Download - Experimental Statistics - week 9
1
Experimental StatisticsExperimental Statistics - week 9 - week 9Experimental StatisticsExperimental Statistics - week 9 - week 9
Chapter 17:
Models with Random Effects
Chapter 18:
Repeated Measures
2
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
3
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
5
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
7
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
8
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
9
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
10
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
11
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
12
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
13
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
14
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
15
Mixed-Effects Example Results and Conclusions:
16
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
17
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
18
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
19
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
20
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
21
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
22
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
23
24
Results:
25
Residual Diagnostics – 1-factor Repeated Measures Data
26
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
27
( )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
28
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;
29
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
30
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
31
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
32
33
Diagnostic Plots for 2-Factor Repeated Measures Data