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Lecture 16

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ADM2304 Lecture 16: 16
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University of Ottawa

Administration

ADM2304

Tony Quon

Fall

Description

Candidate for one-way ANOVA as we have one factor – gasoline type – and one
response variable – gas mileage
But a potentially confounding factor that impacts gas mileage is vehicle type
If we did a completely randomized design, our samples for each factor may have
different types of vehicle which would bring into question any conclusion we might draw
One way of controlling for this is to divide the population of cars up by size and
randomly assign one car from each size to each gasoline type (essentially creating a
2nd factor)
In this way we ensure that the size of the car does not have an impact on the outcome
This is called a randomized block design
How Does this Extend Matched Pairs?
Suppose there were only two gasoline types – regular and premium
The matching would be based on vehicle type so that we would make sure that each
sample has equivalent vehicle types in it so that we can be sure that the vehicle type is
not affecting the results
Thus, each SUV in the sample given regular gasoline would be “matched” with an SUV
in the sample given premium gasoline
Rather than “before” and “after”, we have matching based on characteristics (in this
case vehicle type)
The randomized block design extends this to more than two populations
The model:
We proceed essentially as if this were a two-factor analysis with no
interaction term
Thus, our model is:
X =ij +α + β +εi j ij
We divide the total variation SSTotal into variation due to the treatment (column), SSB,
the variation due to the block (row), SSA and the random variation due to “noise”, SSE
SSTotal = SSB + SSA + SSE
Degrees of freedom are:
N - 1 for SSTotal
b - 1 for SSB
a - 1 for SSA
(a - 1)(b - 1) for SSE
where:
a is the number of blocks – that is the number of observations in each treatment level
b is the number of treatment levels
We are only interested in the main effect of the column variable Hypothesis Test:
Hypotheses:
H0: Main effects due to treatments are all the same (i.e., gasoline has no impact on gas
mileage)
Ha: Main effect due to some treatment is different from another (i.e., at least one
gasoline does improve gas mileage over another)
We compare the variation between the treatments means, MSB, with the overall
variation, MSE by calculating F = MSB/MSE
We reject the null hypothesis if the F-statistic is larger than the critical value Fα with b -
1 and (b - 1)(a - 1) degrees of freedom
We could of course also do a test to determine if the mean effect of all the block levels
is equal but we would expect to reject that hypothesis as we speciﬁcally chose the
blocks because we thought they would impact on the results
Bonferroni Method for Paired Comparisons
We use the Bonferonni margin of error for examining pairs of treatment means:
⎛ 1 1 ⎞
t* MSE ⎜ + ⎟
⎝ a a ⎠
where a is the number of observations comprising each treatment mean and
t* represents the t-value with /2J probability in the right tail
Example: Tyco Valves
Tyco uses a manufacturing resource planning system to reduce lead-time for
manufacturing. It wishes to determine whether lead-times differ according to the type of
valve
Tyco recognizes that the day of the week the valve is ordered may have an impact on
the lead time (especially if it is near the weekend)
Thus they look to do a randomized block design to determine the impact of valve type
on lead time while controlling for any day of the week effect
Valve Type
Day Safety Butterfly Clack Slide Poppet NeedleBlock
Means
Monday 1.6 2.2 1.3 1.8 2.5 0.8 1.7 ANOVA Table
Tuesday 1.8 2.0 1.4 1.5 2.4 1.0 1.68 SourceFD SS MS F P-value
Wednesday 1.0 1.8 1.0 1.6 2.0 0.8 1.37
Day 4 0.77667 0.19417 4.90.006
Thursday 1.8 2.2 1.4 1.6 1.8 0.6 1.57

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