355 lines
11 KiB
FortranFixed
355 lines
11 KiB
FortranFixed
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SUBROUTINE SWILK (INIT, X, N, N1, N2, A, W, PW, IFAULT)
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C
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C ALGORITHM AS R94 APPL. STATIST. (1995) VOL.44, NO.4
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C
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C Calculates the Shapiro-Wilk W test and its significance level
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C
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C IFAULT error code details from the R94 paper:
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C - 0 for no fault
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C - 1 if N1 < 3
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C - 2 if N > 5000 (a non-fatal error)
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C - 3 if N2 < N/2, so insufficient storage for A
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C - 4 if N1 > N or (N1 < N and N < 20)
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C - 5 if the proportion censored (N-N1)/N > 0.8
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C - 6 if the data have zero range (if sorted on input)
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C
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INTEGER N, N1, N2, IFAULT
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REAL X(*), A(*), PW, W
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REAL C1(6), C2(6), C3(4), C4(4), C5(4), C6(3), C7(2)
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REAL C8(2), C9(2), G(2)
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REAL Z90, Z95, Z99, ZM, ZSS, BF1, XX90, XX95, ZERO, ONE, TWO
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REAL THREE, SQRTH, QTR, TH, SMALL, PI6, STQR
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REAL SUMM2, SSUMM2, FAC, RSN, AN, AN25, A1, A2, DELTA, RANGE
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REAL SA, SX, SSX, SSA, SAX, ASA, XSX, SSASSX, W1, Y, XX, XI
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REAL GAMMA, M, S, LD, BF, Z90F, Z95F, Z99F, ZFM, ZSD, ZBAR
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C
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C Auxiliary routines
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C
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REAL PPND, POLY
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DOUBLE PRECISION ALNORM
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C
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INTEGER NCENS, NN2, I, I1, J
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LOGICAL INIT, UPPER
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C
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DATA C1 /0.0E0, 0.221157E0, -0.147981E0, -0.207119E1,
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* 0.4434685E1, -0.2706056E1/
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DATA C2 /0.0E0, 0.42981E-1, -0.293762E0, -0.1752461E1,
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* 0.5682633E1, -0.3582633E1/
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DATA C3 /0.5440E0, -0.39978E0, 0.25054E-1, -0.6714E-3/
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DATA C4 /0.13822E1, -0.77857E0, 0.62767E-1, -0.20322E-2/
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DATA C5 /-0.15861E1, -0.31082E0, -0.83751E-1, 0.38915E-2/
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DATA C6 /-0.4803E0, -0.82676E-1, 0.30302E-2/
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DATA C7 /0.164E0, 0.533E0/
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DATA C8 /0.1736E0, 0.315E0/
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DATA C9 /0.256E0, -0.635E-2/
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DATA G /-0.2273E1, 0.459E0/
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DATA Z90, Z95, Z99 /0.12816E1, 0.16449E1, 0.23263E1/
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DATA ZM, ZSS /0.17509E1, 0.56268E0/
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DATA BF1 /0.8378E0/, XX90, XX95 /0.556E0, 0.622E0/
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DATA ZERO /0.0E0/, ONE/1.0E0/, TWO/2.0E0/, THREE/3.0E0/
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DATA SQRTH /0.70711E0/, QTR/0.25E0/, TH/0.375E0/, SMALL/1E-19/
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DATA PI6 /0.1909859E1/, STQR/0.1047198E1/, UPPER/.TRUE./
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C
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PW = ONE
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IF (W .GE. ZERO) W = ONE
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AN = N
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IFAULT = 3
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NN2 = N/2
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IF (N2 .LT. NN2) RETURN
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IFAULT = 1
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IF (N .LT. 3) RETURN
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C
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C If INIT is false, calculates coefficients for the test
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C
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IF (.NOT. INIT) THEN
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IF (N .EQ. 3) THEN
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A(1) = SQRTH
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ELSE
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AN25 = AN + QTR
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SUMM2 = ZERO
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DO 30 I = 1, N2
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A(I) = PPND((REAL(I) - TH)/AN25,IFAULT)
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SUMM2 = SUMM2 + A(I) ** 2
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30 CONTINUE
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SUMM2 = SUMM2 * TWO
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SSUMM2 = SQRT(SUMM2)
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RSN = ONE / SQRT(AN)
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A1 = POLY(C1, 6, RSN) - A(1) / SSUMM2
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C
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C Normalize coefficients
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C
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IF (N .GT. 5) THEN
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I1 = 3
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A2 = -A(2)/SSUMM2 + POLY(C2,6,RSN)
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FAC = SQRT((SUMM2 - TWO * A(1) ** 2 - TWO *
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* A(2) ** 2)/(ONE - TWO * A1 ** 2 - TWO * A2 ** 2))
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A(1) = A1
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A(2) = A2
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ELSE
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I1 = 2
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FAC = SQRT((SUMM2 - TWO * A(1) ** 2)/
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* (ONE - TWO * A1 ** 2))
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A(1) = A1
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END IF
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DO 40 I = I1, NN2
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A(I) = -A(I)/FAC
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40 CONTINUE
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END IF
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INIT = .TRUE.
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END IF
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IF (N1 .LT. 3) RETURN
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NCENS = N - N1
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IFAULT = 4
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IF (NCENS .LT. 0 .OR. (NCENS .GT. 0 .AND. N .LT. 20)) RETURN
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IFAULT = 5
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DELTA = FLOAT(NCENS)/AN
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IF (DELTA .GT. 0.8) RETURN
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C
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C If W input as negative, calculate significance level of -W
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C
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IF (W .LT. ZERO) THEN
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W1 = ONE + W
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IFAULT = 0
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GOTO 70
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END IF
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C
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C Check for zero range
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C
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IFAULT = 6
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RANGE = X(N1) - X(1)
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IF (RANGE .LT. SMALL) RETURN
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C
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C Check for correct sort order on range - scaled X
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C
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IFAULT = 7
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XX = X(1)/RANGE
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SX = XX
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SA = -A(1)
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J = N - 1
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DO 50 I = 2, N1
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XI = X(I)/RANGE
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CCCCC IF (XX-XI .GT. SMALL) PRINT *,' ANYTHING'
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SX = SX + XI
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IF (I .NE. J) SA = SA + SIGN(1, I - J) * A(MIN(I, J))
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XX = XI
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J = J - 1
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50 CONTINUE
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IFAULT = 0
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IF (N .GT. 5000) IFAULT = 2
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C
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C Calculate W statistic as squared correlation
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C between data and coefficients
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C
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SA = SA/N1
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SX = SX/N1
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SSA = ZERO
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SSX = ZERO
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SAX = ZERO
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J = N
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DO 60 I = 1, N1
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IF (I .NE. J) THEN
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ASA = SIGN(1, I - J) * A(MIN(I, J)) - SA
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ELSE
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ASA = -SA
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END IF
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XSX = X(I)/RANGE - SX
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SSA = SSA + ASA * ASA
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SSX = SSX + XSX * XSX
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SAX = SAX + ASA * XSX
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J = J - 1
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60 CONTINUE
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C
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C W1 equals (1-W) calculated to avoid excessive rounding error
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C for W very near 1 (a potential problem in very large samples)
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C
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SSASSX = SQRT(SSA * SSX)
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W1 = (SSASSX - SAX) * (SSASSX + SAX)/(SSA * SSX)
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70 W = ONE - W1
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C
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C Calculate significance level for W (exact for N=3)
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C
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IF (N .EQ. 3) THEN
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PW = PI6 * (ASIN(SQRT(W)) - STQR)
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RETURN
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END IF
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Y = LOG(W1)
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XX = LOG(AN)
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M = ZERO
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S = ONE
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IF (N .LE. 11) THEN
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GAMMA = POLY(G, 2, AN)
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IF (Y .GE. GAMMA) THEN
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PW = SMALL
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RETURN
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END IF
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Y = -LOG(GAMMA - Y)
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M = POLY(C3, 4, AN)
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S = EXP(POLY(C4, 4, AN))
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ELSE
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M = POLY(C5, 4, XX)
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S = EXP(POLY(C6, 3, XX))
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END IF
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IF (NCENS .GT. 0) THEN
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C
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C Censoring by proportion NCENS/N. Calculate mean and sd
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C of normal equivalent deviate of W.
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C
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LD = -LOG(DELTA)
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BF = ONE + XX * BF1
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Z90F = Z90 + BF * POLY(C7, 2, XX90 ** XX) ** LD
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Z95F = Z95 + BF * POLY(C8, 2, XX95 ** XX) ** LD
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Z99F = Z99 + BF * POLY(C9, 2, XX) ** LD
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C
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C Regress Z90F,...,Z99F on normal deviates Z90,...,Z99 to get
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C pseudo-mean and pseudo-sd of z as the slope and intercept
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C
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ZFM = (Z90F + Z95F + Z99F)/THREE
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ZSD = (Z90*(Z90F-ZFM)+Z95*(Z95F-ZFM)+Z99*(Z99F-ZFM))/ZSS
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ZBAR = ZFM - ZSD * ZM
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M = M + ZBAR * S
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S = S * ZSD
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END IF
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PW = REAL(ALNORM(DBLE((Y - M)/S), UPPER))
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C
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RETURN
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END
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DOUBLE PRECISION FUNCTION ALNORM(X, UPPER)
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C
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C EVALUATES THE TAIL AREA OF THE STANDARDIZED NORMAL CURVE FROM
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C X TO INFINITY IF UPPER IS .TRUE. OR FROM MINUS INFINITY TO X
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C IF UPPER IS .FALSE.
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C
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C NOTE NOVEMBER 2001: MODIFY UTZERO. ALTHOUGH NOT NECESSARY
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C WHEN USING ALNORM FOR SIMPLY COMPUTING PERCENT POINTS,
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C EXTENDING RANGE IS HELPFUL FOR USE WITH FUNCTIONS THAT
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C USE ALNORM IN INTERMEDIATE COMPUTATIONS.
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C
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DOUBLE PRECISION LTONE,UTZERO,ZERO,HALF,ONE,CON,
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$ A1,A2,A3,A4,A5,A6,A7,B1,B2,
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$ B3,B4,B5,B6,B7,B8,B9,B10,B11,B12,X,Y,Z,ZEXP
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LOGICAL UPPER,UP
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C
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C LTONE AND UTZERO MUST BE SET TO SUIT THE PARTICULAR COMPUTER
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C
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CCCCC DATA LTONE, UTZERO /7.0D0, 18.66D0/
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DATA LTONE, UTZERO /7.0D0, 38.00D0/
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DATA ZERO,HALF,ONE,CON /0.0D0,0.5D0,1.0D0,1.28D0/
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DATA A1, A2, A3,
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$ A4, A5, A6,
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$ A7
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$ /0.398942280444D0, 0.399903438504D0, 5.75885480458D0,
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$ 29.8213557808D0, 2.62433121679D0, 48.6959930692D0,
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$ 5.92885724438D0/
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DATA B1, B2, B3,
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$ B4, B5, B6,
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$ B7, B8, B9,
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$ B10, B11, B12
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$ /0.398942280385D0, 3.8052D-8, 1.00000615302D0,
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$ 3.98064794D-4, 1.98615381364D0, 0.151679116635D0,
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$ 5.29330324926D0, 4.8385912808D0, 15.1508972451D0,
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$ 0.742380924027D0, 30.789933034D0, 3.99019417011D0/
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C
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ZEXP(Z) = DEXP(Z)
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C
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UP = UPPER
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Z = X
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IF (Z .GE. ZERO) GOTO 10
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UP = .NOT. UP
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Z = -Z
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10 IF (Z .LE. LTONE .OR. UP .AND. Z .LE. UTZERO) GOTO 20
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ALNORM = ZERO
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GOTO 40
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20 Y = HALF * Z * Z
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IF (Z .GT. CON) GOTO 30
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C
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ALNORM = HALF - Z * (A1- A2 * Y / (Y + A3- A4 / (Y + A5 + A6 /
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$ (Y + A7))))
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GOTO 40
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C
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30 ALNORM = B1* ZEXP(-Y)/(Z - B2 + B3/ (Z +B4 +B5/(Z -B6 +B7/
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$ (Z +B8 -B9/ (Z +B10 +B11/ (Z + B12))))))
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C
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40 IF (.NOT. UP) ALNORM = ONE - ALNORM
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RETURN
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END
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REAL FUNCTION PPND(P, IFAULT)
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C
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C ALGORITHM AS 111 APPL. STATIST. (1977), VOL.26, NO.1
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C
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C PRODUCES NORMAL DEVIATE CORRESPONDING TO LOWER TAIL AREA OF P
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C REAL VERSION FOR EPS = 2 **(-31)
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C THE HASH SUMS ARE THE SUMS OF THE MODULI OF THE COEFFICIENTS
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C THEY HAVE NO INHERENT MEANINGS BUT ARE INCLUDED FOR USE IN
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C CHECKING TRANSCRIPTIONS
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C STANDARD FUNCTIONS ABS, ALOG AND SQRT ARE USED
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C
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C NOTE: WE COULD USE DATAPLOT NORPPF, BUT VARIOUS APPLIED
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C STATISTICS ALGORITHMS USE THIS. SO WE PROVIDE IT TO
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C MAKE USE OF APPLIED STATISTICS ALGORITHMS EASIER.
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C
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REAL ZERO, SPLIT, HALF, ONE
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REAL A0, A1, A2, A3, B1, B2, B3, B4, C0, C1, C2, C3, D1, D2
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REAL P, Q, R
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INTEGER IFAULT
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DATA ZERO /0.0E0/, HALF/0.5E0/, ONE/1.0E0/
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DATA SPLIT /0.42E0/
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DATA A0 / 2.50662823884E0/
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DATA A1 / -18.61500062529E0/
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DATA A2 / 41.39119773534E0/
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DATA A3 / -25.44106049637E0/
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DATA B1 / -8.47351093090E0/
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DATA B2 / 23.08336743743E0/
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DATA B3 / -21.06224101826E0/
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DATA B4 / 3.13082909833E0/
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DATA C0 / -2.78718931138E0/
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DATA C1 / -2.29796479134E0/
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DATA C2 / 4.85014127135E0/
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DATA C3 / 2.32121276858E0/
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DATA D1 / 3.54388924762E0/
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DATA D2 / 1.63706781897E0/
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C
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IFAULT = 0
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Q = P - HALF
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IF (ABS(Q) .GT. SPLIT) GOTO 1
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R = Q*Q
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PPND = Q * (((A3*R + A2)*R + A1) * R + A0) /
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* ((((B4*R + B3)*R + B2) * R + B1) * R + ONE)
|
||
|
|
RETURN
|
||
|
|
1 R = P
|
||
|
|
IF (Q .GT. ZERO)R = ONE - P
|
||
|
|
IF (R .LE. ZERO) GOTO 2
|
||
|
|
R = SQRT(-ALOG(R))
|
||
|
|
PPND = (((C3 * R + C2) * R + C1) * R + C0)/
|
||
|
|
* ((D2*R + D1) * R + ONE)
|
||
|
|
IF (Q .LT. ZERO) PPND = -PPND
|
||
|
|
RETURN
|
||
|
|
2 IFAULT = 1
|
||
|
|
PPND = ZERO
|
||
|
|
RETURN
|
||
|
|
END
|
||
|
|
|
||
|
|
REAL FUNCTION POLY(C, NORD, X)
|
||
|
|
C
|
||
|
|
C
|
||
|
|
C ALGORITHM AS 181.2 APPL. STATIST. (1982) VOL. 31, NO. 2
|
||
|
|
C
|
||
|
|
C CALCULATES THE ALGEBRAIC POLYNOMIAL OF ORDER NORED-1 WITH
|
||
|
|
C ARRAY OF COEFFICIENTS C. ZERO ORDER COEFFICIENT IS C(1)
|
||
|
|
C
|
||
|
|
REAL C(NORD)
|
||
|
|
POLY = C(1)
|
||
|
|
IF(NORD.EQ.1) RETURN
|
||
|
|
P = X*C(NORD)
|
||
|
|
IF(NORD.EQ.2) GOTO 20
|
||
|
|
N2 = NORD-2
|
||
|
|
J = N2+1
|
||
|
|
DO 10 I = 1,N2
|
||
|
|
P = (P+C(J))*X
|
||
|
|
J = J-1
|
||
|
|
10 CONTINUE
|
||
|
|
20 POLY = POLY+P
|
||
|
|
RETURN
|
||
|
|
END
|