7.
Product and Process Comparisons
7.4. Comparisons based on data from more than two processes 7.4.7. How can we make multiple comparisons?


Tukey's method considers all possible pairwise differences of means at the same time 
The Tukey method applies simultaneously to the set of all pairwise
comparisons
The confidence coefficient for the set, when all sample sizes are equal, is exactly 1. For unequal sample sizes, the confidence coefficient is greater than 1. In other words, the Tukey method is conservative when there are unequal sample sizes. 

Studentized Range Distribution  
The studentized range q 
The Tukey method uses the studentized range distribution.
Suppose we have r independent observations y_{1},
..., y_{r} from a normal distribution with mean
and variance
^{2}.
Let w be the range for this set , i.e., the maximum minus
the minimum. Now suppose that we have an estimate
s^{2} of the variance
^{2}
which is based on
degrees of freedom
and is independent of the y_{i}. The studentized
range is defined as


The distribution of q is tabulated in many textbooks and can be calculated using Dataplot 
The distribution of q has been tabulated and appears in many
textbooks on statistics. In addition, Dataplot has a CDF function
(SRACDF) and a percentile function (SRAPPF) for q.
As an example, let r = 5 and = 10. The 95th percentile is q_{.05;5,10} = 4.65. This means: 

Tukey's Method  
Confidence limits for Tukey's method 
The Tukey confidence limits for all pairwise comparisons with
confidence coefficient of at least
1 are:
Notice that the point estimator and the estimated variance are the same as those for a single pairwise comparison that was illustrated previously. The only difference between the confidence limits for simultaneous comparisons and those for a single comparison is the multiple of the estimated standard deviation. Also note that the sample sizes must be equal when using the studentized range approach. 

Example  
Data  We use the data from a previous example.  
Set of all pairwise comparisons 
The set of all pairwise comparisons consists of:
_{2}  _{3}, _{2}  _{4}, _{3}  _{4} 

Confidence intervals for each pair 
Assume we want a confidence coefficient of 95 percent, or .95.
Since r = 4 and n_{t} = 20, the required
percentile of the studentized range distribution is
q_{.05; 4,16}. Using the Tukey method for each of
the six comparisons yields:


Conclusions  The simultaneous pairwise comparisons indicate that the differences _{1}  _{4} and _{2}  _{3} are not significantly different from 0 (their confidence intervals include 0), and all the other pairs are significantly different.  
Unequal sample sizes  It is possible to work with unequal sample sizes. In this case, one has to calculate the estimated standard deviation for each pairwise comparison. The Tukey procedure for unequal sample sizes is sometimes referred to as the TukeyKramer Method. 