Latin Square 1

1
Latin Square Designs

• Lecture Objective
• Introduce basic experimental designs that account
  for two orthogonal sources of extraneous
  variation.
• Terminology
• Square design
• Orthogonal blocks
• Randomizations

2
Examples

• A researcher wishes to perform a yield experiment
  under field conditions, but she/he knows or
  suspects that there are two fertility trends
  running perpendicular to each other across the
  study plots.
• An animal scientists wishes to study weight gain
  in piglets but knows that both litter membership
  and initial weights significantly affect the
  response.
• In a greenhouse, researchers know that there is
  variation in response due to both light
  differences across the building and temperature
  differences along the building.
• An agricultural engineer wishing to test the wear
  of different makes of tractor tire, knows that
  the trial and the location of the tire on the
  (four wheel drive, equal tire size) tractor will
  significantly affect wear.

3
Latin Square Design

• A class of experimental designs that allow for
  two sources of blocking.
• Can be constructed for any number of treatments,
  but there is a cost. If there are t treatments,
  then t2 experimental units will be required.
• If one of the blocking factors is left out of the
  design, we are left with a design that could have
  been obtained as a randomized block design.
• Analysis of a Latin square is very similar to
  that of a RBD, only one more source of variation
  in the model.

4
Cold Protection of Strawberries

• Three different irrigation methods (treatment
  levels) are used on strawberries.
• drip,
• overhead sprinkler and
• no irrigation
• We wish to determine which of these is most
  effective in protecting strawberries from extreme
  cold.
• All strawberries grown through plastic mulch.

5
Field Layout
high
low
Nitrogen Level
none
drip
over
Moisture
none
none
over
drip
drip
over
Moisture and Soil Nitrogen are two sources of
extraneous variation that we wish to
simultaneously control for.
CANAL
Nitrogen Level
high
low
none
drip
over
Moisture
Which design will best allow us to account for
both soil moisture and nitrogen gradients?
none
drip
over
drip
over
none
CANAL
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Advantages and Disadvantages
Advantage - Allows for control of two extraneous
sources of variation. Disadvantages- Requires t
2 experimental units to study t treatments. For
small t, few error degrees of freedom. For large
t, too many error degrees of freedom. Implementat
ion problems. Missing data causes major analysis
problems.
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Constructing a Latin Square Design for t
Treatments

• Treatments designated by first t capital letters
  in the alphabet (A,B,C, etc.)
• Number the levels of blocking factor 1 (call it
  Rows) as R1, R2, Rt.
• Number the levels of blocking factor 2 (call it
  Columns) as C1, C2, Ct.
• Assign the treatment letters in alphabetic order,
  beginning with A, to the t units in the first
  row.
• For the second row, start with the letter B and
  assign treatment letters to the t-th letter then
  follow with A.
• For rows 3 through t, simply shift the treatment
  letters up one at a time, placing the shifted
  letter in the last unit of the row.

8
Basic Square
9
Randomization
Get a random ordering of the rows.
1 2 3 4 replaced by 2 1 4 3
Reorder the rows according to randomization.
10
Randomization
Get a random ordering of the columns.
1 2 3 4 replaced by 4 2 3 1
Reorder the columns according to randomization.
Two Blocking Factors Two Randomizations
Two Constraints on Randomization
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LS Linear Model
t number of treatments, rows and
columns. yij(k) observation on the unit in the
ith row, jth column given the kth treatment. The
indicator k is in parenthesis to remind us that
there are only t2 units, hence if we sum over i
and j we are effectively summing over k. m the
general mean common to all experimental
units. ri the effect of level i of the row
blocking factor. Usually assumed
N(0,sr2). ?j the effect of level j of the
column blocking factor. Usually assumed
N(0,sn2). tk the effect of level k of treatment
factor. eij(k) component of random variation
associated with observation ij(k). Usually
assumed N(0,se2).
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LS Analysis of Variance
13
Sums of Squares
Experimental error response differences between
two experimental units that have experienced the
same treatment. In this case though, the b
replicates for each treatment are spread across
the t row and t column blocks in a specific
fashion. Even more so than with randomized
blocks designs, the variability among treatment
replicates includes the row and column block
effects.
14
Latin Square Means Squares and F Statistics
We reject the null hypothesis of no main effect
if the value of the F-statistic is greater than
the 100(1-a)th percentile of the F distribution
with degrees of freedom specified above.
15
LS Example
The strawberry irrigation cold protection study
data are given below. The effectiveness of the
three irrigation methods was measured by the
weight of the frozen fruit, with lower weights
representing more effective protection. The study
question is Which irrigation method provided
the most protection?
16
Latin Square in SAS
Data strawb input row column irrig weight
_at__at_ datalines 1 1 drip 51 1 2 over 119 1 3
none 60 2 1 none 98 2 2 drip 43 2 3 over
31 3 1 over 99 3 2 none 87 3 3 drip 49
run proc glm class row column irrig model
weight row column irrig title 'Strawberry
Irrigation Latin Square Exp' run
Sum of
Source DF Squares
Mean Square F Value Pr gt F Model
6 5840.000000 973.333333
1.20 0.5205 Error
2 1621.555556 810.777778 Corrected
Total 8 7461.555556
R-Square Coeff Var Root MSE
weight Mean 0.782679
40.23037 28.47416 70.77778 Source
DF Type I SS Mean
Square F Value Pr gt F row
2 817.555556 408.777778
0.50 0.6648 column 2
2616.222222 1308.111111 1.61
0.3826 irrig 2
2406.222222 1203.111111 1.48
0.4026 Source DF Type
III SS Mean Square F Value Pr gt F row
2 817.555556
408.777778 0.50 0.6648 column
2 2616.222222 1308.111111
1.61 0.3826 irrig 2
2406.222222 1203.111111 1.48
0.4026
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Latin Square in SPSS
Input Data Analyze gt General Linear Model gt
Univariate
Note You must use a custom model and
only ask for main effects.
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SPSS Ouptut
19
Latin Square in Minitab
Student edition of Minitab will not do a three
factor Latin Square design directly. We can
trick it into making all the computations we need
but then we have to put it all back together.
Start by doing all three two factor ANOVAs.
Analysis of Variance for weight Source
DF SS MS F P row
2 818 409 0.41 0.691 col
2 2616 1308 1.30
0.368 Error 4 4028 1007 Total
8 7462 Analysis of Variance for
weight Source DF SS MS
F P row 2 818
409 0.39 0.703 irrig 2 2406
1203 1.14 0.407 Error 4
4238 1059 Total 8
7462 Analysis of Variance for weight Source
DF SS MS F P col
2 2616 1308 2.15
0.233 irrig 2 2406 1203
1.97 0.253 Error 4 2439
610 Total 8 7462
Note that the individual Main effect sums of
squares for the three factors are the same across
all models. Now we would collect these sums of
square and their associated degrees of freedom,
add them up and subtract from the total sums of
squares (and df) to get the correct error sums of
squares and df.
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Dummy Variable Latin Square in Minitab
First define mutually orthogonal linear
combinations for the set of levels for each
factor. Next, fit the linear regression of the
response to all the dummy variables created using
the coefficients of the linear combinations.
Collect the sums of squares for each factor from
the set of dummy variables created for this
factor.
Note Dummy variable irr_1 is orthogonal to
irr_2, similarly row_1 is orthogonal to row_2 and
col_1 is orthogonal to col_2.
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ANOVA and Sums of Square
Analysis of Variance Source DF
SS MS F P Regression 6
5840.0 973.3 1.20 0.521 Residual Error 2
1621.6 810.8 Total 8
7461.6 Source DF Seq SS irr_1
1 1734.0 irr_2 1 672.2 row_1
1 4.2 row_2 1
813.4 col_1 1 1944.0 col_2 1
672.2
Note Correct Error SS.
Irrigation SS 1734.0 672.2 2406.2
Row SS 4.2 813.4 817.6
Col SS 1944.0 672.2 2616.2
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Latin Square in Excel
Excel is not suited to the analysis of data from
a Latin Square Design. The Data Analysis Tool
does not handle three factor designs directly. We
can define the dummy variables as was done with
Minitab and run the multiple regression, but
Excel does not provide the Type II (partial) sums
of squares. You have to extract them back from
the statistics the tool provides. An alternative
would be to program the Latin Square analysis
directly in Excel. This works fine for a one-off
analysis but does not work well when you have
multiple responses to analyze.
23
If you are clever you can extract the Type III
sums of squares using the t-statistic and the
mean square residual since