Trailing-Edge
-
PDP-10 Archives
-
decuslib20-05
-
decus/20-0137/advaov/advaov.rno
There are 2 other files named advaov.rno in the archive. Click here to see a list.
.LEFT MARGIN 0
.RIGHT MARGIN 72
.SPACING 1
.TAB STOPS 5 31 57
.TITLE LIBRARY PROGRAM _#1.9.8
.SKIP 3
.CENTER
WESTERN MICHIGAN UNIVERSITY
.CENTER
COMPUTER CENTER
.SKIP 2
LIBRARY PROGRAM##_#1.9.8
.NOFILL
CALLING NAME:##ADVAOV
PROGRAMMED BY:##RUSSELL R. BARR III
PREPARED BY:
STATISTICAL CONSULTANT: MICHAEL R. STOLINE
APPROVED BY: JACK R. MEAGHER
DATE: JULY, 1976
.SKIP 3
.CENTER
ADVANCED ONE-WAY ANALYSIS OF VARIANCE
.BREAK
.SKIP 2
TABLE OF CONTENTS
-----------------
.BREAK
.TAB STOPS 4 9
.SKIP 1
SECTION
1.0 PURPOSE
2.0 NOTATION AND AUTOMATIC OUTPUT
2A. NOTATION
2B. AUTOMATIC OUTPUT
3.0 DATA ENTRY METHODS
3A. DATA ENTRY METHOD 1##(RAW#DATA)
3B. DATA ENTRY METHOD 2(BREAKDOWN#VARIABLE)
3C.##DATA ENTRY METHOD 3##(MEANS#AND#STANDARD#DEVIATIONS)
4.0 OPTION DESCRIPTION AND USE
.TAB STOPS 4 9 17
4A. VAR TEST THE EQUALITY OF PAIRS OF VARIANCES.
4B. TREND LINEAR,QUADRATIC,CUBIC,QUARTIC,AND QUINTIC
COMPONENTS OF THE MEANS ARE GIVEN IN A TREND
ANALYSIS. THIS ANALYSIS CAN ONLY BE USED FOR
EQUI-SPACED MEANS AND BALANCED SAMPLES.
4C. TTEXC T-VALUES AND INDIVIDUAL CONFIDENCE INTERVALS
FOR ALL DIFFERENCES OF PAIRS OF MEANS.
4D. TTAPP APPROXIMATE INDIVIDUAL
CONFIDENCE INTERVALS FOR ALL DIFFERENCES OF PAIRS
OF MEANS. THIS OPTION IS USED INSTEAD OF 'TTEXC'
IF THE POPULATION VARIANCES ARE NOT EQUAL.
4E. SIMTES A SIMULTANEOUS TESTING OPTION. THE USER MAY SELECT
EITHER SCHEFFE, TUKEY, NEWMAN-KEULS, DUNCAN, OR
LEAST SIGNIFICANT DIFFERENCE PROCEDURE.
4F. SIMEST A SIMULTANEOUS ESTIMATION OPTION. THE USER MAY
SELECT THE SCHEFFE, TUKEY, OR BONFERRONI PROCEDURES.
4G. COMPAR THE T-VALUE AND CONFIDENCE INTERVALS ARE PRODUCED
FOR A USER SPECIFIED LINEAR EXPRESSION OR
COMPARISON OF THE MEANS.
4H. COLAOV A COLLAPSING AOV OPTION. THE USER FORMS NEW
GROUPINGS OF THE ORIGINAL VARIABLES. AN AOV
TABLE IS PRODUCED.
4I. TRANS TRANSFORM THE ORIGINAL VALUES OF THE CURRENT DATA
SET. THE TRANSFORMED DATA IS NOT TRANSFORMED.
4J. ORIG RETURN CURRENT DATA SET TO UNTRANSFORMED STATE.
4K. DATA ALLOWS THE ENTRY OF A NEW DATA SET.
4L. HELP TYPES THIS TEXT.
4M. EXIT PRESERVES OR PRINTS RESULTS AND RETURNS TO MONITER.
(OR FINI)
.TAB STOPS 4 9
5.0 EXAMPLES
6.0 PROGRAM DESCRIPTION AND USE
6A. LIST OF THE PROGRAM GENERATED QUESTIONS AND STATEMENTS
WITH EXPLANATIONS
6B. SAMPLE TERMINAL JOB RUN
7.0 LIMITATIONS
8.0 REFERENCES
9.0 EXAMPLE AND SECTION INDEX
.TAB STOPS 5 31 57
.SKIP 2
.INDEX ^^SECTION 1.0\\
SECTION#1.0##PURPOSE
.BREAK
--------------------
.FILL
.SKIP 1
ADVAOV IS A VERY COMPREHENSIVE AND HIGHLY INTERACTIVE STATISTICAL PROGRAM
WHICH INCORPORATES MOST OF THE DATA ANALYSIS FEATURES COMMONLY USED IN THE
ANALYSIS OF ONE-WAY ANALYSIS OF VARIANCE (AOV) DATA.
.SKIP 1
ASSUME THROUGHOUT THIS DESCRIPTION THAT THERE ARE K EXPERIMENTAL GROUPS.
.SKIP 1
THE USER AUTOMATICALLY OBTAINS AS OUTPUT:
.BREAK
(I)####A ONE-WAY AOV TABLE (AN F STATISTIC IS CALCULATED WHICH IS
.BREAK
#########USED TO TEST THE EQUALITY OF THE K POPULATION MEANS),
.BREAK
(II)###DESCRIPTIVE STATISTICS (MEANS AND STANDARD DEVIATIONS FOR
.BREAK
#########EACH OF THE K GROUPS), AND
.BREAK
(III)##BARTLETT'S TEST STATISTIC (WHICH IS USED TO TEST THE
.BREAK
#########EQUALITY OF THE K POPULATION VARIANCES).
.SKIP 1
A MORE DETAILED DESCRIPTION OF THIS FEATURE IS CONTAINED IN SECTION 2.0.
.SKIP 1
ONE OF THREE DIFFERENT DATA ENTRY METHODS MAY BE CHOSEN:
.BREAK
.SKIP 1
(I)####DATA ENTRY METHOD 1--(RAW DATA FOR EACH GROUP IS ENTERED),
.BREAK
(II)###DATA ENTRY METHOD 2--(A BREAKDOWN VARIABLE IS SPECIFIED
.BREAK
#########WHICH IS USED TO CONSTRUCT THE GROUPS.) AND
.BREAK
(III)##DATA ENTRY METHOD 3--(PARTIALLY PROCESSED DATA IS ENTERED--
.BREAK
#########SAMPLE SIZES, MEANS, AND STANDARD DEVIATIONS FOR EACH
.BREAK
#########GROUP).
.SKIP 1
.FILL
MORE DETAILS ABOUT THE USE OF THESE METHODS IS CONTAINED IN SECTION
3.0.
.SKIP 1
THE USER MAY CONTINUE THE ANALYSIS OF THE DATA BY ELECTING ONE OR MORE OF
SEVERAL DATA OPTIONS AVAILABLE. USING THESE OPTIONS THE USER MAY:
.SKIP 1
(I)####TEST THE EQUALITY OF PAIRS OF POPULATION VARIANCES,
.BREAK
(II)###TEST THE EQUALITY OF PAIRS OF POPULATION MEANS USING ONE OF
.BREAK
#########SEVERAL PROCEDURES AVAILABLE,
.BREAK
(III)##OBTAIN A TREND ANALYSIS OF THE ASSUMED EQUI-SPACED
.BREAK
##########
#########POPULATION MEANS,
.BREAK
(IV)###OBTAIN ESTIMATES, CONFIDENCE INTERVALS, AND TESTS FOR
.BREAK
#########SPECIFIED LINEAR FUNCTIONS OR CONTRASTS OF THE POPULA-
.BREAK
#########TION MEANS,
.BREAK
(V)####PERFORM SIMULTANEOUS TESTS ON THE POPULATION MEANS. THE
.BREAK
#########USER MAY CHOOSE ONE OF THE PROCEDURES: SCHEFFE, TUKEY,
.BREAK
#########NEWMAN-KEULS, DUNCAN, OR LEAST SIGNIFICANT DIFFERENCE
.BREAK
(VI)###SIMULTANEOUSLY ESTIMATE OR OBTAIN MULTIPLE COMPARISONS OF
.BREAK
#########ALL PAIRWISE DIFFERENCES OF THE POPULATION MEANS, THE
.BREAK
#########USER MAY CHOOSE ONE OF THE PROCEDURES: SCHEFFE, TUKEY,
.BREAK
#########OR BONFERRONI,
.BREAK
(VII)##OBTAIN AN AOV TABLE FOR A COLLAPSED OR REDEFINED SET
.BREAK
#########OF TREATMENT GROUPS. (E.G., FOR K=4 GROUPS AN AOV TABLE
.BREAK
#########MAY BE OBTAINED ON 3 COLLAPSED GROUPS WHERE:
.BREAK
####################NEW GROUP 1 = OLD GROUP 1
.BREAK
####################NEW GROUP 2 = OLD GROUP 2
.BREAK
####################NEW GROUP 3 = OLD GROUPS 3 AND 4),
.BREAK
(VIII)#OBTAIN AN AOV TABLE FOR TRANSFORMED DATA (ONLY APPLICABLE
.BREAK
#########FOR DATA ENTERED BY DATA ENTRY METHOD 1 OR 2). THE USER
.BREAK
#########MAY ELECT EITHER A SQUARE ROOT, ARC-SIN, NATURAL
.BREAK
#########LOGARITHM, OR RANK TRANSFORMATION. THE OPTIONS ABOVE
.BREAK
#########[(I)-(VII)] ARE ALSO APPLICABLE TO TRANSFORMED DATA.
.SKIP 1
.FILL
OTHER OPTIONS ARE ALSO AVAILABLE WHICH DO NOT ANALYZE OR PROCESS DATA,
BUT WHICH CAN BE HELPFUL. THESE INCLUDE OPTIONS WHICH ALLOW THE USER TO:
.SKIP 1
(IX)###TYPE OUT THE OPTIONS TEXT DESCRIPTION,
.BREAK
(X)####RETURN (TRANSFORMED) DATA BACK TO ITS ORIGINAL (UNTRANS-
.BREAK
#########FORMED) FORM,
.BREAK
.BREAK
(XI)###ENTER NEW DATA (ELIMINATING OLD DATA),
.BREAK
(XII)##EXIT FROM THE PROGRAM.
.FILL
.SKIP 1
A MORE DETAILED DESCRIPTION OF THESE OPTIONS AND THEIR USE IS FOUND IN SECTION 4.0.
EXAMPLES ILLUSTRATING MANY OF THE OPTION AND DATA ANALYSIS FEATURES
ARE CONTAINED IN SECTION 5.0.
SECTION 6.0 CONTAINS THE STANDARD PROGRAM QUESTIONS AND ANSWERS PLUS
A SAMPLE BATCH RUN.
SECTION 7.0 INCLUDES A DESCRIPTION OF THE LIMITATIONS OF ADVAOV.
SECTION 8.0 CONTAINS NINETEEN REFERENCES REFERRED TO IN THIS TEXT. SECTION 9.0 CONTAINS AN EXAMPLE AND SECTION INDEX.
.SKIP 2
.INDEX ^^SECTION 2.0\\
SECTION#2.0##NOTATION#AND#AUTOMATIC#OUTPUT
.BREAK
------------------------------------------
.SKIP 1
.INDEX ^^SECTION 2A\\
2A##NOTATION
.BREAK
------------
.BREAK
ASSUME THAT THERE ARE K EXPERIMENTAL GROUPS OR TREATMENTS BEING ANALYZED
IN THE ONE-WAY AOV.
.SKIP 1
ASSUME THAT THE SAMPLE SIZES FOR THE K GROUPS ARE N(1), N(2),...,N(K)
RESPECTIVELY.
.SKIP 1
LET X(I,J) BE THE JTH OBSERVATION IN THE ITH EXPERIMENTAL GROUP. THE
DATA CAN BE REPRESENTED IN THE FOLLOWING ARRAY:
.SKIP 1
.TEST PAGE 10
.NOFILL
GROUP OR TREATMENT:
###1 ###2############... ###K
.NOFILL
------- ------- -------
X(1,1) X(2,1)##########... X(K,1)
X(1,2) X(2,2)##########... X(K,2)
. . .
. . .
. . .
X(1,N(1)) X(2,N(2))#######... X(K,N(K))
--------- --------- ---------
.FILL
.SKIP 1
LET N=N(1)+N(2)+...+N(K), THE TOTAL SAMPLE SIZE.
.BREAK
.BREAK
LET##X(1),#X(2),...,X(K) BE THE SAMPLE MEANS AND
S(1),S(2),...,S(K) BE THE SAMPLE STANDARD DEVIATIONS FOR THE K GROUPS
RESPECTIVELY, WHERE:
.SKIP 1
.NOFILL
.TEST PAGE 3
###########N(I)
X(I)= X(I,J)/N(I) #####AND
###########J=1
.SKIP 1
.TEST PAGE 3
########N(I)###############2##########1/2
S(I) = (X(I,J) - X(I)) /(N(I)-1) .
########J=1
.SKIP 1
LET #(1), #(2),..., #(K) BE THE K POPULATION MEANS AND #(1), #(2),...,
#(K) BE THE K POPULATION STANDARD DEVIATIONS.
.SKIP 1
.INDEX ^^SECTION 2B\\
2B##AUTOMATIC#OUTPUT
--------------------
.BREAK
####(DESCRIPTIVE#STATISTICS,#AOV#TABLE,#AND#BARTLETT'S
.BREAK
####TEST)
.BREAK
THE AUTOMATIC OUTPUT PRODUCED FOR EACH SET OF DATA ENTERED CONSISTS OF:
.SKIP 1
(I)####DESCRIPTIVE STATISTICS (MEANS--X(I), STANDARD DEVIATIONS--
.BREAK
#########S(I), SAMPLE SIZES--N(I), FOR EACH OF THE K GROUPS).
.SKIP 1
(II)###A ONE-WAY AOV TABLE, WHICH IS USED TO TEST THE EQUALITY
.BREAK
#########OF THE K POPULATION MEANS, I.E.
.SKIP 1
.CENTER
H0:##(1) = ##(2) = ... = ##(K)
.SKIP 1
THE OUTPUT CONSISTS OF THE SUMS OF SQUARES, MEAN SQUARES, TOTAL, AND
THE F-VALUE
.SKIP 1
.CENTER
F=MSB/MSE =
.SKIP 1
.CENTER
(MEAN SQUARE BETWEEN GROUPS)/(MEAN SQUARE WITHIN GROUPS).
.SKIP 1
WHERE:
.TEST PAGE 5
.SKIP 1
.TEST PAGE 5
###########K################2#######K
.BREAK
.NOFILL
### N(I).[X(I)-X]## #####N(I)X(I)
###################################I=1
#####MSB#=######--------------,#X=##----------
###########I=1#######K-1################N
.SKIP 2
.TEST PAGE 4
###########K###N(I)#############2
AND##MSE#= ### ### [X(I,J)-X(I)]##
###########I=1 J=1##--------------
#########################N-K
.SKIP 1
.FILL
F = MSB/MSE HAS AN F DISTRIBUTION WITH K-1 AND N-K DEGREES OF FREEDOM
WHEN THE NULL HYPOTHESIS IS TRUE AND IS USED TO TEST THE HYPOTHESIS OF THE
EQUALITY OF THE K MEANS. SPECIFICALLY REJECT THE NULL HYPOTHESIS OF THE EQUALITY OF THE K MEANS AT LEVEL ## IF F EXCEEDS THE UPPER ## POINT
OF THE F DISTRIBUTION WITH K-1 AND N-K DEGREES OF FREEDOM.
.SKIP 1
AN F PROBABILITY VALUE P IS ALSO GIVEN. IF P <= ##, THEN CONCLUDE THAT
THE K POPULATION MEANS ARE SIGNIFICANTLY DIFFERENT AT THE SIGNIFICANCE LEVEL ##.
.SKIP 1
(III)##BARTLETT'S TEST STATISTIC -- B -- IS USED
.BREAK
#########TO TEST THE EQUALITY OF THE K POPULATION STANDARD DEVIA-
.BREAK
#########TIONS, I.E.
.SKIP 1
.CENTER
H0: ## (1) = ##(2) = ... = ## (K)
.SKIP 1
B HAS AN APPROXIMATE CHI-SQUARE DISTRIBUTION WITH K-1 DEGREES OF FREEDOM
WHEN H0 IS TRUE.
.SKIP 1
A CHI-SQUARE PROBABILITY VALUE P IS GIVEN WHICH HAS K-1 DEGREES OF FREEDOM.
IF P <= ##, THEN WE CONCLUDE THAT THE K POPULATION STANDARD DEVIATIONS
ARE SIGNIFICANTLY DIFFERENT AT THE SIGNIFICANCE LEVEL ##.
.SKIP 1
CAUTION:
.BREAK
BANCROFT [3] RECOMMENDS CHOOSING ## = .25 WHEN USING BARTLETT'S TEST.
GENERALLY, IF P < .25 (P<.05) FOR BARTLETT'S TEST, INTERPRET WITH (EXTREME) CAUTION THE ONE-WAY AOV RESULTS FOR TESTING HO:##(1) = ##(2) = ...
= ###(K), UNLESS N(1) = N(2) = ... = N(K) (SAMPLES ARE BALANCED). IT IS
GENERALLY AGREED THAT THE BEST SAFEGUARD AGAINST THE FAILURE OF THE
EQUALITY OF POPULATION STANDARD DEVIATIONS IS TO DESIGN YOUR EXPERIMENT
WITH BALANCED OR NEARLY BALANCED SAMPLES. SEE CHAPTER 10 OF SCHEFFE [4].
.SKIP 1
A FORMULA FOR BARTLETT'S STATISTIC IS:
.TEST PAGE 5
.NOFILL
.SKIP 2
#############################K#################2
#########(N-K)LOG #(MSE) - ### [N(I)-1].LOG# (S#(I))
#################E#########I=1#############E
#####B = ---------------------------------------------, WHERE
C
.SKIP 1
.TEST PAGE 7
K
1 1
################ ----#-#----
N(I) N-K
I=1
C = 1 + -------------
3(K-1)
.SKIP 2
.INDEX ^^SECTION 3.0\\
SECTION#3.0 DATA ENTRY METHODS
-------------------------------
.FILL
.SKIP 1
THE USER MAY ELECT ONE OF THREE METHODS FOR ENTERING DATA. FOR ANY
METHOD THE USER MUST SPECIFY K, THE NUMBER OF GROUPS OR LEVELS.
.SKIP 1
.INDEX ^^SECTION 3A\\
3A##DATA#ENTRY#METHOD#1
.BREAK
-----------------------
.SKIP 1
.BREAK
(THE RAW DATA FOR EACH GROUP IS ENTERED)
.BREAK
SPECIFICALLY, THE USER SUPPLIES THE NUMBER OF GROUPS K; THE GROUP SAMPLE
SIZES N(1), N(2), ... , N(K) (10 PER LINE SEPARATED BY COMMAS); THE N
DATA POINTS X(I,J) FOR J = 1,2, ... , N(I) AND I = 1,2, ... , K. THE
N(1) DATA POINTS FOR GROUP 1 ARE ENTERED FIRST (ONE PER LINE), FOLLOWED
BY THE N(2) DATA POINTS FOR GROUP 2, ETC.
.SKIP 1
.INDEX ^^EXAMPLE 3.1#(METH 1)\\
EXAMPLE 3.1:
.BREAK
FOR K = 3 GROUPS AND SAMPLE SIZES 6,4, AND 3 AND THE DATA:
.TEST PAGE 10
.NOFILL
.SKIP 1
GROUP 1 GROUP 2 GROUP 3
------- ------- -------
1 1 1
0 4 9
1 6 6
3 2 -------
4 -------
0
-------
N(1)=6 N(2)=4 N(3)=3
.SKIP 1
THE DATA IS ENTERED AS FOLLOWS USING DATA ENTRY METHOD 1:
.SKIP 2
WHICH METHOD OF DATA ENTRY?(1,2,OR 3)
TYPE "HELP" FOR EXPLANATION
1
.SKIP 1
HOW MANY GROUPS? 3
.SKIP 1
FORMAT: (F - TYPE ONLY)
.SKIP 2
ENTER SAMPLE SIZES(10 PER LINE)
6,4,3
.TEST PAGE 7
.SKIP 1
ENTER DATA FOR GROUP 1
1
0
1
3
4
0
.TEST PAGE 5
.SKIP 1
ENTER DATA FOR GROUP 2
1
4
6
2
.SKIP 1
.TEST PAGE 4
ENTER DATA FOR GROUP 3
1
9
6
.SKIP 1
.SKIP 1
.INDEX ^^SECTION 3B\\
3B DATA ENTRY METHOD 2
-----------------------
.FILL
(WITH EACH RAW DATA OBSERVATION ENTERED, A BREAKDOWN VARIABLE
OBSERVATION IS ALSO ENTERED WHICH DETERMINES THE GROUP MEMBERSHIP OF THE RAW DATA OBSERVATION).
.SKIP 1
.BREAK
THE USER SUPPLIES K, THE NUMBER OF GROUPS, AND THE K BREAKDOWN LIMITS:
B(1)<B(2)<...<B(K).
USING THIS METHOD ONLY TWO COLUMNS OF DATA ARE ENTERED. THE USER SPECIFIES
WHICH OF THE TWO VARIABLES (COLUMNS) IS TO BE THE BREAKDOWN VARIABLE:
VARIABLE 1 (COLUMN 1) OR VARIABLE 2 (COLUMN 2). THE USER THEN ENTERS
THE K BREAKDOWN LIMITS SEPARATED BY COMMAS (10 PER LINE): B(1),B(2),...,B(K).
.SKIP 1
LET X(I) AND Y(I) BE THE ITH RAW DATA POINT AND ITH BREAKDOWN
VARIABLE OBSERVATION RESPECTIVELY.
.SKIP 1
THE DATA IS ENTERED AS TWO COLUMNS (ASSUMING VARIABLE 2 (COLUMN 2) HAS
BEEN DESIGNATED AS THE BREAKDOWN VARIABLE) AS:
.TEST PAGE 7
.SKIP 1
.TAB STOPS 10 25
.NOFILL
X(1),Y(1) OBS 1
X(2),Y(2) #OBS 2
. .
. .
. .
X(N),Y(N) OBS N
_^Z
.SKIP 1
NOTE THAT:
.TAB STOPS 5 15
.SKIP 1
.FILL
(I)####THE PAIRS OF OBSERVATIONS ARE ENTERED IN THE ORDER:
.BREAK
(A)##(DATA, BREAKDOWN), IF VARIABLE 2 IS THE BREAKDOWN
.BREAK
#######VARIABLE
.BREAK
(B)##(BREAKDOWN, DATA), IF VARIABLE 1 IS THE BREAKDOWN
.BREAK
#######VARIABLE,
.BREAK
(II)###_^Z (CONTROL Z) SIGNALS THE END OF DATA ENTRY, AND
.BREAK
.TEST PAGE 12
(III)##THE K GROUPS ARE DEFINED AS FOLLOWS USING THE K BREAK-
.BREAK
#########DOWN LIMITS B(1)<B(2)<...<B(K): IF THE JTH OBSERVATION
.BREAK
#########X(J) IS SUCH THAT:
.BREAK
(1)##Y(J)<=B(1), THEN X(J) BELONGS TO GROUP 1,
.BREAK
(2)##B(1)<Y(J)<=B(2), THEN X(J) BELONGS TO GROUP 2,
.BREAK
######.
.BREAK
######.
.BREAK
######.
.BREAK
(K)##B(K-1)<Y(J)<=B(K), THEN X(J) BELONGS TO GROUP K.
.BREAK
(K+1)##IF Y(J) > B(K), THEN X(J) IS NOT CLASSIFIED
.BREAK
#######INTO ANY ONE OF THE K GROUPS.
.SKIP 1
THEREFORE, ANY DATA POINT WHOSE BREAKDOWN DATA OBSERVATION EXCEEDS THE
LARGEST BREAKDOWN LIMIT IS NOT CLASSIFIED INTO ANY GROUP.
.SKIP 1
HENCE, IN SUMMARY, TO USE DATA ENTRY METHOD 2, THE USER SUPPLIES:
.BREAK
.SKIP 1
(I)####K--THE NUMBER OF GROUPS,
.BREAK
(II)###THE K BREAKDOWN LIMITS: B(1) < B(2) <...< B(K),
.BREAK
(III)##THE BREAKDOWN VARIABLE COLUMN (1 OR 2),
.BREAK
(IV)###THE RAW DATA (X) AND THE BREAKDOWN VARIABLE OBSERVATION
.BREAK
#########(Y) FOR ALL N SAMPLE DATA POINTS AS:
.NOFILL
.TEST PAGE 7
.SKIP 1
X(1),Y(1)
X(2),Y(2)
. .
. .
. .
X(N),Y(N) (VARIABLE 2 = BREAKDOWN VARIABLE)
_^Z
.SKIP 1
(V)####_^Z, TO TERMINATE ENTERING OF DATA.
.SKIP 2
.INDEX ^^EXAMPLE 3.2#(METH 2)\\
EXAMPLE 3.2:
.BREAK
.FILL
SUPPOSE THAT AN ANALYSIS OF N=10 OBSERVATIONS ON A VARIABLE FROM A QUES-
TIONNAIRE IS WANTED TO COMPARE VARIOUS GROUPS DEFINED ON THE BASIS OF LEVEL
OF INCOME. ASSUME THE DATA:
.TAB STOPS 12 38 59
.SKIP 1
.TEST PAGE 13
.NOFILL
QUESTIONNAIRE###############VARIABLE ##########INCOME
NUMBER#################OBSERVATION#########LEVEL
############-------------##############-----------#########------
1 7 LOW
2 8 MEDIUM
3 7 HIGH
4 7 MEDIUM
5 8 LOW
6 1 HIGH
7 1 MEDIUM
8 6 HIGH
9 1 LOW
10 3 HIGH
.SKIP 2
.FILL
SUPPOSE THAT AN ANALYSIS OF VARIANCE (AOV) IS WANTED FOR THE CRITERION
VARIABLE OBSERVATION COMPARING THE MEAN DIFFERENCES OF THE THREE INCOME
GROUPS: LOW, MEDIUM, AND HIGH. THIS IS ACCOMPLISHED BY USING A BREAK-
DOWN VARIABLE FOR THE 3 GROUPS WITH BREAKDOWN LIMITS: 1,2,3, WHERE:
.SKIP 1
"LOW" IS CLASSIFIED IF: Y <= 1
.BREAK
"MEDIUM" IS CLASSIFIED IF: 1 < Y <= 2
.BREAK
"HIGH" IS CLASSIFIED IF: 2 < Y <= 3
.SKIP 1
THE DATA IS ENTERED BY TERMINAL AS FOLLOWS:
.NOFILL
.SKIP 1
WHICH METHOD OF DATA ENTRY?(1,2,OR 3)
TYPE "HELP" FOR EXPLANATION
2
.SKIP 1
HOW MANY GROUPS? 3
.SKIP 1
FORMAT: (F - TYPE ONLY)
.SKIP 2
WHICH IS THE BREAKDOWN VARIABLE?(1 OR 2) 1
.SKIP 1
ENTER BREAKDOWN LIMITS(10 PER LINE)
1,2,3
.TEST PAGE 12
.SKIP 1
ENTER DATA
1,7
2,8
3,7
2,7
1,8
3,1
2,1
3,6
1,1
3,3
_^Z
.SKIP 1
NUMBER OF REJECTED SAMPLES IS 0
.SKIP 2
.FILL
NOTE THAT VARIABLE 1 IS THE BREAKDOWN VARIABLE AND VARIABLE 2
CONTAINS THE RAW DATA AND THAT A BREAKDOWN VALUE OF Y=1 IS CLASSIFIED INTO
THE LOW GROUP, Y=2, INTO THE MEDIUM GROUP, AND Y=3, INTO THE HIGH
GROUP, RESPECTIVELY.
.SKIP 1
THE DESCRIPTIVE DATA OUTPUT FOR THIS DATA IS:
.SKIP 1
.TEST PAGE 9
.NOFILL
.CENTER
*** DESCRIPTIVE DATA ***
.SKIP 1
.TAB STOPS 5 19 30 48 62
GROUP SAMPLE SIZE MEAN STD. DEV. VARIANCE
------------------------------------------------------------------------
1 3 5.333 3.786 14.333
.SKIP 1
2 3 5.333 3.786 14.333
.SKIP 1
3 4 4.250 2.754 7.583
.FILL
.SKIP 2
IT IS OBSERVED THAT THE LOW GROUP MEAN IS 5.333 (BASED ON 3 OBSERVA-
TIONS), THE MEDIUM AND HIGH GROUP MEANS ARE 5.333 AND 4.250 RESPECTIVELY.
.SKIP 2
.INDEX ^^SECTION 3C\\
3C##DATA#ENTRY#METHOD#3
.BREAK
-----------------------
.BREAK
(THE K SAMPLE SIZES, SAMPLE MEANS, AND SAMPLE STANDARD DEVIATIONS
ARE ENTERED).
.SKIP 1
.BREAK
FOR DATA ENTRY METHOD 3 THE USER SUPPLIES:
.SKIP 1
.NOFILL
(I)####K--THE NUMBER OF GROUPS,
(II)###THE K SAMPLES SIZES: N(1),N(2),...,N(K)
#########(SEPARATED BY COMMAS--10 PER LINE)
(III)##THE K SAMPLE MEANS: X(1),X(2),...,X(K)
#########(SEPARATED BY COMMAS--10 PER LINE)
(IV)###THE K SAMPLE STANDARD DEVIATIONS:
#########S(1),S(2),...,S(K)
#########(SEPARATED BY COMMAS--10 PER LINE).
.SKIP 1
.FILL
NOTE: DATA ENTRY METHOD 3 IS BEST SUITED FOR SITUATIONS WHERE THE
SAMPLE MEANS AND STANDARD DEVIATIONS HAVE BEEN OBTAINED FROM OTHER
COMPUTER PROGRAM OUTPUT AND FURTHER AOV PROCESSING IS DESIRED.
.SKIP 2
.INDEX ^^EXAMPLE 3.3#(METH 3)\\
EXAMPLE 3.3:
.BREAK
FOR THE DATA OF EXAMPLE 3.1, THE SAMPLE MEANS
AND STANDARD DEVIATIONS FOR THE 3 GROUPS ARE:
.TEST PAGE 7
.SKIP 1
.TAB STOPS 11 27 37 50
.NOFILL
GROUP NUMBER SAMPLE SIZE#####MEAN STANDARD DEVIATION
------------------------------------------------------------------------
1 6 1.500 1.643
.SKIP 1
2 4 3.250 2.217
.SKIP 1
3 3 5.333 4.041
.FILL
.SKIP 1
AN AOV TABLE MAY BE OBTAINED FOR THIS DATA BY USING DATA ENTRY METHOD
3 AS FOLLOWS:
.NOFILL
.SKIP 1
WHICH METHOD OF DATA ENTRY?(1,2,OR 3)
TYPE "HELP" FOR EXPLANATION
3
.SKIP 1
HOW MANY GROUPS? 3
.SKIP 1
FORMAT: (F - TYPE ONLY)
.SKIP 2
ENTER SAMPLE SIZES(10 PER LINE)
6,4,3
.SKIP 1
ENTER THE 3 MEANS
1.500,3.250,5.333
.SKIP 1
ENTER THE 3 STANDARD DEVIATIONS
1.643,2.217,4.041
.SKIP 2
.INDEX ^^SECTION 4.0\\
SECTION#4.0##OPTION#DESCRIPTION#AND#USE
.BREAK
---------------------------------------
.SKIP 1
AFTER THE DATA IS ENTERED AND THE AUTOMATIC OUTPUT IS LISTED, THE USER
MAY CONTINUE THE DATA ANALYSIS BY ELECTING ONE OR MORE OF SEVERAL DATA
OPTIONS AVAILABLE. THE USER IS ASKED:
.SKIP 1
.CENTER
"WHICH OPTION? (TYPE "HELP" FOR EXPLANATION)"
.SKIP 1
THE USER MAY RESPOND USING ONE OF THE THIRTEEN RESPONSES: VAR, TREND,
TTEXC, TTAPP, SIMTES, SIMEST, COMPAR, COLAOV, TRANS, ORIG, DATA, HELP,
AND EXIT (OR FINI).
.SKIP 1
.INDEX ^^SECTION 4A\\
4A##VAR
-------
.BREAK
(TESTS THE EQUALITY OF ALL PAIRS OF POPULATION VARIANCES)
.SKIP 1
PURPOSE:
.BREAK
THIS OPTION IS USED TO TEST THE HYPOTHESIS:
.CENTER
H0:####(I) = ####(J),
AGAINST THE ONE-SIDED ALTERNATIVE HYPOTHESIS:
.CENTER
H1:####(I)#>#####(J),
WHERE ###(I) AND ###(J) DENOTE THE NUMERATOR AND DENOMINATOR
GROUP I AND J POPULATION VARIANCES RESPECTIVELY. TEST OUTPUT IS GIVEN
FOR ALL PAIRS (I,J), WHERE I <> J.
.SKIP 1
OUTPUT:
.BREAK
FOR EACH PAIR (I,J), THE FOLLOWING OUTPUT IS GIVEN:
.TAB STOPS 5 31
.SKIP 1
.NOFILL
(I)####AN F VALUE = F(I,J) = S (I)/S (J) = VAR A/VAR B,
(II)###A PROBABILITY VALUE = P(I,J), AND
(III)##NUMERATOR AND DENOMINATOR DEGREES OF FREEDOM N(I)-1 AND
#########N(J)-1, WHERE S (I) AND S (J) ARE THE GROUP I AND J
#########SAMPLE VARIANCES RESPECTIVELY.
.FILL
.SKIP 1
OUTPUT#DESCRIPTION#AND#USE:
.BREAK
F(I,J) HAS AN F DISTRIBUTION WITH N(I)-1 AND N(J)-1 DEGREES OF FREEDOM
WHEN THE NULL HYPOTHESIS H: ###(I) = ###(J) (POPULATION VARIANCES I
AND J ARE EQUAL) IS TRUE.
.SKIP 1
THE PROBABILITY VALUE P(I,J) IS THE PROBABILITY THAT AN F DISTRIBUTION WITH
N(I)-1 AND N(J)-1 DEGREES OF FREEDOM EXCEEDS THE OBSERVED F(I,J) VALUE.
THE HYPOTHESIS:
.CENTER
H0: ###(I) = ### (J)
.CENTER
H1: ####(I) > ### (J)
MAY BE TESTED AT AN ##-LEVEL OF SIGNIFICANCE BY EITHER:
.SKIP 1
(I)####COMPARING THE OBSERVED F(I,J) VALUE TO THE UPPER ##-POINT
.BREAK
#########OF THE F DISTRIBUTION WITH N(I)-1 AND N(J)-1 DEGREES
.BREAK
#########OF FREEDOM OR
.BREAK
(II)###COMPARING THE PROBABILITY VALUE P(I,J) TO ##.
.BREAK
###########(A)##IF P(I,J) > ##, THEN ACCEPT H0:###(I)=###(J)
.BREAK
################AT LEVEL ##.
.BREAK
###########(B)##IF P(I,J) < ##, THEN REJECT H0:###(I)=###(J)
.BREAK
################AT LEVEL ##.
.SKIP 1
NOTE: THE TWO-SIDED HYPOTHESIS H0:###(I)=###(J) VERSUS H1:###(I)<>###(J)
MAY BE TESTED BY USING OUTPUT FROM BOTH PAIRS (I,J) AND (J,I) AS
FOLLOWS:
.BREAK
REJECT H0: (I) = (J) AND
.BREAK
ACCEPT H1: (I)<> (J) AT SIGNIFICANCE LEVEL ###
.BREAK
IF AND ONLY IF EITHER: PROBABILITY VALUE P(I,J) < ##/2 OR P(J,I) < ##/2.
EQUIVALENTLY
.BREAK
ACCEPT H0: (I) = (J) AND
.BREAK
REJECT H1: (I) <> (J) AT SIGNIFICANCE LEVEL ###
.BREAK
IF AND ONLY IF BOTH P(I,J) > ##/2 AND P(J,I) > ##/2.
.SKIP 1
EXAMPLES:##THE OPTION VAR IS ILLUSTRATED IN (1) EXAMPLE 5.1
.SKIP 2
.INDEX ^^SECTION 4B\\
4B##TREND
.BREAK
---------
.BREAK
(LINEAR, QUADRATIC, CUBIC, QUARTIC, AND QUINTIC COMPONENTS OF THE
MEANS--ONLY VALID FOR EQUI-SPACED MEANS AND BALANCED SAMPLES).
.SKIP 1
PURPOSE:
.BREAK
THE TREND OPTION CAN BE USED IN BALANCED AOV'S WHERE THE K MEANS ARE ASSUMED
TO BE EQUI-SPACED ALONG SOME SCALE (IN REGRESSION ANALYSIS, THE SCALE
WOULD BE AN INDEPENDENT VARIABLE).
.SKIP 4
.TEST PAGE 5
###############X(1)
.BREAK
######################X(2)
.BREAK
###################################X(K)
.BREAK
.SKIP 1
###GROUP########1######2#####...####K
.SKIP 3
THE BETWEEN GROUP SUM OF SQUARES IS DECOMPOSED INTO ITS ORTHOGONAL ONE
DEGREE OF FREEDOM TREND COMPONENTS (LINEAR, QUADRATIC, ETC). THE TREND
COMPONENT SUMS OF SQUARES ARE OUTPUTTED ALONG WITH THE APPROPRIATE F TESTS
FOR TESTING THE SIGNIFICANCE OF THE INDIVIDUAL TRENDS: LINEAR,
QUADRATIC, ETC
.SKIP 1
STATISTICAL DISCUSSION, OUTPUT, AND USE:
.BREAK
WE ASSUME THAT:
.BREAK
N(1) = N(2) = ... = N(K) = M = N/K AND THAT
.SKIP 1
X = THE GRAND SAMPLE MEAN = [X(1) + X (2) + ... + X (K)]/K
.SKIP 1
.BREAK
THE SUM OF SQUARES BETWEEN GROUPS,
.BREAK
.TEST PAGE 3
#############K#############2
.BREAK
#####SSB = ### M [X(J) - X] , WITH K-1 DEGREES OF FREEDOM
.BREAK
############I=1
.BREAK
IS DECOMPOSED INTO THE SUM OF THE ONE DEGREE OF FREEDOM ORTHOGONAL
TREND COMPONENTS WHICH INCLUDE:
.BREAK
.NOFILL
(I)####A LINEAR COMPONENT, SS1,
(II)###A QUADRATIC COMPONENT, SS2, IF K>=3,
(III)##A CUBIC COMPONENT, SS3, IF K>=4,
(IV)###A QUARTIC COMPONENT, SS4, IF K>=5,
(V)####A QUINTIC COMPONENT, SS5, IF K>=6, AND
(VI)###AN ORTHOGONAL SUM OF SQUARES DUE TO "DEPARTURES OF
.BREAK
#########GROUPS FROM QUINTIC" WITH K-6 DEGREES OF FREEDOM,
.BREAK
#########SSD5, IF K>=7.
.SKIP 1
.FILL
THE FOLLOWING TABLE SHOWS THE DECOMPOSITION OF SSB (SUM OF SQUARES BETWEEN
GROUPS) INTO THE SUM OF THE ORTHOGONAL TREND COMPONENTS FOR SEVERAL VALUES
OF K
.TEST PAGE 11
.SKIP 2
.CENTER
TABLE 4B.1
.SKIP 1
.TAB STOPS 5 11 21 30 37 44 51 60
#K #######LINEARITY QUAD CUBIC#QUARTIC#QUINTIC##DEPARTURES
.BREAK
########FROM QUINTIC
.SKIP 1
------------------------------------------------------------------
.NOFILL
K=2 SSB= SS1
K=3 SSB= SS1 + SS2
K=4 SSB= SS1 + SS2 + SS3
K=5 SSB= SS1 + SS2 + SS3 + SS4 +
K=6 SSB= SS1 + SS2 + SS3 + SS4 + SS5
K>=7 SSB= SS1 + SS2 + SS3 + SS4 + SS5 + SSD5
.SKIP 2
.FILL
THE TREND COMPONENTS ARE CALCULATED BY GENERATING TREND COEFFICIENTS.
FOR EXAMPLE, THE LINEAR COMPONENT:
.SKIP 1
.NOFILL
#################################################2
.TEST PAGE 4
#####M[CI(1).X(1)+CI(2).X(2)+...+CI(K).X(K)]
SS1 =####------------------------------------------- ,
######################2########2############2
################(CI(1)) +(CI(2)) +...+(CI(K))
.SKIP 1
.FILL
WHERE CI(1),CI(2),...,CI(K) ARE THE LINEAR TREND COEFFICIENTS.
THE LINEAR TREND COEFFICIENTS AND THE OTHER HIGHER ORDER TREND COEFFICIENTS (FOR QUADRATIC, CUBIC, ETC) ARE TABLED IN SEVERAL
SOURCES INCLUDING:
.BREAK
.SKIP 1
(A)##SNEDECOR AND COCHRAN [2], TABLE A17 (PG 572)
.BREAK
(B)##WINER [5], TABLE C.10 (PG 878).
.SKIP 1
STATISTICAL TESTS FOR TESTING THE SIGNIFICANCE OF THE TREND COMPONENTS
ARE INCLUDED IN THE OUTPUT. SPECIFICALLY, THE TEST FOR THE LINEAR TREND
IS BASED ON THE STATISTIC: F1 = SS1/MSE, WHICH HAS AN F DISTRIBUTION WITH
1 AND N-K DEGREES OF FREEDOM WHEN THE NULL HYPOTHESIS (LINEAR TREND
EFFECT IS ZERO) IS TRUE. THE MEAN SQUARE ERROR OR WITHIN TERM IS GIVEN BY:
.BREAK
.SKIP 1
.TEST PAGE 3
#######K####M###############2
.BREAK
MSE = ### ### [X(I,J)-X(I)] /(N-K)
.BREAK
######I=1##J=1
.BREAK
.SKIP 1
AND IS INCLUDED IN THE AOV TABLE.
.SKIP 1
SPECIFICALLY, IF F1 EXCEEDS THE UPPER ##-POINT OF THE F DISTRIBUTION WITH
1 AND N-K DEGREES OF FREEDOM, THEN THE LINEAR TREND COMPONENT IS DECLARED
SIGNIFICANTLY DIFFERENT FROM ZERO AT LEVEL ##.
.SKIP 1
A PROBABILITY VALUE, P1, IS GIVEN FOR F1, WHICH CAN BE USED DIRECTLY
TO TEST FOR THE LINEAR TREND. IF
.BREAK
(I)####P1 < ##, THEN DECLARE THE LINEAR TREND SIGNIFICANT AT
.BREAK
#########LEVEL ##, OR
.BREAK
(II)###IF P1 > ##, THEN DECLARE THE LINEAR TREND NON-SIGNIFI-
.BREAK
#########CANT AT LEVEL ##.
.BREAK
.SKIP 1
THE TESTS FOR SIGNIFICANCE OF THE OTHER TREND COMPONENTS (QUADRATIC,
CUBIC, QUARTIC, AND QUINTIC) ARE SIMILARLY DEFINED AND PERFORMED.
.SKIP 2
.TEST PAGE 13
.NOFILL
.CENTER
TABLE 4B.2
.SKIP 1
.TAB STOPS 5 22 47
#####TREND##############F-VALUE#########F - DEGREES OF FREEDOM
--------------------------------------------------------------------
.BREAK
.SKIP 1
LINEAR F1= SS1/MSE (1,N-K)
QUADRATIC F2= SS2/MSE (1,N-K)
CUBIC F3= SS3/MSE (1,N-K)
QUARTIC F4= SS4/MSE (1,N-K)
QUINTIC F5= SS5/MSE (1,N-K)
.TEST PAGE 3
DEPARTURE OF #####SSD5
GROUPS FROM FD5= ----##### (K-6,N-K)
QUINTIC ####(K-6)MSE
.SKIP 2
NOTE THAT IF K>=7, THEN A SINGLE TEST OF THE SIGNIFICANCE OF THE TREND
COMPONENTS (HIGHER THEN QUINTIC) IS OBTAINED BY USE OF THE F-VALUE:
.SKIP 1
.TEST PAGE 3
######SSD5
.BREAK
FD5 = -----##### ,
.BREAK
######(K-6)MSE
.SKIP 2
WHICH HAS AN F DISTRIBUTION WITH K-6 AND N-K DEGREES OF FREEDOM WHEN THE
NULL HYPOTHESIS (HIGHER THAN QUINTIC COMPONENTS ARE ZERO) IS TRUE.
.SKIP 1
NOTE:
.BREAK
AN ##-LEVEL EXPERIMENTWISE TESTING PROCEDURE (PROBABILITY OF AT LEAST
ONE TYPE I ERROR) FOR THE TREND COMPONENTS IS OBTAINED AS FOLLOWS:
.SKIP 1
STEP 1--PERFORM THE F TEST FOR TESTING H:##(1)=##(2)=...=##(K) BY
OBTAINING F = MSB/MSE.
.BREAK
.FILL
(I)####IF F IS NON-SIGNIFICANT AT LEVEL ##, THEN STOP ALL FURTHER
.BREAK
#########TESTING AND DECLARE ALL TREND COMPONENTS NON-SIGNIFICANT
.BREAK
#########AT LEVEL ##.
.BREAK
(II)###IF F IS SIGNIFICANT AT LEVEL ##, THEN#GO TO STEP 2.
.SKIP 1
.BREAK
STEP 2--PERFORM THE TREND ANALYSIS TESTS INDIVIDUALLY FOR LINEARITY,
QUADRATIC, CUBIC, ETC, EACH AT AN ##-LEVEL OF SIGNIFICANCE. REPORT AS
SIGNIFICANT ONLY THOSE TREND TESTS THAT ARE INDIVIDUALLY SIGNIFICANT AT
LEVEL ##.
.SKIP 1
EXAMPLES:
.BREAK
THE OPTION TREND IS ILLUSTRATED IN EXAMPLE 5.2
.SKIP 2
.TAB STOPS 5 14
.INDEX ^^SECTION 4C\\
4C##TTEXC
.BREAK
---------
.BREAK
(T VALUES AND INDIVIDUAL CONFIDENCE INTERVALS FOR ALL DIFFERENCES OF PAIRS OF MEANS.)
.BREAK
.SKIP 1
PURPOSE AND OUTPUT:
.BREAK
THIS OPTION PROVIDES A STATISTICAL ANALYSIS OF THE MEAN DIFFERENCES:
.TAB STOPS 5 14
##(I) - ## (J), FOR ALL PAIRS (I,J) WHERE 1<=I<J<=K.
.BREAK
.TAB STOPS 15 20
.TEST PAGE 3
K(K-1)
.BREAK
------###SUCH PAIRS ##.
.BREAK
###2
.BREAK
SPECIFICALLY PROVIDED FOR EACH SUCH PAIR (I,J) IS:
.BREAK
.TAB STOPS 5 14
(I)####AN ESTIMATE X(I) - X(J) OF ##(I) - ##(J),
.BREAK
(II)###A T TEST STATISTIC T(I,J) WHICH IS USED TO TEST THE NULL
.BREAK
#########HYPOTHESIS H0:##(I)=##(J) AGAINST VARIOUS ALTERNATIVE
.BREAK
#########HYPOTHESES. THE USER MAY SPECIFY EITHER A TWO-SAMPLE OR
.BREAK
#########A POOLED SAMPLE ERROR TERM FOR THE T STATISTIC AND
.BREAK
(III)##A 100 (1-##)% CONFIDENCE INTERVAL FOR ##(I)-##(J), WHERE
.BREAK
###########IS USER SPECIFIED.
.BREAK
.SKIP 1
NOTE:
.BREAK
THESE STATISTICAL PRODECURES ARE BASED ON THE CLASSIC ASSUMPTION IN THE ONE-WAY
AOV THAT:
.BREAK
##(1) = ##(2) =...= ## (K) (HOMOGENEITY OF VARIANCE).
.BREAK
IF THIS ASSUMPTION IS NOT MET (BARTLETT'S TEST MAY BE SIGNIFICANT), THEN
THE USER IS ADVISED TO USE THE APPROXIMATE PROCEDURES GIVEN IN THE OPTION:
TTAPP, ESPECIALLY IF THE SAMPLE SIZES N(1),N(2),...,N(K) ARE UNEQUAL
(UNBALANCED).
.SKIP 1
OUTPUT DESCRIPTION AND USE
.BREAK
A T-STATISTIC T(I,J) IS USED TO TEST THE NULL HYPOTHESIS H0:##(I)=##(J) FOR EACH PAIR OF MEANS ##(I) AND ##(J).
.SKIP 1
THE USER MAY SPECIFY THAT :
.BREAK
(A)##T(I,J) HAS A TWO-SAMPLE ERROR TERM, IN WHICH CASE--
.BREAK
.NOFILL
.TEST PAGE 6
###################X(I) - X(J)
T(I,J) = ##########-----------
######################2#############2
#############(N(I)-1)S#(I)+(N(J)-1)S#(J)#####1######1
#############---------------------------##(#----#+#----#)
#####################N(I)+N(J)-2############N(I)###N(J)
.SKIP 1
.FILL
(WHERE T(I,J) HAS N(I)+N(J)-2 DEGREES OF FREEDOM.) , OR THAT
.BREAK
(B)##T(I,J) HAS A POOLED SAMPLE ERROR TERM, IN WHICH CASE
.SKIP 1
.NOFILL
.TEST PAGE 6
#####################X(I)-X(J)
T(I,J) = #######---------
############-------------------- .
###################1######1
#############MSE##----#+#----
##################N(I)###N(J)
.SKIP 1
(T(I,J) HAS N-K DEGREES OF FREEDOM AND MSE IS THE MEAN SQUARE
WITHIN TERM FROM THE AOV TABLE.)
.FILL
.SKIP 1
FOR EACH T(I,J) VALUE, A PROBABILITY VALUE P(I,J) IS GIVEN. IN ALL CASES
.CENTER
P(I,J)=PR###T##>##T(I,J)###=
.SKIP 1
THE PROBABILITY THAT A T DISTRIBUTION WITH EITHER N(I) + N(J) - 2
OR N-K DEGREES OF FREEDOM EXCEEDS (IN ABSOLUTE VALUE) THE OBSERVED T(I,J)
VALUE.
.SKIP 1
THE PROBABILITY VALUE P(I,J) MAY BE USED DIRECTLY TO TEST VARIOUS HYPOTHESES
ABOUT ##(I) AND ##(J) AS FOLLOWS:
.BREAK
.SKIP 1
(I)####(TWO-SIDED CASE)
.BREAK
.CENTER
H0:##(I) = ##(J)
.CENTER
#H1:##(I) <> ##(J)
.BREAK
#########IF P(I,J) < ##, THEN REJECT H0:##(I) = ##(J)
.BREAK
#########AT SIGNIFICANCE LEVEL ##.
.BREAK
#########IF P(I,J) > ##, THEN ACCEPT H0: ##(I) = ##(J)
.BREAK
#########AT SIGNIFICANCE LEVEL ##.
.BREAK
(II)###(ONE-SIDED CASE I) (I<J)
.CENTER
H0:##(I) = ##(J)
.CENTER
H1:##(I) < ##(J)
#########IF X(I)-X(J) < 0 AND IF P(I,J) < 2##,
THEN REJECT
.BREAK
#########H0:##(I) = ###(J) AT SIGNIFICANCE LEVEL ##; OTHERWISE
.BREAK
#########ACCEPT H0 AT LEVEL ##.
.SKIP 1
(III)##(ONE-SIDED CASE II) (I<J)
.CENTER
H0:##(I)#=###(J)
.CENTER
H1:##(I) > ##(J)
#########IF X(I)-X(J) > 0 AND IF P(I,J) < 2##, THEN REJECT
.BREAK
#########H0:##(I)=##(J) AT SIGNIFICANCE LEVEL ##; OTHERWISE ACCEPT
.BREAK
#########HO AT LEVEL ##.
.SKIP 1
FOR EACH PAIR OF MEANS ##(I) AND ##(J) (I<J) A 100(1-##)% CONFIDENCE INTERVAL
FOR ##(I) - ##(J) IS GIVEN, WHICH HAS THE FORM
.NOFILL
.SKIP 1
.TEST PAGE 5
(A)
.BREAK
.BREAK
#########################################2#############2
.BREAK
################################(N(I)-1)S#(I)+(N(J)-1)S#(J)###1####1
X(I)-X(J) + (T##,N(I)+N(J)-2####---------------------------##----#----
.BREAK
####################################N(I) + N(J)-2############N(I)#N(J)
.SKIP 1
.CENTER
########(IF THE USER SPECIFIES THE TWO SAMPLE ERROR TERM), OR
.SKIP 1
.TEST PAGE 5
(B)
.BREAK
#########################
#############X(I)-X(J)+(T##,N-K)######MSE ###1######1
.BREAK
###########################################----#+#----
.BREAK
###########################################N(I)###N(J)
.SKIP 1
.CENTER
########(IF THE USER SPECIFIES THE POOLED SAMPLE ERROR TERM).
.SKIP 1
.FILL
T##/2,DF IS THE UPPER ##/2 POINT OF THE T-DISTRIBUTION WITH (DF) DEGREES OF
FREEDOM. THE USER MAY SELECT ANY TWO DIGIT POSITIVE NUMBER FOR THE
CONFIDENCE PROBABILITY (ENTERING 90 WILL YIELD 90% CONFIDENCE INTERVALS).
A RETURN WILL AUTOMATICALLY PRODUCE 95% CONFIDENCE INTERVALS.
.SKIP 1
EXAMPLE: THE OPTION TTEXC IS ILLUSTRATED IN EXAMPLE 5.4
.SKIP 2
.INDEX ^^SECTION 4D\\
4D##TTAPP
.BREAK
---------
.BREAK
(APPROXIMATE T-VALUES AND APPROXIMATE INDIVIDUAL CONFIDENCE INTERVALS
FOR ALL DIFFERENCES OF PAIRS OF MEANS. THIS OPTION IS USED INSTEAD OF
"TTEXC" IF THE POPULATION VARIANCES ARE NOT EQUAL.)
.SKIP 1
PURPOSE AND USE
.BREAK
THIS OPTION PROVIDES FOR AN APPROXIMATE STATISTICAL ANALYSIS OF ALL
MEAN DIFFERENCES: ##(I)-##(J) (I<J) WHEN THE CONDITIONS REQUIRED
FOR AN EXACT ANALYSIS (GIVEN IN OPTION: TTEXC) ARE NOT MET. THIS ANALYSIS
IS RECOMMENDED WHEN THE HOMOGENEITY OF VARIANCE ASSUMPTION:
.CENTER
##(1) = ##(2) =...=##(K)
IS NOT MET. (BARTLETT'S TEST IS SIGNIFICANT).
.SKIP 1
SPECIFICALLY PROVIDED FOR EACH PAIR (I,J) IS
.SKIP 1
(I)####AN ESTIMATE X(I)-X(J) OF ##(I)-##(J)
.BREAK
(II)###AN APPROXIMATE T STATISTIC TA(I,J) WHICH IS
.BREAK
#########USED TO APPROXIMATELY TEST THE NULL HYPOTHESIS
.BREAK
#########H0:##(I)=##(J) AGAINST VARIOUS ALTERNATIVES.
.BREAK
(III)##AN APPROXIMATE 100(1-##)% CONFIDENCE INTERVAL FOR ##(I)-
.BREAK
###########(J), WHERE ## IS USER SPECIFIED.
.SKIP 1
.FILL
THE APPROXIMATE T-TESTS AND CONFIDENCE INTERVALS USED IN THIS
OPTION ARE BASED ON METHODS DUE TO SATTERTHWAITE [6] WHICH IS DESCRIBED
IN WINER [5] (PAGE 42).
.SKIP 1
NOTE:
.BREAK
##IF THE POPULATION VARIANCES ARE NOT HOMOGENEOUS BUT THE SAMPLE
SIZES ARE BALANCED (OR NEARLY BALANCED), THEN SCHEFFE [4] (CHAPTER 10)
RECOMMENDS THE F TEST IN THE AOV TABLE AND THE T-TESTS GIVEN IN THE
TTEXC OPTION ARE APPROXIMATELY VALID.
.SKIP 1
IT IS ONLY RECOMMENDED THAT THE APPROXIMATE PROCEDURES IN THE OPTION
TTAPP BE USED IF BOTH OF THE FOLLOWING CONDITIONS HOLD:
.SKIP 1
(I)####THE K POPULATION VARIANCES ARE NOT EQUAL AND
.BREAK
(II)###THE SAMPLE SIZES ARE GREATLY UNBALANCED.
.SKIP 1
OUTPUT DESCRIPTION AND USE
.BREAK
THE APPROXIMATE T-STATISTIC TA(I,J) USED TO TEST THE NULL HYPOTHESIS
H0:##(I)=##(J) IS DEFINED FOR EACH PAIR (I,J) WITH I<J AS:
.SKIP 1
.TEST PAGE 6
#####################X(I) - X(J)
.BREAK
TA(I,J) = ###--------------#############,
.BREAK
################2#######2
.BREAK
####################S#(I) # S#(J)
.BREAK
###############-----#+#-----
.BREAK
###############N(I)####N(J)
.SKIP 1
WHICH HAS AN APPROXIMATE T DISTRIBUTION WITH DEGREES OF FREEDOM APPROXIMATED BY:
.SKIP 1
####################2#######2###2
.BREAK
.TEST PAGE 10
###################S#(I)###S#(J)
.BREAK
###################-----#+#-----
.BREAK
###################N(I)####N(J)
.BREAK
######DF(I,J) =####-------------
.BREAK
####################2####2###2####2
.BREAK
###################S#(I)####S#(J)
.BREAK
###################-----###-----
.BREAK
###################N(I)####N(J)
.BREAK
#################-------#+#------
.BREAK
##################N(I)-1###N(J)-1
.BREAK
.SKIP 1
FOR EACH TA(I,J) VALUE, A PROBABILITY VALUE P(I,J) IS GIVEN, WHERE
.SKIP 1
.CENTER
P(I,J) = PR[##T# > #TA(I,J)]## =
.SKIP 1
THE PROBABILITY THAT A T DISTRIBUTION WITH DF(I,J) DEGREES
OF FREEDOM EXCEEDS (IN ABSOLUTE VALUE) THE OBSERVED TA(I,J) VALUE.
.SKIP 1
NOTE:
.BREAK
SATTERTHWAITE [6] SHOWED THAT THE APPROXIMATE DEGREES OF FREEDOM EXPRESSION
DF(I,J)
.SKIP 1
(A)##EXCEEDS THE MINIMUM OF N(I)-1, N(J)-1 AND
.BREAK
(B)##IS LESS THAN N(I) + N(J)-2
.SKIP 1
THE PROBABILITY VALUE P(I,J) MAY BE USED DIRECTLY TO TEST VARIOUS HYPOTHESES
ABOUT ##(I) AND ##(J) IN A MANNER EXACTLY ANALOGOUS TO THAT DESCRIBED IN
OPTION C: TTEXC FOR THE EXACT T METHODS. BRIEFLY, USE P(I,J) DIRECTLY
FOR TWO-SIDED TESTS AND P(I,J)/2 FOR ONE-SIDED TESTS AS DESCRIBED FOR THE
OPTION: TTEXC.
.SKIP 1
FOR EACH PAIR OF MEANS ##(I) AND ##(J) (I<J), A 100(1-##)% APPROXIMATE
CONFIDENCE INTERVAL FOR ##(I)-##(J) IS GIVEN BY:
.SKIP 1
.NOFILL
.TEST PAGE 4
################################2#######2
###############################S#(I)###S#(J)
X(I)-X(J) + ##T####,DF(I,J)#####-----#+#-----
################################N(I)####N(J)
.SKIP 1
.FILL
T##/2,DF(I,J) IS THE UPPER ##/2 POINT OF THE T-DISTRIBUTION WITH DF(I,J)
DEGREES OF FREEDOM.
.SKIP 1
THE USER MAY SELECT ANY TWO DIGIT POSITIVE NUMBER FOR THE CONFIDENCE
PROBABILITY LEVEL (ENTERING#99 WILL YIELD 99% CONFIDENCE INTERVALS). A RETURN
AUTOMATICALLY PRODUCES 95% CONFIDENCE INTERVALS.
.SKIP 1
EXAMPLE:##THE OPTION TTAPP IS ILLUSTRATED IN EXAMPLE 5.1
.SKIP 2
.INDEX ^^SECTION 4E\\
4E##SIMTES
.BREAK
----------
.BREAK
(A SIMULTANEOUS TESTING OPTION. THE USER MAY SELECT EITHER THE SCHEFFE,
TUKEY, NEWMAN-KEULS, DUNCAN OR LEAST SIGNIFICANT DIFFERENCE PROCEDURES.)
.SKIP 1
THE PURPOSE OF THIS OPTION IS TO ALLOW THE USER TO PERFORM A SIMULTANEOUS
TESTING PROCEDURE TO FIND DIFFERENCES BETWEEN THE K MEANS: ##(1),##(2),...,##(K).
.SKIP 1
THE USE OF A SIMULTANEOUS TEST IS OFTEN MOTIVATED BY THE FOLLOWING SITUATION.
IF THE F TEST IN THE AOV TABLE IS STATISTICALLY SIGNIFICANT AT LEVEL ## AND
HENCE H0:##(1) = ##(2) = ... = ##(K) IS REJECTED AT LEVEL ##,
THE QUESTION ARISES:
.SKIP 1
.CENTER
WHAT CAUSED H0:##(1)=##(2)=...=##(K) TO BE REJECTED?
.SKIP 1
THERE ARE SEVERAL DATA ANALYSIS TECHNIQUES, CALLED SIMULTANEOUS TESTING
PROCEDURES, THAT CAN BE USED TO FURTHER ANALYZE THE DATA IN THE HOPE
OF SHEDDING LIGHT UPON THIS QUESTION. THE OPTION, SIMTES, ALLOWS THE USER
TO CHOOSE ONE OF FIVE POPULAR, BUT DIFFERENT (FOLLOW-THRU) OR SIMULTANEOUS
TESTING PROCEDURES. THESE ARE THE:
.SKIP 1
(I)####SCHEFFE,
.BREAK
(II)###TUKEY,
.BREAK
(III)##NEWMAN-KEULS,
.BREAK
(IV)###DUNCAN, AND
.BREAK
(V)####LEAST SIGNIFICANT DIFFERENCE (LSD) PROCEDURES.
.BREAK
#########(PROTECTED AND UNPROTECTED)
.SKIP 1
ALL PROCEDURES SHARE IN COMMON THE MULTIPLE DECISION FEATURE THAT FOR EACH
DISTINCT PAIR OF MEANS ##(I) AND ##(J) IT IS DECIDED THAT EITHER:
.SKIP 1
(A)####(I) = ##(J) OR
.BREAK
(B)####(I) <> ##(J) AT SOME SIGNIFICANCE LEVEL ##.
.SKIP 1
OUTPUT:
.BREAK
ALL FIVE PROCEDURES SHARE THE FOLLOWING OUTPUT FEATURES:
.BREAK
(1)##THE ORDERED MEANS: Y(1) < Y(2) < ... < Y(K) AND
.BREAK
(2)##GROUP NUMBERS FOR THE ORDERED MEANS:##G(1),G(2),...,G(K). G(I) IS
THE GROUP NUMBER OF THE ITH ORDERED MEAN Y(I), WHICH IS THE ITH SMALLEST
OF THE SAMPLE MEANS: X(1),X(2),...,X(K).
.BREAK
(3)##THE ORDERED MEAN DIFFERENCES: Y(J) - Y(I), WHERE I < J. THE
ORDERED DIFFERENCES ARE OUTPUTTED IN AN UPPER TRIANGULAR MATRIX.
.BREAK
(4)TEST VALUES ARE GIVEN WHICH ARE USED TO DETERMINE WHETHER EACH PAIR
OF MEANS ##(I) AND ##(J) ARE SIGNIFICANTLY DIFFERENT OR NOT.
.SKIP 1
TEST VALUES ARE GIVEN FOR THE 5% AND 1% SIGNIFICANCE LEVELS FOR ALL FIVE
PROCEDURES AND ARE GIVEN FOR THE 10% SIGNIFICANCE LEVEL FOR ONLY THE
SCHEFFE AND LEAST SIGNIFICANT DIFFERENCE (LSD) METHODS.
.SKIP 1
ESSENTIALLY, IF THE DIFFERENCE OF THE MEANS X (I) - X(J) EXCEEDS A GIVEN
TEST VALUE, THEN ##(I) AND ##(J) ARE DECLARED SIGNIFICANTLY DIFFERENT.
.SKIP 1
IN THE BALANCED CASES THE TEST VALUES ARE OUTPUTTED; THE TEST
VALUES ARE NOT OUTPUTTED IN THE UNBALANCED CASES.
.SKIP 1
(5)##AN UPPER TRIANGULAR MATRIX TABLE IS GIVEN WHICH YIELDS THE RESULTS
OF THE SIMULTANEOUS TEST PROCEDURE SELECTED FOR EACH PAIR OF MEANS. THE
ENTRIES IN THE TABLE CONSIST OF:
.NOFILL
.SKIP 1
#####* - IF THE PAIR OF MEANS IS SIGNIFICANTLY DIFFERENT AT 10%
####** - IF THE PAIR OF MEANS IS SIGNIFICANTLY DIFFERENT AT 5%
###*** - IF THE PAIR OF MEANS IS SIGNIFICANTLY DIFFERENT AT 1%
(BLANK)- IF THE PAIR OF MEANS ARE NON-SIGNIFICANT AT 10%
.SKIP 1
.SKIP 1
.FILL
DISCUSSION#OF#THE#SIMULTANEOUS#TESTING#PROCEDURES
-------------------------------------------------
.BREAK
.TAB STOPS 5 15
.SKIP 1
REMARK 1 (UNBALANCED SAMPLES)
.BREAK
----------------------------
.BREAK
THE TUKEY, NEWMAN-KEULS, AND DUNCAN PROCEDURES CAN ONLY BE USED IN THE BALANCED CASES.
.SKIP 1
REMARK 2 (SIZE OF TEST VALUES)
.BREAK
-----------------------------
.BREAK
THE SIZE OF THE TEST VALUES DEPENDS UPON WHICH OF FIVE PROCEDURES IS BEING
USED. FOR A FIXED SIGNIFICANCE LEVEL, THE SIZE RANGES FROM LARGEST TO
SMALLEST FOR THE SCHEFFE, TUKEY, NEWMAN-KEULS, AND DUNCAN PROCEDURES.
HENCE SCHEFFE'S PROCEDURE IS MOST CONSERVATIVE - PRODUCING FEWER SIGNIFICANT DIFFERENCES THAN THE OTHER PROCEDURES.
.BREAK
.SKIP 1
REMARK 3 (PROTECTED AND UNPROTECTED LSD)
.BREAK
---------------------------------------
.BREAK
THE SCHEFFE, TUKEY, NEWMAN-KEULS AND DUNCAN SIMULTANEOUS TESTING
PROCEDURES ARE VALIDLY PERFORMED WHETHER OR NOT THE PRELIMINARY F-TEST
ON THE EQUALITY OF THE K POPULATION MEANS IS SIGNIFICANT
OR NOT.
.SKIP 1
THE LSD TEST PROCEDURE IS OBTAINED BY PERFORMING TWO SAMPLE T-TESTS
(USING A POOLED ESTIMATE OF THE POPULATION VARIANCE) ON ALL PAIRS OF
MEANS.
.SKIP 1
IF THE TWO SAMPLE T-TESTS ARE RUN REGARDLESS OF THE OUTCOME OF THE
PRELIMINARY F-TEST THEN THIS PROCEDURE IS CALLED THE: UNPROTECTED
LSD PROCEDURE. IF THE TWO SAMPLE T-TESTS ARE RUN ONLY IF THE PRELIMINARY
F-TEST IS SIGNIFICANT, THEN THIS PROCEDURE IS CALLED THE: PROTECTED LSD
PROCEDURE.
.SKIP 1
REMARK 4 (ERROR RATES)
.BREAK
---------------------
.BREAK
THERE ARE TWO COMMON ## LEVEL ERROR RATES USED IN SIMULTANEOUS TESTING.
.SKIP 1
COMPARISONWISE ## ERROR RATE: IF THE SIMULTANEOUS TEST HAS AS AN ##
LEVEL COMPARISONWISE TEST ERROR RATE, THEN EACH OF THE PAIRWISE
DECISIONS (HYPOTHESES) ABOUT THE MEAN DIFFERENCES HAS AN ##-LEVEL TYPE I
ERROR RATE.
.SKIP 1
EXPERIMENTWISE ## ERROR RATE: IF THE SIMULTANEOUS TEST HAS AN ##-LEVEL
EXPERIMENTWISE ERROR RATE, THEN THE PROBABILITY OF MAKING AT LEAST ONE
TYPE I ERROR IN ALL THE PAIRWISE DECISIONS ABOUT THE MEAN DIFFERENCES
IS LESS THAN OR EQUAL TO ##.
.SKIP 1
THE SCHEFFE, TUKEY, AND PROTECTED LSD PROCEDURES MAINTAIN AN EXPERI-
MENTWISE ERROR RATE. THE UNPROTECTED LSD IS THE ONLY PROCEDURE THAT STRICTLY MAINTAINS A COMPARISONWISE ERROR RATE. THE DUNCAN AND NEWMAN-KEULS
PROCEDURES ARE INTERMEDIATE; THAT IS, THEY ARE NOT AS CONSERVATIVE AS A
STRICT EXPERIMENTWISE PROCEDURE, BUT NOT AS LIBERAL AS A COMPARISONWISE
PROCEDURE.
.SKIP 1
REMARK 5 (NOTATION FOR THE TESTING PROCEDURES)
.NOFILL
---------------------------------------------
ASSUME THAT:
.SKIP 1
(I)####N = N(1) + N(2) + ... + N(K)
(II)###MSE = MEAN SQUARE WITHIN (ERROR TERM IN THE AOV TABLE)
(III)##F(##,K-1,N-K) IS THE UPPER ## POINT OF THE F DISTRIBUTION
WITH K-1 AND N-K DEGREES OF FREEDOM
(IV)###Q(##,K,N-K) IS THE UPPER ## POINT OF THE STUDENTIZED
.BREAK
RANGE DISTRIBUTION WITH K AND N-K DEGREES OF FREEDOM.
(TABLES FOR Q(##,K,N-K) FOR ## = 5% AND 1% ARE FOUND IN
TABLE I OF MILLER [7] AND SCHEFFE [4].)
(V)####D(##,K,N-K) IS THE UPPER ## POINT OF DUNCAN'S NEW
MULTIPLE RANGE TEST WITH K AND N-K DEGREES OF FREEDOM.
(TABLES FOR D(##,K,N-K) FOR ##=5% AND#1% ARE FOUND IN
TABLE V OF MILLER [7]).
(VI)###T(##/2,N-K) IS THE UPPER ##/2 POINT OF THE T-DISTRIBUTION
WITH N-K DEGREES OF FREEDOM.
(VII)###X# = ABSOLUTE VALUE OF X,
(VIII)#IF THE SAMPLES ARE BALANCED, LET M=N(1)=N(2)=...=N(K).
.SKIP 1
REMARK 6 (TESTING PROCEDURES)
----------------------------
.FILL
SCHEFFE METHOD:
.BREAK
LET B(##) = ## (K-1).MSE.F(##,K-1,N-K). FOR EACH PAIR ##(I) AND ##(J),
.SKIP 1
IF ##X(I) - X(J) < B(##) ## 1/N(I) + 1/N(J) ,
.BREAK
THEN DECLARE ##(I) = ##(J).
.SKIP 1
IF ##X(I) - X(J) ## > B(##) ## 1/N(I) + 1/N(J) ,
.BREAK
THEN DECLARE ##(I)<>##(J).
.BREAK
.SKIP 1
COMMENT:
.BREAK
IN THE BALANCED CASES THE TEST VALUES B(##) ## 2/M ARE OUTPUTTED FOR
## = 1%, 5%, AND 10%. SCHEFFE'S TEST RESULTS ARE GIVEN FOR UNBALANCED
CASES, BUT NO TEST VALUES ARE OUTPUTTED.
.SKIP 1
.SKIP 1
TUKEY METHOD:
.BREAK
LET C(##) = Q(##,K,N-K) ## MSE/M. FOR EACH PAIR ##(I) AND ## (J),
.SKIP 1
IF ##X(I) - X(J)## < C(##), THEN DECLARE ##(I) = ##(J).
.SKIP 1
IF ##X(I) - X##(J)## > C(##), THEN DECLARE ##(I) <> ##(J).
.SKIP 1
COMMENT:
.BREAK
TEST VALUES C(##) ARE OUTPUTTED FOR ##=1% AND 5% ONLY. TUKEY'S TEST
IS ONLY VALID IN THE BALANCED CASE.
.SKIP 2
NEWMAN-KEULS PROCEDURE:
.BREAK
THE TEST VALUES USED IN THE NEWMAN-KEULS PROCEDURE ARE:
.SKIP 1
NK(L,##) = Q[##,L,N-K].##MSE/M
.SKIP 1
DEFINED FOR L=2,3,...,K AND ## LEVELS OF 1% AND 5%. FOR EACH (I,J)
ORDERED PAIR OF MEANS Y(I) AND Y(J) WITH J>I, LET L = J-I+1 = THE NUMBER
OF MEANS SEPARATING THE ORDERED PAIR OF MEANS Y(I) AND Y(J) WITH J>I.
.SKIP 1
.TAB STOPS 5 10
THE NEWMAN-KEULS TEST PROCEDURE IS ACCOMPLISHED AS FOLLOWS:
.SKIP 1
DECLARE THE (I,J) ORDERED PAIR OF MEANS SIGNIFICANTLY DIFFERENT
.BREAK
AT LEVEL ## IF AND ONLY IF BOTH OF THE FOLLOWING HOLD:
.SKIP 1
(1)##Y(J) - Y(I) > NK[L,##] = NK[J-I+1,##] AND
.BREAK
(2)##Y(J') - Y(I') > NK[J'-I'+1,##] FOR ALL J'>=J AND ALL I'>=I.
.SKIP 1
(THE RANGE OF EACH AND EVERY SUBSET OF SIZE L' MEANS CONTAINING BOTH
Y(I) AND Y(J) MUST EXCEED THE L' RANGE CRITICAL POINT: NK[L',##].)
.SKIP 1
COMMENT:
.BREAK
THE NEWMAN-KEULS PROCEDURE IS ONLY VALID FOR BALANCED SAMPLES. FOR FURTHER
COMMENT ABOUT THE NEWMAN-KEULS PROCEDURE SEE THE COMMENTS FOR DUNCAN'S
PROCEDURE.
.SKIP 1
DUNCAN'S PROCEDURE:
.BREAK
THE TESTING PROCEDURE FOR THE DUNCAN METHOD IS IDENTICAL TO THE
NEWMAN-KEULS PROCEDURE, EXCEPT FOR THE TEST VALUES. THE TEST VALUES
USED IN DUNCAN'S PROCEDURE ARE:
.SKIP 1
DU(L,##) = D[##,L,N-K]###MSE/M
.SKIP 1
DEFINED FOR L = 2,3,...,K AND ## LEVELS OF 1% AND 5%. FOR THE (I,J)
ORDERED PAIR OF MEANS Y(I) AND Y(J) WITH J>I, THE DUNCAN PROCEDURE IS PERFORMED AS FOLLOWS:
.SKIP 1
DECLARE THE (I,J) ORDERED PAIR OF MEANS SIGNIFICANTLY DIFFERENT
.BREAK
AT LEVEL ## IF AND ONLY IF BOTH OF THE FOLLOWING HOLD:
.BREAK
(1)##Y(J)-Y(I) > DU[L,##] = DU[J-I+1,##] AND
.BREAK
(2)##Y(J')-Y(I') > DU[J'-I'+1,##] FOR ALL J' >= J AND I' <= I.
.SKIP 1
COMMENT:
.BREAK
THE DUNCAN PROCEDURE IS ONLY VALID FOR BALANCED SAMPLES.
.SKIP 1
THE PHILOSOPHY OF THE NEWMAN-KEULS AND DUNCAN PROCEDURES IS IDENTICAL,
EXCEPT FOR THE TEST VALUES. FOR BOTH PROCEDURES THE PAIR OF MEANS
##(I) AND ##(J) IS SIGNIFICANTLY DIFFERENT AT LEVEL ## IF AND ONLY IF THE
RANGE OF EACH SUBSET OF SAMPLE MEANS (HAVING L' MEANS) CONTAINING
X(I) AND X(J) EXCEEDS NK[L',##] (FOR THE NEWMAN-KEULS) OR DU[L',##] (FOR
THE DUNCAN PROCEDURE).
.SKIP 1
THE DIFFERENCE BETWEEN THE ##-LEVEL NEWMAN-KEULS AND THE ##-LEVEL DUNCAN
PROCEDURE IS THAT THE EFFECTIVE SIGNIFICANCE LEVEL FOR ORDERED MEANS
SEPARATED BY L MEANS IS:
.SKIP 1
##L=## FOR THE NEWMAN-KEULS PROCEDURE
.BREAK
AND
.BREAK
##L=1-(1-##)### FOR THE DUNCAN PROCEDURE.
.SKIP 1
LEAST SIGNIFICANT DIFFERENCE (LSD) PROCEDURES:
.BREAK
THE TWO LSD PROCEDURES ARE THE PROTECTED LSD AND UNPROTECTED LSD.
.SKIP 1
PROTECTED LSD: THIS IS A SEQUENTIAL SIMULTANEOUS TESTING PROCEDURE, WHERE
STAGE 1 TESTING IS CARRIED OUT. IF STAGE 1 TESTING IS SIGNIFICANT, THEN
STAGE 2 TESTING IS PERFORMED. SPECIFICALLY, THE ##-LEVEL PROTECTED LSD IS ACCOMPLISHED AS FOLLOWS:
.SKIP 1
STAGE 1: IF F=MSB/MSE > F(##,K-1,N-K), THEN GO TO STAGE 2. IF
.BREAK
F= MSB/MSE < F(##,K-1,N-K), THEN DECLARE ##(I) = ##(J)
.BREAK
FOR ALL PAIRS (I,J) AND STOP FURTHER TESTING.
.BREAK
STAGE 2: FOR EACH PAIR ##(I) AND ##(J), IF X(I) - X(J) <
.BREAK
##MSE ###1/N(I) + 1/N(J) . T(##/2,N-K), THEN DECLARE
.BREAK
##(I) = ##(J). IF #X(I) - X(J)# > ##MSE###1/N(I)+1/N(J)
.BREAK
##T(##/2,N-K) THEN DECLARE ##(I)<>##(J).
.SKIP 1
COMMENT:
.BREAK
THIS IS CALLED A "PROTECTED" TEST SINCE A PRELIMINARY F TEST IS
RUN AT THE FIRST STAGE. STAGE 1 PROTECTS AGAINST "EXPERIMENTWISE" ERROR.
STAGE 2 IS A NON-SIMULTANEOUS TESTING PROCEDURE, WHERE ALL PAIRWISE TESTS
ON THE MEAN DIFFERENCES ARE "INDIVIDUAL" TESTS, EACH RUN AT LEVEL ##.
.SKIP 1
UNPROTECTED LSD: THE ##-LEVEL UNPROTECTED LSD SIMULTANEOUS
TEST IS ACCOMPLISHED BY PERFORMING THE STAGE 2 INDIVIDUAL TESTS DESCRIBED
FOR THE ##-LEVEL PROTECTED LSD TEST. HENCE, STAGE 2 TESTING IS PERFORMED REGARDLESS OF THE OUTCOME OF THE STAGE 1 F TEST.
.SKIP 1
COMMENT:
.BREAK
BOTH THE PROTECTED AND UNPROTECTED LSD TESTS MAY BE PERFORMED FOR UN-
BALANCED SAMPLES. FOR BALANCED SAMPLES THE TEST VALUES:
.BREAK
.SKIP 1
##2(MSE)/M ##.## T(##/2,N-K)
.SKIP 1
ARE OUTPUTTED FOR ## = 1%, 5%, AND 10%. LSD TEST RESULTS ARE GIVEN FOR
UNBALANCED CASES, BUT NO TEST VALUES ARE OUTPUTTED.
.SKIP 1
REMARK 7 (WHICH SIMULTANEOUS TESTING PROCEDURE TO USE?)
.BREAK
-------------------------------------------------------
.BREAK
CLEARLY THERE IS NO SATISFACTORY RESOLUTION OF THE QUESTION "WHICH SIMULTANEOUS PROCEDURE TO USE?". MANY RESEARCHERS MAKE THIS DECISION BASED
ON ERROR RATES. FOR CONTROL OF THE "EXPERIMENTWISE" ERROR RATE
THE SCHEFFE, TUKEY, AND PROTECTED LSD PROCEDURES ARE RECOMMENDED. FOR
CONTROL OF THE "INDIVIDUAL TEST" ERROR RATE, THE UNPROTECTED LSD IS EXACTLY
DESIGNED FOR THIS PURPOSE. THE NEWMAN-KEULS, DUNCAN, AND PROTECTED LSD
PROCEDURE AFFORD CONTROL, TO A LESSER EXTENT, OF THE
"INDIVIDUAL TEST" ERROR RATE.
.SKIP 1
IN MAKING A CHOICE OF A SIMULTANEOUS TESTING PROCEDURE IT IS OFTEN
MOST IMPORTANT TO BE ABLE TO COMPARE YOUR RESULTS TO PREVIOUSLY
REPORTED SIMILAR EXPERIMENTAL RESULTS. IN THESE CASES IT IS
WISE TO CONSIDER USING THE SAME SIMULTANEOUS TESTING PROCEDURE WITH THE
SAME ERROR BASIS (EXPERIMENTWISE OR INDIVIDUAL) AND THE SAME ERROR RATE AS THE PREVIOUS WORK.
.SKIP 1
CARMER AND SWANSON [8] RECENTLY REPORTED THE RESULTS OF A SIMULATION STUDY
OF TEN SIMULTANEOUS TESTING PROCEDURES (FIVE OF WHICH ARE INCLUDED IN
ADVAOV) AND ON THE BASIS OF THIS STUDY RECOMMEND TWO PROCEDURES FOR USE:
.SKIP 1
THE PROTECTED LSD AND A BAYSIAN PROCEDURES,
.BREAK
THE WALLER-DUNCAN [9] (NOT INCLUDED IN ADVAOV).
.SKIP 1
ASIDE FROM ANY OTHER CONSIDERATIONS, THE PROTECTED LSD IS RECOMMENDED
FOR USE FOR THE FOLLOWING REASONS:
.SKIP 1
(I)####ITS PHILOSOPHY IS CONSISTENT WITH CONTROLLING BOTH
.BREAK
"EXPERIMENTWISE" AND "INDIVIDUAL" ERROR RATES. THE
.BREAK
STAGE 1 F-TEST AFFORDS AN "EXPERIMENTWISE" CONTROL, BUT
.BREAK
AFTER THAT (IF THE F-TEST IS SIGNIFICANT) A "COMPARISON-
.BREAK
WISE" OR "INDIVIDUAL" TEST DECISION STANCE IS TAKEN,
.BREAK
(II)###IT HOLDS THE "EXPERIMENTWISE" ##-RATE ALMOST AS
.BREAK
WELL AS ANY COMPETITOR, SEE [8],
.BREAK
(III)##ITS STATISTICAL POWER (ABILITY TO DETECT TYPE II
.BREAK
ERRORS) IS NEARLY AS GOOD AS ANY COMPETITOR, SEE [8],
.BREAK
(IV)###IT CAN READILY BE USED IN THE UNBALANCED CASES, WHERE
.BREAK
MANY OTHER PROCEDURES CAN NOT BE USED, AND
.BREAK
(V)####IT PROBABLY SHARES THE ROBUSTNESS FEATURES OF
.BREAK
THE F AND T DISTRIBUTIONS WITH RESPECT TO NON-NORMALITY
.BREAK
AND HETEROGENEITY OF THE POPULATION VARIANCES.
.SKIP 2
.TEST PAGE 3
REMARK 8 (REFERENCES)
.BREAK
---------------------
.BREAK
FURTHER DISCUSSIONS OF SIMULTANEOUS TESTING PROCEDURES ARE FOUND IN:
.SKIP 1
.NOFILL
(1)##WINER [5], CHAPTER 3,
(2)##BANCROFT [3], CHAPTER 8,
(3)##MILLER [7], CHAPTERS 1 AND 2,
(4)##FRYER [17], CHAPTER 7.4, AND
(5)##KIRK [18], CHAPTER 3.
.FILL
.SKIP 1
EXAMPLES:
THE OPTION SIMTES IS ILLUSTRATED IN:
.SKIP 1
(1)##EXAMPLE 5.3 (NEWMAN-KEULS PROCEDURE)
.BREAK
(2)##EXAMPLE 5.4 (PROTECTED LSD PROCEDURE).
.SKIP 2
.INDEX ^^SECTION 4F\\
4F##SIMEST
.BREAK
----------
.BREAK
(A SIMULTANEOUS ESTIMATION OPTION. THE USER MAY SELECT THE SCHEFFE,
TUKEY, OR BONFERRONI PROCEDURES.)
.SKIP 1
PURPOSE:
.BREAK
THIS OPTION ALLOWS THE USER TO OBTAIN MULTIPLE COMPARISONS (SIMULTANEOUS CONFIDENCE INTERVALS) FOR ALL PAIRWISE DIFFERENCES ##(I) - ##(J) OF
THE K MEANS: ##(1),##(2),...,##(K). THE USER MAY SPECIFY ONE OF
THREE MULTIPLE COMPARISON PROCEDURES:
.SKIP 1
(1)##SCHEFFE
.BREAK
(2)##TUKEY
.BREAK
(3)##BONFERRONI AND
.SKIP 1
ONE OF THREE SIMULTANEOUS CONFIDENCE PROBABILITIES:
.SKIP 1
(1)##99%
.BREAK
(2)##95%
.BREAK
(3)##90% (EXCEPT FOR THE TUKEY PROCEDURE).
.SKIP 1
ALL THREE PROCEDURES MAY BE USED FOR BALANCED AND UNBALANCED DATA SITUATIONS.
.SKIP 1
SPJOTVOLL AND STOLINE [10] HAVE SHOWN HOW TUKEY PROCEDURES IN
THE BALANCED CASE CAN BE EXTENDED FOR USE IN UNBALANCED AOV SITUATIONS.
SUCH PROCEDURES ARE CALLED "EXTENDED TUKEY" PROCEDURES AND ARE INCLUDED
IN THIS OPTION.
.SKIP 1
DESCRIPTION:
.FILL
IT IS CONVENIENT TO DISTINGUISH BETWEEN AN "INDIVIDUAL" CONFIDENCE INTERVAL
AND A "SIMULTANEOUS" CONFIDENCE INTERVAL FOR A PARAMETER. FOR THIS
PURPOSE, LET A(I,J) BE THE EVENT THAT THE (I,J) MEAN DIFFERENCE
##(I) - ##(J) (J>I) IS "TRAPPED" IN THE INTERVAL:
.SKIP 1
DL(I,J) < ##(I) - ##(J) < DU(I,J),
.SKIP 1
WHERE DL(I,J) IS THE LOWER CONFIDENCE LIMIT AND DU(I,J) IS THE UPPER
CONFIDENCE LIMIT. A 95% "INDIVIDUAL" CONFIDENCE INTERVAL FOR "TRAPPING" ##(I) - ##(J) IS EXPRESSED:
.SKIP 1
.CENTER
PR[DL(I,J) <= ##(I) - ##(J) <= DU(I,J)] = PR[A(I,J)] >= . 95.
.SKIP 1
THE 95% "SIMULTANEOUS" CONFIDENCE INTERVALS FOR "TRAPPING" ALL PAIRWISE
MEAN DIFFERENCES ##(I) - ##(J) (J > I) OF ##(1), ##(2),..., ##(K) IS
EXPRESSED:
.SKIP 1
.CENTER
PR[DL(I,J) <= ##(I) - ##(J) <= DU(I,J):1<=I<J<=K] =
.SKIP 1
.CENTER
PR[A(1,2) AND A(1,3) AND ... AND A(K-1,K)] >= .95
.SKIP 1
THE ESSENTIAL DIFFERENCE BETWEEN THE INDIVIDUAL AND SIMULTANEOUS CONFIDENCE
INTERVAL SITUATIONS (ABOVE) IS THAT:
.SKIP 1
(1)##IN THE INDIVIDUAL CASE, .95 IS THE PROBABILITY OF THE SINGLE
.BREAK
EVENT OCCURRING (##(I)-##(J) BEING "TRAPPED"), AND
.BREAK
(2)##IN THE SIMULTANEOUS CASE, .95 IS THE PROBABILITY THAT ALL
.BREAK
EVENTS OCCUR SIMULTANEOUSLY (THE ##(I) - ##(J) ARE "TRAP-
.BREAK
PED" FOR ALL 1 <= I < J <= K).
.SKIP 1
THE INDIVIDUAL CONFIDENCE INTERVALS FOR THE DIFFERENCES ##(I) - ##(J)
ARE GIVEN IN THE OPTION TTEXC.
.SKIP 1
METHODS:
.BREAK
CONSIDER THE DEFINITIONS AND NOTATIONS GIVEN FOR THE F,T, AND STUDENTIZED
RANGE Q DISTRIBUTIONS GIVEN IN REMARK 5 OF SECTION 4E: SIMTES.
THE SPECIFIC MULTIPLE COMPARISON PROCEDURES FOR THE THREE METHODS ARE:
.SKIP 1
SCHEFFE METHOD:
.BREAK
LET A(1-##) = ###(K-1).MSE.F(##,K-1,N-K). THE 100(1-##)% SCHEFFE SIMUL-
TANEOUS CONFIDENCE INTERVALS FOR ALL PAIRWISE COMPARISON ##(I) - ##(J)
OF THE MEANS ##(1),##(2),...,###(K) ARE GIVEN BY:
.SKIP 1
.CENTER
DL(I,J) = X(I)-X(J) -####1/N(I) + 1/N(J) .A(1-##),
.CENTER
(LOWER SCHEFFE CONFIDENCE LIMIT)#####AND
.SKIP 1
.CENTER
DU(I,J) = X(I) - X(J) +####1/N(I) + 1/N(J) . A(1-##),
.CENTER
(UPPER SCHEFFE CONFIDENCE LIMIT).
.SKIP 1
FOR ALL 1<= I < J <= K. [1-## = 90%, 95%, OR 99%]
.SKIP 1
SCHEFFE'S METHOD OF MULTIPLE COMPARISON IS DESCRIBED IN SCHEFFE [4],
WINER [5], MILLER [7], AND BANCROFT [3], AND KIRK [18].
.SKIP 1
TUKEY METHOD:
.BREAK
LET C(1-##) = Q(##,K,N-K) MSE. THE 100(1-##)% (EXTENDED) TUKEY
SIMULTANEOUS CONFIDENCE INTERVALS FOR ALL PAIRWISE COMPARISONS ##(I) -
##(J) OF THE MEANS ##(1),##(2),...,##(K) ARE GIVEN BY:
.SKIP 1
.CENTER
DL(I,J) = X(I) - X(J) - C(1-##)/###MINIMUM (N(I),N(J) ,
.CENTER
(LOWER (EXTENDED) TUKEY CONFIDENCE LIMIT)#######AND
.SKIP 1
.CENTER
DU(I,J) = X(I) - X(J) + C(1-##)/###MINIMUM (N(I),N(J)
.CENTER
(UPPER (EXTENDED) TUKEY CONFIDENCE LIMIT)
.SKIP 1
FOR ALL 1 <= I < J <= K [1-## = 95% AND 99%].
.SKIP 1
FOR THE BALANCED CASE DL(I,J) AND DU(I,J) BECOME X(I) - X(J) + OR - C(1-##)/###M, WHICH IS DESCRIBED IN SCHEFFE [4], WINER [5], MILLER [7]
BANCROFT [3], AND KIRK [18]. THE UNBALANCED CASE (EXTENDED TUKEY) IS
DISCUSSED IN SPJOTVOLL AND STOLINE [10].
.SKIP 1
BONFERRONI METHOD
.BREAK
THIS METHOD IS SOMETIMES CALLED THE DUNN-BONFERRONI METHOD.
.SKIP 1
LET B(1-##) = T[##/(K(K-1)),N-K].###MSE. THE 100(1-##)% BONFERRONI SIMULTANEOUS CONFIDENCE INTERVALS FOR ALL PAIRWISE COMPARISONS ##(I) - ##(J) OF THE MEANS ##(1),##(2),...,##(K) ARE GIVEN BY:
.SKIP 1
.CENTER
DL(I,J) = X(I) - X(J) - ###1/N(I) + 1/N(J) . B(1-##),
.CENTER
(BONFERRONI LOWER CONFIDENCE LIMIT)#####AND
.SKIP 1
.CENTER
DU(I,J) = X(I) - X(J) + ###1/N(I) + 1/N(J) . B(1-##),
.CENTER
(BONFERRONI UPPER CONFIDENCE LIMIT)
.SKIP 1
FOR ALL 1 <= I < J <= K [1-## = 90%,95%, AND 99%].
.SKIP 1
BONFERRONI'S METHOD OF MULTIPLE COMPARISON IS DESCRIBED IN MILLER
[7], DUNN [11], AND KIRK [18].
.SKIP 1
COMMENT
.BREAK
THE SCHEFFE METHOD DEPENDS ON THE F DISTRIBUTION; THE TUKEY AND EXTENDED
TUKEY METHODS DEPEND ON THE STUDENTIZED RANGE Q DISTRIBUTION TABLED
IN SCHEFFE [4] AND MILLER [7]; THE BONFERRONI METHODS DEPEND ON
"SPECIAL" UPPER ##-POINTS OF THE T-DISTRIBUTION TABLED IN TABLE II
OF MILLER [7].
.SKIP 1
INPUT:##
THE USER SPECIFIES THE METHOD: SCHEFFE, TUKEY, OR BONFERRONI
AND THE SIMULTANEOUS PROBABILITY LEVEL 90% (EXCEPT FOR TUKEY), 95%, AND
99%.
.SKIP 1
OUTPUT:
.BREAK
FOR EACH PAIR (I,J) WITH 1<= I < J <= K, THE FOLLOWING OUTPUT IS GIVEN:
.SKIP 1
(I)####X(I) - X(J) (ESTIMATE OF ##(I) - ##(J)
.BREAK
(II)###DL(I,J) (LOWER CONFIDENCE LIMIT)
.BREAK
(III)##DU(I,J) (UPPER CONFIDENCE LIMIT)
.SKIP 1
(WHICH MULTIPLE COMPARISON METHOD TO USE?)
.BREAK
THE FOLLOWING RULES MAY BE USED TO ASSIST THE USER IN
DECIDING WHICH MULTIPLE COMPARISON PROCEDURE TO USE.
.SKIP 1
RULE 1
.BREAK
ONLY USE THE SCHEFFE PROCEDURE IF SIMULTANEOUS COMPARISONS
ARE WANTED FOR OTHER CONTRASTS OF ##(1), ##(2),...,#(K) IN ADDITION TO THE
PAIRWISE COMPARISONS. THE SCHEFFE SIMULTANEOUS CONFIDENCE INTERVALS
FOR GENERAL CONTRASTS MAY BE OBTAINED BY USING THE OPTION: COMPAR.
.SKIP 1
RULE 2
.BREAK
IF PRIMARY INTEREST IS IN PAIRWISE COMPARISONS AND IF THE SAMPLES ARE
BALANCED (N(1) = N(2) = ... = N(K)), THEN THE#TUKEY METHOD OF MULTIPLE
COMPARISON IS RECOMMENDED.
.SKIP 1
IN THIS CASE THE TUKEY METHOD PRODUCES SLIGHTLY NARROWER CONFIDENCE INTERVALS
THAN DOES BONFERRONI'S METHOD. BOTH THE TUKEY AND BONFERRONI METHODS
PRODUCE CONSIDERABLY NARROWER SIMULTANEOUS CONFIDENCE INTERVALS
THAN DOES THE SCHEFFE METHOD FOR PAIRWISE COMPARISONS. FOR A FURTHER
DISCUSSION OF THIS POINT SEE MILLER [7].
.SKIP 1
RULE 3
.BREAK
IF PRIMARY INTEREST IS IN PAIRWISE COMPARISONS AND IF THE SAMPLES ARE UNBALANCED, THEN THE METHOD RECOMMENDED (EXTENDED TUKEY OR BONFERRONI)
DEPENDS ON THE AMOUNT OF UNBALANCE IN THE AOV MODEL. A ROUGH RULE
OF THUMB FOR DETERMINING WHICH PROCEDURE TO USE HAS BEEN DEVELOPED BY
EMMERT[12]. URY[13] DISCUSSES THIS ISSUE ALSO.
.SKIP 1
LET U = UNBALANCE =
.SKIP 1
.TEST PAGE 3
MAXIMUM [N(1),N(2),...,N(K)]#####LARGEST SAMPLE SIZE
.BREAK
----------------------------##=##--------------------
.BREAK
MINIMUM [N(1),N(2),...,N(K)]#####SMALLEST SAMPLE SIZE
.BREAK
.SKIP 1
WHICH MEASURES THE AMOUNT OF UNBALANCE IN THE AOV MODEL. NOTE THAT IF:
.SKIP 1
U = 1; (THE MODEL IS BALANCED)
.BREAK
U > 1; (THE MODEL IS UNBALANCED)
.SKIP 1
IT IS RECOMMENDED IN [12] AND [13] THAT IF U < 1.25 (SMALL TO
MODEST UNBALANCE), THEN USE THE (EXTENDED) TUKEY METHOD. IF U > 1.25
(MODEST TO LARGE UNBALANCE), THEN USE THE BONFERRONI METHOD.
.SKIP 1
CAUTION:
.BREAK
THE ABOVE RULE IS ONLY AN APPROXIMATION AND SHOULD BE USED WITH CARE.
.SKIP 1
EXAMPLES: THE OPTION SIMEST IS ILLUSTRATED IN EXAMPLE 5.4 (EXTENDED TUKEY).
.SKIP 2
.INDEX ^^SECTION 4G\\
4G##COMPAR
.BREAK
----------
.BREAK
(THE T-VALUE AND CONFIDENCE INTERVALS ARE PRODUCED FOR A USER SPECIFIED
LINEAR EXPRESSION OR COMPARISON OF THE MEANS.)
.SKIP 1
PURPOSE:
.BREAK
FOR A USER SPECIFIED LINEAR FUNCTION OF ##(1),##(2),...,##(K), SAY
.SKIP 1
.CENTER
C(##) = C(1).##(1) + C(2).##(2) + ... + C(K).##(K) ,
.SKIP 1
THE COMPAR OPTION PROVIDES:
.SKIP 1
(1)##AN ESTIMATE C(X) OF C(##),
.BREAK
(2)##A TEST OF THE HYPOTHESIS:
.CENTER
#H0: C(##) =##0#####
.BREAK
.CENTER
H1: C(##) <>#0, AND
.BREAK
(3)##95% INDIVIDUAL AND 95% SCHEFFE SIMULTANEOUS CONFIDENCE
.BREAK
INTERVALS FOR C(##).
.SKIP 1
INPUT:
.BREAK
ENTER THE LINEAR COEFFICIENTS: C(1),C(2),...,C(K); ONE AT A TIME
(10 PER LINE).
.SKIP 1
OUTPUT AND USE:
.BREAK
ESTIMATE OF C(##)
.BREAK
THE ESTIMATE OF C(##) IS:
.SKIP 1
.CENTER
C(X) = C(1).X(1) + C(2).X(2) +...+ C(K).X(K)
.SKIP 1
THE STANDARD ERROR OF C(X) IS:
.SKIP 1
.TEST PAGE 4
###################2#########2#############2
.BREAK
#############(C(1))##+#(C(2))########(C(K))
.BREAK
SE = ###MSE.#------- -------#+...+#-------
.BREAK
##############N(1)######N(2)##########N(K)
.BREAK
.SKIP 1
WHERE MSE IS THE MEAN SQUARE WITHIN TERM OBTAINED#FROM THE AOV TABLE AND HAS N-K DEGREES OF FREEDOM.
.SKIP 1
A TEST OF THE HYPOTHESIS:
.BREAK
H0: C(##) = 0
.BREAK
H1: C(##)<> 0
.BREAK
CAN BE PERFORMED BY USING A T-VALUE:
.BREAK
T = C(X)/SE
.BREAK
AND A PROBABILITY VALUE P. BOTH T AND P ARE OUTPUTTED. P IS THE PROBA-
BILITY THAT A T-DISTRIBUTION WITH N-K DEGREES OF FREEDOM EXCEEDS THE OB-
SERVED T-VALUE T IN ABSOLUTE VALUE. HENCE, THE ##-LEVEL TESTING
PROCEDURE IS:
.BREAK
IF P > ##, THEN ACCEPT H0: C(##) = 0,
.BREAK
IF P < ##, THEN REJECT H0: C(##) = 0.
.SKIP 1
A 95% INDIVIDUAL CONFIDENCE INTERVAL FOR C(##)
.BREAK
THE 95% INDIVIDUAL CONFIDENCE INTERVAL FOR C(##) HAS THE FORM:
.BREAK
C(X) + T(.025,N-K).SE,
.BREAK
WHERE T(.025,N-K) IS THE UPPER .025 POINT OF THE T DISTRIBUTION WITH
N-K DEGREES OF FREEDOM.
.SKIP 1
A 95% SIMULTANEOUS SCHEFFE CONFIDENCE INTERVAL FOR C(##)
.BREAK
THE 95% SCHEFFE SIMULTANEOUS CONFIDENCE INTERVAL FOR C(##) IS GIVEN,
WHICH HAS ONE OF TWO FORMS:
.SKIP 1
(I)####C(X) + SE.###K.F(.05,K,N-K) ,
.BREAK
WHICH IS THE 95% SCHEFFE SIMULTANEOUS CONFIDENCE
.BREAK
INTERVAL FOR ALL LINEAR EXPRESSIONS OF ##(1),##(2),...,
.BREAK
##(K), (C(1) + C(2) + ... + C(K) <> 0), OR
.BREAK
(II)###C(X) + SE. ##(K-1) F(.05,K-1,N-K) ,
.BREAK
WHICH IS THE SCHEFFE SIMULTANEOUS CONFIDENCE INTERVAL
.BREAK
FOR ALL CONTRASTS OF ##(1),##(2),...,##(K), (C(1) + C(2)
.BREAK
+ ... + C(K) = 0).
.SKIP 1
A TEST IS MADE (INTERNALLY) OF THE CONTRAST CONDITION: C(1) + C(2) +
_... + C(K) = 0. F(##,K-1,N-K) IS THE UPPER ## POINT OF THE F DISTRIBUTION WITH K-1 AND N-K DEGREES OF FREEDOM.
.SKIP 1
REFERENCES FOR THE SCHEFFE SIMULTANEOUS CONFIDENCE INTERVALS FOR ALL LINEAR EXPRESSIONS AND ALL CONTRASTS OF ##(1),##(2),...,##(K) ARE SCHEFFE
[4] AND MILLER [7].
.SKIP 1
.INDEX ^^EXAMPLE 4.1#(COMPAR)\\
EXAMPLE 4.1
.BREAK
.BREAK
AS AN EXAMPLE OF THE USE OF THIS OPTION, CONSIDER THE CASE OF K=6 MEANS:
##(1),##(2),...,##(6) AND THE STATISTICAL ANALYSIS OF THE LINEAR FUNCTIONS:
.SKIP 1
(1)####(1) - ##(2) -- THE PAIRWISE COMPARISON OF MEANS ##(1)
.BREAK
AND ##(2).
.SKIP 1
.TEST PAGE 4
(2)####(1) + ... ##(5) #########-- A CONTRAST COMPARING THE AVERAGE
.BREAK
#####----------------- - ##(6)#####OF THE FIRST FIVE MEANS WITH THE
.BREAK
##############5####################SIXTH MEAN, WHICH COULD BE A CON-
.BREAK
###################################TROL GROUP.
.SKIP 1
.TEST PAGE 3
(3)####(1) + ... + ##(6) -- THE LINEAR EXPRESSION WHICH IS THE
.BREAK
#####------------------ #####AVERAGE OF THE SIX MEANS.
.BREAK
##############6
.SKIP 1
THE COMPAR OPTION IS USED AS FOLLOWS FOR EACH OF THE THREE LINEAR FUNCTION
EXAMPLES:
.SKIP 1
FOR (1)####(1) - ##(2)
.SKIP 1
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
.NOFILL
COMPAR
.SKIP 1
ENTER THE COEFFICIENT WEIGHTS ONE AT A TIME
SEPARATED BY COMMAS (NO MORE THAN 10 PER LINE).
1,-1,0,0,0,0,
OR
ENTER THE COEFFICIENT WEIGHTS ONE AT A TIME
SEPARATED BY COMMAS (NO MORE THAN 10 PER LINE).
1,-1
.SKIP 1
.TEST PAGE 3
FOR (2) ####(1) + ... + ##(5)
##########------------------- - ##(6)
###################5
.SKIP 1
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
COMPAR
.SKIP 1
ENTER THE COEFFICIENT WEIGHTS ONE AT A TIME
SEPARATED BY COMMAS (NO MORE THAN 10 PER LINE).
_.2,.2,.2,.2,.2,-1
.SKIP 1
.TEST PAGE 3
FOR (3)#####(1) + ... + ##(6)
##########-------------------
###################6
.SKIP 1
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
COMPAR
.SKIP 1
ENTER THE COEFFICIENT WEIGHTS ONE AT A TIME
SEPARATED BY COMMAS (NO MORE THAN 10 PER LINE).
_.167,.167,.167,.167,.167,.167
.SKIP 2
EXAMPLES: THE OPTION COMPAR IS ILLUSTRATED IN:
.SKIP 1
(1)##EXAMPLE 5.2
(2)##EXAMPLE 5.3
.SKIP 2
.INDEX ^^SECTION 4H\\
4H##COLAOV
----------
.FILL
(A COLLAPSING AOV OPTION. THE USER FORMS NEW GROUPINGS OF THE ORIGINAL
GROUPS. AN AOV TABLE IS PRODUCED.)
.SKIP 1
PURPOSE:
.BREAK
THE COLLAPSING AOV OPTION IS A VERY FLEXIBLE OPTION WHICH ALLOWS THE USER
TO OBTAIN AN AOV TABLE FOR VARIOUS USER SPECIFIED
SUBSETS AND REGROUPINGS OF THE ORIGINAL K GROUPS.
ALSO OUTPUTTED ARE THE MEANS, STANDARD DEVIATIONS, AND VARIANCES
OF THE NEWLY SPECIFIED GROUPS. IN ADDITION, T-TESTS AND ACCOMPANYING PROBABILITY VALUES FOR ALL PAIRWISE GROUPS ARE OUTPUTTED.
THESE T-TESTS USE THE POOLED MEAN SQUARE ERROR TERM OBTAINED FROM THE
COLLAPSED AOV TABLE. TEST PROCEDURES ARE CARRIED OUT EXACTLY IN THE MANNER
DESCRIBED IN THE OPTION: TTEXC AND IN SECTION 2B -- AUTOMATIC OUTPUT.
.SKIP 1
.INDEX ^^EXAMPLE 4.2#(METH 3,COMPAR)\\
EXAMPLE 4.2:##AS AN EXAMPLE, SUPPOSE THAT THERE ARE ORIGINALLY K=5 GROUPS
CALLED G1,G2,G3,G4, AND G5 AND THAT A SEPARATE AOV ANALYSIS IS WANTED FOR
EACH OF THE 3 NEWLY DEFINED GROUP SITUATIONS:
.SKIP 1
NEW GROUPS
.SKIP 1
(1)##G1,G2,G3############[AOV ON THE FIRST THREE GROUPS]
.BREAK
(2)##(G1,G2),G3,(G4,G5)##AOV ON THREE NEW GROUPS:
.BREAK
####################NEW GROUP 1 = G1,G2
.BREAK
####################NEW GROUP 2 = G3, AND
.BREAK
####################NEW GROUP 3 = G4,G5
.BREAK
(3)##(G1,G2,G3,G4),G5####AOV ON TWO NEW GROUPS:
.BREAK
####################NEW GROUP 1 = G1,G2,G3,G4
.BREAK
####################NEW GROUP 2 = G5
.SKIP 1
THE AOV TABLES FOR THE THREE SITUATIONS ARE OBTAINED BY THREE SEPARATE
USES OF THE COLAOV OPTION AS FOLLOWS:
.SKIP 1
.TEST PAGE 5
.TAB STOPS 30 45
#####FOR SITUATION 1:####NEW GROUPS#####OLD GROUPS
.NOFILL
#########################----------#####----------
1 1
2 2
3 3
.SKIP 1
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
COLAOV
.SKIP 2
COLLAPSING AOV
.SKIP 1
HOW MANY NEW GROUPS? 3
.SKIP 1
ENTER NUMBER OF GROUPS FOR EACH OF THE NEW GROUPS(10 PER LINE)
1,1,1
.SKIP 1
SPECIFY THE GROUP NUMBERS FOR THE NEW GROUP 1 (10 PER LINE)
1
.SKIP 1
SPECIFY THE GROUP NUMBERS FOR THE NEW GROUP 2 (10 PER LINE)
2
.SKIP 1
SPECIFY THE GROUP NUMBERS FOR THE NEW GROUP 3 (10 PER LINE)
3
.TEST PAGE 5
.SKIP 2
#####FOR SITUATION 2:####NEW GROUPS#####OLD GROUPS
#########################----------#####----------
1 1,2
2 3
3 4,5
.SKIP 1
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
COLAOV
.SKIP 2
COLLAPSING AOV
.SKIP 1
HOW MANY NEW GROUPS? 3
.SKIP 1
ENTER NUMBER OF GROUPS FOR EACH OF THE NEW GROUPS(10 PER LINE)
2,1,2
.SKIP 1
SPECIFY THE GROUP NUMBERS FOR THE NEW GROUP 1 (10 PER LINE)
1,2
.SKIP 1
SPECIFY THE GROUP NUMBERS FOR THE NEW GROUP 2 (10 PER LINE)
3
.SKIP 1
SPECIFY THE GROUP NUMBERS FOR THE NEW GROUP 3 (10 PER LINE)
4,5
.TEST PAGE 4
.SKIP 1
#####FOR SITUATION 3:####NEW GROUPS#####OLD GROUPS
#########################----------#####----------
1###########1,2,3,4
2 5
.SKIP 1
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
COLAOV
.SKIP 2
COLLAPSING AOV
.SKIP 1
HOW MANY NEW GROUPS? 2
.SKIP 1
ENTER NUMBER OF GROUPS FOR EACH OF THE NEW GROUPS(10 PER LINE)
4,1
.SKIP 1
SPECIFY THE GROUP NUMBERS FOR THE NEW GROUP 1 (10 PER LINE)
1,2,3,4
.SKIP 1
SPECIFY THE GROUP NUMBERS FOR THE NEW GROUP 2 (10 PER LINE)
5
.SKIP 2
.FILL
NOTE:
.BREAK
A PORTION OF THE COLLAPSED AOV ANALYSIS FOR SITUATION 3 COULD BE OBTAINED
BY USING OPTION COMPAR UPON THE LINEAR CONTRAST:
.SKIP 1
.TAB STOPS 5 15
.TEST PAGE 3
##(1) + ##(2) + ##(3) + ##(4)
.BREAK
----------------------------- - ##(5).
.BREAK
######################4
.SKIP 1
EXAMPLES:##THE OPTION COLAOV IS ILLUSTRATED IN EXAMPLE 5.3.
.SKIP 2
.INDEX ^^SECTION 4I\\
4I##TRANS
.BREAK
---------
.BREAK
(TRANSFORM THE ORIGINAL VALUES OF THE CURRENT DATA SET. THE TRANSFORMED
DATA IS NOT TRANSFORMED)
.SKIP 1
PURPOSE AND OUTPUT:
.BREAK
THE USER CHOOSES ONE OF FOUR TRANSFORMATIONS:
.SKIP 1
(1) SQUARE-ROOT
.BREAK
(2)##ARC-SIN
.BREAK
(3)##NATURAL LOGARITHM
.BREAK
(4)##RANK.
.SKIP 1
THE RAW DATA IS THEN TRANSFORMED ACCORDING TO THE TRANSFORMATION SELECTED
AND THE FOLLOWING OUTPUT ANALYSES FOR THE TRANSFORMED DATA ARE AUTOMATICALLY
OBTAINED:
.SKIP 1
(1)##THE MEANS, STANDARD DEVIATIONS, AND VARIANCES OF THE
.BREAK
TRANSFORMED DATA,
.BREAK
(2)##THE AOV TABLE (FOR TESTING THE EQUALITY OF THE TRANSFORMED
.BREAK
POPULATION MEANS), AND
.BREAK
(3)##BARTLETT'S TEST STATISTIC (USED TO TEST THE EQUALITY OF THE
.BREAK
OF THE TRANSFORMED POPULATION STANDARD DEVIATIONS).
.SKIP 1
ANY OPTION MAY BE APPLIED TO THE TRANSFORMED DATA (INCLUDING VAR, TREND,
TTEXC, TTAPP, SIMTES, SIMEST, COMPAR, AND COLAOV), EXCEPT ANOTHER
TRANSFORMATION. TRANSFORMED DATA MAY NOT BE TRANSFORMED.
IF DATA ANALYSES ARE WANTED ON ANOTHER TRANSFORMATION OF THE ORIGINAL
DATA, THEN USE THE OPTION ORIG. THIS OPTION TRANSFERS CONTROL BACK
TO THE ORIGINAL DATA ENTERED. THE USER MAY NOW SELECT ANOTHER TRANSFORMATION
TO BE APPLIED TO THE ORIGINAL DATA BY USING THE OPTION TRANS OR CONTINUE
PROCESSING THE ORIGINAL DATA BY SELECTING SOME OTHER OPTION.
.SKIP 1
THE OPTION TRANS MAY BE APPLIED ONLY TO DATA ENTERED BY DATA METHODS 1
AND 2, SINCE THE TRANSFORMATION: T(I,J) = TRAN[X(I,J)] IS APPLIED ONLY
TO RAW DATA X(I,J) (J=1,2,...,N(I); I=1,2,...,K) AND NOT TO THE SAMPLE
MEANS. X(I);I=1,2,...,K.
.SKIP 1
A TRANSFORMATION IS MOST OFTEN EMPLOYED TO CORRECT FOR THE HETEROGENEITY
OF THE POPULATION VARIANCES AND TO REDUCE THE EFFECT OF NON-NORMALITY OF
THE RAW DATA.
.SKIP 1
IN SOME EXPERIMENTAL CASES IT IS KNOWN A PRIORI THAT THE GROUP POPULATION
MEANS ##(I) ARE RELATED TO THE GROUP POPULATION STANDARD DEVIATIONS
##(I) IN A KNOWN FUNCTIONAL FORM: ##(I) = F(##(I)). IN SOME OF THESE
CASES (WHERE THE FUNCTION OF F IS KNOWN), A SPECIFIC
TRANSFORMATION IS RECOMMENDED TO REMOVE THE HETEROGENEITY EFFECT OR TO STABLIZE
THE POPULATION STANDARD DEVIATIONS.
.SKIP 2
TRANSFORMATION (DESCRIPTION, USE, AND EXAMPLES)
.BREAK
-----------------------------------------------
.BREAK
.BREAK
CASE 1 (SQUARE-ROOT TRANSFORMATION)
.BREAK
----------------------------------
.BREAK
IF THE RAW DATA IS COUNT DATA (ESPECIALLY OF RARE EVENTS), THEN THE UNDERLYING
DISTRIBUTION IS OFTEN POISSON, WHERE THE RELATIONSHIP BETWEEN ##(I) AND
##(I) IS:
.SKIP 1
##2
.BREAK
###(I) ## CONSTANT . ##(I)
.BREAK
(THE POPULATION VARIANCE IS PROPORTIONAL TO THE POPULATION MEAN.)
.SKIP 1
A SQUARE-ROOT TRANSFORMATION TENDS TO STABLIZE THE VARIANCE IN THESE
EXPERIMENTAL SITUATIONS. THE SQUARE ROOT
TRANSFORMATION APPLIED IN OPTION TRANS IS:
.SKIP 1
T(I,J) = TRAN(X(I,J)) = ###X(I,J) .
.SKIP 1
THE SQUARE-ROOT TRANSFORMATION MAY BE APPLIED ONLY TO NON-NEGATIVE DATA;
OTHERWISE AN ERROR MESSAGE IS PRODUCED.
.SKIP 1
CASE 2
(ARC-SIN TRANSFORMATION)
.BREAK
.BREAK
-------------------------------
.BREAK
IF THE RAW DATA IS PROPORTION OR PERCENTAGE DATA (BINOMIALLY DISTRIBUTED), THEN THE RELATIONSHIP BETWEEN ##(I) AND ##(I) IS:
.SKIP 1
##2
.BREAK
###(I) ## CONSTANT . ##(I)[1-##(I)].
.SKIP 1
AN ARC-SIN TRANSFORMATION TENDS TO STABILIZE THE HETEROGENEOUS VARIANCES
IN THESE SITUATIONS. THE ARC-SIN TRANSFORMATION PERFORMED IN OPTION
TRANS IS:
.SKIP 1
T(I,J) = TRAN (X(I,J)) = ARC-SIN ## X(I,J)
.SKIP 1
THE ARC-SIN TRANSFORMATION MAY BE APPLIED ONLY TO DATA X IN THE RANGE
0 <= X <= 1, (THE RANGE OF A PROPORTION); OTHERWISE AN ERROR MESSAGE
IS GIVEN.
.SKIP 1
.BREAK
CASE 3 (NATURAL LOGARITHM TRANSFORMATION)
.BREAK
-----------------------------------------
.BREAK
IN MANY EXPERIMENTAL SITUATIONS IT IS EITHER KNOWN OR OBSERVED THAT AS
THE GROUP MEAN INCREASES, THE GROUP STANDARD DEVIATION INCREASES IN A
DIRECT PROPORTION. HENCE THE RELATIONSHIP BETWEEN ##(I) AND ##(I) IS
.SKIP 1
##(I) = CONSTANT . ##(I)
.SKIP 1
IN SUCH SITUATIONS THE DATA IS NON-NORMAL, POSITIVELY SKEWED, AND HAS A
CONSTANT GROUP BY GROUP COEFFICIENT OF VARIATION, I.E., CONSTANT =
##(I)/##(I).
.SKIP 1
A NATURAL LOGARITHM TRANSFORMATION IS VERY OFTEN USED IN SUCH EXPERIMENTAL
SITUATIONS TO REMOVE THE NON-NORMALITY AND TO STABILIZE THE VARIANCES. THE NATURAL LOGARITHM TRANSFORMATION APPLIED IN TRANS
IS:
.SKIP 1
T(I,J) = TRAN (X(I,J)) = LOG##(X(I,J))
.BREAK
############################E
.SKIP 1
THE NATURAL LOGARITHM TRANSFORMATION MAY BE APPLIED ONLY TO STRICTLY
POSITIVE DATA.
.SKIP 1
THE FOLLOWING TABLE SUMMARIZES THE MAIN POINTS CONCERNING THE SQUARE-ROOT
ARC-SIN, AND NATURAL LOGARITHM TRANSFORMATIONS.
.SKIP 1
.TEST PAGE 8
.CENTER
TRANSFORMATION TABLE
.CENTER
--------------------
.TAB STOPS 13 24 36 45 58
UNDERLYING RANGE OF RELATIONSHIP PROPERTIES
.NOFILL
DESCRIPTION DISTRIBU- TRANSFORMA- OF VALUES##BETWEEN####OF UNDERLYING
TION TION OF X OF X ## AND ## DATA
----------------------------------------------------------------------
SQUARE POISSON ###X X >= 0 #####C.## 1-COUNT DATA
ROOT 2-POSITIVE
####INTEGER
3-POISSON
.TEST PAGE 6
-----------------------------------------------------------------------
ARC-SIN BINOMIAL ARC-SIN##X 0<=X<=1 ###C.#(1-#) 1-PROPORTIONS
----------------------------------------------------------------------
NATURAL POSITIVELY LOG#(X) X>=1 ####C.## 1-POSITIVELY
LOGARITHM SKEWED ###E SKEWED
2-NON-NORMAL
.SKIP 1
REMARK 1
.BREAK
DISCUSSIONS OF THESE TRANSFORMATIONS ARE FOUND IN:
.TAB STOPS 5 15
(1) WINER [5] (SECTION 5.21),
.BREAK
(2) SNEDECOR AND COCHRAN [2] (SECTION 11.14 - 11.17),
.BREAK
(3) FRYER [17] (SECTION 9.4),
.BREAK
(4) OSTLE [14] (SECTION 9.4),
.BREAK
(5) KIRK [18] (SECTION 2.7).
.SKIP 1
.TEST PAGE 2
REMARK 2
.BREAK
DATA EXAMPLES ILLUSTRATING THE TRANSFORMATIONS ARE FOUND IN:
(1) SNEDECOR AND COCHRAN [2] (TABLE 11.15.1, PAGE 326)
.BREAK
(SQUARE ROOT TRANSFORMATION)
.BREAK
(2) SNEDECOR AND COCHRAN [2] (TABLE 11.16.1, PAGE 328)
.BREAK
(ARC-SIN TRANSFORMATION)
.BREAK
(3) SNEDECOR AND COCHRAN [2] (TABLE 11.17.1, PAGE 329)
.BREAK
(NATURAL LOGARITHM TRANSFORMATION).
.FILL
.SKIP 1
THIS NATURAL LOGARITHM EXAMPLE IS INCLUDED IN THIS DOCUMENTATION AS EXAMPLE 5.5.
.SKIP 1
CASE 4
.BREAK
(RANK TRANSFORMATION)
.BREAK
---------------------
.BREAK
THE RANK TRANSFORMATION REPLACES EACH OBSERVATION X(I,J) WITH ITS RANK
Y(I,J) IN THE COMBINED SAMPLE OF N=N(1) + N(2) + ... + N(K) OBSERVATIONS.
SPECIFICALLY:
.SKIP 1
.CENTER
Y(I,J) = T(I,J) = TRAN(X(I,J)) = RANK [X(I,J)]
.SKIP 1
A RANK VALUE OF Y(I,J) INDICATES THAT OBSERVATION X(I,J) IS THE Y(I,J)TH
SMALLEST OBSERVATION IN THE COMBINED SAMPLE. RANKS FOR TIED DATA SCORES
ARE AVERAGED.
.SKIP 1
IN ADDITION TO THE DESCRIPTIVE DATA, AOV TABLE, AND BARTLETT'S TEST
CALCULATED FOR THE RANK TRANSFORMED DATA, THE KRUSKAL-WALLIS H STATISTIC
IS OUTPUTTED, WHERE:
.SKIP 1
.TEST PAGE 3
####SS GROUPS###SUM OF SQUARES GROUPS###(N-1).SUM OF SQUARE GROUPS
.BREAK
H = ---------#=#---------------------#=#--------------------------
.BREAK
####MS TOTAL####MEAN SQUARE TOTAL########SUM OF SQUARES TOTAL
.SKIP 1
.FILL
A PROBABILITY VALUE P IS ALSO OUTPUTTED FOR THE KRUSKAL-WALLIS STATISTIC. THE KRUSKAL-WALLIS STATISTIC H IS USED TO TEST THE HYPOTHESIS
H0: ##(1) = ##(2) = ... = ##(K) (EQUALITY OF THE K RANK TRANSFORMED MEANS).
H HAS AN APPROXIMATE CHI-SQUARE DISTRIBUTION WITH K-1 DEGREES OF FREEDOM
WHEN HO IS TRUE. SPECIFICALLY, IF P < ##, REJECT HO AND IF P > ##, THEN ACCEPT H0: (EQUALITY OF THE RANK TRANSFORMED MEANS) AT AN ##-
LEVEL OF SIGNIFICANCE.
.SKIP 1
REMARK#3#--#A DISCUSSION OF THE RANK TRANSFORMATION AND THE KRUSKAL-WALLIS
H STATISTIC IS FOUND IN SIEGEL [19] (PAGE 184--194).
.SKIP 1
EXAMPLES:##THE OPTION TRANS IS ILLUSTRATED IN:
.NOFILL
.SKIP 1
(1) EXAMPLE 5.5 (NATURAL LOGARITHM)
.BREAK
(2) EXAMPLE 5.6 (RANK TRANSFORMATION).
.SKIP 1
.SKIP 1
.INDEX ^^SECTION 4J\\
4J##ORIG
.BREAK
(RETURN CURRENT DATA SET TO UNTRANSFORMED STATE).
.FILL
.BREAK
------------------------------------------------
.BREAK
.BREAK
THE DATA ANALYSIS OPTIONS OF VAR, TREND, TTEXC, TTAPP, SIMTES, SIMEST,
COMPAR, AND COLAOV MAY BE APPLIED TO TRANSFORMED DATA OBTAINED THROUGH
USE OF THE OPTION TRANS.
.SKIP 1
THE OPTION ORIG TRANSFERS CONTROL BACK TO THE ORIGINAL UNTRANSFORMED DATA SO THAT EITHER:
.SKIP 1
.NOFILL
(I)####FURTHER PROCESSING OF THE ORIGINAL DATA MAY CONTINUE OR
.BREAK
(II)###ANOTHER TRANSFORMATION MAY BE APPLIED TO THE ORIGINAL DATA.
.SKIP 1
EXAMPLE:##THE OPTION ORIG IS ILLUSTRATED IN EXAMPLE 5.5.
.SKIP 2
.INDEX ^^SECTION 4K\\
4K##DATA
.BREAK
(ALLOWS THE ENTRY OF A NEW DATA SET).
-------------------------------------
.BREAK
.BREAK
WHEN ALL PROCESSING OF THE ORIGINAL DATA HAS ENDED AND AN ANALYSIS OF A
NEW, DISTINCT SET OF DATA IS WANTED, THEN USE OF THE OPTION DATA ALLOWS
THE NEW DATA TO BE ENTERED. THE ORIGINAL OR OLD DATA IS LOST WHEN DATA
IS USED.
.SKIP 1
.INDEX ^^SECTION 4L\\
4L##HELP
.BREAK
(TYPES THIS TEXT)
.BREAK
-------------------
.BREAK
.FILL
.BREAK
THE OPTION HELP LISTS THE 13 OPTION NAMES AND A SHORT DESCRIPTION FOR EACH OPTION.
.SKIP 2
.INDEX ^^SECTION 4M\\
4M##EXIT (OR FINI)
.BREAK
(PRESERVES OR PRINTS RESULTS AND RETURNS TO MONITOR)
.BREAK
-----------------------------------------------------
.BREAK
.BREAK
TO EXIT THE PROGRAM ADVAOV USE OPTION EXIT
.SKIP 3
.INDEX ^^SECTION 5.0\\
SECTION 5.0##EXAMPLES
.BREAK
---------------------
.BREAK
THE FOLLOWING SIX EXAMPLES ILLUSTRATE SOME, BUT NOT ALL, OF THE STATISTICAL
PROCEDURES THAT ARE POSSIBLE TO ACCOMPLISH USING ADVAOV.
.SKIP 1
.INDEX ^^EXAMPLE 5.1#(METH 3,VAR,TTAPP)\\
EXAMPLE 5.1
.BREAK
-----------
.BREAK
THIS EXAMPLE ILLUSTRATES:
(1)##DATA ENTRY METHOD 3
.BREAK
(2)##THE OPTION: VAR
.BREAK
(3)##THE OPTION:TTAPP
.SKIP 1
SOURCE:##THIS EXAMPLE IS TAKEN FROM SNEDECOR AND COCHRAN [2] (EXAMPLE 10.12.1, PAGE 278)
.SKIP 1
.SKIP 1
.FILL
.SPACING 1
.LEFT MARGIN 0
.RIGHT MARGIN 70
.TAB STOPS 5 15
THE DATA IN THIS EXAMPLE ARE THE NUMBER OF DAYS SURVIVED BY MICE EACH OF
WHICH HAS BEEN INNOCULATED WITH ONE OF THREE STRAINS OF TYPHOID ORGANISMS:
##9D, 11C, OR DSC 1. THE SAMPLE SIZES FOR THE THREE GROUPS ARE: 31,
60, AND 133 RESPECTIVELY. HENCE THIS IS AN EXTREMELY UNBALANCED EXPERIMENT.
.SKIP 1
THE DATA FOR THIS EXPERIMENT:
.SKIP 1
.TEST PAGE 10
.TAB STOPS 35 45 55
STRAIN:############################9D 11C DSC1
.BREAK
------- ---- ---- ----
.SKIP 1
SAMPLE SIZE 31 60 133
.SKIP 1
MEAN NUMBER OF DAYS
.BREAK
#####SURVIVED 4.03 7.37 7.80
.SKIP 1
VARIANCE OF THE NUMBER
.BREAK
#####OF DAYS SURVIVED 1.90 5.86 6.64
.SKIP 1
STANDARD DEVIATION OF THE
.BREAK
#####NUMBER OF DAYS SURVIVED 1.38 2.42 2.58
.SKIP 2
PURPOSE:
.BREAK
(1) IT IS TO BE DETERMINED IF THERE ARE SIGNIFICANT DIFFERENCES IN THE
MEAN NUMBER OF DAYS SURVIVED FOR THE THREE GROUPS OF INNOCULATED MICE.
IT WILL BE SHOWN THAT BARTLETT'S STATISTIC APPLIED TO THIS DATA IS
SIGNIFICANT.
.FILL
.SKIP 1
(2) THE OPTION VAR IS USED TO HELP DETERMINE WHICH PAIRS OF POPULATION
VARIANCES ARE SIGNIFICANTLY DIFFERENT, AS INDICATED BY BARTLETT'S TEST.
.SKIP 1
(3) THE OPTION TTAPP IS USED TO ANALYZE DIFFERENCES BETWEEN PAIRS OF
MEAN SURVIVAL DAYS. THIS APPROXIMATE PROCEDURE IS WARRANTED SINCE BARTLETT'S
TEST IS SIGNIFICANT AND THE SAMPLE SIZES ARE VERY UNBALANCED.
.SKIP 1
METHOD:
.BREAK
THE DATA IS INPUTTED INTO ADVAOV USING DATA ENTRY METHOD 3 AS FOLLOWS:
.SKIP 2
WHICH METHOD OF DATA ENTRY?(1,2,OR 3)
.NOFILL
TYPE "HELP" FOR EXPLANATION
3
.SKIP 1
HOW MANY GROUPS? 3
.SKIP 1
FORMAT: (F - TYPE ONLY)
.SKIP 1
ENTER SAMPLE SIZES(10 PER LINE)
31,60,133
.SKIP 1
ENTER THE 3 MEANS
4.03,7.37,7.80
.SKIP 1
ENTER THE 3 STANDARD DEVIATIONS
1.38,2.42,2.58
.SKIP 1
THE FOLLOWING AUTOMATIC OUTPUT IS OBTAINED:
.TEST PAGE 7
.TEST PAGE 7
.SKIP 1
.CENTER
*** DESCRIPTIVE DATA ***
.SKIP 1
.TAB STOPS 5 18 31 44 57
###GROUP######SAMPLE#SIZE######MEAN#######STD. DEV.#####VARIANCE###
----------------------------------------------------------------------
1 31 4.030 1.380 1.904
2 60 7.370 2.420 5.856
3 133 7.800 2.580 6.656
.SKIP 2
BARTLETT'S TEST STATISTIC VALUE IS 14.447
WHICH HAS A CHI-SQUARE PROBABILITY VALUE
OF 0.001 WITH 2 DEGREES OF FREEDOM.
.SKIP 2
.CENTER
*** AOV TABLE ***
.SKIP 1
.TAB STOPS 5 19 32 40 52 63
SOURCE ###SS DF ##MS ##F F-PROB
-----------------------------------------------------------------------
GROUPS 360.825 ##2 180.412 31.118 .000
WITHIN GR 1281.304 221 ##5.798
TOTAL 1642.129 223
.SKIP 2
.FILL
BARTLETT'S TEST IS OBSERVED TO BE SIGNIFICANT AT THE 1% LEVEL. TO SHED
LIGHT ON WHY BARTLETT'S TEST OF THE HOMOGENEITY OF VARIANCES WAS REJECTED
WE FURTHER ANALYZE THE DATA USING THE OPTION VAR AS FOLLOWS:
.SKIP 1
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
.BREAK
VAR
.SKIP 1
THE RATIOS OF VARIANCES ARE USED TO
DETERMINE IF THE NUMERATOR POPULATION
POPULATION VARIANCE IS SIGNIFICANTLY GREATER THAN
THE DENOMINATOR POPULATION VARIANCE. THE
RATIOS HAVE AN F DISTRIBUTION WHEN
THE POPULATION VARIANCES ARE EQUAL.
.TEST PAGE 10
.SKIP 1
.CENTER
*** VAR OPTION ***
.SKIP 1
.NOFILL
VAR A#####VAR B#####VAR A/VAR B############PROBABILITY
-----------------------------------------------------------------------
.TAB STOPS 7 17 28 39 49
1 2 0.325 0.999 WITH D.F. ( 30, 59)
1 3 0.286 1.000 WITH D.F. ( 30, 132)
2 1 3.075 0.001 WITH D.F. ( 59, 30)
2 3 0.880 0.707 WITH D.F. ( 59, 132)
3 1 3.495 0.000 WITH D.F. ( 132, 30)
3 2 1.137 0.293 WITH D.F. ( 132, 59)
.SKIP 2
.SKIP 1
.FILL
FROM OPTION VAR IT IS CONCLUDED THAT THE REASON THAT THE HYPOTHESIS
HO: ##(1) = ##(2) = ##(3) WAS REJECTED BY BARTLETT'S TEST IS THAT
BOTH THE SECOND AND THIRD VARIANCES ARE SIGNIFICANTLY LARGER THAN THE FIRST
VARIANCE AT A SIGNIFICANCE LEVEL LESS THAN 1%.
.SKIP 1
SINCE BARTLETT'S TEST IS SIGNIFICANT AT THE 1% LEVEL AND ALSO SINCE THE
SAMPLE SIZES ARE VERY UNBALANCED, THE F-VALUE OF 31.118 IN THE AOV TABLE
SHOULD BE INTERPRETED WITH CAUTION. A PROPER DATA ANALYSIS TECHNIQUE
UNDER THESE CIRCUMSTANCES IS THE STATISTICAL EXAMINATION OF ALL PAIRWISE
DIFFERENCES OF THE THREE MEANS USING APPROXIMATE TWO-SAMPLE T'S AS GIVEN IN
THE OPTION TTAPP.
.SKIP 1
THIS IS DONE AS FOLLOWS:
.SKIP 1
.NOFILL
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
TTAPP
.SKIP 1
TYPE A TWO-DIGIT NUMBER WHICH WILL BE
THE CONFIDENCE LEVEL FOR THE CONFIDENCE INTERVALS
FOR EACH OF THE DIFFERENCES BETWEEN PAIRS OF
MEANS. TYPING A RETURN AUTOMATICALLY GIVES
A 95% CONFIDENCE LIMIT.
.SKIP 1
.SKIP 1
APPROXIMATE TWO-SAMPLE T-VALUES AND 95%
INDIVIDUAL CONFIDENCE INTERVALS FOR PAIRS OF MEAN
DIFFERENCES.
.SKIP 1
THE PROBABILITY ASSOCIATED WITH EACH T-VALUE
IS CORRECT FOR A TWO-TAILED TEST. A ONE-TAILED TEST MAY
MAY BE OBTAINED BY HALVING THE PROBABILITY VALUE GIVEN.
.TEST PAGE 6
.SKIP 1
#############TWO SAMPLE#################MEAN#########95. % IND.
GROUP##GROUP##T-VALUE#####DF###PROB##DIFFERENCE###CONF. INTERVALS
---------------------------------------------------------------------
.TAB STOPS 3 10 15 26 31 40 49
1 2 -8.375 88 0.000 -3.340 (###-4.133,###-2.547)
1 3 -11.291 85 -0.000 -3.770 (###-4.434,###-3.106)
2 3 -1.119 120 0.265 -0.430 (###-1.191,####0.331)
.SKIP 2
.FILL
CLEARLY THE INTERPRETATION FROM THIS APPROXIMATE TWO-SAMPLE ANALYSIS IS
THAT THE MEAN NUMBER OF DAYS SURVIVED BY MICE INNOCULATED WITH 9D (GROUP 1)
IS SIGNIFICANTLY LOWER THAN THE MEAN NUMBER OF DAYS SURVIVED BY MICE INNOCULATED
WITH 11C AND DSC1 (GROUPS 2 AND 3). (##<= 1%)
.SKIP 2
.INDEX ^^EXAMPLE 5.2#(METH 1,TREND,COMPAR)\\
EXAMPLE 5.2
.BREAK
-----------
.BREAK
THIS EXAMPLE ILLUSTRATES:
.TAB STOPS 5 15
.SKIP 1
(1) DATA ENTRY METHOD 1
.BREAK
(2) THE TREND OPTION
.BREAK
(3) THE COMPAR OPTION
.SKIP 1
SOURCE: OSTLE [14] (EXAMPLE 11.15, PAGE 314).
.SKIP 1
THE DATA BELOW ARE YIELDS (CONVERTED TO BUSHELS/ACRE) OF A CERTAIN GRAIN
CROP IN A FERTILIZER TRIAL EXPERIMENT.
.SKIP 1
.TEST PAGE 11
.CENTER
LEVEL OF FERTILIZER
.CENTER
-------------------
.SKIP 1
#####NO###########10 LBS########20 LBS########30 LBS########40 LBS
.NOFILL
##TREATMENT######PER PLOT######PER PLOT######PER PLOT######PER PLOT
-----------------------------------------------------------------------
.TAB STOPS 6 21 35 49 63
20 25 36 35 43
25 29 37 39 40
23 31 29 31 36
27 30 40 42 48
19 27 33 44 47
.SKIP 1
PURPOSE:
.FILL
.BREAK
(1) IT IS TO BE DETERMINED IF THERE ARE ANY SIGNIFICANT DIFFERENCES
IN YIELD DUE TO DIFFERENCES IN THE FIVE FERTILIZER LEVELS.
.SKIP 1
(2) IF THERE IS A SIGNIFICANT DIFFERENCE FOUND IN (1), THEN A TREND
ANALYSIS IS WANTED ON THE EQUI-SPACED FERTILIZER LEVEL GROUPS.
.SKIP 1
(3) A T-TEST IS WANTED COMPARING THE (NO-FERTILIZER) GROUP AGAINST ALL
OTHER GROUPS (WHICH HAVE SOME FERTILIZER APPLIED).
.SKIP 1
METHOD:
.BREAK
THE DATA IS INPUTTED INTO ADVAOV AS FOLLOWS:
.SKIP 1
.NOFILL
WHICH METHOD OF DATA ENTRY?(1,2,OR 3)
TYPE "HELP" FOR EXPLANATION
1
.SKIP 1
HOW MANY GROUPS? 5
.SKIP 1
FORMAT: (F - TYPE ONLY)
.SKIP 1
ENTER SAMPLE SIZES(10 PER LINE)
5,5,5,5,5
.TEST PAGE 6
.SKIP 1
ENTER DATA FOR GROUP 1
20
25
23
27
19
.TEST PAGE 6
.SKIP 1
ENTER DATA FOR GROUP 2
25
29
31
30
27
.TEST PAGE 6
.SKIP 1
ENTER DATA FOR GROUP 3
36
37
29
40
33
.TEST PAGE 6
.SKIP 1
ENTER DATA FOR GROUP 4
35
39
31
42
44
.TEST PAGE 6
.SKIP 1
ENTER DATA FOR GROUP 5
43
40
36
48
47
.SKIP 2
THE STANDARD OUTPUT FOR THIS DATA IS:
.SKIP 2
.TEST PAGE 9
.NOFILL
.CENTER
***##DESCRIPTIVE DATA##***
.SKIP 1
#####GROUP#####SAMPLE SIZE#####MEAN#######STD. DEV.#######VARIANCE
-----------------------------------------------------------------------
.TAB STOPS 8 21 31 44 59
1 5 22.800 3.347 11.200
2 5 28.400 2.408 5.800
3 5 35.000 4.183 17.500
4 5 38.200 5.263 27.700
5 5 42.800 4.970 24.700
.SKIP 1
BARTLETT'S TEST STATISTIC VALUE IS #####2.591
WHICH HAS A CHI-SQUARE PROBABILITY VALUE
OF #####0.629 WITH ###4 DEGREES OF FREEDOM.
.TEST PAGE 7
.SKIP 2
.CENTER
***##AOV TABLE##***
.SKIP 1
.TAB STOPS 5 19 31 38 47 60
#####SOURCE##########SS########DF#######MS#######F########F-PROB
--------------------------------------------------------------------
GROUPS 1256.560 #4 314.140 18.075 .000
.NOFILL
WITHIN GR #347.600 20 #17.380
TOTAL 1604.160 24
.SKIP 2
.FILL
BARTLETT'S TEST STATISTIC IS CLEARLY NON-SIGNIFICANT AND
THE F-VALUE OF 18.075 IS SIGNIFICANT AT 1%. HENCE, THE FERTILIZER MEAN
YIELD LEVELS ARE SIGNIFICANTLY DIFFERENT.
.SKIP 1
A PLOT OF THE SAMPLE MEANS INDICATES THAT POSSIBLY THE DIFFERENCES IN YIELD LEVELS ARE DUE TO A LINEAR TREND OF YIELD AS FERTILIZER LEVEL INCREASES.
.SKIP 1
.NOFILL
.TEST PAGE 14
(BUSHELS/ACRE)
.TAB STOPS 15 20 25 30 35 40
########45 BU .
.SKIP 1
########40 BU .
.SKIP 1
########35 BU .
.SKIP 1
########30 BU .
.SKIP 1
########25 BU .
.SKIP 1
########20 BU .-------------------------------
0 10 20 30 40
.CENTER
LBS OF FERTILIZER/PLOT
.SKIP 3
.FILL
SINCE THERE ARE AN EQUAL NUMBER OF OBSERVATIONS PER GROUP AND SINCE THE
GROUPS ARE EQUALLY SPACED ALONG THE LBS OF FERTILIZER PER PLOT AXIS,
THE TREND OPTION CAN BE VALIDLY USED AS FOLLOWS FOR THIS DATA:
.SKIP 1
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
.NOFILL
TREND
.SKIP 2
THE FOLLOWING TREND ANALYSIS ASSUMES THAT THE GROUPS MEANS ARE
EQUALLY SPACED.
.SKIP 1
.TEST PAGE 12
.CENTER
AOV - TREND ANALYSIS
.TAB STOPS 5 19 31 37 49 60
.SKIP 1
#####SOURCE#########SS########DF#########MS########F########F-PROB
-----------------------------------------------------------------------
GROUPS 1256.560 #4 #314.140 18.075 .000
LINEAR 1240.020 #1 1240.020 71.348 .000
QUADRATIC ##10.414 #1 ##10.414 #0.599 .448
CUBIC ###0.080 #1 ###0.080 #0.005 .947
QUARTIC ###6.046 #1 ###6.046 #0.348 .562
WITHIN GR #347.600 20 ##17.380
TOTAL 1604.160 24
.FILL
.SKIP 1
THE TREND ANALYSIS SHOWS THAT LINEAR TREND COMPONENT, 1240.02, ACCOUNTS
FOR THE 'LION'S SHARE' OF THE SUM OF SQUARES BETWEEN FERTILIZER
TREATMENTS, 1256.56, AND THAT THE LINEAR TREND COMPONENT IS THE ONLY
SIGNIFICANT TREND COMPONENT.
.SKIP 1
HENCE FERTILIZER LEVEL HAS A SIGNIFICANT (ALMOST TOTALLY LINEAR) EFFECT
UPON YIELD.
.SKIP 1
DOES THE APPLICATION OF SOME FERTILIZER HAVE A SIGNIFICANT EFFECT COMPARED
TO NO FERTILIZER?
.SKIP 1
TO SHED LIGHT ON THIS QUESTION WE COMPARE:#####0 (MEAN YIELD OF THE PLOT WITH NO FERTILIZER) AND (###10 + ## 20 + ## 30 + ## 40)/4 (AVERAGE YIELD OF THE
PLOTS WITH SOME FERTILIZER APPLIED).
.SKIP 1
LET THE DIFFERENCE OF THESE BE:
.SKIP 1
.NOFILL
.TEST PAGE 4
##############=#####-##[#####+######+######+#####]
############D#####0#######10#####20#####30#####40
########################-------------------------
####################################4
.SKIP 1
.FILL
.FILL
TO STATISTICALLY ANALYZE THE DIFFERENCE ##D, WHICH IS A CONTRAST OF ##0,
##10, ##20, ##30, AND ##40 WE USE THE OPTION COMPAR:
.SKIP 1
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
.NOFILL
COMPAR
.SKIP 1
ENTER THE COEFFICIENT WEIGHTS ONE AT A TIME
SEPARATED BY COMMAS (NO MORE THAN 10 PER LINE).
1,-.25,-.25,-.25,-.25
.TEST PAGE 7
.SKIP 1
THE ESTIMATE FOR LINEAR CONTRAST
.SKIP 1
.TAB STOPS 5 22 38 54
#1.0000000#* 22.8000000#+ -0.25000000#* 28.4000000#+
.SKIP 1
-0.2500000#* 35.0000000#+ -0.25000000#* 38.2000000#+
.SKIP 1
-0.2500000#* 42.8000000
.SKIP 1
IS###-13.3000000
WITH T-VALUE#####-6.381
AND PROBABILITY ##0.000
95% IND. CONF. LIMITS (#####-17.650,#####-8.950)
AND 95% SCHEFF'E SIMULTANEOUS
CONFIDENCE LIMITS (#####-20.355,#####-6.245)
FOR ALL LINEAR CONTRASTS
.SKIP 1
.FILL
THE ESTIMATE FOR THIS DIFFERENCE ##D IS -13.3 WITH A T-VALUE = -6.381 WHICH
IS SIGNIFICANT AT #< 1%.
.SKIP 1
.INDEX ^^EXAMPLE 5.3#(METH 3,SIMTES,COMPAR)\\
EXAMPLE 5.3
.BREAK
-----------
.NOFILL
THIS EXAMPLE ILLUSTRATES:
(1)##DATA ENTRY METHOD 3
(2)##THE OPTION: SIMTES (NEWMAN-KEULS)
(3)##THE OPTION: COMPAR
.SKIP 2
SOURCE
BURR [15] (EXAMPLE 12.4.2, PAGE 343)
.SKIP 2
.FILL
THE DATA GIVES THE LOSS IN WEIGHT OF DISKS OF ALUMINUM BRONZE SUSPENDED
IN SULPHURIC ACID. THE FIVE EXPERIMENTAL CONDITIONS DIFFER
IN THE ADDITION OF METALLICS
.SKIP 1
.TEST PAGE 9
.NOFILL
.TAB STOPS 2 16 28 40 52 64
###GROUP #NONE SILVER SILICON SILVER SILICON
#.36% #.27% #.87% #.50%
--------------------------------------------------------------------
.SKIP 1
SAMPLE SIZE 10 10 10 10 10
.SKIP 1
MEAN 31.80 30.13 30.10 32.58 31.83
.SKIP 1
STAND. DEV. #1.087 #1.444 #2.238 #1.082 #1.281
.SKIP 1
.FILL
PURPOSE:
.BREAK
----------
.BREAK
(1)##IT IS TO BE DETERMINED IF THERE EXIST SIGNIFICANT
DIFFERENCES IN THE 5 EXPERIMENTAL CONDITIONS.
.SKIP 1
(2)##IF THERE DO EXIST DIFFERENCES IN (1), THEN THE 5% NEWMAN-KEULS
SIMULTANEOUS TESTING PROCEDURE IS TO BE RUN ON THE FIVE MEANS AS
WAS DONE IN BURR [15] FOR THIS DATA.
.SKIP 1
(3)##A T-TEST IS WANTED COMPARING THE MEAN OF THE TWO SILVER CONDITIONS
AGAINST THE MEAN OF THE TWO SILICON CONDITIONS. A T-TEST IS ALSO
WANTED COMPARING THE TWO HIGH CONCENTRATIONS AGAINST THE TWO LOW CONCENTRATIONS.
.SKIP 1
(4)THIS DATA IS ENTERED INTO THE ADVAOV USING THE DATA ENTRY METHOD 3:
.SKIP 1
.NOFILL
WHICH METHOD OF DATA ENTRY? (1,2,OR 3)
TYPE "HELP" FOR EXPLANATION
3
.SKIP 1
HOW MANY GROUPS? 5
.SKIP 1
FORMAT: (F - TYPE ONLY)
.SKIP 1
ENTER SAMPLE SIZES(10 PER LINE)
10,10,10,10,10
.SKIP 1
ENTER THE 5 MEANS
31.8,30.13,30.1,32.58,31.83
.SKIP 1
ENTER THE 5 STANDARD DEVIATIONS
1.087,1.444,2.238,1.082,1.281
.SKIP 1
.SKIP 1
SOME OF THE STANDARD OUTPUT FOR THE DATA IS:
.SKIP 1
BARTLETT'S TEST STATISTIC VALUE IS 7.031
WHICH HAS A CHI-SQUARE PROBABILITY VALUE
OF 0.134 WITH 4 DEGREES OF FREEDOM.
.TEST PAGE 9
.SKIP 1
.CENTER
***##AOV#TABLE##***
.SKIP 1
.TAB STOPS 5 19 31 38 47 60
SOURCE##########SS########DF#######MS#######F########F-PROB
.NOFILL
-----------------------------------------------------------------------
GROUP #49.775 #4 12.444 5.612 .001
.SKIP 1
WITHIN GR #99.783 45 #2.217
.SKIP 1
TOTAL 149.558 49
.SKIP 2
.FILL
BARTLETT'S TEST OF THE HOMOGENEITY OF VARIANCES IS NON-SIGNIFICANT AT
## = 10%. THE F-TEST VALUE 5.612 IS SIGNIFICANT AT ## = 1%.
.SKIP 1
IN [15], THIS DATA WAS ANALYZED USING THE NEWMAN-KEULS SIMULTANEOUS
TESTING PROCEDURE AT ## = 5%. THIS TESTING PROCEDURE IS CARRIED OUT WITH
ADVAOV USING OPTION SIMTES AS FOLLOWS:
.NOFILL
.SKIP 1
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
SIMTES
.SKIP 1
SIMULTANEOUS TESTING PROCEDURE
.SKIP 1
SELECT ONE OF THE FIVE TESTING PROCEDURES.
TYPE:
1 FOR SCHEFFE
2 FOR TUKEY
3 FOR NEWMAN-KEULS
4 FOR DUNCANS
5 FOR LEAST SIGNIFICANT DIFFERENCE
3
.TEST PAGE 13
.SKIP 1
.CENTER
NEWMAN-KEULS
.CENTER
SIMULTANEOUS TESTING PROCEDURE
.SKIP 1
.CENTER
THE ORDERED MEANS
.CENTER
-----------------
.SKIP 3
.TAB STOPS 19 30 41 52 63
ORDERED MEANS #####1 #####2 #####3 #####4 #####5
.SKIP 1
GROUP # #####3 #####2 #####1 #####5 #####4
.SKIP 1
MEAN 30.100 30.130 31.800 31.830 32.580
.SKIP 2
.TEST PAGE 9
ORDERED MEAN DIFFERENCES
------------------------
.SKIP 1
##2 ##3 ##4 ##5
.SKIP 1
1 0.030 1.700 1.730 2.480
2 1.670 1.700 2.450
3 0.030 0.780
4 0.750
.SKIP 1
A PAIR OF MEANS IS SIGNIFICANTLY DIFFERENT USING
THE NEWMAN-KEULS PROCEDURE AT THE 1% (5%) LEVEL
OF SIGNIFICANCE ONLY IF THE RANGE (OR DIFFERENCE)
OF EACH AND EVERY ORDERED PAIR OF MEANS CONTAINING
THE ORIGINAL PAIR OF MEANS AND SEPARATED BY I MEANS
IS GREATER THAN THE 1%(5%) CRITICAL TEST VALUE
FOR I MEANS.
.TEST PAGE 7
.SKIP 1
.CENTER
CRITICAL TEST VALUE
.CENTER
I############1%############5%
.CENTER
----------------------------------
.CENTER
2#########1.790#########1.341
.CENTER
3#########2.041#########1.613
.CENTER
4#########2.193#########1.776
.CENTER
5#########2.302#########1.891
.TEST PAGE 8
.SKIP 2
ORDERED MEAN TEST RESULTS
-------------------------
.SKIP 1
2 3 4 5
####1 ***
####2 ***
####3
####4
.SKIP 1
CODE * SIGNIFICANT AT 10 PERCENT
** SIGNIFICANT AT 5 PERCENT
*** SIGNIFICANT AT 1 PERCENT
(BLANK) NON-SIGNIFICANT FOR LEVEL
LESS THAN 10 PERCENT
.SKIP 2
.FILL
THE INTERPRETATION OF THE NEWMAN-KEULS 5% TEST FOR THE ORDERED MEANS:
.NOFILL
###1 ###2 ###3 ###4 ###5
SILICON SILVER #NONE SILICON SILVER
#.27% #.36% #.50% #.87%
-------------------------------------------------------------------
.SKIP 1
MEANS #30.10 #30.13 #31.80 #31.83 #32.58
.SKIP 1
.FILL
IS THAT ONLY THE PAIRS (1,5) AND (2,5) ARE SIGNIFICANTLY DIFFERENT
AT THE 5% LEVEL (ALSO AT THE 1% LEVEL). SPECIFICALLY ONLY THE PAIRS:
.SKIP 1
#####(SILICON/.27% AND SILVER/.87%) AND (SILVER/.36% AND SILVER/.87%)
.BREAK
.SKIP 1
ARE DIFFERENT AT THE 5% TESTING LEVEL USING THE NEWMAN-KEULS PROCEDURE.
.SKIP 1
A POPULAR METHOD USED TO ILLUSTRATE THE RESULTS OF A SIMULTANEOUS TEST
IS TO UNDERLINE GROUPS OF ORDERED MEANS WHICH ARE NOT SIGNIFICANTLY DIFFERENT.
.SKIP 1
FOR THE DATA OF THIS EXAMPLE WE HAVE:
.NOFILL
.SKIP 1
.TEST PAGE 4
SILICON SILVER #NONE SILICON SILVER
##.27% #.36% ##.50% #.87%
------------------------------
----------------------------------------
.FILL
.SKIP 1
PAIRS OF MEAN WHICH SHARE A LINE IN COMMON ARE NOT SIGNIFICANTLY DIFFERENT
AT THE 5% SIGNIFICANCE LEVEL USING THE NEWMAN-KEULS TESTING PROCEDURE.
.SKIP 1
FOR FURTHER INFORMATION ABOUT THE 'UNDERLINING' SEE WINER [5],
MILLER [7], AND BURR [15], AND KIRK [18].
.SKIP 1
TO COMPARE THE SILVER CONCENTRATIONS AGAINST THE SILICON CONCENTRATIONS THE CONTRAST:
.SKIP 1
.NOFILL
.TEST PAGE 3
#######################(2) + ##(4)###[##(3) + ##(5)]
################D#=#-------------- - ---------------#####=
##########################2##################2
.SKIP 1
.TEST PAGE 3
##[SILVER/.36%] + #[SILVER/.87%]###[#(SILICON/.27%) + #(SILICON/.50%)]
------------------------------- - ----------------------------------
################2#####################################2
.FILL
.SKIP 1
IS DEFINED, WHICH COMPARES THE AVERAGE OF THE TWO SILVER CONCENTRATIONS AGAINST THE AVERAGE OF THE TWO SILICON CONCENTRATIONS.
.SKIP 1
THE#DIFFERENCE#### IS STATISTICALLY ANALYZED USING THE OPTION COMPAR:
.BREAK
#################D
.NOFILL
.SKIP 1
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
COMPAR
.SKIP 1
ENTER THE COEFFICIENT WEIGHTS ONE AT A TIME
SEPARATED BY COMMAS (NO MORE THAN 10 PER LINE).
0,.5,-.5,.5,-.5
.SKIP 1
THE ESTIMATE FOR THE LINEAR CONTRAST
.SKIP 1
.TAB STOPS 5 22 38 53
#0.0000000 * 31.8000000 + 0.5000000 * 30.1300000 +
.SKIP 1
-0.5000000 * 30.1000000 + 0.5000000 * 32.5800000 +
.SKIP 1
-0.5000000 * 31.8300000
.SKIP 1
IS 0.3900001
WITH T-VALUE 0.828
AND PROBABILITY 0.412
95% IND. CONF. LIMITS ( -0.559, 1.339)
AND 95% SCHEFFE SIMULTANEOUS
CONFIDENCE LIMITS ( -1.122, 1.902)
FOR ALL LINEAR CONTRAST
.SKIP 1
.FILL
THE ESTIMATE FOR THE MEAN DIFFERENCE OF SILVER AND SILICON IS .39
WITH A T-VALUE OF 0.828 AND HAS A NON-SIGNIFICANT PROBABILITY VALUE
OF .412.
.SKIP 1
TO COMPARE THE TWO HIGH CONCENTRATIONS AGAINST THE TWO LOW CONCENTRATIONS
THE CONTRAST:
.SKIP 1
.TEST PAGE 3
######################(2) + ##(3)###[##(4) + ##(5)]
.NOFILL
################D = --------------- - ----------------###=
###########################2#################2
.SKIP 1
.TEST PAGE 3
##[SILVER/.36%] + #[SILICON/.27%]####[SILVER/.87%] + #[SILICON/.50%]
-------------------------------- - ----------------------------------
################2#####################################2
.SKIP 1
IS DEFINED.
.SKIP 1
THE OPTION COMPAR APPLIED TO THIS CONTRAST YIELDS:
.SKIP 1
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
COMPAR
.SKIP 1
ENTER THE COEFFICIENT WEIGHTS ONE AT A TIME
SEPARATED BY COMMAS (NO MORE THAN 10 PER LINE).
0,.5,.5,-.5,-.5
.SKIP 1
THE ESTIMATE FOR THE LINEAR CONTRAST
.SKIP 1
.TAB STOPS 5 22 38 53
#0.0000000 * 31.8000000 + #0.5000000 * 30.1300000 +
.SKIP 1
#0.5000000 * 30.1000000 + -0.5000000 * 32.5800000 +
.SKIP 1
-0.5000000 * 31.8300000
.SKIP 1
IS -2.0899999
WITH T-VALUE -4.438
AND PROBABILITY 0.000
95% IND. CONF. LIMITS ( -3.039, -1.141)
AND 95% SCHEFFE SIMULTANEOUS
CONFIDENCE LIMITS ( -3.602, -0.578)
FOR ALL LINEAR CONTRAST
.SKIP 1
.FILL
THE ESTIMATE FOR THE DIFFERENCE BETWEEN THE LOW AND HIGH CONCENTRATIONS IS -2.09 WITH A T-VALUE OF -4.438, WHICH IS SIGNIFICANT
AT 1%. THE CONCLUSION IS THAT THE SILVER VERSUS SILICON DIFFERENCES ARE
NON-SIGNIFICANT, WHEREAS THE HIGH VERSUS LOW CONCENTRATION DIFFERENCES ARE SIGNIFICANT.
.SKIP 1
.NOFILL
.INDEX ^^EXAMPLE 5.4#(METH 1,TTEXC,SIMTES)\\
.TEST PAGE 4
EXAMPLE 5.4
-----------
THIS EXAMPLE ILLUSTRATES:
(1) DATA ENTRY METHOD 1
(2) THE OPTION TTEXC
(3) THE OPTION SIMEST ((EXTENDED) TUKEY METHOD OF
#######MULTIPLE COMPARISON IN AN UNBALANCED AOV.)
(4) THE OPTION SIMTES (PROTECTED LSD).
.SKIP 1
SOURCE:
.FILL
THIS EXAMPLE IS TAKEN FROM BROWNLEE [16] (TABLE 10.2, PAGE 315).THE EXPERIMENT INVOLVES DETERMINATIONS OF THE GRAVITATIONAL CONSTANT
USING THREE DIFFERENT MATERIALS: GOLD, PLATINUM AND GLASS. THE DATA
COLLECTED IS:
.TEST PAGE 9
.SKIP 1
.NOFILL
.TAB STOPS 16 32 46
##############GOLD#########PLATINUM#########GLASS
##############-------------------------------------
83 61 78
81 61 71
76 67 75
78 67 72
79 64 74
72 -- --
--
.SKIP 1
.FILL
PURPOSE:
.BREAK
--------
.BREAK
(1) IT IS TO BE DETERMINED IF THERE ARE ANY SIGNIFICANT DIFFERENCES IN THE
POPULATION MEAN DETERMINATIONS OF GOLD, PLATINUM, AND GLASS.
.SKIP 1
(2)##95% INDIVIDUAL CONFIDENCE INTERVALS ARE WANTED FOR EACH
OF THE THREE DIFFERENCES:
.SKIP 1
##(GOLD) - ## (PLATINUM)
.BREAK
##(GOLD) - ##(GLASS)
.BREAK
##(PLATINUM) - ##(GLASS)
.SKIP 1
(3)##95% SIMULTANEOUS CONFIDENCE INTERVALS ARE WANTED FOR THE THREE
DIFFERENCES IN (2).
.SKIP 1
(4)##A 95% SIMULTANEOUS TEST PROCEDURE IS TO BE RUN.
.SKIP 2
METHOD:
.BREAK
-------
.BREAK
THE DATA IS ENTERED INTO ADVAOV USING THE DATA ENTRY METHOD 1:
.SKIP 1
.NOFILL
WHICH METHOD OF DATA ENTRY?(1,2,OR 3)
TYPE "HELP" FOR EXPLANATION
1
.SKIP 1
HOW MANY GROUPS? 3
.SKIP 1
FORMAT: (F - TYPE ONLY)
.SKIP 1
ENTER SAMPLE SIZES(10 PER LINE)
6,5,5
.SKIP 1
ENTER DATA FOR GROUP 1
83
81
76
78
79
72
.SKIP 1
ENTER DATA FOR GROUP 2
61
61
67
67
64
.SKIP 1
ENTER DATA FOR GROUP 3
78
71
75
72
74
.SKIP 1
THE DESCRIPTIVE DATA, BARTLETT'S TEST, AND AOV TABLE FOR THIS EXPERIMENT ARE:
.TEST PAGE 6
.SKIP 1
.CENTER
***##DESCRIPTIVE DATA##***
.TAB STOPS 7 19 29 41 53
.SKIP 1
####GROUP####SAMPLE SIZE#####MEAN#####STD. DEV.####VARIANCE
.NOFILL
---------------------------------------------------------------
1 6 78.167 3.869 14.967
2 5 64.000 3.000 #9.000
3 5 74.000 2.739 #7.500
.SKIP 2
BARTLETT'S TEST STATISTIC VALUE IS 0.540
WHICH HAS A CHI-SQUARE PROBABILITY VALUE
OF 0.763 WITH 2 DEGREES OF FREEDOM.
.TEST PAGE 7
.SKIP 1
.CENTER
***##AOV TABLE##***
.SKIP 1
.TAB STOPS 5 19 31 38 47 60
####SOURCE##########SS########DF#######MS#######F########F-PROB
-----------------------------------------------------------------------
.SKIP 1
GROUPS 565.104 #2 282.552 26.082 .000
WITHIN GR 140.833 13 #10.833
TOTAL 705.938 15
.SKIP 1
.FILL
SINCE BARTLETT'S TEST IS NON-SIGNIFICANT, THE F-VALUE 26.082 IS SEEN TO
BE SIGNIFICANT AT 1%. HENCE THERE IS A SIGNIFICANT DIFFERENCE BETWEEN
GOLD, PLATINUM AND GLASS DETERMINATIONS. THE 95% INDIVIDUAL CONFIDENCE
INTERVALS FOR THE THREE MEAN DIFFERENCES ARE OBTAINED USING THE OPTION
TTEXC.
.SKIP 1
.NOFILL
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
TTEXC
.SKIP 2
TYPE A TWO-DIGIT NUMBER WHICH WILL BE THE
CONFIDENCE LEVEL FOR THE CONFIDENCE INTERVALS
FOR EACH OF THE DIFFERENCES BETWEEN PAIRS OF
MEANS. TYPING A RETURN AUTOMATICALLY GIVES
A 95% CONFIDENCE LIMIT.
.SKIP 1
TYPE:
1 FOR INDIVIDUAL ERROR
2 FOR GROUP MEAN SQUARE ERROR
2
.SKIP 1
EXACT TWO SAMPLE T-VALUES AND 95. PER CENT
INDIVIDUAL CONFIDENCE INTERVALS FOR PAIRS OF MEAN
DIFFERENCES.
THE PROBABILITY ASSOCIATED WITH EACH T-VALUE IS
CORRECT FOR A TWO-TAILED TEST. A ONE-TAILED TEST MAY
BE OBTAINED BY HALVING THE PROBABILITY VALUES GIVEN.
.TAB STOPS 3 9 15 26 32 41 51
.TEST PAGE 6
.SKIP 1
#############TWO SAMPLE###################MEAN############95.% IND.
GROUP-GROUP T-VALUE DF PROB####DIFFERENCE ###CONF INTERVALS
-----------------------------------------------------------------------
1 2 #7.108 13 0.000 #14.167 (###9.859,###18.474)
1 3 #2.091 13 0.057 ##4.167 (##-0.141,####8,474)
2 3 -4.804 13 0.000 -10.000 (#-14.499,###-5.501)
.FILL
.SKIP 1
THE USE OF THE EXACT METHOD GIVEN IN TTEXC, OPPOSED TO THE APPROXIMATE
METHOD IN TTAPP, IS WARRANTED SINCE BARTLETT'S TEST IS NON-
SIGNIFICANT. THE POOLED MEAN SQUARE ERROR TERM IS CHOSEN IN PREFERENCE
TO THE TWO-SAMPLE OR INDIVIDUAL ERROR TERMS.
.SKIP 1
THE SIMULTANEOUS 95% CONFIDENCE INTERVALS FOR THE THREE MEAN DIFFERENCES
ARE OBTAINED USING OPTION: SIMEST. THE (EXTENDED) TUKEY METHOD OF
MULTIPLE COMPARISON IS USED IN PREFERENCE TO THE BONFERRONI METHOD SINCE:
.SKIP 1
.NOFILL
#####MAXIMUM SAMPLE SIZE###6###
#####-------------------#=#-#=#1.20 AND
#####MINIMUM SAMPLE SIZE###5
.SKIP 1
.FILL
1.20 IS LESS THAN 1.25, WHICH IS AS RECOMMENDED IN RULE 3 OF SECTION 4F.
THE 95% SIMULTANEOUS TUKEY CONFIDENCE INTERVALS ARE OBTAINED USING THE
OPTION SIMEST AS FOLLOWS:
.SKIP 1
.NOFILL
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
SIMEST
.SKIP 1
SIMULTANEOUS ESTIMATION PROCEDURE
.SKIP 1
SELECT ONE OF THE THREE ESTIMATION PROCEDURES
.SKIP 1
TYPE:
1 FOR SCHEFFE
2 FOR TUKEY
3 FOR BONFERRONI
2
.SKIP 1
SELECT ONE OF THE THREE CONFIDENCE PROBABILITIES
TYPE:
1 FOR 99%
2 FOR 95%
3 FOR 90%
2
.TEST PAGE 7
.SKIP 1
TUKEY SIMULTANEOUS ESTIMATION PROCEDURE
.SKIP 1
GROUP-GROUP#####MEAN DIFFERENCE#####95% LOWER LIMIT####95% UPPER LIMIT
----------------------------------------------------------------------
.TAB STOPS 3 9 21 41 57
1 2 #14.167 ##8.669 19.664
1 3 ##4.167 #-1.331 #9.664
2 3 -10.000 -15.498 -4.502
.SKIP 1
.FILL
THE ESTIMATES AND 95% INDIVIDUAL AND SIMULTANEOUS TUKEY CONFIDENCE FOR THE
THREE DIFFERENCES ARE:
.TEST PAGE 7
.NOFILL
.TAB STOPS 2 30 42 56
.SKIP 1
########################################################95% EXTENDED
########################################################### TUKEY
######DIFFERENCE ESTIMATE####95% IND. CI##SIMULTANEOUS CI
----------------------------------------------------------------------
.NOFILL
##(GOLD) - ##(PLATINUM) #14.167 (9.9,18.5) (8.7,19.7)
##(GOLD) - ##(GLASS) ##4.167 (-.1,8.5) (-1.3,9.7)
##(PLATINUM) - ##(GLASS) -10.000 (-14.5,-5.5) (-15.5,-4.5)
.SKIP 1
.FILL
THE LSD AND SCHEFFE PROCEDURES ARE THE ONLY SIMULTANEOUS TESTING PROCEDURES
AVAILABLE FOR USE IN ADVAOV IN THIS UNBALANCED SITUATION.
THE PROTECTED LSD PROCEDURE IS CHOSEN FOR USE IN THIS INSTANCE SINCE, AS NOTED IN REMARK 7 OF SECTION 4E,
IT IS GENERALLY PREFERABLE TO OTHER PROCEDURES.
.SKIP 1
THE 5% PROTECTED LSD PROCEDURE IS RUN USING THE OPTION SIMTES AS FOLLOWS:
.NOFILL
.SKIP 1
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
SIMTES
.SKIP 1
SIMULTANEOUS TESTING PROCEDURE
.SKIP 1
SELECT ONE OF THE FIVE TESTING PROCEDURES.
TYPE:
1 FOR SCHEFFE
2 FOR TUKEY
3 FOR NEWMAN-KEULS
4 FOR DUNCANS
5 FOR LEAST SIGNIFICANT DIFFERENCE
.BREAK
5
.TEST PAGE 11
.SKIP 1
.CENTER
LEAST SIGNIFICANT DIFFERENCE
.CENTER
SIMULTANEOUS TESTING PROCEDURE
.SKIP 1
.CENTER
THE ORDERED MEANS
.SKIP 1
.SKIP 1
ORDERED MEANS #####1 #####2 #####3
.SKIP 1
GROUP #####2 #####3 #####1
.SKIP 1
MEAN 64.000 74.000 78.167
.SKIP 2
.TEST PAGE 5
ORDERED MEAN DIFFERENCES
.SKIP 1
###2 ###3
#####1 10.000 14.167
#####2 #4.167
.SKIP 2
.TEST PAGE 5
ORDERED MEAN TEST RESULTS:
.SKIP 1
###2 ###3
#####1 ##*** ###***
#####2 ###*
.SKIP 1
CODE * SIGNIFICANT AT 10 PERCENT
** SIGNIFICANT AT #5 PERCENT
*** SIGNIFICANT AT #1 PERCENT
(BLANK) NON-SIGNIFICANT FOR LEVEL
LESS THAN 10 PERCENT
.FILL
.SKIP 1
THE 5% PROTECTED LSD IS PERFORMED IN STAGES. SINCE THE STAGE 1 F-VALUE
OF 26.082 GIVEN IN THE AOV TABLE IS
SIGNIFICANT AT 5%, THE STAGE 2 TESTING GIVEN IN THE SIMTES OPTION OUTPUT
FOR THE 5% LSD PROCEDURE IS USED.
.SKIP 1
THE ORDERED MEANS ARE:
.TAB STOPS 10 22 33
.SKIP 1
###1 ###2 ###3
.NOFILL
GROUP: PLATINUM #GLASS #GOLD
.SKIP 1
MEAN: 64.000 74.000 78.167
.SKIP 1
.FILL
FROM THE ORDERED MEAN TEST RESULTS IT IS SEEN THAT THE MEANS FOR:
#####PLATINUM AND GOLD ARE SIGNIFICANTLY DIFFERENT,
.BREAK
#####PLATINUM AND GLASS ARE SIGNIFICANTLY DIFFERENT, AND
.BREAK
#####GLASS AND GOLD ARE NON-SIGNIFICANT
.BREAK
AT A LEVEL OF 5%.
.SKIP 2
.INDEX ^^EXAMPLE 5.5#(METH 1,TRANS,ORIG)\\
EXAMPLE 5.5
.BREAK
-----------
.BREAK
THIS EXAMPLE ILLUSTRATES:
.NOFILL
.TAB STOPS 5 13
(1) DATA ENTRY METHOD 1
(2) THE OPTION TRANS (NATURAL LOGARITHM)
(3) THE OPTION ORIG
.SKIP 1
.FILL
SOURCE:
.BREAK
THE DATA FOR THIS EXAMPLE COMES FROM SNEDECOR AND COCHRAN [2] (TABLE 11.17.1,
PAGE 329).
THE DATA YIELD ESTIMATED NUMBERS OF 4 TYPES (GROUPS) OF PLANKTON CAUGHT IN
EACH OF 12 HAULS FOR EACH GROUP.
.SKIP 1
SNEDECOR AND COCHRAN ANALYZED THIS DATA UTILIZING HAUL LEVEL (1,2,...,12)
AS A FACTOR IN A TWO-WAY AOV. HAUL LEVEL IS IGNORED IN THIS DATA ANALYSIS
AND ONLY PLANKTON GROUP NUMBER IN CONSIDERED.
.SKIP 1
.TEST PAGE 17
.CENTER
ESTIMATED NUMBER OF PLANKTON
.SKIP 1
.TAB STOPS 7 20 33 47
GROUP GROUP GROUP GROUP
.NOFILL
##I #II #III #IV
---------------------------------------------
895 1520 43300 11000
540 1610 32800 8600
1020 1900 28800 8260
470 1350 34600 8900
428 980 27800 9830
620 1710 32800 7600
760 1930 28100 9650
537 1960 18900 6060
845 1840 31400 10200
1050 2410 39500 15500
387 1520 29000 9250
497 1685 22300 7900
.SKIP 1
.FILL
.NOFILL
MEANS 670.75 1701.25 30775 9395.8
.SKIP 1
STD.
DEV. 233.9 356.5 6688.7 2326.0
.SKIP 2
.FILL
PURPOSE:
.BREAK
THE PRIMARY GOAL OF THIS EXPERIMENT IS TO DETERMINE IF THERE ARE SIGNIFICANT
DIFFERENCES IN THE ESTIMATED MEAN NUMBERS OF PLANKTON BETWEEN THE FOUR GROUPS. THE STANDARD DEVIATIONS ARE NOT EQUAL. IN FACT, THE STANDARD
DEVIATIONS TEND TO INCREASE AS THE GROUP MEAN INCREASES. A LOGARITHM
TRANSFORMATION OF THE RAW DATA WILL BE UTILIZED, IF BARTLETT'S TEST ON
THE ORIGINAL TEST DATA IS SIGNIFICANT. STATISTICAL INFERENCES WILL BE
DRAWN FROM THE TRANSFORMED AOV TABLE, IN SUCH AN EVENT.
FINALLY, THE OPTION ORIG IS USED TO RETURN CONTROL OF ADVAOV TO THE ORIGINAL
UNTRANSFORMED DATA.
.SKIP 1
METHOD:
.BREAK
THE DATA IS ENTERED INTO ADVAOV USING DATA ENTRY METHOD 1:
.SKIP 1
.NOFILL
WHICH METHOD OF DATA ENTRY?(1,2,OR 3)
TYPE "HELP" FOR EXPLANATION
1
.SKIP 1
HOW MANY GROUPS? 4
.SKIP 1
FORMAT: (F - TYPE ONLY)
.SKIP 1
ENTER SAMPLE SIZES(10 PER LINE)
12,12,12,12
.SKIP 1
ENTER DATA FOR GROUP 1
895
540
1020
470
428
620
760
537
845
1050
387
497
.SKIP 1
ENTER DATA FOR GROUP 2
.SKIP 1
THE DATA FOR GROUPS 2, 3, AND 4 ARE ENTERED SIMILARLY.
.SKIP 1
BARTLETT'S TEST STATISTIC VALUE IS 101.834
WHICH HAS A CHI-SQUARE PROBABILITY VALUE
OF 0.000 WITH 3 DEGREES OF FREEDOM.
.FILL
.SKIP 1
BARTLETT'S TEST FOR EQUALITY OF THE POPULATION STANDARD DEVIATIONS IS SOUNDLY
REJECTED (AT 1%) HENCE THE ACCOMPANYING AOV TABLE IS DELETED
AND IGNORED.
.SKIP 1
THE NATURAL LOGARITHM TRANSFORMATION IS APPLIED TO THE RAW DATA AND
THE ACCOMPANYING TRANSFORMED DATA ANALYSIS IS OBTAINED:
.SKIP 1
.NOFILL
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
TRANS
.SKIP 1
ENTER TRANSFORM NUMBER
TYPE "HELP" FOR EXPLANATION
3
.SKIP 1
DATA BEING PROCESSED
.SKIP 1
.TEST PAGE 9
.CENTER
TRANSFORMATION BY LOGARITHM
.CENTER
*** DESCRIPTIVE DATA ***
.SKIP 1
.TAB STOPS 7 19 29 41 53
####GROUP####SAMPLE SIZE #MEAN#####STD. DEV.####VARIANCE
---------------------------------------------------------------------
1 12 #6.453 0.346 0.120
2 12 #7.417 0.225 0.051
3 12 10.312 0.226 0.051
4 12 #9.123 0.228 0.052
.SKIP 1
BARTLETT'S TEST STATISTIC VALUE IS 3.218
WHICH HAS A CHI-SQUARE PROBABILITY VALUE
OF 0.359 WITH 3 DEGREES OF FREEDOM.
.SKIP 1
.TEST PAGE 7
.CENTER
*** AOV TABLE ***
.SKIP 1
.TAB STOPS 5 19 31 38 47 60
SOURCE ##SS DF ##MS ###F#######F-PROB
----------------------------------------------------------------------
GROUPS 106.938 #3 35.646 521.569 .000
WITHIN GR ##3.007 44 #0.068
TOTAL 109.945 47
.FILL
.SKIP 1
NOTE THAT BARTLETT'S TEST ON THE LOG TRANSFORMED DATA IS CLEARLY NON-
SIGNIFICANT. THE ESTIMATES FOR THE TRANSFORMED STANDARD DEVIATIONS ARE
NOTED TO BE MORE HOMOGENEOUS THAN THE CORRESPONDING STANDARD DEVIATIONS
FOR THE UNTRANSFORMED DATA.
.SKIP 1
THE TRANSFORMED AOV TABLE YIELDS AN F-VALUE OF 521.6 WHICH IS
CLEARLY SIGNIFICANT (AT 1%). HENCE, IT IS CONCLUDED THERE ARE SIGNIFICANT
DIFFERENCES OF THE MEAN ESTIMATED NUMBERS OF PLANKTON BETWEEN THE FOUR GROUPS.
.SKIP 1
FINALLY, TO TRANSFER CONTROL OF ADVAOV BACK TO THE ORIGINAL DATA FOR FURTHER
PROCESSING, THE OPTION ORIG IS USED AS FOLLOWS:
.SKIP 1
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
.BREAK
ORIG
.SKIP 1
THE ORIGINAL DATA IS RESTORED.
.SKIP 1
.SKIP 1
.INDEX ^^EXAMPLE 5.6#(METH 1,TRANS)\\
.INDEX ^^EXAMPLE 5.6#(KRUSKAL-WALLIS H)\\
EXAMPLE 5.6
.BREAK
-----------
.BREAK
THIS EXAMPLE ILLUSTRATES:
.NOFILL
(1) DATA ENTRY METHOD 1
(2) THE OPTION TRANS (RANK)
(3) THE KRUSKAL-WALLIS H STATISTIC
.SKIP 1
SOURCE:
.FILL
THE DATA FOR THIS EXAMPLE IS TAKEN FROM SIEGEL [19] (TABLE 8.7, PAGE 190)
THE DATA OBSERVATIONS ARE BIRTH WEIGHTS IN TENTHS OF A POUND OF EIGHT
LITTERS OF POLAND CHINA PIGS.
.SKIP 1
.TEST PAGE 15
.NOFILL
.CENTER
LITTER
.CENTER
------
.TAB STOPS 10 17 24 31 38 45 52 59
.SKIP 1
1 2 3 4 5 6 7 8
--------------------------------------------------
20 35 33 32 26 31 26 25
28 28 36 33 26 29 22 24
33 32 26 32 29 31 22 30
32 35 31 29 20 25 25 15
44 23 32 33 20 12
36 24 33 25 21 12
19 20 29 26
33 16 34 28
28 32
11 32
.SKIP 1
.FILL
PURPOSE:
.BREAK
-------
.BREAK
THE PURPOSE OF THIS EXAMPLE IS TO DEMONSTRATE HOW THE RANK TRANSFORMATION
OF OPTION TRANS OF ADVAOV MAY BE UTILIZED TO OBTAIN STATISTICAL ANALYSES
OF RANK TRANSFORMED DATA AND TO ALSO ILLUSTRATE THE KRUSKAL-WALLIS
H STATISTIC. SPECIFICALLY, IT IS DESIRED TO TEST
THE HYPOTHESIS OF THE EQUALITY OF THE EIGHT LITTERS MEANS USING RANKED,
INSTEAD OF RAW DATA.
.SKIP 1
METHOD:
.BREAK
--------
.BREAK
THE DATA IS INPUTTED USING DATA ENTRY METHOD 1 AS FOLLOWS:
.SKIP 1
.NOFILL
WHICH METHOD OF DATA ENTRY?(1,2,OR 3)
TYPE "HELP" FOR EXPLANATION
1
.SKIP 1
HOW MANY GROUPS? 1
.SKIP 1
FORMAT: (F - TYPE ONLY)
.SKIP 1
ENTER SAMPLE SIZES(10 PER LINE)
10
.SKIP 1
ENTER DATA FOR GROUP 1
20
28
33
32
44
36
19
33
28
11
.SKIP 1
THE OTHER SEVEN GROUPS ARE ENTERED SIMILARLY.
.SKIP 1
THE DESCRIPTIVE DATA AND BARTLETT'S TEST USING THE RAW DATA:
.SKIP 1
.TEST PAGE 12
.CENTER
*** DESCRIPTIVE DATA ***
.SKIP 1
.TAB STOPS 7 19 29 41 53
####GROUP#####SAMPLE SIZE #MEAN#####STD. DEV.####VARIANCE
--------------------------------------------------------------------
1 10 28.400 9.536 90.933
2 #8 26.625 7.050 49.696
3 10 31.800 2.741 #7.511
4 #8 29.750 3.196 10.214
5 #6 23.667 3.830 14.667
6 #4 29.000 2.828 #8.000
7 #6 19.833 6.274 39.367
8 #4 23.500 6.245 39.000
.SKIP 1
BARTLETT'S TEST STATISTIC VALUE IS 18.921
WHICH HAS A CHI-SQUARE PROBABILITY VALUE
OF 0.008 WITH 7 DEGREES OF FREEDOM.
.SKIP 1
.FILL
BARTLETT'S TEST IS SIGNIFICANT AT 1% AND SINCE THE SAMPLE SIZES ARE UNBALANCED,
THE AOV TABLE FOR THE RAW DATA IS DELETED AND IGNORED.
.SKIP 1
THE ANALYSIS OF THE DATA BY RANKS IS CARRIED OUT AS FOLLOWS USING THE OPTION
TRANS:
.SKIP 1
.NOFILL
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
TRANS
.SKIP 1
ENTER TRANSFORM NUMBER
TYPE "HELP" FOR EXPLANATION
4
.SKIP 1
DATA BEING PROCESSED
.SKIP 1
.TEST PAGE 14
.CENTER
TRANSFORMATION BY RANKS
.CENTER
*** DESCRIPTIVE DATA ***
.SKIP 1
.TAB STOPS 7 19 29 41 53
####GROUP####SAMPLE SIZE #MEAN#####STD. DEV.####VARIANCE
-----------------------------------------------------------------------
1 10 31.700 20.743 430.289
2 #8 27.063 19.362 374.888
3 10 41.400 #9.454 #89.378
4 #8 34.688 11.139 124.067
5 #6 17.583 #9.599 #92.142
6 #4 30.500 #8.276 #68.500
7 #6 11.917 #8.297 #68.842
8 #4 18.000 12.363 152.833
.SKIP 1
BARTLETT'S TEST STATISTIC VALUE IS 11.843
WHICH HAS A CHI-SQUARE PROBABILITY VALUE
OF 0.106 WITH 7 DEGREES OF FREEDOM.
.SKIP 1
.TEST PAGE 7
.CENTER
*** AOV TABLE ***
.SKIP 1
.TAB STOPS 5 18 31 38 47 60
SOURCE##########SS DF ##MS ##F#######F-PROB
---------------------------------------------------------------------
GROUPS #4911.396 #7 701.628 3.494 .004
WITHIN GR #9638.604 48 200.804
TOTAL 14550.000 55
.SKIP 1
THE KRUSKAL-WALLIS H-STATISTIC IS 18.565
WHICH HAS A CHI-SQUARE PROBABILITY VALUE
OF 0.010 WITH 7 DEGREE OF FREEDOM.
.SKIP 1
.FILL
.FILL
NOTE THAT BARTLETT'S TEST ON THE RANK TRANSFORMED DATA IS NOT SIGNIFICANT
AT 10%. THE ACCOMPANYING F-VALUE IN THE AOV TABLE IS 3.494
WHICH IS SIGNIFICANT AT 1%. ALSO THE KRUSKAL-WALLIS H STATISTIC IS 18.565
WHICH IS BARELY SIGNIFICANT AT 1% AND IS CERTAINLY SIGNIFICANT AT 5%. IT IS CONCLUDED THAT THERE ARE SIGNIFICANT MEAN LITTER WEIGHT DIFFERENCES
USING THIS RANK TRANSFORM ANALYSIS.
.SKIP 2
.INDEX ^^SECTION 6.0\\
SECTION 6.0 PROGRAM DESCRIPTION AND USE
.BREAK
---------------------------------------
.BREAK
.INDEX ^^SECTION 6A\\
6A##PROGRAM GENERATED QUESTIONS _& STATEMENTS WITH EXPLANATIONS
.BREAK
--------------------------------------------------------------
.SKIP 1
6.1##OUTPUT? (FOR HELP TYPE HELP)--
.BREAK
.SKIP 1
.BREAK
6.2##INPUT? (FOR HELP TYPE HELP)--
.SKIP 1
"OUTPUT?" AND "INPUT?" DEFINE WHERE THE USER INTENDS TO WRITE HIS OUTPUT
FILE AND FROM WHERE THE USER EXPECTS TO READ HIS INPUT DATA. SEE NOTE (2)
BELOW FOR OTHER INPUT OPTIONS.
.SKIP 1
THE PROPER RESPONSE TO EACH OF THESE QUESTIONS CONSISTS OF THREE BASIC
PARTS: A DEVICE, A FILENAME AND A PROJECT-PROGRAMMER NUMBER.
.SKIP 1
THE GENERAL FORMAT FOR THESE THREE PARTS IS AS FOLLOWS:
.SKIP 1
.CENTER
DEV:FILE.EXT[PROJ,PROG]
.SKIP 1
1)##DEV:##ANY OF THE FOLLOWING DEVICES ARE APPROPRIATE WHERE INDICATED:
.NOFILL
.SKIP 1
.TEST PAGE 16
.TAB STOPS 8 25 46
####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
.SKIP 1
DEVICES MAY BE SPECIFIED BY LOGICAL OR PHYSICAL NAMES.
.BREAK
THE DEVICE LIST COLUMN HAS PHYSICAL NAMES.
.SKIP 1
INPUT MAY NOT BE DONE FROM THE LINE PRINTER NOR MAY OUTPUT GO TO THE
CARD READER.
.SKIP 1
.FILL
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.
.SKIP 1
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.
.SKIP 1
.NOFILL
EXAMPLE
-------
.SKIP 1
OUTPUT? LPT:/2
INPUT? DSK:DATA.DAT[71171,71026]
.SKIP 1
.FILL
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]
.SKIP 1
DEFAULTS:
.BREAK
---------
.FILL
.SKIP 1
1)##IF NO DEVICE IS SPECIFIED BUT A FILENAME IS SPECIFIED THE DEFAULT
.BREAK
DEVICE WILL BE DSK:
.SKIP 1
2)##IF NO FILENAME IS SPECIFIED AND A DISK OR DECTAPE IS USED THE
.BREAK
DEFAULT ON INPUT WILL BE FROM INPUT.DAT; ON OUTPUT IT WILL
.BREAK
BE OUTPUT.DAT.
.SKIP 1
3)##IF THE PROGRAM IS RUN FROM THE TERMINAL AND NO SPECIFICATION IF
.BREAK
GIVEN (JUST A CARRIAGE RETURN) BOTH INPUT AND OUTPUT DEVICE
.BREAK
WILL BE THE TERMINAL
.SKIP 1
4)##IF THE PROGRAM IS RUN THROUGH BATCH AND NO SPECIFICATION IS GIVEN,
.BREAK
(A BLANK CARD) THE INPUT WILL BE CDR: AND THE OUTPUT DEVICE
.BREAK
DEVICE WILL BE LPT:
.SKIP 1
5)##IF NO PROJECT-PROGRAMMER NUMBER IS GIVEN, THE USER'S OWN NUMBER
.BREAK
WILL BE ASSUMED.
.SKIP 1
NOTE:
.BREAK
-----
.FILL
.BREAK
1)##IF LPT: IS USED AS AN OUTPUT DEVICE MULTIPLE COPIES MAY BE
.BREAK
OBTAINED BY SPECIFYING LPT:/N WHERE N REFERS TO THE NUMBER OF
.BREAK
COPIES DESIRED.
.SKIP 1
2)##THE FOLLOWING TWO OPTIONS ARE NOT APPLICABLE FOR THE FIRST DATA
.BREAK
SET, I.E., IT IS APPLICABLE ONLY WHEN THE PROGRAM BRANCHES
.BREAK
BACK TO "INPUT?".
.BREAK
.SKIP 1
(A)##SAME OPTION
.BREAK
#####UPON RETURNING TO "INPUT?", IF THE SAME DATA FILE IS TO
.BREAK
#####BE USED AGAIN, SIMPLY ENTER "SAME", OTHERWISE, EITHER USE
.BREAK
#####THE FINISH OPTION OR ENTER ANOTHER FILE NAME ETC.
.BREAK
#####THE SAME OPTION MAY NOT BE USED WITH INPUT FROM TTY:.
.SKIP 1
(B)##FINISH OPTION
.BREAK
#####THE USER MUST ENTER "FINISH" TO BRANCH OUT OF THE
.BREAK
#####PROGRAM, FAILURE TO DO SO MIGHT RESULT IN LOSING THE
#############ENTIRE OUTPUT FILE.
.SKIP 1
6.3##ENTER ID, ELSE RETURN
.BREAK
.SKIP 1
YOU MAY ENTER A LINE WHICH WILL BE INSERTED AT THE HEAD OF YOUR
OUTPUT OR JUST TYPE <RETURN>. THE NEXT QUESTION IS 6.4.
.SKIP 1
6.4##WHICH METHOD OF DATA ENTRY?(1,2,OR 3)
.BREAK
#####TYPE "HELP" FOR EXPLANATION
.BREAK
.SKIP 1
THE 3 METHODS OF DATA ENTRY ARE EXPLAINED IN SECTION 3.0. THE NEXT QUESTION
IS 6.5.
.SKIP 1
6.5##HOW MANY GROUPS?
.SKIP 1
ENTER THE NUMBER OF GROUPS. THE NEXT QUESTION IS 6.6.
.SKIP 1
6.6##FORMAT:(F - TYPE ONLY)
.SKIP 1
THERE ARE 3 OPTIONS AVAILABLE FOR THE FORMAT, NAMELY:
.SKIP 1
(A)##STANDARD FORMAT OPTION
.BREAK
#####UNLESS OTHERWISE SPECIFIED, THE PROGRAM ASSUMES THE STANDARD
OPTION. FOR METHOD 1, THE DATA IS ENTERED 1 PER LINE. FOR METHOD 2, ONE
DATA ITEM AND ITS CORRESPONDING BREAKDOWN VALUE IS ENTERED SEPARATED
BY A COMMA, ONE EACH PER LINE. FOR METHOD 3 THE DATA (MEANS, ETC.) ARE
ENTERED 10 PER LINE.
.SKIP 1
TO USE THIS OPTION, SIMPLY ENTER <RETURN> ON TERMINAL JOBS OR USE A BLANK
CARD FOR BATCH JOBS.
.SKIP 1
(B)##OBJECT TIME FORMAT OPTION
.BREAK
#####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 3 LINES FOR THE
FORMAT, EACH LINE BEING 80 COLUMNS LONG.
.SKIP 1
NOTE THAT THE FORMAT SPECIFICATION LIST MUST USE THE FLOATING POINT (F-TYPE)
NOTATION AND MUST CONTAIN SPECIFICATION FOR EACH OF THE VARIABLES. THE
SPECIFICATIONS FOR THE FORMAT ITSELF ARE THE SAME AS FOR THE FORTRAN IV
FORMAT STATEMENT. (FOR COMPLETE DESCRIPTION, SEE DECSYSTEM-10 MATHEMATICAL
LANGUAGES HANDBOOK, SECTION I FORTRAN, CHAPTER 5, SECTION 5.1.1.).
.SKIP 1
(C)##SAME OPTION
.BREAK
#####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##"
.SKIP 1
THE NEXT QUESTION IS 6.7.
.SKIP 1
6.7##ENTER SAMPLE SIZES(10 PER LINE)
.BREAK
.SKIP 1
ENTER THE SAMPLE SIZES SEPARATED BY COMMAS (10 PER LINE). IF YOUR RESPONSE
TO QUESTION 6.2 WAS "TTY:" OR A <RETURN> AND YOUR RESPONSE TO QUESTION 6.4
WAS "1", THE NEXT QUESTION IS 6.8. IF YOUR RESPONSE TO QUESTION 6.4
WAS "3", THE NEXT QUESTION IS 6.12. IF NEITHER OF THE ABOVE IS TRUE THE STATEMENT;
.SKIP 1
DATA IS BEING READ
.SKIP 1
WILL BE TYPED AND THE NEXT QUESTION WILL BE 6.14.
.SKIP 1
6.8 ENTER DATA FOR GROUP XX
.SKIP 1
GROUP NUMBERS ("XX") WILL BE PRESENTED CONSECUTIVELY. ENTER THE DATA
IN THE FORM PRESCRIBED BY YOUR FORMAT ENTERED IN QUESTION 6.6. (SEE
SECTION 3A.) AFTER THE DATA FOR THE LAST GROUP IS ENTERED, THE NEXT QUESTION
WILL BE 6.14.
.SKIP 1
6.9##WHICH IS THE BREAKDOWN VARIABLE?(1 OR 2)
.BREAK
.SKIP 1
ENTER THE NUMBER OF THE BREAKDOWN VARIABLE (SEE SECTION 3B.) THE
NEXT QUESTION IS 6.10.
.SKIP 1
6.10##ENTER THE BREAKDOWN LIMITS(10 PER LINE)
.SKIP 1
ENTER THE BREAKDOWN LIMITS, SEPARATED BY COMMAS, 10 PER LINE. IF
YOUR RESPONSE TO QUESTION 6.4 WAS "TTY:" THE NEXT QUESTION IS 6.11. OTHERWISE;
.SKIP 1
DATA IS BEING READ
.SKIP 1
WILL BE TYPED AND THE NEXT QUESTION WILL BE 6.14.
.SKIP 1
6.11 ##ENTER DATA
.SKIP 1
ENTER YOUR DATA IN THE FORM PRESCRIBED BY YOUR FORMAT ENTERED IN QUESTION
6.6. ON THE LINE AFTER YOUR LAST DATA LINE, ENTER A "_^Z" (CONTROL-Z).
THE NEXT QUESTION WILL BE 6.14.
.SKIP 1
6.12##ENTER THE XX MEANS
.SKIP 1
ENTER THE MEANS SEPARATED BY COMMAS ACCORDING TO THE FORMAT SPECIFIED
BY YOU IN QUESTION 6.6. XX IS THE NUMBER OF GROUPS ENTERED
IN QUESTION 6.5.
(SEE SECTION 3.C.) THE NEXT QUESTION IN 6.13.
.SKIP 1
6.13##ENTER THE XX STANDARD DEVIATIONS
.SKIP 1
ENTER THE STANDARD DEVIATIONS, SEPARATED BY COMMAS ACCORDING TO THE FORMAT
SPECIFIED BY YOU IN QUESTION 6.6. THE NEXT QUESTION IS 6.14.
.SKIP 1
6.14##WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
.SKIP 1
ENTER THE NAME OF THE OPTION DESIRED. SEE SECTION 4.0 FOR AN EXPLANATION
OF THE VARIOUS OPTIONS. TYPING "HELP" YIELDS A BRIEF DESCRIPTION OF EACH OPTION. AFTER EACH OPTION IS COMPLETE THE NEXT
QUESTION WILL BE 6.14(THIS QUESTION).
.SKIP 1
ANY TIME YOU WISH TO RETURN TO THIS QUESTION FROM WITHIN AN OPTION, SIMPLY
TYPE "_^Z". IF YOU TYPE "_^Z" IN RESPONSE TO THIS QUESTION OR ANY OTHER QUESTION IN SECTION
6, IT WOULD BE THE SAME AS IF YOU TYPED "DATA" (E.G. THE NEXT QUESTION IS
6.2.).
.SKIP 2
.INDEX ^^SECTION 6B\\
6B##SAMPLE TERMINAL JOB RUN
.BREAK
---------------------------
.BREAK
.SKIP 1
.INDEX ^^EXAMPLE OF TERMINAL RUN\\
THE FOLLOWING IS THE TOTAL RUN FOR THE EXAMPLE 5.3.
.SKIP 1
.NOFILL
_.R ADVAOV
.LITERAL
WMU ADVANCED ANALYSIS OF VARIANCE
OUTPUT? (FOR HELP TYPE HELP)--TTY:
INPUT? (FOR HELP TYPE HELP)--TTY:
ENTER ID, ELSE RETURN
BURR[15] (EXAMPLE 12.4.2,PAGE 343)
WHICH METHOD OF DATA ENTRY?(1,2,OR 3)
TYPE "HELP" FOR EXPLANATION
3
HOW MANY GROUPS? 5
FORMAT: (F-TYPE ONLY)
STD
ENTER SAMPLE SIZES(10 PER LINE)
10,10,10,10,10
ENTER THE 5 MEANS
31.8,30.13,30.1,32.58,31.83
ENTER THE 5 STANDARD DEVIATIONS
1.087,1.444,2.238,1.082,1.281
WMU ADVANCED ANALYSIS OF VARIANCE PROGRAM
08:54 2-Apr-75
BURR[15] (EXAMPLE 12.4.2,PAGE 343)
.END LITERAL
.TEST PAGE 13
.LITERAL
*** DESCRIPTIVE DATA ***
GROUP SAMPLE SIZE MEAN ##STD. DEV. VARIANCE
---------------------------------------------------------------------
1 10 31.800 1.087 1.182
2 10 30.130 1.444 2.085
3 10 30.100 2.238 5.009
4 10 32.580 1.082 1.171
5 10 31.830 1.281 1.641
BARTLETT'S TEST STATISTIC VALUE IS 7.031
WHICH HAS A CHI-SQUARE PROBABILITY VALUE
OF 0.134 WITH 4 DEGREES OF FREEDOM.
.END LITERAL
.TEST PAGE 9
.LITERAL
*** AOV TABLE ***
SOURCE SS DF MS F F-PROB
--------------------------------------------------------------------
GROUPS 49.775 4 12.444 5.612 .001
WITHIN GR 99.783 45 2.217
TOTAL 149.558 49
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
SIMTES
SIMULTANEOUS TESTING PROCEDURE
SELECT ONE OF THE FIVE TESTING PROCEDURES.
TYPE:
1 FOR SCHEFFE
2 FOR TUKEY
3 FOR NEWMAN-KEULS
4 FOR DUNCANS
5 FOR LEAST SIGNIFICANT DIFFERENCE
3
.END LITERAL
.TEST PAGE 13
.LITERAL
NEWMAN-KEULS
SIMULTANEOUS TESTING PROCEDURE
THE ORDERED MEANS
ORDERED MEANS 1 2 3 4 5
GROUP # 3 2 1 5 4
MEAN 30.100 30.130 31.800 31.830 32.580
.END LITERAL
.TEST PAGE 11
.LITERAL
ORDERED MEAN DIFFERENCES
2 3 4 5
1 0.030 1.700 1.730 2.480
2 1.670 1.700 2.450
3 0.030 0.780
4 0.750
A PAIR OF MEANS IS SIGNIFICANTLY DIFFERENT USING
THE NEWMAN-KEULS PROCEDURE AT THE 1% (5%) LEVEL
OF SIGNIFICANCE ONLY IF THE RANGE (OR DIFFERENCE)
OF EACH AND EVERY ORDERED PAIR OF MEANS CONTAINS
THE ORIGINAL PAIR OF MEANS AND SEPARATED BY I MEANS
IS GREATER THAN THE 1%(5%) CRITICAL TEST VALUE
FOR I MEANS.
.END LITERAL
.TEST PAGE 7
.LITERAL
CRITICAL TEST VALUE
I 1% 5%
-----------------------------
2 1.790 1.341
3 2.041 1.613
4 2.193 1.776
5 2.302 1.891
.END LITERAL
.TEST PAGE 11
.LITERAL
ORDERED MEANS TEST RESULTS
2 3 4 5
1 ***
2 ***
3
4
CODE * SIGNIFICANT AT 10 PERCENT
** SIGNIFICANT AT 5 PERCENT
*** SIGNIFICANT AT 1 PER CENT
(BLANK) NON-SIGNIFICANT FOR
LEVEL LESS THAN 10 PER CENT
TUKEY SIGNIFICANCE TEST - .1 - IS NOT IMPLEMENTED
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
COMPAR
ENTER THE COEFFICIENT WEIGHTS ONE AT A TIME
SEPARATED BY COMMAS (NO MORE THAN 10 PER LINE).
0,.5,-.5,.5,-.5
THE ESTIMATE FOR THE LINEAR CONTRAST
.END LITERAL
.TEST PAGE 5
.LITERAL
0.0000000 * 31.8000000 + 0.5000000 * 30.1300000 +
-0.5000000 * 30.1000000 + 0.5000000 * 32.5800000 +
-0.5000000 * 31.8300000
IS 0.3900001
WITH T-VALUE 0.828
AND PROBABILITY 0.412
95% IND. CONF. LIMITS ( -0.559, 1.339)
AND 95 % SCHEFFE SIMULTANEOUS
CONFIDENCE LIMITS ( -1.122, 1.902)
FOR ALL LINEAR CONTRAST
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
COMPAR
ENTER THE COEFFICIENT WEIGHTS ONE AT A TIME
SEPARATED BY COMMAS (NO MORE THAN 10 PER LINE).
0,.5,.5,-.5,-.5
THE ESTIMATE FOR THE LINEAR CONTRAST
.END LITERAL
.TEST PAGE 5
.LITERAL
0.0000000 * 31.8000000 + 0.5000000 * 30.1300000 +
0.5000000 * 30.1000000 + -0.5000000 * 32.5800000 +
-0.5000000 * 31.8300000
IS -2.0899999
WITH T-VALUE -4.438
AND PROBABILITY 0.000
95% IND. CONF. LIMITS ( -3.039, -1.141)
AND 95 % SCHEFFE SIMULTANEOUS
CONFIDENCE LIMITS ( -3.602, -0.578)
FOR ALL LINEAR CONTRAST
WHICH OPTION?(TYPE "HELP" FOR EXPLANATION)
EXIT
END OF EXECUTION
CPU TIME: 2.81 ELAPSED TIME: 16.88
EXIT
.END LITERAL
.SKIP 2
.INDEX ^^SECTION 6C\\
6C##SAMPLE BATCH SETUP
.BREAK
------------------------------
.BREAK
.SKIP 1
.FILL
IN THE FOLLOWING BATCH JOB SETUP, EACH LINE REPRESENTS ONE CARD, EACH CARD STARTING
IN COLUMN 1. DO NOT INCLUDE THE COMMENTS AT THE RIGHT. SEE COMPUTER CENTER USERS GUIDE _#7 OR THE DECSYSTEM-10 USERS HANDBOOK FOR OTHER BATCH SYSTEM COMMANDS.
.SKIP 1
.NOFILL
-----------------------------------------------------------------
.TAB STOPS 30
$JOB [_#_#_#,_#_#_#] ;_#_#_#,_#_#_# REPRESENTS THE USER'S
;PROJECT-PROGRAMMER NUMBER
$PASSWORD _#_#_#_# ;_#_#_#_# REPRESENTS THE USERS PASSWORD
_.R ADVAOV ;START ADVAOV
######[COMMANDS TO ADVAOV]
(EOF) ;"END OF FILE" CARD AVAILABLE FROM
;COMPUTER CENTER
.BREAK
-----------------------------------------------------------------
.FILL
.SKIP 2
.INDEX ^^SECTION 7.0\\
SECTION 7.0##LIMITATIONS
.NOFILL
.BREAK
----------------
.BREAK
.TAB STOPS 8 12
.SKIP 1
1)##THE MAXIMUM NUMBER OF GROUPS (K) IS 20.
.SKIP 1
2)##IF RANKS TRANSFORMATION IS USED THE FORMULA 35840>=22000+N*I
.BREAK
MUST ALSO BE SATISFIED.*
.FOOTNOTE 3
.SKIP 1
--------------------
.BREAK
* 35840 WORDS IS 35K OF CORE, THE NORMAL AMOUNT AVAILABLE TO THE USER.
!
WHERE I=2, IF DATA ENTRY METHOD 1 IS USED OR I=3 IF DATA ENTRY
.BREAK
METHOD 2 IS USED (DATA ENTRY METHOD 3 MAY NOT BE USED WITH ANY
.BREAK
TRANSFORMATION.) N = THE SUM OF THE NUMBER OF OBSERVATIONS
.BREAK
IN ALL GROUPS.
.SKIP 1
3)##THERE IS NO LIMIT ON THE NUMBER OF DATA ITEMS PER GROUP EXCEPT
.BREAK
WHEN RANKS TRANSFORMATION IS USED AS SHOWN IN (2).
.SKIP 1
4)##THE MAXIMUM NUMBER OF FORMAT LINES IS 2.
.PAGE
.INDEX ^^SECTION 8.0\\
SECTION 8.0##REFERENCES
.BREAK
-----------------------
.BREAK
.FILL
[1]##LEONE, F. AND JOHNSON, N. (1964):##"STATISTICS AND EXPERIMENTAL
DESIGN IN ENGINEERING AND THE PHYSICAL SCIENCES", VOL. 1, JOHN WILEY
AND SONS, INC., N.Y.
.SKIP 1
[2]##SNEDECOR, G. AND COCHRAN, W. (1968):##"STATISTICAL METHODS", SIXTH
EDITION, THE IOWA STATE UNIVERSITY PRESS, AMES,IOWA.
.SKIP 1
[3]##BANCROFT, T.A. (1968): ##"TOPICS IN INTERMEDIATE STATISTICAL METHODS",
VOL. 1, THE IOWA STATE UNIVERSITY PRESS, AMES, IOWA.
.SKIP 1
[4]##SCHEFFE, H. (1959): ##"THE ANALYSIS OF VARIANCE", JOHN WILEY AND
SONS, INC., N.Y.
.SKIP 1
[5]##WINER, B.J. (1971): #"STATISTICAL PRINCIPLES IN EXPERIMENTAL DESIGN",
SECOND EDITION, MCGRAW-HILL BOOK COMPANY.
.SKIP 1
[6]##SATTERTHWAITE, F.E. (1946): AN APPROXIMATE DISTRIBUTION OF ESTIMATES
OF VARIANCE COMPONENTS. BIOMETRICS BULLETIN, 2, 110-114.
.SKIP 1
[7]##MILLER, R.G. JR. (1966): "SIMULTANEOUS STATISTICAL INFERENCE",
MCGRAW-HILL BOOK COMPANY.
.SKIP 1
[8]##CARMER, S.G. AND SWANSON, M.R. (1973): AN EVALUATION OF TEN PAIRWISE MULTIPLE COMPARISON PROCEDURES BY MONTO CARLO METHODS, J. AM. STATIST.
ASSOC., (MARCH 1973), V. 68, NU. 341, PAGES 66-74.
.SKIP 1
[9]##WALLER, R.A. AND DUNCAN, D.B., (1969):##A BAYES RULE FOR THE
SYMMETRIC MULTIPLE COMPARISONS PROBLEM, J. AM. STATIST. ASSOC. (DECEMBER
1969), V.64, NU. 328, PAGES 1484-503.
.SKIP 1
[10]##SPJOTVOLL, E. AND STOLINE, M.R. (1973): AN EXTENSION OF THE T-METHOD
OF MULTIPLE COMPARISON TO INCLUDE CASES WITH UNEQUAL SAMPLE SIZES, J. AM. STATIST.
ASSOC. (DECEMBER 1973), V. 68, NU. 344, PAGES 975-978.
.SKIP 1
[11]##DUNN, O.J. (1961) MULTIPLE COMPARISONS AMONG MEANS. JOURNAL OF THE
AMER. STATIST. ASSOC. 56, PAGES 52-64.
.SKIP 1
[12]##EMMERT, W. (1974): THE EXTENDED TUKEY PROCEDURE COMPARED
TO THE SCHEFFE AND BONFERRONI PROCEDURES, SPECIALIST THESIS, WESTERN MICHIGAN
UNIVERSITY, KALAMAZOO, MI.
.SKIP 1
[13]##URY,H.K. (1976). A COMPARISON OF FOUR PROCEDURES FOR MUITIPLE
COMPARISONS AMONG MEANS(PAIRWISE CONTRASTS) FOR ARBITARY SAMPLE
SIZES, TECHNOMETRICS 18,89-97.
.SKIP 1
[14]##OSTLE, B. (1963): "STATISTICS IN RESEARCH", SECOND EDITION, THE
IOWA STATE COLLEGE PRESS, AMES, IOWA.
.SKIP 1
[15]##BURR, I.W. (1974): "APPLIED STATISTICAL METHODS", ACADEMIC PRESS,
NEW YORK AND LONDON.
.SKIP 1
[16]##BROWNLEE, K.A. (1965): "STATISTICAL THEORY AND METHODOLOGY IN SCIENCE
AND ENGINEERING", SECOND EDITION, JOHN WILEY AND SONS, INC., N.Y.
.SKIP 1
[17]##FRYER, H.C. (1966): "CONCEPTS AND METHODS OF EXPERIMENTAL STATISTICS",
ALLYN AND BACON, INC., BOSTON.
.SKIP 1
[18]##KIRK, R.E., (1966): "EXPERIMENTAL DESIGN: PROCEDURES FOR THE BEHAVIORAL
SCIENCES," WADSWORTH PUBLISHING COMPANY, INC., BELMONT, CA.
.SKIP 1
[19]##SIEGEL, S. (1956): "NON-PARAMETRIC STATISTICS", MCGRAW-HILL BOOK
COMPANY.
.PAGE
.INDEX ^^SECTION 9.0\\
SECTION 9.0##EXAMPLE AND SECTION INDEX
.BREAK
--------------------------------------
.BREAK
.SKIP 1
.PRINT INDEX