SEQUENTIAL PROCEDURES IN PROBIT ANALYSIS
by
TADEPALLI VENKATA NARAYANA
Special report to THE UNITED STATES AIR FORCE
under Contract AF 18(600)-83 monitored by the
Office of Scientific Research.
Institute of Statistics
Mimeograph Series No.82
October, 1953
SEQUENTIAL PROCEDURES IN PROBIT ANALYSIS
by
TADEPALLI VENKATA NARAYANA
A thesis submitted to the Faculty of
the University of North Carolina in
partial fulfillment of the requirements
for the degree of Doctor of Philosophy
in the Department of Statistics
Chapel Hill
1953
Approved by:
Adviser
ii
A C K NOW LED GEM E N T
I wish to express my deep gratitude to Professor N. L.
Johnson for suggesting the problem and constantly guiding me
throughout the preparation of this dissertation.
I wish also to thank the Institute of Statistics and the
U. S. Air Force for financial assistance which made this study
possible.
T. V. Narayana
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENT
ii
INTRODUCTION
iv
Chapter
I.
STATm1~NT
OF PROBLEM
1
II.
APPLICATION OF THE MOOD AND DIXON HETHOD
TO RANKITS
14
III.
THE ALTERNATIVE METHODS AND THE MOOD AND
DDCON METHOD IN THE CASE OF SMALL SAMPLES
23
THE ASYMPTOTIC PROPERTIES OF THE DIFFERENT
SEnUENTIAL PROCEDURES
48
ESTIMATION OF THE ED,O IN SAMPLES OF
MODERATE SIZES
79
APPENDIX
97
IV.
v.
BIBLIOGRAPHY
107
INTRODUCTION
The standard technique of Probit Analysis enables us to estimate the "median effective dose" (ED50) in a biological assay.
When a simple normalizing transformation for the doses is available, the normalizing measure of "dosage" has a normally distributed tolerance.
The problem of estimating the mean and standard
deviation of this distribution is solved by the probit transformation of the experimental results.
Often in practice, it is possible to apply the "doses" at
certain discrete points only, rather than on a continuous scale.
When we have no a priori information of the position of the ED50,
a wide range of "levels" should be used in order to be sure to
bracket the ED50 point.
our test €ubjects at
The~problem
th~se
now arises how to allocate
various levels to estimate the ED50
with accuracy.
For quantal
responses, Mood and Dixon discuss a sequential
method, known as the "up and down" method, which automatically
restricts testing to the dosage levels near the mean.
nique, suggested by them, is as follows:
The tech-
At any stage of the
experiment we move to the next higher or the next lower level of
dosage according as the preVious result was a "failure" or a
"success", the first test being done at the level we believe is
closest to the mean.
The dosage levels chosen for the tests form
in effect a Markov chain, the result of the (n - l)~ observation
v
th
alone influencing where the n. is made.
Chapter I is devoted to the statement of the problem and a
resume of Mood and Dixon's results.
process of Robbins and Monro
to our problem.
A stochastic approximation
is briefly considered with reference
We consider in Chapter II the application of the
Mood and Dixon method to rankits, where tolerance is supposed to
be rectangularly distributed rather than normally.
In Chapter III we introduce and study two alternative techniques of procedure, which atm -- similar to that of Mood and Dixon
-- at concentrating testing near the mean.
called
The first of these,
the "l-rule", takes into account at the nth observation
all the previous observations on the level on which we made the
nth trial.
The total number of successes and failures on this
level is used, together with the result of the nth observation in
order to decide where the next trial is made.
The other technique
discussed here, the "3-rule", is similar to the "l-rule", but
takes into account the results of all previous observations on any
particular level, as well as of the observations on the two neighbouring levels, namely, the ones Just above and below it, in order
to come to a decision where the next trial is made.
The performances of the Mood and Dixon method, l-rule and 3rule, are compared in small samples.
The probability of taking
the nth trial at a certain level can, in fact, be directly calculated when n is small for all the three above methods.
vi
Theoretically this is possible for any finite sample size, however
large; but practically it is extremely laborious if n > 10.
The asymptotic properties of the Mood and Dixon method and
the 1- and 3-rules are investigated in Chapter IV.
The actual
asymptotic probability distribution, i.e., the exact stationary
distribution for the Mood and Dixon method exists and has been obtained for the following cases
(2)
(1) Rankit Analysis (Chapter II)
A truncated normal population.
For a comparison of the effi-
ciency of estimation by the various methods, only the asymptotic
"proportions" of observations which fallon the various levels are
relevant.
These proportions are obtained exactly for the I-rule
and a method for getting the lower bound for the efficiency of
estimation by the 3-rule in certain cases is indicated.
Methods of analyzing data obtained by the 1- and 3-rules in
samples of moderate sizes are considered in Chapter V.
Experimen-
tal work on samples of sizes 20 and 50 enables us to make a comparison between the various methods.
sults is given in this chapter.
A short summary of the re-
CHAPTER I.
STATElI1ENT OF PROBLEM.
The problem with which we are concerned is with reference to
the estimation of the dose corresponding to the ED50 from a series
of tests for quantaI responses.
Essentially, we are dealing with
the case when observations are taken on individuals rather than on
groups of individuals.
The question of how to allocate the avail-
able supply of test subjects to different doses in order to maximize the precision of the estimate has been discussed at considerable length in the literature.
A technique has recently been pro-
posed by Mood and Dixon ~8_7 which can be used in certain
situations as an alternative to the probit technique developed by
Bliss and Fisher.
This technique, known as the "up and down"
method, was developed for explosive sensitivity experiments in the
Explosive Research Laboratory at Bruceton, Pennsylvania.
We study
this technique in more detail and introduce and study two new
techniques which might be applied in similar situations.
In analyzing quantal responses we are often dealing with a
continuous variable which cannot be measured in practice.
As an
example, to test the sensitivity of explosives to shock, it is
found convenient to drop a weight on specimens of the same explosive mixture from various heights.
Depending on the particular
specimen chosen and the height from which a constant weight is
dropped, some of the specimens will detonate and some will not.
2
We assume that with every specimen of the explosive is associated
a "critical height" and that the specimen will or will not detonate
according as the weight is dropped on it from a greater or lesser
height.
We can, before an experiment, choose from previous infor-
mation, certain "levels" or heights from which we assume the
weight to be dropped and we usually fix these levels throughout
the experiment.
We can now select some one or other of these
levels and determine whether the critical height for a given specimen 1s less than or greater than the selected height.
As is well-known, in true sensitivity experiments it is at most
possible to make one observation on a given specimen.
Once a
weight has been dropped on a specimen, and no detonation occurs,
this specimen cannot again be used, since it is materially altered.
The explosive 1s packed.
same situation arises.
Even in other fields of research the
In testing an insecticide and obtaining
"all or none" data, a bona fide result cannot be obtained from a
second test on the same insect in case it had survived the first
test.
The insect might have built up a resistance to the insecti-
cide or, on the other hand, it may have been weakened.
MOOD AND DIXON'S METHOD
We shall for illustrative purposes continue the example of
explosives and indicate Mood and Dixon's method of analyzing
the data.
The sequential procedure used by Mood and Dixon is
3
as follows:
After test heights or levels, which are usually uni-
formly spaced, have been chosen
(a priori knowledge helps us fix
these levels), a certain level is selected at random and the first
specimen is tested at this level -- say, dropping a weight on a
specimen of explosive mixture from a certain height.
The second
specimen is tested at the next higher or next lower level according as the first specimen did not or did detonate.
In general, a
certain specimen is tested at the next higher or next lower level
according as the previous one did not or did detonate.
We record the sequence of detonations and non-detonations as
below, x denoting a detonation and 0 a non-detonation.
Typical
se~uence
of detonations (XIS) and non-detonations
(O's) artificially obtained by using random sampling numbers.
x
x
x
0
o
x
0
x
x
o
x
o
x
0
0
x
0
x
x
0
x
x
x
o
0
0
o
We observe that the level at which we test a particular specimen
depends only on the result of the specimen just prior to it and
this, as will be
Markov chain.
d~scussed
later, is an example of a simple
4
The assumptions underlying the Mood and Dixon method are as
follows:
1)
The natural variate should be transformed to a new variable
which is distributed normally.
well-know~
In most fields of research, it is
that with enough experience with the data we are deal-
ing With, one can specify the nature of the distribution function
of the random variable under question.
Often, it can be assumed
in dosage-mortality eXPeriments of this type, that the logarithm
of the height (dosage-concentration) is normally distributed.
2)
We should have some idea of the standard deviation of the
normally distributed transformed variate.
When the levels or
heights used in the experiment are equally spaced, with a common
distance d between them, Mood and Dixon suggest ~8_7 that d
should be chosen approximately equal to the standard deviation.
For the statistical analysis however Mood and Dixon show that even
if the interval used is less than twice the standard deViation,
simple analysis is possible.
3)
Since large sample theory is used, the size of the sample
must be "large", for the analysis to be applicable at alL Mood
and Dixon suggest that this analysis be restricted to sample sizes
of over fifty observations.
MATHEMATICS OF THE MOOD AND DIXON METHOD.
Under the above assumptions let h be the height in an
5
explosives experiment, so that
rough estimate of
y
= log
h
2
N(~, 0 )
is
and d, a
is the distance between testing heights.
0,
The experiment is performed as described, the first specimen being
tested at the level which is closest to the anticipated mean.
By
the very nature of experimentation, the number of non-detonations
at any given level can differ from the number of detonations at
the next higher level
n
i
by~
most one.
detonations have occurred at the i
Yi
= Yo
.:t
i
ide
Let m.1. non-detonations and
th
level
= 0,
1, 2, ...
Yo being the initial level corresponding to height hOe
Let
and
The likelihood of obtaining such a sample is
00
(1.1)
where Pi
(1.2)
L(n, m
IT
o = k i=-oo
y )
= probability
1
=-
o j2';.
of detonation at the i
r
- 00
th
level
dt
=1
- q.1.
6
and k is independent of I.l. and
2
(J.
Now
=0
(1.3)
or 1
Hence either of the sets (n ) or (m ) summarizes practically all
i
i
the information given by the sample.
The smaller of N or M is
chosen for the analysis (N ~ M say).
Now M - N is expected to be
small (1.3).
However, in the case where the initial level was
rather poorly chosen, a certain number of observations have to be
expended to get to the region of the mean and these observations
contribute little to a precise location of the mean.
This portion
of the information is thus neglected and for simplicity of analyais, the likelihood function to be maximized is taken as
LI
(1.4)
(n,m
Applying the principle of maximum likelihood for the estimation
of I.l. and
2
(J
,
the derivatives of log L' are equated to zero:
= Ln.
(1.5)
(1.6)
~
a 1og
da
L'
=~
x .
n
(1i
lZ1 1
~-l
-
_
x. z1
-!--)
Pi
=0
7
where x i is the standardized variable
1
=-0[2iC
e
1 Yi - 1..1 2
- 2(
0)
Set
2
, the ordinate of the N(!J.., a ) variate at
w = 1
0
1-1
=IT
j=O
qj
Pj
i> 0
wi
1
=IT
j=-i
Now
i
th
~i + m~
Pj
qj
= expected
i < 0
number of observations on the
_. C(n i +l )
level "':"
q.
~
e: (n.)
~
=
Close approximations for the roots of the maximum likelihood equationa can be obtained when d < 20.
8
Consider a(u)
z ()
x
=
qrxr -
d
z(x + -)
CJ
p(x + ~)
CJ
d
where u = x + 2CJ'
This expression 1s nearly linear in u and similarly,
(x +~) z(x + ~)
f3 (u) = xz ~ x ~ _ __..;CJ_ _-:----.;.o_
q X
p(x + !!)
a
1s nearly quadratic in u.
Thus setting
(1.7)
(1.8)
we have
(1.9)
(1.10)
Thus Mood and Dixon show that the maximum likelihood estimates of
1.1.
and CJ are
(1.11)
y being the normalized height corresponding to the lowest level
9
on which the less frequent event (between detonations and nondetonations) occurs, the plus sign being used when the analysis
is based on non-detonations and vice-versa.
(1.12)
"a = the
sample standard deviation
,
The second derivatives of log L provide variances and covariances
of the estimates
aand ~.
VARIOUS METHODS OF ANALYZING QUANTAL DATA AND THEIR
EFFICIENCY.
The method now widely adopted for the statistical analysis of
quantal assay data is the probit technique attributed to Gaddum and
Bliss.
A detailed exposition of the probit method, its history and
its applications in various branches of biological assay may be
found in D. J. Finney's book "Probit Analysis"
Recently M. S. Bartlett
Ll-7 has
L5J.
advocated an inverse samp-
ling procedure which is useful for the estimation of a high or a
low percentage point.
A modified probit technique proposed by
him may be applied in quantal response investigations where subjects are tested one at a time and each can be classified as
responding or not responding before the next is tested.
Bartlett's
10
inverse sampling procedure could be adopted in explosive
sen-
sitivity experiments for the estimation of high (or low) percentage points.
A summary of the results of Mood and Dixon, and various
methods for the statistical analysis of sensitivity data obtained
from explosive experiments have been given by D. R. Westgarth
L12J in
an unpublished thesis at the London University.
The
sequential method of obtaining data proposed by Mood and Dixon
is used in ~12_7 as a pilot test preltmi~ary to the standard
probit technique or the inverse sampling method of Bartlett.
further alternative method considered in
L12J
A
is to apply the
sequential Mood and Dixon procedure for both pilot and main experiments.
A statistical analysis of the various methods and
numerical examples to illustrate them are included in Westgarth's
thesis.
For ordinary probit analysis, M. S. Bartlett ~1_7 considers
the case where we divide the sample into five equal groups and
test these groups at y
the estimate of
~
= 0,
~a,
!2a.
The ratio of the variance of
by the Mood and Dixon method to the variance of
the estimate of ~ by probit analysis is in such a case 71
Mood and Dixon ~a_7.
groups at y
= ~ 1/2
a,
When the sample is tested
~
l'
cf
in six equal
3/2 a, ~ 5/2 a this ratio becomes
58 ./. . However these comparisons, as Mood and Dixon L-aJ point
out, are not fair "unless there 1s considerable uncertainty as to
11
the general location of the mean."
When we have some a priori
infor.mation regarding the position of the mean and we could 10cate it to within
a of its true position, a method of obtain-
ing data and applying standard probit technique is as follows:
We divide the sn;n:ole into a nUI'lber of
une~.·J.8.l
larger grou:9s at the levels which are
pre8~~bly
mean.
erOu:9s
and test the
closer to the
This increases the efficiency of the probit technique.
B~t
in obtaining
d~ta
by the Mood and Dixon method
test~ng
is automatically concentrated near the mean -- even if we did
not feel a great assurance about the true value of the mean.
Thus, Mood and Dixon's method obviates this difficulty of choice
of numbers of observations to be tested at the levels nearer the
mean, when we are uncertain about the actual position of the
mean.
Howe,rer we use in the Mood e.nd Dixon method only the very
last observation in determining where the next observation is
made.
The information contained in the observations
f~om
1, ... ,
m-l does not play any role in fixing t~e (m + l)st observation.
Intuitively it appears advantageous to use all the m
p~evious
ob-
servations to decide upon a rule for determining where the (m + l)st
observation
s~ould
be taken.
By using all information at hand,
Le., taking into consideration the results of observations from
1 ... m at the (m + l)st trial, we might conceivably concentrate
testing closer to the mean and possibly get a better estimate of
it.
12
One of the disadvantages af the Mood and Dixon method is that
if we start at a level rather far removed from the mean, a large
number of observations are ignored in the analysis.
The informa-
tion contained in these observations which lead us to the region
of the mean is, as we have seen, neglected.
The effective sample
size may thus be very considerably decreased and the assumption
that only a small portion of the information is being neglected
might prove rather inadequate -- particularly in small samples.
Robbins and Monro
~9_7 describe a stochastic approximation
th
process by which the m observation would actually converge to .
the mean with probability 1 for large m.
However the heights or
levels at which we make the observations are not fixed beforehand
but must be chosen arbitrarily by this method.
This stochastic
approximation process too uses only the last observation in determining where the next observation is to be made.
Hodges and
Lehmann ~6_7 have shown that in a situation where the levels of
experimentation can be chosen at will and we follow the stochastic
approximation process of Robbins and Monro, the last observation
summarizes all the effective information in the sample, and the
best
Robbins-~1onro
scheme cannot be improved.
Dan Teichrcew has
made an experimental investigation of the Robbins and Monro
The stochastic approximation process of Robbins and Monro
possesses the attractive property that when m is large, the mth
l~
observation converges to the mean with probability 1.
However
this method is not applicable in our case since the levels at
which we conduct our testing are fixed at the start of the experiment and there is no freedom of choice of the levels of
experimentation at any other levels except those chosen beforehand.
We shall thus attempt to find a process which would permit
us to take observations on the level or levels closest to the
mean, since this would appear to lead to better efficiency in
estimation.
In particular, if the mean happens to lie on one of
the levels, we should like such a process to take most of the
observations on this level.
CHAPl'ER II.
APPLICATION OF THE MOOD AND DIXON METHOD TO RANKITS.
We shall now apply the Mood and Dixon method to the case uf
rankits, where tolerance is assumed to be distributed rectangularly rather than normally.
Let us therefore assume that there are
exactly (n + 1) levels 0, 1, .•• , n, the probability of explosion
at the k
th
level being!, k
n
= 0,
1, ••. , n.
The probability of
the next experiment being done at the (k - 1)
th
k
level is -, and
n
the probability of its being performed at the level (k + 1) is
k
1 - -.
n
The experiment is thus seen to be a Markov chain with the
stochastic matrix of order
n + 1
,
/ 0
1
, -1
0
I!
n
I
0
2
n
1
0
0
0
- -n1
0
0
0
o
o
o
o
o
o
2
1 - -n
0
-n1
1
o /
This stochastic matrix characterizes the "Ehrenfest Model of Diffusion" which represents diffusion with a central force.
ct'.
[),4J
15
Following Feller
L-4_7,
we can investigate the ergodic pro-
perties of this chain, and if
-{Uk}
is the stationary distri-
bution, we find, assuming we had started at the level k or any
level differing from it by 2 (like k-2, k-4, etc.), that after 2m
trials, where m is large, the probabilities
~-l'
uk+l ' uk+ are
3
all zero and after (2m + 1) trials, m being large, the probabilities
~-2' ~, ~+2
are all zero.
The limiting distribution, after a
long number of trials,is as shown below, depending on the level we
start with and on whether an even or odd number of trials have
elapsed.
2m + 1
2m
~+1=0
u =0
k
u. =0
K-2
'it-2
u.
K-3
=0
~-3
Further,
uk
= (1
-
k - 1)
n-
~-l +
k + 1
n
~+l
k = 1, ••
0'
n-l
16
u
u
11
n-l
= --11
n
Z
k=l
~
=1
.
So that, after 2m trials,
~-2 +
uk +
~+2 + •••
=1
..• uk _l
= uk+l =... =0.
and after (2m + 1) trials
•.. + ~-l + ~+l + ••.
=1
~-2
= uk = uk+2 =.•. =0 .
The required solution is, therefore,
Let x
m
th
denote the m observation in an experiment using the Mood
and Dixon method in the case of rankits.
RECURRENCE RELATION FOR
We can then obtain the
E(xm).
Let us assume that we have initially started at level j, the
first observation being denoted by xl'
Using the Mood and Dixon
method the secor..d. 'Jbservation x2 will be taken at level (j - 1)
with probability
Let x
m
.J.n or at level
(J "" 1) with probability 1 _
.J.n .
th
denote the m observation taken at the level k (say) in
such an experiment.
Then
k
x +l is taken either at level (k + 1) with probability 1 - n
m
-Jr 3.t LlI(ol (l.- - 1) with probability ~ •
n
Hence setting
f.]
m
'(x)::
m
we ha.ve
(2.1)
l.t.
C'In+, ::
1'1
em
(f-%;).
The solution of this difference equation is)btained by c,.msidering
the corresponding homogeneous equation,
t
m+l
= t
(1
2)
m - n
·
which yiel<1s for its solution
f
(2.2)
The value
t m ::
C
m
= k(l
_ ~)m-l
n
is a solution ,yf (2.1) if
C'
-
1
-t.
C(1
-
2
n)'
i.€. ,
18
if
c
= 2n .
Thus, we obtain
as the solution of (2.1), where the constant k is determj.ned from
the initial condition
E. 1 -k
j .
= (j
- ~)
and
n
+2
(2.4)
is the expected value of x •
m
Thus as m becomes sufficiently large, whatever fi.xed number
of levels we start with, we expect ~ m to equal ~.
This is in
keeping with the intuitive ideas regarding simple Markov chains
with a stationary distribution.
Whatever initial position we may
start with, we expect after a large number of trials
that the in-
fluence of this initial level of observation will gradually wear
off.
The probability of the mthobservation being on a certain level,
when m is large, is independent of the initial level
j
we start
19
with.
This chain has thus the property of being ergodic.
RECURRENCE RELATION FOR V(Xm).
We obtain now, in a similar way, the value of V(x )
m
= Fm
(say).
th
Given that the m observation is taken at level k, let us
2
m+
consider the condition expectation of x l '
t
2
(x 1
m+
I
x )
m
= (k
~)
+ 1)2(1 -
= k2 (1
n
+ (k _ 1)2
~
n
4
- -) + 2k + 1.
n
Hence
(2.5)
Now
and
2
E (xm+
1)
F
E(xm2 ) (1
=
- V(x
m+l -
m+1
~)
n
2 )
C(xm+1
) -
= Vexm) = t
m
F
-
(x2 ) _
m
+ 2
-
t
(x ) + 1
m
t m+l
2
't m2.
Therefore
(2.6)
F
m+1 +
t
2 1
m+
= FmL'1
-
~n - 7 +
t.
2(1 - .:t) + 2
m
n
'!. m +
1
20
or
(2.7)
F
m+l
=>
F
/-1 -
m-
l!n-7 +
c ,
m
where
Solving (2.7) as before,
where the arbitrary constant A is determined from the initial
condition F
l
= O.
Thus
or
A
. n)2
n
= ( J"2 - '4 •
Hence
is the value of Var (x ).
m
21
T~li
fram (2.8), as m
>
00,
2
~
n 1:.r;:1
•
n
• • Fill ~ 4:
2
4::7
- n _7 +
m~
as
fiT"
1 -
4n
-1.
00.
EVALUATION OF COY (Xm' xm+1 ) •
With the
sa~e
notation as before
c. (xmxm+1 I xm)
~
= k(lt + 1) ;-1 -
-
= k;-1
-
n -
7 + k(k
_ 1) k
n
g)
7
n -
+ k(l -
or
c(x ).
(2.11)
IT.
Now cov(xmx11l+1)
and
.~
2
= E(xmxm+ 1)
'f!2
m
C(xm': = V(x
) + c
\ ill
-
t
= Fm +
ill
e- m+1
-e2
(,' In •
Therefore,
(2.12) cov(x x
m m+.....,)
= FillL-l - gnJ· +£. m·
2(1 -g)
n
-et
m m+1 +
em
22
Hence as m ~
n
+ -2
==
cov(xrnxm+ 1) - l
00,
~'+
rl -
L
2
g
J
n
+
~
;--1
,+.-
- g
7
n -
and x
1 ap-
n - 2
4
Thus as m ---;>
00,
the correlation bet.. .Teen x
proaches the value ~2.
n
ill
m+
CHAPrER III.
THE ALTERNATIVE METHODS AND THE MOOD AND DIXON
METHOD IN THE CASE OF SMALL SAMPLES.
We now propose two alternative methods which are based on
sequential procedures like the Mood and Dixon method.
These se-
quential procedures can be applied to either probits or rankite
the arguments used in this chapter being essentially the same
whether developed for probits or rankits.
For purposes of numeri-
cal calculation we use a N(O, 1) population, truncated at both
tails at a distance 3 from the mean) since th& chance of observations falling outside the range
(-3, 3) in a N(O,
very small indeed, using any of the above methods.
1) population is
We could, in
fact, deal with a N(O, 1) population, truncated at both tails at
any finite distance from the mean, and the arguments still apply
with no modification at all while the numerical values are only
very slightly affected.
THE
l-RULE.
The first method we propose arises very naturally in the consideration of a problem of this type.
When viewed asymptotically,
this method appears a reasonable one to follow, since it achieves
a marked concentration of observations on the levels closest to
the mean.
as follows:
We call this method the "l-rule" and i t may be stated
24
Let us assume that at the mth trial we obtain a detonation
th
(x) on the k
level. We consider all the observations we have obtained on this level, and if the number of x's (i.e., number of
detonations) exceeds the number of O's (i.e., number of nondetonations) on this level, we decide to move down.
However, if
the number of O's exceeds the number of x's (including the x at
th
the m trial) we decide to stay on this level. If the number of
x's and O's is equal on this level, we proceed to take a further
observation on the k th level, and decide on the basis of this observation to move up or down.
A similar procedure is followed if
the last observation at the mth trial were a O. Thus, at any
stage of the experiment, we consider the total number of
XIS
and
the total number of O's on the level k say where we made our last
trial.
Let these be n and n2 • The l-rule may then be expressed
l
as follows:
Result of Last Trial.
Case 1
"Success" i.e. x.
"Failure" or O.
Case 2 n < n2 •
l
Case 3 nl = n2 •
Level at which next trial
is made.
k - 1.
k.
Success
k.
Failure
k + 1.
Success
k.
Failure
k.
25
Thus at any stage of the experiment, if nl
= n2 ,
we take a
further observation on that level and move up or down on the
basis of this further observation.
It should be noted that the re-
sult of the last trial is included either in the number of
(n )
l
XIS
or in the number of O's (n2 ) on our level k of experimentation.
THE 3-RULE.
The other alternative procedure may be called the "3-rule".
It is similar to the l-rule but uses at every stage, all the previous results on the two neighbouring levels as well as those on
the level used at that stage.
For the 3-rule we thus use the re-
sults of observations on the three following levels:
the level on
which the last observation was made and the two levels just above
and below the level on which the last observation was taken.
We can formally state the 3-rule as follows:
Let us assume
that at the mth observation we make a trial on the k th level. Consider all observations on the levels (k - 1), k and (k + 1).
Let
the number of detonations and non-detonations on these three levels
be a , b ; a 2 , b and a , b respectively.
l
2
l
3
3
In evaluating a
2
and
th
b2 we include as before the result of the m observation as well.
Case 1.
If the last observation were a detonation we take the next
26
observation at level (k - 1); while if the last observation were
a non-detonation we make the next observation at level k.
Case 2.
In this case,
w~
move to level (k + 1) if the last observation
were a non-detonation, but take a further observation on level k,
if the last observation were a detonation.
Case 3.
In this case from the observations in the immediate neighborhood of the last observation, we have no idea whether to move up,
stay on the same level or move down.
So we just consider a 2 and
b , the observations on the particular level we are experimenting
2
and follow the
1-rule~
If a 2 > b2 , we take our next observation at level (k - 1) or
k according as we obtain an x or a 0 at the last trial.
If
a 2 < b 2 , the next observation is taken at level (k + 1) or k according as we obtain a 0 or an x at the last trial, while, if a
2
= b ,
we stay on level k for our next observation irrespective of the
result of the last observation.
THE METHODS COMPARED IN SMALL SAMPLES.
To illustrate the procedures, let us assume we have a model,
2
27
which in the limit leads to the Ehrenfest model of diffusion.
The number of levels taken for the numerical examples l whether for
probits or rank1ts l is 11.
(In the truncated normal population l
we take levels to be uniformly spaced at a distance of .6 1 the
middle level
5 taken to coincide with the mean l so that the number
of levels is 11 in the normal case as well.)
bered from the top 10 1 91 ••• , 1 1 0
coincides with the mean.
k is thus kilO (k
=0
1
60
The levels are num-
that the middle level "5"
The probability of detonation at level
1, ••• , 10) in the rankit case.
We shall
therefore, following any of the above methods, automatically move
up at level 0 and down at level 10 1 since we are certain to obtain
a non-detonation at level 0 and. a detonation at level 10 always.
However, we formally introduce 2 further levels, level "-1" and
level "11 11 below the 0 level and above the level 10 respectively,
for we can now strictly follow the 3-rule as stated above.
Either
following the l-rule or the 3-ru1e l we are certain never to take
an observation on the levels "-1" and "11".
We introduce these
levels merely as the adjoining levels of 0 and 10 1 since 1n the
3-rule we have to take into account, when an observation 1s made
on level k say,
lI
all observations on levels (k - 1), k and (k + 1)".
This would otherwise break down for the values k
=0
and 10, since
in the former case the level (k - 1) would be undefined l while in
the latter level (k + 1) is undefined.
With this convention we can now cumpute the probabilities for
28
the Mood and Dixon method as well as for the alternative methods
where the second, third, •••
observations will be made, given
that we start at a certain level
j
say.
The probability distribu-
tions of the observations at trials 1, ••• , 10 on the various levels
following the Mood and Dixon method, when we are assumed to start
at level 7, is given as Table I in this chapter.
In order to ob-
tain the corresponding probability distributions by our alternative
methods for purposes of comparison, we need to stUdy the patterns
which lead to different procedures for the Mood and Dixon method
and the two alternative methods proposed.
in more detail below.
We
indica~e
These will be discussed
in detail how Table II -- giVing
the probability distributions at trials 1, ••. , 10 on the various
levels following the '-rule, known that we start at level 7
is
obtained; while Table III, for the l-rule in the same case, is obtained in a similar fashion.
Tables IV, V, VI are the rankit
probabilities for the Mood and Dixon method, '-rule and l-rule
respectively, had we started at the level 5 rather than level 7.
The remaining Tables VII - XII deal with the truncated normal case,
starting at levels 7 and 5 and follOWing the three different
methods.
THE EVALUATION OF
II
RANKIT II PROBABILITIES FOR TRIALS
1 TO 10 IN THE CASE OF THE 3-RULE, WITH THE
START AT LEVEL 1.
Having obtained the Mood and Dixon probabilities known that
we start at level 1 (Table I), we note that for the first four
trials, i.e., for m = 1, 2, 3, 4, the 3-ru1e is identical with the
Mood and Dixon method.
Let us now consider the two following pat-
terns after four observations have been made, which we might possibly obtain, proceeding either by the Mood and Dixon method or the
3-rule.
( i)
Level 1
6
5
(11)
x
x
9
0
x
o
o
8
Level 1
x
o
In (i) we have thus obtained two detonations ut levels 1 and 6 respectively, followed by two non-detonations at levels 5 and 6 respectively.
By the 3-rule we now have to take our next observation at
level 6; while by the Mood and Dixon method our next observation
would have to be made at level 1.
The probability of obtaining
Similarly, had we obtained the pattern (ii), the probability of
30
which is (7o80 9x 8x )
= .0432
.
we shall stay at level 8 Ltself
for
the fifth trial by the 3-rule, instead of coming down to level
as is expected by the Mood and Dixon method.
7
By actual enumera-
tion of all possible patterns which would arise with four observations, we find that (i) and (ii) are the only patterns in which we
would move in a way contrary to the level indicated by the last
observation, had we been following the "up-and-down" method.
We
shall call a pattern with this property a "fixed" pattern, 1n so
much as the result of the last observation does not decide the actual level to move to, but the pattern itself determines the next
level, which is different from the level indicated by the last observation according to the "up-and-down" rule.
Noting that a pat-
tern such as
o
10
o
9
o
8
7
o
cannot exist, since by our hypothesis a detonation 1s certain at
the level 10, we give a short table, Table A, of the nunilier of possible patterns and fixed patterns for each value of m from 1 to 9,
proceeding by the 3-rule.
Table B gives the number of possible
patterns and fixed patterns for m = 1, ••• , 9 proceeding by the 1rule.
e.
We assume a ranklt set-up with ten levels, the first
31
observation being at level 7.
Table A.
Table of number of possible patterns N and the number of fixed
patterns P for the 3-rule in the first nine trials.
m
1
2
3
4
5
6
7
8
9
N
2
4
8
15
30
56
110
208
408
P
0
0
0
2
2
3
9
27
49
Table B.
Table of number of possible patterns N' and the number of fixed
patterns pI for the l-rule in the first nine trials.
m
1
2
3
4
5
6
7
8
9
N'
2
4
8
15
30
59
114
224
443
p'
0
0
2
4
8
18
;7
7;
151
It will be observed that if we had a normal distribution with an
infinity of levels on either side of the mean, the number of possible patterns with m observations is 2m and among these patterns
the fixed patterns help us in evaluating the probabilities of
st
where, i.e., at what level, the (m + 1)
trial will be made.
Thus since for m
= 1,
2, 3, the number of fixed patterns for
the 3-rule is zero, the probabilities of taking an observation at
32
the different levels for the first four trials will be exactly
the same as for the Mood and Dixon method.
By the Mood and Dixon
method we have a probability .3976 for taking the fifth observation
at level 7.
The two fixed patterns with m = 4, contribute proba-
bilities of .0840 from level 7 to level 6 and .04;2 from level 7
to level 8 respectively.
a probability
Thus for the 3-rule we shall only have
.3976 - .0840 - .0432
observation at level 7.
= .2704 of taking the fifth
Correspondingly we have probabilities
.0840 and .0432 of taking the fifth observation at levels 6 and 7
respectively, by the 3-rule.
The probabilities at the other
levels remain unchanged from those of the Mood and Dixon method.
The probability distribution at the fifth trial having been
obtained, we may then obtain the uncorrected probability distribution at the sixth trial by using the appropriate probabilities
of detonation and non-detonation at the various levels.
These
values which we obtain at trial 6 are corrected for the fixed
patterns at m = 5, to give the 3-rule probability distribution on
the various levels at the sixth trial.
Once an enumeration of
these fixed patterns is made, it is fairly simple to compute the
th
probabilities of taking the m observation at any level, by making the appropriate corrections for the fixed patterns with (m-l)
observations.
However when m> 9, the process becomes rather
tedious, for as yet no method has been determined which gives the
fixed patterns for any value of m.
Complete enumeration for a
33
value of m like 50, involving for a normal distribution 250 pattern
would be impossible and until an exact mathematical formula for the
number of fixed patterns is obtained, the probabilities at any stage
cannot be easily calculated.
It should be noted that in an experiment
i'lfith the 3-rule starting at level 7, these patterns as well
3.S
the
process of evaluation of probabilities at each level, carryover for
a truncated normal population too -- except, of course, that the
numerical probabilities of detonation and non-detonation at each level
are now different.
Tables I - XII now follow.
Table I
A
Trial 1
T...evels
Table of Probahilj.ties by the Mood Dixon "liiethod for a Ranldt
Set Up with 11 Levels Starting at Level 1
2..
J
.06
9
8
6
5
4
3
2
1
0
5
.006
10
7
4
.0480
.52
1
.7
.298159
.484512
.453427
.489431
.338688
.02520
•36,23?
.172166
.137760
.0840
.11691)0
.411030
.29400
.210
.003057
.134184
.51352
10
.030566
.:j3Sy12
.4704
9
.003730
.16248
.574
8
.°3'7296
.3916
.l.t.2
7
.00480
.210
.3
6
.045864
.060.35'9
.0096 7 7
.00504
.000,04
.OOO9 tJC
34
Table II
A Table of Probabilities by the 3-rule for a, Rankit set-up with 11
1eve1e. start level 7.
4
Tria,l 1 2 3
6
8
10
7
5
9
Level
.06
9
8
7
6
5
4
3
2
1
0
.00480 .001404 .002858 .000955 .001202
.006
10
.210 .0432
.3
1
.0480 .01404 .028584 .011628 .014868 .009168
.11892 .040860 .082531 .038909 .054459
.2704 .06816 .206752 .082358 .164491 .103850
.52
.574 .0840 .36148 .129612
.7
.42
.2950~
.177205 .243283
.4704 .11340 .358008 .166975 .297115 .229539
.210
.29400 .091980 .261324 .140288 .221352
.0840
.137760 .050904 .137592 .082414
.02520
.045864 .018900 .049533
.00504
.009677 .004234
.000504
.000968
35
Ta.ble III
A Table of ~obBbi1itie8 for B. Rank1t eet-up
wi th 11 levels by the l-rule eta.rt1ng a.t level 7
Trial
Level
1
2
3
.06
9
8
6
5
4
3
2
1
0
5
6
7
8
9
10
.00060 .001440 .000348 .000198 .000438
.006
10
7
4
.0168 .02304 .005424 .007877 .010535 .004531
.126 .1332 .03708 .076032 .085558 .038934 .048236
.3
. ')2 .252 .1360 .32160 .213184 .136128 .231293 .186728
1
.406 .3780 .15316 .339192 .337070 .203359 .299878
.7
.2520 .31752 .121968 .264600 .312409 .192338
.42
.12180 .191t040 .071148 .145429 .199517
.210
.0840
.04;680 .086184 .029971 .054613
.010584 .026460 .008679
.02520
.001411 .004990
.005040
.000504
.000050
~
~~N*
~
Tria.l
Level
Table of Probabilities for a Rankit set-up with 11 levels
starting a.t level 5 by the Mood a.nd Dixon method
1
2
3
4
5
10
.012
8
.06
6
e
.5
1
.6
.018547
.23072
.086611
.233062
. 413984
.50496
10
.001855
.084336
•4208
.528
9
.001680
.0180
. 44
8
.01680
.224
.2
7
*
7
.0012
9
5
6
.411534
.496781
In the ta.bles marked with an asterisk where we start at level 5,
the proba.bility distributions for tria.ls 1, •.. , 10 on the levels
below the mean level 5 have been omitted since the ta.ble repea.ts
itself. By symmetry, the proba.bility distribution on level 4
is the same a.S that on level 6, the proba.bility distribution
on level 3 is the same as tha.t on level 7 and so on.
e
37
Table V~lA Table of Probabilities for a Rankit set-up with 11 levels
by the 3-rule starting at level 5
Trial
Level
1
2
3
4
6
5
10
7
.0012
.2
7
6
.5
.6
1
5
.44
10
.000338
.001793
.02544
.003384
.017928
.006729
.0780
.01152
.076920
.029484
.054840
.224
.0672
.18368
.075744
.149197
.099732
.084
.3032
.12096
.270344
.152069
.240735
.360
.1008
.31680
.142128
.301968
.1923L~1
.06
8
9
.002544
.012
9
8
Table VI~lA Table of Probahilities for a Rankit set-up with 11 levels
by the I-rule starting at level 5
Trial
Level
1
2
3
4
5
7
.00336
.007848
.002410
.00e915
.0252
.04284
.014040
.022010
.0.31774
.104
.1380
.0!.+928
.095088 .121971
.'~0724eo
.29
.294
.1142
.24882
.274184
.163219
.225977
.30
.180
.4428
.31140
.217440
.379555
.333426
.06
.2
6
.5
1
.6
10
.000202
.012
7
9
.000612
9
8
8
.000120
.0012
10
5
6
38
Table VII
A
Trial
L3v81
Table of Probabilities by the Mood and Dixon method for a
Normal Population starting at level 7
1
2
3
4
5
6
7
8
9
10
10
8
7
6
5
4
3
2
1
0
.0016
.0039'
9
.114
1
.0262
.0441
.2161
.3525
.886
.0009
.6341
.6435
.0?06
.1727
.1575
.7000
.6943
.42515
.0880 5
.0187
.5030
.5386
.3218
.0007
.4899
.7007
.4617
.1260
.0100
.4749
.1405
.0165
.0147
.0003 5
.0005
3;
Table VIII
A Table of Probabilities by the 3-ru1e for a Normal Population starting at level 7
Trial
1
2
3
6
4
7
8
9
10
Level
10
.0039'
9
8
7
6
.5
4
3
2
1
0
.0441
.114
.3.52,
1
.886
.0016
.0002
.000,' .0001
.0001
.0001
.0038
.0157
.0034' .0068
.0017
.0030
.1243
.0277.5 .0762
.0303 .0460
.0316
.6341 .0880' .3404 .1470
.643.5
.6943
.3218
.1808
.4994 .2689
.42.51' .1471
.0880'
.277.5
.1260
.0100
.3.528
.44.59
.3.573
02037
.3009
.0488 '.1104 .0627
.0147
.0003.5
.18,8 .2312
.00.59
.0131
.000,
.0002
40
Table IX
A Table of Probabilities by the I-rule for a Normal Population starting at level 7
Trial
1
2
3
4
5
6
7
8
9
10
Level
10
.0039 5
9
.114
8
7
6
5
4
3
2
1
0
1
.0165
.0002
.0003
.0185
.0019
.3525 .1252 .0709
.886
.1324 .0580
.3218
.0044 .0008
.0!.~33
.0011
.0675 .0346
.3987
.2823 .2034 .3264
.4674 .4738 .2147
.4934 .4826 .3197
.5366 .3550 .1867
.6435
.0037
.2935 .0998 .2199 .2859
.08805
.0310 .0753 5 .0185 .0305
.0100
.00145 .0072 .0016
.0003 5
.0002
.1949
41
Table X*
A Table of Probabilities by the Mood and Dixon method for a
Normal Population starting at level 5
Trial
1
2
3
6
5
4
7
8
9
10
Level
10
.0006
.0005
9
.0156
8
.1368
7
6
.5
1
5
.1488
.4826
.4844
.7264
.0176
.0174
.1476
.0006
.1490
.4824
.7011
.7037
.0176
.4824
.7008
~'-
Table XI"
A Table of Probabilities by the 3-rule for a Normal Population
starting at level 5
Trial
1
2
3
4
5
6
7
8
9
10
Level
10
.0002
.0005
.0001
.0174
.0048
.0081
.1476
.0374
.1206
.0143 5 .0045
.0433 5 .0841
.0502
.0880
.3812
.1185
.3487
.1609
.2992
.5276
.1279
.5111
.1869
.4999
.2848
.0156
8
.1368
7
6
5
.0006
.0005
9
.5
1
.4844
.7264
42
Table XII
..
.,~
A Table of Probabilities by the I-rule for a Normal Population starting at level
Trial
1
2
3
4
5
6
5
7
8
10
9
Level
10
8
.0023
.0069
.0013
.0011
.0034
.0482
.0797
.0192
.0342
.0550
.0239
.3 0 28
.3193
.1202
.2589
.3009
.1657
.2337
.3632
.2638
.5958
.4298
.3266
.5563
.4778
.0156
.1368
7
6
5
.0002
.0005
9
.5
1
.7264
. The Various'Methods Compared. in Small Samples
We are now in a position to compare how the probability distributions of the first few observations on the different levels vary when we
use the Mood and Dixon method, the I-rule and the 3-rule.
From a hand-
ful of results on samples of ten we do not expect to draw general conclusions, but it is of interest to observe how the methods differ in
small samples.
From Tables VII - XII we may evaluate for the first 10 trials the
expected number of observations which fallon levels 0 --- 10, by each of
the above methods for a truncated normal population.
Tables XIII and
XIV below give the expected numbers of observations falling on each
43
trial level in samples of 10 by the various methods assuming (1)
we have started at level 7 (2) we have started at level 5.
The
last column in Tables XIII and XIV gives the weight function
w == z 2/pq at each level.
Probit Analysis
The values of ware taken from Finney's
["5_7.
Table XIII
Method
Mood and Dixon
3-rule
1-ru1e
w
.0044'
.062
Level
10
.015
9
.00715
.0066
8
.2236
.1609
.180
1
1.8988
.19255
1.68865
1.8844
.310
6
2.1900 5
3.1751
.558
3.0901
1.15145
3.0951
.631
4
3.0516
2.7385
1.68355
1.4158
.558
3
• 33~.5
.4359 5
2
.Oh12
1
.0008 5
5
0
,5
.0437
.2434
.0202 5
.180
.0010 5
.00055
.062
.310
.015
44
~t­
Table XIV
Mood and Dixon
Method
3-rule
I-rule
w
Level
10
9
.0017
.0019
.0007
.062
8
.0682
.0647'
.0306
.180
7
.,822
.6200'
.3970
.370
,
2.4318
2.3809
2.201,
.,,8
3.8320
3.8646
4.7397
.637
6
We notice that the alternative methods have the advantage that,
even for very small values of m, they may be expected to approach
the mean more rapidly than the Mood and Dixon method.
Further, in
the case of the normal distribution, which is the important case
in practice, the I-rule achieves a marked concentration of
observations near the mean levels.
From this point of view,
there appears to be little difference between the Mood and
Dixon method and the 3-rule in the case of small samples.
Let us now define the function I(x) as follows:
For any method
we multiply the expected number of observations falling on each
level by the appropriate weight function at that level and sum
this quantity over all our trial levels. Let us denote this
sum by I.(x) i=1,2,3 for the Mood and Dixon method, 3-rule and
~
I-rule respectively.
The function rex) is precisley Sbw in the
notation of standard probit analysis; and we know that for
45
probit analysis, if m the estimated log 1050 is nearly equal to the
mean value of the dosages used in the
eA~eriment,
may be taken to be inversely proportional to Snw
the variance of m
L-5_7.
We may
thus expect I(x) to furnish a basis of comparison between the
sequential procedures used, and the values of I.(x) evaluated
~
from Tables XIII and XIV are as follows:
(1) Start at level 7.
II(x)
= 5.27
1 2(x) ... 5.33
1 (x)
3
(2)
= 5.35
Start at level 5.
Il(x)
= 5.61
12(x) ... 5.60
1 (x) ... 5.78
3
For the case of rankits, we similarly prepare Tables XV and XVI
giving the expected number of observations fallinc on the various
trial levels in samples of 10 following the sequential procedures
under consideration.
lj.6
Table XV
;IGthod
Mood and Dixon
3-rule
I-rule
Level
10
.017587
.017219
.009024
9
.175862
.186288
.128207
8
.923622
.888879
.845040
7
2.551151
2.416011
2.996933
6
2.717977
2.564633
2.816659
5
1.864343
2.055437
1.880835
4
1.207920
1.218944
.941934
3
.393926
.492670
.298448
2
.131423
.139497
.070923
1
.014717
.018951
.011441
0
.001472
.001472
.000554
Table XVI~~
~iethod
~Jlood
and Dixon
3-rule
I-rule
Level
10
.004735
.005875
.002134
9
.047347
.065481
.028533
8
.308947
.310764
.195864
7
.887782
.999553
.780759
6
2.186318
2.111308
2.110400
5
3.129741
3.014037
3.764621
47
The weight function in the case of rankits is w
= l/pq.
We
observe a difference between the case of probits and rankits:
the case of probits, the weight function w
= z2
for
/pq has its
maximum value at the levels closest to the mean and at the levels
further away from the mean the value of
z~/pq steadily diminishes.
On the other hand, in the rankit case the weight function w
increases as we move to levels farther from the mean.
= l/pq
The rankit
case has been used up to now for illustrative purposes, and this
difference in weight function suggests that the case of rankits
might well require a different type of procedure for efficiency
in estimation than the sequential procedures we have been
discussing,
which aim at concentration of observations
near the mean levels.
A complete enumeration of the fixed patterns in the case
of the 3-rule for the first nine observations assuming that we
had started at level 7 for a set-up involving 11 levels is given
in the Appendix.
CHAPTER IV
THE ASYMPrOTIC PROPERTIES OF THE DIFFSRENT SE0UENTIAL PROCEDURES
We have hitherto been considering the Mood and Dixon procedure,
the I-rule and the J-rule for the case of finite samples.
We shall
henceforward restrict ourselves to the case of a normal distribution
which is the important case in practice -- for investigating the asymptotic properties of the three
se~ential
idea of their accuracy of estimation.
procedures and obtaining rome
The considerations which follow
and the arguments developed can be applied with slight modifications
for investigating the asymptotic properties of the various procedures
in a rankit set-up as well.
We shall at the outset make the followD1g simplifying assumptions:
(1)
1,re shall consider only the case where the mcan does not
actually lie on one of our Ittrial" levels.
This is a reasonable
assumption to make as far as practical considerations go, since we
shall not in the vast majority of cases, when the mean is unknowq,
choose a level which coincides with the mean.
unlikely to occur.
Such an event is rather
Further, we can generally expect, certainly in the
case of the Uood and Dixon method and the I-rule, that our results
which hold true when the moan is arbitrarily close to a level but not
exactly on it, will also hold llith slight and obvious modifications to
the limiting case as well, when the mean exactly falls on a level.
49
(2)
In the case of the 3-rule we,furthe~exclude the case
when the ,mean is halfway between two levels for the same considerations.
(3)
We restrict ourselves, as before, to the case of a trun-
cated normal distribution as this would alter the numerical values
only slightly and simplify our arguments.
The Mood and Dixon Method
Let us assume, as before, we have 11 levels of experimentation,
numbered 10, 9, ••• , 0 from top to bottom.
Following our assumptions,
let us suppose that the middle level 5 coincides with the value -.1 in
a N(O,l) population, and that the distance between the uniformly spaced
levels is .6.
We are thus, in effect, considering a normal population
truncated above 2.9 and below 3.1, the middle level 5 coinciding with
-.1.
The probabilities of detonation and non-detonation from levels
10 to 0 are then as shown in Table C below:
50
Table C
pt.
~
Level
= probability
of detonation
~.~ = probability
of non-detonation
10
1
9
.99112
.00888
8
.95718
.04282
1
.86582
.13418
6
.69246
.30154
5
.46051
.53949
4
.24168
.15832
3
.09610
.90390
2
.02783
.91211
1
.00525
.99475
0
0
1
0
Let p 10' P 9 ••• PO be the probabilities of taking an observation at the levels
10, 9, ••• , 0 after a long number of trials. As is
to be expected, depending on the number of trials olapsed and the level
at which our initial trial is made the two sets of probabilities
P 10' P 8' p 6' p 4' p 2' Po and P 9' p 7' p 5' p 3' PI will alternatively
vanish.
The levels chosen for the trials form, as pointed out before,
a Markoff chain with the transition values being the different probabilities of detonation and non-detonation at the various levels given
in Table C.
51
Consider the stationary distribution
PIa' ••• , Po. Since
this is a stationary value, we know that after exactly two trials, tho
same set of probabilities will be repeated, provided a long number of
trials have elapsed.
Now noting that we can get to n certain of these
levels 6 (say) in two trials in only one of the four following ways:
(1)
Start at level 6 and obtain the sequence XO.
(2)
Start at level 6 and obtain the sequence OX.
(3)
Start at level 8 and obtain the sequence XX.
(4)
Start at level 4 and obtain the sequence 00.
we have the equation
For the two probabilities
p 10 and P a we have simpler equa-
tions since thoy can be reached in two trials only in the following
ways,
Level 10
Level
a
(1)
Start at lovel 10 and obtain the sequence XO.
(2)
Start at level 8 and obtain the sequence 00.
(1) Start at level o and obtain the sequence OX.
(2)
Start a t level 2 and obtain the sequence XX.
Hence the equation for PIa' for example, is
We thus have five independent equations connocting tho values
••• , P 0 together with the condition
PIa'
52
Po + P2 + ••• + P 10
=
1
Solving them, using the numerical values in Table C, we find
the values of P 10' ••• , Po correct to five decimals as follows:
(4.1)
•••
P 10
=
.00001
',p 8
=
.02692
p6
=
.54057
p4
=
.42136
p2
=
.01114
p
=
.00000
o
The alternat e set of probabilities
p 9' •.• , P 1 can be com-
puted either directly or more simply from (4.1).
places of docimals are:
(4.2)
P9
=
.00116
P7
=
.19201
P5
=
'.69386
p
=
.11266
=
.00031
3
P1
Their values to five
53
l'1e can now evaluat e r(x) for the Mood and Dixon method for our
numerical example taking the values of the weight function w
from Finnay's Probit Analysis
£
= z2 Jpq
5 _7. Tho value of rex) for our
example is
THE l-RULE
From assumption (1) made at the start of· this Chapter, it follows
that the mean lies between two levels.
Let us assume that tho levels
above, the true mean are numbered 1, 2, 3, ••• and the levels below the
mean -1, -2, -), ••••
From assumption (3) wo have only a finite number
of levels on either side of the mean both for the rectangular and normal
cases.
We shall now sh9w that asymptotically all observations in the
case of the l-rule are confined only to the two closest levels on
either side of the mean, with a probability as near to 1 as we please.
To simplify th3 argumonts we shall assume hereafter that in tho case
where the moan doos not lie halfway between levels 1 and -1, the mean
is situated closer to lovel -1.
The contrary case, whero the mean is
closer to level 1 may be discussed in a similar fashion.
An
experi-
ment with the I-rule whore the mean lies exactly halfway between two
levels may be called a "symmetrical" I-rule experiment, while an experiment where the mean does not lie halfway between two levels may be
referred to as an "unsymmotrical ll I-rule experiment.
S4
TheorQffi 1.
11ith a probability approaching unity as ncar as we please
all observations fall after a large nlwber of trials on levols 1 3nd
-1 in tho case of the I-rule.
Let us assume that we have made a largo number N of trials.
~~
sh211 show that asymptotically as N approaches infinity, the prob-
ability that an observation will be taken on lovel 2 say, is as noar
to
z~ro
as we pleaso.
The probability that an observation will be
taken on any other level except 1 and -1 will be similarly zero.
The proof follows as a simple consequence of the strong law
of large numbers.
cf. J. V. Uspensky ~ 11
i, i being positive.
_7.
Consider any level
For the level i, the probability of obtaining
an X at any observation made on this level is Pi> ~.
Thus, for
every level i, there exists a number n. such that the relative fre~
quency of X's on this lovel when n
observations have been taken on
i
level i, differs from p. by less than an arbitrary e, with the prob-
-
ability 1.
~
Further, we know that if more than n. observations are
~
taken on level i, the difference in absolute value between p i and the
th
relative frequency of XIS on the i
level will continue to remain loss
than e.
The value of n. will vary according as tho value of p. and
~
~
the level i chosen, but for every i such an n. exists by the strong
~
law of large num\)Qrs.
li ence after at most n
observations on level i
i
we shall bo constrained whonever we get an explosion on theith level
to move down with a probability approaching 1 as near as we pleaso.
Similarly, a non-explosion on level i, after n. trials have been made
~
on this level, will lead
US
to stay on tho same i th level with a prob-
55
ability arbitrarily close to 1, for the number of
XIS
on this level
is with a probability arbitrarily close to 1 greater than the number
Thorefore, after at most N = 2(n + n + n ••• ) trials
l
l
2
3
have beon made on the levels i > 0 i.e. 1, 2, 3, ••• we know that
of zeros.
with a probability as near unity as we please, whatever level i
(i > 0) we may take an observation on, should we obtain a non-detonation we stay on level i and for every detonation we move
do~m.
Since
this holds for lovel 1 also, we shall not move above level 1 with a
probability as near to unity as we please, for we shall at most stay
on level 1 should a non-detonation occur on it.
With a probability
arbitrarily close to unity we find, similarly that after
N2
=
2(n_ l + n_ 2 + n_ + ••• ) trials have been made on the levels i < 0
3
i.e. i = -1, -2, -3, ••• we shall not move below level -1 and taking
N > N + N2 our theorem is proved.
l
Tho exact proportions of observations falling on the two levels
1 and -1 can be evaluated as follows:
Let PI > .5 and p -1 < .5 be the probabilities of detonation
on tho tHO lovels closest to the mean, 1 and -1, in an experiment with
the I-rule.
All observations will be confined to these two levels
asymptotically.
fa lUng on them.
Let Q and
QI
be the proportiona
Then
e
+
et =
1,
and
of observations
56
since an observation can only be made on 113vel 1 in one of the two follo't-ling ways:
(i) We get a non-detonation on levelland stay thore, the
probability of which is (l-P ) (ii) We get a non-detonation on level -1
l
~nd move up, for which the probability is (l-p_ ). From (4.3)
l
Hence
e
=
ef =
(The limiting case where the mean is situated halfway between two
lev-~ls
i. e. midway between 1 and -1, offers no difficulty;
in this case and 0
= e' = 1/2 for a
s~~etrical
PI=q-l
1-rule experiment.)
Now the relat ive frequency of detonations on both levels together in
a series of experiments on levels 1 and -1 with the l-rulo will be
a PI
+
af
p.1 ; and the relative frequency of non-detonations eql+O'q_l"
Thus the relative frequency of detonations will be proportional to
Plq-I+PlP-1
= PI
qlq-I+P1q.l = q.l.
and the relative frequency of non-detonations to
Hence in an lIunsymmetrical" I-rule experiment
like the one we are considering, the number of detonations will ex·
57
ccad the numbor of non-detonations, since P 1 > q-l' provided a
sufficiently
l~rgc
number of trials arc made.
We shall state this result as Theorem 2.
Theorem 2.
In an experiment using the I-rule, asymptotically there
will be proportions of observations
q-l
on level 1 and
Pl+q-l
p1_
P l+q-l
on level -1 and the limiting overall proportion. of
Xts
will be
• Hence with a probability as near to unity as we please,
thero will bv '1n infinite excess of Xrs if PI > q-l and the I-rule
is used.
THE 3-RULE
Under our assumptions we shall investigate the Asymptotic proportios of the 3-rule.
Theorom 3.
Asymptotically all observations in the C3se of the 3-rule
are confinod to tho three levels closest to the mean 1, -1, -2 with a
probability approaching unity as near as wo please.
Theorom 3 may be established by using
Lemma 1.
~
series of simple lemmas.
In an Jxporiment with the 3-ru13, :li'ter a long number of
tri91s, observations can at most bo confined to tho four closest levels
to tho moan 2, 1, -1, -2
35
we pleaso.
with a probability approaching unity as noar
58
The proof of Lemma 1 is similar to that used for Thoorem 1.
Under our assruuptions that the
me~n
lies botween levels 1 and -1,
closer to lovel -1, Lemma 1 may be easily established.
Lemma 2.
Given three series of binomial trials with limiting rela-
tive frequencies
a,~,v
'1nd probabilities p a'
Pv
p~,
respectively,
tho limiting rolntive frequency of the combined series of trials exists
l.nd is equal to
The proof follows from the fact that we know that there exists
an NO such that, when N > NO trialshavo beon made,
I~ --I
<0,
I~-pl
<&
'/:v _vi
<&
and such that
< 6 , /;
where p"
a'
pII
Ii
ll'
p
v
~
- PB(
<
I)
arc the observed relat.ive frequGncics in the
first, socond and third binomial series,
n , nA , n
a
I-'
v
are tho numbers
of times we conduct the first, second and 'third binomial serias -iIi Ntrials and
e;
.6 arbitrarily small quantities greater than zero.
59
Lmnma 3.
In an experiment with the 3-rule, if
c£, ~,
y, 0 are tho limit-
ing proportions of observations falling on levels 2, 1, -1, -2, then at
leqst one of tho quantities
a,o vanishes.
If possible let us aSStml8 that
~~, ~l y.~
6 are the proportions of
observations which fallon the levels 2, 1, -1, -2 respectively, and
that a,
~
number of
, y ~ 6 are all > O.
XIS
Let 1: 1 be the proportion of the total
on the three levels 2, 1, -1 to the total number of ob-
servations on those three levels, and lot 1:_1 be the similar proportion when considering the three levels 1, -1, -2.
According to
tho 3-rul0, it is possible to obtain an observation on level 2, if
°
on lovel 1. We can thus oband only if 1: ~ 1/2 and we obtain a
1
tain an observation on level 2 only if 1:1 assumes a value < 1/2. Thorofore if
c£
> 0, we know that since in a largo num1)or of trials we obtain
a proportion a of the observations on level 2, ~l < 1/2 infinitoly
often.
After moving to level 2, as long as 1: < 1/2 we continue ob1
servations on levels 2 and 1.
Since P2 and Pl' the probabilities of
detonation on thoso levels are both> 1/2, we shall obtain an excess
of Xl s and move down below 1 with a jXobability as ncar to unity as
we plcase in a finite number of trials.
Thus sooner or later L bol
comes > 1/2 and we obtain a X on levelland we shall move down. Thus
as
c£
~ 0,
L
l
< 0.
Now a ,
~
1/2 infinitely often and ~l > 1/2 infinitely often.
, y. , 6 being the proportion of observations falling on
60
levels 2,1,-1,-2 respectively, the proportion of the total number of
XIS on tho three levols 2,1,-1 to the total number of observations on
P2CL +
PIAt' + P -I'..,
This is tho limiting value of the proportion of the number of
XI s on levJls 2.,1, -1 to t he total number of observations on these
"
levels and hence must be the limit of Zl.
> 1/2 and < 1/2 infinitely often.
But Zl assumes values
Thus this limit itself must be
1/2 so that
Similarly
Pl~ +
P-1Y + P_2 6
~ + Y
+ 6'
1
= '2
=~
+ y + 5'
61
or subtracting,
Le.
(2 p 2 -l)a
= a(2 P-2 -1).
But 2 P 2 > 1 and 2 p -2 .< 1 so that
a(a positive quantity)
which is absurd, unless either a or 5
a
= 6(a
negative qumtity)
0
From Lernna 3, it follows that we cannot have limiting
proportions
4
quantities
aJ~ ,y.)o
a,p.,y,t)
of observations on levels 2,,1,-1,-2 with all
> O.
However, 1.mder our assumptions that the
mean lies between levels 1 and -1, being closer to level -1, it
follows from the very nature of the 3-rule (cf Theorem 2) that at
least 3 of the quantities of
a.,f3.,y.,5 must bo greater than zero.
We shall now examine the possibility whether observations could be
confined to the three levels 2-.,1,-1, i.e. a
Lemma
4.
Lemma
3, we cannot have
~ 'Y
> 0 and 5 = 0
Under our assumptions and using the samo notation as
a.
f3
'Y > 0, 5
=0
For, if possible, let
observations be confined to tile 3 levels 2,.1, -1 so that a
~
"'( > 0
_OJ
and
8' ::
0
Then, after a long number of trials when we obtain an
X on level -1, we stay on level -1, since we cannot reach level -2.
Henc8, the value of
~-1
(in which we of course use 5
=0
) is
always < 1/2 and consequently whenever we obtain a 0 on level -1
we shall move to 1.
Further the limit i: l
= P 1a+ P
2~+
P-1 Y
should tend to 1/2 just as in the previous argument, since it
assumes values> 1/2 and < 1/2 infinitely ofton.
i:
1
>
At some stage when
1/2 we shall thus experiment on the levels 1 and -1 with the
rule that on a X at level 1 we move down; at a 0 on level 1 we stay
on this level.
Noting that 1:_ < 1/2 al ways, we shall thus be
1
using when i: > 1/2, the I-rule.
l
But by Theorem 2, we know th~t
with a nrobability as close to unity as we please, we obtain an
indefinitely large excess of detonations (X1s) when we proceed by
the I-rule on levels 1 and -1.
Hence we shall find that after a
stage in our experiments, i: will continue to remain> 1/2 and
1
i:_
1
steadily increases and, ultimately after a sufficisnt1y large
nurabor of trials, since i:_ will exceed 1/2 with a probability
l
as close to 1 as we please, we will have to move down to level -2.
This contradicts our assumption that all observations are confined
to levels 2,1, -1 and our lemma is proved.
By similar
~rguments
it can be proved that oscillation cannot
take place on all four levels 2,1,-1,-2, and we have thus established
Theorem 3.
After a sufficiently large number of trials with the
3-rule, we shall be experimenting on the three closest levels
to the mean, with a probability approaching as near to unity as
we please.
From this form of statement of Theorem 3 it is clear that if
the mean lies closer to level -1 than level 1 the three levels on
which all observations are confined arc 1,·1,-2; whereas, if the
mean were closer to level 1 than level -1, all observations are
confined to levels 2,1,-1.
For simplicity, we shall continue to
nSSlrme hereafter that the mean is closer to level -1.
X-events and O-events.
He now define an X-event and a O-event in experimenting with the
3-rule.
An
X-event starts when we first pass from the middle levo1
-1 down to the lower level -2.
Prior to an X-event occuring we
shall be taking observations on the top and middle levels only.
However, since we are bound to obtain an excess of XIS while we
experiment on the top and middle levels, a stage will soon come when
we shall have to pass below the middle level -1, according to t h3
3-rule.
The X-event starts with th3 first trial on level -2 and
continues while we take observations on the bottom and middle
levels.
Thus as the X-event continues we obtain eventually an
excess of O's and by our 3-rule the stage arrives when we shall cross
the middle level after which
level.
WG
shall take an observation on the top
The first observation on the top level now constitutes the
start of an O-event which event continues till we pass the middle level
again towards the lower level.
The first ohbervation which is an X
on the middle level after which we crossthe middle level towards the
lower level is the last trial in the O-event.
An
X-event is obviously followed by a. O-event and preceeded by
a O-event.
Asymptotically, X-events and O-events will alternate in-
finitely often in an experiment with the 3-rule, the probability of
such an alternation not taking place being arbitrarily small after a
sufficiently large number of trials.
the fact that
~-l
This is merely an expression of
the proportion of X's on tho 3 levels 1, -1, -2 to
the total nQmher of observations on these levels takes values > ~ and
>~ infinitely often.
Theorem
4.
With a probability approaching unity as near as we plcaso,
X-events and O-events have finite lengths.
This follows as a Corollary from Theorem 2.
The top level has
probability PI for an X and tha middle level P-l for an X.
As
PI > q-l by assumption, i f we experiment on the top and middle levels
alone, we know that we
sh~ll
obtain an arbitrarily large excess of
X's provided a sufficiently large number of observations are made.
In fact, the rolative frequencies of X's and O's we shall obtain in
65
N trials whore N is large, will be
,
Hence, we expect an indefinitely large excess of XiS if we continue
oxporimentation on the top and middle levels long enough; and we are
bound to cross tho middle level according to the 3-rule before a finita
number of trials.
x2
and Xl events.
An X event can be split up into two distinct types of events
according as the excess of XiS when the previous O-event terminates
is 1 or 2.
There cannot be more than an excess of 2 XiS or less than
an excess of zero XiS when a O-event terminates.
This is obvious from
tho way the 3-rule operates and our definition of O-event.
If at a
certain point we have an X on the middle level but no excess of XiS on
the three levels together, tho O-event has not yet terminated; for at
least one more t rial has to be made on tho middle level before we move
down and thus tho O-event is still in progress.
If we had an excess
of thrGe or more XiS bofore we move down, and an X on the middle level,
this implies that before an even number of trials we should have had
an excess of at least one X with an X on the middle level; so that the
O-event must have already terminated and we l-lould have moved down. 1"3'0
shall, therefore, never move down with more than an excess of two XiS
or less than an excess of zero XiS, whenever a zero event terminates.
66
A zero event which terminates with an excess of 2 XIS gives rise to an
2
X event and a zero event which terminates with an excess of one X gives
rise to an Xl event.
We could similarly define 00 and 01 events.
All possible series which give rise ~ an X2 event
Let us assume we have a 00 event.
All series of this 00 event
which have the form as below:
T
Series I
M
x
x
x
X
o
X
-X
X
o
X
0
X
••••
0
i.e. all 0 series which do not include a 0 on the top level will obviously lead to X2 events. Further in a 00 event, i f at any stage we
obtain a 0 on the top level, we could not obtain a X2 event.
For in
o
a 0 event we have no excess of zeros when we start at the top level.
st
Let the 1
trial on the top level be a zero. Then since whenever we
get a X on the top level we move down, we can
almo~~
reach level -1
with no excess of X's.
Now if we obtain an X on the middle level we
0
obtain a Xl event following the 0 event. However, if we obtain a 0
on the middle lovel we shall again return to the same situation where
we have an excess of I zero and we experiment at the top level.
This
excess of 1 zero can almost be neutralized by experimenting at the top
level,whjle, in order for an X2 event to occur we need an excess of I X
at the top level to be followed by a X on the TT'iddle level.
Thus the
series I (above) includes all possible series constituting a 00 event
which can give rise to a X2 event, and any 0 event which includes a
67
zero
011
the top level can only give rise to a Xl event.
Obviously a 01 event is never followed by a X2 event, for in a
1
0 event we can never come d own from the top lovel with an excess of
I X.
Honce, a 04.
0
vent is -9.lways followed by a Xl Gvent.
A 00 event
is followed by a X2 event with probability p (which is the probability
of a 00 event in SeriesI occurring)and by a Xl event with probability
l-p.
Cons ide ring similarly the series II of
X+
evonts
on
the middle and bottom levels which give rise to a O~ event
o
M
Series II
X
eo·
o
X
0
X
000
0
.....
and denoting by q the probability of all such sories,
A Xl evant is followed by a
01
event with probaQility.q; and by a
0° event with probability l-q.
A X2 event is always followed by a 00 event.
Theorem
50
o 01 ,X1 and X2 events form a l'1arkov chain 'ttlith
Thusthe 0,
the Transition matrix:
i
10
0
l-p
0
1
q
0
0
0
I
I
V:q
0
P \
o ,
I
:)
68
~e
run tho
can now ovaluato the probabilities with which in the long
4 types
of events can occur.
Let them be
~,
u2 ' u ' u
3
Then
kUk
=
1,
and
•
From the transition matrix,
Solving these oquations, wo obtain
1-q
~
==
9(1-qp)
q(l-p)
u
2
==
2(1-qp)
(4.5)
1-p
u3
==
u4
=
2(1-qp)
p(l-q)
2(1-qp)
4
69
l\To
shall now consider a series which constitulBes a 01 GVGl'lt.
We knovl
that a 01 event is followed by a X~ event always, so that the last
1
trial which tcrminat es a 0 event is an X on the middle level. Let
M, T , M
TO be t he numbers of XIS and 0 I S on the middle and top
x xO'
.
levels respectively. when a 01 event has occurred. Let N and NO be
x
the total number of X's and O's on both top and middle lovels in a
01 event.
Since at the end of a 01 event we obtain a Xl event,
Further when we have terminated the 01 event, we must have come down
one more time from the top level than we have gone up from the middle
love1, since at a 0 1 event we start at the top level.
(4.6)
T
x
= M0
+
1
x +
M
Therefore
Now
x =
N
T
x
by definition•
•
••
70
•
(4.7)
Mx "" TO + I
•
= M + M or in a 01 event the number
x
x
O
of observations which fallon the top and middle levels is always the
Using (4.6) and (4.7)
To + T
same.
Arguing in a similar way, the number of observations which
fallon tho bottom and middle levels in a x2 event is always the same.
We have already seen that the only possible series which give
2
rise to a X event are the follOWing series constituting a 00 event.
x
x
X
X
X
X
o
o X
X
0
X
....
All these series together have the probability p of occurring, and
these series too give rise to equal numbers of observations on the top
and ~ddle levels.
The remaining series, when a 00 Gvent gives rise to
a Xl event can in a similar way be sho1fn to givG rise only to the
following possible sots of observations on tho top and middle levels.
T
2
J
4
5
•••••
M
I
2
3
4
71
Theorem 6. During any 0 event, the only series which give rise to unequal numbers of trials on tho top and middle levels are the 00 series
which load to Xl events.
The top level in
such a case will have
exactly one more observation than tho middle level.
proportion
(l-p)(l-q)
2(1-pq)
Thus only in a
of cases, which represents exactly the prob-
ability of a 00 event boing followed by a Xl event, do we expect to
have one moro observation on the top level than on the middlo level.
In the remaining
p(l-q)
q(l-p)
+
2(1-qp)
2(1-qp)
0
(Prob. of 0
event being .
followed byX2
event)
p+q-2pq
=
(Prob. of
01 event
occurring)
2(1-pq)
1
= -2
(I-pH l-q)
-2(I-pq)
proportion of casos of 0 events, wo shall obtain an equal number of observations on the top and middle levels.
Arguing similarly for X events,
wo
find that we obtain
an
equal
number of observations on the middle and bottom levels in a proportion
p(l-q)
+
q(l-p)
2(I-qp)
2(1-qp)
,
(Prob. of X2 .~vcnt
occm.rr1ng' .
of cases of X events
( Prob. of XJ:
event being
followed -by
oJ. yvent.)
p+q-2pq
=
2(1-pq)
72
and in the remaining
p+q .. 2pq
1
2
,...
(l-p)(l-q)
2(1-pq)
2(1-pq)
proportion of cases we expect one observation more to fallon the
bottom level than the middle level.
Bounds for the value of rex) by the 3-ru1e
Utilizing Theorem 6 we shall now set bounds for the value of rex)
in the case of the 3-ru1e for our nLwerica1 example.
When an 0 event occurs, the distribution of the numbers of
observations on the top and middle levols is as follows:
T
1
2
3
4
M
1
2
3
4
T
2
3
4
M
1
2
3
eithor
...
or
Thus, given that a
4
...
0 event has occured, the expected numbers of
observations on the top level t (T
I 0)
and the middle level
73
. ,(M I 0) satisfy the following inequality
1
<~
(M
r
C (T I 0) = ~
Let
Similarly
t (:a I X)
=
0)
<!
(T
I
0)
C. (M I
<I
where 0 < 9 < 1
(M ( 0) ... 9.
~ (M / X) + 9 1
0) + 1.
where 0 < 9 1 < 1.
The table below gives tho , closest levels to the mean in our
example with their corresponding probits and weight function, and
the expected proportions of observations falling on them:
Probit
w
1
5.5
.581
(rf
Middle Level -1
4"9
.634
O
X
(M + M )/
Top level
Bottom level -2
where MO
Proportion of observations
O
+ Q)/L-2(M +
(!If
.532
= t(M' 0) and
L-2(MO
MX.
~)
...
+ Gl)
~ (M
t
X).
Q~]
+
Q
+
MX)
+
Q +
_7
01
74
Let us consider the function
= 1.215 M
° + 1.166 M x + .581 Q + .532
2( M 0+ M x) + Q +
where
°<
~ <
1,
°
< 9 1 < 1,
M
QI
Ql
° and Mare > 1.
x
This represents I(x) for our numerical oxamp1e and is a continuous
function throughout the range of the variables we consider,sihoe
the denominator can never assume the val110 0.
The numerator and
denominator are simple linear functions of tho variab1os.
;g
= L-2( MO+ MX)+O~QI_7.581 -~lo215MO+1.66M x+&551Q+532~lnZ< 0
L-2(M O+M x) + 0+91_72
whatever the values of M 0, Mx, 9 1
may be.
75
Similarly
O
whatevor fixod values M ,
while
NX,
g may have)
~2( ~+ ~)+Q+OI_71.215 • 2 ~1.2l5MO+ •••+.532Q'_7
2". •
•
»- -..7
4A '
whatever ).Ix. 0. 0' may be.
o
-.098M ••
004~ + .102QI
4
•
*
0;
L
2
_7
Given any fixc4 va1iji of ~, the value o! ~ can be decre~.ed
by
> 0
76
increasing 8 to 1.
increasing S' to its maximum value 1.
o
decreasing M to its minimum possible value 1.
(iii)
e
At
o=1
= 1, 8' = 1, M
>
O.
Hence giving JIf the value 1, we get the lowest value for ¢ under our
conditions as
3
¢ = ~.634
+ .532 + .581
_7 = .582
(approximately)
.\n upper bound for the value of l(x) could be gotten as fellows:
For every observation which falls on the middle level we know that
at least one observation falls either on the top or lower levels.
Hence at most 50 per cent of the observations in a 3-rule experiment
can fallon the middle level.
The remaining 50 per cent can at best,
for the greatest value of l(x), fallon the levell, which is closer to
the mean than -2.
s = 8'
This corresponds to the values
.. 0, MO
= 00,
llr any finite value in our function
C;.
77
The upper bound for the value of r(x) is for the case of
the 3-rule
1.215 = .607 (nearly).
2
~e
shall conclude this chapter with a table of values of
r(x) for the various sequential procedures under consideration.
The level -1 coincides in all cases with the value ... 1 for a
N(O, 1) population, and different heights d were used between the
equally spaced levels.
~s
Method
of rex) for the different
Mood and Dixon
d
seq~e.Etial proce'!.~
3-rulo
I-rule
Lower Bound
Upper Bound
.3
.593
.617
.631
.631
.4
.579
Q608
.625
.626
.5
.565
.596
.618
.620
.6
.552
.582
0607
.611
.7
.537
.565
.596
.605
.8
.523
.546
.,,83
.595
.9
.511
.525
.569
.586
1
.497
.503
.553
.575
78
Thus, asymptotically the I-rule has the greatest concontration
of observations about the mean.
The lower bound for the value of
I(x) in the case of the 3-rule is greater than the value of I(x)
for the Mood and Dixon method, throughout the range of values of
d considered.
CHAPTER V
EST~1ATION
OF THE ED50 IN SAMPLES OF MODERATE SIZES
In the two preceding chapters, we have shown how to obtain in
small samples up to size JD the exact probabilities of taking the mth
observ~tion
en a given level following the Mood and Dixon method, the
I-rule and the ]-rule, and we have investigated the asymptotic behaviour
of these sequential procedures.
However, we have up till now not con~
sidered the problem of estimation of the ED,O in samples of moderate
sizes obtained by the 1- or 3-rules.
To apply for a sample of size ,0
or even 20 , the exact method we have used for samples of size 10 would
prove rather tedious; on the other hand, the asymptotic formulas obtained might prove grossly inadequate for samples of these sizes.
We
shall therefore in this chapter consider methods of estimation of the
ED,O from data o1:tained by the 1- and 3-rules and attempt, in a few
particular cases at le,st, to
com~~re
the efficiency of estimation by
these methods with that of the Mood and Dixon method, by the use of
numericJl eX1mples.
THE 3-RULE
1~Testgarth
L-
12
_7
considers the following estimate of the mean
in samples of size 20:
An experiment is performed using the sequential procedure proposed by Mood and Dixon for a sample of size 20. Let r be the number
i
th
of observations on tho i
level. The mean of the frequency distribution obtained by Mood and Dixon's method is itself taken as an estimate
80
of the mean of the original distribution.
Zir.
1\
IJ.w
= -
J.
20
This method of analyzing the data usually introduces inaccuracies when the sample is small,
But in a test with only 20 observa-
tions any other alternative procedure, too, might prove inadequat e.
How-
ever, this method of analysis involves little computation and it is
suggested ~ 12
_7
that this method be used with pilot sequential tests
for obtaining a rough estimato of IJ..
A study of the Tables for samples
of size 10 given in Chapter III suggests an alternative method.
If ob-
servntions were taken by the J-rule, we may, in small samples, use the
mean of the frequency distribution of the observations as a rough estim'.l.te of IJ..
Example 1.
50 samples of size 20 were obtained artificially using
Smith's Tables of Random Sampling Numbers L- 7 _7.
The s~e numbors ~ 21 st and 22n~housands_7 were used for obtaining
Kondnll and
B~bington
estimates of the mean of a N(O,l) population in 50 samples of size 20
by (1) the Mood and Dixon method (2) the 3-rule.
Th~ height interval
used for the experiment was,a and the various levels chosen for experimentation are listed below:
81
Loval
(arbitr~y
Normal
scale)
Random numb8rs corresponding to a
detonation
Equiv~lent
Deviate
3
2.2
00 - 98
2
1.4
00 - 91
1
.6
00 - 72
0
-.2
00
- 42
-1
-l.
00 - 15
-2
-1.8
00 - 3
-3
-2.6
00 - 0
For o1ch s'.mlple of sizG 21" taken by oithor m8thod, tho same inttinl
level of Gxporimentntion was chosen at random using
L- 7 _7.
From a
typiCl1 sequence of 20 observations obtained by 8ach of tho above
methods we illustrate b'llow tho estimation of \-L.
(1)
3
2
1
0
-1
-2
-3
The Mood and Dixon lVluthod.
X
X
X
X
X
0
X
0
0
X
0
X
0
X X X X
000
82
Tho Mood and Dixon estimate is b3sod, in this case, on the
numbor of non-detonations and using equ3tion (1.11)
~. and D
• ]v1
Wcstg~rthls
= - 6/8
~ 1/2
= - .25
mQthod applied to tho above d2ta yields
1'\
IJ.
w
=
..
20
Using tho sarno random numbers and the same initi81
arenow obtained by the 3-rule.
1eve~
The sequence of observations is shown
below:
.3
2
1
o
-1
X
X
X
X
X
X
o
0X
0
0
o
X
X
-3
Thusl ~.3-rUlo
X
000
-2
"-
observations
.00
X
83
In tho 50 sQmples of size 20 obtQined in this fashion using the
Mood and Dixon mothod and the 3-rule, the moan and variance of the 50
~'s by
(1) the Mood and Dixon method (2) Westgarth1s method (3) the
3-rulo arc calculated and given below in arbitrary units:
i\
Variance ~
Mothod
Mean I.l.
Mood and Dixon
.196
.1$66
1Nostgarth
,192
.1482
3-rulo
.205
The estimates of the true mean 0 with their standard deviation
from the 50 samples aro obtained as follows:
~. and D =
(1~
w
= .192
I.l.w
1\
1.l. 3 - rul o
=
= - .0464
x .8 - ,2
(1,4
::;:
j .1482
x
.205 x .8 - .2 =
(1"
1.l. 3 - rul e
.0432
•
= ).1566 x.8 = .317
M :'\nd D
"
I.l.
= -
.196 x.8 -.2
. M
=
•
•
.8 = .308
•
- .0360
J.1683 x
.8 =
.328
84
Estimation with the l-rule.
Wo know thatasymptotically all observations are confined in the
case of the l-rule to the two levels closest to the mean on eithor sido
of it.
Lot the probabilities of detonation at these levels 1 and 0
say, be expressed in percontages as y and x.
Then for every y observations on 0 we hnve
level so that y > 50 > x).
100-x observations on 1.
Hance out of every 100 + y - x observations,
y fallon 0 and 100-x on 1.
y
1
x
o
deton~tions
(y being at tho upper
Lot us now calculato tho total number of
100-x
100+y-x
y
100+y-x
on both the lovels whon a total of say M observations nro
taken on the two levels together.
We expect out of these M observaMy
M(lOO-x)
to fallon l'lnd
tions,
to
f~ll
on O.
Tho
100+y-x
100+y-x
M(lOO-x)y
expected numbor of dotonations on '1' is
and on '0',
( 100+y-x)100
My
lOO+y-x
-100x
The overall number of detonations on these two
85
l'1y
and the overall numbor of non-detonations is
lovels is
100+y-x
M(lOO-x)
We shall consider as in Chapter IV the case
simil:lrly
100+y-x
where the lovel 0 is closer to the mean, the opposite case where the
level 1 is closer to the mean being handled similarly.
vic have then,
y > 100 - x and tho excess of detonations we expect to obtain jn M obscrvations is
M(y + x - 100)
(5.1)
100 + y - x
Tho true moan is at 50 and in tho scale represented by taking
the level with a probability of detonations y as 1, and the level with
a probability of detonations x as 0, we may reasonably take the moan as
50 - x
y - x
The mean of the M obsorv3tions taken by tho I-rule, when M is
100-x
and we need to apply to this me3n the corroc-
l'lrge, is at
100+y-x
86
100-x
50-x
lOO+y-x
y-x
tion f3ctor (c.f.)
to estimate the true monn.
This c.f. may be written as
50 (y+x-100)
(lOO+y-x) ( y-x)
50
Thus if we multiply (5.1) by
and usc this as the c.f. on the
(y-x)M
mean of the obser'rations obtained by the l-rule we expect to obtain
an esti!l13te of the menno
In prnctice, however, y and x are unknown.
itsolf to provide estimates of y and x.
ber of
observ~tions
\~on
We might usc the dnta
a sufficiently large
n~~-
nrc at hand we might expect to got fairly
y,
accurate estimates
Q of y and x which could be used for the evalua50
and thus obtain tho c.f. This is hardly the case
tion of
y-x
in s~mplos of size 50, especially with fine levels.
Even with a rough estimate of cr, which is
gener~lly
needed for
all the methods usod in practice, tho question of the bost choico of
the hoight interval between levels arises.
The concentration of ob-
sorvations ne8r the mean lovels is, as we have noticed, quito marked
when using tho l-rule.
using
:'1.
However, with a sample of 50 obsorvations and
hoight interval of
S1Y
.5
between tho levels of Gxperimenta-
87
tion, we find that a few non-responses at a high lovel or vice versa
in the first few observations prevents a rapid approach to the mean.
Estimates of the moan from small samples by the I-rule using the mean
of tho observations as an estimate of tho true mean tend to be rather
poor.
It would be preferable to modify the I-rule in some way, so
that we would bo assured in the first ten or fifteen trials a fairly
even distribution of observations around the mean.
In the examples
below, we use the 3-rule to start with for the first few observations,
and then continue the exporiment using tho 1-ru1e.
Example 2.
Estimation by the 1-rule in samples of size 50.
Fo110w-
ing the same procedure used in ~xamp10 1,30 samples of size 50 were
taken (1) by the Mood and Dixon method (2) A modified 1-ru1e.
The
height interval used is 1 and the various 1evo1s chosen for experimentation are given below:
Level
N.E.D.
Random Numbers corresponding to a detonation
3
2.25
00 - 98
2
1.25
00 - 88
1
.,
.25
0
-.75
00 - 22
-1
-1. 75
00 - 03
-2
-2.75
00 - 00
-
00 -
57
88
The modified I-rule which was used is as follows:
For each sample of
50 tho first 15 observations wore obtained using the 3-rule.
is now followed for the remaining 35 observations.
The l-rul<J
For tho estimation
of tho mean only the last 35 observations taken by the.l-rule are
utilized.
The first 15 observations arc used only in so far as they
affect the remaining observations in following the I-rule.
Let n be the number of detonations and m the number of nondetonations in the last 35 observation. Since the height interval used
50
in the experiment is (J, the factor ----- will vary for the case of a
y-x
50
5°
normal distribution from
to
• These values corres-
38.30
34.13
pond to the extreme cases when the moan is exactly halfway between the
two levels of experimentation used in the I-rule, and when it falls on
one or other of the two levels.
Hence
~
is expected to vary in
y-x
such a case from 1.31 to 1.46.
-/-y = 57,
x = 22, y-x: 35
50
In our oxample 2, it is exactly __
35
-7.
Thus the c.f. applied to the mean of the last 35 obsorvations is
50 ( m-n )
3S );
We give below a typical sOqlence of observations by the modified I-rule
and the
estim~tion
of the mean from it.
89
3
2
. X
1
o
X
0
X
o
o
x x x
X
XO
0
x
0
01
'.7
o
ox
xxo
x
o
0 0 0 0 X
0
-1
-2
3
2
x
1
o
X0
oX x
X
0
O
o
0
Jlo.
0 0 X
o
o
0
X
0
•
-1
-2
The estimation of
~
was done as follows:
Numbers of
detonations
Numbors of
non-detonations
Total numbors
of observations
Level 1
11
10
21
Level 0
3
11
14
m = 21
10
c.f. = 'E
35
n
=-14
(m-n)
•
"
~l-rulo
=>
7
21
= "E
+
10
"E - .8857
•
90
The 30 s:lmples of 50 by each of the abovo methods were obtainod
using the 71st, 72nd and 73rd thousands of ~7_7.
the samples of size 50 we start at
uxperimontation,
both the methods.
r~dom
For each of
at one of our lovols of
the same random initial lovel being used for
The estimates of \.l. in the 30 samples of size
50 by tho I-rule and the Mood and Dixon method are giv0n in the
table below.
S.1mp1e Number
Initial Level
"'-'M
1\
mdD
\.l.l-rule
1
3
.6304
.5143
2
0
.7000
.5429
3
-1
.8333
.8857
4
2
.7083
.4743
5
1
.5400
.7714
6
0
.6200
.8000 .
7
1
1.0000
1.1143
8
2
.7000
.6286
9
2
.8600
.7429
10
0
.9400
.7714
11
0
.maS
.8286
91
Sample Number
Initial Level
1\
1\
liM and D
iiI-rule
12
0
.875
.7429
13
0
.7917
.7143
14
-1
1.0417
1.0571
15
0
.66
16
2
1.125
17
2
.7083
.9714
18
1
1.02
.9714
19
1
.66
.8
20
2
.75
.6857
21
2
.66
.6571
22
0
.4583
.6286
23
2
.875
.9714
24
1
.42
.6286
25
2
.75
.4743
26
2
.82
.6286
27
-1
.7174
.8571
28
-1
.82
.9000
29
1
.86
.8729
30
1
.94
.9143
.8
.8857
92
The estilllDtes of the true mean .00 with their variances from the
samples of 50 arc given below:
1\
~ and D = .0277
Var ~ ::; ,,02659
~ I-rule
Var IJ.
= .0245
1\
= .02692.
The agreement between the two methods used is very good indeed.
Summary
We thus notice that the 3-rule and the I-rule may be used to
provide estimates of IJ. in small samples.
In the case of the I-rule,
the method of estimation proposed here depends on having a prior knowledge of the approximate value of cr, as, effectively, do all other
methods.
However, i f an approximate value of cr is not available, the
value of lIy_x ll for the c.f. may be estimated from the data itself.
With fine levels, particularly in the case where one of the trials
levels is chosen rather close to the mean, a sample of 50 may prove
too small for obtaining an accurate estimate of the c.f.
A sampling
experiment for this case, where one of the trial levels is rather close
93
to the mean and the value of y-x is estimated from the data, is described
below.
In a recent paper published by Brom11ee, Hodges and Rosenblatt
L- 2 _7,
the question of using rather large "steps" at first to reach
the area of the mean and the possibility of using smaller steps later
has been investigated for the Mood and Dixon method.
They suggest the
use of wide intervals for the first few observations and a change to
smaller intervals with the first change of sign i. e. if the first observation were a detonation we use the wide intervals until the first
non-detonation is obtained and then change to smaller intervals.
We
shall use this subdivision of intervals for the modified I-rule in our
example 3.
Example 3.
The hoight interval used for the estimation of the raean in
a N(O,l) population is
.5.
The levels chosen for the experiment with
random numbers corresponding to a detonation are listed below.
Arbitrary Scale
N.E.D.
5
2.47
1.97
1.47
.97
.47
- .03
- .53
-1.03
-1.53
-2.03
4
3
2
1
0
-1
-2
-3
-4
Random numbers corresponding
to a detonation
00
00
00
00
00
00
00
00
00
00
-
99
97
92
83
68
48
29
15
06
02
94
3a samples of size So were obtained using
Dixon ~ethod (2) a modified I-rule.
modified I-rule is as follows:
-/-7-7,
by (1) the Hood and
The procedure adopted for the
A height interval 1 is used until we
obtain the first change of sign, and thereafter a height interval 1/2
is used.
The first IS observations arc taken by the 3-rule, the re-
maining 3S being taken by the I-rule.
A typical sequence detained by
this method and the estimation of I.l. from it is given below:
4
3
2
X
1
a
X
~l
a
-2
ax
X
X
a
0
alax
X
x-a
a
x
a a
X
-3
-4
2
1
a
-1
X
0
aa0
0
aX
X
0
xa
x
x0
a
X
a
0
aaX
X
Xx x
a
X
a a
95
Levels 4, 2, 0, -2, -4 were used at the start of the experiment,
and the chango to a height interval 1/2 is made after the first chango
of sign. (3 rd observation).
The result of the last 35 observations can be shown as follows:
Level
Number of
detonations
1
5
2
a
7
13
-1
3
5
n=15
m=20
-
(m-n)
=5
Number of
non-detonations
•
If, in such an experiment, all 35 observations were confined to
just two levels, the value of y-x is easily calculated from the data,
and the c. f. used for the mean of the observations is
+
..22
y-x
(m-
S) .
3
However, if the last 35 observations were confined to three
levels as in the example above, we calculate y' and
1
of detonations on the two extreme levels and t a,ce
Xl
.
100
the percentages
yl-x r
(m-n)
~
as
th
e
c.f.
Thus in our example
y' = 5/7 = .714
x'
and
= 5/14 = .357
~ = .3680 (in arbitrary units)
The 30 values of ~ were calculated for the 30 samples of 50 by
96
both the Mood and Dixon method and the modified I-rule, using the
random nwnbers and the same initial level for each sample.
sam~
The mean
and variance of these 30 values of ~ (in arbitrary units) are given be-
"~
and D
= .0435 •
~l-rule = .1194 •
Vail'
~ = .1054 •
Var "
I-L
=
.1328 •
=
.0263 •
=
.0332 •
Converting from arbitrary units
~H and
1\
I-Ll-rule
D
1\
.0082
Var I-L
= .0297
Var I-L
=
I'
The Mood and Dixon estimate is closer to the true mean and has a
smaller variance as well.
Thus the I-rule can be used only when the following two conditions are satisfied (1) A prior estimate of a is available (2) The mean
should not coincide with or lio too close to one of our trial levels,
but should approximately be situatod halfway between them.
97
APPENDIX
~fu
givG below a complete list of the fixed patterns for the values
...
of m=l, 2,
9, for a rankit set-up with 11 levels, the first observa-
tion being at level 7. (C.f.
Chapter III)
p=2.
m=4.
o
0
o
x
x
x
X
x
p=2.
m=5.
x
X
o
o X
X
o
0
o
o
p=3.
m=6.
X
X
X
0
0
0
X
X
X
X
X
0
X
0
0
X
0
0
98
rrF7.
p=9.
x
x x
x
x
o
x
o
x
x
x
0
x
o
x
x
o
0
o
o
o
x
x
x
x
x
o x
0
x
o
0 0
o
x
o
0
x
0
x
0
o
x x
o
o
o xx
x
o
o
o
x
x
o
o x
x
x
0
o
o
99
m=8
x
x
o
p==27.
x
X
x
x
x
x
x
x
x
x x
x
0
o
x
o
x
0
0
0
o
x
x
x
x
0
x
0
x
0
0
X
x
x
x
0
0
0
0
X
0
0
X
0
0
X
X
0
X
0
0
0
0
X
X
0
X
X
X
X
0
0
0
0
0
X
X
0
0
100
x
x
x
0
x x x
o 0 o x
o
o
0
0
x x
o x
x
o
o
o
x x
x
0
0
x
o x
o x
x
0
o
x
x
x
0
x
o
o
x
x
x
o
o
o
x
0
x
X
X
0
x x
o x
0
x
x
o
x
o
x
0
x
x x
o
0
o
x x
o
0
x
o
0
0
o
x
x x
o
o x
0
o
o x x
o
0
o
101
x x x
o
0
0
x
o
X
x
x
0
x
o
o
o
x
x
0
x
x
X
0
x
0
0
x
0
x
o
x
x
x
0
x
x
x
X
0
X
0
0
X
X
0
0
0
0
m=9
X
X
0
X
X
X
0
p=49
0
X
X
0
X
X
0
o x
X
0
0
0
x x
o
x x
0
0
0
x x x
0
0
X
0
X
o
X
X
x
x x
000
c
0
o
0
x
x
x
0
o
102
x
x x
o
0
x
x
o
x
x
0
o
x
x
X
0
x x
o o 0
x
o
x
x
o o
0
0
X
0
x
x
x
0
o x
0
o
0
.
X
o o
x
x x
x
0
x
o
o
x
x
o
x
0
o
o
o
0
x
x
0
0
x x
0
x
x
0
0
x
o x
x
x
x
x
0
o
0
IaJ
x
x
x x
x
x x
o x
x
o x
x
o
x
x
x
x
a
o
0
x
x
o
x
a
o
x
x x
a a x
x
x
x
x
x
o
x
0
o
x
x
0
o
a
x
x a
o
o
x
o
x
0
o
o
o
x
x
o x
o
x
x a
o
x
x
o x x
x
0
x
o
o
x
a
a
104
x
x
o x
o x
x
x
o
0
x
o
x
x
o
X
o x x
0
X
0
o
o
X
o
x
x
x
0
x
0
o
0
x
x
o o
o
0
o
x
x
x
0
X
o
o
0
0
x
X
x
0
x x
x
o
o
o
X
0
o
0
X
X
0
0
X
X
X
o
0
o
x
0
o
x
o
o
o
X
x
o
X
0
x
x
X
x x o
o o
x
x
o
o
0
X
x
x
X
x
X
X
o
o
105
x
X
X
x
X
X
x x
o
0
x
X
0
X
0
0
X
0
X
X
0
X
o
X
X
0
0
x
x
o
x
0
X
o
X
X
x
0
o
o
X
0
X
X
x
0
X
X
o
o
o
X
x
0
o
o
X
X
0
o
0
0
x
X
X
0
0
o
x
x
0
0
o
X
x
o
o
X
0
X
X
106
x
x
o x x
o
x
o
0
o
o
x
o
o
o
0
x
x
0
X
0
o
x
x
x
x x
o
x
o
o
o
x
o
X
X
107
BIBLIOGRAPHY
~1_7 Bartlett, M. S.,
,~ modified probit technique for small
probabilities", Journal of the Royal
Statistical Society, Supplement VIII
(1946), 113-117.
£:2_7 Brownlee,
liThe up-and-down method with small
K. A.,
samples lt , Journal of the American
Hodges, J. L., Jr.,
and Rosenblatt,Murray, Statistical Association, XLVIII
(1953), 262-277.
~3_7
Ehrenfest, P. and T.,
"Uber zwei bekannte Einwande gegen das
Boltzmannsche H-Theorem", Physikalische
Zeitschrift, VIII (1907), 311-314.
An Introduction to Probability Theory
and its Applications, New York, John
Wiley and Sons, Inc., 1950.
Probit Analysis, Cambridge, University
Press, 1952.
~6
-
7 Hodges,
J. L., Jr.,
'~symptotic Properties of the Robbinsand Lehmann, Erich L., Monro Process", The Annals of Mathematical Statistics, XXIV (1953) 141 (Abs.)
r7- 7 Babington
Kendall, M. G. and
Smith, B.
ra 7 Hood, A. and
- - Dixon, i1. J.
-
1'1.
Tables of Random Sampling Numbers,
Cambridge, University Press, 1939.
'lA method for obtaining and analyzing
sensitivity data", Journal of the
American Statistical Association,
XLIII, (1948) 109-126.
/-9 7 Robbins, Herbert and
-
-
Monro, Sutton,
"A stochastic approximation method II,
The Annals of Mathematical Statistics,
XXII (1951) 400-407.
"Private Communication" (unpublished).
108
~11_7
Uspensky, J.
v.,
Introduction to Mathematical Probability,
New York, McGraw-Hill Book Company, Inc.,
1937.
IITwo Problems in Efficiency", (1948)
(Unpublished Thesis submitted to the
University of London for the M.Sc.
degree).