Download Huang, Qishen, Johnson, Norman L. and Kotz, Samuel; (1988).Modified Dorfman-Sterrett Screening (Group Testing) Procedures and the Effects of Faulty Inspection."

MODIFIED OORFMAN-STERRETI SCREENING (GROUP TESfING) PROCEDURES
AND TIIE EFFECTS OF FAULlY INSPECfION
Qishen Huang
University of Maryland
College Park
Norman L. Johnson
University of North Carolina
at Chapel Hill
Samuel Kotz
University of Maryland
College Park
Abstract
Continuing previous work on effects of errors in inspection on
group sampling schemes, a modification of Dorfman-Sterrett schemes is
studied.
The modification consists of reversion to group sampling when
a specified number of decisions of nonconformance have occurred in the
course of inspection of individual items.
KEY WORDS:
Average sample number; inspection errors; quality control;
screening
1.
INTRODUCfION
Dorfman (1943) screening procedures are schemes designed to reduce
the expected amount of testing needed to identify nonconforming (NC)
items in a lot.
They can be applied when it is possible to apply a
single test to a group of n items, that will indicate whether there is
at least one NC item present; for example, the items might be bottles
of a liquid product; samples from a bottles can be mixed and the
mixture tested for presence of a contaminant.
The most prominent
application has been in blood testing, which has been extensively
covered in the work of M. Sobel (Sobel (1968) and the references
thereon) .
If there is no indication of presence of a NC item all n items in
the group are declared to be conforming (C).
If an NC item is
indicated, each item in the group is tested individually.
The total
number of tests needed is either 1 or (n+1). as compared with n tests
if plain individual testing is used.
If the proportion (w) of NC items
in the lot is small, the probability that only 1 test is needed is
high, with consequent reduction in the expected number of tests.
Assuming that lot size is large enough for a binomial distribution to
be used for the number (Y) of NC items in the group of n items, the
reduction would be from n tests to 1+n{1-(1-w)n}
~ 1+n2w, provided
there are no inspection errors.
Sterrett (1957) introduced a modification, aimed at effecting
further reduction in the expected number of tests.
This modification
calls for reversion to group testing whenever an item is declared NC on
individual test.
Thus, if the M-th item so tested is the first one
- 2 -
declared NC, and M ~ n-2, Dorfman screening is applied to the remaining
(as yet untested) (n-M) items.
In the original proposal, there was no
limit to the number of times this could occur - whenever an individual
test gives a NC decision and there are at least two items not yet
tested indiVidually.
Practical considerations may limit the number of
reversions allowed; if up to k reversions are permitted, we have a
k-stage Dorfman-Sterrett procedure.
A further modification, which will be studied in the present
paper, requires that reversion to Dorfman screening is not undertaken
until the g-th NC declaration (g
~
2) occurs.
This modification might
be expected to be more effective than Sterrett's when w is somewhat
higher, (though still quite low) because g NC items have been removed,
making it less likely that any remain among the items again subjected
to Dorfman screening.
Effects of inspection errors on Dorfman screening procedures have
been described by Johnson, Kotz and Rodriguez (1988); and effects on land 2-stage Dorfman-Sterrett procedures have been described by the same
authors (1987 - henceforth referred to as JKR).
In the present paper
we study effects of inspection errors on modified Dorfman-Sterrett
procedures, restructing our analysis to I-stage procedures.
2.
NOTATION
As in earlier studies, we wish to evaluate three measures of
performance:
PC(C):
probability of correct classification of C items;
PC(NC): probability of correct classification of NC items;
•
- 3 -
E:
expected total number of tests (group and individual).
These will depend on w and n.
a subscript.
As in
To indicate dependence on n we add
JKR, we will present tables of values, PC{C) ny
I '
PC{NC) n Iy and En Iy of these three quantities conditional on fixed
values of the number (Y=y) of NC items among the n items in the group.
To obtain overall values, these values must be averaged with respect to
the appropriate distributions of Y.
If lot sizes are large
(effectively infinite) these are
For PC{C) ny
I:
binomial with parameters (n-l,w).
For PC{NC) ny
I : 1 + [binomial with parameters (n-l,w)].
For Eny
I :
binomial with parameters (n,w).
(For finite lot sizes, appropriate hypergeometric distributions are
relevant.)
As in
JKR, also, we first consider situations conditional not only
on Y=y, but also on the order (M=m) in the individual test sequence at
which the second NC declaration occurs, and the number (T=t) of truly
NC items among these first M items tested.
To evaluate PC{C) ny
I and PC{NC) ny
I we will use the relations
PC{C)nIY = E{C)nly/{n-y ); PC{NC)nIY = E{NC)nly/Y
I ' E{NC) ny
I denote the expected numbers of C, NC items,
where E{C) ny
respectively, that are identified correctly.
To indicate conditioning
on M=m, T=t and J=j in addition to Y=y, the set of subscripts nlm,t,y,j
wi 11 be used.
A group containing at least one NC item will be called a NC group;
if a group is not NC, it is a C group.
Inspection quality will be defined in terms of
- 4 -
PO(po):
probability that a NC(C) group is declared NC (assumed to
be the same whatever the value of n
pep' ):
~
2).
probability that a NC(C) item is declared NC when tested
individually.
The event (M=m) n (T=t) can be split into subevents according to
the number of truly C items among those declared NC.
this number by J.
We will denote
(The order in which these events occur is not
relevant.)
We define
P(m,t,jln,y) = probability that M=m, T=t and J=j, given Y=y.
Also
P(m,tln,y) =
l P(m,t,jln,y) = probability that M=m and T=t, given Y=y.
j
(Limits for j are given in (1), below.)
It is assumed that the result of any test (group or individual) is
independent of the result(s) of any other testes).
3.
ANALYSIS
For y
>0
P(m,t,jln,y) = Pr[group test gives NC result]
x Pr[t NC's and (m-t) C's in first n items tested) n «g-j) NC's
and j C's declared NC)]
x Pr[g-th item declared NC is m-th item tested, given g NC
decisions in first m test]
= PO x (~)-1 (~=~)(~)(g~j)(m~t)pg-j(l_P)t-g+jp·j(l_p,)m-t-j
x (gm-1)
(max(O,y-n+m) ~ t ~ min(y,m);
max(O,g-t)
~
j
~
min(g,m-t»
(1)
- 5 -
If y=O, Po is replaced by Po and only j::g is relevant.
If y=n, only
j=O is relevant.
For 2
~
m
~
n-2
E(C) n Im, t ,y, j = m-t-j + (n-y-m+t){I-h(t,y)p'}
E(NC) n Im, t ,y, j = g-j + (y-t)p_D
lr
En Im, t ,y = m+2 + (n-m)h(t,y)
JPO
where
h(t,y) = lpo
if t
<y
if t = y'
(Note that En I
' does not depend on j.)
m, t
,y,J
Also
E(C) n In-l, t,y,j = n-y-j-(t+l-y)p'
E(C) n In,y,y,j = n-y-j
E(NC) n In -1 , t ,y, j = g-j + (y-t)p
E(NC) n In,y,y, j
= g-j
En In-l,t,y = En In,y,y = n+l
and
In calculating E(C) ny
I ' E(NC) ny
I and Eny
I we have to take into
account
(a) cases when fewer than g NC declarations occur and also (b) cases
when group-testing leads to immediate declaration that all items in the
group are C.
We have (for limits for t and j see (1»:
E(C) n Iy =
n
~ L\ L\ P(m,t,jln,y)E(C) n I
' +
m, t
,y,J
m::g t j
+ p
g-1
h j
h j .
.
(n-y-j)( y )(n-y)p - (l-p)y- + p,J(I_p,)n-y- J
o h=O j=O
h-j
j
~
h
~
+(n-y){I-h(n-y,n)}
- 6 -
n
E(NC) n Iy = !
\ \ P(m,t,jln,y)E(NC) n Im, t ,y, j +
LL
rn--g t j
For y
•
>0
n
En Iy = \L L
\ P(m,tln,y)En Im, t ,y
rn--g t
g-1
+ npo
h
2 2 (h:j)(n-~-Y)ph-j(l_p)Y-h+jp·j(l_p·)n-2-y-j + I-po
h=O j=O
•
4.
TABLES
Tables are presented here for the case g=2 only.
Further tables
are in preparation for a later paper, in which the question of choice
of value of g will be addressed, among other topics.
Values of E6 1y' PC(NC)6IY = E(NC)6IY/y and
E(C)6Iy/(6-y)
3 respectively.
PC(C)6IY =
for all relevant values of yare given in Tables 1,2 and
They are comparable with values in Tables 1-6 of JKR
according to the scheme.
Table
Tables in .JKR
1
(E
)
6Iy
1
(I E6Iy) ;
2
(PC(NC)6IY)
2
(1 PC(NC)6IY) ; 5
(2PC (NC)6IY)
3
(PC(C)6IY)
3
(I PC (C)6IY;
(2PC (NC)6IY)
4
6
(2E6Iy)
- 7 -
In the
JKR
tables. the prefix (1 or 2) refers to the number of
stages in the (nonmodified) Dorfman-Sterrett procedure.
For example. with pO=p--o.75 and
Y E61y
E
1 61y
E
2 61y
1.59
5.36
5.30
5.56
5.63
5.62
5.69
1.52
4.78
5.44
5.47
5.44
5.40
5.37
1.52
4.56
5.31
5.54
5.51
5.43
5.36
0
1
2
3
4
5
6
PC(NC)6IY
.533
.500
.538
.479
.467
.459
.433
.514
.497
.485
2 PC (NC)6IY
1PC (NC)6IY
.612
.570
PO=p'=O.10. we find
.531
.491
.456
.429
.409
PC(C)6IY
1PC (C)6IY
.987
.924
.938
.935
.934
.935
.992
.949
.938
.938
.939
.940
.395
The expected numbers of tests are somewhat greater for the
modified procedure. but the probabilities of correct decision for NC
items (PC(NC»
are substantially greater.
There is a relatively slight
decrease in the probability of correct decision for C items. especially
by comparison with the 2-stage unmodified Dorfman Sterrett procedure.
(However. comparison with 2-stage procedures is perhaps unwarranted.)
The somewhat irregular progression of values of E
and PC(C)6IY
61y
with respect to y should be noted.
2PC(C), but PC(NC)
Note also that PC(C)
> 1PC(NC) > 2PC(NC).
< 1PC(C) <
These features are associated
with increasing use of group tests. leading to increased chances of C
decision without individual testing.
Tables 1A. 2A and 3A give values for n=8. but only for y
~
6.
These exhibit similar patterns to those for n=6.
5.
A SPECIAL CASE
For the case of lots containing a large (effectively infinite)
number of items. it is possible to evaluate PC(NC) directly. in an
2 PC (C)6
.994
.951
.947
.945
.946
.947
- 8 -
elegant, elementary fashion, without finding PC(NC) ny
I first.
This
approach exploits the fact that, given a particular NC item, d, say in
•
a group of n items, the probability of proceeding to individual
inspection as a result of a 'NC group' decision is PO' whatever the
status of the other (n-l) items.
Hence, we can evaluate performance on
individual testing by regarding the number of NC decisions on the other
(n-l) items as a binomial variable with parameters (n-l) and w =
wp+(I-w)p' (the probability that a randomly chosen item will receive a
NC decision on individual test).
The probability that there would be h NC decisions among the other
(n-l) items, and a (correct) NC decision for d, supposing each item is
tested individually is
n-l
h
n-l-h
p( h ) w (l-w)
= Ph'
•
say.
Given this event, the conditional probability that d will actually be
declared NC is
1
g
if h
g
g+1 + n(n-l) +
{
g
1
g+1
g
Lp
<g
if h=g
n(n-l)f 0
g
g
h+l + (1 - h+l)PO
if h
>g
Note that the correct decision is reached without reversion to group
testing if (a)
~
is among the first g NC decisions 2r (b) with h=g, it
is the last item tested and the immediately preceeding item is NC
(necessarily the g-th NC).
The probability of (b) is
n-2) / {(n-l)}
( g-1
n g
_g_
= n(n-l)
- 9 -
Hence
This method cannot be applied to evaluate PC(C). because. given a
particular C item. the probability of proceeding to individual testing
does depend on the status of the other (n-l) items in the set.
Given
that individual testing starts. therefore. these (n-l) items cannot be
regarded as randomly chosen from the lot.
ACKNOWLEDGMENTS
The authors acknowledge stimulating discussions with Mr. Luo Zhen
and Mr. Wang Xi Bing of the Department of Mathematics. Nankai
University. Tianjin. China.
REFERENCES
Dorfman. R. (1943) "The detection of defective members of large populations,"
Ann. Math. Statist .• 14. 436-440.
Johnson. N.L.. Kotz. S. and Rodriguez. R.N. (1987) "Dorfman-Sterrett (group
testing) schemes and the effects of faul ty inspection." Inst. Statist.
Univ. North Carolina. Mimeo Series No. 1722.
Johnson. N.L.. Kotz. S. and Rodriguez. R.N. (1988) "Statistical effects of
imperfect inspection sampling. III" • .I. Qual. Technol.. 20. (to appear).
Sobel, M. (1968) "Binomial and hypergeometric group testing" Stud. Scient.
Math. Hungarica. 3. 19-42.
'"
Sterrett. A. (1957) "On the detection of defective members of large
populations." Ann. Math. Statist .• 28. 1033-1036.
- 10 -
TABLE 1 VALUES OF E 1y
6
0.75
PO=p=
0.90
0.95
•
Y
PO=P'= 0.25
0.10
0.05
0.25
0.10
0.05
0.25
0.10
0.05
0
1
2
3
4
5
6
2.43
5.29
5.41
5.59
5.62
5.62
5.59
1.59
5.36
5.30
5.56
5.63
5.62
5.59
1.30
5.42
5.26
5.55
5.63
5.63
5.59
2.43
6.15
6.42
6.88
6.97
6.97
6.96
1.59
6.21
6.14
6.83
6.96
6.97
6.96
1.30
6.29
6.03
6.80
6.96
6.98
6.96
2.43
6.44
6.77
7.38
7.47
7.47
7.46
1.59
6.49
6.41
7.32
7.47
7.48
7.46
1.30
6.57
6.28
7.29
7.47
7.48
7.46
TABLE 2
Y
1
2
3
4
5
6
PO=P=
PO=P'=
VALUES OF PC(NC)6IY
0.90
0.75
0.10
0.05
0.25
0.10
0.05
0.25
0.10
0.05
.513
.612
.570
.538
.514
.497
.485
.672
.711
.784
.779
.770
.764
.759
.778
.802
.787
.773
.765
.759
.828
.810
.790
.774
.765
.759
.785
.887
.884
.879
.876
.873
.836
.896
.889
.880
.876
.873
.883
.900
.890
.881
.876
.873
.520
.506
.494
.485
.588
.546
.517
.498
.485
TABLE 3
Y
0
1
2
3
4
5
0.95
0.25
.530
VALUES OF PC(C)6IY
0.95
PO=P=
PO=P'= 0.25
0.75
0.10
0.05
0.25
0.10
0.05
0.25
0.10
0.05
.906
.987
.924
.938
.935
.934
.935
.997
.961
.969
.967
.967
.906
.987
.902
.928
.916
.915
.916
.997
.949
.906
.987
.894
.924
.997
.945
.963
.954
.954
.954
.827
.844
.837
.837
.838
•
0.90
.968
.779
.810
.790
.788
.789
.965
.958
.958
.958
.760
.798
.771
.770
.770
.908
.908
.908
e
•
•
- 11 -
TABLE 1A VALUES OF E 1y
8
0.75
0.90
0.95
Y
PO=p=
p6=p'= 0.25
0.10
0.05
0.25
0.10
0.05
0.25
0.10
0.05
0
1
2
3
4
5
6
2.43
5.62
5.82
6.27
6.54
6.68
6.73
1.71
5.86
5.20
5.82
6.23
6.62
6.73
1.39
6.29
5.00
5.62
6.24
6.59
6.73
2.43
6.75
7.19
8.26
8.62
8.70
8.69
1. 71
6.76
5.79
7.76
8.55
8.74
8.73
1.39
7.26
5.16
7.56
8.50
8.73
8.74
2.43
7.16
7.70
9.07
9.40
9.44
9.42
1.72
7.06
5.98
8.64
9.40
9.50
9.48
1.39
7.58
5.15
8.39
9.38
9.50
9.48
TABLE 2A
Y
1
2
3
4
5
6
VALUES OF PC(NC}8IY
Po=P=
p6=p'= 0.25
0.75
0.90
0.10
0.05
0.25
0.10
.509
.711
.777
.772
.765
.760
.757
.778
.796
.782
.771
.763
.758
.612
.672
.515
.558
.580
.505
.529
.493
.493
.482
.539
.495
.495
.484
.495
.486
.479
0.95
0.05
0.25
0.10
0.05
.828
.785
.883
.881
.877
.874
.872
.836
.893
.886
.875
.875
.873
.883
.898
.888
.880
.876
.873
.806
.787
.773
.764
.759
TABLE 3A VALUES OF PC(C}8IY
0.75
Y
Po=P=
PO=P'= 0.25
0.90
0.10
0.05
0.25
0.10
0.05
0.25
0.10
0.05
0
1
2
3
4
5
6
.867
.820
.842
.839
.839
.841
.842
.982
.915
.936
.935
.935
.936
.937
.996
.954
.968
.968
.867
.762
.982
.805
.926
.916
.915
.916
.916
.996
.937
.964
.958
.958
.958
.958
.867
.739
.793
.771
.770
.771
.771
.982
.874
.924
.996
.931
.963
.954
.954
.954
.954
.967
.968
.968
.790
.789
.790
.791
.885
0.95
.909
.908
.908
.908