Latin Square Design

1
Latin Square Design

• If you can block on two (perpendicular) sources
  of variation (rows x columns) you can reduce
  experimental error when compared to the RBD
• More restrictive than the RBD
• The total number of plots is the square of the
  number of treatments
• Each treatment appears once and only once in each
  row and column

2
Advantages and Disadvantages

• Advantage
• Allows the experimenter to control two sources of
  variation
• Disadvantages
• The experiment becomes very large if the number
  of treatments is large
• The statistical analysis is complicated by
  missing plots and misassigned treatments
• Error df is small if there are only a few
  treatments
• This limitation can be overcome by repeating a
  small
• Latin Square and then combining the
  experiments -
• a 3x3 Latin Square repeated 4 times
• a 4x4 Latin Square repeated 2 times

3
Useful in Animal Nutrition Studies

• Suppose you had four feeds you wanted to test on
  dairy cows. The feeds would be tested over time
  during the lactation period
• This experiment would require 4 animals (think of
  these as the rows)
• There would be 4 feeding periods at even
  intervals during the lactation period beginning
  early in lactation (these would be the columns)
• The treatments would be the four feeds. Each
  animal receives each treatment one time only.

4
The Latin Square Cow
A simple type of crossover design
Are there any potential problems with this design?
5
Uses in Field Experiments

• When two sources of variation must be controlled
• Slope and fertility
• Furrow irrigation and shading
• If you must plant your plots perpendicular to a
  linear gradient

• Practically speaking, use only when you have more
  than four but fewer than ten treatments
• a minimum of 12 df for error

6
Randomization

• First row in alphabetical order ? A B C D
  E
• Subsequent rows - shift letters one position
• 4 3 5 1 2
• A B C D E 2 C D E A B A B D C E
• B C D E A 4 A B C D E D E B A C
• C D E A B 1 D E A B C B C E D A
• D E A B C 3 B C D E A E A C B D
• E A B C D 5 E A B C D C D A E B
• Randomize the order of the rows ie 2 4 1 3 5
• Finally, randomize the order of the columns ie 4
  3 5 1 2

7
Linear Model

• Linear Model Yij ? ?i ?j ?k ?ij
• ? mean effect
• ßi ith block effect
• ?j jth column effect
• ?k kth treatment effect
• ?ij random error
• Each treatment occurs once in each block and once
  in each column
• r c t
• N t2

8
Analysis

• Set up a two-way table and compute the row and
  column means and deviations
• Compute a table of treatment means and deviations
• Set up an ANOVA table divided into sources of
  variation
• Rows
• Columns
• Treatments
• Error
• Significance tests
• FT tests difference among treatment means
• FR and FC test if row and column groupings are
  effective

9
The ANOVA
10
Means and Standard Errors
Standard Error of a treatment mean
Confidence interval estimate
Standard Error of a difference
Confidence interval estimate
t to test difference between two means
11
Oh NO!!! not Missing Plots

• If only one plot is missing, you can use the
  following formula

Yij(k) t(Ri Cj Tk)-2G (t-1)(t-2)

• Where
• Ri sum of remaining observations in the ith row
• Cj sum of remaining observations in the jth
  column
• Tk sum of remaining observations in the kth
  treatment
• G grand total of the available observations
• t number of treatments

• Total and error df must be reduced by 1
• Alternatively use software such as SAS
  Procedures GLM, MIXED, or GLIMMIX that adjust for
  missing values

12
Relative Efficiency

• To compare with an RBD using columns as blocks
• RE MSR (t-1)MSE
• tMSE
• To compare with an RBD using rows as blocks
• RE MSC (t-1)MSE
• tMSE
• To compare with a CRD
• RE MSR MSC (t-1)MSE
• (t1)MSE

13
Numerical Examples
To determine the effect of four different sources
of seed inoculum, A, B, C, and D, and a control,
E, on the dry matter yield of irrigated alfalfa.
The plots were furrow irrigated and there was a
line of trees that might form a shading gradient.
Are the blocks for shading represented by the
row or columns? Is the gradient due to irrigation
accounted for by rows or columns?
14
Data collection
A B D C E 33.8 33.7 30.4 32.7 24.4
D E B A C 37.0 28.8 33.5 34.6 33.4
C D A E B 35.8 35.6 36.9 26.7 35.1
E A C B D 33.2 37.1 37.4 38.1 34.1
B C E D A 34.8 39.1 32.7 37.4 36.4
15
ANOVA of Dry Matter Yield
16
Report of Statistical Analysis
I source A B C D None SE Mean Yield 35.8 35.0 35.
7 34.9 29.2 0.78 a a a a b LSD2.4

• Differences among treatment means were highly
  significant
• No difference among inocula. However,
  inoculation, regardless of source produced more
  dry matter than did no inoculation
• Blocking by irrigation effect was useful in
  reducing experimental error
• Distance from shade did not appear to have a
  significant effect

17
Relative Efficiency

• To compare with an RBD using columns as blocks
• RE MSR (t-1)MSE
• tMSE
• To compare with an RBD using rows as blocks
• RE MSC (t-1)MSE
• tMSE
• To compare with a CRD
• RE MSR MSC (t-1)MSE
• (t1)MSE

(21.85(5-1)3.07)/(5x3.07)2.22 (2.22-1)100
122 gain in efficiency by adding rows
(4.14(5-1)3.07)/(5x3.07)1.07 (1.07-1)100 7
gain in efficiency by adding columns
(21.854.14(5-1)3.07)/(6x3.07)2.08 (2.08-1)100
108 gain CRD would require 2.085 or 11 reps