ST 524
Completely Randomized Design with Subsampling
NCSU - Fall 2008
Completely Randomized Design with subsamples
Example (ST&D, p 159)
Experiment to analyze the effect of hours of daylight and night temperature on the stem growth of mint
plants: 6 treatments (combinations of temperature and daylight) were randomly assigned to 3 pots each;
pots are nested under each level of treatment, and 4 plants were measured from each pot, subsamples are
nested within each level of pot and treatment.
• Plants randomly assigned to pots: 4 per pot
• Treatment randomly assigned to pots: 3 pots per treatment
• Experimental unit: pot
• Subsample: plants within each pot.
• Since every level of the nested factor does not appear with every level of the treatment factor, no
interaction between these two factors.
Sources of Variation
• Variation among the subsamples (plants) from the same experimental unit (pot) and treatment
Sampling Error
• Variation among experimental units (pots) treated alike: pots within treatments
Experimental Error
Data: One-week stem growth
Low Temp
Plant No
1
2
3
4
Pot total = Yij.
8 hs
12 hs
16 hs
8 hs
T1
Pot number
1
2
3
3.5 2.5 3.0
4.0 4.5 3.0
3.0 5.5 2.5
4.5 5.0 3.0
T2
Pot number
1
2
3
5.0 3.5 4.5
5.5 3.5 4.0
4.0 3.0 4.0
3.5 4.0 5.0
T3
Pot number
1
2
3
5.0 5.5 5.5
4.5 6.0 4.5
5.0 5.0 6.5
4.5 5.0 5.5
T4
Pot number
1
2
3
8.5 6.5 7.0
6.0 7.0 7.0
9.0 8.0 7.0
8.5 6.5 7.0
T5
Pot number
1
2
3
6.0 6.0 6.5
5.5 8.5 6.5
3.5 4.5 8.5
7.0 7.5 7.5
T6
Pot number
1
2
3
7.0 6.0 11.0
9.0 7.0 7.0
8.5 7.0 9.0
8.5 7.0 8.0
32
22
33
15 17.5 11.5 18
Treatment total = Yi..
Treatment mean =
High Temp
Y i..
14 17.5 19
21.5 22
28
12 hs
28
26.5 29
16 hs
27
44.0
49.5
62.5
88.0
77.5
95.0
3.7
4.1
5.2
7.3
6.5
7.9
35
Additive Linear Model
Yijk = μ + τ i + ε ij + δ ijk
t
•
Treatment is a Fixed Effect
∑τ
i =1
i
=0
ε ij ~ iidN ( 0, σ ε2 )
•
Pot is a Random effect , nested on treatments
•
Plant is a Random effect, nested on pots δ ijk ~ iidN 0, σ δ ,
•
(
ε ij
and
δ ijk
2
)
are independent random effects
Analysis of Variance - Expected Mean Squares
Sum of Squares Decomposition
Tuesday September 9, 2008 CRD Analysis of variance with subsampling
1
ST 524
Completely Randomized Design with Subsampling
t
r
s
t
∑∑∑
NCSU - Fall 2008
t
r
s
(Yijk − Y ... ) 2 =rs ∑ (Y i.. − Y ... ) 2 + s ∑∑ (Y ij . − Y i.. ) 2 + ∑∑∑ (Yijk − Y ij . ) 2
i =1 j =1 k =1
i =1
i =1 j =1
i
j
k =1
SS due to Sampling Error
Variation among samples within pots
SS due to Treatment Effects
SS due to Experimental Error
Variation among pots within treatments
Analysis of Variance Table
Sources
Treatment
Sum of
Squares
df
Pots(Treatment)
Sampling Error
35.9284722
σ δ + 4σ + ( 4 × 3)
6 × ( 3 − 1) =12
25.8333333
2.1527778
σ δ2 + 4σ e2
6 × 3 × ( 4 − 1) = 54
50.4375000
0.9340278
σ δ2
6 × 3 × 4 − 1 = 71
255.9131944
Plants(Pots)
Corrected total
E(MS)
179.6423611
6-1 = 5
Experimental Error
Mean
Square
2
2
e
∑τ
2
i
i
( 6 − 1)
Dependent Variable: growth
Source
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
17
205.4756944
12.0868056
12.94
<.0001
Error
54
50.4375000
0.9340278
Corrected Total
71
255.9131944
R-Square
Coeff Var
Root MSE
growth Mean
0.802912
16.70696
0.966451
5.784722
Source
DF
Type I SS
treatment
pot(treatment)
5
12
179.6423611
25.8333333
1
Mean Square1
F Value
35.9284722
2.1527778
38.47
2.30
Pr > F
<.0001
0.0186
Note that each mean square is on a per-observation (plant) basis. Means should be as well on a per-observation basis.
proc GLM data=a;
class treatment pot;
model growth= treatment pot (treatment);
random pot (treatment)/test;
run;
Tuesday September 9, 2008 CRD Analysis of variance with subsampling
2
ST 524
Completely Randomized Design with Subsampling
NCSU - Fall 2008
Test of Hypotheses
Tests of Hypotheses for Mixed Model Analysis of Variance
Dependent Variable: growth
Source
DF
Type III SS
Mean Square
F Value
Pr > F
5
179.642361
35.928472
16.69
<.0001
Error
Error: MS(pot(treatment))
12
25.833333
2.152778
Source
DF
Type III SS
Mean Square
F Value
Pr > F
pot(treatment)
12
25.833333
2.152778
2.30
0.0186
Error: MS(Error)
54
50.437500
0.934028
treatment
•
Treatments
H o :τ1 = τ 2 = τ 3 = τ 4 = τ 5 = τ 6 = 0
H1 : At least one τ i ≠ 0 , i = 1, 2," , 6
Fcalc =
•
35.9285
= 16.69
2.1528
p-value <0.0001 Reject Ho
Experimental error
H o : σ ε2 = 0
H1 : σ ε2 > 0
Fcalc =
2.1528
= 2.30
0.9340
p-value = 0.0186
Reject Ho
Variance Components Estimation
∧
Var(among subsamples units within pot and treatment):
∧
Var(among experimental units):
σ ε2 =
σ δ2
= 0.9340
2.1528 − 0.9340
= 0.3047
4
Variance among plants within pots
Variance among pots within treatments
Proc VARCOMP
Type 1 Estimates
proc varcomp Method= Type1;
class treatment pot;
model growth= treatment pot(treatment)/fixed=1;
run;
Variance Component
Var(pot(treatment))
Var(Error)
Estimate
0.30469
0.93403
Proc MIXED – Mixed Model: Treatments are Fixed effects, Pots(Treatment) are random effects,
Plants(pots and treatment) are random effects.
proc mixed data=a;
class treatment pot;
model growth = treatment;
random pot (treatment);
lsmeans treatment;
run;
Tuesday September 9, 2008 CRD Analysis of variance with subsampling
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ST 524
Completely Randomized Design with Subsampling
NCSU - Fall 2008
The Mixed Procedure
Covariance Parameter
Estimates
Cov Parm
pot(treatment)
Residual
Plants(pot and treatment)
Estimate
0.3047
0.9340
Fit Statistics
-2 Res Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
207.7
211.7
211.9
213.5
Type 3 Tests of Fixed Effects
Num
DF
5
Effect
treatment
An Observation, Yijk
Den
DF
12
F Value
16.69
Pr > F
<.0001
= μ + τ i + ε ij + δ ijk
Variance of an observation
Var (Yijk ) = σ ε2 + σ δ2
estimated by 0.3047 + 0.9340 = 1.2387
Pot mean
Y ij . = μ + τ i +
∑ε
k
s
ij
+
∑δ
ijk
k
s
= μ + τ i + ε ij +
δ ij .
s
Variance of a pot mean
( )
(
)
Var Y ij . = Var μ + τ i + ε ij + δ ij . = σ ε2 +
σ δ2
, estimated by
s
Experimental Error MS
= 0.5382
4
Treatment mean
Y i.. = μ + τ i +
∑ε
jk
rs
ij
+
∑δ
ijk
jk
rs
= μ +τi +
ε i.
r
+
δ i..
rs
Variance of a treatment mean
⎛
ε i. δ i.. ⎞ σ ε2 σ δ2
Experimental Error MS
, estimated by
= 0.1794
Var Y i.. = Var ⎜ μ + τ i + +
+
⎟=
r
rs
r
sr
3× 4
⎝
⎠
( )
Standard error of a treatment mean =
( )
Var Y i... = 0.1794 = 0.4236
Tuesday September 9, 2008 CRD Analysis of variance with subsampling
4
ST 524
Completely Randomized Design with Subsampling
NCSU - Fall 2008
Least Squares Means2
Effect
treatment
treatment
treatment
treatment
treatment
treatment
treatment
T1
T2
T3
T4
T5
T6
Estimate
Standard
Error
DF
t Value
Pr > |t|
3.6667
4.1250
5.2083
7.3333
6.4583
7.9167
0.4236
0.4236
0.4236
0.4236
0.4236
0.4236
12
12
12
12
12
12
8.66
9.74
12.30
17.31
15.25
18.69
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
2
Note: Treatment LSMEANS from PROC MIXED.
Testing null hypothesis for Experimental Error - Covtest in PROC MIXED
The Mixed Procedure
Model Information
Data Set
WORK.A
Dependent Variable
growth
Covariance Structure
Variance Components
Subject Effect
pot(treatment)
Estimation Method
REML
Residual Variance Method
Profile
Fixed Effects SE Method
Model-Based
Degrees of Freedom Method
Containment
Class Level Information
Class
treatment
pot
proc mixed data=a covtest;
class treatment pot;
model growth = treatment;
random intercept/ subject= pot (treatment);
lsmeans treatment;
run;
Levels
6
3
Values
T1 T2 T3 T4 T5 T6
1 2 3
σ δ2
Dimensions
Covariance Parameters
Columns in X
Columns in Z Per Subject
Subjects
Max Obs Per Subject
2
7
1
18
4
and
σ ε2
6 treatments + intercept
3 pots*6 treatments = 18
Number of Observations
Number of Observations Read
Number of Observations Used
Number of Observations Not Used
72
72
0
Convergence criteria met.
H o : σ ε2 = 0
Covariance Parameter Estimates
Cov Parm
Subject
Intercept
Residual
pot(treatment)
H1 : σ ε2 > 0
Estimate
Standard
Error
Z
Value
Pr Z
0.3047
0.9340
0.2243
0.1798
1.36
5.20
0.0871
<.0001
Fit Statistics
-2 Res Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
207.7
211.7
211.9
213.5
H o : σ δ2 = 0
H1 : σ δ2 > 0
Type 3 Tests of Fixed Effects
Effect
treatment
Num
DF
Den
DF
F Value
Pr > F
5
12
16.69
<.0001
Tuesday September 9, 2008 CRD Analysis of variance with subsampling
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