1

Experimental StatisticsExperimental Statistics - week 8 - week 8Experimental StatisticsExperimental Statistics - week 8 - week 8

Chapter 17:

Mixed Models

Chapter 18:

Repeated Measures

2

2-Factor Random Effects Model

ijk i j ij ijky

Assumptions:

2(0,2. ) i N :2-- normally distributed with mean 0 and variance

6. 's, 's, 's and are independent rv'si j ij ij

1. is overall mean

2(0,3. ) j N :2(0,4. ) ij N :

2(0,5. ) ijk N :

Sum-of-Squares obtained as in Fixed-Effects case

3

Expected Mean Squares for

2-Factor ANOVA with Random EffectsRandom Effects:

A

B

AB

Error

2 2 2n bn

2

Expected MS

2 2 2n an

2 2n

4

To Test:2

0

2

: 0

: 0a

H

H

we use F = MSA/MSAB

20

2

: 0

: 0a

H

H

we use F = MSB/MSAB

we use F = MSAB/MSE

20

2

: 0

: 0a

H

H

Note: Test each of these 3 hypotheses (no matter

whether Ho: is rejected)

5

2-Factor Random Effects ANOVA Table

 

 

Source SS df MS F 

Main Effects

A SSA a 1  

B SSB b1

Interaction

AB SSAB (a 1)(b1)

Error SSE ab(n 1)   Total TSS abn  

/( 1)MSB SSB b

/ ( 1)MSE SSE ab n

/MSA MSAB

/( 1)( 1)MSAB SSAB a b

/MSB MSAB

/( 1)MSA SSA a

/MSAB MSE

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Estimating Variance Components2-Factor Random Effects Model

2ˆMSAB MSE

n

2ˆ MSE

2ˆMSA MSAB

bn

2ˆMSB MSAB

an

(note error on page 986)

7

2ˆ

8

DATA one;INPUT operator filter loss;DATALINES;1 1 16.21 1 16.81 1 17.11 2 16.61 2 16.91 2 16.8 . . .4 1 14.94 2 15.44 2 14.64 2 15.94 3 16.14 3 15.44 3 15.6;PROC GLM; CLASS operator filter; MODEL loss=operator filter operator*filter; TITLE ‘2-Factor Random Effects Model'; RANDOM operator filter operator*filter/test;RUN;

1 2 3 4 16.2 15.9 15.6 14.91 16.8 15.1 15.9 15.2 17.1 14.5 16.1 14.9

16.6 16.0 16.1 15.42 16.9 16.3 16.0 14.6 16.8 16.5 17.2 15.9

16.7 16.5 16.4 16.13 16.9 16.9 17.4 15.4 17.1 16.8 16.9 15.6

Operator

Filter

Filtration Process:Response - % material lost through filtrationA – Operator (randomly selected) (a = )B – Filter (randomly selected) (b = )

n =

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2-Factor Random Effects Model General Linear Models ProcedureDependent Variable: LOSS Sum of MeanSource DF Squares Square F Value Pr > FModel 11 16.60888889 1.50989899 8.16 0.0001Error 24 4.44000000 0.18500000Corrected Total 35 21.04888889

R-Square C.V. Root MSE LOSS Mean 0.789062 2.664175 0.4301163 16.144444

Source DF Type III SS Mean Square F Value Pr > FOPERATOR 3 10.31777778 3.43925926 18.59 0.0001FILTER 2 4.63388889 2.31694444 12.52 0.0002OPERATOR*FILTER 6 1.65722222 0.27620370 1.49 0.2229 Source Type III Expected Mean SquareOPERATOR Var(Error) + 3 Var(OPERATOR*FILTER) + 9 Var(OPERATOR)FILTER Var(Error) + 3 Var(OPERATOR*FILTER) + 12 Var(FILTER)OPERATOR*FILTER Var(Error) + 3 Var(OPERATOR*FILTER)

SAS Random-Effects Output(Filtration Data)

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Tests of Hypotheses for Random Model Analysis of VarianceDependent Variable: LOSS Source: OPERATORError: MS(OPERATOR*FILTER) Denominator Denominator DF Type III MS DF MS F Value Pr > F 3 3.4392592593 6 0.2762037037 12.4519 0.0055Source: FILTERError: MS(OPERATOR*FILTER) Denominator Denominator DF Type III MS DF MS F Value Pr > F 2 2.3169444444 6 0.2762037037 8.3885 0.0183Source: OPERATOR*FILTERError: MS(Error) Denominator Denominator DF Type III MS DF MS F Value Pr > F 6 0.2762037037 24 0.185 1.4930 0.2229

SAS Random-Effects Output – continued

“../test” option

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Filtration Problem Results and Conclusions

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Total Variability – 1-factor ModelBased on the 1-factor random effects model,

2 2 2Y

ij i ijy

it follows that the total variability in Y can be expressed as

As a result, an estimate of the proportion of variability explained by the random factor A can be estimated using

2

2 2

ˆ

ˆ ˆ

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For the operator data example (1-factor random effects example from Thursday’s lecture)

2

2 2

ˆ 28.91 28.91.78

28.91 8.32 37.23ˆ ˆ

i.e. 78% of the variability in Y is explained by the operator to operator variability

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Total Variability – 2 factor model

2 2 2 2 2Y

For better appreciation of the role of the individual factors, it is helpful to express each variance as the proportion of total variability it explains. The proportion of variability explained by

In 2-factor random effects model, we expressed the total variability in Y as

Factor A =

Factor B =

2

2 2 2 2

ˆ

ˆ ˆ ˆ ˆ

2

2 2 2 2

ˆ

ˆ ˆ ˆ ˆ

2

2 2 2 2

ˆ

ˆ ˆ ˆ ˆ

Factor AB =

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2-Factor Mixed Effects Model

ijk i j ij ijky

Assumptions:

1

02. a

ii

6. , 's and 's are independent rv'sj ij ijk

1. is overall mean

2(0,3. ) j N :

2(0,4. ) ij N :

2(0,5. ) ijk N :

Sum-of-Squares obtained as before

fixed random

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Expected Mean Squares for

2-Factor ANOVA with Mixed Effects Effects:

A

B

AB

Error

2 2 2

11

a

ii

nbn

a

2

SAS Expected MS

2 2 2n an

2 2n

(fixed)

(random)

Book’s Expected MS

2 2 2

11

a

ii

nbn

a

2 2an

2 2n

2

17

To Test:

0 1 2: 0

: 0 at least one a

a i

H

H

use F =

20

2

: 0

: 0a

H

H

SAS uses F =

use F =

20

2

: 0

: 0a

H

H

Mixed-Effects Model

Again: Test each of these 3 hypotheses as in random-effects case.

MSAMSAB

MSBMSAB

MSABMSE

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2-Factor Mixed-Effects ANOVA Table(using SAS Expected MS)

 

 

Source SS df MS F 

Main Effects

A SSA a 1  

B SSB b1

Interaction

AB SSAB (a 1)(b1)

Error SSE ab(n 1)   Total TSS abn  

/( 1)MSB SSB b

/ ( 1)MSE SSE ab n

/MSA MSAB

/( 1)( 1)MSAB SSAB a b

/MSB MSAB

/( 1)MSA SSA a

/MSAB MSE

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Estimating Variance Components2-Factor Mixed-Effects Model

2ˆMSAB MSE

n

2ˆ MSE

2ˆMSB MSAB

an (based on SAS Expected MS)

Note: A is a fixed effect

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7.50 7.08 6.15 7.42 6.17 5.52

1 5.85 5.65 5.48 5.89 5.30 5.48 5.35 5.02 5.98

7.58 7.68 6.17 6.52 5.86 6.20

2 6.54 5.28 5.44 5.64 5.38 5.75 5.12 4.87 5.68

7.70 7.19 6.21 6.82 6.19 5.66

3 6.42 5.85 5.36 5.39 5.35 5.90 5.35 5.01 6.12

(F)ullMilitaryInspect.

(R)educedMilitaryInspect.

(C)ommercial

Inspector

Response – fatigue of mechanical part

A – type of inspection (a = )

B – inspector (randomly selected) (b = )

n =

Product Inspection

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DATA one;INPUT insp$ level$ fatigue;DATALINES;1 F 7.50 1 F 7.42 1 F 5.85 1 F 5.89 . . .2 C 5.683 C 6.213 C 5.663 C 5.363 C 5.903 C 6.12; PROC GLM; CLASS insp level; MODEL fatigue= level insp level*insp; TITLE 'Mixed-Effects Model';

RANDOM insp level*insp/test; RUN;PROC MEANS mean var; CLASS level; VAR fatigue;RUN;

Mixed-Effects Data

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Mixed-Effects Model The GLM ProcedureDependent Variable: fatigue Sum of Source DF Squares Mean Square F Value Pr > F Model 8 2.70711111 0.33838889 0.53 0.8282 Error 36 23.11448000 0.64206889 Corrected Total 44 25.82159111

R-Square Coeff Var Root MSE fatigue Mean 0.104839 13.35141 0.801292 6.001556

Source DF Type III SS Mean Square F Value Pr > F level 2 2.58739111 1.29369556 2.01 0.1481 insp 2 0.02523111 0.01261556 0.02 0.9806 insp*level 4 0.09448889 0.02362222 0.04 0.9973

SAS Mixed-Effects Output

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Mixed-Effects Model The GLM Procedure Source Type III Expected Mean Square level Var(Error) + 5 Var(insp*level) + Q(level) insp Var(Error) + 5 Var(insp*level) + 15 Var(insp) insp*level Var(Error) + 5 Var(insp*level)

Mixed-Effects Model The GLM Procedure Tests of Hypotheses for Mixed Model Analysis of Variance Dependent Variable: fatigue Source DF Type III SS Mean Square F Value Pr > F level 2 2.587391 1.293696 54.77 0.0012 insp 2 0.025231 0.012616 0.53 0.6229

Error 4 0.094489 0.023622 Error: MS(insp*level)

Source DF Type III SS Mean Square F Value Pr > F insp*level 4 0.094489 0.023622 0.04 0.9973

Error: MS(Error) 36 23.114480 0.642069

SAS Mixed-Effects Output - Continued

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Multiple Comparisons for Fixed Effect (Inspection Level)

-- Use MSAB in place of MSE

1 2(y y2 marginal means and ) are declared

to be significantly different (using LSD) if

1 2 22

( ) | | α/MSAB

y y tN

where ▪ N denotes the # of observations involved in the computation of a marginal mean ▪ v denotes the df associated with AB interaction

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The MEANS Procedure Analysis Variable : fatigue N level Obs Mean Variance ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ C 15 5.8066667 0.0981810 F 15 6.3393333 0.8208638 R 15 5.8586667 0.7405410 ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ

SAS Mixed-Effects Output –

Output from PROC Means

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Mixed-Effects Example Results and Conclusions:

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Repeated Measures Designs

Setting:1. Random sample of “subjects”

2. Each subject is measured at t different time points

3. Interested in the effect of treatment over time

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Repeated Measures with a Single Factor

Time

11 21 1

12 22 2

1 2

1 2

1

2

...

...

...

t

t

n n tn

t

y y y

y y y

n y y y

Subject

:ijy

ith time period

jth subject

Reading for

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Single Factor Repeated Measures Designs

• single factor repeated measures model is similar to the randomized complete block model - i.e. 2 factors (subject and time) with one observation cell - since there is only one observation per cell, we cannot estimate an interaction term

• typically: - subject is a random effect - time is a fixed effect

ij i j ijy

time subject

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ANOVA Table for Repeated Measure Design with Single Factor

Source SS df MS EMS F 

Between subjects SSP n 1 MSP MSP/MSE  Time SSA a 1 MSA MSA/MSE

Error SSE (n 1)(a 1) MSE

Total TSS an  

2 2

11

t

ii

na

2 2a

2

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Data – 5 subjects take tablet

-- blood samples taken .5, 1, 2, 3, and 4 hours after ingestion

Goal: understand rate at which medicine enters blood

Time

Subject .5 1 2 3 4

1 50 75 120 60 30

2 40 80 135 70 40

3 55 75 125 85 50

4 70 85 140 90 40

5 60 90 150 95 50

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Dependent Variable: conc Sum of Source DF Squares Mean Square F Value Pr > F Model 8 26442.00000 3305.25000 66.60 <.0001 Error 16 794.00000 49.62500 Corrected Total 24 27236.00000

R-Square Coeff Var Root MSE conc Mean 0.970847 8.985333 7.044501 78.40000

Source DF Type III SS Mean Square F Value Pr > F subject 4 1576.00000 394.00000 7.94 0.0010 time 4 24866.00000 6216.50000 125.27 <.000

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The GLM Procedure t Tests (LSD) for conc NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate. Alpha 0.05 Error Degrees of Freedom 16 Error Mean Square 49.625 Critical Value of t 2.11991 Least Significant Difference 9.4449

Means with the same letter are not significantly different.

t Grouping Mean N time

A 134.000 5 2

B 81.000 5 1 B B 80.000 5 3

C 55.000 5 0.5

D 42.000 5 4

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35

Results:

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Residual Diagnostics – 1-factor Repeated Measures Data

37

Two-Factor Repeated Measure Data (p.1033)

Data – 10 subjects (5 take tablet, 5 take capsule)

-- blood samples .5, 1, 2, 3, and 4 hours after ingestion

Goal: compare blood concentration patterns of the two methods of administration

Time

Subject .5 1 2 3 4

1 50 75 120 60 30

2 40 80 135 70 40

3 55 75 125 85 50

4 70 85 140 90 40

5 60 90 150 95 50

Time

Subject .5 1 2 3 4

1 30 55 80 130 65

2 25 50 75 125 60

3 35 65 85 140 85

4 45 70 90 145 80

5 50 75 95 160 90

Tablet Capsule

38

( )ijk i j i k ik ijky

2-Factor with Repeated Measure -- Model

type subject within type

timetype by time interaction

NOTE: type and time are both fixed effects in the current example

39

PROC GLM; CLASS type subject time; MODEL conc=type subject(type) time type*time; TITLE 'Repeated Measures – 2 factors'; OUTPUT out=new r=resid; MEANS type time/LSD; RANDOM subject(type)/test;

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The GLM ProcedureDependent Variable: conc Sum ofSource DF Squares Mean Square F Value Pr > FModel 17 57720.50000 3395.32353 110.87 <.0001Error 32 980.00000 30.62500Corrected Total 49 58700.50000

R-Square Coeff Var Root MSE conc Mean 0.983305 6.978545 5.533986 79.30000

Source DF Type III SS Mean Square F Value Pr > F

type 1 40.50000 40.50000 1.32 0.2587subject(type) 8 3920.00000 490.00000 16.00 <.0001time 4 34288.00000 8572.00000 279.90 <.0001type*time 4 19472.00000 4868.00000 158.96 <.0001

2-Factor Repeated Measures – ANOVA Output

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2-factor Repeated Measures

Source Type III Expected Mean Square

type Var(Error) + 5 Var(subject(type)) + Q(type,type*time) subject(type) Var(Error) + 5 Var(subject(type)) time Var(Error) + Q(time,type*time) type*time Var(Error) + Q(type*time)

The GLM Procedure Tests of Hypotheses for Mixed Model Analysis of Variance

Dependent Variable: conc Source DF Type III SS Mean Square F Value Pr > F* type 1 40.500000 40.500000 0.08 0.7810 Error 8 3920.000000 490.000000Error: MS(subject(type))* This test assumes one or more other fixed effects are zero. Source DF Type III SS Mean Square F Value Pr > F subject(type) 8 3920.000000 490.000000 16.00 <.0001* time 4 34288 8572.000000 279.90 <.0001 type*time 4 19472 4868.000000 158.96 <.0001 Error: MS(Error) 32 980.000000 30.625000

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NOTE: Since time x type interaction is significant, and since these are fixed effects we DO NOT test main effects

– we compare cell means (using MSE)

.0252 2(30.6250)

(32) 2.042 7.1475 5

MSELSD t

.5 1 2 3 4 C 37 63 85 140 76 T 55 81 134 80 42

Cell Means

43

44

Diagnostic Plots for 2-Factor Repeated Measures Data