RCB - Example
• An accounting firm wants to select training program for its auditors
who conduct statistical sampling as part of their job.
• Three training methods are under consideration: home study,
presentations by local staff and training sessions at head office.
• 30 auditors were placed into one of 10 groups of 3 based on time
since graduation.
• Within each group, one auditor was randomly allocated to each of
the 3 training methods a proficiency exam was administered at the
end of the training period.
• The results of the proficiency test are given in the following table:
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• As we can see on average, training at the head office tends to
produce the highest proficiency scores.
• The blocks appear to be quite heterogeneous with respect to
proficiency.
• The sums of squares and the ANOV are…
• Test that the 3 training methods result in the same mean proficiency
scores…
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Linear Contrasts
• As we have seen previously, the ANOVA will tell us only if there is
an overall difference between the treatment means.
• In order to test hypotheses about the equality of specific treatment
means, we use linear contrasts.
• The contrasts, the corresponding hypothesis and the test statistic are
the same as before.
• Hypothesis: H0 : c1μ1 + c2μ2 + · · · + caμa = 0
Ha : c1μ1 + c2μ2 + · · · + caμa ≠ 0
• Test statistic is:

a
Fobs 
MS

2
cY
i 1 i i 
a
2
E
i 1 i

c /b
~ F1, a 1b1
• So P-value = P(F(1,(a-1)(b-1)) > Fobs)
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Tests Concerning Block Means
• From an examination of the expected mean squares, it would seem
reasonable to test :
H0: β1 = β2 = ... = βb = 0
by comparing MSBl / MSE to the critical F value.
• However, when we applied randomization, we did so only within
blocks.
• Some authors argue that because of the restriction on randomization
that we imposed, this F test would be a test of the equality of block
plus restriction effect.
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Increased Precision Due to Blocking
• In designing the study, we decided to use blocks and impose a
restriction on randomization.
• We did this because we believed that by controlling the nuisance
factor we would decrease haphazard error.
• We might be interested in knowing how much we gained by
blocking.
• The test for equality of block means is open to interpretation, and
even it wasn’t, it would not help quantify any gains due to blocking.
• Several authors have proposed a measure of the relative gain in
efficiency due to inclusion of the blocking factor.
• Their measure takes into account the fact that in the completely
randomized (CR) design there are more degrees for freedom for
estimating error variability.
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• It is given by
2

dfBL  1dfCR  3  CR
R
 2
dfBL  3dfCR  1  BL
2
• In this expression,  CR
is the experimental error variance of the
2
completely randomized design, and  BL is the corresponding
quantity from the randomized complete block design.
• In order to estimate R, we must first estimate
2
2
and  BL .
 CR
• We can use MSE from the randomized block design to estimate
 BL2 .
2
• Further, it can be shown that an unbiased estimator of  CR
is
b  1MS BL  ba  1MS E
2
 CR

ab  1
where MSBLis the mean square for blocks from the randomized
complete block design.
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Example
• Consider the example of the accounting firm that was interested in
selecting the best of 3training methods.
• The study used time since graduation as a blocking variable.
• To find the efficiency of this design relative to the completely
randomized design, we need the following…
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RCB ANOVA Using SAS
• The SAS procedure that we will use to conduct the analysis of
variance in the randomized complete block design is GLM.
• In fact, we will use the same commands as in the two factor fixed
effect model.
• As usual, start by inputting the data…
• The ANOVA for the previous example is produced as follows:
proc glm data = example ;
class years program ;
model score=years program ;
run ;
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Assessing Model Adequacy
• As in the case of the one factor model, we made a number of
assumptions about the data that we must verify.
• Once the model has been fit, the residuals should be examined to
assess the validity of the normality assumption.
• Similarly, the variance should be examined within each treatment
and within each block to ensure homogeneity.
• The same tools can be used here as in the one factor case:
scatterplots, box plots, and normal probability plots.
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Replication and Interaction
• The model we used for the randomized complete block design
assumes that the size and direction of the treatment effect is the
same within each block.
• We would need more than 1 observation per treatment within each
block to look for interaction of treatments with blocks.
• We will handle this case in the same manner as in the two-factor
design.
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Estimating Model Parameters
• One method of estimating the model parameters involves
minimizing the squared distances from the “fitted” model to the
observed response values.
• That is, we need to find the values of μ, τi, and βj which minimize
Q   Yij     i   j 
a
b
2
i 1 j 1
• Differentiate Q with respect to each of the parameters, set resulting
equations to 0, and solve for the parameter values….
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Exercise
• What is the distribution of the least squares estimates?
• Caution: Y and Yi are correlated!
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Latin Square Design
• Recall: the purpose of blocking was to minimize variability due to
known & controllable nuisance factor.
• There may be more than one nuisance factor that is known &
controllable.
• We will now examine the case where there are 2 such nuisance
factors.
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Example
• Output of 4 industrial processes to be compared.
• The processes usually only run on Mon, Wed, Fri, and Sat. We can
run several processes on each day.
• Processes are affected by external conditions such as weather which
vary day to day.
• The external conditions also vary by time of day.
• In studying the 4 processes, we need to ensure each process runs in
each time slot on each day.
• The 4 processes form the treatments in this study, while day and
time of day are 2 nuisance factors which are known & controllable.
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• One possible way to run this study is described in the following
table
• This allocation ensures that there is a balance with respect to day of
the week and time of day. That is, each process is run on each day
and each process is run in each time slot.
• This is an example of a Latin Square design.
• Each of the nuisance factors must have the same number of levels as
the factor being studied.
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Randomization
• For a given number of treatments, a, there are several possible Latin
squares.
• In order to randomize the experiment, a Latin square would ideally
be chosen from among all possible squares.
• As the number of treatments increases, so does the number of
possible Latin squares.
• In general,
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• For example, in the case where there are 3 treatments the possible
Latin squares are:
• It would be tedious and time consuming to write out all possible
Latin squares, and for a greater than 3 this is not feasible.
• So if we can’t do this then how do we randomly select a Latin
square?
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Standard Latin Squares
• A standard Latin square is one which has both the first row and the
first column in numeric (or alphabetic) order.
• For example,
• To randomize, start with the standard Latin square and randomly
permute the rows and columns.
• This is still difficult when the number of treatments is large.
• Cyclic squares, in which treatments always follow each other in the
same order are often used in large designs.
• Disadvantages: order effect, know which treatment comes next.
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The Model for Latin Square
• We will assume that the effect of the treatment is the same
regardless of the level of the 2 nuisance factors, i.e., we assume that
the factors do not interact with each other.
• The model is then….
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Notation
• We will use the “dot” notation with 3 subscripts…
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Sources of Variation
• Possible sources of variability in this model, include:
 Differences in treatment means
 Differences in means for each row level
 Differences in means for each column level
 Random variability
• The total sum of squares can be partitioned in the usual manner:
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Hypothesis Testing
• Although this design involves 3 factors, only one is of primary interest.
• The other 2 are nuisance factors and are included in the design to
minimize experimental error.
• The hypothesis about the equality of treatment means can be tested in a
manner similar to that used in the CR and randomized complete block
designs.
• The hypothesis of interest is:
H0: μ1 = μ2 =....= μa versus Ha: not all μi equal or
H0:τ1 = τ2 = ... = τa =0 versus Ha: not all τi = 0
• To begin, construct the analysis of variance table. It is given in the next
slide…
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Example
• A medical researcher was interested in comparing four formulas that
were fed to newborn infants.
• The outcome of interest was the weight gain (in ounces per day) after
one week.
• The study was designed in such a way that each formula was to be fed
to each infant for 1 week.
• Four infants participated in the study, and 4 weeks were needed to
complete the study
• Although primary interest was in the difference in mean weight gain
between the 4 formulas, two nuisance factors which could affect the
outcome were identified.
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• These were:
(a) infants, i.e. some infants may gain weight more quickly than
others.
(b) week i.e. the week determines the age of the infant, which could
affect outcome and week could also be an indicator for external
conditions i.e. in week x there was a flu virus circulating which
caused weight loss.
• The following table indicates Latin square that was randomly
selected to determine when infants would receive formulas 1-4
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• The study was conducted and the results are:
• The mean weight gain for each formula is:
F1: 0.9825
F2: 1.0100
F3: 1.0450
F4: 1.1650
• Do these data provide any evidence that there is a difference in
mean weight gain between the 4 formulas?
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Solution
• Before proceeding to the analysis of variance, examine the sample
means so we know what to expect.
• We blocked on 2 factors in the belief that there was considerable
variability within blocks, and less variability between blocks.
• Does blocking on these 2 variables appear to have been worthwhile?
• One of the blocking factors was infant; there seems to be quite a bit
a difference in average weight gain for the 4 infants, so perhaps this
was worthwhile including in the design.
• The second blocking factor was week; the mean weight gain for
weeks 2-4 do not appear to differ very much; however, the weight
gain in week 1 was quite a bit less, so perhaps it was a good idea to
have included this in the study design.
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• The factor of primary interest was the type of formula that was fed
to the infants.
• Although there are some differences between the formulas, these
differences aren’t as pronounced as the differences between infants.
• Need to conduct the ANOVA to determine whether the differences
are significant.
• Start by calculating the sums of squares…
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Latin Squares Using SAS
• To obtain the ANOVA for the previous example using SAS, input the data
data example;
input infant week formula wtgain ;
cards
1 1 2
1 2 3
1 3 4
.....
run ;
;
0.4
1.11
1.16
;
• Conduct the ANOVA using PROC GLM
proc glm data = example ;
class formula infant week ;
Model wtgain=formula infant week ;
run ;
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Advantages / Disadvantages of Latin Square Design
• One of the advantages of the Latin square design is that the use of 2
blocking variables can greatly reduce experimental error.
• The total number of experimental units required is relatively small,
which makes this design very practical for pilot or preliminary
studies.
• One of the drawbacks of using a Latin square design is that the
number of levels of each nuisance factor must equal the number of
treatments.
• In some cases, the complexity of the randomization can be a
disadvantage.
• The degrees of freedom for estimating σ2 are relatively small,
meaning that we will not have a very precise estimate.
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Relative Efficiency
• The Latin square is a design one level more complicated that the
randomized complete block design.
• It is worth asking whether the added complexity of the Latin square
paid off in terms of increased precision and power.
• Consider the simpler randomized complete block design where
control of the row factor is part of the design, and column effects
will (hopefully) be averaged out by randomization.
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• The relative efficiency of the Latin square design compared to this
blocked design is
2
 RCB
R 2
 LS
• The denominator can be estimated by MSE from the Latin square
design.
• The numerator can be estimated by
ˆ
2
RCB
MS col  a  1ME E

a
where MSCol is the mean square for the column facture in the Latin
square design, and MSE is also from the Latin square design.
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Example
• In the infant formula example, we might wonder whether there was
any gain in efficiency by blocking on weeks and infants, as
compared to just blocking on infants.
• An estimate of the relative efficiency is obtained by first calculating
the numerator
2
ˆ RCB

MS col  a  1ME E 0.2141  3  0.0521

 0.0926
a
4
• So the relative efficiency is R = 0.0926 / 0.0521 = 1.78.
• Had we decided to use a design which blocked on infant and
randomized order of formula within each infant, we would have
required approximately twice the number of observations to obtain
the same precision.
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Replicated Latin Squares
• In general, even when the number of treatments is large, residual
degrees of freedom in Latin Square remains relatively small.
• When there are only two treatments (a = 2) there are no degrees of
freedom for error.
• It may be desirable to replicate the experiment in order to increase
precision.
• That is, we want to increase the number of experimental units and
still maintain the balance.
• To do this we must add one or more Latin squares to the experiment.
• Depending on the study, there may be more than one way to add
replicates.
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Example
• The output from manufacturing process depends on settings of the
machine used.
• Three settings will be studied to determine if there is a difference in
mean output.
• It is believed that output may also be affected by the particular
machine and by the person operating the machine.
• In order to control for these 2 nuisance factors, and Latin square
design will be used…
Machine 1
Machine 2
Machine 3
Operator 1
S3
S1
S2
Operator 2
S2
S3
S1
Operator 3
S1
S2
S3
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• If we wish to increase size of experiment and to add more
experimental units, we can do it in several ways by adding more
squares.
• We can either use the same operators and machines, or if possible by
including new operators and machines.
• Several options are presented…
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Example
• Retraining program to teach automobile repair skills to individuals.
• Three incentive methods are being tested for use in the retraining
program.
• It is believed that age and level of formal education also have an
impact on outcome of training.
• There are 18 study participants available to participate in this
experiment.
• Two Latin squares with the same rows and columns were used to
design the
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• The study was conducted, and achievement scores were :
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Graeco-Latin Squares
• The Latin square design was used to control for 2 nuisance factors.
• When there is a 3rd nuisance factor to be included in the design of
an experiment, Graeco- Latin squares are used.
• A Graeco-Latin square consists of superimposing 2 orthogonal
Latin squares.
• Consider 2 Latin squares, one with Latin letters and one with Greek
letters.
• The 2 squares are said to be orthogonal if each Latin letter appears
exactly once with each Greek letter.
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Example
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