Randomized Complete Block and Repeated Measures (Each
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Transcript Randomized Complete Block and Repeated Measures (Each
Randomized Complete Block and Repeated
Measures (Each Subject Receives Each
Treatment) Designs
KNNL – Chapters 21,27.1-2
Block Designs
• Prior to treatment assignment to experimental units, we
may have information on unit characteristics
• When possible, we will create “blocks” of homogeneous
units, based on the characteristics
• Within each block, we randomize the treatments to the
experimental units
• Complete Block Designs have block size = number of
treatments (or an integer multiple)
• Block Designs allow the removal of block to block
variation, for more powerful tests
• When Subjects are blocking variable, use Repeated
Measures Designs, with adjustments made to Block
Analysis (in many cases, the analysis is done the same)
Randomized Block Design – Model & Estimates
Blocks made based on specified categories: age, gender, day of week, etc: Fixed Effects
Model: Yij i j ij
i 1,..., nb ; j 1,..., r
where: overall mean
i effect of i block (typically row effect) with:
th
nb
i 1
i
0
j effect of j treatment (typically column effect) with:
th
ij ~ N 0, 2
independent
Yij ~ N i j , 2
independent
Least Squares Estimators:
^
^
Y
^
i Y i Y
^
^
^
^
j Y j Y
Y ij i j Y Y i Y Y j Y Y i Y j Y
^
eij Yij Y ij Yij Y i Y j Y
r
j 1
j
0
Analysis of Variance
nb
nb
Block (Row) Sum of Squares: SSBL r Y i Y
i 1
2
E MSBL 2
df BL nb 1
r i2
i 1
nb 1
r
r
Treatment (Column) Sum of Squares: SSTR nb Y j Y
j 1
nb
r
2
dfTR r 1
Error (Block Trt) Sum of Squares: SSBL.TR Yij Y i Y j Y
i 1 j 1
df BL.TR nb 1 r 1
E MSTR 2
nb 2j
j 1
r 1
2
E MSBL.TR 2
Testing for Treatment Effects (Rarely interested in Block Effects, Except to reduce Experimental Error):
H 0 : 1 ... r 0 (No Treatment Differences) H A : Not all j 0
Test Statistic: F *
ANOVA (RCBD)
Source
Blocks
Treatments
Error=Blks*Trts
Total
MSTR
MSBL.TR
Rejection Region: F * F 0.95; r 1, nb 1 r 1
df
n_b-1
r-1
(n_b-1)(r-1)
(n_b)r-1
SS
SSBL
SSTR
SSBL.TR
SSTO
MS
F*
P-Value
MSBL
(Printed by Programs) (Printed by Programs)
MSTR
F*=MSTR/MSBL.TR
P(F(r-1,(nb-1)(r-1))>F*)
MSBL.TR
RBD -- Non-Normal Data Friedman’s Test
• When data are non-normal, test is based on ranks
• Procedure to obtain test statistic:
Rank r treatments within each block (1=smallest, r=largest)
adjusting for ties
Compute rank sums for treatments (R•j ) across blocks
H0: The r populations are identical
HA: Differences exist among the r group means
12
r
2
T .S . : X
R 3nb ( r 1)
j 1 j
nb r ( r 1)
2
F
R.R. : X
2
F
2
, r 1
P val : P ( X )
2
2
F
Checking Model Assumptions
• Strip plots of residuals versus blocks (equal variance
among blocks – all blocks received all treatments)
• Plots of residuals versus fitted values (and
treatments – equal variances)
• Plot of residuals versus time order (in many lab
experiments, blocks are days – independent errors)
• Block-treatment interactions – Tukey’s test for
additivity
Comparing Treatment Effects (All Pairs)
Tukey's Method:
HSD jj ' q 1 ; r , nb 1 r 1
MSBL.TR
nb
Conclude j j ' if Y j Y j ' HSD jj '
Simultaneous Confidence Intervals: Y j Y j ' HSD jj '
Bonferroni's Method: r ( r 1) / 2 # of Pairs of Treatment Means
2MSBL.TR
BSD jj ' t 1 ; r (r 1) / 2; nb 1 r 1
nb
2
Conclude j j ' if Y j Y j ' BSD jj '
Simultaneous Confidence Intervals: Y j Y j ' BSD jj '
Extensions of RCBD
• Can have more than one blocking variable
Gender/Age among Human Subjects
Region/Size among cities
Observer/Day among Reviewers (Note: Observers are really
subjects, same individual)
• Can have more than one replicate per block, but prefer
to have equal treatment exposure per block
• Can have factorial structures run in blocks (usual
breakdown of treatment SS). Problems with many
treatments (non-homogeneous blocks).
Main Effects
Interaction Effects
Relative Efficiency
• Measures the ratio of the experimental error variance for
the Completely Randomized Design (r2) to that for the
Randomized Block Design (b2)
• Computed from the Mean Squares for Blocks and Error
• Represents how many observations would be needed per
treatment in CRD to have comparable precision in
estimating means (standard errors) as the RBD
r2
E 2
b
sr2 nb 1 MSBL nb r 1 MSBL.TR
E 2
sb
nb r 1 MSBL.TR
^
Sometimes the efficiency is modified to reflect differences in Error df:
^ '
E
df 2 1 df1 3 ^
E
df 2 3 df1 2
where df1 df CRD r nb 1 and df 2 df RBD r 1 nb 1
Repeated Measures Design
• Subjects (people, cities, supermarkets, etc) are
selected at random, and assigned to receive each
treatment (in random order)
• Unlike block effects, which were treated as fixed,
subject effects are random variables (since the
subjects were selected at random)
• Measurements on subjects are correlated, however
conditional on a subject being selected, they are
independent (no carry-over effects or order effects)
• The analysis is conducted in a similar manner to
Randomized Complete Block Design
Repeated Measures Design – Model
Subjects Randomly Selected and Assigned to Each Treatment: Random Effects
Model: Yij i j ij i 1,..., s; j 1,..., r
where overall mean
i effect of i th subject (typically row effect) with: i ~ N 0, 2 independent
r
j effect of j treatment (typically column effect) with: j 0
th
j 1
ij ~ N 0, 2 independent
,
independent
Y , Y i i '
Y 2 Y , Y
Yij ~ N j , 2 2
Yij , Yij ' 2
j j'
2
ij
2 Yij Yij ' 2 Yij 2
2 Y j Y j'
2 2
s
i' j'
2
ij '
ij
s2 Y j Y j'
ij '
2 MSTR.S
s
2
2 2 2 2 2 2
s Y j Y j'
2 MSTR.S
s
Repeated Measures Design – ANOVA
s
Subjects Sum of Squares: SSS r Y i Y
i 1
2
df S s 1
E MSS 2 r 2
r
r
Treatment Sum of Squares: SSTR s Y j Y
j 1
2
dfTR r 1
s
r
E MSTR 2
Error (Subject by Treatment) Sum of Squares: SSTR.S Yij Y i Y j Y
i 1 j 1
dfTR.S r 1 s 1
j 1
r 1
2
E MSTR.S 2
Within Subjects Sum of Squares: SSW Yij Y i
Testing for Treatment Effects: H 0 : 1 ... r 0
H A : Not all i 0
s
r
i 1 j 1
Test Statistic: F *
s 2j
MSTR
MSTR.S
^
Relative Efficiency: E
2
SSTR SSTR.S
Rejection Region: F * F 1 ; r 1, r 1 s 1
s 1 MSS s r 1 MSTR.S
sr 1 MSTR.S
^
Completely Randomized Design needs n s E replicates per treatment for same s Y j Y j '
Comparing Treatment Effects (All Pairs)
Tukey's Method:
HSD jj ' q 1 ; r , r 1 s 1
MSTR.S
s
Conclude j j ' if Y j Y j ' HSD jj '
Simultaneous Confidence Intervals:
Y
j
Y j ' HSD jj '
Bonferroni's Method: r ( r 1) / 2 # of Pairs of Treatment Means
2 MSTR.S
BSD jj ' t 1 ; r (r 1) / 2; r 1 s 1
s
2
Conclude j j ' if Y j Y j ' BSD jj '
Simultaneous Confidence Intervals:
Y
j
Y j ' BSD jj '
Within-Subject Variance-Covariance Matrix
Common Assumptions for the Repeated Measures ANOVA
• Variances of measurements for each treatment are equal: 12 ... r2
• Covariances of measurements for each pair treatments are the same
Note: These will not hold exactly for sample data, should give a feel if reasonable
Population Structure
1r 2 jj '
2 r jj ' 2
12 12
2
2
21
r1 r 2
2
r
Y
s
where: s 2j
i 1
Sample Structure
ij
Y j
s 1
jj '
jj '
jj ' jj '
2
Y
s
2
s jj '
s12
^
s
21
sr1
i 1
ij
Y j Yij ' Y j '
s 1
s12
s22
sr 2
s1r
s2 r
2
sr