Tukey-Kramer Procedure

Tukey and Kramer proposed a procedure for pairwise testing of means in a one-way analysis of variance with unequal sample sizes. The procedure is routinely applied after a significant overall F test, although the F test is not required. In Tukey's honestly significant difference (HSD) procedure, a single critical difference, CD, is calculated for each pair of means. That critical difference uses critical values from the Studentized range statistic. In particular,

where

q1-α(k, dfE) is the Studentized range statistic at Level A for k means and dfE,

dfE is the error degrees of freedom,

MSE is the error mean square, and

N is the common sample size.

In the Tukey-Kramer procedure, a different CD is required to evaluate the significance of the difference between each pair of means that are based on different sample sizes. The pairs are investigated independent of all other pairs. Critical differences are all based on the Studentized range distribution. The formula for the Tukey-Kramer CD testing means X¯iX¯j is

where

Ni is mean i (i = 1,…, k) and

Nj is mean j (j = 1,…, k), but j ≠ i.

Anthony J. Hayter proved that the Tukey-Kramer is conservative, that is, the probability of one or more Type I error never exceeding α even if applied without a significant F test. However, if a significant F test is required, then the Tukey-Kramer CD becomes [Page 1017]

The CD has the value that would be used only with k − 1 means. The resulting Hayter-Fisher version of the Tukey-Kramer procedure will be more powerful than the original Tukey-Kramer.

Illustrative Example

Consider the following data, in which four treatment groups are being compared to a control. Lower scores indicate better performance. The MSWG for these data is 3.0326, with dfWG = 179.

	Control	Treat 1	Treat 2	Treat 3	Treat 4
Mean	4.05	3.52	3.18	3.15	2.24
N	42	24	38	40	40

The analysis of variance (ANOVA) for these data would give F = 16.48 > 2.43 = F.95(4,160) > F.95(4,179) = CV. Therefore, we reject the full null hypothesis at the .05 level and proceed to pairwise testing.

The CD for the Control (N = 42) and Treatment 1 (N = 24) group means is obtained from

Applying the same calculations to the Control (N = 42) and Treatment 2 (N = 38) group means produces the CD

Applying the same calculations to the Control (N = 42) and Treatment 3 (N = 40) group means produces the CD Applying the same calculations to the Treatment 1 (N = 24) and Treatment 2 (N = 38) group means produces the CD Applying the same calculations to the Treatment 1 (N = 24) and Treatment 3 (N = 40) group means produces the CD

Applying the same calculations to the Treatment 2 (N = 38) and Treatment 3 (N = 40) group means produces the CD

Applying the same calculations to the Treatment 3 (N = 40) and Treatment 4 (N = 40) group means produces the CD

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Applying the same calculations to the Treatment 2 (N = 38) and Treatment 3 (N = 40) group means produces the CD

Applying the same calculations to the Treatment 3 (N = 40) and Treatment 4 (N = 40) group means produces the

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