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 