DOE 72b The Latin Square Design

1
DOE 7-2b The Latin Square Design
The RCBD just discussed in DOE 7-1 7-2a looked
at the problem of one nuisance factor (variations
in batch quality in the vascular grafts case
study).
2
DOE 7-2b The Latin Square Design
The RCBD just discussed in DOE 7-1 7-2a looked
at the problem of one nuisance factor (variations
in batch quality in the vascular grafts case
study). What happens when you must deal with two
nuisance factors that you must somehow average
out in your experiment design?
3
DOE 7-2b The Latin Square Design
The RCBD just discussed in DOE 7-1 7-2a looked
at the problem of one nuisance factor (variations
in batch quality in the vascular grafts case
study). What happens when you must deal with two
nuisance factors that you must somehow average
out in your experiment design? Example You
produce rocket fuel for jet pilots ejector
seats. There is inherent variation in the raw
materials and in the skill of the operators who
mix the solid fuel. Nuisance factors 1) raw
materials 2) operators.
4
DOE 7-2b The Latin Square Design
The RCBD just discussed in DOE 7-1 7-2a looked
at the problem of one nuisance factor (variations
in batch quality in the vascular grafts case
study). What happens when you must deal with two
nuisance factors that you must somehow average
out in your experiment design? Example You
produce rocket fuel for jet pilots ejector
seats. There is inherent variation in the raw
materials and in the skill of the operators who
mix the solid fuel. Nuisance factors 1) raw
materials 2) operators. Lets assume that there
are five formulations (treatments) of rocket fuel
that you wish to test.
5

• The design answer is to
• test each formulation once per batch of raw
  materials
• have each operator prepare each formulation
  exactly once.
• We will assume that there are 5 operators
  comment on 5.

6

• The design answer is to
• test each formulation once per batch of raw
  materials
• have each operator prepare each formulation
  exactly once.
• We will assume that there are 5 operators
  comment on 5.
• Lets number the raw material batches 1 -5 and
  the operators 1-5. The five fuel formulations
  will be denoted by the letters A, B, C, D and E.

7

• The design answer is to
• test each formulation once per batch of raw
  materials
• have each operator prepare each formulation
  exactly once.
• We will assume that there are 5 operators
  comment on 5.

8
This design arrangement goes by the name of a
Latin Square. If the formulations are replaced
by their experimental values (burn times under
testing), then we get a table of experimental
observations
9
With two nuisance factors (raw materials,
operators) the scope for random variation is
increased
Where yijk is the observation in row i, column k
and treatment j.
10
With two nuisance factors (raw materials,
operators) the scope for random variation is
increased
Where yijk is the observation in row i, column k
and treatment j. µ is the overall mean (very
large sample) ai is the ith row effect t j is the
jth treatment effect, and ß k is the kth column
effect.
11
j
i
k
12
Just as with the RCBD, a slight adjustment to our
earlier ANOVA equations will allow to apply them
to the Latin Square case.
13
Just as with the RCBD, a slight adjustment to our
earlier ANOVA equations will allow to apply them
to the Latin Square case. The total sum of
squares (SST) of the N p2 observations now has
contributions from the partitioning into rows,
columns, treatments and error
14
Just as with the RCBD, a slight adjustment to our
earlier ANOVA equations will allow to apply them
to the Latin Square case. The total sum of
squares (SST) of the N p2 observations now has
contributions from the partitioning into rows,
columns, treatments and error
where in our first pass at ANOVA DOE 5-1 2,
we merely dealt with