1 experimental design and statistical analyses of data lesson 5: mixed models nested anovas...
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1
Experimental design and statistical analyses of data
Lesson 5:Mixed modelsNested anovas
Split-plot designs
2
Randomized block design
• All treatments are allocated to the same experimental units
• Treatments are allocated at random
B C B
A B D
D A A
C D C
Blocks (b = 3)
Treatments (a = 4)
3
Dy1
Treatments
Patient
A B C D Average
1
2
3
Average
Cy1
Ay2
Ay3
Cy2By2 Dy2
By3 Cy3 Dy3
Ay By
1y
2y
3y
Cy Dy y
55443322110 xxxxxy
Blocks (patients) Treatments (drugs)
Ay1 By1
4
An alternative way of writing a GLM
jiijy
Response of patient j receiving drug i
Overall mean Effect of drug i
Effect of patient j
Residual
αi = μi - μ
βj = μj - μ
5
jiijy ˆˆˆˆ
Predicted value of y
αi = μi - μ
βj = μj - μ
yyyii
yy jj Response of patient j receiving drug i
6
Treatments
Patient
A B C D
1 5.17 5.21 4.91 4.74 5.008
2 6.23 7.34 6.18 6.31 6.515
3 4.93 4.55 4.64 4.61 4.683
5.443 5.700 5.243 5.220 5.402iy
jy
402.5ˆ y042.0402.5443.5ˆ yyAA298.0402.5700.5ˆ yyBB
158.0402.5243.5ˆ yyCC182.0402.5220.5ˆ yyDD
0ˆ i
i
7
Treatments
Patient
A B C D
1 5.17 5.21 4.91 4.74 5.008
2 6.23 7.34 6.18 6.31 6.515
3 4.93 4.55 4.64 4.61 4.683
5.443 5.700 5.243 5.220 5.402iy
jy
402.5ˆ y
394.0402.5008.5ˆ11 yy
113.1402.5515.6ˆ22 yy
719.0402.5683.4ˆ33 yy
0ˆ j
j
8
402.5ˆ y042.0402.5443.5ˆ yyAA298.0402.5700.5ˆ yyBB
158.0402.5243.5ˆ yyCC182.0402.5220.5ˆ yyDD
394.0402.5008.5ˆ11 yy
113.1402.5515.6ˆ22 yy
719.0402.5683.4ˆ33 yy
Effects of drugs
Effects of patients
Ex: Patient 2 receiving treatment C:
357.6113.1158.0402.5ˆˆˆ 22 CC yy
9
Consider the two questions:
• Are the three patients different?
• Are patients in general different?
• In the first case, ”patients” is considered as a fixed factor
• In the second case, ”patients” is considered as a random factor
10
”Patients” is a random effect:
jiy
βj is assumed to be iid ND(0,σb2)
0
Value of
Pro
babi
lity
of
β
If patients are randomly chosen, βj will be a stochastic variable
i.e. independently and identically normally distributed with zero mean and variance σ²b
11
V(y) = V(μ + αi + βj + ε) = V(μ)+ V(αi)+ V( βj)+ V(ε)
= σa2 + σb
2 + σ2
Variances
Variance due to drug (factor a)Variance due to patient (factor b)
Residual variance
12
Both factors are fixed
V(y) = V(μ + αi + βj + ε) = V(μ)+ V(αi)+ V( βj)+ V(ε)
= σa2 + σb
2 + σ2
V(y) = σ2
banyVyV
2
/)()(
Variance of a single observation:
Variance of an average:
13
”Patients” is a random factor (mixed anova)
V(y) = V(μ + αi + βj + ε) = V(μ)+ V(αi)+ V( βj)+ V(ε)
= σa2 + σb
2 + σ2
V(y) = σb2 + σ2 Variance of a single observation:
Variance of an average: baabab
yV bb
/)( 22
22
14
Both factors are random
V(y) = V(μ + αi + βj + ε) = V(μ)+ V(αi)+ V( βj)+ V(ε)
= σa2 + σb
2 + σ2
V(y) = σa2 +σb
2 + σ2 Variance of a single observation:
Variance of an average:
baabbaba
yV baba
/)( 222
222
15
Source df MS E[MS] F
Drugs
Patients
Error
a-1
b-1
(a-1)(b-1)
MSa
MSb
MSe
Total ab-1
Expected Means Squares
16
Expected Mean Squares
E[MSa] = bσa2 + σ2
E[MSb] = aσb2 + σ2
E[MSe] = σ2
df = a-1
df = b-1
df = (a-1)(b-1)
H0: αA = αB = αC = αD = 0 → σa2 = 0 →
e
a
MS
MSF
H0: β1 = β2 = β3 = 0 → σb2 = 0 →
e
b
MS
MSF
17
Source df MS E[MS] F
Drugs
Patients
Error
a-1
b-1
(a-1)(b-1)
MSa
MSb
MSe
bσa2 + σ2
aσb2 + σ2
σ2
MSa/Mse
MSb/MSe
Total ab-1
18
Source df MS E[MS] F
Drugs
Patients
Error
3
2
6
0.149
3.824
0.117
bσa2 + σ2
aσb2 + σ2
σ2
MSa/Mse
MSb/MSe
Total 11
19
Hvis ”patients” is a random factor, σb2 is estimated from
E[MSb] = aσb2 + σ2 →
a
MSMS
a
sMSs ebb
b
22
927.04
117.0824.32
bs
V(y) = σb2 + σ2 = 0.927+0.117 = 1.044Variance of a single observation:
Variance of the average:
0.3190.0100.30912
117.0
3
927.0)(
22
bab
yV b
20
How to do it with SAS
21
DATA eks5_1;
INPUT pat $ treat $ y; /* indlæser data */
CARDS; /* her kommer data. Kan også indlæses fra en fil */
1 A 5.17
2 A 6.23
3 A 4.93
1 B 5.21
2 B 7.34
3 B 4.55
1 C 4.91
2 C 6.18
3 C 4.64
1 D 4.74
2 D 6.31
3 D 4.61
;
PROC GLM; /* procedure General Linear Models */
TITLE 'Eksempel 5.1'; /* medtages hvis der ønskes en titel */
CLASS pat treat; /* pat og treat er klasse (kvalitative) variable */
MODEL y = pat treat;
RANDOM pat; /* Patienter er en tilfældig faktor */
RUN;
22
Eksempel 5.1 8 13:18 Monday, November 5, 2001 General Linear Models Procedure Dependent Variable: Y Source DF Sum of Squares Mean Square F Value Pr > F Model 5 8.09475000 1.61895000 13.80 0.0031 Error 6 0.70401667 0.11733611 Corrected Total 11 8.79876667 R-Square C.V. Root MSE Y Mean 0.919987 6.341443 0.34254359 5.40166667 Source DF Type I SS Mean Square F Value Pr > F PAT 2 7.64831667 3.82415833 32.59 0.0006TREAT 3 0.44643333 0.14881111 1.27 0.3666 Source DF Type III SS Mean Square F Value Pr > F PAT 2 7.64831667 3.82415833 32.59 0.0006TREAT 3 0.44643333 0.14881111 1.27 0.3666
MSb
MSa
MSe
23
Eksempel 5.1 18
09:00 Friday, November 16, 2001
General Linear Models Procedure
Source Type III Expected Mean Square
PAT Var(Error) + 4 Var(PAT)
TREAT Var(Error) + Q(TREAT)
24
Nested designs
25
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
1 2 3 1 2 3 1 2 3 1 2 3
A B C DFactor A (drug)
Factor B (patient)
Replicate
Model: kijijjiijky )()(
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
A B C DFactor A (drug)
Factor B (patient)
Replicate
Model: kijjiiijky )()(
1 2 3 1 2 3 1 2 3 1 2 3
Patient j is the same for all drugs
Patient j is not the same for all drugs
Patients are said to be nested within drugs
Replicates can also be regarded as nestedwithin drugs and patients
26
Rules for finding the EMS(after Dunn and Clark)
1. For each effect, write down every possible variance component containing every letter of the effect name. For example, in a two way design with r replicates per cell, the EMS for factor A includes σa
2, σab2 and σ(ab)e
2, but not σb2
2. For any nested factor add in parentheses to the effect name the name(s) of the factor within it is nested e.g if B is nested in A, σ(a)b
2 is the variance of β(i)j.
3. For the coefficient of each variance component, use all letters not in the subscripts of the variance component
4. For each variance component, look at any subscripts outside parentheses that are not in the effect name; if any of these letters corresponds to a fixed effect, omit that variance component
27
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
1 2 3 1 2 3 1 2 3 1 2 3
A B C D
Two-way anova (A and B fixed)Factor A (drug)
Factor B (patient)
Replicate
Model: kijijjiijky )()(
Interaction between drug and patient
Residual of the kth replicate nested within drug i and patient j
28
Model: kijijjiijky )()(
0i
i 0j
j i j
ij 0)(
(1) For each effect, write down every possible variance component containing every letter of the effect name. For example, in a two way design with r replicates per cell, the EMS for factor A includes σa
2, σab2 and σ(ab)e
2, but not σb2
σa2 + σab
2 + σ(ab)e 2Factor A:
σb2 + σab
2 + σ(ab)e 2Factor B:
σab2 + σ(ab)e 2Factor AB:
σ(ab)e 2Residual:
29
Model: kijijjiijky )()(
0i
i 0j
j i j
ij 0)(
σa2 + σab
2 + σ(ab)e 2
σb2 + σab
2 + σ(ab)e 2
σab2 + σ(ab)e 2
Factor A:
Factor B:
Factor AB:
(2) For any nested factor add in parentheses to the effect name the name(s) of the factor within it is nested e.g if B is nested in A, σ(a)b
2 is the variance of β(i)j.
σ(ab)e 2Residual:
30
Model: kijijjiijky )()(
0i
i 0j
j i j
ij 0)(
brσa2 + rσab
2 + σ(ab)e 2Factor A:
arσb2 + rσab
2 + σ(ab)e 2Factor B:
rσab2 + σ(ab)e 2Factor AB:
(3) For the coefficient of each variance component, use all letters not in the subscripts of the variance component
σ(ab)e 2Residual:
31
Model: kijijjiijky )()(
0i
i 0j
j i j
ij 0)(
brσa2 + rσab
2 + σ(ab)e 2
arσb2 + rσab
2 + σ(ab)e 2
rσab2 + σ(ab)e 2
Factor A:
Factor B:
Factor AB:
(4) For each variance component, look at any subscripts outside parentheses that are not in the effect name; if any of these letters corresponds to a fixed effect, omit that variance component
σ(ab)e 2Residual:
32
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
1 2 3 1 2 3 1 2 3 1 2 3
A B C D
Two-way anova (A and B fixed)Factor A (drug)
Factor B (patient)
Replicate
Model: kijijjiijky )()(
0i
i 0j
j i j
ij 0)(
2)(
222)ˆ( rababbayV abryV 2)(
Source df MS E[MS] F
A
B
AB
Error
a-1
b-1
(a-1)(b-1)
ab(r-1)
MSa
MSb
MSab
MSe
brσa2 +r σab
2+ σ2
arσb2 + r σab
2+ σ2
r σab2+ σ2
σ2
MSa/MSe
MSb/MSe
MSab/MSe
33
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
1 2 3 1 2 3 1 2 3 1 2 3
A B C D
Two-way anova (A fixed, B random)Factor A (drug)
Factor B (patient)
Replicate
Model: kijijjiijky )()(
0i
i
2)(
222)ˆ( rababbayV
Source df MS E[MS] F
A
B
AB
Error
a-1
b-1
(a-1)(b-1)
ab(r-1)
MSa
MSb
MSab
MSe
brσa2 +r σab
2+ σ2
arσb2 + r σab
2+ σ2
r σab2+ σ2
σ2
MSa/MSab
MSb/MSe
MSab/MSe
i
ij 0)( βj is ND(0, σb2) (αβ)ij is ND(0; σab
2(1-1/a))
abrarabrb
yV bb /)( 22
22
NB!
34
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
1 2 3 1 2 3 1 2 3 1 2 3
A B C D
Two-way anova (A and B random)Factor A:
Factor B:
Replicate
Model: kijijjiijky )()(
2)(
222)ˆ( rababbayV
Source df MS E[MS] F
A
B
AB
Error
a-1
b-1
(a-1)(b-1)
ab(r-1)
MSa
MSb
MSab
MSe
brσa2 +r σab
2+ σ2
arσb2 + r σab
2+ σ2
r σab2+ σ2
σ2
MSa/MSab
MSb/MSab
MSab/MSe
βi is ND(0, σb2) (αβ)ij is ND(0; σab
2)αi is ND(0, σa2)
abrrarbryV abba /)( 2222
35
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
A B C D
Nested anova (A fixed, B random)Factor A (drug)
Factor B (patient)
Replicate
Model: kijjiiijky )()(
2)(
2)(
2)ˆ( rabbaayV
Source df MS E[MS] F
A
B(A)
Error
a-1
a(b-1)
ab(r-1)
MSa
MS(a)b
MSe
brσa2 +r σ(a)b
2+ σ2
rσ(a)b2 + σ2
σ2
MSa/MS(a)b
MS(a)b/MSe
MSe
β(i)j is ND(0, σ(a)b2)
1 2 3 1 2 3 1 2 3 1 2 3
0i
i
abrrabrab
yV baba /)( 22
)(
22)(
36
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
A B C D
Nested anova (A and B random)Factor A (doctor)
Factor B (patient)
Replicate
Model: kijjiiijky )()(
2)(
2)(
2)ˆ( rabbaayV
Source df MS E[MS] F
A
B(A)
Error
a-1
a(b-1)
ab(r-1)
MSa
MS(a)b
MSe
brσa2 +r σ(a)b
2+ σ2
rσ(a)b2 + σ2
σ2
MSa/MS(a)b
MS(a)b/MSe
MSe
β(i)j is ND(0, σ(a)b2)
abrrbrabraba
yV baabaa /)( 22
)(2
22)(
2
αi is ND(0, σa2)
1 2 3 1 2 3 1 2 3 1 2 3
37
40% 20% 0% Four level nested anova
Tree (b = 2 )
Replicate (r = 2)
Model: rijkkijjiiijky )()()(
2)(
2)(
2)(
2)ˆ( rabccabbaayV
β(i)j is ND(0, σ(a)b2)
abcrrcrabcrabcab
yV cabbacabba /)( 22
)(2
)(
22)(
2)(
Leaf (c = 3 )
1 2 1 2 1 2
1
1 2 3
1 2 1 2 1 2
2
1 2 3
1 2 1 2 1 2
1
1 2 3
1 2 1 2 1 2
2
1 2 3
1 2 1 2 1 2
1
1 2 3
1 2 1 2 1 2
2
1 2 3
Treatment (a = 3)
γ(ij)k is ND(0, σ(ab)c2)0
ii
38
Sourcec df MS E[MS] F
Treatments
Trees
Leaves
Error
a-1
a(b-1)
ab(c-1)
abc(r-1)
MSa
MS(a)b
MS(ab)c
MSe
bcrσa2 +cr σ(a)b
2+ r σ(ab)c2 +σ2
cr σ(a)b2+ r σ(ab)c
2 +σ2 r σ(ab)c
2 +σ2
σ2
MSa/MS(a)b
MS(a)b/MS(ab)c
MS(ab)c/MSe
MSe
MS(ab)c = rs(ab)c2 + s2 →
r
MSMSs ecab
cab
)(2
)(
MS(a)b = cr s(a)b2+ r s(ab)c
2 +s2 = cr s(a)b2 + MS(ab)c
→cr
MSMSs cabba
ba)()(2
)(
MSa = bcrsa2 +cr s(a)b
2+ r s(ab)c2 +s2 = bcrsa
2 +MS(a)b →bcr
MSMSs baa
a)(2
2
)(2
)(2
)(2)ˆ( rabccabbaayV
22)(
2)(
2)ˆ( ssssyV cabbaa
39
How do it with SAS
40
PROC GLM;
CLASS treat tree leaf disc;
MODEL Nitro = treat tree(treat) leaf(tree treat);
/* treatment is a fixed factor, while trees and leaves are random */
RANDOM tree(treat) leaf(tree treat);
/* gives the expected means squares */
RUN;
DATA nested; /* Nested anova (eks 6-4 in the lecture notes) */
INFILE 'H:\lin-mod\eks6x.prn' firstobs =2 ;
INPUT treat $ tree $ leaf $ disc $ Nitro ;
41
General Linear Models Procedure
Dependent Variable: NITRO
Source DF Sum of Squares Mean Square F Value Pr > F
Model 17 134.04000000 7.88470588 8.00 0.0001
Error 18 17.75000000 0.98611111
Corrected Total 35 151.79000000
R-Square C.V. Root MSE NITRO Mean
0.883062 3.271932 0.99303127 30.35000000
Source DF Type I SS Mean Square F Value Pr > F
TREAT 2 71.78000000 35.89000000 36.40 0.0001
TREE(TREAT) 3 36.04666667 12.01555556 12.18 0.0001
LEAF(TREAT*TREE) 12 26.21333333 2.18444444 2.22 0.0618
Source DF Type III SS Mean Square F Value Pr > F
TREAT 2 71.78000000 35.89000000 36.40 0.0001
TREE(TREAT) 3 36.04666667 12.01555556 12.18 0.0001
LEAF(TREAT*TREE) 12 26.21333333 2.18444444 2.22 0.0618
NB!These values are based on MSe as the error term, which is wrong!
42
PROC GLM;
CLASS treat tree leaf disc;
MODEL Nitro = treat tree(treat) leaf(tree treat);
/* treatment is a fixed factor, while trees and leaves are random */
RANDOM tree(treat) leaf(tree treat);
/* gives the expected means squares */
RUN;
DATA nested; /* Nested anova (eks 6-4 in the lecture notes) */
INFILE 'H:\lin-mod\eks6x.prn' firstobs =2 ;
INPUT treat $ tree $ leaf $ disc $ Nitro ;
43
General Linear Models Procedure
Source Type III Expected Mean Square
TREAT Var(Error) + 2 Var(LEAF(TREAT*TREE)) + 6 Var(TREE(TREAT))
+ Q(TREAT)
TREE(TREAT) Var(Error) + 2 Var(LEAF(TREAT*TREE)) + 6 Var(TREE(TREAT))
LEAF(TREAT*TREE) Var(Error) + 2 Var(LEAF(TREAT*TREE))
44
PROC GLM;
CLASS treat tree leaf disc;
MODEL Nitro = treat tree(treat) leaf(tree treat);
/* treatment is a fixed factor, while trees and leaves are random */
RANDOM tree(treat) leaf(tree treat);
/* gives the expected means squares */
TEST h=treat e= tree(treat);
/* tests for the difference between treatments with MS for tree(treat) as denominator */
TEST h= tree(treat) e=leaf(tree treat);
/* tests for the difference between trees with MS for leaf(tree treat) as denominator*/
45
General Linear Models Procedure
Dependent Variable: NITRO
Tests of Hypotheses using the Type III MS for TREE(TREAT) as an error term
Source DF Type III SS Mean Square F Value Pr > F
TREAT 2 71.78000000 35.89000000 2.99 0.1933
Tests of Hypotheses using the Type III MS for LEAF(TREAT*TREE) as an error term
Source DF Type III SS Mean Square F Value Pr > F
TREE(TREAT) 3 36.04666667 12.01555556 5.50 0.0130
46
PROC GLM;
CLASS treat tree leaf disc;
MODEL Nitro = treat tree(treat) leaf(tree treat);
/* treatment is a fixed factor, while trees and leaves are random */
RANDOM tree(treat) leaf(tree treat);
/* gives the expected means squares */
TEST h=treat e= tree(treat);
/* tests for the difference between treatments with MS for tree(treat) as denominator */
TEST h= tree(treat) e=leaf(tree treat);
/* tests for the difference between trees with MS for leaf(tree treat) as denominator*/
MEANS treat / Tukey Dunnett('Control') e= tree(treat) cldiff;
/* finds possible significant differences between treatments and the control and the other treatments */
RUN;
47
Tukey's Studentized Range (HSD) Test for variable: NITRO
NOTE: This test controls the type I experimentwise error rate.
Alpha= 0.05 Confidence= 0.95 df= 3 MSE= 12.01556
Critical Value of Studentized Range= 5.910
Minimum Significant Difference= 5.9134
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
TREAT Confidence Between Confidence
Comparison Limit Means Limit
20% - 40% -3.663 2.250 8.163
20% - Control -2.513 3.400 9.313
40% - 20% -8.163 -2.250 3.663
40% - Control -4.763 1.150 7.063
Control - 20% -9.313 -3.400 2.513
Control - 40% -7.063 -1.150 4.763
48
Dunnett's T tests for variable: NITRO
NOTE: This tests controls the type I experimentwise error for
comparisons of all treatments against a control.
Alpha= 0.05 Confidence= 0.95 df= 3 MSE= 12.01556
Critical Value of Dunnett's T= 3.866
Minimum Significant Difference= 5.4714
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
TREAT Confidence Between Confidence
Comparison Limit Means Limit
20% - Control -2.071 3.400 8.871
40% - Control -4.321 1.150 6.621
49
PROC NESTED;
CLASS treat tree leaf;
VAR Nitro;
RUN;
50
Coefficients of Expected Mean Squares
Source TREAT TREE LEAF ERROR
TREAT 12 6 2 1
TREE 0 6 2 1
LEAF 0 0 2 1
ERROR 0 0 0 1
Sourcec df MS E[MS] F
Treatments
Trees
Leaves
Error
a-1
a(b-1)
ab(c-1)
abc(r-1)
MSa
MS(a)b
MS(ab)c
MSe
bcrσa2 +cr σ(a)b
2+ r σ(ab)c2 +σ2
cr σ(a)b2+ r σ(ab)c
2 +σ2 r σ(ab)c
2 +σ2
σ2
MSa/MS(a)b
MS(a)b/MS(ab)c
MS(ab)c/MSe
MSe
51
Nested Random Effects Analysis of Variance for Variable NITRO
Degrees
Variance of Sum of Error
Source Freedom Squares F Value Pr > F Term
TOTAL 35 151.790000
TREAT 2 71.780000 2.987 0.1933 TREE
TREE 3 36.046667 5.501 0.0130 LEAF
LEAF 12 26.213333 2.215 0.0618 ERROR
ERROR 18 17.750000
Variance Variance Percent
Source Mean Square Component of Total
TOTAL 4.336857 5.213333 100.0000
TREAT 35.890000 1.989537 38.1625
TREE 12.015556 1.638519 31.4294
LEAF 2.184444 0.599167 11.4930
ERROR 0.986111 0.986111 18.9152
Mean 30.35000000
Standard error of mean 0.99847105
r
MSMSs ecab
cab
)(2
)(
cr
MSMSs cabba
ba)()(2
)(
bcr
MSMSs baa
a)(2
52
The problem of pseudoreplication
53
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
1 2 3 1 2 3 1 2 3
A B C
Two-way anova (A fixed, B random)
Factor A (drug)
Factor B (patient)
Replicate
18 measurements
If we want to increase the power of the analysis, we may e.g. double thenumber of measurements
But be careful about what you do!
54
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
1 2 3 1 2 3 1 2 3
A B C
1 2
1
1 2
2
1 2
3
1 2
4
1 2
5
1 2
6
A
1 2
1
1 2
2
1 2
3
1 2
4
1 2
5
1 2
6
1 2
1
1 2
2
1 2
3
1 2
4
1 2
5
1 2
6
CB
Design 1
Design 2
Both experiments have 36 measurements
3 experimental units/treatment
6 experimental units/treatment
Pseudoreplicates
Design 2 is best because it uses 6 experimental units/treatment
55
40% 20% 0%
Four level nested anova
Tree (b = 2 )
Replicate (r = 2)
Leaf (c = 3 )
1 2 1 2 1 2
1
1 2 3
1 2 1 2 1 2
2
1 2 3
1 2 1 2 1 2
1
1 2 3
1 2 1 2 1 2
2
1 2 3
1 2 1 2 1 2
1
1 2 3
1 2 1 2 1 2
2
1 2 3
Treatment (a = 3)
Trees are the experimental units(2 replicates/treatment)Pseudoreplicates
56
Split-plot designs
• Three types of fertilizers
• Two types of soil treatment
• Interactions between fertilizers and soil treatment
57
A1
A2
Block 3
A2
A1
Block 1
A2
A1
Block 4
A1
A2
Block 2
2 whole-plots within each block
Soil treatments
58
A1
A2
Block 3
A2
A1
Block 1
A2
A1
Block 4
A1
A2
Block 2
Fertilizer treatments
3 sub- plots within each whole-plot
59
Analysis of whole-plots
Factor df MS E[MS] FSoil treatment (A)
Block (B)
Soil*Block (AB)
Error
a-1 = 1
b-1 =3
(a-1)(b-1) = 3
0
MSa
MSb
MSab
bσa2+σab
2
aσb2+ σab
2
σab2
MSa/MSab
MSb/MSab
Total ab-1 = 7
kijijjiijky )()(
0i
i βj is ND(0, σb2)
Effect of soil treatInteraction betweensoil and block
Effect of block
Interaction term serves as error term
60
Analysis of sub-plots
Factor df MS E[MS] FWhole plots
Fertilizer (C)
Soil*Fertilizer (AC)
Block*Fert. (BC)
Soil*Block*Fert. (ABC)
Error
ab-1 = 7
c-1 = 2
(a-1)( c-1) = 2
(b-1)(c-1) = 6
(a-1)(b-1)(c-1) = 6
0
MSc
MSac
MSbc
MSabc
abσc2+σabc
2
bσac2+ σabc
2
aσbc2+ σabc
2
σabc2
MSc/MSabc
MSac/MSabc
MSbc/MSabc
Total abc-1 = 23
0k
k (βγ)jk is ND(0, σbc2(1-1/c))
jkikkijjiijky )()()(ˆ
0)( i k
ik
Effect of fertilizer
Interaction between soil treatment and fertilizer
Interaction between block and fertilizer
61
Analysis of sub-plots
Factor df MS E[MS] FWhole plots
Fertilizer (C)
Soil*Fertilizer (AC)
Block*Fert. (BC)
Soil*Block*Fert. (ABC)
Error
ab-1 = 7
c-1 =2
(a-1)( c-1) =2
(b-1)(c-1) = 6
(a-1)(b-1)(c-1) = 6
0
MSc
MSac
MSbc
MSabc
abσc2+σ2
bσac2+ σ2
bσbc2+ σ2
σ2
MSc/MSe
MSac/MSe
MSbc/MSe
Total abc-1 = 23
0k
k (βγ)jk is ND(0, σbc2(1-1/c))
jkikkijjiijky )()()(ˆ
0)( i k
ik
62
Analysis of sub-plots
Factor df MS E[MS] FWhole plots
Fertilizer (C)
Soil*Fertilizer (AC)
Block*Fert. (BC)
Soil*Block*Fert. (ABC)
Error
ab-1 = 7
c-1 =2
(a-1)( c-1) =2
(b-1)(c-1) = 6
(a-1)(b-1)(c-1) = 6
0
MSc
MSac
MSbc
MSabc
abσc2+σ2
bσac2+ σ2
σ2
MSc/MSe
MSac/MSe
Total abc-1 = 23
0k
k (βγ)jk is ND(0, σbc2(1-1/c))
jkikkijjiijky )()()(ˆ
0)( i k
ik
63
How do it with SAS
64
DATA SplitPlt;
/* Example 6-8 in the lecture notes */
/* block = block effect (random factor) */
/* soil = effect of soil treatment (whole-plot effect) */
/* fert = effect of fertilizer (subplot effect) */
/* yield = dependent variable */
INFILE 'h:\lin-mod\eks6-8.prn';
INPUT soil $ block $ fert $ yield;
PROC GLM;
TITLE 'Split plot - full model';
CLASS block soil fert;
MODEL yield= block soil block*soil fert soil*fert block*fert ;
RANDOM block; /* declare block as a random effect */
TEST h = soil e = block*soil; /* tests effect of wholeplot */
TEST h = block e = block*soil; /* tests effect of blocks */
RUN;
65
Split plot - full model
The GLM ProcedureDependent Variable: yield Sum of Source DF Squares Mean Square F Value Pr > F Model 17 32796.58333 1929.21078 3.24 0.0764 Error 6 3575.41667 595.90278 Corrected Total 23 36372.00000 R-Square Coeff Var Root MSE yield Mean 0.901699 15.02223 24.41112 162.5000 Source DF Type III SS Mean Square F Value Pr > F block 3 588.33333 196.11111 0.33 0.8050 soil 1 7848.16667 7848.16667 13.17 0.0110 block*soil 3 3740.83333 1246.94444 2.09 0.2027 fert 2 10950.75000 5475.37500 9.19 0.0149 soil*fert 2 462.58333 231.29167 0.39 0.6942 block*fert 6 9205.91667 1534.31944 2.57 0.1373 Tests of Hypotheses Using the Type III MS for block*soil as an Error Term Source DF Type III SS Mean Square F Value Pr > F soil 1 7848.166667 7848.166667 6.29 0.0870 block 3 588.333333 196.111111 0.16 0.9185
Sub-plot effects
NB! These P-values cannot be used!
Instead use these whole-plot results
Whole-plot effects
66
PROC GLM;
TITLE 'Split plot - reduced model
block*fert omitted';
CLASS block soil fert;
MODEL yield= block soil block*soil fert soil*fert;
RANDOM block;
TEST h = soil e = block*soil; /* tests effect of wholeplot */
TEST h = block e = block*soil; /* tests effect of blocks */
RUN;
67
Split plot - reduced modelblock*fert omitted
The GLM Procedure Dependent Variable: yield Sum of Source DF Squares Mean Square F Value Pr > F Model 11 23590.66667 2144.60606 2.01 0.1224 Error 12 12781.33333 1065.11111 Corrected Total 23 36372.00000 R-Square Coeff Var Root MSE yield Mean 0.648594 20.08372 32.63604 162.5000 Source DF Type III SS Mean Square F Value Pr > F block 3 588.33333 196.11111 0.18 0.9051 soil 1 7848.16667 7848.16667 7.37 0.0188 block*soil 3 3740.83333 1246.94444 1.17 0.3615 fert 2 10950.75000 5475.37500 5.14 0.0244 soil*fert 2 462.58333 231.29167 0.22 0.8079
68
PROC GLM;
TITLE 'Split plot - reduced model
block*fert and soil*fert omitted';
CLASS block soil fert;
MODEL yield= block soil block*soil fert;
RANDOM block;
TEST h = soil e = block*soil; /* tests effect of wholeplot */
TEST h = block e = block*soil; /* tests effect of blocks */
MEANS soil /TUKEY e= block*soil CLM CLDIFF; /* confidence limits for wholeplot effects */
MEANS fert /TUKEY CLM CLDIFF; /* confidence limits for subplot effects */
RUN;
69
Split plot - reduced model block*fert and soil*fert omitted
97Dependent Variable: yield Sum of Source DF Squares Mean Square F Value Pr > F Model 9 23128.08333 2569.78704 2.72 0.0457 Error 14 13243.91667 945.99405 Corrected Total 23 36372.00000 R-Square Coeff Var Root MSE yield Mean 0.635876 18.92739 30.75702 162.5000 Source DF Type III SS Mean Square F Value Pr > F block 3 588.33333 196.11111 0.21 0.8896 soil 1 7848.16667 7848.16667 8.30 0.0121 block*soil 3 3740.83333 1246.94444 1.32 0.3079 fert 2 10950.75000 5475.37500 5.79 0.0147
70
The GLM Procedure Tukey's Studentized Range (HSD) Test for yield NOTE: This test controls the Type I experimentwise error rate. Alpha 0.05 Error Degrees of Freedom 3 Error Mean Square 1246.944 Critical Value of Studentized Range 4.50067 Minimum Significant Difference 45.879 Comparisons significant at the 0.05 level are indicated by ***. Difference Simultaneous soil Between 95% Confidence Comparison Means Limits 2 - 1 36.17 -9.71 82.05 1 - 2 -36.17 -82.05 9.71
71
The GLM Procedure Tukey's Studentized Range (HSD) Test for yield NOTE: This test controls the Type I experimentwise error rate. Alpha 0.05 Error Degrees of Freedom 14 Error Mean Square 945.994 Critical Value of Studentized Range 3.70139 Minimum Significant Difference 40.25 Comparisons significant at the 0.05 level are indicated by ***. Difference Simultaneous fert Between 95% Confidence Comparison Means Limits 3 - 2 15.38 -24.87 55.62 3 - 1 51.00 10.75 91.25 *** 2 - 3 -15.38 -55.62 24.87 2 - 1 35.63 -4.62 75.87 1 - 3 -51.00 -91.25 -10.75 *** 1 - 2 -35.63 -75.87 4.62