Latin Square -1

1
Latin Square Designs (15.4)

• 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.
• Two restrictions on randomization.

4
Cold Protection of Strawberries

• Three different irrigation methods (treatment
  levels) are used on strawberries
• drip,
• overhead sprinkler,
• no irrigation.
• We wish to determine which of these is most
  effective in protecting strawberries from extreme
  cold.
• All strawberries grown through plastic mulch.
• Measure weight of frozen fruit (lower values
  indicate more protection).

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

• Advantages
• Allows for control of two extraneous sources of
  variation.
• Analysis is quite simple.
• Disadvantages
• Requires t2 experimental units to study t
  treatments.
• Best suited for t in range 5 ? t ? 10.
• The effect of each treatment on the response must
  be approximately the same across rows and
  columns.
• Implementation problems.
• Missing data causes major analysis problems.

7
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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Latin Square Linear Model A Three-Way AOV
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
specifying i and j effectively determines the
treatment 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), a random effect. ?j the effect of
level j of the column blocking factor. Usually
assumed N(0,sn2), a random effect. tk the
effect of level k of treatment factor, a fixed
effect. eij(k) component of random variation
associated with observation ij(k). Usually
assumed N(0,se2).
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Latin Square Analysis of Variance
13
Sums of Squares
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Experimental Error
Experimental error response differences between
two experimental units that have experienced the
same treatment. In this case though, the
replicates for each treatment are spread across
the t row and t column blocks in a specific
fashion. Even more so than with randomized
block designs, the variability among treatment
replicates includes the row and column block
effects. In similar fashion as for RCBDs, the
specific latin square layout will filter out the
extraneous (row col) sources of variability
when performing comparisons of treatment means.
(Show on board) Note that this would not have
been the case if the experiment had erroneously
been laid out as a CRD or RBD
15
Latin Square Mean 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.
16
Latin Square 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?
17
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
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Latin Square in Minitab
Stat gt ANOVA gt General Linear Model
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MTB ANOVA and Sums of Squares
General Linear Model weight versus row, column,
irrig Factor Type Levels Values row
fixed 3 1, 2, 3 column fixed 3 1,
2, 3 irrig fixed 3 drip, none,
over Analysis of Variance for weight, using
Adjusted SS for Tests Source DF Seq SS Adj SS
Adj MS F P row 2 817.6 817.6
408.8 0.50 0.665 column 2 2616.2 2616.2
1308.1 1.61 0.383 irrig 2 2406.2 2406.2
1203.1 1.48 0.403 Error 2 1621.6 1621.6
810.8 Total 8 7461.6 S 28.4742 R-Sq
78.27 R-Sq(adj) 13.07
22
Latin Square with R
gt straw lt- read.table("Data/latin_square.txt",hea
derTRUE) gt straw.lm lt- lm(weight factor(row)
factor(column) factor(irrig),
datastraw) gt anova(straw.lm) Analysis of
Variance Table Response weight
Df Sum Sq Mean Sq F value Pr(gtF) factor(row)
2 817.56 408.78 0.5042 0.6648 factor(column)
2 2616.22 1308.11 1.6134 0.3826 factor(irrig)
2 2406.22 1203.11 1.4839 0.4026 Residuals
2 1621.56 810.78
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Medical Example of a Latin Square
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Randomized, controlled, double-blinded, NICE!
Design extracts out differences due to time and
patients!