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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
Statistical Analysis

Professor Lynne Stokes

Department of Statistical Science

Lecture 18

Random Effects

fixed vs random factors
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

fixed factors effects
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

random factors effects
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

random factors effects1
Random Factors (Effects)

One-Factor Model

yij = m + ai + eij

Main Effects

Random ai

Variability =

Estimate Variance

Components

sa2 , s2

skin swelling measurements
Skin Swelling Measurements

Factors

Laboratory animals (Random)

Location of the measurement: Back, Ear (Fixed)

Repeat measurements (2 / location)

automatic cutoff times
Automatic Cutoff Times

Factors

Manufacturers: A, B (Fixed)

Lawnmowers: 3 for each manufacturer (Random)

Speeds: High, Low (Fixed)

}

MGH Table 13.6

random factor effects
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

random effects model assumptions all factors random
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

skin color measurements
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

two factor random effects model main effects only

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

two factor random effects model

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

two factor model differences

Random Effects

mij = m

Variance Components

Two-Factor Model Differences

Fixed Effects

Mean

mij = m + ai + bj + (ab)ij

Change the Mean

Variance

expected mean squares
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
expected mean squares1
Expected Mean Squares

One Factor, Fixed Effects

yij = m + ai + eij

i = 1, ... , a ; j = 1, ... , r

eij ~ NID(0,se2)

expected mean squares2

Sum of Squares

Expected Mean Squares

One Factor, Fixed Effects

expected mean squares3
Expected Mean Squares

One Factor, Fixed Effects

Sum of Squares

E{MSA)=se2a1 = a2 = ... = aa

expected mean squares4
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
expected mean squares5
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

expected mean squares6
Expected Mean Squares

One Factor, Random Effects

Sum of Squares

expected mean squares7
Expected Mean Squares

One Factor, Random Effects

Sum of Squares

E{MSA)=se2sa2 = 0

skin color measurements1
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)

expected mean squares8
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)
expected mean squares balanced random effects models
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

expected mean squares balanced experimental designs
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

expected mean squares balanced experimental designs1
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

two factors fixed effects
Two Factors, Fixed Effects

yijk = m + ai + bj + (ab)ij + eijk

expected mean squares balanced experimental designs2
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

two factors fixed effects1
Two Factors, Fixed Effects

yijk = m + ai + bj + (ab)ij + eijk

Step 3a

two factors fixed effects2
Two Factors, Fixed Effects

yijk = m + ai + bj + (ab)ij + eijk

Step 3b

two factors fixed effects3
Two Factors, Fixed Effects

yijk = m + ai + bj + (ab)ij + eijk

Step 3c

expected mean squares balanced experimental designs3
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

two factors fixed effects4
Two Factors, Fixed Effects

yijk = m + ai + bj + (ab)ij + eijk

Step 4a

two factors fixed effects5
Two Factors, Fixed Effects

yijk = m + ai + bj + (ab)ij + eijk

Step 4b

expected mean squares balanced experimental designs4
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.

two factors fixed effects6
Two Factors, Fixed Effects

yijk = m + ai + bj + (ab)ij + eijk

Step 5

expected mean squares balanced experimental designs5
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.

two factors fixed effects7
Two Factors, Fixed Effects

yijk = m + ai + bj + (ab)ij + eijk

Step 6a

two factors fixed effects8
Two Factors, Fixed Effects

yijk = m + ai + bj + (ab)ij + eijk

Step 6b: MSAB

two factors fixed effects9
Two Factors, Fixed Effects

yijk = m + ai + bj + (ab)ij + eijk

Step 6c

two factors fixed effects10
Two Factors, Fixed Effects

yijk = m + ai + bj + (ab)ij + eijk

Step 6b: MSB

two factors fixed effects11
Two Factors, Fixed Effects

yijk = m + ai + bj + (ab)ij + eijk

Step 6c

two factors fixed effects12
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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