1. Stat 470-8 Today: Greco-Latin Squares, some examples Assignment 2: –2.8, –2.15 (2-way ANOVA only) –2.16 –2.20 (use only transformations discussed in class) –2.27 –AND 2.25 & 2.26

2. Example: A Martindale wear tester is a machine used for testing the quality types of cloth, and has the property that it can test 4 pieces of cloth simultaneously Experiment (Box, Hunter and Hunter, 1978) was run to compare the wear quality of 4 different types of cloth (A-D) Specimens of th 4 different types of cloth, whose wearing quality is to be compared, are mounted into 4 position holders (1- 4) on the machine 4 sheets of emery paper is cut into quarters and used to cause the wearing of the cloth The response is the material loss (in tenths of a mg)

3. Data

4. Two Blocking Variables; Two Factors Can have 2 blocking factors and 2 experiment factors Experiment design approach superimposes 2 latin squares

5. Graeco-Latin Squares Design Situation: –Have 2 blocking factors - one for rows and one for columns –Have 2 experimental factors –Each factor has k levels –Design is arranged so that each pair of level settings of the experiment factors appears exactly one time in each row and each column –The levels of the two blocking factors are assigned at random to the columns and rows

6. Model: i, j=1,2,…,k l indicates the index for the Latin letter in the (i,j) th cell m indicates the index for the Latin letter in the (i,j) th cell

7. Notes No interaction, since interactions cannot be estimated in an un- replicated experiment Usual assumptions apply Can do multiple comparisons in usual way, but must adjust the degrees of freedom

8. Hypotheses For the factor corresponding to the Latin letter For the factor corresponding to the Greek letter

9. ANOVA Table

10. Can use the Graeco-Latin Squares design for situation with 3 blocking variables and 1 experiment factor

11. Example: A Martindale wear tester is a machine used for testing the quality types of cloth, and has the property that it can test 4 pieces of cloth simultaneously Experiment (Box, Hunter and Hunter, 1978) was run to compare the wear quality of 4 different types of cloth (A-D) Specimens of th 4 different types of cloth, whose wearing quality is to be compared, are mounted into 4 position holders (1- 4) on the machine 4 sheets of emery paper is cut into quarters and used to cause the wearing of the cloth The response is the material loss (in tenths of a mg)

12. Data

13. ANOVA Table

14. Multiple Comparisons

16. Example 1 An engineer is conducting an experiment on eye focus time. He is interested in the effect of the distance of an object from the eye on the focus time Four different distances are of interest (4, 6, 8 and 10 feet) There are five individuals available to participate in the experiment Design an experiment to investigate the impact of distance on focus time (includes treatments, experimental units, randomization and the model to be fit) What is the name of your experiment design?

17. Example 2 A chemist wishes to run an experiment to see how the standing time of the chemicals (2, 4, 6, 8 and 10 minutes), acid concentrations (50, 100, 150, 200 and 250 ionic parts per million) and also the catalyst concentration (5, 20, 15, 20 and 25 % of volume) impact the yield of a chemical process There are 5 batches of raw material available, each which can be split into 5 sub-batches Design an experiment to investigate the impact of distance on focus time (includes treatments, experimental units, randomization and the model to be fit) What is the name of your experiment design?

18. Example 3 Parking is a problem at the University of Michigan, particularly in the Church Street lot There are two types of parking permit for the Church Street lot: Blue and Gold parking There are always lots of empty gold parking spots in the lot…too many if you ask me The would be more parking for your instructor if there were fewer gold spots Design an experiment to estimate the number of empty gold spots as a function of the time of day