Tukey–Kramer method

The Tukey method, named for John Tukey is a single-step multiple comparison procedure which applies simultaneously to the set of all pairwise comparisons

$\mu _{i}-\mu _{j}$

The confidence coefficient for the set, when all sample sizes are equal, is exactly 1-α. For unequal sample sizes, the confidence coefficient is greater than 1-α. In other words, the Tukey method is conservative when there are unequal sample sizes.

Studentized range distribution

The Tukey method uses the studentized range distribution. Suppose we have r independent observations y₁, ..., y_r from a normal distribution with mean μ and variance σ². Let w be the range for this set; i.e., the maximum minus the minimum. Now suppose that we have an estimate s² of the variance σ² which is based on ν degrees of freedom and is independent of the y_i. The studentized range is defined as

$q_{r,\nu }=w/s$

The distribution of q has been tabulated and appears in many textbooks on statistics. In addition, the R programming language offers a cumulative distribution function (ptukey) and a quantile function (qtukey) for q.

Tukey's method

The Tukey confidence limits for all pairwise comparisons with confidence coefficient of at least 1 - α are

${\bar {y}}_{i\cdot }-{\bar {y}}_{j\cdot }\pm {\frac {q}{\sqrt {2}}}q_{\alpha ;r;N-r}{\hat {\sigma }}_{\epsilon }{\sqrt {\frac {2}{n}}}\quad i,j=1,\ldots ,r\quad i\neq j$

Notice that the point estimator and the estimated variance are the same as those for a single pairwise comparison. 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.

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 Tukey-Kramer Method.

Comparison with the Scheffé method

If only pairwise comparisons are to be made, the Tukey-Kramer method will result in a narrower confidence limit, which is preferable. In the general case when many or all contrasts might be of interest, the Scheffé method tends to give narrower confidence limits and is therefore the preferred method.

References

• Tukey's method

 This article incorporates public domain material from the National Institute of Standards and Technology