expected Mean SquaresBalanced 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 yijk = m + ai + bj + (ab)ij + eijk Step 3a

  31. Two Factors, Fixed Effects yijk = m + ai + bj + (ab)ij + eijk Step 3b

  32. Two Factors, Fixed Effects yijk = m + ai + bj + (ab)ij + eijk Step 3c

  33. Expected Mean SquaresBalanced 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 yijk = m + ai + bj + (ab)ij + eijk Step 4a

  35. Two Factors, Fixed Effects yijk = m + ai + bj + (ab)ij + eijk Step 4b

  36. Expected Mean SquaresBalanced Experimental Designs 5. Notation a. f = Qfactor(s) for fixed main effects and interactions b. f = sfactor(s)2 for random main effects and interactions List eachfparameter in a column on the same line as its corresponding model term.

  37. Two Factors, Fixed Effects yijk = m + ai + bj + (ab)ij + eijk Step 5

  38. Expected Mean SquaresBalanced Experimental Designs 6. MS = Mean Square, C = Set of All Subscripts for the Corresponding Model Term a. Identify the f parameters whose model terms contain all the subscripts in C (Note: can have more than those in C) b. Multipliers for each f : -- Eliminate all columns having the subscripts in C -- Eliminate all rows not in 6a. -- Multiply remaining constants across rows for each f c. E(MS) is the linear combination of the coefficients from 6b and the corresponding f parameters; E(MSE) = se2.

  39. Two Factors, Fixed Effects yijk = m + ai + bj + (ab)ij + eijk Step 6a

  40. Two Factors, Fixed Effects yijk = m + ai + bj + (ab)ij + eijk Step 6b: MSAB

  41. Two Factors, Fixed Effects yijk = m + ai + bj + (ab)ij + eijk Step 6c

  42. Two Factors, Fixed Effects yijk = m + ai + bj + (ab)ij + eijk Step 6b: MSB

  43. Two Factors, Fixed Effects yijk = m + ai + bj + (ab)ij + eijk Step 6c

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

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