.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(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##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) #########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 #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) 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= .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 . 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 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 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