Some Notes on the Application of
Sequential Methods in the Analysis of Variance
." r
1
By
N. L. Johnson
Institute of statistics
Mimeograph Series No. 61
Limited Distribution
November, 1952
1. This research was supported in part by THE UNITED STATES
AIR FORCE under Contract AF 18(600)-83 monitored by the Office of
Scientific Research.
-- 1
SUMI·fARY
Sequential tests of linear
model are studied.
hypot~eses
in the systematic linear
Methods of overcoming difficulties in the construction
of tests when there is a random model are considered.
1.
INTRODUCTION
The original methods L-l_7 of constructj.ng seq.uential tests reqUired problems to be formulated as discrimination between two simple
hypotheses.
In cases where composite hypotheses were involved a more
or less arbitrary weighting function was introduced so that the problem could in effect be reduced to discrimination between simple hypotheses.
Recent work by Barnard
L-2_7 and COX L-'-7 has
made it possible to extend
the sphere of application of sequential tests to a number of cases where
composite hypotheses are to be compared.
Barnard refers to unpublished
work by C. M. Stetn on this problem and there is a remark in
L-4_7 which
implies that both Stein and M. A. Girshick approach the problem from the
same angle as Cox.
It is the purpose of these notes to discuss some points of detail arising in the application of sequential methods to the particular
type of COMposite hypotheses associated with the analysis of variance.
Tests of the general linear hypothesis in systematic (parametric) models
will be discussed first, followed by a discussion of component of variance
models for simple special cases.
- 2 2.
THE GENERAL LINEAR HYPOTHESIS
,
It will be helpful to start with a brief reSUMe of the general
linear hypothesis and its likelihood ratio test in the case of samples
It is asaumed that
(i)
(11)
where
Q
= (Ql'
x
= (xl.'
•••• ,
) a:':"e N 1:!'ldeper.dent Normal variables
N
7.
g:<.~) =Qe'
•••• ,
Qs
= (Q1'
)
•• ·Qs-ql Qs-q+l' ••• Qs )
= (e(l)' e(2»
c
(s
=
=
< N)
(C (1)' C(2) )
where C 1s partitioned between the (s_q)th and (s_q+l)th columns, as also
is
Q.
(1:'.i) var xi
= a2 (1 = 1, .•• , N)
(iY) The Q's and a2 are unknown parameters; the
constants, defined by the design of the
CIS
are known
experi~ent.
The hy?otheeis to be tested is
Ho : Q(?)
= (0,0, .•••0) = a
A likelihood ratio criterion
monotonic function of G
Sa
= ~b/Sa'
fo~
this test 1s provided by any
where
is the m5.n1J.num of (x - QC') (x - go') 'with
respect to
Q
-- 3 -
e.
It can be shown
L4_7 L-7_7
function of G satisfies
1
l:2~
(1)
peG
I Q(2)
0- )
= e
peG
that the probability density
1
f O)M('2(N-s+q),
2'
~o
where
= Q(2)C(2)(I
1
1
-1 '
t
- C(l)(C(l)C(l»
1
~i 2~G(1+G)- )
,
C(1»C(2)Q(2)
and
is a confluent
h~~ergeometric
3.
function.
SEQtTENTIAL ANALYSIS FOR THE SYSTEMATIC MODEL
Now consider a sequential form of experiment in which successive
XIS,
or sets of x's, are measured until a decision is reached.
As the
experiment is continued, so will C grow by the addition of further rows.
The way in which it.is decided to obtain each successive observation (or
set of observations) will determine the numerical values of the
the successive rows of C.
CIS
in
In these Dotes only those cases where the de-
sign is predetermined (i.e., where the c's are not random variables depending on the results of earlier measurements) will be considered, although it would appear, intuitively, that determination of the
CIS
on
the basis of results already observed would lead to improved procedures.
At each stage in the experiment a value
(The superscript
L-N_7
aL-N_7 may be
calculated.
means 'pertaining to sample size N', and will be
omitted when confusion is not likely to be incurred by such omission).
The distributions of the corresponding random variables will depend on
8(2)0- 1
sequence
t~rough
the paraneters
aL- N_7
~L-N_7.
It might be hoped to use the
in a test to discriminate between the hypotheses
- 4-
8(2)°
HJ
(Evidently by taking ,11
=0
-1
=
1\
~J
(j
and c-hoOsing
= 1,2)
2 suitably a sequential
test discriminating between H and H could be compared with the likel
2
lihood ratio test of Ho ).
Such a sequential test could certainly be obtained if
.... ,J-N_? = p(J-N_7t e(2)0-1)f(aL- 2_7, .... aL-N_7)
f(aL- 2_ 7, .• •r;L.-N_7) does not depend on 8 (2) 0- 1 • Cox £·3_7 gives
(2) p(aL- 2_7,
where
conditions under which (2) is true.
{oL- Nn-71
If it is possible to pick out a SUbsequence
such
that each of the terms in the corresponding subsequence {>.L-Nn_7 t
.
J
depend only on the same scalar function of 8(2)0 -1 then it can be shown
that Cox·s conditions apply, and so
7 •••OLrNn-7 ,8(2)0
I
-1)
1-,
rN
p (~
rN
7
Since p(G'- 0- , e (2) 0-
1
)
= P(OLr Nn-7.,8(2)0-1) f ( OLrN 1-7,···OLrNn-7
L-Nn-J7 which
.
depends only on '"
is itself re-
quired to depend only on some scalar function ¢( 9(2)0-1) of 9(2)0 -1
we may write
j-N
7
j-N
p(GL- 1-, ••.• GL-
7l
n- , ¢ )
7
= p(GI-/-Nn-'
j-N
7
n-7)
rN
¢)f{Gl- 1-, • HOL
It follows that
(a) all hypotheses (about 8(2)0-1) which specify the same value
for
¢ will
be, for present purposes, equivalent
(b) a sequential test for discriminating between the hypotheses
H' : ¢ = ¢' and H" : ¢
•
= ¢"
will be specified by instructions of the form
- 5 -
(I)
"Accept H' i f
Accept H" if
~::n=!-.L¢.2 ~ J _
p(C.... n7f¢,)
1 - a
p (cJ-Nn_7J ~")
p
(oL-Nn-7, ¢' )
Otherwise take a further set of Nn+l - Nn
accordance with the p:..-est::ribed patte'l:'n"
in
observatio~s
provided this sequence ot operations terminates with probability
one when either H' or H" is true.
~L-Nn-7
The 'prescrihed pattern' will, of course, be such that
depends or.ly on ¢(0(2)o-1).
4. L!lY1ITING FORM OF THE TEST
Decision to take a further set of observations will be made if
<
@
1 - a
I, {JI,l)
peG f ¢')
p (G
,./
......
1 - t3
where we have, for convenience, dropped the superscript
L-N_7 .
From (1) this is equivalent to
(2)
A+
~(~"-~') <: log M(~(Nn-s+q) 1 ~q; ~~"y2)
1
1
1
2
- log M(2'(Nn -s+q), "2q; {2~' y )
where A
= log
t3
----; B
1 - a
= log
1 - t3
; y
1
1- B~{~"-~
1
)
= GI()l+G 2
a
The application of tests of this type (for q> 1) requires
tables of M(X,Y;u) rather more extensive than are known to the author
at present.
He can, however, make certain deductions about inequalities
- 6 (2)1 incidentally showing that in certain important cases; the sequence
does I indeed, terminate with probability one when either H' or H" is
true.
It will be assumed from now on that ¢" > ¢, and "''' >
1
'" t.
As
1
u increases from 0 to 1, L-log M(X" Y; '2"'''u) - log M(X I Y; r.\'u)_7
increases from 0 to L-log M(X I '¥'; ~"''') - log M (XI Y;
(if X
and Y > 0).
>0
Hence (2) is equivalent to
Q(Nnl a"t3,"" ,~.")
(3)
Q and Gbeing
!"'l )_7
< G <. G(Nn'O:,~,,,,,
,,,,")
fixed numbers defined by the quantities in brackets.
If A~("'''-'''')~O then Q ... 0 while if B~("'l1_"") ~ log M(X,y;~",,,)
(N.B.
1
- log M(X,Y;'2"")
then G
=
00).
Now consider the special case where successive sets of k observations are taken, so that Nn = kn, and each set is arranged in the
same pattern, so that identical sets of rows are added to C at each stage.
In t~is case it 1s possible to take ¢=",L-k-l/k and then ",L- Nn-7 ...
",L-kn.J
= nk¢
is a function of
¢ only.
,
centrality per unit observation).
(4)
(¢ may be thought of as 'non-
(2) now becomes
A+~nk(¢II_¢t) < log M(!(nk-S+q) I ~q; ~nk¢"y2)
- log M(~(Dk-S+q),
Now PerrOn
l..-a 7 has
- r (' 1
}u ~I1k¢II)< B+!nk(¢lI_¢')
shown that
Iu
M(X,Y;u) ... ~~ e~
2/7"
1 - ~Y
(xu)1
1
e
r.:r.:
2tXu
"'?5
(l+O(X ~) )
,
Hence from (4) if n is large and ¢ ., 0, we have
1
(5)
-,A+~nk(¢"_¢t)+0(n-2)< !(q-l) 109(¢t/¢")+~(¢1I_¢')y2+
1
nL-k{ k-(s-q)n-~1_72 (J¢"- J¢') y
,1
<. B-ink (¢"-¢' )+O(n- '2 )
- 7 Remembering that ¢,,;> ¢') (5) may be rearranged to 8ive
3
(6)
2. + {q.. l)log(¢"!¢')+4A
+ o(n- 2)
nk(¢"-¢' )
<
1
y2. +
(l-.!-q) 2
4
.J¢"+,I¢'
n
y
3
<: 2. + (q-l)log(¢"/f) +4B + O(~ 2)
nk(¢"-¢' )
This implies
4
1Y2. +
Y - 2.
<1
,
J¢" +J¢'
I
As n tends to infinity, y terds to
and y2 tends to ¢(l+¢)-l in probability.
¢
(7)
1 +
for either
¢ = ¢,
or
¢
+
4
J¢"+ J¢'
¢ = ¢I\
1
O(n-)
1
:2
¢ (l+¢)
-2
In general we shall not have
i¢
J(l+¢)
=
2.
so that the sequence (I) will terminate
with probability one in either case.
It is of interest to note that large samples are not likely to
be reached unless (7) is satisfied, at any rate approximately.
1
(8)
1
¢2(1+¢) '"'2
From (7)
1
= 2 L- f l~( J¢"+ J¢,;2}"2
-1_7(
Since 0 ~ ¢I(l+¢) ~ 1, (8) cannot be satisfied if
j¢"+ )¢,)-l
J¢"+ j¢' >- 4.
Hence
unless ¢" (and so also ¢') is small enough it is likely that large sample
sizes will be reached.
("Large" is here used in the sense of "large
.,1
compared to the average sample size when ¢=¢' or ¢=¢").
that i t both ¢" and ¢' are small (8) gives
J¢ =: ~( J¢" +J¢')
We may note
- 8 which 1s not unexpected on intuitive grounds.
The argument above applies to the case
If ¢,
=0
A1
then
in~tead
nk¢"+0(n -2)
~
o.
(5) we obtain
of
..;.
¢,
-< i(q-l)lOg
k¢"+ink¢"y2+nk {l-(s-q)/nkl 2 ,J¢lI y
1
~
BJnk¢"+0 (n-2 )
c
leading finally to (7) with ¢'
5.
1
= o.
ONE·WAY CLASSIFICATION
The special case of one-way classification into k groups will
now be con.sidered.
At ee,ch ate.ge we decide whether or not to take a
further set of k observations, one from each group.
The usual systematic
moCl.el
(t=l, ••• k,; i=l, ••• n; E b t
t
is included in the
ge~eral
linear model described in section 2.
in the notation of section 3, Nn
(t
= 1,
.• k) implies q
=k
oL- nk_ 7 = n
- 1.
= nk,
Hence the sequence
J-- 2o.J,
t
b~
~ E(Xti _
t i
/
/el,
¢
=
J·~3n.) . . . ma~r be used in a sequential
Ht
H"
and constructed as in (I).
x~)2
~ b~/ka2 •
t
test based on discrimination between
and
=0
Further
~(~t. - X.. )2
= n E
Then,
s = k and the hypothesis bt
t
>.,L-nk_7
= 0)
Eb~/1~a2 = ¢II
·96 • lWIDOM MODEL IN THE ONE·WAY CLASSIFICATION
It is well-known that sometimes a model different from (9) may
be used.
This is
where the u's are Normal variables, each 'with expected value zero
and standard deviation cr , mutually independent and also independent
R
of the z's.
This model is often called the 'random' or 'component of
variance I model.
tn this case
(11)
2
"
C'
systematic model.
oL- 3n_7, ...
tion between
(l+no +G'-
2
whereo =OR/ ° ,
!],ihis depends only on
rnk
1
7 2"(kn-l)
-)
8 ,
which plays the part played by ¢ in the
2n_ 7,
Cox's result shows that the sequence
aL·
may be used in a sequential test based on discrimina-
HR:
~
=8
,
and
HR: S =
~'
II
provided the procedure
specified terminates with probability one.
the procedure will be
(II)
"Accept H~ i f f (G, 8',
Accept
H;~ i f
.s ll) £ 13/ (l-a)
f(G, $\', f").) (l-(:3)/a
Otherwise measure one further item in each of the k groupsll
1
where
f(G, """, ~. ")
=
'")
'1
+n ~
( l+n is:, ,
~(n-l)
G
l+n I; •
( l+n
t
II
+G'\~(kn_l)
+G J1
Under this procedure sampling will be continued whenever
.·10 /where (12) Q
R
€
= n(
G".!_
GR = n( 6"£-
5')(1-.!)-1.1;
8')(1-€)-1_1
(.!+n Ii' )k(n-l)Alm-l)
= (-P_)~kn-l)
l-a
l~
;
~~~
.€ = (!-f3) 21 (kn.l) (l+n E; '.)(1m-l)
a:
l+n g,"
lim Q = L-2A+(k-l)log(
R
n---+ 00
lim
D~OO
so that
6R
~"1 8')_7 I k( 8,-1. S"-J.)
= L-2B+(k-l)log( Sill f,')_7 /
lim ..QR
n .... 00·
{:
lim
n--+
00
G
R
00
&' = 0
QR -=
(the c~se considered by cox)
.
lim
n--+
0.0
G
R
2
models in which the quantity Ebt/ko
S. ({_1 1k ).
2
=~
0
a
The random model can be regarded as
.
k( S,-l- S,-l)
-
lim
n.......
On the other hand it
S '> 0
S"~
It can easily be shown that, if
~
mixture ot systematic
is distributed as
For any systematic model such that
lim
~oo
!!R
< rJ
~ lim
n~oo
GR
<
there is a non-zero probability that Q
J-nk_7 <.. GR tor all n. If
R
0 there will be a non-zero proportion ot systematic models, in
S {:
the mixture constituting the random model, for which this is the case.
Hence, the sequential procedure outlineiabove tor
not conclude with probability one unless
NoW consider the test based on
minate with probability one both when
~.
G")o
~'
~
= O.
This will ter-
0 will
= o.
r
tJ">
$= B
II
and when
& = 0,
and
- 11 -
-
QR and GR for
so can be used as a sequential test.
Some values of
case are given in Tables Ia and Ib.
As might be expected the procedure
is not very practicable for small values of k.
this
For example, in neither
of the cases covered by the tables is it possible to obtain a decision
&...
in favor of H' (
0) with n
~
60 if k ... 2.
In such eases it will
probably be a good scheme to curtail testing at some convenient value
of n and 'accept
8'" 0' if no previous decision has been reached.
7.
ALTERNATIVE PROCEDURES
An alternative method of procedure is to keep n constant and
to decide, at each stage, whether to choose another
take a sample of n from it.
~roup
at random and
This method, which may sometimes be
practicable, has the advantage that it has in general a. probability of
»'
one of concluding even when based on a. non-zero value for
the ratio peG
f r, ") /p{G I (..") vill have the same mathematical
as in (II) it follows that the appropriate values of
given by (12).
I.
Further, in the case when
&' ...
QR'
-
Since
e~pression
GR are those
0 tables such as Table
I may be used in carrying out the test, proceeding along the rows of the
table (i.e., increasing k) instead of down the columns (i.e., increasing
n) •
By analogy with the case of samples of fixed size it would be expeeted that this latter procedure should give, in general, a lower average sample number (ot individuals) than the method based on increasing
n.
It is interesting to note
L-9-7
that, while it 1s impossible
- 12 -
to construct a fixed size sample test using G as criterion if
r0
S'
unless k exceeds a certain minimum value depending on Ct, f3 and
1: " / :" ,
Q
Cl
for a sufficiently large k a fixed sample size test!! available.
This suggests that there should be sequential tests, having the required properties, available for a somewhat wider range of values of k.
8. FURTHER ALTERNATIVE PROCEDURES
Owing to the method of construction of our teats it 1s not
possible to use Wald's approximate formula for average sampling number,
which appHes only when the sequence of random variables used in the
test are mutually
indepe~dent.
Barnard's approach
L-2_7
seems to be
useful in this connection as it shows that the test can be considered
as a limiting case of a series of sequential tests of standard for.m.
However, this approach leads to rather intractable formulae.
It, therefore, seems worthwhile to note that the hypotheses dis•.
cussed in earlier sections can be subjected to sequential tests of
standard type based on sequences of independent random variables.
This
was pointed out to me by A. G. Baker, but the idea has also been used
byO. J. carpenter(Jc.)·j.
In the systematic case such a test can be constructed as follows.
Successive samples of size m(
~
2) are taken from each of the k groups.
A value, g, of G is calculated separately for each such sample of size
km
and the sequence of
indepen~ent
g's so obtained used in a sequential
procedure constructed in the usual manner.
Such a procedure will not, however, lead to independent g's in
the random modeL
If a new set of k groups
is chosen at random each
- 13 time, and an observation taken from each, then the successive gls will be independent:· . ~ we denote the i th .value obtained by gi the procedure will be
(III)
"After the nth set of km observationsj Accept HI: 5 = 0' if
l+m 8 I+S
2A.
1
!: log l+m i: "+g 'lon-l
1=1
1
n
Accept H": j;
= ~"
n
l+m
1:1 loS l+m
8 1 +g i
nk(m-1)
l+m ~
loS l+tIl
+ lon-1
if
2B
5 "+gi ~ kiii:!
nlt(m-1)
l+m G I
+ -lon-l
log l+m GIi
otherwise take a further sa:mp1e of n from each group.
I'
In th1s case, using WaldIe approximate formula for average
sample number, the approximate expected
n~ber
of individuals to be
observed is
mk {A(l-a)+Ba'
Em,k( 6 .)
mk
B(l-t3)+A~
Em,k (8")
where
\)t( S)
=
r-"
L·log
(J
it HI is true
if H" is true
p(g 1£2
1
l+m l\"
p(gTrT 16.. 7 2 k(m-l) log l.;m-tr
.t(lon-l) ~L-l
l+m 0 I+g
v
og l+m o"+g
+ 2
I 0_
\,.' 7
The leading terms of a useful approximate expression for
E¥(
8 ),
Em,k( ~) =
(obtained by the method of statistical differentia.ls) are
~k(m-l)
log
~::
r,'
+
~(lon-l)L-log{RI fR")
3
_ 8m (m-l)k(k_l) (km-2k+l}l{"'-/~•
3 (lon+l) (km+3)
R" 3
.p ..
~x -~
1)317
R13-
- 14 where
R'
= km
- 1 + m(k-l)
S
+ m(m-l}k 8
R"
= km
- 1 + m(k-l)
8
+ m(m-l)k I,.!\. "
Some calculations in the case
when $
=
S'
= 0,
S = 0,
I
I
~\;"
=1
indicate that,
the average sampling number will be minimized by
taking m = 4.
REFERENCES
Wald, A.
~equent1a1Ana1ys!~.
New York: John Wiley,
19 47.
Barnard, G.A. ":The Frequency Justification of Certain sequential
Tests" Biometrika, VoL 39(1952), pp. 144-150.
1-3_7 Cox, D. R.
~. ~.
""Sequential Tests for Composite Hypotheses'"
Soc., Vol. 48(1952), pp. 290~299.
Proe.
Arnold, K. J. Introduction to Tables to Facilitate Sequential
t-tests
National Bureau of-Standards:-A.M.S. 7 (1951)
/-5 7 Ko1odzieczy~, S. '~On an Im'Oortant Class of Statistical Hypotheses"
- -' Biometrika, Vol. 27(1935),
165-190.
pp.
L-6_7
David, F. N. and N. L. Johnson "A Method of Investigating the
Effect of Nonnormality and Heterogeneity of Variance on Tests of
the General Linear Hypothesis" An~la13 of Math. stat. Vol 22(1951},
pp. 382-392.
-- ---- ----
L-7_7
Heu, P.L. "Analysis of Variance from the Power Function Standpoint" Biometrika, Vol. 32(1941) pp. 62-69.
Perron, O. '~Uber das Verhalten einer Ausgearteten Hypergeometrischan
Reihe be:!. unbegrenzten Wachstum eines Parameters" :!.. fur reine~
aUl;. Hath. Vol. 151(1921), pp. 63-78.
L-9_7
Johnson, N.L. "Alternative Systems in the Analysis of Variance"
Vol. 35(19 1+8) pp. 80-89.
Bi0E:~!rika,
Ph.D. Dissertation, Iowa State College
- 15 TABLE la
o II
=1
J
0'
= 0,
0:
= 13 = 0.05
(i) Values of QR
n'\k
2
3
4
5
2
-
3
4
5
- .037
- .066
6
- .080
7
- .015 .086
8
- .026.088
9
- .03; .088
10
- .038 .087
11 - .041 .086
12 - - .04; .084
15 - - .045 .018
20 - .004 .044 .069
;0 - .011 .039 .056
60 - .013 .028 .037
6
.086
.126
.137
.1;8
.135
.131
.126
.121
.117
.112
.101
.086
.067
.042
7
.086
.180
.194
.190
.181
.172
.162
.154
.146
.139
.133
.117
.098
.076
.047
8
.204
.256
.249
.232
.215
.200
.187
.175
.165
.156
.148
.129
.107
.082
.050
9
.;08
.319
.292
.265
.242
.222
.206
.192
.180
.170
.161
.139
.115
.087
.052
10
.398
.372
.;a8
.292
.264
.240
.221
.206
.192
.181
.171
.147
.121
.090
.054
11
.478
.417
.359
.315
.282
.256
.234
.217
.202
.190
.179
.153
.125
.094
.056
12 15 20 ;0 60 00
.549 .720 .:1181.1461.4141.732
.455 .545 .64C .744 ~855 .974
.;85 .444 .;05 .569 .636 .707
.33t~ .378 .423 .469 .517 .566
.297 .331 .:~6 .403 .439 .477
.268 .297 .32~ .355 .385 .416
.245 .269 .29~ .319 .,44 .370
.226 .247 .269 .£90 .312 .335
.211 .~~9 .248 .267 .287 .;06
.197 .2:4 .231 .248 .265 .283
.186 .201 .216 .232 .248 .263
.159 .171 .183 .195 .207 .220
.130 .138 .147 .156 .165 .174
.096 .102 .108 .114 .120 .126
.057 .060 .063 .066 .069 .072
-R
(11) Values of G
\k
2
;
4
5
6
7
8
9
o
1
2
5
o
5· 6
- ~.314
-7J+57 3.962 2.909 2.403
4.6J02.376 1.761~ 1.479 1.315
221.41.440 1J.58 1.014 .926
1.4651.042 .873 .783 .724
1.100 .823 .705 .6 h o .599
.883 .682 095 .545 .514
.739.;85 .516 .477 .451
.6;6 .513 A57 .424 .404
.559 .457 .11.11 .383 .366
.500 .414 .373 .350 .33'
.380 .323 :i$6 .280 .270
.274 .239 .222 .212 .206
.178 .l£o .151 .146 .143
.089 .084 .08.1 .079 .078
2 :
3.
4
7
8
9
12.456 8.377 6.542
2.106 1.9111.773
1.2081.1331.078
.867 .825 .793
.685 .657 .635
.571 .550 .534
.492 .476 .463
.434 .421 .411
.369 .378 .370
.354 .3" .337
.324 .317 .311
.262 .257 .252
.201 .198 .195
.140 .138 .137
.077 .076 .076
10 11 12 15 20 30 60
5.5004.828 4.3593.537 2.918 2.435 2049
1.6711. 591 1.528 1.398 1.279 1.169 1.068
1.0351.001 .974' .915 .866 .806 .738
.768 .748 .731 .696 .662 .629 .597
.618 .604 .593 .569 .545 .522 .499
.522 .511 .503 .485 .468 .450 .433
.454 .446 .439 .425 .411 .397 .384
.403 .396 .391 .;80 .;68 .357 .;46
.;63 .358 .354 .344 .334 .325 .316
.332 .327 .323 .315 .307 .299 .291
.306 .302 .298 .291 .284 .277 .270
.249 .21~6 .244 .239 .234 .229 .224
.193 .l91 .190 ,187 .183 .180 .177
.136 .135 .131~ .133 .131 .129 .127
.076 .075 .075 .074 .074 .073 .073
00
1.732
.974
.707
.566
.477
.416
.370
.335
.306
.283
.263
.220
.174
.126
.072
- 16 TABLE 1b
n\k 2
2
3
4
5
6
7
8
9
10
11
12
15
20
30
60
4
3
6
5
-
-
•
-
-
-
-
-
-
.001
.0'9
.015
.020
.028
.032
.010 .032
.013 .025
-
·
- - -
•
-
·
·
-
.018
.034
.044
.050
.054
.056
.057
.058
.055
.047
.033
0' = 0, () = t3 = 0.01
5 " -- 1 ,
7
.022
.057
.074
.082
.086
.087
.087
.086
.085
.080
.071
.058
.038
(i) Values of QR
8
10 11
9
.057 .168
.030 .104 .167 .221
.088 .143 .189 .228
.110 .153 .188 .218
.117 .152 .181 .206
.119 .149 .173 .193
.118 .143 .164 .182
.115 .138 .156 .171
.112 .132 .1~9 .162
.109 .127 .142 .154
.106 .122 .136 .147
.096 .109 .120 .129
.084 .093 .101 .108
.066 .073 .078 .082
.042 .046 .048 .050
-
(11) Values of
,k 2
3
4
5
6
7
8
~
3
-
57D58 5.470
) 7.645 &548
) 2.8871.675
7 1.964 i.255
1-
i 1.490 1.007
1
)
L
)
5
~~
v
o
1.203
1.009
.870
.765
.564
.394
.249
.120
.843
.727
.640
.572
.436
.315
.208
.104
14.9116.572 4.459 3.499 2852
3.150 2.342 1.9341.6891.524
1.792 1.457 1.263 1.139 1.052
1.2711.070 .950 .870 .814
.990 .852 .767 .710 .669
.814
.694
'.606
.539
.486
.377
.278
.185
.096
.711
.612
.539
.482
.437
.344
.256
.173
.091
.647
.561
.496
.446
.406
.322
.242
.165
.088
.603
.525
.467
.421
.384
.306
.232
.159
.086
.571
.500
.446
.403
.368
.295
.224
.155
.084
12
.216
.269
.261
.243
.226
.210
.196
.184
.174
.164
.156
.136
.113
.086
.052
15
.405
.382
.339
.302
.273
.249
.229
.213
.199
.187
.177
.152
.125
.094
.056
20
.635
.507
.421
.363
.321
.289
.263
.242
.225
.210
.198
.169
.137
.102
.060
30
.909
.646
.510
.427
.371
.330
.298
.272
.2'2
.234
.219
.185
.149
.110
.064
60
1.273
.800
.605
.495
.423
.372
.334
.303
.279
.258
.241
.202
.162
.118
.068
00
1·732
.974
.707
.566
.477
.416
.370
.335
'.,06
.283
.263
.220
.174
.126
.072
GR
9
10
11 12 15 20
30
60
00
47.04419.17512.5339.5495.9994.1573.0322.2751.732
2.599 2.352 2.171 2.031 1.757 1.520 1. 314 1.134 .974
1.407 1.319 1.2511.1961.083 .978 .881' .791 ·.707~·
.988 .939 .900 .868 .801 .738 .678 .620 .566
.766 .738 .711 .690 .644 .600 .558 .517 .477
.638 .613 .594 .578 .543 .510 .478 .446 .416
.546
.480
.429
.389
.356
.286
.219
.152
.083
.527
.464
.416
.378
.347
.279
.214
.149
.082
.512
.452
.406
.369
.339
.274
.210
.147
.081
.500
.442
.397
.361
.332
.269
.207
.145
.080
.472 .446
.419 .398
.~78 .360
.345 .329
.318 .304
.259 .249
.201 .194
.141 .137
.078 .077
.420
.376
.342
.314
.290
.239
.187
.133
.075
.395
.355
.324
.298
.277
.229
.181
.130
.074
.370
.335
.306
.283
.263
.220
.174
.126
.072