prepared by hanadi. an p ×p latin square has p rows and p columns and entries from the first p...
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An p ×p Latin Square has p rows and p columns and entries from the first p letters such that each letter appears in every row and in every column.
Latin square design used to eliminate two sources of variability (i.e. allows blocking in two directions).
Advantages:
1 .You can control variation in two Directions .2 .Hopefully you increase efficiency as compared
to the RCBD .
Disadvantages : The number of levels of each blocking variable
must equal the number of levels of the treatment factor. The Latin square model assumes that there are no interactions between the blocking variables or between the treatment variable and the blocking variable.
The null hypotheses can be considered are:
1 (there is no significant difference in treatment means .
2 (there is no significant difference in row means.
3(there is no significant difference in column means.
Check on assumptions (Normality, Constant variance) perform the ANOVA for the Latin-Square design,
Click through Analyze _ General Linear Model Univariate select dependent variable for the Dependent Variable box. Select row and columns to the Fixed Factor(s) box. click Model in the upper right hand corner.
In that dialogue box put the circle for Custom and then click treatments, row, columns over to the right hand box. In the middle, click the down arrow to Main Effects. click off the arrow in the box labeled Include Intercept in Model. Then hit Continue. For multiple comparisons, click Post Hoc and select the factors for performing multiple comparison procedure, and check on thebox for selecting the method for comparisons (Tukey), and then click Continue and hit OK.
Compare
2 -If , we reject there is exist differences between rows means
)i.e. the row factor is important selection.If , we reject there is significant differences between columns means (i.e. the column factor is important)
.
Hreject then we, 0
differaremeanstreatmentstheor
meanstreatmentbetweensdifferenceistherethatmeansFFIf tablet
tabler FF
tableC FF
0H
0H
If we decide there is no significance difference between rows means or columns means then we can ignore this factor and back to analyze RCBD.
There are four cars available for this comparative study of tire performance. It is believed that tires wearing out in a different rate at different location of a car. Tires were installedin four different locations: Right-Front (RF), Left-Front (LF), Right-Rear (RR) and Left-Rear (LR). The measurements of the wearing of tires in this investigation are listed in the following table from a Latin Square Design setting. Three factors are considered in this study. They are tireposition, car and the different tires studied in this investigation?Construct ANOVA , analyze and comment?
Between-Subjects Factors
Value LabelN
car14
24
34
44
poisition1RF4
2LF4
3RR4
4LR4
tire1A4
2B4
3C4
4D4
Tests of Between-Subjects Effects
Dependent Variable:tirewear
SourceType III Sum of SquaresdfMean SquareFSig.
Corrected Model3629.500a9403.2781613.111.000
Intercept46225.000146225.000184900.000.000
car2850.5003950.1673800.667.000
poisition133.500344.500178.000.000
tire645.5003215.167860.667.000
Error1.5006.250
Total49856.00016
Corrected Total3631.00015
a. R Squared = 1.000 (Adjusted R Squared = .999)
Multiple Comparisons
tirewear
Tukey HSD
(I) car(J) car
Mean
Difference (I-J)Std. ErrorSig.
95% Confidence Interval
Lower
Bound
Upper
Bound
12-5.0000*.35355.000-6.2239--3.7761-
3-10.2500*.35355.000-11.4739--9.0261-
4-34.7500*.35355.000-35.9739--33.5261-
215.0000*.35355.0003.77616.2239
3-5.2500*.35355.000-6.4739--4.0261-
4-29.7500*.35355.000-30.9739--28.5261-
3110.2500*.35355.0009.026111.4739
25.2500*.35355.0004.02616.4739
4-24.5000*.35355.000-25.7239--23.2761-
4134.7500*.35355.00033.526135.9739
229.7500*.35355.00028.526130.9739
324.5000*.35355.00023.276125.7239
Based on observed means.
The error term is Mean Square(Error) = .250.
*. The mean difference is significant at the .05 level.
Multiple Comparisons
tirewear
Tukey HSD
(I) poisition(J) poisition
Mean
Difference (I-J)Std. ErrorSig.
95% Confidence Interval
Lower
Bound
Upper
Bound
RFLF-2.0000*.35355.005-3.2239--.7761-
RR-5.7500*.35355.000-6.9739--4.5261-
LR-7.2500*.35355.000-8.4739--6.0261-
LFRF2.0000*.35355.005.77613.2239
RR-3.7500*.35355.000-4.9739--2.5261-
LR-5.2500*.35355.000-6.4739--4.0261-
RRRF5.7500*.35355.0004.52616.9739
LF3.7500*.35355.0002.52614.9739
LR-1.5000*.35355.021-2.7239--.2761-
LRRF7.2500*.35355.0006.02618.4739
LF5.2500*.35355.0004.02616.4739
RR1.5000*.35355.021.27612.7239
Based on observed means.
The error term is Mean Square(Error) = .250.
*. The mean difference is significant at the .05 level.