..
"
ON COMPARING TWO GAMMAS FROM ONE SAMPLE
by
Dana Quade
Depa~tment of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics MimeD Series No. 1111
April 1977
.
ON COMPARING TWO GAMMAS FROM ONE SAMPLE
by
Dana Quade
University of North Carolina
March 1977
Let there be given
abIes
U , U , U , U ,
4
l
2
3
N observations on 4 ordered-categorical variwith
M , M , M , M categories respectively.
2
4
l
3
It is desired to compare the value of Goodman and Kruskalts gamma G
12
calculated from variables
from variables
~
U ,
4
with
there are
U
3
and
M M M M
l 2 3 4
M M
l 2
and
U ,
4
U
2
with the gamma
G
34
calculated
The 4-way contingency table of
Ul , U , U ,
2
3
cells, may be arranged as a 2-way table in which
columns representing the
cation and there are
classification.
U
l
M M
3 4
Xl
and
X
2
rows representing the· X
3
Let the row totals be
Rl""'~
M
cross-classifiand
X
4
cross-
and let the column
3 4
Cl"",C M M' Then
1 2
calculated from the CIS, and
. totals be
Rls,
G12
and its standard error
S12
can be
G
34
and its standard error
S34
from the
in accordance with well-known methods.
however, we need the standard error
To compare
S of the difference
G
12
(G
and
12
-G
34
G ,
34
).
We have
but this involves the coyari.ance of
G12
and
G ,
34
for which no formula
has been published\
A comprehensive methodology which may be used for obtaining
S was
first presented by Grizzle, Starmer, and Koch (1969), and extended by
'oj
Forthofer and Koch (1973); it
~s
known as lta,nalysis of categoricaJ data
by weighted least squares ll ~ or more popularly "GSK analysis It.
approach begins by rewriting the
tWQ~wa.y
Specifically, if the table has entries
The GSK
contingency table as a
i
for
Fij
c
I, .•. ,r
vectQr~
and
j
= I, ..• , c and
LLF ij
k
= l, .•. ,rc
GI, = FII •. G2 = FI2 ' ...• ,G c = Flc' Gc+l = G21"'~' Grc =
expresses the statistic to be studied by means of comb ina-
F
rc
Next
= Nt then the vector 'G has elements Gk for
where
GSK
tions of three operators applied to the vector
G:
(simple linear)
Y = AG
with
VC!)
= A[V(G)]A'
(logarithmic)
Y = Alri(Q.)
(exponential)
In these formulas, the operator
the vector
"In"
transforms each element
Gk
of
G into its natural logarithm,
In(G ), and the operator
k
t1exptl converts each element G into its antilogarithm expCG ), A
k
k
is any matrix having as many columns as G has elements ,and AI: is its
transpose.
D is the diagonal matrix whose diagonal elements are the
G
elements of Q, and D is the diagonal matrix whose diagonal elements
Q
are the antilogarithms of the elements of
the asymptotic variance matrix
(~lso
G,
Finally,
des~red,
denotes
called variance-covariance matrix)
of the random vector indicated within the parentheses,
can be compounded as
VC.)
The three operators
For example, one may study
y = C exp (Bln (Ag) ) ,
-3-
..
for which the variance matrix may be derived in stages as i;ollows:
,.
Let
r1 = AG
Let
!2
Let
Y = C exp(!2)
-3
=
,
then
BIn (Y;))
,
V(!l)
then
then
;:::
A[V(~)]A 1
,
V(!2~'
!
... BD-Y1 [V(Y- 1)]D-IBI
Y
l
l
V(!3) ~ CDy [V (Y2) ] Dy C ~
2
2
then, substituting back, we find
!3 = C exp (BIn C:!l)) = C exp (BIn (AG)) ,
which is
Y
and
. These formulas are more fully explained, and illustrated, in the papers
cited above.
In applying this method to the problem at hand we shall first assume,
for convenience of exposition, that the 4 original variables
Xl' X2 , X3 ,
X all have 3 categories. Then the two-way contingency table has 9 rows
4
and 9 columns; examples are given as Table 1, containing the tooth data
from Davis and Quade (1968), and Table 2, containing data for a crosslagged panel analysis from Brenner (1976).
The vector
G derived from
such a table has 81 elements; for the tooth data
G = (24,0,0,0,0,0,0,0,0,0,4,0,5, ... ,0,0,69)\
and for the Brenner data
G = (55, 27, 2, 6, 3, 0,
° ° 0,
J
J
5P
'"
°1
J
J
4)
I
~4-
TABLE 1
Numbers of Carious Second Molars in Different Parts of the Mouth
derived from Davis and Quade (1968), Table 1
U
1
= Left
Side
U = Right Side
2
U
U
3
4
Upper
Lower
Jaw
Jaw
0
1
2
0
2
1
0
1
2
0
1
2
0
1
2
Total
0
24
-
-
-
-
-
-
-
-
24
1
-
4
-
5
-
-
.-
-
-
9
2
-
-
-
-
26
-
-
-
-
26
0
-
3
-
6
-
-
-
-
-
9
1
-
-
4
-
10*
-
0
-
-
14
2
-
-
-
-
-
13
-
7
-
20
0
-
-
-
-
5
-
-
-
-
5
1
-
-
-
-
-
9
-
7
-
16
2
-
-
-
-
-
-
-
-
69
69
24
7
4
11
41
22
0
14
69
192
Total
Notes:
- - impossible event with tooth data
o = possible
event which did not happen to occur
* = the sum of 5 occurrences each of Outcomes #7 and #10 in Table 1
of Davis and Quade
-5-
TABLE 2
Dat~
0
F
Banel Analysis
0
M
R
0
F
R
0
F
R
0
F
Total
R
55
27
2
6
3
0
0
0
0
93
0
5
6
3
1
6
2
0
0
0
23
M
0
0
0
0
0
0
0
0
0
0
R
33
40
5
7
15
4
0
1
1
106
0
4
16
1
2
25
3
0
1
0
52
M
0
0
0
0
2
0
0
0
0
2
R
19
25
9
2
5
4
1
1
1
67
0
1
9
4
0
11
3
0
1
3
32
M
0
0
0
0
0
1
0
1
4
6
117 123
24
18
67
17
1
5
9
381
U
2
R
Cross~lagged
R
U
l
U
3
for
U
4
Total
Notes:
U
l
=
How often felt depressed during' week before first interview
U
2
=
How often had psychophysiologic problems during week before
second interview
U
3
=
How often had psychophysiologic problems during week before
first interview
U
4
R
=
How often felt depressed during week before second interview
= Rarely
o = Occasionally
F
= Frequently
M = Mostly
The first stage of the GSK analysis is to apply a linear operator
which generates the row and column totals of the two-way contingency table.
Such an operator is determined by the matrix A, of lS rows and Slcolumns, as
follows:
J
1 1 1 1 1 1 1 1 1 :0 0 0 0
I
A
=
o0
0 0 0
:1 1 1 1 1 1 1 1 1
I
I
0
I
I
7x9
I
0
Sxg
I
I
J _________________
-----------------
I
I
I
Ig
where
Ig
indicates the
matrix of k
Z
lSx1
J
I
I
I
I
I
J
I
I
I·
A G
I
I
I
I
J
J
I
1J _____
~I
1 1
1 1 1 1 1 1____
1____________
J
I
I
.
J
J
J
____ L
F
_
I
=[
F'
.
1
k09
I
D
: c
]
-
..!.
Z ZI
N --
g
indicates a null
Then we obtain the vector
=
DR
~
)
Z has the form
VeZ)
0
8x9
J
J
gxg identity matrix, and
lSx81 -Slxl
The variance matrix of
J
I
I
I
I
1
Ig
rows and 9 columns.
=
1
-7-
where the matrix
F is the original two-way contingency table (and
is its transpose), and
DR
and
DC
FI
are diagonal matrices containing as
their diagonal elements the row totals and column totals, respectively,
of
F.
The next stage of the GSK analysis is to apply alogarithmic operator
which generates the individual products which are summands of the numbers
of concordant and discordant pairs.
Y
36xl
Here
0
In(Z)
B
= 36x18
l8xl
Specifically, we take
where
B
=
[+H-}
is a null matrix of 18 rows and 9 columns, and Q is the matrix
of 18 rows and 9 columns given as Table 3.
Then Y is a vector of 36
elements as follows:
•
r
i
¥
Y
l
= In(RlRS)
Y
lO
= In (R 3R4)
Y
19
= In(ClC S)
Y
28
= In(C 3C4 )
Y2
= In(Rl R6)
Y
ll
= In(R 3RS)
Y
20
= In(C l C6 )
Y29
= In(C 3CS)
Y
3
Y
4
= In(RI R8)
Y
12
= In(R 3R7)
Y2l
= In(CI C8)
Y
30
= In(C 3C7)
= In (R l Rg)
Y
13
= In(R 3R8)
Y22
= In(ClC 9 )
Y
3l
= In(C 3C8)
Y
S
= In(R 2R6)
Y
14
= In(R 2R4 )
Y
23
= In(C 2C6 )
Y
32
= In(C 2C4 )
Y
6
= In(R 2Rg )
Y
lS
= In(Rl 7)
Y24
= In(C 2C9 )
Y
33
= In(Cl 7)
Y
7
= In(R 3R8)
= In(R6R7)
Y2S
= In(C 3C8)
Y
34
= In (C 6C7)
Y
8
Yg
= In(R 3Rg )
Y
16
Y
17
= In(R 6R8)
Y26
= In(C 3C9 )
= In(C 6C8)
= In (RSRg )
Y
18
= In(R SR7)
Y27
= In(CSC g )
Y
3S
Y
36
= In(C SC7)
-9-
Note that the elements in the first column above are the logari thms of the
products required for the concordant pairs in
G , the second column for
34
the discordant pairs in G , the third column for the concordant pairs
34
in G ' and the fourth column for the discordant pairs in G ' The
IZ
IZ
variance matrix of Y, using the general formula, is
where
of
D is the diagonal matrix whose diagonal elements are the elements
Z
Z. A computational complication arises, by the way, if any element of
the vector Z is zero (which happens for both our examples), since then
-1
DZ
and
In(~
writing down
do not exist.
However, the same result is obtained if in
DZ the zero elements are replaced by arbitrary
non~zero num~
bers, since these numbers will all be multiplied by zero elements from
~
and thus have no effect on
Vet)
V(!J.
The GSK analysis continues 'with application of an exponential operator
which generates the numerator and denominator of each of
G
IZ
Specifically, we take
where
c=
[~-~-~]
o
1
1
and
H
H is the following matrix of Z rows and 18 columns;
H
=
111111111
[1 1 1 1 1 1 1 1 1
Then the elements of
X are
-1 -1 -1 ",,1 -1 -1
1
1
1
1
1
1
~l
~l
1
1
-:]
and
G ,
34
-10-
•
Xl
=
(number of pairs of observations which are concordant with
respect to
V
and
3
V ) -4
(number of pairs of observations
which are discordant with respect to
=
numerator of
and
VI
V )
2
G '
34
and similarly
X
2
=
denominator of
X
3
=
numerator of
X
4
=
denominator of
G
12
The variance matrix of !'
V(!)
Oy
,
G
l2
from the general formula, is
= COy[V(Y)]OyCI
CD BO- 1 [V(Z)]O-lB ' D C'
Y Z
Z
Y
=
where
G ,
34
is the diagonal matrix whose diagonal elements are the anti-
logarithms of the elements of
Y.
The final stages of the GSK analysis involve one further application each
of the logarithmic and exponential operators.
W [1
i~l = 0
-1
0]
o
o
We take
1 -1
and
exp(W)
The asymptotic variance matrix of
[1
Q 0
o
-1
0
0
1
G12
0] -1
-1
Ox
and
G34
[V(X)]D~l
is then
1
0
-1
0
0
1
0 -1
0Q
where
and
D is the diagonal matrix with diagonal elements
X
D
Q
the diagonal matrix with diagonal elements
(Xl' X ' X ' X )
2
4
3
CG34,G12)'
Upon
performing some of the indicated multiplications, we find
and the variance matrix
Xl
1
X - 2
x2
2
0
where
"cov"
0
1/X
0
is the covariance of
can be calculated.
1
0
2
2
0
-X/X 2
[V (X)]
Xl
0
1/X
1
4
X - 2
2
x4
4
0
-Xi X4
0
G
12
and
The correlation between
=
G ,
34
G
12
[S;4 c;v]
cov
S12
Thus the required
and
G
34
S2
is of course
The listing of a Fortran IV program written to accomplish the above
analysis, and its output when given as data the two examples presented
above, form an Appendix to this paper.
(Example 1:
The results were:
tooth data)
G
34
= .5076
± .0838
G12
= ,8479
± ,0407
G12
.,-
G
34
:;
.. 3403 ± ,0790
~12",
(Example 2;
Brenner data)
G
34
Gl2
Gl2
,2613 ± ,0838
~
,5359 ± . 0725
;::
G
34
~
;::
,2746 ± ,1030
A test of the hypothesis that
G
12
and
G
34
estimate identical
population values can be obtained by taking the ratio of their
differ~
ence to its standard error as a normal deviate; thus;
(Example 1):
Zl;:: .3403/.0790 ;:: 4,31
(Example 2):
Zl;:: ,2746/.1030 ;:: 2,67 ,
In both examples the difference is significant.
An alternative approach would be to use the general computer program
GENCAT of Landis et al (1976) which was especially developed to perform
GSK analyses.
Unfortunately, the present problem, involving a contingency
table of 81 cells, is already too large for a straightforward application
of GENCAT, although it can be managed by using the Hraw data option".
Let us consider briefly the modification which would be necessary
should the numbers of categories of
Xl' X , X and X , namely M , M ,
3
4
2
I
2
3. In this general case the matrix F
M and M , not all be equal to
3
4
has M M columns and M M rows, and the vector G derived from it
l 2
3 4
has M M M M elements, The matrix A has the form
l 2 3 4
I
e
A
;::
[
HI
I
I
I
...
...
I
I
I
I
HM M
3 4
-~-~-----~------I
I
I
I
r
I
I
I
1
)
-13-
•
where each
.
is the identity matrix of
I
is a matrix of
"
tains only
Its,
Z will have
columns and
Ml M2
(M M + M M )
l 2
3 4
given earlier.
Q
2
row contains two
e
(M -1) (M
1
2
~
1)
l's
1 IS
two
l's
(M - 1)
4
l' s
~
1)/2
rows and
and the remainder
rows and M M
3 4
and the remainder
H
k
Then the vector
F
followed
B will have the form
and the remainder
M (M - 1)M (M -1)/2
3 3
4 4
and
and its variance matrix has the same formula as
M (M -1)M (~12
l l
2
has
ot.s,
elements, namely the row totals of
F,
The matrix
columns~
rows such that its k ....th row con . .
M;-5 M4
and the remaining rows contain only
by the column totals of
Here
rows and
M M2
l
0 t s,
and the remainder
M M
l 2
Ots,
O!s;
columns, where every
and every column contains
similarly,
l
columns, where every row contains
(M - 1) x
3
and every column contains
0 IS,
may be obtained by analogy with the
has
Q
The exact forms of
3x3
Q
1
and
Q
2
case treated above.
Then
where
HI
columns and
H
2
and
HI
H
2
each have two rows, but
has
Its,
has
M (M - 1)M (M - 1) /2
2 2
l l
columns; in each of
the second half of the first row consists of
the matrix consists of
.
M (}13 . . 1)M CM4 - 1) /2
3
4
H
2
-l's
HI
and
and. the rest of
The remainder of the analysis is unchanged •
ACKNOWLEDGMENTS
I wish to thank Dr, Berthold 8renner (Center for Epidemiologic
Studies~
Division of Biometry and Epidemiology, NIMH) for bringing this problem to my
attention and supplying the illustrative data of Table 2; and also Mr, Donald
S. Rae (Division of Computer Systems, ADAMHA) for converting my program from
an obscure dialect into standard Fortran IV,
REFERENCES
Berthold Brenner, (1976):
Personal Communication.
C.E. Davis and Dana Quade, (1968); On comparing the correlations within
two pair of variables, Biometrics ~, 987~995.
Ronald N. Forthofer and Gary G. Koch, (1973); An analysis for compounded
functions of categorical data, Biometpics ~, l43~157.
James E. Grizzle, C. Frank Starmer, and Gary G. Koch, (1969)1 Analysis
of categorical data by linear models, Biometpics 25, 489~504.
J. Richard Landis, William M. Stanish, Jean L. Freeman, and Gary G, Koch,
1976: A computer program for the generalized chi-square analysis of
categorical data using weighted least squares (GENCAT), Computep Programs in Biomedicine ~, 196-231,
;:
(e
FORTRAN IV 31
0001
0002
0003
0004
0005
0006
0007
0008
0009
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
00:';
0026
0027
0028
0029
0030
0031
0032
0033
0034
0035
0036
0037
0038
0039
0040
0041
0042
0043
...
i'
RELEASE 2.0
MAIN
•
DA T1;: = 7711 6
"
(e
09/14/33
C SLGAM1
REAL*8 N.Z(~8).VZ(18.18).F(9.9)
DIMENSION 0(18.9)
REAL*8 H1 1 2.9). H2(2.9). VX(4.4). X(4). Y(4).EXPY(36)
REAL*8 R.C,RI.RJ.CI.CJ.W
DATA 0/
-1 .. 1 .. 1 .. 1 .• 0 .. 0 .. 0 .. 0 .. 0 .. 0 .. 0 .. 0 .. 0 .. 0 .. 0 .. 0 .. 0 .. 0 .. 0 .• 0 .• 0 .. 0 .•
-1 .. 1 .. 0 .. 0 .. 0 .. 0 .. 0 .. 0 .. 0 .. 1 .. 1 .. 0 .. 0 .. 0 .. 0 .. 0 .. 0 .. 0 .. 0 .. 0 .. 0 .. 0 ..
- O. . 1 . . 1 . . 1 . . 1 . . 0 .. O. . O.. O. . O.. O. . O.. 0 .. O.. 0 .• 0. . 1 .• 1 .. O. • 1 .. O. . O..
-0 .. 1 .. 0 .. 0 .. 0 .. 0 .. 1 .. 0 .. 0 .• 0 .. 0 .. 0 .. 0 .• 0 .. 1 .• 0 .. 1 .• 0 .. 0 .. 0 .. 0 .. 0 ..
-0 .. 1 .. 0 .. 1 .. 0 .. 0 .. 1 .. 0 .. 0 .. 0 .• 0 .• 0 .. 0 .. 0. ,0. ,0. ,0 .. 1 .• 1. ,0 .. 0 .• 0 .•
-0 .. 0 .. 0 .. 0 .. 0 .• 0 .. 0 .. 0 .. 0 .. 1. ,0 .. 0 .• 1 .• 1. ,0 .. 1 .• 0 .• 0 .. 1. ,0 .• 0 .• 0 .•
-1 .. 0 .. 0 .. 0 .. 0 .. 0 .. 1 .. 0 .. 0 .• 0 .• 1 .• 0 .. 0.,0 .. 0 .• 1 .. 0 .• 1.,0.,1 .. 1.,0 .•
-0 .. 0 .. 0 .• 0 .. 0 .. 0 .• 0 .• 0./
NX=NX + 1
DIMENSION A(15)
READ(5.111.END=999) A
111 FORMAT(20X,15A4)
DO 66 I=1.9
READI5.112.END=999) (F(I,J).J=1.9)
112
FORMAT(16F5.0)
66
CONTINUE
WRITE(6.166) A
166 FORMAT( '1' .15A4/)
N=O
WRITE{6.201 )
201 FORMAT(' ORIGINAL DATA MATRIX:'/)
DO 202 1= 1 .9
202 WRITE( 6.203) (F( 1. J). J=1 .9)
203 FORMAT(9F8.0)
C CALCULATE VECTOR Z OF ROW + COLUMN TOTALS OF F
DO 3 I =1.9
R=O
C=O
DO 2 J=1.9
R=R+F(~.J)
C=C+F(J.I)
2
CONTINUE
Z(I)=R
Z( I+9}=C
3
N=N+R
C CALCULATE VARIANCE MATRIX VZ OF VECTOR Z
DO 4 1= 1 .9
RI=ZII)
CI=ZII+9)
DO 4 J=1.9
RJ=ZIJ}
CJ=ZIJ+9)
VZ(I,J)=-RI*RJ/N
VZ(I+9.J+9) = -CI*CJ/N
VZ(J+9.I} = F(I.J}-RI*CJ/N
4
VZ{I.J+9) = VZ{J+9.I)
DO 99 1=1.18
VZ(I.I) = VZ(I,I) + Z{I)
PAGE 0001
(~
FORTRAN IV G1
0044
0045
0046
0047
0048
0049
0050
0051
0052
0053
0054
0055
0051'>
005/
0058
0059
0060
0061
0062
0063
0064
0065
0066
0067
0068
0069
0070
0071
0072
0073
0074
0075
0076
0077
0078
0079
0080
0081
0082
0083
0084
0085
0086
008',
0088
0089
0090
0091
0092
0093
-
(4t
' -----,'
RELEASE 2.0
MAIN
DATE = 77116
99 CONTINUE
C CALCULATE VECTOR EXPIY)
DO 6 1=1.18
R=l
C=l
DO 5 u=1.9
IFIQII.u).LE.O) GO TO 5
R=R~Z(u)
~
C=C""Z(U+9)
5
CONTINUE
EXPY(I)=R
6
EXPYlI+18)=C
PRE- AND POST-MULTIPLY MATRIX VZ BY INV(DZ)
DO 8 1=1.18
DO 7 J= 1 .18
W=Z(I)~Z(U)
IFlW.LE.O) GO TO 7
W=VZlI.u)/W
7
VZl!'u)=W
8
CONTINUE
C CALCULATE VECTOR X
DO 11 1=1.2
DO 11 u= 1 .9
R=O
10
11
C=O
D010K=1.18
W=H( I ,K)"'QlK.u)
R=R+EXPYIK)"'W
C=C+EXPYIK+18)*W
CONTINUE
H1lI.u)=R
H2(I.u)=C
CONTINUE
DO 13 1=1.2
R=O
C=O
DO 12 u= 1,18
R=R+HII.u)~EXPYlu)
12
C=C+HII.u)"'EXPy{u+18)
XlI)=R
13
XII+2)=';
C CA~CULATE VARIANCE MATRIX VX OF VECTOR X
DO 13 1=1.2
DO 15 J=1.2
R=O
c=o
14
W=O
DO 14 K= 1 .9
D014L=1.9
R=R :. H11!'K)*VZIK.L)*H1(J.L)
C=C + H2II.K)*VZIK+9.L+9)*H2(u.L)
W=W + H1II.K)*VZIK.L+9)*H2(u.L)
VXI!,u)=R
VX(I+2.u+2)=C
09/14/33
PAGE 0002
-,
...
1-..(
(e
FORTRAN IV G1
RELEASE 2.0
REAL FJNCT10N
0001
0002
0007
H*4(I.~)
H=1.
1F( 1-1 )9.10.10
0003
0004
0005
0006
H
10
IF ( J - 10 ) 9 . 1 " • 1 1
11
H= -1 .
9
RETURN
END
•
DATE
= 77116
~
09/14/33
PAGE 0001
\.
(e
->'
j
j,~
(e
OUTPUT FOR TOOTH DATA
.
•
ORIGINAL DATA MATRIX:
24.
O.
O.
O.
O.
O.
O.
O.
O.
O.
4.
O.
3.
O.
O.
O.
O.
O.
O.
O.
O.
O.
O.
5.
O.
O.
O.
26.
4.
O.
O.
O.
O.
GAMMA
6.
O.
O.
O.
O.
O.
O.
10.
o.
5.
O.
O.
SE GAMMA
o.
O.
o.
O.
o.
13.
O.
9.
O.
O.
O.
o.
O.
O.
O.
o.
O.
o.
GAMMA/5E GAMMA
GAMMA' .2
0.84787
0.04069
20.83501
GAMMA 3.4
0.50755
0.08382
6.05526
DIFF(G12-G34)
0.34031
0.07897
4.30939
COR(G12.G34)
0.35849
O.
o.
o.
o.
o.
O.
o.
o.
O.
O.
7.
O.
7.
O.
O.
O.
O.
69.
~
(e
"
OU~PUT
\;
•
..
..
....
•
--
FOR BRENNER DATA
ORIGINAL DATA MATRIX:
55.
5.
O.
33.
4.
O.
19.
1.
O.
."
27.
6.
O.
40.
16.
O.
25.
9.
O.
2.
3.
O.
5.
1.
o.
9.
4.
O.
GAMMA
6.
1.
o.
7.
2.
O.
2.
O.
O.
3.
6.
O.
15.
25.
2.
5.
11 .
O.
SE GAMMA
O.
2.
O.
4.
3.
O.
4.
3.
1.
o.
O.
o.
O.
o.
o.
O.
O.
1.
1-
O.
O.
1.
1.
1.
1.
O.
O.
G.AMMA/SE GAMMA
GAMMA 1.2
0.53585
0.0:251
7.38954
GAMMA 3.4
0.26127
0.08381
3.11737
DI FF( G12-G34;'
0.27458
0.10296
2.66681
COR(G12.G34)
0.17'834
o.
O.
O.
1.
O.
O.
1.
3 .•
4.
...
<e