Chapter 4: Analysis of Variance

SNKMC

Performs Student-Newman-Keuls multiple comparison test.

Required Arguments

YMEANS — Vector of length NGROUP containing the means.   (Input)

SEMEAN — Effective estimated standard error of a mean.   (Input)
In fixed effects models, SEMEAN equals the estimated standard error of a mean. For example, in a one-way model

            where s2 is the estimate of σ2 and n is the number of responses in a sample mean. In models with random components, use

            where SEDIF is the estimated standard error of the difference of two means.

DFSE — Degrees of freedom associated with SEMEAN.   (Input)

ALPHA — Significance level of test.   (Input)
ALPHA must be in the interval [0.01, 0.10].

IEQMNS — Vector of length NGROUP 1 indicating the size of groups of means declared to be equal.   (Output)
IEQMNS(I) = J indicates the I-th smallest mean and the next J 1 larger means are declared equal. IEQMNS(I) = 0 indicates no group of means starts with the I-th smallest mean.

Optional Arguments

NGROUP — Number of groups under consideration.   (Input)
Default: NGROUP = size (YMEANS,1).

IPRINT — Printing option.   (Input)
Default: IPRINT = 0.

IPRINT                  Action

0                             No printing is performed.

1                              Printing is performed.

FORTRAN 90 Interface

Generic:          CALL SNKMC (YMEANS, SEMEAN, DFSE, ALPHA, IEQMNS [,…])

Specific:                             The specific interface names are S_SNKMC and D_SNKMC.

FORTRAN 77 Interface

Single:            CALL SNKMC (NGROUP, YMEANS, SEMEAN, DFSE, ALPHA, IPRINT, IEQMNS)

Double:                              The double precision name is DSNKMC.

Description

Routine SNKMC performs a multiple comparison analysis of means using the Student-Newman-Keuls method. The null hypothesis is equality of all possible ordered subsets of a set of means. This null hypothesis is tested using the studentized range for each of the corresponding subsets of sample means. The method is discussed in many elementary statistics texts, e.g., Kirk (1982, pages 123125).

Comments

Workspace may be explicitly provided, if desired, by use of S2KMC/DS2KMC. The reference is:

CALL S2KMC (NGROUP, YMEANS, SEMEAN, DFSE, ALPHA, IPRINT, IEQMNS, WK, IWK)

The additional arguments are as follows:

WK — Vector of length NGROUP containing YMEANS in ascending order.   (Output)

IWK — Work vector of length 2 * NGROUP.

Example

A multiple comparisons analysis is performed using data discussed by Kirk (1982, pages 123125). In the output, means that are not connected by a common underline are declared different.

 

      USE UMACH_INT

      USE SNKMC_INT

 

      IMPLICIT   NONE

      INTEGER    IEQMNS(4), IPRINT, N, NOUT

      REAL       ALPHA, DFSE, S2, SEMEAN, SQRT, YMEANS(5)

      INTRINSIC  SQRT

!

      DATA YMEANS/36.7, 48.7, 43.4, 47.2, 40.3/

!

      CALL UMACH (2, NOUT)

      S2     = 28.8

      N      = 10

      SEMEAN = SQRT(S2/N)

      DFSE   = 45.0

      ALPHA  = .01

      IPRINT = 1

      CALL SNKMC (YMEANS, SEMEAN, DFSE, ALPHA, IEQMNS, IPRINT=IPRINT)

      WRITE (NOUT,99999) IEQMNS

99999 FORMAT (' IEQMNS = ', 4I3)

      END

Output

 

 Group           1           5           3           4           2
 Mean        36.70       40.30       43.40       47.20       48.70

        AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
                    BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
                                CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

 IEQMNS =   3  3  3  0



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