# Example Problem

# Graeco-Latin Square Design of Experiments

[Same Latin-Square Problem with another factor (generally blocking factor) added to it]

Many car companies provide maintenance services to new cars for a certain period of time, e.g., 2 years or 24000 miles. A car manufacturer is trying to reduce the cost of maintenance during this service period covered by the manufacturer. This will also reduce the cost of ownership for five years. Due to the better engine materials and quality lubricating oil, the oil change schedule has changed from 3000 miles to about 10,000 miles. In the past, the tire rotation schedule was about every 6000 miles. A tire manufacturer is testing if they can also provide tires that only need rotation every 10,000 miles. The tire manufacturer sells tire in all weather conditions and for all types of car brands. Therefore, to study the rotation effect on the tire wear, they have chosen four different regions: (1) cold weather like in Canada, (2) hot and humid weather like in Florida/Louisiana, (3) Medium cold and humidity weather like in Midwest, (4) medium heat and humidity like in California, and four different models: (1) Base, (2) LT, (3) LTZ, and (4) LIMITED from a car brand. If there is no significant difference in thread wears between the front and rear tires within the 10,000 miles, then the tire manufacturer can claim that their tires do not need to be rotated before 10,000 miles. The percentage of thread-wear-life worn out after 10,000 miles were measured from 4 cars equipped with new tires that are provided below. Data is provided both in Table 6.

Now, the car manufacturer is running a similar study, including one more variable of tire brands. From the car manufacturer point, they would like to provide the best tire possible to their customer without adding cost. Therefore, they are testing the following four competitive tire brands, include, (1) Road Hog (α), (2) Road Horse (β), (3) Road Elephant (γ), and (4) The Maverick (δ). The percentage of thread-wear-life worn out after 10,000 miles were measured from 4 cars equipped with new tires are provided below.

Table 6. Graeco-Latin Data

# Four Steps

# Latin Square Design of Experiments

### Step # 1. Hypothesis

As the interest of a Latin Square design is the treatment factor, the hypothesis is written for the treatment factor, the Position of the tire in this case.

### Step # 2. Method

Graeco-Latin Square Design of Experiment. Data is analyzed using Minitab version 19.

### Step # 3. Analysis and Results

The analysis result is shown in Figure 10.

Figure 10. Graeco-Latin Square Design Analysis Output

We reject the null hypothesis because of *p*-value (0.001) is smaller than the level of significance (0.05).

### Step # 4. Contextual Conclusion

Statistically, at least one of the tire positions is significantly different with respect to the tire thread wear. [rewrite the accepted hypothesis. The alternative is accepted in this case]

Similarly, statistically, at least one of tire brands is significantly different with respect to the tire tread wear.

# Post-hoc Pairwise Comparison Analysis

When the ANOVA result is observed to be significant, **post-hoc **pairwise comparisons analysis is performed to determine the best or the worst level of the factor. The post-hoc analysis output is provided in Figure 11 and Figure 12. Tire position A (front driver side) and B (front passenger side) are observed to be worn more than the rear two tires (Figure 11). Therefore, the tire rotation must be done earlier than the tested 10,000 miles as the tires wear significantly more in the front than the rear. The Maverick Tire brand shows significantly least amount of thread wear (Figure 12.).

Figure 11. Post-hoc analysis for Tire Position for Graco-Latin Square Design

Figure 12. Post-hoc analysis for Tire Brand for Graco-Latin Square Design