CALL SEQUENCE... t2test U1BAR VU1 V1BAR VV1 COV1 U2BAR VU2 V2BAR VV2 COV2 \ DU DV AMP FVALUE KVALUE DVALUE MASK INPUT OUTPUT \ [ -a VAL1 -lx VAL2 -ly VAL3 -kind VAL4 ] DESCRIPTION... t2test - CONDUCT A TWO SAMPLE HOTELLING'S T**2 STATISTICAL TEST. AUTHORS - F. Zwiers and F. Majaess, November 06 1987. LAST REVISION: $Header: Code-beta-03-de-r.diag_6.3.1-et-cdf2rpn_2.2.1 @ Thu, 19 Jan 2017 16:44:52 -0500 (Bernard Dugas) $ PURPOSE - CONDUCT A TWO SAMPLE HOTELLING'S T**2 STATISTICAL TEST. UPON RETURN, THE PROGRAM REPORTS THE COMPUTED STATISTICAL VALUES IN FVALUE AS WELL AS RETURNING THE K-VALUE AND D-VALUE WHICH ARE RESPECTIVELY THE TRANSFORMED SIGNIFICANCE LEVELS AND THE ACCEPTANCE/REJECTION FLAG. ALSO RETURNED ARE THE DIFFERENCES EXPERIMENT-CONTROL OF THE TWO COMPONENTS IN DU AND DV AS WELL AS THE AMPLITUDE OF THE VECTOR DIFFERENCE IN AMP. NOTE - 1) TESTS ARE NOT PERFORMED WHEN ONE OR MORE OF THE "POOLED VARIANCES" AND/OR THE CORRELATION COEFFICIENT IS APPROXIMATELY EQUAL TO THE NOISE INDUCED BY THE PACKING ALGORITHM. 2) INTERPRETATION OF VU1, VV1, VU2 AND VV2 FILES IS CONTROLED BY INPUT PARAMETER IVAR. 3) MASK FILE NEED NOT BE SPECIFIED IN THE PROGRAM CALLING SEQUENCE. (SEE OUTPUT FILES SECTION BELOW). INPUT FILES... CONTROL DATASET STATISTICS FOR W1(U1,V1): ========================================= U1BAR = FIELDS OF MEANS OF THE 'U1-CONTROL' VARIABLE. SUM( U1(I) ) / N1, I=1,...,N1. (IF (IVAR.EQ.0) THEN (VU1 = THE VARIANCE OF U1. SUM( (U1(I)-U1BAR)**2 ) / N1, I=1,...,N1. (OTHERWISE, (VU1 = THE UNBIASED ESTIMATES OF THE STANDARD DEVIATIONS OF U1. ( SQRT( SUM( (U1(I)-U1BAR)**2 ) / (N1-1) ), I=1,...,N1. V1BAR = SAME AS U1BAR EXCEPT FOR THE 'V1-CONTROL' VARIABLE. VV1 = SAME AS VU1 EXCEPT FOR THE 'V1-CONTROL' VARIABLE. COV1 = COVARIANCE OF THE CONTROL COMPONENTS. SUM( (U1(I)-U1BAR) * (V1(I)-V1BAR) ) / N1, I=1,...,N1. EXPERIMENT DATASET STATISTICS FOR W2(U2,V2): ============================================ U2BAR = FIELDS OF MEANS OF THE 'U2-EXPERIMENT' VARIABLE. SUM( U2(J) ) / N2, J=1,...,N2. (IF (IVAR.EQ.0) THEN (VU2 = THE VARIANCE OF U2. SUM( (U2(J)-U2BAR)**2 ) / N2, J=1,...,N2. (OTHERWISE, (VU2 = THE UNBIASED ESTIMATES OF THE STANDARD DEVIATIONS OF U2. ( SQRT( SUM( (U2(J)-U2BAR)**2 ) / (N2-1) ), J=1,...,N2. V2BAR = SAME AS U2BAR EXCEPT FOR THE 'V2-EXPERIMENT' VARIABLE. VV2 = SAME AS VU2 EXCEPT FOR THE 'V2-EXPERIMENT' VARIABLE. COV2 = COVARIANCE OF THE EXPERIMENT COMPONENTS. SUM( (U2(J)-U2BAR) * (V2(J)-V2BAR) ) / N2, J=1,...,N2. OUTPUT FILES... DU = FIELDS OF MEAN DIFFERENCES FOR THE FIRST COMPONENTS COMPUTED AS DU=U2BAR-U1BAR. DV = FIELDS OF MEAN DIFFERENCES FOR THE SECOND COMPONENTS COMPUTED AS DV=V2BAR-V1BAR. AMP = FIELDS OF THE AMPLITUDE OF THE MEAN DIFFERENCES COMPUTED AS AMP=SQRT(DU**2+DV**2). FVALUE = FIELDS OF 'F-VALUES' STANDARDIZED DIFFERENCES OF MEANS. USING N1,N2, DU,DV, VU1,VU2, COV1,COV2 AS DEFINED ABOVE, THIS FIELD IS COMPUTED AS FOLLOWS: F = FACT3*T22 FACT3 = (N1+N2-2-1)/(2*(N1+N2-2)) T22 = FACT2*SDIF/(1-ROP**2) FACT2 = N1*N2/(N1+N2) SDIF = SU**2 - 2*ROP*SU*SV + SV**2 SU = DU/S1P SV = DV/S2P S1P = SQRT( S1P2 ) S1P2 = FACT1*(N1*VU1+N2*VU2) FACT1 = 1/(N1+N2-2) S2P = SQRT( S2P2 ) S2P2 = FACT1*(N1*VV1+N2*VV2) ROP = FACT1*(N1*COV1+N2*COV2)/(S1P*S2P) T22 IS THE TWO SAMPLE HOTELLING'S T**2 STATISTICAL VALUE. KVALUE = FIELDS OF 'K-VALUES' OF TRANSFORMED SIGNIFICANCE LEVELS. COMPUTED AS K IN: P = ALPHA/(5**(K-1)) OR K = LOG (ALPHA/P)+1 5 WHICH INDICATES THAT THE OBSERVED STATISTIC IS SIGNIFICANT AT THE ALPHA/(5**(K-1)) SIGNIFICANCE LEVEL. THUS K=1 INDICATES THAT THE OBSERVED STATISTIC IS JUST SIGNIFICANT AT THE ALPHA SIGNIFICANCE LEVEL. IF PLOTTED WITH UNIT CONTOUR INTERVALS, SUCCESSIVE CONTOURS WILL ENCLOSE REGIONS WHERE LOCALLY IT IS FIVE TIMES AS UNLIKELY THAT VALUES OF THE OBSERVED STATISTICS ARE CONSISTENT WITH THE NULL HYPOTHESIS THAN IN REGIONS OUTSIDE THE NEXT LOWER CONTOUR. DVALUE = FIELDS OF 'D-VALUES' COMPUTED AS: __ ! 0 IF THE NULL HYPOTHESIS IS ACCEPTED D = ! ! 1 IF THE NULL HYPOTHESIS IS REJECTED -- MASK = FIELDS OF 'MASK-VALUES' COMPUTED AS: __ ! 1 IF THE TEST WAS CONDUCTED MASK = ! ! 0 IF THE TEST WAS NOT CONDUCTED -- THE MASK FILE IS NOT RETURNED IF THE PROGRAM IS NOT CALLED WITH OUTPUT FILE MASK. INPUT PARAMETERS... ALPHA (a) = THE SIGNIFICANCE LEVEL OF THE TEST. N1 (lx) = THE NUMBER OF OBSERVATIONS IN THE CONTROL DATA SET, N2 (ly) = THE NUMBER OF OBSERVATIONS IN THE EXPERIMENT DATA SET. THE ACTUAL VALUE USED ARE RATHER NX AND NY DEFINED AS N1 = INT( AN1 ) AND N2 = INT( AN2 ). IF N1 >= 1 , N2 >= 1 AND N1+N2 > 3 THEN N1 AND N2 FROM THE INPUT CARD ARE USED IN THE COMPUTATION OF THE NUMBER OF DEGREES OF FREEDOM. OTHERWISE, THE N1 AND N2 VALUES NEEDED IN THE COMPUTATION ARE OBTAINED FROM THE RECORD LABELS OR FROM THE IP3 DESCRIPTORS, DEPENDING ON THE FILE TYPE OF THE U1BAR AND U2BAR FILES. IVAR (kind) = A FLAG USED TO DETERMINE THE CONTENTS OF VU1, VV1, VU2 AND VV2 FILES. THAT IS : IF (IVAR.EQ.0) THEN THE VU1, VV1, VU2 AND VV2 FILES CONTAIN THE VARIANCES OF U1, V1, U2 AND V2, RESPECTIVELY. OTHERWISE, VU1, VV1, VU2 AND VV2 FILES CONTAIN THE UNBIASED ESTIMATES OF THE STANDARD DEVIATIONS OF U1, V1, U2 AND V2 RESPECTIVELY. READ(5,5010) ALPHA,AN1,AN2,IVAR 5010 FORMAT(10X,E10.0,2F5.0,I5) EXAMPLE OF INPUT CARD... *T2TEST. 5.E-2 10. 5. 1 0123456789012345678901234567890123456789012345678901234567890 1 2 3 4 5 6 7 EXIT CONDITIONS... 0 NO PROBLEMS FATAL CONDITIONS... 1 PREMATURE EOF ON STDIN 2 PREMATURE EOF ON UNIT 11 AND/OR 16 3 FIRST U1BAR AND U2BAR NOT THE SAME SIZE 4 UNABLE TO READ FIRST U1BAR 5 FOUND ILLEGAL N1 VALUE 6 MISSING U2BAR ARRAY 7 FOUND ILLEGAL N2 VALUE 8 FOUND N1+N2 SMALLER OR EQUAL TO 3 9 U1BAR AND/OR U2BAR NOT THE SAME SIZE AS FIRST U1BAR 10 MISSING V1BAR ARRAY 11 MISSING V2BAR ARRAY 12 V1BAR AND/OR V2BAR NOT THE SAME SIZE AS FIRST U1BAR 13 MISSING VU1 ARRAY 14 VU1 NOT THE SAME SIZE AS U1BAR 15 MISSING VU2 ARRAY 16 VU2 NOT THE SAME SIZE AS U2BAR 17 MISSING VV1 ARRAY 18 VV1 NOT THE SAME SIZE AS U1BAR 19 MISSING VV2 ARRAY 20 VV2 NOT THE SAME SIZE AS U2BAR 21 MISSING COV1 ARRAY 22 COV1 NOT THE SAME SIZE AS U1BAR 23 MISSING COV2 ARRAY 24 COV2 NOT THE SAME SIZE AS U2BAR 25 MEMORY ALLOCATION PROBLEM NON-FATAL CONDITIONS... 101 VERY SMALL DENOMINATORS IN T**2 STATISTIC 102 NAMES OF MEAN FIELDS DO NOT MATCH