- Email: [email protected]
A pocket calculator program for Duncan's New Multiple Range Test and analysis of variance
Cornput. B,ol. ,Mrd. Vol. 14, No. 3. pp 357-362, Pnnted in Great Britain.
1984
c
0010 4825’84 s3.00+ Ml 1984 Pcrgamon Press Ltd
A POCKET CALCULATOR PROGRAM FOR DUNCAN’S NEW MULTIPLE RANGE TEST AND ANALYSIS OF VARIANCE RICHARD Departments
of Medicine,
Biochemistry of Medicine,
A. LAWRENCE*
and Molecular Biology, Louisiana Shreveport, LA 71130, U.S.A.
(Receioed
13
State University
School
April 1983)
Abstract-A program for the TI-59 calculator to be used in analysis of variance and testing of significance ofdifferences between each mean and every other mean by Duncan’s New Multiple Range Test is presented. The test is both simple and powerful for data involving more than two treatment groups in a completely random design. Up to six groups with any number of replicates may be analyzed with this program. TI-59 calculator Multiple comparison
Duncan’s
New Multiple Range Test
Analysis
of variance
test
A problem that often confronts the biological investigator is the analysis of an experiment involving three or more treatment groups. Analysis of variance will indicate if there are significant differences among treatment groups, but often it is desirable to compare each treatment mean with every other treatment mean. Successive applications of Student’s t-test or the least significant difference (LSD.) to such a problem is inappropriate. Cochran and Cox [l] have indicated, for example, that in the extreme case when only the difference between the highest and lowest means is compared using the f-test or the L.S.D., the difference will be substantial even when no effect is present. It can be shown that with three treatment means, the t value calculated for the largest difference will be greater than the tabulated 5% level about 13% of the time, with six treatments 40%, with ten treatments 60’4, and with twenty treatments 90%. Thus, a test thought to be made at the 5% level is actually made at the 13% level with three groups, the 40% level with six groups, etc. Duncan [2] developed a test using a set of significant differences of increasing size depending on the closeness of the means after ranking. The program presented here uses Kramer’s modification [3] of Duncan’s New Multiple Range Test to allow for unequal replication in the treatment groups. However, when unequal replication is used the test is only approximate [4]. Following the computation of the analysis of variance, the procedure is: 1. Multiply S = error mean square by the appropriate tabulated Studentized significant ranges to obtain a set of intermediate significant ranges. 2. Rank the means in order from smallest to largest. Test the differences between means in the following order: largest minus smallest, largest minus second smallest . . . largest minus second largest, then second largest minus smallest, second largest minus second smallest and so on down to second smallest minus smallest. Each difference is compared to the corresponding intermediate significant range multiplied by [l/2 (l/r, + l/r,)]“’ where r1 and r2 are the number of replications in the two treatment groups being compared. 3. If the difference exceeds this least significant range then the difference is declared significant at the chosen level with one exception. No difference can be declared significant if
* Present address: The Audie L. Murphy Merton Minter Boulevard. San Antonio,
Memorial Veterans TX 78284. U.S.A. 351
Hospital.
Pulmonary
Disease Section,
11 lE, 7400
RICHARDA. LAWRENCE
358
Table 1. Effect of paraquat dose on 24 h urinary paraquat excretion
________________~_
____ -- ---.. ___-
PQ Dose
PQ Excreted
mglkg B.W.
p moles/ml
20
54 + 11* A
30
76210 101 +
40
l
____.-_- I_______-__
Mean+SEM
9 EC
123 + 4 C
50 --__---
AB
.____- _.__.._
Meansnotfollowedbythesameletterare
significantly different by Duncan'sNew Multiplerangetestat p c.05.
Table 2. .:8. I’ ,:i i3 ._a
Group
1 ‘3 13, 4 1 2 ::: ,
Group
14. 4!:1?~674i~l’~ 4
4
5; II
14 '::' _!_I1 .::; :z1 7,
!3’EM
5 5 E
Analysis
of Variance
1 1. 5 2 15 6 1I:I’~ '7. .-, c; Jl.
F Treatment d.f. Error d.f.
2~:3~4. I;I~::.Error Sum of Squares ~:&'~:~4i. ~51~:2 Treatment SS Total SS 5 43 iIl:3 . :~~y~:~ Studentized Significant Ranqes
‘3 L
Differences
101.
45
Mean
,>7e .
C’
I
4
Bet. Means
a
Duncan’s
new multiple
359
range test
Table 3. Press
Enter i.
Initialize
2.
Enter data for treatment group
4.
Repeat steps 2 and 3 for each treatment qroup.+ (If display flashes go to step 9)
5.
Compute
6.
Rank the means and :4 (rI + r2)llz for
SD and qroup
analysis
compute each comparison$
Enter significant studentized ranges (R) in order from largest to smallest 5
8.
Compute differences between and test for significancef
9.
If display flashes during data entry, reset raw data pointer by pressing D' and continue
pointer, + If raw
recording
4 For
A
1
Xi1
A
2
Xi2
XiJ
A
j
the
the
-
K’S
means
-
keyboard
or from
statistics
intermediate on magnetic
(or side).
XiJ
6
Mean, SEW
C
F, Treatment d.f., Error d.f., Error S.S., Treatment S.S., total S.S.
10
E
-
R's as
A'
-
Differences in order tested. Sig. differences followed by asterisks
magnetic manual
cards except
following that
B prints
instructions
D' resets
data
base
and
each
treatment
group
should
the
raw data
pointer
here
reset
Entered
-
cards,
To do this
SD,
D
D'
of the applied
is recorded
card,
mean,
SD,
and
in data
the raw
data
SEM.
be recorded
on
by pressing
a
D' after
data.
steps
the
column.
from
ST-06
B' compiles data
separate
$ These
be entered
program
Xi1
of variance
as
Entered
Xi2
SEM
7.
may
Data
t of
Compute mean, for treatment
entry
Printed
16
Entries
3.
* Data
Displayed
E'
may
table Read
the
R opposite
The
R in the
take
from
LSJ,
across the next
up to 3', minutes find
to the
degrees
column
protection previous
error
on
freedom
by the
desired
is the
depending of
headed
level
column
each
from
number
in the
second
the number
second
R, etc.,
ANOVA
of qroups column. back
of
treatment
output
in the
being
tested.
This
is the
to the column
groups. left
hand
Choose first
headed
R.
by 2.
it is included in a larger subset with a nonsignificant range, A more detailed discussion can be found in Steel and Torrie [IS]. This program computes the analysis ofvariance for up to six groups of any size followed by Duncan’s New Multiple Range Test. The user is required to enter the Studentized significant ranges (R) in order from largest to smallest from a table and the execution of two of the program segments requires up to 3.5 min each depending on the number of treatment groups to be analyzed. The program is written for the TI-59 calculator with print cradle and uses subroutines from the applied statistics library. It uses all 480program locations and all 60 data registers of the default partition when six treatment groups are analyzed. The program also calculates the standard deviation and standard error of the mean for each treatment group. A data summary from a sample problem is presented in Table 1 and the sample output using this program for this problem in Table 2. Differences between means are given in the order tested (see Step 2 above) and differences that are significant are followed by an asterisk. The user should make sure that no difference declared significant is contained within a larger subset with a nonsignificant range. Raw data or the intermediate data base can be recorded
360
RICHARDA. LAWRENCE
Table 4.
Duncan’s
new multiple
Table 4 continued
range
test
36
362
RICHARDA. LAWRENCE
on magnetic cards for future use as detailed in the applied statistics module manual for data entry program 06. Stepwise instructions for running the program are given in Table 3 and the program listing in Table 4. Studentized significant ranges can be found in Steel and Torrie [5] and other general statistics texts or in [2].
SUMMARY A program for the TI-59 calculator to implement Duncan’s New Multiple Range Test is presented. This test allows comparison of every mean with every other mean following analysis of variance in an experiment with completely random design involving more than two treatment groups. All of the necessary statistics are calculated for the analysis ofvariance and Duncan’s test and the data may be stored on magnetic cards for future use. This test is available in statistical packages such as SAS and SPSS if analysis of more than six groups is required. The results for the sample problem presented here agree with results obtained from SPSS.
Acknowkdgements-The author wishes to thank Dr. Thomas for testing the sample problem with SPSS.
J. Prihoda
for helpful suggestions
and comments
and
REFERENCES 1. W. G. Cochran and G. M. Cox, Experimental Designs, 2nd ed. John Wiley, New York (1957). 2. D. B. Duncan, Multiple range and multiple F tests, Biometrics 11, 1 (1955). Biometrics 3. C. Y. Kramer, Extension of multiple range tests to group means with unequal numbers ofreplication. 12, 307 (1956). 4. T. A. Bancroft, Topics in Intermediate Statistical Methods VI. Iowa State Press, Ames, IA (1968). New York (1960). 5. R. G. D. Steel and J. H. Torrie, Principles and Procedures ofStatistics. McGraw-Hill.
About the Author-RlCHmD A. LAWRENCE received his B.S. degree from Brigham Young University in 1971, and his Ph.D. degree from the University of Wisconsin, Madison in 1975. Dr. Lawrence then did postdoctoral studies at the University of Texas Health Science Center in Dallas and joined the faculty of the Department of Medicine at LSU Medical Center in Shreveport in 1978. Dr. Lawrence is presently Research Assistant Professor of Medicine at The University ofTexas Science Center in San Antonio. His major research interests are in the area of nutritional factors protecting against pulmonary oxygen toxicity. He is a member of the American Institute of Nutrition, the American Thoracic Society and Sigma Xi. He is an author of numerous scientific publications.
1984
c
0010 4825’84 s3.00+ Ml 1984 Pcrgamon Press Ltd
A POCKET CALCULATOR PROGRAM FOR DUNCAN’S NEW MULTIPLE RANGE TEST AND ANALYSIS OF VARIANCE RICHARD Departments
of Medicine,
Biochemistry of Medicine,
A. LAWRENCE*
and Molecular Biology, Louisiana Shreveport, LA 71130, U.S.A.
(Receioed
13
State University
School
April 1983)
Abstract-A program for the TI-59 calculator to be used in analysis of variance and testing of significance ofdifferences between each mean and every other mean by Duncan’s New Multiple Range Test is presented. The test is both simple and powerful for data involving more than two treatment groups in a completely random design. Up to six groups with any number of replicates may be analyzed with this program. TI-59 calculator Multiple comparison
Duncan’s
New Multiple Range Test
Analysis
of variance
test
A problem that often confronts the biological investigator is the analysis of an experiment involving three or more treatment groups. Analysis of variance will indicate if there are significant differences among treatment groups, but often it is desirable to compare each treatment mean with every other treatment mean. Successive applications of Student’s t-test or the least significant difference (LSD.) to such a problem is inappropriate. Cochran and Cox [l] have indicated, for example, that in the extreme case when only the difference between the highest and lowest means is compared using the f-test or the L.S.D., the difference will be substantial even when no effect is present. It can be shown that with three treatment means, the t value calculated for the largest difference will be greater than the tabulated 5% level about 13% of the time, with six treatments 40%, with ten treatments 60’4, and with twenty treatments 90%. Thus, a test thought to be made at the 5% level is actually made at the 13% level with three groups, the 40% level with six groups, etc. Duncan [2] developed a test using a set of significant differences of increasing size depending on the closeness of the means after ranking. The program presented here uses Kramer’s modification [3] of Duncan’s New Multiple Range Test to allow for unequal replication in the treatment groups. However, when unequal replication is used the test is only approximate [4]. Following the computation of the analysis of variance, the procedure is: 1. Multiply S = error mean square by the appropriate tabulated Studentized significant ranges to obtain a set of intermediate significant ranges. 2. Rank the means in order from smallest to largest. Test the differences between means in the following order: largest minus smallest, largest minus second smallest . . . largest minus second largest, then second largest minus smallest, second largest minus second smallest and so on down to second smallest minus smallest. Each difference is compared to the corresponding intermediate significant range multiplied by [l/2 (l/r, + l/r,)]“’ where r1 and r2 are the number of replications in the two treatment groups being compared. 3. If the difference exceeds this least significant range then the difference is declared significant at the chosen level with one exception. No difference can be declared significant if
* Present address: The Audie L. Murphy Merton Minter Boulevard. San Antonio,
Memorial Veterans TX 78284. U.S.A. 351
Hospital.
Pulmonary
Disease Section,
11 lE, 7400
RICHARDA. LAWRENCE
358
Table 1. Effect of paraquat dose on 24 h urinary paraquat excretion
________________~_
____ -- ---.. ___-
PQ Dose
PQ Excreted
mglkg B.W.
p moles/ml
20
54 + 11* A
30
76210 101 +
40
l
____.-_- I_______-__
Mean+SEM
9 EC
123 + 4 C
50 --__---
AB
.____- _.__.._
Meansnotfollowedbythesameletterare
significantly different by Duncan'sNew Multiplerangetestat p c.05.
Table 2. .:8. I’ ,:i i3 ._a
Group
1 ‘3 13, 4 1 2 ::: ,
Group
14. 4!:1?~674i~l’~ 4
4
5; II
14 '::' _!_I1 .::; :z1 7,
!3’EM
5 5 E
Analysis
of Variance
1 1. 5 2 15 6 1I:I’~ '7. .-, c; Jl.
F Treatment d.f. Error d.f.
2~:3~4. I;I~::.Error Sum of Squares ~:&'~:~4i. ~51~:2 Treatment SS Total SS 5 43 iIl:3 . :~~y~:~ Studentized Significant Ranqes
‘3 L
Differences
101.
45
Mean
,>7e .
C’
I
4
Bet. Means
a
Duncan’s
new multiple
359
range test
Table 3. Press
Enter i.
Initialize
2.
Enter data for treatment group
4.
Repeat steps 2 and 3 for each treatment qroup.+ (If display flashes go to step 9)
5.
Compute
6.
Rank the means and :4 (rI + r2)llz for
SD and qroup
analysis
compute each comparison$
Enter significant studentized ranges (R) in order from largest to smallest 5
8.
Compute differences between and test for significancef
9.
If display flashes during data entry, reset raw data pointer by pressing D' and continue
pointer, + If raw
recording
4 For
A
1
Xi1
A
2
Xi2
XiJ
A
j
the
the
-
K’S
means
-
keyboard
or from
statistics
intermediate on magnetic
(or side).
XiJ
6
Mean, SEW
C
F, Treatment d.f., Error d.f., Error S.S., Treatment S.S., total S.S.
10
E
-
R's as
A'
-
Differences in order tested. Sig. differences followed by asterisks
magnetic manual
cards except
following that
B prints
instructions
D' resets
data
base
and
each
treatment
group
should
the
raw data
pointer
here
reset
Entered
-
cards,
To do this
SD,
D
D'
of the applied
is recorded
card,
mean,
SD,
and
in data
the raw
data
SEM.
be recorded
on
by pressing
a
D' after
data.
steps
the
column.
from
ST-06
B' compiles data
separate
$ These
be entered
program
Xi1
of variance
as
Entered
Xi2
SEM
7.
may
Data
t of
Compute mean, for treatment
entry
Printed
16
Entries
3.
* Data
Displayed
E'
may
table Read
the
R opposite
The
R in the
take
from
LSJ,
across the next
up to 3', minutes find
to the
degrees
column
protection previous
error
on
freedom
by the
desired
is the
depending of
headed
level
column
each
from
number
in the
second
the number
second
R, etc.,
ANOVA
of qroups column. back
of
treatment
output
in the
being
tested.
This
is the
to the column
groups. left
hand
Choose first
headed
R.
by 2.
it is included in a larger subset with a nonsignificant range, A more detailed discussion can be found in Steel and Torrie [IS]. This program computes the analysis ofvariance for up to six groups of any size followed by Duncan’s New Multiple Range Test. The user is required to enter the Studentized significant ranges (R) in order from largest to smallest from a table and the execution of two of the program segments requires up to 3.5 min each depending on the number of treatment groups to be analyzed. The program is written for the TI-59 calculator with print cradle and uses subroutines from the applied statistics library. It uses all 480program locations and all 60 data registers of the default partition when six treatment groups are analyzed. The program also calculates the standard deviation and standard error of the mean for each treatment group. A data summary from a sample problem is presented in Table 1 and the sample output using this program for this problem in Table 2. Differences between means are given in the order tested (see Step 2 above) and differences that are significant are followed by an asterisk. The user should make sure that no difference declared significant is contained within a larger subset with a nonsignificant range. Raw data or the intermediate data base can be recorded
360
RICHARDA. LAWRENCE
Table 4.
Duncan’s
new multiple
Table 4 continued
range
test
36
362
RICHARDA. LAWRENCE
on magnetic cards for future use as detailed in the applied statistics module manual for data entry program 06. Stepwise instructions for running the program are given in Table 3 and the program listing in Table 4. Studentized significant ranges can be found in Steel and Torrie [5] and other general statistics texts or in [2].
SUMMARY A program for the TI-59 calculator to implement Duncan’s New Multiple Range Test is presented. This test allows comparison of every mean with every other mean following analysis of variance in an experiment with completely random design involving more than two treatment groups. All of the necessary statistics are calculated for the analysis ofvariance and Duncan’s test and the data may be stored on magnetic cards for future use. This test is available in statistical packages such as SAS and SPSS if analysis of more than six groups is required. The results for the sample problem presented here agree with results obtained from SPSS.
Acknowkdgements-The author wishes to thank Dr. Thomas for testing the sample problem with SPSS.
J. Prihoda
for helpful suggestions
and comments
and
REFERENCES 1. W. G. Cochran and G. M. Cox, Experimental Designs, 2nd ed. John Wiley, New York (1957). 2. D. B. Duncan, Multiple range and multiple F tests, Biometrics 11, 1 (1955). Biometrics 3. C. Y. Kramer, Extension of multiple range tests to group means with unequal numbers ofreplication. 12, 307 (1956). 4. T. A. Bancroft, Topics in Intermediate Statistical Methods VI. Iowa State Press, Ames, IA (1968). New York (1960). 5. R. G. D. Steel and J. H. Torrie, Principles and Procedures ofStatistics. McGraw-Hill.
About the Author-RlCHmD A. LAWRENCE received his B.S. degree from Brigham Young University in 1971, and his Ph.D. degree from the University of Wisconsin, Madison in 1975. Dr. Lawrence then did postdoctoral studies at the University of Texas Health Science Center in Dallas and joined the faculty of the Department of Medicine at LSU Medical Center in Shreveport in 1978. Dr. Lawrence is presently Research Assistant Professor of Medicine at The University ofTexas Science Center in San Antonio. His major research interests are in the area of nutritional factors protecting against pulmonary oxygen toxicity. He is a member of the American Institute of Nutrition, the American Thoracic Society and Sigma Xi. He is an author of numerous scientific publications.