# Our Definition of Tukey’s Test Statistic

Suppose $r$ independent observations denoted: ${ Y_1 }, \cdots ,{ Y_r } \mathop \sim\limits^{ iid } N\left( { \mu ,{ \sigma ^2 } } \right)$ Let $W = range(Y) = max(Y_{ i })-min(Y_{ i })$

Now, suppose that we have an estimate $s^2$ of the variance $\sigma^2$ which is based on $ν$ degrees of freedom and is independent of the $Y_i$ $\left(i = 1,\cdots,r \right)$. $v$ is usually derived from analysis of variance.

Then, the Tukey’s Test Statistic is:

## Ranges: Internalized

Let $W=X_{ n }-X_{ 1 }$

Internally Studentized Range: Population $\sigma^2$ is unknown

where $s = \left( \frac{ 1 }{ \left( { n - 1 } \right) }{ \sum\limits_{ i = 1 }^n { { { \left( { { X_i } - \bar X } \right) }^2 } } } \right)^{ 1/2 }$

## Ranges: Externalized

Let $W=X_{ n }-X_{ 1 }$

Externally Studentized Range: Population $\sigma^2$ is unknown AND an independent estimator $s_{ v }^2$ of $\sigma^2$ is available with degrees of freedom $v$.

The dependence of the distribution $W$ on unknown $\sigma$ can be removed by studentization. So, $S_{ v }^{ 2 }$ is changable to $vS_{ v }^{ 2 }/\sigma^2 \sim \chi_{ v }^2$, independent of $W$.

## Ranges: Both

Let $W=X_{ n }-X_{ 1 }$

Externally and Internally Studentized Range:

where $\tilde{ s } = \left( \frac{ 1 }{ \left( { n - 1 + v } \right) }{ \sum\limits_{ i = 1 }^n { { { \left( { { X_i } - \bar X } \right) }^2 } } } +vs_{ v }^2 \right)^{ 1/2 }$

# Our use

We will be using the Externally Studentized Range….

Specifically, our definition resembles….

## Tukey’s Studentized Range Test Statistic ANOVA

Let ${ Y_{ ij } } \mathop \sim\limits^{ iid } N\left( { 0 ,{ \sigma ^2 } } \right)$, where $j=1,\cdots,n$ and $i=1,\cdots,k$ be independent observations in a balanced one-way ANOVA with $k$ treatments.

Then $\bar{Y}_1, \cdots , \bar{Y}_k$ are the sample averages and $S^2$ is the independent and unbiased estimator of $\sigma^2$ based on $v = k\left( { n-1 } \right)$.

Let $W$ be the range of $\bar{ Y }_i$.

## Example - Data

Consider a dosage regiment with repeated measures:

Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Row Total
Dose 1 27.0 26.2 28.8 33.5 28.8 144.3
Dose 2 22.8 23.1 27.7 27.6 24.0 125.2
Dose 3 21.9 23.4 20.1 27.8 19.3 112.5
Dose 4 23.5 19.6 23.7 20.8 23.9 111.5
Col Total 95.2 92.3 100.3 109.7 96.0 493.5

## Tukey’s 95% C.I. pairwise comparison process

All pairwise differences $\mu_{ i }-\mu_{ j }$ are given by $100 \times \left({ 1-\alpha } \right)$% C.I.

Note: Always start with the largest mean and smallest mean pair, if the result is not significant, then the result will hold for all means between the largest and smallest.

## Largest vs. Smallest

$q_{ .05 ;4,20 - 4 } = qtukey(0.95,4,20-4) =$ 4.046093

$6.56 \pm 4.8788568$

## In R:

So, only 3-1 and 4-1 are significant.

# That’s All folks!

That just about covers the Tukey Test Statistic…. The only bits left are an alternate formulation and the references used to construct this post.

## Tukey’s Test Statistic Rewritten

Let ${ X_1 }, \cdots ,{ X_m } \mathop \sim\limits^{ iid } N\left( { 0 ,{ \sigma ^2 } } \right)$, where $n \ge 2$, and let $Z$ be $\chi^{ 2 }$ with $n$ degrees of freedom.

In the case of $m=2$, $q$ closely resembles the two sample $t$ test statistic.

That is, $X_{ 1 }$ and $X_{ 2 }$ are are taken to be the standardized sample means of the two samples and $Z/n$ is the pooled sample variance, $S_p^2 = \frac{ { S_1^2\left( { { n_1 } - 1 } \right) + S_2^2\left( { { n_2 } - 1 } \right) } }{ { { n_1 } + { n_2 } - 2 } },{ \text{ } }{ S_E } = { S_p }\sqrt { \frac{ 1 }{ { { n_1 } } } + \frac{ 1 }{ { { n_2 } } } }$

# References

• David, H. A.. “Studentized Range.” Encyclopedia of statistical sciences. New York: Wiley, 2006. 1-3. Print.

• Falk, Michael, and Frank Marohn. “The One-Way Analysis of Variance.” Foundations of statistical analyses and applications with SAS. Basel: Birkhauser Verlag, 2002. 193-194. Print.

• Harter, H. Leon, and N. Balakrishnan. “The Studentized Range of Samples from a Normal Population.” Tables for the use of range and studentized range in tests of hypotheses. Boca Raton, Fla.: CRC Press, 1998. 52-53. Print.

• Hochberg, Yosef. “Studentized Range.” Encyclopedia of Biostatistics. Hoboken, NJ: John Wiley & Sons, Ltd, 2005. 1-3. Print.

• Lowry, Richard. “Ch 14: One-Way Analysis of Variance for Independent Samples.” Concepts & Applications of Inferential Statistics. 2000. 1-3. Print.