Tukey's range test

1. The observations being tested are independent within and among the groups.
2. The groups associated with each mean in the test are normally distributed.
3. There is equal within-group variance across the groups associated with each mean in the test (homogeneity of variance).

Tukey's test is based on a formula very similar to that of the t-test. In fact, Tukey's test is essentially a t-test, except that it corrects for family-wise error rate.

The formula for Tukey's test is

${\displaystyle q_{s}={\frac {Y_{A}-Y_{B}}{SE}},}$

where Y_A is the larger of the two means being compared, Y_B is the smaller of the two means being compared, and SE is the standard error of the sum of the means.

This q_s value can then be compared to a q value from the studentized range distribution. If the q_s value is larger than the critical value q_α obtained from the distribution, the two means are said to be significantly different at level ${\displaystyle \alpha :0\leq \alpha \leq 1}$.[3]

Since the null hypothesis for Tukey's test states that all means being compared are from the same population (i.e. μ₁ = μ₂ = μ₃ = ... = μ_k), the means should be normally distributed (according to the central limit theorem). This gives rise to the normality assumption of Tukey's test.

The Tukey method uses the studentized range distribution. Suppose that we take a sample of size n from each of k populations with the same normal distribution N(μ, σ²) and suppose that ${\displaystyle {\bar {y}}}$_min is the smallest of these sample means and ${\displaystyle {\bar {y}}}$_max is the largest of these sample means, and suppose S² is the pooled sample variance from these samples. Then the following random variable has a Studentized range distribution.

${\displaystyle q={\frac {{\overline {y}}_{\max }-{\overline {y}}_{\min }}{S{\sqrt {2/n}}}}}$

This value of q is the basis of the critical value of q, based on three factors:

1. α (the Type I error rate, or the probability of rejecting a true null hypothesis)
2. k (the number of populations)
3. df (the number of degrees of freedom (N – k) where N is the total number of observations)

The distribution of q has been tabulated and appears in many textbooks on statistics. In some tables the distribution of q has been tabulated without the ${\displaystyle {\sqrt {2}}}$ factor. To understand which table it is, we can compute the result for k = 2 and compare it to the result of the Student's t-distribution with the same degrees of freedom and the same α. In addition, R offers a cumulative distribution function (ptukey) and a quantile function (qtukey) for q.

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

${\displaystyle {\bar {y}}_{i\bullet }-{\bar {y}}_{j\bullet }\pm {\frac {q_{\alpha ;k;N-k}}{\sqrt {2}}}{\widehat {\sigma }}_{\varepsilon }{\sqrt {\frac {2}{n}}}\qquad i,j=1,\ldots ,k\quad i\neq j.}$

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. ${\displaystyle {\widehat {\sigma }}_{\varepsilon }}$ is the standard deviation of the entire design, not just that of the two groups being compared. It is possible to work with unequal sample sizes. In this case, one has to calculate the estimated standard deviation for each pairwise comparison as formalized by Clyde Kramer in 1956, so the procedure for unequal sample sizes is sometimes referred to as the Tukey–Kramer method which is as follows:

${\displaystyle {\bar {y}}_{i\bullet }-{\bar {y}}_{j\bullet }\pm {\frac {q_{\alpha ;k;N-k}}{\sqrt {2}}}{\widehat {\sigma }}_{\varepsilon }{\sqrt {{\frac {1}{n}}_{i}+{\frac {1}{n}}_{j}}}\qquad }$

where n _i and n _j are the sizes of groups i and j respectively. The degrees of freedom for the whole design is also applied.

Both ANOVA and Tukey–Kramer tests are based on the same assumptions. However, these two tests for k groups (i.e. μ₁ = μ₂ = ... = μ_k) may result in logical contradictions when k> 2, even if the assumptions hold. It is possible to generate a set of pseudorandom samples of strictly positive measure such that hypothesis μ₁ = μ₂ is rejected at significance level ${\displaystyle 1-\alpha >0.95}$ while μ₁ = μ₂ = μ₃ is not rejected even at ${\displaystyle 1-\alpha =0.975}$.[4]

• Family-wise error rate
• Newman–Keuls method

• Montgomery, Douglas C. (2013). Design and Analysis of Experiments (Eighth ed.). Wiley. Section 3.5.7.

• NIST/SEMATECH e-Handbook of Statistical Methods: Tukey's method