DISCLAIMER

ADJUSTP is provided "as is" without warranty of any kind.
The entire risk as to the quality, performance, and fitness
for intended purpose is with you. You assume responsibility
for the selection of the program and for the use of results
obtained from that program.

DESCRIPTION

There are 8 p-value adjustment algorithms available. They include
one-step, step-down and step-up adjustment methods.

1. One-Step Bonferroni
2. One-Step Sidak [References 1 and 2]
3. Step-Down Holm [3]
4. Step-Down Sidak [1,2,4 and 5]
5. Step-Down Finner [4 and 5]
6. Step-Up Hommel [6 and 7]
7. Step-Up Hochberg [8]
8. Step-Up Simes [9]

All the results have been tested with Multiplicity program (2.0) ,
by Barry W. Brown, The University of Texas M D Anderson Cancer Center.

ALGORITHMS

The following nomenclature will be used:

n     = number of p-values
p(i)  = ith smallest p-value
p#(i) = adjusted value of p(i)

In one-step adjustment methods p-values are compared to a
predetermined value that is a function of alpha, the significance
level, and n, the number of p-values.

- One-step Bonferroni:
  p#(i) = n*p(i)

- One-Step Sidak:
  p#(i) = 1-(1-p(i))^n

In step-down methods p-values are examined in order, from smallest
to largest. Once a p-value is found that is large according to a 
criterion based on alpha and the p-value's position in the list, that
p-value and all larger p-values are accepted. 

- Step-down Holm:
  p#(i) = (n-i+1)*p(i)

- Step-down Sidak:
  p#(i) = 1-(1-p(i))^(n-i+1)

- Step-down Finner:
  p#(i) = 1-(1-p(i))^(n/i)

In step-up methods p-values are examined in order, from largest to
smallest. Once a p-value is found that is small according to a
criterion based on alpha and the p-value's position in the list,
that p-value and all smaller p-values are rejected.

- Step-up Hommel:
  p#(i) = n*Cn*p(i)/i
  Cn = 1+1/2+ ... +1/n

- Step-up Hochberg:
  p#(i) = (n-i+1)*p(i)

- Step-up Simes:
  p#(i) = n*p(i)/i

See [10] for a general reference on p-value adjustment algorithms.

REFERENCES

 1. Sidak Z (1967) "Rectangular Confidence Regions for the Means
    of Multivariate Normal Distributions" American Statistical
    Association, 62, 626-633.
 2. Sidak Z (1971) "On probabilities of rectangles in multivariate 
    Student distributions: their dependence on correlations" Ann Math 
    Statist, 42, 169-175.
 3. Holm S (1979) "A Simple Sequentially Rejective Multiple Test
     Procedure" Scandinavian Journal of Statistics, 6, 65-70.
 4. Finner H (1990) "Some New Inequalities for the Rnad Distribution
    With Application to the determination of Optimum Significance
    Levels of Multiple Range Tests" Journal of the American
    Statistical Association, 85, 191-194.
 5. Finner H (1993) "On A Monotonicity Problem in Step-Down Multiple
    Test Procedures" Journal of the American Statistical Association,
    88, 920-923.
 6. Hommel G (1988) "A stagewise rejective multiple test procedure
    based on a modified Bonferroni test" Biometrika, 75, 383-386.
 7. Falk RW (1989) "Hommel's Bonferroni-type inequality for unequally
    spaced levels" Biometrika, 76, 190-191.
 8. Hochberg Y and Benjamini Y (1990) "More powerful procedures for
    multiple significance testing" Statistics in Medicine, 9, 811-818.
 9. Simes RJ (1986) "An improved Bonferroni procedure for multiple
    tests of significance" Biometrika, 73, 751-754.
10. Westfall PH and Young SS (1993), Resampling-Based Multiple
    Testing: examples and methods for p-value adjustment. 
    New York: John Wiley and Sons.

**********************************************************************
* BEGINNING OF SYNTAX

* Create dataset with p-values. Replace by your own *.
DATA LIST LIST / pvalue(F9.4).
BEGIN DATA
0.0728
0.0023
0.3829
0.0041
0.0101
0.4557
END DATA.

* SOME AUXILIARY VARIABLES *.
COMPUTE id = $CASENUM.
FORMAT id (F2.0).
SORT CASES BY  pvalue (A) .
COMPUTE pos = $CASENUM.
FORMAT pos (F2.0).
* Calculate the number of p values *.
PRESERVE.
SET ERRORS=NONE RESULTS=NONE.
RANK pvalue /n into N /PRINT = NO.
RESTORE.

* ADJUSTED P-VALUES *.
* (1) ONE STEP METHODS *.
COMPUTE bonferr = MIN(1,pvalue*n).
COMPUTE sidak = 1-(1-pvalue)**n.
* (2) STEP-DOWN METHODS *.
COMPUTE holm = MIN(1,(n-pos+1)*pvalue).
IF (holm LT LAG(holm)) holm = LAG(holm).
COMPUTE downsidk = 1-(1-pvalue)**(n-pos+1).
IF (downsidk LT LAG(downsidk)) downsidk = LAG(downsidk).
COMPUTE finner = 1-(1-pvalue)**(n/pos).
IF (finner LT LAG(finner)) finner = LAG(finner).
* (3) STEP-UP METHODS *.
COMPUTE cn = cn+1/pos .
LEAVE cn. /* With thanks to Richard Ristow for showing me the use of LEAVE *.
SORT CASES BY pos(D).
IF cn LT LAG(cn) cn = LAG(cn).
COMPUTE hommel = MIN(1,cn*n*pvalue/pos).
IF (hommel GT LAG(hommel)) hommel = LAG(hommel).
COMPUTE hochberg = (n-pos+1)*pvalue.
IF (hochberg GT LAG(hochberg)) hochberg = LAG(hochberg).
COMPUTE simes = n*pvalue/pos.
IF (simes GT LAG(simes)) simes = LAG(simes).
EXECUTE.
DELETE VARIABLES pos,n,cn.

* FINAL REPORT *.
FORMAT bonferr to simes (f9.4).
VARIABLE LABELS id 'Nr.' /pvalue 'Original p-value'
        /bonferr 'One-step Bonferroni' /sidak 'One-step Sidak'
        /holm 'Step-down Holm' /downsidk 'Step-down Sidak'
        /finner 'Step-down Finner' /hommel 'Step-up Hommel'
        /hochberg 'Step-up Hochberg' /simes 'Step-up Simes'.
SORT CASES BY id (A).
SUMMARIZE
 /TABLES = pvalue bonferr TO simes
 /FORMAT = LIST NOCASENUM TOTAL
 /TITLE = 'Original & Adjusted p-values'
 /MISSING = VARIABLE
 /CELLS = NONE.
********************************************************************************
* END OF SYNTAX.