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Appendix 10: Post Hoc Tests 1
Post hoc tests in SPSS are available in more than one procedure, including ONEWAY  and GLM . Notation
The following notation is used throughout this appendix unless otherwise stated:
k  Number of levels for an effect
n i  Number of observations at level i
x i  Mean at level i
s i  Standard deviation of level i
v i  Degrees of freedom for level i, n i −1
s pp  Square root of the mean square error
()()
x x n ij i j n i k i i k i −−===∑∑∑2
1111 ε
Experimentwise error rate under the complete null hypothesis α  Comparisonwise error rate r
Number of steps between means f  Degrees of freedom for the within-groups mean square
()n i
i k −=∑11
v i j , Absolute difference between the i th and j th means ||x x i j − k * k k −1205
/                                                  1 These algorithms apply to SPSS 7.0 and later releases.
Appendix 10
2Q i j ,
s n n pp i j 1211+
n h  Harmonic mean of the sample size n k
n h i i k =−≤≤∑1
1
Q h  s n pp h /
Studentized Range and Studentized Maximum Modulus
Let x x x r 12,,,K  be independent and identically distributed N (,)µσ. Let s m be an
estimate of σ with m  degrees of freedom, which is independent of the x i ;@
, and
ms m 222/~σχ. Then the quantity S x x x x s r m r r m
,max(,,)min(,,)=−11K K  is called the Studentized range . The upper-ε critical point of this distribution is
denoted by S r m ε,,.  The quantity
M x x s r m r m
,max(||,,||)=1K  is called the Studentized maximum modulus . The upper-ε critical point of this
distribution is denoted as M r m ε,,.
Methods
The tests are grouped below according to assumptions about sample sizes and
variances.
Equal Variances
The tests in this section are based on the assumption that variances are equal.
Appendix 10
3
Waller-Duncan t  Test
The Waller-Duncan t  test statistic is given by
v x x t w F q f S n i j i j B ,||,,,)/=−≥(2
where t w F q f B (,,,) is the Bayesian t value that depends on w  ( a measure of the relative seriousness of a Type I error versus a Type II error), the F  statistic for the one-way ANOVA,
F MS MS treat error
= and
S MS error 2=
Here f k n =−()1 and  q k =−1. MS error  and MS treat  are the usual mean squares in the ANOVA table.
Only homogeneous subsets are given for the Waller-Duncan t test . This method is for equal sample sizes. For unequal sample sizes, the harmonic mean n h  is used instead of n .
Constructing Homogeneous Subsets
comparisonsFor many tests assuming equal variances, homogeneous subsets are constructed using a range determined by the specific test being used. The following steps are used to construct the homogeneous subsets:
1. Rank the k  means in ascending order and denote the ordered means as
x x k ()(),,1K .
2. Determine the range value, R k f ε,,, for the specific test, as shown in “Range
Values” below. 3. If
x x Q R k h k f ()()−>1ε,,, there is a significant  range and the ranges of the two sets of k −1 means {x x k ()(),,11K −} and {x x k ()(),,2K } are compared with Q R h k f ε,,−1. SPSS continues to examine smaller subsets of means as long as the previous subset has a significant range.
Appendix 10
4 For some tests, Q i j , is used instead of Q h , as indicated under specific tests
described in “Range Values” below.
4. Each time a range proves nonsignificant , the means involved are included in a
single group—a homogeneous subset.
Range Values
Range values for the various types of tests are provided below.
Student-Newman-Keuls (SNK)
R S r f r f εε,,,,=
Tukey’s Honestly Significant Difference Test (TUKEY)
R S r f k f εε,,,,=
The confidence intervals of the mean difference are calculated using Q i j , instead of Q h . Tukey’s b (TUKEYB)
R S S r f r f k f εεε,,,,,,=+2
Duncan’s Multiple Range Test (DUNCAN)
R S r f r f r εα,,,,= where αεr r =−−−111() Scheffé Test (SCHEFFE)
R F f r f εε,,()(,)=−−−21k 1k 1
Appendix 10    5
The confidence intervals of the mean difference are calculated using Q i j , instead of Q h . Hochberg’s GT2 (GT2)
R M r f k f εε,,,,*=2
The confidence intervals of the mean difference are calculated using Q i j , instead of Q h .
Gabriel’s Pairwise Comparisons Test (GABRIEL) The test statistic and the critical point are as follows:
|,,*x x s n n M i j pp i j k f −≥+|(1212ε (1)
For homogeneous subsets, n h  is used instead of n i  and n j .
The confidence intervals of the mean difference are calculated based on equation
(1).
Least Significant Difference (LSD), Bonferroni, and Sidak
For the least significant difference, Bonferroni, and Sidak tests, only pairwise confidence intervals are given. The test statistic is
x x Q R i j k f i j −>,,,ε
where the range, R k f ε,,, for each test is provided below.
Least Significant Difference (LSD)
R F f r f αα,,(,)=−211

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