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WESTERN MICHIGAN UNIVERSITY
COMPUTER CENTER
LIBRARY PROGRAM #1.9.2
CALLING NAME: TWOAOV
PREPARED BY: SAM ANEMA
PROGRAMMED BY: EVA GAINES AND SAM ANEMA
MODIFIED BY: RUSSELL R. BARR III
STATISTICAL CONSULTANT: MICHAEL R. STOLINE
APPROVED BY: JACK R. MEAGHER
DATE: AUGUST 1972
TWO-WAY ANALYSIS OF VARIANCE
(UNBALANCED CASE)
TABLE OF CONTENTS
1.0 PURPOSE AND SUMMARY
2.0 USE OF THE EXACT PROCEDURE AND INTERPRETATION OF RESULTS
3.0 DATA ENTRY OPTION
4.0 EXAMPLES
5.0 SAMPLE BATCH RUN
1.0 PURPOSE AND SUMMARY
THIS PROGRAM IS DESIGNED TO GIVE THE USER AN EXACT ANALYSIS OF AN R X C CROSS-
CLASSIFICATION MODEL (TWO-WAY ANOVA) IN THE UNBALANCED CASE.
FACTOR 2
1 2 C
--------------------------------------------------------------
1 X(1,1,I) X(1,2,I) .......... X(1,C,I)
I=1,...,N(11) I=1,...,N(12) I=1,...,N(1C)
--------------------------------------------------------------
2 X(2,1,I) X(2,2,I) .......... X(2,C,I)
I=1,...,N(21) I=1,...,N(22) I=1,...,N(2C)
--------------------------------------------------------------
FACTOR . . .......... .
1 . . .
. . .
--------------------------------------------------------------
R X(R,1,I) X([,2,I) .......... X(R,C,I)
I=1,...,N(R1) I=1,...,N(R2) I=1,...,N(RC)
--------------------------------------------------------------
UNBALANCE MEANS THAT THE RC SAMPLE SIZES N(IJ) ARE UNEQUAL OR DISPROPORTIONATE.
SOME COMBINATIONS OF THE SAMPLE SIZES N(IJ) MAY BE ZERO. (SOME CELLS MAY BE
EMPTY.) THIS EXACT PROCEDURE FOLLOWS THAT OUTLINED IN SECTIONS 1.7, 1.8, AND
1.9 OF T.A. BANCROFT'S BOOK, "TOPICS IN INTERMEDIATE STATISTICAL METHODS",
VOLUME 1, IOWA STATE UNIVERSITY. THE USER MAY SELECT ONE OF THREE METHODS OF
ENTERING THE
N = SUM N(IJ) DATA POINTS X(IJK).
I,J=R,C
THESE METHODS ARE EXPLAINED IN DETAIL IN SECTION 3.0.
CONSIDER, FOR EXAMPLE, THE FOLLOWING 2 X 2 DESIGN:
FACTOR 2
-------------------------
22 67
1 90 72
41 26
94
FACTOR 1 -------------------------
44 82
2 13 8
33
-------------------------
THE PROGRAM YIELDS THE FOLLOWING OUTPUT FOR THIS DATA:
FACTOR 2
LEVEL - - - - - - - 1 2
.
. MEAN - - - 42.00 54.57
. .
SIZE 3.00 4.00
1 58.86 MEAN 51.00 64.75
ST DEV 35.09 28.37
FACTOR SIZE 2.00 3.00
1 2 36.00 MEAN 28.50 41.00
ST DEV 21.92 37.64
PRELIMINARY ANOVA
SOURCE DF SS MS F PROB
CELLS 3.00 2035.42 678.47 0.66 0.598
1 IGNORING 2 1.00 1523.81
2 IGNORING 1 1.00 460.95
WITHIN 8.00 8191.25 1023.91
TOTAL 11.00 10226.67
LEAST SQUARES ANOVA
SOURCE DF SS MS F PROB
CELLS 3.00 2035.42
1 ELIMINATING 2 1.00 1573.36 1573.36 1.54 0.250
2 ELIMINATING 1 1.00 510.50 510.50 0.50 0.500
1 BY 2 1.00 1.10 1.10 0.00 0.975
WITHIN 8.00 8191.25 1023.91
TOTAL 11.00 10226.67
WEIGHTED MEANS ANOVA
SOURCE DF SS MS F PROB
CELLS 3.00 2035.42 678.47
1 1.00 1509.93 1509.93 1.47 0.259
2 1.00 486.40 486.40 0.48 0.510
1 BY 2 1.00 1.10 1.10 0.00 0.975
WITHIN 8.00 8191.25 1023.91
TOTAL 11.00 10226.67
IN GENERAL, THE USER OBTAINS THE FOLLOWING OUTPUT:
(I) THE RC CELL MEANS AND STANDARD DEVIATIONS.
(II) A PRELIMINARY ANOVA TABLE (USED IN TESTING EQUALITY OF THE RC
POPULATION CELL MEANS).
(III) A LEAST SQUARES ANOVA TABLE (USED TO TEST FOR SIGNIFICANT R X C
INTERACTION AND MAIN ROW AND COLUMN EFFECTS, IF INTERACTION IS NOT
SIGNIFICANT).
(IV) A WEIGHTED MEANS ANOVA TABLE (USED TO TEST MAIN ROW AND COLUMN
EFFECTS, IF INTERACTION IS PRESENT OR SIGNIFICANT).
BEFORE INTERPRETING THE RESULTS FROM THIS PROGRAM THE USER IS CAUTIONED TO
READ SECTION 2.0. CARE AND PRUDENCE MUST BE EXERCISED IN ANALYZING DATA IN AN
UNBALANCED CASE.
A DESIGN WHICH HAS ONLY ONE ENTRY PER CELL FOR ALL CELLS IS PERMISSIBLE. THE
MAIN EFFECTS ARE TESTED USING THE INTERACTION MEAN SQUARE AS THE ERROR MEAN
SQUARE. A DESIGN HAVING ONLY ONE LEVEL ON ONE OF THE TWO FACTORS IS
PERMISSIBLE.
2.0 USE OF THE EXACT PROCEDURE AND INTERPRETATION OF RESULTS
THE FOLLOWING PROCEDURE IS RECOMMENDED IN ANALYZING THESE RESULTS OF THIS
PROGRAM FOR AN UNBALANCED TWO-WAY ANOVA.
STEP 1 A PRELIMINARY TEST FOR THE EQUALITY OF THE NON-EMPTY CELL MEANS IS
GIVEN IN THE PRELIMINARY ANOVA TABLE. AN F STATISTIC IS CALCULATED
F = CELL MS/WITHIN MS
A PROBABILITY = PROB IS ALSO GIVEN FOR THIS F. LET THE SIGNIFICANCE LEVEL
CHOSEN BY THE USER BE ALPHA (USUALLY .05, .01, OR .001).
IF PROB <ALPHA THEN THE TEST IS SIGNIFICANT AND PROCEED TO STEP 2.
IF PROB>ALPHA THEN THE TEST IS NOT SIGNIFICANT. IN THIS CASE THE USER IS URGED TO
EITHER IGNORE OR TO INTERPRET WITH CAUTION ANY SUBSEQUENT RESULTS FROM EITHER
THE LEAST SQUARES OR WEIGHTED MEANS ANOVA'S.
STEP 2 USING THE LEAST SQUARES ANOVA TEST FOR ROW X COLUMN INTERACTION.
(I) IF THE INTERACTION TEST IS NON-SIGNIFICANT OR IF NO INTERACTION IS
POSITED IN THE MODEL, THEN THE MODEL:
MU(IJ) = MU + A(I) + B(J) IS VALID, AND
THE TESTS FOR SIGNIFICANT ROW AND COLUMN EFFECTS ARE COMPLETED USING
THE LEAST SQUARES ANOVA.
(II) IF THE INTERACTION TEST IS SIGNIFICANT OR IF INTERACTION IS KNOWN
TO BE IN THE MODEL, THEN THE MODEL:
MU(IJ) = MU + A(I) + B(J) +(AB)(IJ) IS VALID, AND
THE TESTS FOR SIGNIFICANT ROW AND COLUMN EFFECTS ARE COMPLETED
USING THE WEIGHTED MEANS ANALYSIS.
(III) IF THE INTERACTION TEST IS USED AS A PRELIMINARY TEST TO DETERMINE
WHETHER TO USE THE LEAST SQUARES OR WEIGHTED MEANS ANOVA'S FOR THE
ROW AND COLUMN EFFECTS, IT IS RECOMMENDED THAT THE SIGNIFICANCE
LEVEL OF THIS PRELIMINARY TEST BE SET AT ALPHA =.25
WE MAKE THE FOLLOWING GENERAL REMARKS:
1. IF ONE OR MORE CELLS ARE EMPTY, THEN IT IS NOT POSSIBLE TO OBTAIN A
WEIGHTED MEANS ANOVA, SINCE THE CALCULATIONS FOR THIS ANALYSIS REQUIRE
EVALUATING THE 1/N(IJ)'S. IN THIS CASE THE LEAST SQUARES ANOVA IS USED
WHETHER INTERACTION IS SIGNIFICANT OR NOT. CARE SHOULD BE EXERCISED.
2. THERE ARE TWO POTENTIAL ERRORS POSSIBLE WHEN TESTING FOR MAIN EFFECTS:
ERROR I - USING THE LEAST SQUARES ANOVA WHEN INTERACTION IS PRESENT,
AND
ERROR II - USING THE WEIGHTED MEANS ANOVA WHEN INTERACTION IS ABSENT.
ERROR I IS MORE SERIOUS, SINCE THE TEST WILL BE BIASED (NOT AT THE
TRUE SIGNIFICANCE LEVEL INDICATED).
ERROR II IS LESS SERIOUS, SINCE THE TESTS ARE STILL UNBIASED, BUT
INEFFICIENT.
3. THE LEAST SQUARES ANOVA IS ALSO CALLED THE FINAL ANOVA IN BANCROFT'S
BOOK AND IS OBTAINED BY THE METHOD OF FITTING CONSTANTS.
4. IN THE CASE OF A BALANCED OR PROPORTIONAL ANOVA, THEN THE LEAST SQUARE
AND WEIGHTED MEANS ANOVA'S ARE IDENTICAL.
5. FOR A GENERAL DISCUSSION OF THE UNDERLYING STATISTICAL CONCEPTS
INVOLVED WITH THE LEAST SQUARES AND WEIGHTED MEANS ANOVA AND IN
PARTICULAR FOR AN EXPLANATION OF THE CONCEPTS OF "1 IGNORING 2" AND
"1 ELIMINATING 2", SEE EITHER:
(I) T.A. BANCROFT, "TOPICS IN INTERMEDIATE STATISTICAL
METHODS", CHAP. 1.
(II) SNEDECOR & COCHRAN, "STATISTICAL METHODS", CHAPTER 16,
SECTION 7.
(III) S.R. SEARLE, "LINEAR MODELS", CHAPTER 7.1-7.2.
(IV) B.J. WINER, "STATISTICAL PRINCIPLES IN EXPERIMENTAL
DESIGN", SECTION 5.23.
3.0 DATA ENTRY OPTIONS
THE PROGRAM IS INITIALIZED BY TYPING R TWOAOV.
THE FIRST TWO QUESTIONS TO BE ASKED ARE "OUTPUT?" AND "INPUT?"
THE PROPER RESPONSES TO EACH OF THESE QUESTIONS CONSISTS OF THREE BASIC PARTS:
A DEVICE, A FILENAME, AND A PROJECT-PROGRAMMER NUMBER.
THE GENERAL FORMAT FOR THESE THREE PARTS IS AS FOLLOWS:
DEV:FILE.EXT[PROJ,PROG]
1) DEV: ANY OF THE FOLLOWING DEVICES ARE APPROPRIATE WHERE INDICATED:
DEVICE LIST DEFINITION STATEMENT USE
TTY: TERMINAL INPUT OR OUTPUT
DSK: DISK INPUT OR OUTPUT
CDR: CARD READER INPUT ONLY
LPT: LINE PRINTER OUTPUT ONLY
DTA0: DECTAPE 0 INPUT OR OUTPUT
DTA1: DECTAPE 1 INPUT OR OUTPUT
DTA2: DECTAPE 2 INPUT OR OUTPUT
DTA3: DECTAPE 3 INPUT OR OUTPUT
DTA4: DECTAPE 4 INPUT OR OUTPUT
DTA5: DECTAPE 5 INPUT OR OUTPUT
DTA6: DECTAPE 6 INPUT OR OUTPUT
DTA7: DECTAPE 7 INPUT OR OUTPUT
MTA0: MAGNETIC TAPE 0 INPUT OR OUTPUT
MTA1: MAGNETIC TAPE 1 INPUT OR OUTPUT
INPUT MAY NOT BE DONE FROM THE LINE PRINTER NOR MAY OUTPUT GO TO THE CARD
READER.
2) FILE.EXT IS THE NAME AND EXTENSION OF THE FILE TO BE USED. THIS PART OF
THE SPECIFICATION IS USED ONLY IF DISK OR DECTAPE IS USED.
3) [PROJ,PROG] IF A DISK IS USED AND THE USER WISHES TO READ A FILE IN
ANOTHER PERSON'S DIRECTORY, HE MAY DO SO BY SPECIFYING THE PROJECT-
PROGRAMMER NUMBER OF THE DIRECTORY FROM WHICH HE WISHES TO READ. THE
PROJECT NUMBER AND THE PROGRAMMER NUMBER MUST BE SEPARATED BY A COMMA AND
ENCLOSED IN BRACKETS. OUTPUT MUST GO TO YOUR OWN AREA.
EXAMPLE:
OUTPUT? LPT:/2
INPUT? DSK:DATA.DAT[71171,71026]
IN THE EXAMPLE, TWO COPIES OF THE OUTPUT ARE TO BE PRINTED BY THE HIGH SPEED
LINE PRINTER. THE INPUT DATA IS A DISK FILE OF NAME DATA.DAT IN USER DIRECTORY
[71171,71026].
DEFAULTS:
1) IF NO DEVICE IS SPECIFIED BUT A FILENAME IS SPECIFIED THE DEFAULT DEVICE
WILL BE DSK:
2) IF NO FILENAME IS SPECIFIED AND A DISK OR DECTAPE IS USED THE DEFAULT
ON INPUT WILL BE FROM INPUT.DAT; ON OUTPUT IT WILL BE OUTPT.DAT.
3) IF THE PROGRAM IS RUN FROM THE TERMINAL AND NO SPECIFICATION IS GIVEN (JUST
A CARRIAGE RETURN) BOTH INPUT AND OUTPUT DEVICES WILL BE THE TERMINAL.
4) IF THE PROGRAM IS RUN THROUGH BATCH AND NO SPECIFICATION IS GIVEN, (A
BLANK CARD) THE INPUT DEVICE WILL BE CDR: AND THE OUTPUT DEVICE WILL BE LPT:
5) IF NO PROJECT-PROGRAMMER NUMBER IS GIVEN, THE USER'S OWN NUMBER WILL BE
ASSUMED.
NOTE: (1) IF LPT: IS USED AS AN OUTPUT DEVICE, MULTIPLE COPIES MAY BE
OBTAINED BY SPECIFYING LPT:/N WHERE N REFERS TO THE NUMBER OF COPIES
DESIRED.
(2) THE FOLLOWING TWO OPTIONS ARE NOT APPLICABLE FOR THE FIRST DATA SET,
I.E., IT IS APPLICABLE ONLY WHEN THE PROGRAM BRANCHES BACK TO THE
BEGINNING UPON FIRST COMPLETION OF AN ANALYSIS.
(A) SAME OPTION
IF THE SAME DATA FILE IS TO BE USED AGAIN, SIMPLY ENTER
"SAME<CR>"; OTHERWISE, EITHER USE THE FINISH OPTION OR ENTER
ANOTHER FILENAME ETC.
(B) FINISH OPTION
THE USER MUST ENTER "FINISH<CR>" TO BRANCH OUT OF TWOAOV
PROGRAM WHEN THE USER HAS GIVEN THE INPUT AND OUTPUT
SPECIFICATIONS. THE PROGRAM WILL REQUEST A FORMAT.
THERE ARE 3 OPTIONS AVAILABLE FOR THE FORMAT; NAMELY:
(A) STANDARD FORMAT OPTION
UNLESS OTHERWISE SPECIFIED, THE PROGRAM ASSUMES THE STANDARD
OPTION. THE STANDARD FORMAT FOR EACH METHOD OF DATA ENTRY
IS GIVEN WITH THE DESCRIPTION OF THE METHOD.
TO USE THIS OPTION, SIMPLY TYPE IN "<CR>" ON TERMINAL JOBS OR
USE A BLANK CARD FOR BATCH JOBS OR ENTER "STD<CR>".
(B) OBJECT TIME FORMAT OPTION
IF THE DATA IS SUCH THAT A USER'S OWN FORMAT IS REQUIRED,
SIMPLY ENTER A LEFT PARENTHESIS FOLLOWED BY THE FIRST FORMAT
SPECIFICATION, A COMMA AND THE SECOND SPECIFICATION, ETC. WHEN
YOU FINISH ENTER A RIGHT PARENTHESIS, AND THEN A CARRIAGE
RETURN. THERE CAN BE A MAXIMUM OF ONE LINE 80 COLUMNS FOR THE
FORMAT.
THE SPECIFICATIONS FOR THE FORMAT ITSELF ARE THE SAME AS FOR
THE FORTRAN IV FORMAT STATEMENT.
(C) SAME OPTION
THE SAME OPTION IS APPLICABLE ONLY TO JOBS THAT USE MORE THAN
ONE DATA FILE. IF AN OBJECT TIME FORMAT WAS USED ON A DATA SET
AND THE SUCCEEDING DATA SET UTILIZES THE SAME FORMAT, SIMPLY
ENTER "SAME<CR>".
THE USER MAY ELECT ONE OF THREE METHODS TO ENTER THE DATA:
X(IJK),I=1,2,...,R (R LEVELS OF FACTOR 1, THE ROW
FACTOR)
J=1,2,...,C (C LEVELS OF FACTOR 2, THE COLUMN
FACTOR)
K=1,2,...,N(IJ) (N(IJ) = THE (I,J)TH CELL SIZE.)
THEREFORE, CELL (I,J) HAS THE N(IJ) DATA POINTS:
X(IJ1)
X(IJ2)
.
.
.
X(IJN(IJ))
FOR METHOD 1 THE USER SUPPLIES:
(I) R AND C, THE LEVEL NUMBERS FOR FACTOR 1 AND 2
RESPECTIVELY,
(II) THE RC SAMPLE SIZES N(IJ), AND
(III) THE N = SUM N(IJ) X(IJ) DATA POINTS.
THIS IS DONE IN GENERAL AS FOLLOWS:
.R TWOAOV<CR>
WMU TWO-WAY ANALYSIS OF VARIANCE
OUTPUT? TTY:<CR>
INPUT? TTY:<CR>
FORMAT
STD<CR>
ENTER IDENTIFICATION IF DESIRED
EXAMPLE OF METHOD 1<CR>
ENTER INPUT METHOD DESIRED. (1,2,OR 3)
1<CR>
NUMBER OF LEVELS ON FACTOR 1 = R
NUMBER OF LEVELS ON FACTOR 2 = C
ENTER CELL SIZES
N(11),N(12),...,N(1C)
N(21),N(22),...,N(2C)
.....................
N(R1),N(R2),...,N(RC)
OBSERVE THAT ALL OF THE SAMPLE SIZES N(IJ) FOR THE C CELLS OF
THE FIRST ROW OF THE DESIGN ARE ENTERED IN THE FIRST ROW
SEPARATED BY COMMAS, THE SAMPLE SIZES N(2J) OF THE C CELLS
OF THE SECOND ROW, ETC. UNTIL R ROWS OF SAMPLE SIZES ARE
ENTERED.
ENTER DATA FOR CELL (1,1)
X(111),X(112),...,X(11N(11))
ENTER DATA FOR CELL (1,2)
X(121),X(122),...,X(12N(12))
ENTER DATA FOR CELL (R,C)
X(RC1),X(RC2),...,X(RCN(RC))
CONSIDER THE FOLLOWING EXAMPLE OF A 2 X 2 DESIGN:
FACTOR 2
----------------------------------
N(11)=3 N(12)=2
1 (22)
(22) (27)
FACTOR 1 (23) (26)
-----------------------------------
N(21)=4 N(22)=1
2 (17) (16) (20)
(19) (18)
-----------------------------------
.R TWOAOV<CR>
WMU TWO-WAY ANALYSIS OF VARIANCE
OUTPUT? TTY:<CR>
INPUT? TTY:<CR>
FORMAT
STD<CR>
ENTER IDENTIFICATION IF DESIRED
REAL EXAMPLE OF METHOD 1<CR>
ENTER INPUT METHOD DESIRED. (1,2,OR 3)
1<CR>
NUMBER OF LEVELS ON FACTOR 1 = 2<CR>
NUMBER OF LEVELS ON FACTOR 2 = 2<CR>
ENTER CELL SIZES
3,2<CR>
4,1<CR>
ENTER DATA FOR CELL (1,1)
22,22,23<CR>
ENTER DATA FOR CELL (1,2)
27,26<CR>
ENTER DATA FOR CELL (2,1)
17,16,19,18<CR>
ENTER DATA FOR CELL (2,2)
20<CR>
METHODS 2 AND 3 ARE EASILY DESCRIBED BY THE USE OF THE SAME
ILLUSTRATIVE EXAMPLE.
METHOD 2
WMU TWO-WAY ANALYSIS OF VARIANCE
OUTPUT? TTY:<CR>
INPUT? TTY:<CR>
FORMAT
STD<CR>
ENTER IDENTIFICATION IF DESIRED
EXAMPLE OF METHOD 2<CR>
ENTER INPUT METHOD DESIRED. (1,2,OR 3)
2<CR>
ENTER DATA
*1,1<CR>
22<CR>
22<CR>
23<CR>
*1,2<CR>
27<CR>
26<CR>
*2,1<CR>
17<CR>
16<CR>
19<CR>
18<CR>
*2,2<CR>
20<CR>
*<CR>
RULES FOR METHOD 2
(I) THE USER TYPES *I,J. (AN ASTERISK FOLLOWED BY THE
DESIGNITION FOR CELL (I,J).) THE DATA LINE WHICH
BEGINS WITH AN ASTERISK SPECIFIES THE CELL INTO
WHICH THE DATA FOLLOWING IT WILL FALL.
(II) ENTER THE DATA FOR CELL (I,J) IN THE FOLLOWING
LINES ONE ENTRY PER LINE UNTIL ALL N(IJ) DATA
POINTS HAVE BEEN PRINTED.
(III) ENTER AN ASTERISK FOLLOWED BY A DIFFERENT I,J
COMBINATION. CONTINUE UNTIL ALL DATA IS ENTERED.
(IV) CELLS MAY BE SUBMITTED IN ANY ORDER.
(V) DATA MUST BE SUBMITTED ONE ENTRY PER LINE.
(VI) AN ASTERISK FOLLOWED BY A RETURN SIGNALS THE END
OF THE DATA.
(VII) IF THE USER SUBMITS THE FORMAT, IT MUST HAVE ONE
FLOATING (F) FIELD.
METHOD 3 (USING THE SAME DATA)
.R TWOAOV
WMU TWO-WAY ANALYSIS OF VARIANCE
OUTPUT? TTY:<CR>
INPUT? TTY:<CR>
FORMAT
STD<CR>
ENTER IDENTIFICATION IF DESIRED
EXAMPLE OF METHOD 3<CR>
ENTER INPUT METHOD DESIRED. (1,2,OR 3)
3<CR>
ENTER DATA
1,1,22<CR>
1,1,22<CR>
1,1,23<CR>
1,2,27<CR>
1,2,26<CR>
2,1,17<CR>
2,1,16<CR>
2,1,19<CR>
2,1,18<CR>
2,2,20<CR>
<CR>
RULES FOR METHOD 3
(I) THE DATA MAY BE SUBMITTED IN ANY ORDER.
(II) THE FIRST NUMBER ON EACH DATA LINE IS THE LEVEL
FOR FACTOR 1; THE SECOND IS THE LEVEL FOR FACTOR 2;
AND THE THIRD IS AN ENTRY IN THAT CELL.
(III) A RETURN INDICATES THE END OF THE DATA.
(IV) IF THE USER SUBMITS THE FORMAT, IT MUST HAVE TWO
INTEGER (I) FIELDS FOLLOWED BY A FLOATING (F)
FIELD.
IN EACH OF THE ABOVE CASES THE COMPUTER WILL LOOK FOR
COMPLETELY EMPTY ROWS AND COLUMNS AND OMIT THEM FROM THE
ANALYSIS.
LIMITATIONS:
(I) THE MAXIMUM NUMBER OF LEVELS ON ANY FACTOR IS 12.
(II) THE MAXIMUM NUMBER OF ENTRIES PER CELL USING METHOD
1 IS 400; OTHERWISE IT IS UNLIMITED.
(III) THE NUMBER OF LEVELS ON FACTOR 1 AND FACTOR 2
IS DETERMINED BY 4*N1+5*N2+2*N1*N2+NM*N2
WHERE
N1 = NO. OF LEVELS ON FACTOR 1
N2 = NO. OF LEVELS ON FACTOR 2
NM = MAXIMUM OF N1 AND N2
4.0 EXAMPLES
WE CONSIDER SEVERAL EXAMPLES HERE TO ILLUSTRATE THE USE OF THIS PROGRAM.
EXAMPLE 1 CONSIDER AGAIN THE EXAMPLE IN SECTION 1.0.
SINCE F=0.66 IS NOT SIGNIFICANT FOR ANY ALPHA<=.598, THE LEAST SQUARES AND
WEIGHTED MEANS ANALYSES ARE IGNORED.
EXAMPLE 2 FOR THE DATA OF THE 2 X 2 EXAMPLE USED IN SECTION 3.0:
FACTOR 2
--------------------------
22
1 22 27
23 26
FACTOR 1 --------------------------
17 16
2 19 18 20
--------------------------
WE HAVE:
FACTOR 2
LEVEL - - - - - - - 1 2
.
. MEAN - - - 19.57 24.33
. .
SIZE 3.00 2.00
1 24.00 MEAN 22.33 26.50
ST DEV 0.58 0.71
FACTOR SIZE 4.00 1.00
1 2 18.00 MEAN 17.50 20.00
ST DEV 1.29 0.00
PRELIMINARY ANOVA
SOURCE DF SS MS F PROB
CELLS 3.00 115.83 38.61 37.57 0.000
1 IGNORING 2 1.00 90.00
2 IGNORING 1 1.00 47.62
WITHIN 6.00 6.17 1.03
TOTAL 9.00 122.00
LEAST SQUARES ANOVA
SOURCE DF SS MS F PROB
CELLS 3.00 115.83
1 ELIMINATING 2 1.00 66.88 66.88 65.07 0.000
2 ELIMINATING 1 1.00 24.50 24.50 23.84 0.003
1 BY 2 1.00 1.33 1.33 1.30 0.298
WITHIN 6.00 6.17 1.03
TOTAL 9.00 122.00
WEIGHTED MEANS ANOVA
SOURCE DF SS MS F PROB
CELLS 3.00 115.83 38.61
1 1.00 61.65 61.65 59.99 0.000
2 1.00 21.33 21.33 20.76 0.004
1 BY 2 1.00 1.33 1.33 1.30 0.298
WITHIN 6.00 6.17 1.03
TOTAL 9.00 122.00
THE PRELIMINARY F=37.57 AND IS SIGNIFICANT FOR ALPHA<=.001. THEREFORE, WE ARE
ENTITLED TO PROCEED WITH THE INTERACTION TEST IN THE LEAST SQUARES ANOVA.
NOW F INTERACTION = 1.30 AND IS NOT SIGNIFICANT FOR ANY ALPHA<=.298. THEREFORE,
WE CONCLUDE THAT INTERACTION IS NON-SIGNIFICANT AT ALPHA<=.25 AND PROCEED WITH
TESTING MAIN EFFECTS USING THE LEAST SQUARES ANOVA. BOTH ROW AND COLUMN ARE
SIGNIFICANT AT ALPHA=.01.
EXAMPLE 3 (INTERACTION PRESENT, USE OF WEIGHTED MEANS) CONSIDER THE FOLLOWING
DATA IN THE 2 X 2 DESIGN:
1 2
---------------------------------
1 12 14 9
13 8
---------------------------------
2 11
9 10
---------------------------------
.R TWOAOV
WMU TWO-WAY ANALYSIS OF VARIANCE
OUTPUT? TTY:<CR>
INPUT? TTY:<CR>
FORMAT
STD<CR>
ENTER IDENTIFICATION IF DESIRED
EXAMPLE 3<CR>
ENTER INPUT METHOD DESIRED. (1,2,OR 3)
3<CR>
ENTER DATA
1,1,12<CR>
1,1,13<CR>
1,1,14<CR>
1,2,9<CR>
1,2,8<CR>
2,1,9<CR>
2,2,11<CR>
2,2,10<CR>
<CR>
FACTOR 2
LEVEL - - - - - - - 1 2
.
. MEAN - - - 12.00 9.50
. .
SIZE 3.00 2.00
1 11.20 MEAN 13.00 8.50
ST DEV 1.00 0.71
FACTOR SIZE 1.00 2.00
1 2 10.00 MEAN 9.00 10.50
ST DEV 0.00 0.71
PRELIMINARY ANOVA
SOURCE DF SS MS F PROB
CELLS 3.00 28.50 9.50 12.67 0.016
1 IGNORING 2 1.00 2.70
2 IGNORING 1 1.00 12.50
WITHIN 4.00 3.00 0.75
TOTAL 7.00 31.50
LEAST SQUARES ANOVA
SOURCE DF SS MS F PROB
CELLS 3.00 28.50
1 ELIMINATING 2 1.00 0.57 0.57 0.76 0.432
2 ELIMINATING 1 1.00 10.37 10.37 13.83 0.020
1 BY 2 1.00 15.43 15.43 20.57 0.011
WITHIN 4.00 3.00 0.75
TOTAL 7.00 31.50
WEIGHTED MEANS ANOVA
SOURCE DF SS MS F PROB
CELLS 3.00 28.50 9.50
1 1.00 1.71 1.71 2.29 0.205
2 1.00 3.86 3.86 5.14 0.086
1 BY 2 1.00 15.43 15.43 20.57 0.011
WITHIN 4.00 3.00 0.75
TOTAL 7.00 31.50
THE PRELIMINARY ANOVA IS SIGNIFICANT AT ALPHA=.05; HENCE WE ARE ENTITLED TO TEST
FOR INTERACTION. THE TEST FOR INTERACTION YIELDS AN F=20.57 WHICH IS
SIGNIFICANT FOR ALPHA=.05. HENCE, WE CONCLUDE THAT INTERACTION IS IN THE MODEL
AND PROCEED TO TEST FOR MAIN EFFECTS USING THE WEIGHTED MEANS ANOVA.
NOTE THAT NEITHER FACTOR 1 NOR FACTOR 2 MAIN EFFECTS ARE SIGNIFICANT AT ALPHA=.05.
(IF THE WEIGHTED MEANS ANOVA HAD BEEN IGNORED AND THE LEAST SQUARES ANOVA
USED, WE WOULD HAVE FALSELY CONCLUDED THAT FACTOR 2 WAS SIGNIFICANT AT ALPHA=.02.)
EXAMPLE 4 (MISSING CELL DATA)
THIS EXAMPLE IS TAKEN FROM TABLE 1.10 ON PAGE 20 OF BANCROFT'S BOOK, "TOPICS
IN INTERMEDIATE STATISTICAL METHODS."
FACTOR 2
----------------------------
1 22 -1
25 40
18
----------------------------
2 41 23
41 13
----------------------------
29
FACTOR 1 3 20
37
4 49
50 61
----------------------------
5 55
----------------------------
NOTE THAT CELLS (3,2) AND (5,2) ARE EMPTY. THE ANALYSIS FOR THIS DATA IS:
.R TWOAOV
WMU TWO-WAY ANALYSIS OF VARIANCE
OUTPUT? TTY:<CR>
INPUT? TTY:<CR>
FORMAT
STD<CR>
ENTER IDENTIFICATION IF DESIRED
EXAMPLE 4<CR>
ENTER INPUT METHOD DESIRED (1,2,OR 3)
3<CR>
ENTER DATA
1,1,22<CR>
1,1,25<CR>
1,2,-1<CR>
1,2,40<CR>
1,2,18<CR>
2,1,41<CR>
2,1,41<CR>
2,2,23<CR>
2,2,13<CR>
3,1,29<CR>
3,1,20<CR>
3,1,37<CR>
4,1,49<CR>
4,1,50<CR>
4,2,61<CR>
5,1,55<CR>
<CR>
EXAMPLE 4
FACTOR 2
LEVEL - - - - - - - 1 2
.
. MEAN - - - 36.90 25.67
. .
SIZE 2.00 3.00
1 20.80 MEAN 23.50 19.00
ST DEV 2.12 20.52
FACTOR SIZE 2.00 2.00
1 2 29.50 MEAN 41.00 18.00
ST DEV 0.00 7.07
SIZE 3.00 0.00
3 28.67 MEAN 28.67 0.00
ST DEV 8.50 0.00
SIZE 2.00 1.00
4 53.33 MEAN 49.50 61.00
ST DEV 0.71 0.00
SIZE 1.00 0.00
5 55.00 MEAN 55.00 0.00
ST DEV 0.00 0.00
PRELIMINARY ANOVA
SOURCE DF SS MS F PROB
CELLS 7.00 3213.77 459.11 3.53 0.049
1 IGNORING 2 4.00 2572.30
2 IGNORING 1 1.00 473.20
WITHIN 8.00 1041.67 130.21
TOTAL 15.00 4255.44
LEAST SQUARES ANOVA
SOURCE DF SS MS F PROB
CELLS 7.00 3213.77
1 ELIMINATING 2 4.00 2249.06 562.26 4.32 0.037
2 ELIMINATING 1 1.00 149.96 149.96 1.15 0.315
1 BY 2 2.00 491.51 245.76 1.89 0.213
WITHIN 8.00 1041.67 130.21
TOTAL 15.00 4255.44
NO WEIGHTED MEANS ANALYSIS CAN BE PERFORMED WITH EMPTY CELLS
INPUT? (TYPE HELP IF NEEDED)--FINISH<CR>
THE PRELIMINARY ANOVA TABLE YIELDS AN F=3.53 WHICH IS BARELY SIGNIFICANT AT
ALPHA=.05.
THE LEAST SQUARES ANOVA IS VALID AND THE F INTERACTION VALUE = 1.89 WHICH IS
NOT SIGNIFICANT FOR ALPHA<.213.
THE MAIN EFFECTS TESTS FOR FACTOR 1 IS SIGNIFICANT AND THAT FOR FACTOR 2 IS NOT
SIGNIFICANT AT ALPHA<.05.
NOW, SINCE THE INTERACTION TEST IS SIGNIFICANT AT ALPHA=.25 WE SHOULD THEN USE A
WEIGHTED MEANS ANOVA. HOWEVER, SINCE THERE IS MISSING CELL DATA, WE HAVE NO
WEIGHTED MEANS ANOVA. HENCE, INTERPRET WITH SOME CAUTION THE SIGNIFICANT
FACTOR 1 MAIN EFFECT.
EXAMPLE 5 (2 X 2 DESIGN WITH ONE DATA POINT PER CELL) CONSIDER THE FOLLOWING
2 X 2 DESIGN:
FACTOR 2
---------------------------------
1 1 3
FACTOR 1 ---------------------------------
2 5 9
---------------------------------
.R TWOAOV<CR>
WMU TWO-WAY ANALYSIS OF VARIANCE
OUTPUT? TTY:<CR>
INPUT? TTY:<CR>
FORMAT
STD<CR>
ENTER IDENTIFICATION IF DESIRED
EXAMPLE 5<CR>
ENTER INPUT METHOD DESIRED. (1,2,OR 3)
3<CR>
ENTER DATA
1,1,1<CR>
1,2,3<CR>
2,1,5<CR>
2,2,9<CR>
<CR>
EXAMPLE 5
FACTOR 2
LEVEL - - - - - - - 1 2
.
. MEAN - - - 3.00 6.00
. .
SIZE 1.00 1.00
1 2.00 MEAN 1.00 3.00
ST DEV 0.00 0.00
FACTOR SIZE 1.00 1.00
1 2 7.00 MEAN 5.00 9.00
ST DEV 0.00 0.00
LEAST SQUARES ANOVA
SOURCE DF SS MS F PROB
CELLS 3.00 35.00
1 ELIMINATING 2 1.00 25.00 25.00 25.00 0.126
2 ELIMINATING 1 1.00 9.00 9.00 9.00 0.205
1 BY 2 1.00 1.00 1.00
TOTAL 3.00 35.00
THE DESIGN IS BALANCED. IN THIS CASE THE WEIGHTED
MEANS AND LEAST SQUARES ANOVA'S ARE IDENTICAL.
INPUT? (TYPE HELP IF NEEDED)--FINISH
END OF EXECUTION
CPU TIME: 1.94 ELAPSED TIME: 26.25
EXIT
THE TEST FOR MAIN EFFECTS IS NOT SIGNIFICANT AT ALPHA=.05 FOR EITHER FACTOR 1 OR 2.
NO TEST FOR INTERACTION IS POSSIBLE, SINCE THE MEAN SQUARE FOR WITHIN IS ZERO.
FOR ALL SUCH CASES THE LEAST SQUARES AND WEIGHTED MEANS ANOVA'S ARE IDENTICAL.
5.0 BATCH SETUP
THE FOLLOWING IS A BATCH JOB SETUP: (EACH LINE REPRESENTS ONE CARD, EACH CARD
STARTING IN COLUMN 1; DO NOT INCLUDE THE COMMENTS AT THE RIGHT.
--------------------------------------------------------------------------------
COMMENTS
$JOB [#,#] JOB CARD; INSERT USER'S PROJECT-
PROGRAMMER NUMBER WITHIN THE BRACKET
$PASSWORD ###### IN PLACE OF THE 6#'S, PUT IN THE
PASSOWRD
$DATA SIGNIFY BEGINNING OF DATA DECK
(DATA CARDS) INSERT THE DATA CARD DECK TO BE ANALYZED
$EOD SIGNIFY THE END OF DATA CARD DECK
.R TWOAOV START THE EXECUTION
(RESPONSES TO LINES 1-24
IN SECTION 3.0 REPEATED
OR NOT) USER'S RESPONSE
(EOF) AN END-OF-FILE CARD (GET THESE AT THE
COMPUTER CENTER)
--------------------------------------------------------------------------------
EXAMPLE:
$JOB [410,410]
$PASSWORD NNNNNN
$DATA
1112
1113
1114
1209
1208
2109
2211
2210
$EOD
.R TWOAOV
LPT:
CDR:
(2I1,F2.0)
BATCH EXAMPLE
3
FINISH
(EOF)