Randomized Block Design

1
Randomized Block Design Two-way ANOVA

• 10.3 Randomized Block Design
• 10.4 Two-Way Analysis of Variance

2
10.3 The Randomized Block Design

• A randomized block design compares p treatments
  (for example, production methods) on each of b
  blocks (or experimental units or sets of units
  for example, machine operators)
• Each block is used exactly once to measure the
  effect of every treatment
• The order in which each treatment is assigned to
  a block should be random
• A generalization of the paired difference design,
  this design controls for variability in
  experimental units by comparing each treatment on
  the same (not independent) experimental units
• Differences in the treatments are not hidden by
  differences in the experimental units (the blocks)

3
Randomized Block Design

• Define
• xij the value of the response variable when
  block j uses treatment i
• the mean of the b response variable observed
  when using treatment i
• the treatment i mean
• the mean of the p values of the response
  variable when using block j
• the block j mean
• the mean of all the bp values of the response
  variable observed in the experiment
• the overall mean

4
The ANOVA Table, RandomizedBlocks
Degrees Sum of Mean F Source of
Freedom Squares Squares Statistic Treatments p-
1 SST MST SST F(trt) MST
p-1 MSE Blocks b-1 SSB M
SB SSB F(blk) MSB b-1
MSE Error (p-1)?(b-1) SSE MSE
SSE
(p-1)(b-1) Total (p?b)-1 SSTO
where SSTO SST SSB SSE
5
Sum of Squares

• SST measures the amount of between-treatment
  variability
• SSB measures the amount of variability due to the
  blocks
• SSTO measures the total amount of variability
• SSE measures the amount of the variability due to
  error
• SSE SSTO SST SSB

6
F Test for Treatment Effects
H0 No difference between treatment effects Ha
At least two treatment effects differ
Reject H0 if F gt Fa or p-value lt
a Fa is based on p-1 numerator and (p-1)?(b-1)
denominator degrees of freedom
7
F Test for Block Effects
H0 No difference between block effects Ha At
least two block effects differ
Reject H0 if F gt Fa or p-value lt
a Fa is based on b-1 numerator and (p-1)?(b-1)
denominator degrees of freedom
8
Estimation of Treatment DifferencesUnder
Randomized Blocks, Individual Intervals
Individual 100(1 - a) confidence interval for
mi? - mh?
ta/2 is based on (p-1)(b-1) degrees of freedom
9
Estimation of Treatment Differences Under
Randomized Blocks, Simultaneous Intervals
Tukey simultaneous 100(1 - a) confidence
interval for mi? - mh?
qa is the upper ? percentage point of the
studentized range for p and (p-1)(b-1) from Table
A.9 (pg. 831)
10
Randomized Block ANOVA Assumptions

• Normality
• The p populations of values of the response
  variable all have normal distributions
• No Interaction between blocks and treatments
• The ranking of treatment means is the same within
  each block. E.g. if we have, for the treatment
  levels, 2 gt 1 gt 3 in block 1 this will be the
  same order in block 2 and block 3, etc.

11
10.4 Two-Way Analysis of Variance
A two factor factorial design compares the mean
response for a levels of factor 1 (for example,
display height) and each of b levels of factor 2
( for example, display width.) A treatment is a
combination of a level of factor 1 and a level of
factor 2
xijk response for the kth experimental unit
(k1,,m) assigned to the ith level of Factor
1 and the jth level of Factor 2
12
Two-Way ANOVA Table
Degrees Sum of Mean F Source of
Freedom Squares Squares Statistic Factor
1 a-1 SS(1) MS(1) SS(1) F(1)
MS(1) a-1
MSE Factor 1 b-1 SS(2) MS(2)
SS(2) F(2) MS(2)
b-1 MSE Interaction (a-1
)(b-1) SS(int) MS(int) SS(int) F(int)
MS(int)
(a-1)(b-1) MSE Error ab(m-1) SS
E MSE SSE
ab(m-1) Total abm-1 SSTO
13
Two-Way ANOVA Assumptions

• Completely randomized experimental design
• Constant variance among all ab treatments
• Normality
• The ab populations of values of the response
  variable all have normal distributions
• Independence
• The samples of experimental units are randomly
  selected, independent samples
• Can have interactions between factor A and factor
  B

14
F Test for Interaction Effect
Must begin with test for Interaction because
subsequent procedure depends on results H0 No
interaction between factor 1 and factor 2 Ha
factor 1 interacts with factor 2
Reject H0 if F gt Fa or p-value lt
a Fa is based on (a-1)(b-1) numerator and ab x
(m-1) denominator degrees of freedom
15
F Test for Main (Treatment) Effects(only if no
interaction exists)
H0 No difference between effects of factor 1
levels Ha At least two levels of factor 1 have
different effects
Reject H0 if F gt Fa or p-value lt
a Fa is based on a-1 numerator and (ab)?(m-1)
denominator degrees of freedom
16
If no interaction exists, repeat F Test for Main
Effects for each Factor
H0 No difference between effects of factor 2
levels Ha At least two levels of factor 2 have
different effects
Reject H0 if F gt Fa or p-value lt
a Fa is based on b-1 numerator and (ab)?(m-1)
denominator degrees of freedom
17
Estimation of Treatment DifferencesUnder Two-Way
ANOVA, Factor 1
Individual 100(1 - a) confidence interval for
mi? - mi?
ta/2 is based on ab(m-1) degrees of freedom
Tukey simultaneous 100(1 - a) confidence
interval for mi? - mi?
qa is the upper ? percentage point of the
studentized range for a and ab(m-1) from Table
A.9 (pg. 831)
18
Estimation of Treatment DifferencesUnder Two-Way
ANOVA, Factor 2
Individual 100(1 - a) confidence interval for
m?j - m?j
ta/2 is based on ab(m-1) degrees of freedom
Tukey simultaneous 100(1 - a) confidence
interval for m?j - m?j
qa is the upper ? percentage point of the
studentized range for b and ab(m-1) from Table
A.9 (pg. 831)
19
If interaction does exist, cannot test for Main
Effects. Conduct One-way ANOVA
H0 No difference between effects of the ab
factor levels Ha At least two of the ab levels
of the factor have different effects
Reject H0 if F gt Fa or p-value lt
a Fa is based on ab-1 numerator and (ab)?(m-1)
denominator degrees of freedom
20
MegaStat Output for Shelf Display Case 1
21
MegaStat Output for Shelf Display Case 2
22
MegaStat Output for Shelf Display Case 3
23
MegaStat Output for Shelf Display Case 4
24
MegaStat Output for Shelf Display Case 5