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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

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 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 (Effects)

One-Factor Model

yij = m + ai + eij

Main Effects

Random ai

Variability =

Estimate Variance

Components

sa2 , s2

Skin Swelling Measurements

Factors

Laboratory animals (Random)

Location of the measurement: Back, Ear (Fixed)

Repeat measurements (2 / location)

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

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)

- 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

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

s

0

Two-Factor Random Effects Model: Main Effects OnlyTwo-Factor Main Effects Model

yijk = m + ai + bj + eijk i = 1, ..., a

j = 1, ..., b

s

0

Two-Factor Random Effects ModelTwo-Factor Model

yijk = m + ai + bj + (ab)ij + eijk i = 1, ..., a

j = 1, ..., b

k = 1, ..., r

mij = m

Variance Components

Two-Factor Model DifferencesFixed Effects

Mean

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

Change the Mean

Variance

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 Squares

One Factor, Fixed Effects

yij = m + ai + eij

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

eij ~ NID(0,se2)

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 Squares

One Factor, Random Effects

yij = m + ai + eij

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

ai ~ NID(0,sa2) , eij ~ NID(0,se2)

Independent

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 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 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 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 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

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

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

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

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.

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 Effects

Under appropriate null hypotheses,

E(MS) for A, B, and AB same as E(MSE)

F = MS / MSE

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