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" "4" "4"
[20] "4"
= c(data[1,], data[2,], data[3,], data[4,]) samples
TheCoatlessProfessor
April 8, 2014
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 $ ^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:
\[q_{ r,v } = \frac{ W }{ s }\]
Let $ W=X_{ n }-X_{ 1 } $
Internally Studentized Range: Population $ ^2 $ is unknown
\[q_{ n,n-1 } = \frac{ W }{ s } = \frac{ X_{ n }-X_{ 1 } }{ s },\]
where $ s = ( )^{ 1/2 } $
Let $ W=X_{ n }-X_{ 1 } $
Externally Studentized Range: Population $ ^2 $ is unknown AND an independent estimator $ s_{ v }^2 $ of $ ^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 $ $ can be removed by studentization. So, $ S_{ v }^{ 2 } $ is changable to $ vS_{ v }^{ 2 }/^2 _{ v }^2 $, independent of $ W $.
Let $ W=X_{ n }-X_{ 1 } $
Externally and Internally Studentized Range:
\[T=\frac{ W }{ \tilde{ s } }=\frac{ X_{ n }-X_{ 1 } }{ \tilde{ s } },\]
where $ = ( +vs_{ v }^2 )^{ 1/2 } $
We will be using the Externally Studentized Range….
Specifically, our definition resembles….
Let \({ Y_{ ij } } \mathop \sim\limits^{ iid } N\left( { 0 ,{ \sigma ^2 } } \right)\), where $j=1,,n $ and $ i=1,,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 $ { Y }_i $.
\[q_{ n,v } = \frac{ W }{ S/\sqrt{ n } } = \frac{ max\left({ \bar{ Y_{ i\cdot } } }\right)-min\left({ \bar{ Y_{ i\cdot } } }\right) }{ \sqrt{ \left({ MS_{ error }/n }\right) } }\]
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 |
[1] "1" "1" "1" "1" "1" "2" "2" "2" "2" "2" "3" "3" "3" "3" "3" "4" "4" "4" "4"
[20] "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 19.3
[16] 23.5 19.6 23.7 20.8 23.9
All pairwise differences $ { i }-{ j } $ are given by $ 100 ({ 1-} ) $% 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.
\[\left( { 28.86 - 22.30 } \right) \pm \frac{ { { q_{ .05 ;4,20 - 4 } } } }{ { \sqrt 2 } } \cdot \sqrt{ \left(116.3/16\right) } \cdot \sqrt { \frac{ 1 }{ { { 5 } } } + \frac{ 1 }{ { { 5 } } } } , \]
$q_{ .05 ;4,20 - 4 } = qtukey(0.95,4,20-4) = $ 4.046093
\(6.56 \pm 4.8788568\)
Tukey multiple comparisons of means
95% family-wise confidence level
Fit: aov(formula = model)
$`factor(dose_type)`
diff lwr upr p adj
2-1 -3.82 -8.698941 1.058941 0.1546383
3-1 -6.36 -11.238941 -1.481059 0.0088837
4-1 -6.56 -11.438941 -1.681059 0.0069985
3-2 -2.54 -7.418941 2.338941 0.4661560
4-2 -2.74 -7.618941 2.138941 0.4026925
4-3 -0.20 -5.078941 4.678941 0.9993976
So, only 3-1 and 4-1 are significant.
That just about covers the Tukey Test Statistic…. The only bits left are an alternate formulation and the references used to construct this post.
Let \({ X_1 }, \cdots ,{ X_m } \mathop \sim\limits^{ iid } N\left( { 0 ,{ \sigma ^2 } } \right)\), where $ n $, and let $ Z $ be $ ^{ 2 } $ with $ n $ degrees of freedom.
\[ q = \frac{ { \mathop { \max }\limits_{ 1 \leqslant i,j \leqslant m } \left| { { X_i } - { X_j } } \right| } }{ { \sqrt { Z/n } } } = \frac{ { { X_{ m:m } } - { X_{ 1:m } } } }{ { \sqrt { Z/n } } }\]
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 = ,{ }{ S_E } = { S_p } $
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.