# What is Hsu's multiple comparisons with the best (MCB)?

Hsu's MCB method is a multiple comparison method that is designed to identify factor levels that are the best, insignificantly different from the best, and those that are significantly different from the best. You can define "best" as either the highest or lowest mean. This procedure is usually used after an ANOVA to more precisely analyze differences between level means.

Hsu's MCB method creates a confidence interval for the difference between each level mean and the best of the remaining level means. If an interval has zero as an end point, there is a statistically significant difference between the corresponding means. Specifically:

Highest is best Lowest is best
Confidence interval contains zero No difference No difference
Confidence interval entirely above zero Significantly better Significantly worse
Confidence interval entirely below zero Significantly worse Significantly better

For this method, you specify the family error rate and the individual error rate is adjusted to achieve it. Hsu's MCB method only compares a subset of all possible pairwise comparisons, unlike Tukey's method which does all comparisons. Therefore, Hsu's MCB method will generate tighter confidence intervals and more powerful tests for any specified family error rate.

## Example of Hsu's MCB method

For example, a memory chip manufacturer randomly samples four production lines to determine which line produces the chips with the fastest response time. The mean response time for each production line is in the following table.

Production line Mean response time N
1 4.85 20
2 10.05 20
3 7.45 20
4 1.20 20

The analyst defines "best" as being the lowest (fastest) mean response time, which is line 4, and uses Hsu's MCB method to identify any production lines that are significantly different from the best. This produces the following confidence intervals [tested level - best of remaining levels].

Production line (compared to best) Lower limit Center Upper limit
1 -1.2 3.65 8.5
2 0 8.85 13.3
3 0 6.25 10.2
4 -8.5 -3.65 1.2

Based on the confidence intervals, the analyst concludes that lines 2 and 3 are producing chips that are significantly slower (higher mean) than line 4 because their confidence intervals are entirely above zero. However, there is no evidence to indicate a significant difference between lines 1 and 4 because their confidence intervals contain zero (no difference). You might investigate the processes of lines 2 and 3 more carefully.

###### Note

When the tested level is significantly better or worse than the comparison level, Hsu's MCB method does not provide a minimum bound on how much better/worse.

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