Week 7
Randomized Block – Fixed Model
Example: Consider three treatments such as three teaching methods or three marketing
techniques.
Using a Single Anova technique, H
0
assumes that all means are equal
μ
1
=
μ
2
=
μ
3
Let us assume that the null is true as the table below indicates. However, if the sum of
the e
ij
are ‘quite’ negative in treatment 1 and ‘quite’ positive in the second treatment, this
would separate the sample means, and H
0
might be falsely rejected (a type one error).
This could happen if the poorer students in general happened to be assigned to
treatment 1, and the better students to treatment two.
Treatment 1
Treatment 2
Treatment 3
Pop.
Means
30
30
30
Error
← 
→ +
What would increase the probability that the sum or the average of the error terms
would deviate meaningfully from zero. Remember assignments are at random. Small
sample sizes would increase the likelihood of that happening. For example, the
probability of five positive error terms in a row is 1 out of 32.
differs from
μ
1
by the average of the e
i1
Imagine another scenario. Imagine that the Null hypothesis of equality of mean is false
as the following table would indicate. If the sum of the e
ij
are ‘quite’ positive in treatment
1 and negative in the second treatment, this would bring the sample means together,
and H
0
might be falsely accepted (a type two error). This could happen if the better
students in general happened to be assigned to treatment 1, and the poorer students
assigned to treatment two.
Treatment 1
Treatment 2
Treatment 3
Pop.
Means
20
30
40
Error
→ +
← 
The next technique that we are about to study is superior to one way Analysis of
Variance in that it reduces the probability that Individuals assigned to the treatments
1
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have a positive or negative bias. And, if successful, it greatly increases the ability to
discover significant differences, by increasing power. This is accomplished by reducing
the variance attributed to error variance.
X
ij
=
μ
+
α
i
+
β
j
+ e
ij
α
i
(block effect) is a new use of
α
To apply, consider three teaching effects with 15 students that first take a quantitative
test before being assigned to a treatment.
The students with the 3 lowest scores on the
quantitative test are assigned to the three different treatments randomly.
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 Summer '10
 chandrasekhar
 Statistics, xij, Treatment Treatment Treatment

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