Lecture 27: Latin Square Analysis
Example: Movie Appeal
First Blocking Factor
Response: How many people,
out of 50, would recommend the
movie to a friend.
Conditions: Four types of
movies; Action, Sci Fi, Comedy
or Drama.
Experimental Units: Audiences
of 50 people.
Day of the week. Because
there are four movies (levels
of the factor of interest) four
days of the week; M, T, W, Th
make up one blocking factor.
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Second Blocking Factor
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Movie Appeal Data
Time of day. Use four
times of day; Early
Matinee, Late Matinee,
Early Evening, Late
Evening.
Monday
Tuesday
Wednesday Thursday
Early
Matinee
Comedy
32
Drama
23
Sci Fi
33
Action
36
Late
Matinee
Sci Fi
26
Action
36
Comedy
31
Drama
22
Early
Evening
Drama
17
Comedy
38
Action
41
Sci Fi
27
Late
Evening
Action
37
Sci Fi
28
Drama
18
Comedy
31
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Movie Appeal Means
Action:
Sci Fi:
Comedy:
Drama:
Overall:
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Estimated Effects
Action:
Sci Fi:
Comedy:
Drama:
37.5
28.5
33.0
20.0
29.75
5
7.75
–1.25
3.25
–9.75
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Lecture 27: Latin Square Analysis
Interpretation
Oneway Analysis of Appeal By Movie
45
40
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More people, on average, will
recommend the Action movie
to a friend.
Fewer people, on average, will
recommend the Drama movie
to a friend.
7.75
3.25
30
–1.25
–9.75
25
20
15
Action
Comedy
Drama
Sci Fi
Movie
7
8
Sources of Variation
Source of variation
Movie
Day of week
Time of day
Error
C. Total
Model
df
3
3
3
6
15
μ – Overall population mean
Mi – Movie effect
Tj – Time of day effect
Dk – Day of week effect
ε – Random error
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Analysis of Variance
Source
df
SS MS
F
Movie
3 669.0 223.0 23.069
Day of week 3 27.5
Time of day
3 20.5
Error
6 58.0 9.667
C. Total
15 775.0
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Test of Hypothesis
Prob>F
0.0011
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H0: All Mi = 0
HA: Some Mi ≠ 0
F = 23.069, P-value = 0.0011
Reject H0 because the Pvalue is so small (< 0.05).
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2
Lecture 27: Latin Square Analysis
Conclusion
Multiple Comparisons
There are some movie
effects that are not zero.
There are some movies that
differ significantly in terms
of average appeal.
Because there are 4 movies,
there are 6 pair-wise
comparisons of means one
should use Tukey’s HSD.
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Multiple Comparisons
Multiple Comparisons
Movie
Action
Comedy
Sci Fi
Drama
2
3.46172 9.667
3.46172 2.19848
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2
4
Mean
37.5 A
33.0 A B
28.5
B
20.0
C
Movies not connected by the same
letter are significantly different.
7.61
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Multiple Comparisons
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JMP
In order to have JMP analyze
the data from a Latin Square
you first have to get the data
into a form JMP understands.
Statistically significant
differences
Action and Sci Fi: 9.0 > 7.61
Action and Drama: 17.5 > 7.61
Comedy and Drama: 13.0 > 7.61
Sci fi and Drama: 8.5 > 7.61
Cases in rows.
Variables in columns.
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Lecture 27: Latin Square Analysis
JMP Data Table
Day
M
Tu
W
Th
⁞
W
Th
Important Note
Time
EM
EM
EM
EM
Movie
Comedy
Drama
Sci Fi
Action
Appeal
32
23
33
36
LE
LE
Drama
Comedy
18
31
The model effect variables
(Day, Time and Movie)
have to be;
Data Type: Character
Modeling Type: Nominal
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Fit Model
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Response Appeal
Analysis of Variance
Y: Appeal
Construct Model Effects
Source
Model
Error
C. Total
Day
Time
Movie
DF
9
6
15
Sum of
Squares Mean Square
F Ratio
79.6667
717.00000
8.2414
9.6667 Prob > F
58.00000
775.00000
0.0092*
Effect Tests
Source
Day
Time
Movie
DF
3
3
3
Sum of
Squares Mean Square
27.50000
20.50000
669.00000
9.1667
6.8333
223.0000
F Ratio Prob > F
0.9483
0.7069
23.0690
0.4746
0.5820
0.0011*
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Complete ANOVA
Source
df
SS MS
F
Day of week 3 27.5
Time of day
3 20.5
Movie
3 669.0 223.0 23.069
Error
6 58.0 9.667
C. Total
15 775.0
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Multiple Comparisons
Prob>F
0.0011
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Effect Details – Movie
LSMeans Tukey HSD
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Lecture 27: Latin Square Analysis
Comment
Mean Squares
For this experiment, the
blocking (nuisance)
variables were not much
different from random error.
Source
df
Day of week 3
Time of day
3
Error
6
SS MS
27.5 9.167
20.5 6.833
58.0 9.667
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Mean Squares
Comment
A mean square quantifies the
amount of variation in the
response due to the source.
Both Day and Time have about
the same (if not smaller) mean
squares as that for random error.
Running a Latin Square design is
supposed to account for the
variability due to nuisance
variables separate from error,
thus making the mean square
error smaller.
This did not happen with the
movie experiment.
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Comment
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Comment
If an experiment like this was
to be done again, you would
not need to use a Latin Square
design because accounting for
Day and Time do not reduce
the size of the mean square
error.
The experiment was
performed as a Latin Square
and so Day and Time should
be included as sources of
variation in the analysis.
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