# 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:

$$q_{ r,v } = \frac{ W }{ s }$$

## Ranges: Internalized

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

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

$$q_{ n,n-1 } = \frac{ W }{ s } = \frac{ X_{ n }-X_{ 1 } }{ s },$$

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 $v$.

$$T=\frac{ W }{ s_v }=\frac{ X_{ n }-X_{ 1 } }{ s_v }$$

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:

$$T=\frac{ W }{ \tilde{ s } }=\frac{ X_{ n }-X_{ 1 } }{ \tilde{ s } },$$

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 $\bar{ Y }_i$.

$$q_{ n,v } = \frac{ W }{ S/\sqrt{ n } } = \frac{ max\left({ \bar{ Y_{ i\cdot } } }\right)-min\left({ \bar{ Y_{ i\cdot } } }\right) }{ \sqrt{ \left({ MS_{ error }/n }\right) } }$$

## 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

## In R

(dose_type = c(rep("1",5), rep("2",5), rep("3",5), rep("4",5)))

##  [1] "1" "1" "1" "1" "1" "2" "2" "2" "2" "2" "3" "3" "3" "3" "3" "4" "4"
## [18] "4" "4" "4"

samples = c(data[1,], data[2,], data[3,], data[4,])

##  [1] 27.0 26.2 28.8 33.5 28.8 22.8 23.1 27.7 27.6 24.0 21.9 23.4 20.1 27.8
## [15] 19.3 23.5 19.6 23.7 20.8 23.9


## ANOVA Table:

model = glm(samples~factor(dose_type))
aov(model)

## Call:
##    aov(formula = model)
##
## Terms:
##                 factor(dose_type) Residuals
## Sum of Squares           140.0935  116.3240
## Deg. of Freedom                 3        16
##
## Residual standard error: 2.69634
## Estimated effects may be unbalanced


## 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.

$$\left( { { { \bar Y }i } - { { \bar Y }j } } \right) \pm \frac{ { { q{ \alpha ;J,N - J } } } }{ { \sqrt 2 } } \cdot { s{ pooled } } \cdot \sqrt { \frac{ 1 }{ { { n_1 } } } + \frac{ 1 }{ { { n_2 } } } } ,{ \text{ } }{ s_{ pooled } } = \sqrt { MSW }$$

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

$$\left( { 28.86 - 22.30 } \right) \pm \frac{ { { q_{ .05 ;4,20 - 4 } } } }{ { \sqrt 2 } } \cdot \sqrt{ \left(116.3/16\right) } \cdot \sqrt { \frac{ 1 }{ { { 5 } } } + \frac{ 1 }{ { { 5 } } } } ,$$

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

$$6.56 \pm 4.8788568$$

## In R:

TukeyHSD(aov(model))

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
##
## Fit: aov(formula = model)
##

# 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.