RANOOMIZED-SEQUENTIAL GROUP TESTING PROCEDURES
Norman L. Johnson
Samuel Kotz
University of North Carolina
University of Maryland
College Park. MD 20742
O1a.pe I Hi II. NC 27599-3260
Wang Qiwen
University of Maryland
College Park. MD 20742
Key words and Phrases: Dorfman and Dorfman-Sterrett procedures:
errors tn testtng: tnspectton sampltng.
ABSTRACf
Dorfman and Dorfman-Sterrett procedures are extended by
introducing a random element in decisions on when to revert to
group testing after individual testing has commenced.
Properties of the proposed procedures. when inspection errors
are present. are investigated. and some useful approximations
are suggested.
INTRODUCTION
In a series of papers (see. e.g .• Johnson et al. (1985).
(1986). (1988) and the bibliographies therein for an
applications-oriented survey of some of the results) the effects
of inspection errors on a number of procedures for detecting
nonconforming (NC) items among a set of n items have been
studied.
- 1 -
- 2 -
The simplest such scheme is individual testing of each
item.
This requires n tests.
The other procedures have been
designed with the aim of reducing the number of tests required.
If inspection is perfect then all procedures ultimately identify
all the NC items and no others.
no longer so.
If errors occur, then this is
In such cases, in addition to the expected number
of tests required, it is necessary to consider the probabilities
of correct assessment, PC(NC) , PC(C) for nonconforming and
conforming (C) items, respectively.
The procedures so far considered include:
1)
Dorfman group testing.
Dorfman (1943) proposed that (if
possible) a single test be applied to all n items as a group to
determine if there is at least one NC item among them.
Only if
the presence of at least one item is indicated, is individual
testing undertaken.
A set containing at least one NC item will
be termed a NC set; a set with no NC items is a C set.
2)
Hierarchical Dorfman group testing.
divided into hI subsets of n
into
~
h2hl~
sub
2
sets of
= ... ).
~
1
The set of n items is
items each; each subset is divided
items each and so on.
(n
= hlnl =
As in 1) the whole set is tested first, but if
the presence of at least one NC item is indicated, each subset
is tested (rather than immediate recourse to individual
testing). The procedure continues (testing each sub i set in any
i-I
sub
set found to contain NC items, and so on) until the
smallest (subk sets) are reached. Only if the latter give NC
indication is individual testing undertaken.
3)
Dorfman-Sterrett group testing.
Sterrett (1957) modified
Dorfman's procedure by requiring reversion to group testing of
the remaining (untested) items after any item is found NC on an
individual test.
There are various further modifications of
this procedures:
•
(a)
there may be a limit on the number of reversions to
group testing:
(b)
reversions may occur only after k() 1) items are found
to be NC.
Combinations of 1). 2) and/or 3) are also possible.
Combining Dorfman-Sterrett with hierarchal classification we
might consider a procedure in which. as in the Dorfman
procedure.
(i)
the whole set of n items is tested for the presence of
at least one NC item. and
(ii)
if the presence of an NC item is indicated. all items
in the first of h subsets. each of size n • are tested
1
individually:
then. depending on the results obtained in (ii). all items
in the second subset are tested individually or the remaining
(n-n 1 ) items are tested as a group (then proceeding as in (i):
and so on).
The dependence on the results in (ii) might be
deterministic - for example. proceed to individual testing of
the next subset if at least k items (among the n
tested individually) are found to be NC.
in the subset
The Dorfman-Sterrett
procedure is a special case of this type with n 1
and k = 1.
1
=1
(so hI
= n)
(The 'subsets' are. in fact. individual items.)
Since we may find z
= O.I •.... (n1-l)
appear to be NC among the n
1
or n
1
items which
tested in the subset it would seem
reasonable to try to take into account the actual value of z in
a more fine-tuned manner.
This can be done by making the
decision to revert to group testing a random one. with the
probability of such reversion depending on z.
In this paper we will study the effects of errors in
inspection on such procedures.
We will confine ourselves to
- 4 -
"ALL' n IrotS C"
FINAL
DECISIONS
-
i_t - J, _J,_t_
1_"(0 © ... @::.@_@'~
.--
)
<5 NC DECISION~
/.....:.-_
I -PZ!'I
SUBSET (2) OF n
1
1
ITEMS,
\PZ!'I
SUBSETS (2) - (h); (n-n1) ITEMS
GROUPTEsr
~©-"
•
t"Al1 (n-n1) ITEMS C".l
FINAL
DECISIONS
(OF ~ ITEMS)
INDIVIDUAL TESTS
~
FINAL
DECISIONS
Figure 1.
RA.Nro.1-SEQUENTIAL PROCEDURE - FLOW CHART
["FINAL DECISIONS" IN
~arES]
©---+ 1 "AU n1
ITEMS C" •
- 5 -
one-stage hierarchies - that is. with no further subdivision
beyond subsets.
Notation.
Some notation will now be introduced.
In accordance with the practice in earlier papers
(see. e.g .• Kotz and Johnson (t982»
we will use p to denote the
probability that a NC decision is (correctly) obtained when
nonconformity is really present. and p' to denote the
probability that this decision is (incorrectly) obtained when
there is no nonconformity.
Subscripts are used to indicate the
situations to which these symbols apply.
The absence of a subscript indicates a reference to
individual testing: the subscript 0 refers to group (or
subgroups) testing.
It is possible to allow all these
probabilities to depend on the size and/or constitution of the
group (or set of subgroups) tested. at the expense of some
formal complexity. but this will not be done here.
(See also
next Section.)
We denote the number of truly NC items in the set of n
items by Y. with Y • Y •...• Y denoting the numbers in
t
2
h
successive subgroups of size n
t
(nth=n).
Assuming the
independence of tests. the number Zj of items found to be NC
when the j-th subgroup is tested individually is distributed as
the sum of two independent binomial variables with parameters
(Yj'p) and (n l - y j •p ·) - conditional on Yj = Yj (j=t •.... h).
We
have
zj y
n -y
z -i
Y -i
n -y -z +i
T
I =Pr[Z =z ]= ~ ( j)( I j)pi p ' j (I-p) j (I-p') I j j
j j i=O i
zj-i
Zj Yj
(Zj = 0.1. .... n t ).
We will allow the probability of resorting to group testing for
the remaining (n-ntj) untested individuals. given an observed
- 6 -
value, Zj' of Zj to depend on j as well as Zj: we will denote it
by P
j.
Zj'
Then the probability that the remaining (n-n1j) items
are subjected to a group test, given that the j-th subgroup is
tested individually is
n
"Y
I
j Yj
=
1
1
::0
Zj-
11'
I
z.J Yj
P
Zj , j
(When no confusion can be caused, the symbol
"Y
j
will be used.)
The values {Pz, j} can be chosen arbitrarily.
constitute the inspection strategy.
They
It is to be hoped that,
by a suitable choice of inspection strategy, good procedures
(that is, economical in testing and reasonably robust to
inspection errors) - can be obtained.
Note that if Pz, j::O for
all Z and j we have the original Dorfman procedure.
It will be assumed that the group of n items has been
chosen at random from a large (effectively infinite) population.
The proportion of NC items in this population will be denoted by
w.
Figure 1 is a flow chart of the procedure.
Analytical
properties will now be discussed.
ANALYSIS
As already mentioned, in this paper we suppose the
probabilities of decision for any group test do not depend on
the size of the set tested.
The probabilities of NC decision
will be denoted by PO (for group testing) and p (for individual
testing) for truly NC items or (sub)sets: for truly C items or
groups the symbols will be
Po and p'
respectively.
We also
suppose that the total number of items is a multiple of the
subset size, so that n=hn .
1
•
- 7 A further assumption is that results of tests are mutually
independent.
We first make calculations conditioned on fixed values
y1 •...• yh of the actual numbers of NC items in the 1st •...• h-th
subsets.
The resulting expressions then need to be averaged
over the joint distribution of y1 •...• yh. with appropriate
weights.
We also introduce the symbols
tj
= Yj+1
+ ... + Yh
and
if
if
t
j
=0
(j = O. 1 •...• h-1 ) ;
here t. is the number of NC items among the last (h-j) subsets.
J
and ~. is the probability that a NC decision will be obtained on
J
testing them as a group.
(Note that to corresponds to the total
number of NC items.)
Given that the j-th subset is tested individually. the
probability that the (j+1)-th subset. also. is tested
individually is
9
j
= Pr[last
(h-j) subsets not tested as a group]
+ Pr[last (h-j) subsets tested as a group. and NC decision
obtained]
=1
- ~j + ~j ~j
(j
= 1 •...• h-1)
.
The probability that the first subset is tested individually is
90 =
~o'
The probability that the j-th subset is tested individually is
j-1
n 9 i • (j = 2 •...• h).
i=O
- 8 -
When the j-th subset is tested individually. there will
certainly be n
I
individual tests. plus. with probability
group test of the remaining (h-j) subsets.
number of tests. given
E
l
=1 +
l.
h
I
~j'
a
Hence the expected
is
j-I
{n
j=I
i=O
a i } (n I +
~j)
with
~h
= O.
(I)
(The '1' on the right hand side of (I) corresponds to the
initial group test of all n items.)
= (yI •...• yh}):
For any given NC item (and given l
pc(NCll)
=
h
I
Pr[NC is in j-th subset] Pr[j-th subset is tested
j=I
individually] p
For any given C item:
PC(Cll}
=
h
I
Pr[C is in j-th subset] {Pr[j-th subset not tested
j=I
individually]
+ Pr[j-th subset tested individually] (I-p'}}
=
h In -y
I
j=I
n~t
j
{I -
0
-1
= 1 - p'(n-tO}
j-I
n ai
i=O
h
I
j=I
+
j-I
(n ai)(I-p'}}
]
i=O
{(n I - y j )(
j-I
n
ail}
(3)
i=O
In order to obtain the unconditioned values. (I). (2) and
(3) must be averaged over all l with appropriate weights.
The probability that! = l (i.e.
Yj=Yj; for j=I •...•h) is
- 9 -
P
l
=
h
n
y
n -y
H (1) w j(l-w) 1 j
j=1
h
= {H
j=1
Yj
n
t
n-t
(I)} w O(I-w)
O.
(4)
Yj
For the expected number of tests. we have
E=!P E.
l l l
(5)
The expected percentage reduction (as compared with simple
individual inspection) in number of tests is
EPR=I00(I-Eln)
For probability of correct classification of NC item.
(5' )
(since pc(NCll) applies to each of to NC items):
PC(NC) = ! to P pc(NCll)/(! to P )
l
l
l
l
(6)
For probability of correct classification of a C item
(since PC(Cll) applies to each of (n-t O) C items)
PC(C) = ! (n-tO)P PC(Cll)/{!(n-tO)P}
l
l
l
l
= ! (n-tO)P PC(Cll)/{n(l-w)}.
l
l
(7)
Tables I and II (extracted from a considerably more
extensive set. available from the second author) provide some
representative values of EPR. PC(C) and PC(NC) for n = 12, a few
combinations of values of the parameters n 1 , PO' PO' p, p' and
Pz, j. Although it is necessary, in any specific case, to take
into account details specific thereto, it is possible to make
- 10 -
some useful general comments.
Even without studying specific numerical values. it can be
seen that
PC(C) ~ 1 - p'
and (11) PC(NC) ~ poP.
(i)
This is because (i) an item can be declared C either on
individual testing Q[ as a result of a group test. and (ii) to
be declared NC an item must be tested individually.
Conversely. any item which is not tested individually mY!!
have been declared C.
We would expect. therefore. that any
procedure that increases the amount of group testing will tend
to increase PC(C). but. unfortunately. to decrease PC(NC).
Of course. numerical values assist in assessing the
relative importance of these two effects. though it is also
desirable to have some idea of relative costs (and also sampling
costs. when taking EPR into consideration).
From Tables I and II. we see that
(a)
EPR is not greatly affected by inspection strategy
(b)
PC(NC) is affected more substantially by inspection
strategy.
P
j
There is an adverse effect when strategies with
decreasing with z are used - more specifically when
z.
PO. j is higb.
(c)
EPR increases as w decreases.
~t
If
is desired to optimize only one of the values PC(C)
and PC(NC) a deterministic strategy. with each P
z.
either 0 or 1.
j
will be nearly optimal. but when some compromise
is needed a probabilistic strategy may be
advan~eous.
We now present studies of a few specific cases.
= 12.
= 2 and inspection strategy
PO• j = 0; P1 • = 1-(0.5)J; P2 • ' =
J
J
n
equal to
We take n
1
1
(J=I •...• h-l)
Table I:
z
Po pI0
.99
.95
.95
.90
.80
.99
.95
.95
.90
.80
a
.01
.05
.10
.10
.20
.01
.05
•10
.10
.20
2
P
pI
.99
•95
.95
.90
.80
.99
.95
.95
.90
.80
h
.01
.05
.05
.10
.20
EPR PC(C) PC(NC)
w=O.Ol
80.7 .9993 .9566
80.8 .9964 .8053
79.3 .9959 .8053
80.4 .9924 .6563
78.2 .9807 .4484
EPR PC(C)
81.1
81.1
79.7
80.7
78.6
.01
.05
.05
.10
.20
w=0.05
46.4 .9970 .9579
50.3 .9861 .8103
49.2 .9856 .8103
54.2 .9742 .6632
58.8 .9527 .4549
=6
0.00 .
1 - (0.50)J
1.00
1.00
0.50
0.00
f1.00 .
Pz j = 1(0.50)J
•
0.00
=1
" = 12'• "1 = 2·•
0.00
0.50
1.00
PC(NC)
EPR PC(C)
PC(NC)
EPR PC(C)
.9993
.9966
.9961
.9928
.9818
.9564
.8028
.8028
.6495
.4337
82.3 .9992
79.4 .9947
75.3 .9927
76.7 .9871
72.6 .9677
.9794
.8930
.8930
.7798
.5569
81.4 .9991 .9796
78.6 .9942 .8959
74.4 .9921 .8959
75.7 .9857 .7884
71.1 .9637 .5776
47.7 .9972
51.4 .9870
50.2 .9865
51.1 .9758
59.8 .9562
.9573
.8064
.8064
.6546
.4387
54.8 .9970 .9780
54.0 .9846 .0874
51.0 .9831 .8874
53.7 .9692 .7715
54.3 .9410 .5487
52.1 .9966 .9787
51.6 .9827 .8918
48.7 .9812 .8918
51.5 .9656 .7822
52.7 .9341 .5708
19.5
26.3
25.4
33.6
44.3
.9585
.8109
.8109
.6610
.4450
31.5
32.4
30.3
34.5
40.4
28.0
29.5
27.7
32.0
38.1
PC(NC)
w=0.10
.99
.95
.95
.90
.80
.01
.05
.10
.10
.20
e
.99
•95
.95
.90
.80
.01
.05
.05
.10
.20
18.1
25.1
24.2
32.3
42.9
.9949 .9596
.9768 .8164
.9765 .8164
.9579 .6718
.9282 .4631
.9953
.9783
.9779
.9606
.9335
e
.9951
.9762
.9750
.9542
.9189
.9764
.8807
.8807
.7615
.5389
.9945 .9775
.9733 .8868
.9723 .8868
.9491 .7744
.9100 .5623
e
Table I (continued):
roOD
0
z =1
2
Po Po
.99
.95
.95
.90
.80
.01
.05
.10
.10
.20
_
_
_
P
P'
.99
.95
.95
.90
.80
.01
.05
.05
.10
.20
.99
.95
.95
.90
.80
.01
.05
.10
.10
.20
.99
.95
.95
.90
.80
.01
.05
.05
.10
.20
.99
.95
.95
.90
.80
.01
.05
.10
.10
.20
.99
.95
.95
.90
.80
.01
.05
.05
.10
.20
w=O.Ol
(0.80)
EPR PC(C) PC(NC)
81.4
81.4
80.0
81.2
79.7
00=0.05
48.7
52.3
51.2
56.2
61.7
00=0.10
20.3
27.2
26.3
34.8
46.7
=2 ;
81.6 .9994
81.6 .9969
80.2 .9965
81.2 .9936
79.7 .9847
.9973 .9565
.9877 .8008
.9873 .8008
.9776 .6414
.9612 .4104
.9570
.8026
.8026
.6439
.4126
h=6
0.00
0.25
1.00
0.00 j
1-(0.75)
1.00
EPR PC(C) PC(NC)
.9994 .9561
.9968 .• 7993
.9964 .7993
.9934 .6394
.9843 .4086
.9955
.9794
.9791
.9633
.9409
n = 12; n1
1.00
0.90
0.80
Pz . = (0•90) jj
,J
-
EPR PC(C) PC(NC)
EPR PC(C) PC(NC)
.9560
.7981
.7981
.6354
.4012
81.2
78.4
74.2
75.4
70.5
.9991 .9797
.9941 .8964
.9920 .8964
.9855 .7901
.9627 .5819
80.5 .9990 .9798
77.7 .9936 .8990
73.3 .9914 .8990
74.4 .9841 .7980
68.9 .9582 .6015
49.3
52.8
51. 7
56.6
62.2
.9975 .9562
.9882 .7986
.9878 .7986
.9784 .6369
.9627 .4021
51.6
51.3
48.3
51.2
52.2
.9966 .9788
.9826 .8927
.9811 .8927
.9654 .7842
.9333 .6459
49.1 .9962 .9793
48.9 .9808 .8966
46.1 .9793 .8966
49.0 .9617 .7940
49.9 .9257 .5964
21.2
27.9
27.0
35.4
47.3
.9957
.9803
.9799
.9648
.9433
27.8 .9945
29.4 .9734
27.5 .9724
31.7 .9454
37.6 .9095
.9564
.7997
.7997
.6381
.4032
.9977
.8880
.8880
.7769
.5675
24.7
26.6
24.8
29.2
35.2
.9939
.9706
.9697
.9440
.8999
.9786
.8935
.8935
.7888
.5898
TABLE II " = 12;
"1=3; h ='4
z
Po
0.95
0.95
0.90
0.80
III
0
1
2
3
pI
0
0.05
0.10
0.10
0.20
PZ1j
P
pI
0.95
0.95
0.90
0.80
0.05
0.05
0.10
0.20
III
.
roo
1.00 j
t
(2/3).j
(1/3)J
0.00
EPR PC(C)
51.2
49.7
53.3
55.7
w = 0.05
.9848
.9842
.9708
.9431
0.00 j
1 - (2/3).
1 - O/3)J
1.00
9lj
0•.7) j
0.5)
PC(NC)
.8452
.8452
.7164
.5171
EPR PC(C)
53.0
51.5
55.2
58.2
PC(NC)
.9862 .8401
.9856 .8401
.9726 .7040
.9502 .4902
EPR PC(C)
50.7
47.8
50.7
51.3
.9822
.9807
.9644
.9305
0.00 j
1-
!0.9l'
1 - 0.7 ~
1 - 0.5)J
PC(NC)
.8938
.8938
.7874
.5848
EPR PC(C)
47.4
44.7
47.3
47.4
PC(NC)
.9797 .8993
.9783 .8993
.9593 .8012
.9187 .6164
"1=4; h = 3
z
p"
0
0.95
0.95
0.90
0.80
:=
0
1
2
3
4
pI
0
0.05
0.10
0.10
0.20
e
r'
OO
1.00 j
pz1j •
P
pI
0.95
0.95
0.90
0.80
0.05
0.05
0.10
0.20
(o.soljj
(0.25)
0.00
EPR PC(C)
51.4
49.7
52.6
53.7
.9840
.9833
.9685
.9367
0.00 j
1 - (0.75).
1 - (0.50)~
1 - (0.25)J
1.00
~0.9)j
(0.15 j
O.O)j
(0.7)j
,(0.6)
PC(NC)
.8632
.8632
.7446
.5513
EPR PC(C)
52.8
51.0
54.1
55.7
.9851
.9843
.9708
.9427
e
PC(NC)
.8603
.0603
.7371
.5335
EPR PC(C)
0.00 j
1-(0.9~.
1 - (0.8 ~
1 - (0.7)~
1 - (0.6)J
~C(NC)
48.8 .9807 .8973
45.9 .9792 .8973
48.7 .9612 .7963
49.0 .9231 .6056
EPR PC(C)
PC(NC)
46.8 .9792 .9003
44.0 .9778 .9003
46.5 .9580 .0042
46.2 .9150 .6251
e
-14with two sets of decision probabilities
(a)
p
(b)
p
o
o
= p = 0.98; p'0 = p' = 0.05;
= 0.9; p = 0.95; p'0 = p' = 0.05.
Values for the Dorfman procedure are also shown the following
table
Dorfman
Random-Sequential
PC(NC)
EPR
PC(C)
PC(NC)
.9539
.9114
.9656
.9452
43.9
20.9
0.0
.9604
.9604
.9604
.8268
.8143
.1913
41.8
25.1
1.5
.9192
Set Ca)
EPR
0.05
0.10
0.20
52.8
30.25
4.25
.9843
.9154
.9509
.9658
56.2
36.4
15.1
.9855
PC(C)
.9556
Set (b)
0.05
0.10
0.20
.9111
.9691
.9683
.9581
.8550
.8550
.8550
In this ~rticular cOmParison it can be seen that the
random-sequential procedure requires substantially less
inspection, on average, and increases PC(C) as compared with the
Dorfman procedure.
However, it leads to a decrease in PC(NC)
which might be of greater importance if the nonconformity is of
serious nature.
Note that the lower values of p and p in (b)
o
result, as is to be expected. in lower probabilities of
detecting NC items by either procedure.
This effect is of
greater magnitude then that resulting from differences between
the procedures.
For large values of PO and p, (such as PO,p
0.98. say). however, the decrease in PC(NC) is slight.
ADDENDA
(I)
We plan to study the effects of departures from our
assumptions on the properties of randomized-sequential
procedures.
In an initial inquiry, we have supposed that the
probability (PO) of a NC decision for group test of the last
~
- 15 (h-j) subsets containing at least one truly NC item is
Poo - k x
(proportion of C items)
= Poo - k[1 - t j /{{h- j )n 1}]
(t j
> 0)
(8)
(j=O.1 •...• h-1)
with k=O.1 for illustration.
(see Hwang (1976».
[If
This reflects a dilution effect
~j=O
we still have arbitrary Po for
probability of a NC decision.]
Table III shows values obtained when n=12. n =2. h=6;
1
POO=O.9S. PO
=O·05. p=O.9S. p'=O.05. for four inspection
strategies and three values (0.01. 0.05. 0.10) of w.
Values obtained with fixed values. 0.88 and 0.90. of Po are
also included for purposes of comparison. to assess the accuracy
of certain rough approximations.
It seems reasonable that a
fixed "equivalent value" for PO' equal to a rough average of (8)
might give useful approximations.
The expected value of
Poo - 0.1[1 - T /{{h-j)n }]
j
depends on j (as well as w).
always
1
Since the initial test (j=O) will
be used we consider using the corresponding expected
value of TO'
of E[TOITO
Even with this restriction. the appropriate values
> 0]
depend on which of EPR. PC{C) and PC{NC) is
being approximated.
For EPR.
E[TOITO
> 0]
E[TOITO
for PC{C).
for PC{NC).
n -1
= nw{1 - (1-w)}
> 0]
= (n-l)w{I-{1-w)
(9.1)
;
n-l -1
} ;
E[TO] = 1 + {n-l)w.
(9.2)
(9.3)
(TO is necessarily greater than zero in the last case.)
These three values do not vary much if w is small and n is not
too small
p " is
o
•
(~3.
say).
In the last case the suggested "equivalent
- 16 -1
Poo - O.I[I-n {I + (n-l)w}]
-1
= Poo
- O.I(I-n
)(I-w).
(10)
For the parameter values used in Table III, the "equivalent PO"
values from (10) are
0.889,
for
w
= 0.01,
0.893 and
0.8975
0.05 and 0.10
respectively.
The approximations are qUite good, but it appears that somewhat
Po
greater (by about 0.007) values for
would give even better
results.
For given w, the smaller the size of group ('n' or (h-j)n )
1
the larger the "equivalent PO". Since smaller groups are used
in the reversions to group testing (in the course of individual
testing) one would expect values greater than those given by
(10) to produce some improvement in approximation.
Taking an
average value (between 12 and the minimum possible, 2) of 7 for
'n' in (10) produces "equivalent PO" values of
0.895,
for w = 0.01,
0.899 and
0.05 and
0.902
0.10
respectively.
Very nearly the same values for approximating values of EPR and
PC(C) are obtained using (9.1) and (9.2), namely:
EPR:
PC(C):
0.895,
0.897
0.895,
for w = 0.01,
(II)
and
0.899
0.896 and
0.05 and
0.898
0.10
respectively.
The analysis of this paper may be extended to situations
in which there are more than one kind (g, say) of
nonconformities.
SPecial consideration has to be given to the
inspection strategy.
In general, the probability of proceeding
to a group test of the remaining subsets (not yet tested
indiVidually) might depend on the vector z of numbers of items
""
with each of the possible ~ combinations of nonconformities.
It might also be necessary to allow for possible
differences in probabilities of correct classification for
•
Table III.
n
(*'Variabler. denotes
Po
z =0
1
2
w
Po
Pz.J.
=
= 12;
n1
= 2;.
h
= 6;
p&
= 0.05;
p
= 0.98;
p
= 0.05
value is 0.98-0.lx(proportion of C items) in group testing)
o .
0
0
1.0
0
1.0
1.0
1-(0.5)J
1.0
EPR PC(C) PC(NC)
EPR PC(C) PC(NC)
EPR PC(C)
1.0
1.0
1.0
PC(NC)
EPR PC(C) PC(NC)
0.01
0.88
0.90
*Variable
77.4 .9933
77.2 - .9932
77.3 .9932
.8615
.8812
.8707
80.1 .9951 .8407
79.9 .9950 .8635
80.0 .9950 .8530
81.1
80.9
81.0
.9956
.9955
.9956
.8332
.8570
.8461
83.2
82.8
83.0
.9975 .6415
.9973 .6888
.9974 .6754
0.05
0.88
0.90
*Variable
49.4 .9802 .8600
48.5 .9798 .8799
48.9 .9800 .•8730
57.3 .9860 .8278
56.4 .9857 .8524
56.7 .9858 .8462
59.7
58.9
59.2
.9876
.9873
.9875
.8161
.8425
.8359
59.9
58.1
58.6
.9904
.9899
.9901
.6415
.6888
.6840
0.10
0.88
0.90
*Variable
28.6
27.1
27.5
38.2
36.7
37.0
41.0 .9807
39.4 .9801
39.7 .9802
.7965
.8255
.8255
39.7
36.7
37.0
.9841
.9832
.9834
.6415
.6888
.6947
e
.9702
.9695
.9697
.8571
.8775
.8752
.9785 .8125
.9779 .8392
.9781 .8393
e
.e
- 18 -
different combinations of nonconformities.
Also. one would need
formulas for correct detection of each possible combination of
nonconformities.
The analysis would follow the same lines as in
this paper. but l would now be a ~xh matrix showing numbers of
each possible combination of nonconformities in each of the h
subsets.
~ombining
The weights for
the conditional
probabilities of correct assignment would be proportional to the
product of P and the number of items with the appropriate
l
combination according to l.
The weights for the expected number
of tests would still be P .
l
REFERENCES
OORFMAN. R. (1943)
populations.
The detection of defective members of large
Ann. Math. Statist .• ........
14. 436-440.
HWANG. F.K. (1916) Group testing with a dilution effect.
Biometrika. 63.
........ 611-613 .
JOHNSON. N.L.. KOTZ. S. and RODRIGUEZ. R.N. (1985. 1986. 1988)
Statistical effects of imperfect inspection sampling: I.
II. III. J. Qual. Technol .• 11. 1-31: 18. 116-138: 20.
98-124.
........................
KOTZ. S. and JOHNSON. N.L. (1982) Errors in inspection and
grading: Distributional aspects of screening and
hierarchal screening. Commun. Statist. - Theor. Meth .• 11.
1991-2016.
........
STERRETT._ A. (1951) On the detection of defective members of
large populations. Ann. Math. Statist .• ........
28. 1033-1036 .
•