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Experimental StatisticsExperimental Statistics - week 9 - week 9Experimental StatisticsExperimental Statistics - week 9 - week 9

Chapter 17:

Models with Random Effects

Chapter 18:

Repeated Measures

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Discussion of CommentsDiscussion of CommentsDiscussion of CommentsDiscussion of Comments

• upset about HW grade– I will drop one HW

• availability of slides

• HW - do by hand

• in-class examples

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4

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

6

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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Results:

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Residual Diagnostics – 1-factor Repeated Measures Data

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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

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( )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

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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

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Diagnostic Plots for 2-Factor Repeated Measures Data