Download Post Hoc Tests - Mathematics and Statistics - Study Notes and more Study notes Mathematical Statistics in PDF only on Docsity!
1
Appendix 10: Post Hoc Tests
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 ni Number of observations at level i xi Mean at level i s (^) i Standard deviation of level i v (^) i Degrees of freedom for level i, ni − 1 spp Square root of the mean square error
( )
x x
n
j ij^ i
n i
k
i i
k
i −
= =
=
2 1 1
1
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
( ni ) i
k − =
1 vi j , Absolute difference between the i th and j th means | x (^) i − xj |
k *^ k^ H k^^ −^1 S/^2
1 These algorithms apply to SPSS 7.0 and later releases.
Qi j , s pp (^) n (^) i nj
% '
&
( 0
)
nh Harmonic mean of the sample size n k n
h i i k
−
≤ ≤
∑
1
1 Qh (^) s (^) pp / nh
Studentized Range and Studentized Maximum Modulus
Let x (^) 1 , x (^) 2 , !, xr be independent and identically distributed N ( μ, σ ). Let sm be an estimate of σ with m degrees of freedom, which is independent of the (^) Y d x (^) i , and ms (^) m^2 / σ 2 ~χ 2. Then the quantity
S
x x x x r m (^) s
r r m
,
max( , , ) min( , , )
1!^ − 1!
is called the Studentized range. The upper-ε critical point of this distribution is denoted by S ε (^) , , r m. The quantity
M
x x r m (^) s
r m
,
max(| |, ,| |) = 1
is called the Studentized maximum modulus. The upper-ε critical point of this distribution is denoted as M (^) ε, , r m.
Methods
The tests are grouped below according to assumptions about sample sizes and variances.
- If x (^) ( ) k − x (^) ( ) 1 > Qh R ε, , k f , there is a significant range and the ranges of the two sets of k − 1 means { x (^) ( ) 1 ,! , x (^) ( k − 1 )} and { x (^) ( ) 2 , !, x (^) ( ) k } are compared with Q (^) h R ε, k − 1 , f. SPSS continues to examine smaller subsets of means as long as the previous subset has a significant range. For some tests, Qi j , is used instead of Qh , as indicated under specific tests described in “Range Values” below.
- 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 ε (^) , , r f = S ε, , r f
Tukey’s Honestly Significant Difference Test (TUKEY)
R ε (^) , , r f = S ε, , k f
The confidence intervals of the mean difference are calculated using Qi j , instead of Qh.
Tukey’s b (TUKEYB)
R
S S
r f
r f k f ε
ε ε , ,
, , , ,
Duncan’s Multiple Range Test (DUNCAN)
R ε (^) , , r f = S α r , , r f where α r = 1 − ( 1 − ε) r −^1
Scheffé Test (SCHEFFE)
R ε (^) , , r f = 2 ( k − 1 ) F 1 (^) −ε( k − 1 , f )
The confidence intervals of the mean difference are calculated using Qi j , instead of Qh.
Hochberg’s GT2 (GT2)
R ε (^) , , r f = 2 M ε, k *, f
The confidence intervals of the mean difference are calculated using Qi j , instead of Qh.
Gabriel’s Pairwise Comparisons Test (GABRIEL)
The test statistic and the critical point are as follows:
| x x s (^) , *, n n i j pp M i j
− |≥ ( + ) k f
2 ε^
For homogeneous subsets, nh is used instead of ni 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 (^) i − x (^) j> Qi j (^) , R ε, , k f
where the range, R (^) ε, , k f , for each test is provided below.
T T
i k
= (^) i 1 ≤ ≤
max {|^ |}
where T x x
s n n
i
i
dd i
0
and s
x x
n
dd
j ij^ i
n i
k
i i
k
i 2
2 0 1
0
= =
=
∑ ∑
∑
.
Dunnett’s One-Tailed t Test (DUNNETTL)
This Dunnett’s one-tailed t test indicates whether the mean at any level is smaller than a reference category.
x x dU s i k v dd (^) n ni
0
,
ε
where dU (^) k v^ ε,^ is the upper 100 ε percentage point of the distribution of
T T
i k
= (^) i 1 ≤ ≤
max { }
Confidence intervals are given only for the difference between the control group and the other groups.
Dunnett’s One-Tailed t Test (DUNNETTR)
This Dunnett’s one-tailed t test indicates whether the mean at any level is larger than a reference category.
x x dL s n n i k v dd i
0
,
ε
where dL^ ε k v^ , is the lower 100 ε percentage point of the distribution of
T T
i k
= (^) i 1 ≤ ≤
max {^ }
Confidence intervals are given only for the difference between the control group and the other groups.
Ryan-Einot-Gabriel-Welsch (R-E-G-W) Multiple Stepdown Procedures
For the R-E-G-W F test and the R-E-G-W Q test, a new significant level, γ (^) r , based on the number of steps between means is introduced:
γ
ε ε r
r k (^) r k
r k
7 8 9 u
( ) if if
/
Note : For homogeneous subsets, the ni and n (^) j are used for the R-E-G-W F test and the R-E-G-W Q test. To apply these methods, the procedures are same as in “Constructing Homogeneous Subsets” on p. 3, using the tests provided below.
Ryan-Einot-Gabriel-Welsch Based on the Studentized Range Test (QREGW)
The R-E-G-W Q test is based on
max ) min , )} / i j R
x (^) i x (^) j n (^) i n (^) j s (^) pp Srr f ,
− ≥ γ
Ryan-Einot-Gabriel-Welsch Procedure Based on an F Test (FREGW)
The R-E-G-W F test is based on
n x n x n
r s
F
i i i i i R
i i R i R
pp rr^ f
2
2
%
'
& &
(
0
) )
%
'
& &
(
0
) )
−
∈ ∈ ∈ −
∑ ∑ /∑
, , ( ) γ ,
where r = j − i + 1 and summations are over R = X i , !, j c.
Unequal Sample Sizes and Unequal Variances
The tests in this section are based on assumptions that variances are unequal and sample sizes are unequal. An estimate of the degrees of freedom is used. The estimator is
Dunnett’s C (C)
R
S s n S s n s n s n r v
k n i i k n j j
i i j j
i j ε
ε ε , ,
− 1 −
2 1
2
2 2
References
Cheng, P. H. & Meng, C. Y. K. (1992), A New Formula for Tail probabilities of DUNNETT’s T with Unequal Sample Sizes, ASA Proc. Stat. Comp ., 177-182.
Duncan, D. B. (1955), Multiple Range and Multiple F tests, Biometrics , 11, 1-42.
Duncan, D. B. (1975), t Tests and Intervals for Comparisons Suggested by the Data, Biometrics , 31, 339-360.
Dunnett, C. W. (1955), A Multiple Comparisons Procedure for Comparing Several Treatments with a Control, JASA , 50, 1096-1121.
Dunnett, C. W. (1980), Pairwise Multiple Comparisons in Homogeneous Variance, Unequal Sample Size Case, JASA , 75, 789-795.
Dunnett, C. W. (1980), Pairwise Multiple Comparisons in the Unequal Variance Case, JASA , 75, 796-800.
Dunnett, C. W. (1989), Multivariate Normal Probability Integrals with Product Correlation Structure, Applied Statistics , 38, 564-571.
Einot, I. and Gabriel, K. R. (1975), A Study of the powers of Several Methods of Multiple Comparisons, JASA , 70, 574-783.
Gabriel, K. R. (1978), A Simple method of Multiple Comparisons if Means, JASA , 73, 724-729.
Games, P.A. and Howell, J.F.(1976), Pairwise Multiple Comparison Procedures with Unequal N’s and/or Variances: A Monte Carlo Study, J. Educ. Statist ., 1, 113-125.
Gebhardt, F. (1966), Approximation to the Critical Values for Duncan’s Multiple Range Test, Biometrics , 22, 179-182.
Hochberg, Y. (1974), Some Generalizations of the T-method in Simultaneous Inference, J. Mult. Anal ., 4, 224-234.
Hochberg, Y. and Tamhane, A. C. (1987), Multiple Comparison Procedures, New York: John Wiley & Sons, Inc.
Hsu, J. C. (1989), Multiple Comparison Procedures, ASA Short Course.
Miller, R. G. (1980), Simultaneous Statistical Inference, 2nd., Springer-Verlag, N.Y.
Milliken, G. A. and Johnson, D. E. (1984), Analysis of messy data, New York: John Wiley & Sons, Inc.
Ramsey, P. H. (1978), Power Differences Between Pairwise Multiple Comparisons, JASA , 73, 479-485.
Ryan, T. A. (1959), Multiple Comparisons in Psychological Research, Psychol. Bull ., 56, 26-47.
Ryan, T. A. (1960), Significance Tests for Multiple Comparison of Proportions, Variances, and Other Statistics, Psychol. Bull ., 57, 318-328.
Scheffe, H. (1953), A method for Judging All Contrasts in the Analysis of Variance, Biometrika, 40, 87-104.
Scheffe, H. (1959), The analysis of Variance, New York: John Wiley & Sons, Inc.
Searle, S. R. (1971), Linear Models, New York: John Wiley & Sons, Inc.
Sidak, Z. (1967), Rectangular confidence regions for the means of multivariate normal distributions, JASA , 62, 626-633.
SAS Institute, Inc. (1990), SAS/STAT User’s Guide, Version 6 4th Edition, SAS Institute Inc., Cary, NC.
Tamhane, A.C. (1977), Multiple Comparisons in Model I One-Way ANOVA with Unequal Variances, Commun. Statist ., A 6, 15-32.
Tamhane, A.C. (1979), A Comparison of Procedures for Multiple Comparisons of Means with Unequal Variances, JASA , 74, 471-480.
