1. Prepared by Hanadi

2. 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.

3. Latin square design used to eliminate two sources of variability (i.e. allows blocking in two directions).

4. 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.

5. The interested model:

6. 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.

7.  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 the box for selecting the method for comparisons (Tukey), and then click Continue and hit OK. 

8. 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)

9. 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.

10. 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 installed in 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 tire position, car and the different tires studied in this investigation? Construct ANOVA, analyze and comment?

16. output

17. Between-Subjects Factors Value LabelN car1 4 2 4 3 4 4 4 poisition1 RF4 2 LF4 3 RR4 4 LR4 tire1 A4 2 B4 3 C4 4 D4 Tests of Between-Subjects Effects Dependent Variable:tirewear SourceType III Sum of SquaresdfMean SquareFSig. Corrected Model 3629.500 a 9403.2781613.111.000 Intercept 46225.0001 184900.000.000 car 2850.5003950.1673800.667.000 poisition 133.500344.500178.000.000 tire 645.5003215.167860.667.000 Error 1.5006.250 Total 49856.00016 Corrected Total 3631.00015 a. R Squared = 1.000 (Adjusted R Squared =.999)

19. 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- 21 5.0000 *.35355.0003.77616.2239 3 -5.2500 *.35355.000-6.4739--4.0261- 4 -29.7500 *.35355.000-30.9739--28.5261- 31 10.2500 *.35355.0009.026111.4739 2 5.2500 *.35355.0004.02616.4739 4 -24.5000 *.35355.000-25.7239--23.2761- 41 34.7500 *.35355.00033.526135.9739 2 29.7500 *.35355.00028.526130.9739 3 24.5000 *.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- LFRF 2.0000 *.35355.005.77613.2239 RR -3.7500 *.35355.000-4.9739--2.5261- LR -5.2500 *.35355.000-6.4739--4.0261- RRRF 5.7500 *.35355.0004.52616.9739 LF 3.7500 *.35355.0002.52614.9739 LR -1.5000 *.35355.021-2.7239--.2761- LRRF 7.2500 *.35355.0006.02618.4739 LF 5.2500 *.35355.0004.02616.4739 RR 1.5000 *.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.