Latin Square Design is a type of experimental design used to eliminate two nuisance sources of variability. It systematically allows for blocking in two directions by arranging treatments in a square layout where treatments occur once in each row and column. This restricts randomization in two ways represented by the rows and columns. The design is useful when there are two or more nuisance factors that may affect the response variable and need to be controlled, such as different locations, times, or equipment. Statistical analysis of a Latin Square Design involves testing for effects of rows, columns, and treatments to determine their significance on the response variable. Replication of Latin Squares adds additional squares to improve precision and check adequacy of the model.

1. Latin Square Design
ARIF RAHMAN
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2. Experiments
Hypothesis/
Conjecture
Design of
Experiment
Conduct
Experiment
Analysis of
Data
Discussion &
Conclusion
Refine
the Model
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3. Elements of Design of
Experiments
1. Conjecture or hypothesis
2. Response variable
3. Factors, levels and ranges
4. Treatments of factors
5. Blockings
6. Tools and methods for experiments and
measurements
7. Effect models (independent or interaction factors)
8. Replication, randomization and local factor
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4. P Diagram
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Signal
Factors
(m)
Noise
Factors
(x)
Control
Factors
(z)
Scaling
Factors
(r)
Response
Variables
(y)
F(x,m,z,r)
Inputs
Controllable Factors
(x)
Uncontrollable Factors
(z)
Output
(y)
F(x,z)

5. Blocking Principles and Nuisance
Factors
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6. Blocking Principles and Nuisance
Factors
 Blocking is a technique for dealing with nuisance factors
 A nuisance factor is a factor that probably has some
effect on the response, but it’s of no interest to the
experimenter…however, the variability it transmits to the
response needs to be minimized
 Typical nuisance factors include batches of raw material,
operators, pieces of test equipment, time (shifts, days,
etc.), different experimental units
 Many industrial experiments involve blocking (or should)
 Failure to block is a common flaw in designing an
experiment (consequences?)
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7. Blocking Principles and Nuisance
Factors
 If the nuisance variable is known and controllable, we
use blocking
 If the nuisance factor is known and uncontrollable,
sometimes we can use the analysis of covariance to
remove the effect of the nuisance factor from the analysis
 If the nuisance factor is unknown and uncontrollable (a
“lurking” variable), we hope that randomization
balances out its impact across the experiment
 Sometimes several sources of variability are combined in
a block, so the block becomes an aggregate variable
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8. Latin Square Design (LSD)
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9. Latin Square Design (LSD)
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10. Latin Square Design (LSD)
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11. Latin Square Design (LSD)
Latin Square Design is a type of
experimental design to eliminate two nuisance
sources of variability by systematically
allowing blocking in two directions. The rows
and columns actually represent two
restrictions on randomization. A p X p latin
square is a square containing p rows and p
columns. There are p Latin letters that
correspond to the treatments and occur once
in each row and column.
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12. Latin Square Design (LSD)
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13. Latin Square Design (LSD)
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14. Latin Square Design (LSD)
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Random Order Experimental Units
1 2 3 4 5 6 7 8 9 10 11 12
A A A A A A B B B B B B
B B C C D D A A C C D D
C D B D B C C D A D A C
D C D B C B D C D A C A
Randomization Alternatives of four Experimental Units
24
!
4
4
4 

P
Random Order Experimental Units
13 14 15 16 17 18 19 20 21 22 23 24
C C C C C C D D D D D D
A A B B D D A A B B C C
B D A D A B B C A C A B
D B D A B A C B C A B A

15. Latin Square Design (LSD)
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Standard Latin Square
A B C D
B C D A
C D A B
D A B C
A B C D
B A D C
C D B A
D C A B
A B C D
B D A C
C A D B
D C B A
A B C D
B A D C
C D A B
D C B A

16. Latin Square Design (LSD)
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17. Statistical Analysis of the LSD
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18. Statistical Analysis of the LSD
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19. Statistical Analysis of the LSD
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20. Model Adequacy Checking
Normal Probability Plot of the Residuals
Plot of Residuals in Time Sequence
Plot of Residuals versus Fitted Values
Plot of Residuals versus Other Variables
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21. an Example: The Rocket
Propellant Experiment
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22. an Example: The Rocket
Propellant Experiment
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23. an Example: The Rocket
Propellant Experiment
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24. an Example: The Rocket
Propellant Experiment
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25. Replication of Latin Squares
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26. Replication of Latin Squares
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27. Replication of Latin Squares
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28. Replication of Latin Squares
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29. Crossover Designs
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30. Crossover Designs
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31. 31
That’s all
... Any Questions ???