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METHODS OF SEQUENTIALLY TESTING COMPOSITE HYPOTHESES
WITH SPECIAL REFERENCE TO THE TWO-SAMPLE PROBLEM
By
W. J. Hall
University of North Carolina
Institute of Statistics Mimeo Series No. 441
August 1965
This research was supported by the National Institutes
of Health under Grant GM-10397.
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
meetings" )
e
SPECIAL REFERENCE TO
WI1~
1~E
TWO-SAMPLE PROBLEM
Correction: pg. 37, line 12. "'l'est 1a" and "'i.'3St Ib" should be "Test lc" and "Test Id"
(Notice of other correctic[i':l, comments, and criticism are invited.)
2~
Addendum to Append:l.x
0
{'.)HELIMINARY REPORT ON SOME CALCULATIONS OF RELATIVE
EFFICIENCY OF THE PERMUTATION TEST
approx. ASN of 'lormal theory SPRT under Hi
=
A~;:fi
approx.
of permutation test under
H~
1.
The RHS is the ratio of '"pect l; d increments in the cUJD\lletive sums defin:in. g the tests
(see pp. 52-3). The hypo-.these:: Hi are normal thec.ry hypotheses about the distribution
of the observed differe"~e Z w!.en sampling in pairs (see bottom of pg. 52). The
parameter b is a 1D&8s:u'e of ~he difference between IIi and H~ (middle of pg. 53).
Tao approximations in OJ are du~ solely to the inaccuracy· of the error probabilities,
due to ignori.ng oversh',..,.t.
T!' e ratio e. is n measure of relative efficiency of the
).
permutation test relat: .VEl to the normal theory test when both tests are valid (Hi true),
It is shown on j)age 53 ·:",:it e::l. tends to 1 as h tends to 0 in fact, ei ::;;
1 - (ih) + 0(h2 ) and that Ul average number of additional observations required by
the permutation test is asymp'tCtically Ci as h tends to OJ ci 1s commonly around two to
six (see ,bottom of p:,~. 51).
Ca. ... (;ui..ntions of ai ai'a in p.i.'og ...' ess by Petel' Nemenyi.
Calc°.llations 1nd:l.cate tha"tr
e
(see below fo ...' I.li.'eliminas.'y calculations)
(1) the eff ...~iency rango'.:S from about '15% or lese (in situations requiring very
small ASN's) m(}noto:dc~lly tow~,rds 100 percent;
(2) the asymptotic
th~ory
is fairly rapidly &pproached;
(3) ·the extra number of observations required by the permutation test is
approxilJiiat.ely Ct or less-not just asymptotically, but over the range of h i.ll,vestigated.
H'
h
(& ~ 1a?2
::;;
eff'iciency
1
- ih
--
...
N= {
:6
0
5!
f3
005)
0.5
.08
,02
=
.749
.903
.981
.995
.50
.875
0980
.995
=
-- -- -
- - - - - - - - - - - - - - -
perm. test
7.1
normal test
5.3
23.5
135.1
e
_ -- -
... 532.4
-
1
--
BiN
21.2
132.5
530.0
2.6
2.4
==
.72
008
.853
.980
.820
.980
--
=\,u
- -
- -
14 ••/
--
135.1
132.5
:III
[
EoN:: .
I!ll:
___•• __J.
HU true
2.0
diff~~rence
-~
(o:
.true
==
(est. of e i )
--(0
eO
-- 0
perm. teat
l
normal test
001)
difference
AI
~
- 1.8
- _. - - 2.3
- - ULO
39.9
9
36
3,,0
3 9
0
229.4
225
4.4
- - -- -
904,,3
900
4.3
EIN =
2.6
t
29 3
•
25
4,3
W. d. HaH
Department of Statistics
Phi~lips Hall, U. N. C.
ChaPfJ1 Hill, N. C. - 27515
2.6
229.5
225
4.5
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METHODS OF SEQUENTIALLY TESTmG COMPOSITE HYPOTHESES
2
WITH SPECIAL REFERENCE TO THE TWO-SAMPLE PROBLEML,
By W. J. Hall
University of North Carolina
Three methods for developing sequential ;Qrobability ratio
tests (SPRT's) for testing composite (parametric and non-parametric) hypotheses
"'-~:
are described in Part I: (I) parameter separation, (2) data reduction, and
(3) conditioning. Method (I) is an elaboration &nd extension of an approach
attributed to Nandi [24] and used in a special context by Girshick [12]; it
is closely related to Fraser-sufficiency [9]. It generalizes naturally in
either of two ways, leading respectively to methods (2) and (3). Method (2)
includes Cox's method [7] and the invari&nce method of Stein [16] and has been
widely used. Method (3) has not been described previously except in an example
of Wald's [31]. A primary contribution of this paper, which is otherwise
largely expository, is to describe this conditioning method in some generality
and to apply it to develop some new tests for the two-sample problem, namely
sequential permutation tests analogous to the non-sequential tests of Pitman
[27] as described by Lehmann and Stein [23]. All three methods are illustrated by tests for the two-sample problem.
Part II of the paper consists of a listing and brief description of a
number of SPRT's for this two-sample problem: three kinds of normal tests,
some sign tests, some permutation tests, and some rank tests. They include
several standard tests, several recent tests, and several. new tests. Some
numerical. comparisons of these tests are planned for a subsequent paper.
1.
Presented at the 125th annual meeting of the American Statistical
Association, September 1965, at Philadelphia.
2.
Research supported by the National Institutes of Health under Grant
GM-10397.
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TAB L E
o F
CON TEN T S
SUMMARY
PART I:
1
THREE METHODS OF SEQUENTIALLY TESTING COMPOSITE HYPOTHESES
Introduction
3
Method (1):
Parameter Separation.
Method (2):
Data Reduction.
Method (3):
Conditioning
PART I I:
7
..
· 18
SEQUENTIAL PROBABILITY RATIO TESTS FOR THE TWO-SAMPLE PROBLEM
Summary of Part II .
· 32
Introduction to the Tests
• 33
Normal Tests
• 36
Normal mean, variance known (Tests la-ld)
36
Normal mean, variance unknown (Tests 2a-2b)
37
Normal mean in standard units-- t-tests (Tests 3a-3d)
Sign Tests
• 38
• 40
Sign test of differences (Test 4)
• 41
Double-dichotomy tests (Tests 5a-5b)
· 42
Permutation Tests (Test 6)
• 43
Rank Tests (Tests 7a-7f) .
• 45
I"
APPENDIX 1:
ASN of the Extended Double-Dichotomy Test
APPENDIX 2:
ASN of the Permutation Test
I
REFERENCES
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• 13
49
· 52
• 54
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PART I:
THREE METHODS OF SEQUENTIALLY TESTnm COMPOSITE HYPOTHESES
Jlf.CRODUCTION
Tlu'oughout we are concerned with a sequentially performed experiment.
We designate the data potentially available at stage n by X , and the accumu.
lated data from the first n stages by X(n)
n
= (Xl ,X2, ... ,Xn ),
n
= 1,2, ••••
We shall assmne the stages are independent, although it would not always be
essential, but there is no need for asstuning identical distributions, or even
identical sample spaces for the successive stages.
A family of probability
:rrodels is assmned for the sequence of da:t a, including t"iro sub-families desig-
11..
1\ as
nated H and
We may alternatively describe H and
O
O
eses about the probability model of the data.
composite hypoth-
We seek methods for constructing valid sequential probability ratio
tests (SPRT's) of these hypotheses.
By
"valid" we mean that the strength
(a,13) is correctly prescribed:
the probability of incorrectly rejecting H
O
is never :rrore than a and the probability of incorrectly accepting H is never
o
more than 13--at least within the limits of approximation that are inherent in
SPRT's due to the possibility of overshooting the stopping boundaries.
By
a
8PRT we mean of course a sequential test based on some kind of a joint likelihood ratio with constant termination boundaries.
lihood ratio (l.r.) at stage n by Rn (the
11.
Designating the joint like-
likelihood in the numerator and
the H likelihood in the denominator), the test is:
O
at stage n (n
after observing the data
= 1,2, ••• ),
continue to the next stage if
terminate and accept H if
O
terminate and reject HO if
:3
B<R
n
<A,
R
<B
,
R
>A
,
nn-
I
and
A
= (l-f3)/o.
and B
= f3/(l-o.).
If R is a bonafide joint l.r., then the
n
test has (approximate) strength (o.,f3) [31].3
The point about which we have so far been vague is:
bonafide l.r. for composite hypotheses?
of this paper.
how can we get a
That is indeed the SUbject of Part I
We shall describe three kinds of situations in which joint l.r. 's,
of some useful sort, may be constructed, and exemplif'y each.
Our examples will
all be related to the two-sample problem, but of course the methods are not
confined to this single area of application.
It will be noted that there is some
overlap anxmg the methods, a nd that nx>re than one of them may be required in a
single problem; nevertheless, many problems are not amenable to these methods.
In brief, the applicability of the three methods may be summa.rized as follows:
~thod (1) is applicable when the l.r., with specific values assigned to the
nuisance parameters so that the two hypotheses are simple, is free of the assigned
values; two sufficient conditions, each utilizing Fraser-sufficiency
given and they generalize respectively to the other two methods.
[9], are
In method (2),
the available data are reduced to a sequence of statistics having probability
IOOdels completely specified by the two hypotheses, and thereby a l.r. can be
constructed; the Stein Theorem of invariance and sufficiency theory [16] (or
Cox's factorization theorem [ 7]) may be useful.
In method. (3), a sequence of
statistics is chosen and fixed in such a way that the hypotheses about the
conditional distributions become simple hypotheses, and hence a conditional l.r.
can be constructed; the resulting conditional SPRT's (CSPRT's) are readily seen
to be valid unconditional tests of the composite hypotheses.
One can
bounds,
instead
on page
assure bounds on the error probabilities, rather than approximate
by using Waldls conservative stopping boundaries A = 1/0. and B = f3
of his recommended ones given above; see paragraph headed strength
586 of [16].
4
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He shall not attempt a complete survey of the methodology of sequential
composite-hypothesis testing. The methods introduced by Wald ([31], Chapter 4)
and by ~rnard [5],
in vlhich nuisance parameters are integrated out of the likelihoods, will not be
discussed at all; familiar applications of them are also covered by Cox's method
or invariance and the latter ,·Till be briefly described (see also [19] and [16]).
Nandi [24] described certain situations, Which we understand to be special cases
of methods
(1)
or (21 in ,~1ich composite hypotheses may be tested.
Girshick([12]
and Section 4.2.4 of [3l])described certain kinds of two-sample problems--of the
"'·ihicl1 of two treatments is the better one" type rather than the "is one treatment
better or is there no difference" type treated in the e.xa.nij?les here--in Which
SPRT's of composite hypotheses are possible; this is included in .method
(1).
Johnson and Leone ([20], Section 16.11) give several examples in which SPRT's
of composite hypotheses nay be constructed; they may be vievred as examples
of "lhat vle call method (2).
Both Johnson [19] and Hall, I'Tijsman and Ghosh [16]
mention many examples not mentioned here.
Large-sample methods of sequentially
testing composite hypotheses have recently been given by Cox [8], but are not
considered here.
Of course, knowledge of a 1. r. R facilitates any generalized sequential
n
probability ratio test (GSPRT)4 [21], and not just a SPRT.
In particular, a
fixed sample size test (FSST)5 may be based on R for fixed n, with specified
n
.
sic;nificance level ex, but not necessarily with specified t3.
A critical ree;ion
4. A GSPRT is a test which may be described
in an identical manner to the
description of a SPRT given earlier, except that the constants A and B may
be chosen in other ways and my vary with the stage index n.
5. A 1. r. FSST is a GSPRT ,dth
. An
Am
= 00,
Bm
=0
= B
n = c n for the fixed sample size n, and
for m 1= n.
5
I
of the form R > c
n -
n
where c
n
level provides such a test.
being based on a l.r.
is chosen to achieve a specified significance
Such a FSST would be
~ po~rerful
in same sense,
We shall not consider GSPRT's here, other than SPRT's--
that is, we confine attention to the case of constant termination boundaries A
and B.
other types of boundaries have been investigated by various authors,
including Armitage [3) and Anderson [1), but largely in special contexts.
Approaches other than those based on a joint l.r. are also omitted from
consideration here (e.g., Sec. 15.3 of Wilks [33), Nemenyi, Adelman and Miller
[25], and Hall [13]), as are all decision-theoretic methods (except as they overlap ,'lith SPRT's).
It should be emphasized that we are not making optimality claims--we are
simply seeking some test with pre-assigned strength (0:,13), and the class of SPRT's
seems a likely place to search.
We state once more that the methods have non-
sequential application as wel); in fact, they have been used extensively there,
and ,.,hat
~Te
--I
describe here rray be viewed as sequential analogs of them.
It will frequently be convenient to use parametric notation and let
index the family of possible models.
We can then characterize H. as
J.
e
r(e) = r.J.
(i=O, 1) •
When e may be written in in the form (r, Tl), ,.,e may then call r the
parameter
2f.
interest and Tl the nuisance parameter.
Thus,
hypotheses to be tested and Tl indexes each of the families
r characterizes
Ha and H:I..
the
We shall use density function notation, tacitly assuming a suitable
dominating measure; in fact,
~re
frequently use conditionaJ. densities as well.
For our purposes of exposition, we shall act as if we are dealing only with
the discrete case so that densities (probability mass functions) and conditional
densities (probabilities) can be assumed and manipulated without concern for
measure-theoretic regularities.
The methods described are certainly valid in
the usual continuous model problems as well; the full extent of their validity
"I'dll not be pursued.
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Before turning to the three methods to be surveyed and exemplified in this
paper,
I'Te
i'lish to remark that monotonicity provides a technique of extending
the validity of tests of simple hypotheses to certain Idnds of composite
hypotheses--a technique which is vTell-known [31] and vThich is also useful in
e2dending tests of composite hypotheses.
Specifically, if the model is
specified by a parameter (vector) e, and we test e
the OC function (probability of accepting
is decreasing in a numerical function
He
r(e),
= eO
el ,
vs. e =
then, if
as a function of e) of the test
the test
llaS
strength (a,~) not only
for testing the hypotheses as given but also for testing the composite hypotheses
r( e)
:.s
r( eo) vs. r( e) ~ r( 0 ).
1
Simple (and well-knOl-m) examples include the
SPRT for testing such hypotheses about the normal mean 0 = Il when the variance
2
0"
is Immm (r( e)
=
e,
eo <
the sequential t-test (e =
in the tiro- sided test).
01' random sampling from a normal population) and
Il/ 0",
r( e) = e in the one-sided test andre 0) =
(Il/ 0")
2
Monotonicity proofs are available i-Then the joint 1. r.
is :monotone in r --both for the case of tests based on independent observations
(Lehlmnn [22]) and more generally (GhOSh [11]).
The latter reference covers
nnny of the common examples of sequential tests derived by use of Cox's theorem
or invariance (method (2», and the former applies to marlJ' examples of vThat
describe as methods
(1)
i'le
and (3) below.
METHOD (1):
PARAMETER SEPARATION
In this section we aSSU1;1e a parametric formulation i'lith
e = (r, 11), and
we aSSUDill further that the parameter space is a product space.
attention to the two values 1'0 and r
l
We can confine
for 1', specified by HO and
1I:I.'
but a
la.rger range is possible too; 11 is the nuisance parameter with the same range
under H and
O
1I:I..
7
I
Suppose
Tj
is fixed at
so that H and
Tj'
o
When is the l.r. free of Tj'?
1\ become
simple hypotheses.
It will be seen that a necessary and sufficient
condition is provided by the following notion of 1 and Tj being separable.
DEFlNITION:
Suppose, for each n and each x , the likelihood. of x factors
n
n
into a function of 1 only times a function of Tj only; we then say that 1
and Tj are separable parameters.
Supressing n from the notation, we thus have
f
where
f
1,Tj
1,Tj
(x)
= g 1 (x)
h (x)
Tj
is the density function of X •
n
Since the joint likelihood is the
product of the marginal likelihoods for the respective stages, the joint 1ike1ihood would likewise factor, and the factor depending only on Tj would cancel out of
the 1.r.
for all (1
On the other hand, if the 1.r. at (11'Tj') and (1 ,Tj') is free of Tj'
0
and 1 under consideration), then we can let h
0
Tj
= f1
for fixed 1
0
and f
= g1 , which" is Tj-free by assumption, so that f 1,Tj = g 1 h;
1,Tj 1 0 ,Tj
Tj
hence, 1 and Tj are separable. Note also that if S = s (X( )) is a sufficient
1
0
If
n
statistic for
e = (1,Tj),
n
, Tj
n
then the separability test could equivalently be applied
to the distribution of S
n
rather than the distribution of X
n
(for every n).
In practice, the choice of nuisance parameter Tj as a function of
fixed by the formulation of the hypotheses
~
and
H:t.
e
is not
Hence, to determine the
applicability of tne method one would ordinarily write down the density function
of X (or of S ) with parameter
n
n
Q,
factor out that part depending solely on 1( e),
and determine if the remaining factor depends on
which (1,Tj) is one-to-one in
e and
e only through some Tj
= Tj( e) for
the range of Tj does not depend on 1.
thus speak of 1 being "separated out of
(JI,
or simply of
II
We
1 being separable."
The consequence of 1 being separable is that the nuisance parameter is
really not a nuisance!
We can act as if Tj were known, assigning it an arbitrary
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value Tj'.
Tj.
The resulting SPRT will have strength (ex, 13), whatever the value of
Since the test is in effect one of simple hypotheses (for Tj' fixed), all
of Hald's methods [31] for investigating its properties are applicable--for
every fixed value of Tj.
ThUS, under conditions of simple random sampling, the
test terminates with certainty, l'laJ.d' s OC and ASN approximation methods are
applicable, and the test has minimal ASN under HO and 11. uniformly in Tj. AlSO,
assuming simple random sampling, the approximate OC (Wald' s method) may be
seen to be free of the nuisance parameter Tj, but the ASN may depend on Tj since
it depends on the distribution of the l.r. and that distribution may depend on
Tj (but see below).
By way of contrast, for a FSST of level ex based on the l.r.
R , the critical number c may depend on Tj so that we cannot conclude either
n
n
that the test is uniformly (in Tj)
~
powerful, nor that its power (or OC) is
Tj-free.
Trivial examples of parameter separation occur vThenever X is of the form
(U, V) vThere U and V are independent and the U-distribution is governed by "'I
and the V-distribution by Tj.
Obviously, Tj (and v) drop out of the l.r., and
might just as well be ignored from the outset.
Other examples are provided by
6
the class of problems considered by Girshick [12] (not pursued here ); we will
give other examples below, relevant to the two-sample problem.
6.
In the notation of Sec. 1.8'of [12], we hold v( e ) - v( e ) fixed, at 2c
l
2
)
)
(say), let Tj(el ,e2 = (v(el + v(e2 ), w(el ) + w(e2 » = (Tjl'~)' and let
Y( e , e ) = sign of (e -e ). Then the joint density of the pair (~,~),
l 2
l'Tith u(x.)
~
= u.,
2
(u
l
2
+ u )/2
l
exp[(~-ul)Yc + 11 Tjl + ~ +
hypotheses to be tested are
v(e ) - v(e ) = 2c or -2c.
2
l
te~t, as described above in
~
= 11
and r(x.)
~
= r.,
~
is found to be
r l + r 2 J. Hence, r and Tj are separable. The
"'I = Yo = +1 VS. "'I = "'1 = -1, or equivalently
1
Girshick noted sameo~ the properties of his
a more general context.
9
I
T'lro sufficient conditions for parameter separation will now be given,
both involving the concept of Fraser-sufficiencY [9].
DEFmrrION:
Suppose X has a probability IOOdel with parameter (,., TJ)
(vdth a product-space range).
Let T
= t(X)
and U = u(X).
Then T is
Fraser-sufficient for r if its distribution depends only on r and if
the conditional distribution of X given T depends only
on~.
Likewise,
U is Fraser-sufficient for TJ if its distribution depends only on TJ and
if the conditional distribution of X given U depends only on
r.
such a statistic Tn = t n (Xn ), or = t n (X( n )), exists which is Frasersufficient for r, then r may certainly be separated, for, writing the density
If
of Xn (or X( n )) as the product of the marginal density of Tn and the conditional
density (probability)7 of Xn given Tn achieves the required factorization.
Then, in this case, the density of X(n) (and of Xn ) is the product of the
density of a statistic Tn depending only on r and a factor free of r. ThuS,
one might just as well confine attention to the statistic T instead of the
. .
n
actual data, and use its l.r. for testing H vs. HI. This situation was desO
cribed by Nandi [24]8. We pursue this conclusion (with or without separability)
as method (2) in the next section.
Since the l.r. is identical with the l.r.
for the statistic Tn' which has a model governed solely by
r,
any GSPRT would
have properties depending solely on r--specifically, the ASN is r-free.
such a test would have minimal ASN under
see page 543 in [16].)
He
and
11. has
(That
been noted by J. K. Ghosh;
Before exemplifying this situation, we turn to a second
sufficient condition f or separability.
7.
8.
In regular continuous cases, we must first replace X (or X(n»b Y a one-to-one
function (T,Y) and the role of conditional density is played by the conditional
density of Y given T.
At least, this is Johnson's' [19] interpretation of Nandi's exposition, but
Nandi is not quite so clear.
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Hate that the definition
f separability is symmetric in the roles of '1
0
Hence, we conclude immediately that if there is a statistic U = u (X ),
n
n n
or = un(X(n»' which is Fraser-sufficient for 1), then '1 and 1) are separable.
and
1).
In this case, the density of X(n) (and of Xn ) is the product of the conditional
density given Un , which depends only on '1, and a factor which is '1-free. Consequently, the l.r. (at X(n»
is identical with the conditional l.r., each
lilcelihood (or rather density) being conditiona1 on a fixed value of U (or
n
This situation has not been previously described (in a sequential
Ul'·"'Un )·
context).
For U to be Fraser-sufficient for
sufficient for
n
1)
1),
it is necessary that it be
for every fixed value of '1 (utilizing the factorization theorem
of sufficiency)--in particular, it must be sufficient for H and for HI.
o
Further consideration of conditional 1.r. 's, conditioned on such U 's, is
n
given under method
(3).
lie thus conclude that the existence of a Fraser-sufficient statistic,
either for the nuisance parameter or for the parameter of interest, implies
the separability of the parameters.
Thus, the separability of r and
1)
is seen
to be a symmetric generalization of the concept of Fraser-sufficiency.
We now
turn to some examples.
Exanwle:
trinomial.
problem "Till appear later.)
(The relevance of this example to the two-sample
Suppose each stage (experiment) results in one
of three outcomes--label them success, failure, and tie (the labels need not
have the connoted significance)--with respective probabilities Pl' P2 and p~
.
(positive ~~d summing to unity).
then
1)
)
If '1
= Pl!P2 is the parameter of interest,
= PI + P2 may be taken to be the nuisance parameter (0 < '1 <
~, 0
< 1) < 1).
It is readily seen that the joint likelihood at stage n may be expressed
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• t he f1rst
.
as '1t( 1+'1 )-u 1)u( 1-1) )n-u where t is the number of successes 1n
n
exPeriments and u is the number of successes and failures (non-ties) in the
11
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first n experiments.
That '1 may be separated is obvious.
.1
Hence, the joint
l.r. for testing (10'~') vs. (1l'~') is
for arbitrary ~'.
It is seen that the "tie" category data do not affect the 1. r. ,
which seems intuitively correct.
The resulting SPRT was introduced by Wald
(see next section), but by a different approach.
to be Fraser-sufficient for
~;
Incidentally, U may be shown
a Fraser-sufficient statistic for 1 apparently
does not exist.
I f we interchange the definition of
~
and 1, we have an example of separa-
tion where T is Fraser-sufficient for the parameter of interest.
Example:
difference between two normal means (variance known).
Suppose each observation consists of a pair (X,Y) of independent normal random
variables with means A and IJ., respectively, and unit variances.
and ~
=
(IJ.+A)/2.
Let 1 =
IJ.- A
The density of a single pair (X, Y) is
f
1, ~
(x,y)
=
(2~)
= (2~)
-1
-1
so that the separation is achieved.
1
exp[-2(X-A)
1
2
1
2
- ~y-IJ.) ]
2
1
2
exp[-4(y-X-1)] exp[-4(x+y-2~) ]
(In fact, letting X'
= Y-X
and Y'
= X+Y,
this problem in terms of (X', Y') falls in the "trivial e~le" category noted
earlier.)
The resulting l.r. depends only on y-x and tl~ resulting SPRT only
on the difference between sample means.
At stage n, S = (T,U) is sufficient for ('1,~) where T =
y-X and
U
= (X
(X and Y are the sample means of the first n X' s and Y' s, respectively.)
+ Y)/2.
Note
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that T and U are independently distributed and T is nornrU >lith mean 1 and
variance 2/n while U is nornrU with mean ~ and variance 1/2n.
Fraser-sufficient for '1 and U is Fraser-sufficient for
12
~.
Hence T is
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This example may also be treated by invariance (method (2); see next section).
That tllere is an overlap between the concepts of invariant sufficiency and
Fraser-sufficiency has been noted elsewhere [16] but not extensively explored.
Example:
~
test.
Fraser [9] gives a non-parametric example of
Fraser-sufficiency in Which the "parameter of interest" is the probability, p,
to the right of the origin in the population (with continuous distribution
function) from which one is sampling, and the "nuisance parameter" consists
of the two conditional distributions, to the right and to the left of the origin.
Suppose one takes one observation from each of tiro populations (with continuous distribution functions) at each stage and wishes to test hypotheses
about p, the probability that the difference between the observations is
positive.
Then separation is applicable (the number of positive differences
being Fraser-sufficient for y
= p),
and the SPRT is seen to be the sequential
sign test (first described by Armitage [2]), applied to
tl~
sequence of diffe-
(See Test 4 in Part II; also see Test 7b.)
rences.
METHOD (2):
DATA REDUCTION
A standard technique of statistical inference in the presence of nuisance
parameters, and in nonparametric theory, is to replace the original data by a
statistic
v~1ose
distribution depends solely on the parameter of interest.
In
non-sequential theory, one need only know the distribution of the statistic
under H to construct a test of level ex.
o
The method is of course applicabl€
sCCJ.ucntiaJ.ly as well, although distributions under both H and
O
lI:L
need to be
lillOI'm.
SequenJ~ially,
first,
ife
let Y
n
two variations may be convenient.ly distinguished.
= Yn (Xn )
In th)
be a statistic based on the stage n data X , with
n
13
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distribution governed solely by "I (since the stages are mutually independent,
the joint distribution of the Yn'S is determined by the margina.ls).
second, we let Yn
= Yn(X(n»
In the
be based on all the accumulated data X(n)
= (~' ••• 'Xn)'
again vlith distribution governed solely by "I--in fact, we must be sure that
= (Yl, ••• ,Yn )
the joint distribution of Yen)
be governed solely by "I for each n.
(For the first, it may be necessary for each stage to represent a group of
observations for such a Y to exist non-trivially).
n
for the hypotheses H vs.
O
and
1\ reduce
1\
to the simple
In the first, a SPRT
is readily constructed (in principle) since ~
h~'Potheses
"1= 'YO vs. "I = "1
1 about the distribution
of the members of the sequence Yl' Y , ••• of independent statistics.
2
The lor.
for each Y is determined, and the joint lor. R obtained by multiplying
n
.
.
together the stage-wise l.r. 'so
n
The second case is quite similar, except that
the Y 's are no longer independent.
Therefore, the joint lor. R requires
n
n
knovTledge of the joint distribution of Yen).
We refer to either of these
--------
variations as data reduction methods.
Cox [7] introduced conditions under which the joint distribution of Yen)
would conveniently factor; under these conditions the joint 1. r. of Y(n) is
identical "lith the 1. r. of Y.
n
A more general version of Cox's Theorem is due
to Stein, and appears in [16], together with an extensive discussion of its use.
To obtain this convenient factorization, an invariance structure is required,
the statistic Y being derived as an invariantly sufficient statistic under
n
some appropriate group of transformations leaving the problem (and the
hypotheses ~ and
~-free.
1\)
invariant.
The distribution of such a statistic is
For further explanation and details, see [16].
Invariance may also be applicable in the first case considered above,
each Y being invariantly sufficient for the data from that stage. This case
n
is also discussed in [16] (see also Tests 3c, 3d, 7c, 7d, 7e, and 7r in Part II).
14
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'He shall now give some examples of the invariance approach, ",ith
relevance in the two-sample
Exa1]Ple:
two normal means (variance knownl.
This example, introduced
in the previous section, may also be treated by invariance.
The difference
beti'leen the sample means is an invariantly sufficient statistic for
COTll."11On shifts in location:
(x,y)
r under
-> (x+a, y+a)[16], and. hence reducing the
data to this difference perrdts construction of a SPRT.
In fact, the observa-
tions need not be in pairs, but can be taken one at a time in an arbitrarJ order
(so long as the order is free of dependence on the previous observations;
see Test 1 in Part II).
With such a sampling scheme,
r and
~
are no longer
generally separable; in fact, neither T nor U is Fraser-sufficient except ''Then
the number of X's, say m, is equal to the number of Y's, say n.
S
= (T,U)
is
still sufficient for (r,~), and the marginal distributions still depend solely
on a single parameter, but the conditional distributions depend on both parameters.
S pecificaJ.ly, with T the difference between the sample means and U the average
of the hro sample means, T eiven U=u is normal ",ith mean
and variance 4/(m+n), and hence not ~-free unless m
= n;
r + 2(m+-n)
(u - ~)
mn
and likewise for the
conditional distribution of U given T (and the same "'"Quld hold if U ",ere replaced
by the grand mean U'
E)~le:
= (DC
+ ~)/(m+n».
two-sample t-tests.
derived, using invariance theory.
A two-sample t-test may be analogously
After obse:,,'V;ing n X's and nY's, the data
are reduced to the usual t-statistic, based on the difference between the
sar:rple means and the pooled (Hithin samples) estimate of the cOrmrK>n but unlmo'l'm
2
cr
(r'l+n-2 degrees of freedon).
only on the parameter
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proble~
hypotheses about
r.
r
Tllis t-statistic has a distribution depending
= (fl- i\) / cr, so that an SPRT may be constructed to test
See Tests 3a and 3b in Part II; a ti"o-sample t-test for
testing against two-sided alternatives
[17].
15
(r=o
vs.
Irl = rl )
has been given by Hajnal
I
Example:
rank tests.
[16], and [26].
Sequential rank tests have been described in [32],
We briefly summarize':. for testing whether t"i'1O samples come
from the same population against the alternative that one population has a
distribution function Which is a power of the distribution function of the
other (Lehmann alternatives), one may reduce the data to the ranks of the
observations in one sample from a combined ranking of the two samples.
Invari-
ance theory guarantees that the original hypotheses will be simple hypotheses
about the distribution of the reduced data.
Use of the Stein Theorem [16],
or alternatively the notion of sequential ~ vectors [26], facilitates construction of the l.r. on which an SPRT may be based.
Wilcoxon, Rhodes and
Bradley [32] used what we called the first variation at the beginning of this
section, thereby preserving independence among the
stage-~dse
statistics
(each stage consisting of several observations from each population).
Their
approach has the advantage that rank sum tests can also be devised, since the
distribution of the rank sum can be worked out under both H and
o
stages.
1\ within
The other approaches have the advantage that they permit sampling
from one population at a time and in an arbitrary order.
These tests are
summarized in Part II (Tests 7a to 7f).
NmT the data reduction met110d is not confined to the method of invariance.
In particular, whenever there is a statistic T which is Fraser-sufficient for
n
"1, method (1) leads to a data reduction to T.
.
n
example of the previous section.
see [22] and [16].)
ExamPle:
See, for example, the sign test
(Sign tests can also be derived by invariance;
We now give an example in which invariance is not applicable.
double dichotomy test.
Suppose (simple random) sampling is done
in pairs from each of two Bernoulli processes.
(ActuaJ.J.y, the data may have
been nnre elaborate, but reduced to two Bernoulli sequences by a preliminary
data reduction.)
Thus, each stage results in one of the four possible outcomes:
16
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SS, SF, FS, FF, the symbols indicating succes s or failure in the respective
Bernoulli trials.
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Wald considered this problem in Chapter 6 of [31].
proposed merging the
~
He
pairs--that is, of replacing the above double
dichotomy with the trichotomy SF, FS, and (SS,FF}, \mich
success, failure, and tie, respectively.
'\ole
now re-label as
The probability model for the
trichotomy has parameters (Pl,P2,P3)' say, \mere
and (Pl' ,P2') are the probabilities of success, respectively, in the t"TO Bernoulli
sequences;
~'
= l-Pi'.
Suppose the parameter of interest is the pair (Pl ,P2)' a function of e
• (Pl ',P2'), but not 1-1 (actually 1-2). Then the merging of tied pairs is
an e;mmple of eliminating a nuisance parameter by data reduction.
Any justi-
fication--other than convenience--for such a reduction is not known; a minimal
sufficient statistic in the reduced data trinomial problem (the numbers of
successes and of failures) is not a function of a minimal sufficient statistic
in the original double dichotomy problem (the numbers of successes in each
of the Bernoulli sequences), so tIlat inference procedures after data reduction
will not be functions of the original sufficient statistic.
some lack of efficiency.
This suggests
(Cox [8] has introduced sequential tests for this problem
"mich are based solely on tIle minimal sufficient statistic at each stage, but
they Ilave only an asymptotic justification.)
If instead we are only interested in the parameter r = Pl /P2 = Pl' ~ '/ ql 'P2 '
(the ~ ratio), a combination of the above data reduction (merging tied pairs)
wi tIl the separation exemplified in the previous section (trinomial example)
leads to vlald' s double dichotonw test [31] (Test 5a in Part II).
17
This test
I
may be viewed as a SPRT of h:,rpotheses about the odds ratio "1 with an arbitrary
value assigned to the nuisance parameter P1 + P2
= P1' %'
+ q1'P2'.
test my (and was by Wald) also be derived by method (3) below.
This
That this
test is a SPRT about simple hypotheses in the trinomial IlDde1 could be exploited
in developing its properties--e.g., approximating the OC or ASN functions.
In
this particular example, it has been found [31] just as convenient to exploit
the conditioning approach from method (3) below.
The simple hypothesis SPRT
approach just alluded to, however, has been useful in a recent extension of
Wald's double dichotomy test [15].
METHOD 3:
CONDITIONmG
We now describe a method which, to our knOWledge, has only been used
sequentially in the double dichotomy problem as treated by Wald [31]; however,
it is a fairly straightforward extension of the conditional test notions that
have been widely used in non-sequential theory.
The most familiar example,
perhaps, is FiSher's exact test for 2x2 tables which is a conditional test,
conditional on fixed narginal totals; but, having the prescribed significance
leYel llhatever the narginal totals, it is a valid unconditional test (valid
in that its significance level is correct).
MOst of the similar tests of
classical Neyman and Pearson theory, and what Lehmann has called tests of
Neyman structure, are all examples of conditiona! tests.
Another important
class of examples to be mimicked here,are the permutation tests introduced
by Pitman [27], and derived by Lehmann and Stein [23] as tests of non-parametric
hypotheses against parametric alternatives.
All of these are well-known in
non-sequential theory (e.g., [22] and [lO])and all frequently have some kind of
optimality.
We shall now describe an analogous technique in sequential theory
18
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and construct CSPRT's--conditional sequential probability ratio tests.
optimal
properties are knolm for these CSPRT's.)
Let U be a statistic based solely on the stage n data X.
n
n
conditional distribution of X given U
n
n
=u
Suppose the
is completely determined by each
of HO and Hl--that is, the conditional distribution depends only on
r.
A
CSPRT of H vs. Hl' which are simple hypotheses about the conditional disO
tribution, is constructed by letting the role of the l.r. R be played by
n
the conditional likelihood ratio (c.l.r.).
For each n, He condition X on
n
U , and since we assume the X 's to be independent, the joint conditional
n
n
=
(~,
••• , Xn )
given U '''.' Un is the product of the conditional
l
densities of X given U. (j=l, ••• ,n). The c.l.r. is this product.
j
J
density of X(n)
For a fixed U-sequence U ,U2' ••• ' an SPRT based on R is a perfectly
l
n
valid SPRT, and hence has the prescribed error probabilities a and
mately).
~
(approxi-
Since these conditional error probabilities do not depend on the
condition, namely the values assumed by the U-sequence, the unconditional error
probabilities are also a and ~ (approximately).
That is, since the conditional
probability of rejecting H ' when H is true and the U-sequence is fixed, is
O
O
approximately a, whatever the fixed values of the U-sequence, so is its expectation for any e-value in H ; thus the unconditional probability of rejecting
O
H ' when e (inH ) is true, is equal to a. Likewise, the second 'kind of error
o
O
is not only conditionally equal to ,~, but unconditionally equal to ~ for any
e-value in
1)..
We thus s'ee that such CSPRT I S are doubly similar--similar both
with respect to H and with respect to H • Of course, the performance of
O
l
the'test may depend on the nuisance parameter ~, but the error probabilities
do not (except for the approximation introduced by overshoot).
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(No
19
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It is important to note that at stage n the data are being conditioned on
a function Un of the stage n data, and not on a function of all the data
available so far (other than Ul, ••• ,Un ); for, having fixed ul, ••• ,U _ at
n l
the previous stage they cannot be "unfixed" at stage n. Thus, we are led to
more comprehensive "fixing" than is done in nonsequential problems.
remark
belo"T.
may
This
be clearer in the context of an example; see the first example
Incidentally, it is because of this extensive conditioning that no
optimal properties of CSPRT's are immediately apparent.
Even conditionally,
they are not known to be optimal in Wald' s sense since they are not based on
a sequence of conditionally identically distributed observations--each observation Xn being conditioned on a value for Un , say u n , so that the conditional
distrilJutions have a parameter u n varying from stage to stage. Also, because
of the extensive conditioning, it may be necessary to sample in groups.
When is this conditioning approach applicable?
What is required is that
there exist a statistic U for which the conditional distribution of X given
n.
n
U = u be completely specified by r = r.(i=O,l); of course, it is also necessary
n
~
that these two conditional distributions not be identical for all values of u.
whichis (i) sufficient
Thus, it is sufficient that there -=eX1=·~s~t -a statistic Un
H ~ H , since, by definition of sufficiency, the conditional
l
O
distributions would then be parameter-free, and (ii) that the probability ~
~ ~~
~ ~ ~ ~ ~
identical.
conditional distributions given this statistic
~
In parametric notation, a statement essentially equivalent to (i)
is that r be partially separable--that is, that, for each n, the density of
X may be written in the form
n
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for then, setting r
= r O or
r l , this factorization implies (by the factorization
theorem of sufficiency theory) the sufficiency of U for fixed
The factor
t e may always be taken to be the density of U and gr the conditional density
(probability) given U; hence, the condition (ii) above is simply that the
above separation of the gr factor be non-trivial--that the factor should
indeed vary vdth r, at least for sorne u-values.
This sufficiency for H
O
and for H , or this non-trivial partial separation of r, are of course not
l
necessary for the conditioning approach to be applicable.
A special case to be noted is that when a statistic U exists which is
n
Fraser-sufficient
for~.
It vms already noted under method
a statistic is sufficient for, H and for
O
not just partially, separable.
Hl~
(1)
that such
and hence r is then completely,
Then, the CSPRT is identical with the SPRT for
any fixed value of the nuisance parameter
c.l.r. is identical with the l.r. for
~
since, as noted earlier, the
arbitrary~.
All examples of method
(1)
based on a Fraser-sufficient U are therefore examples of method (3) as
n
But method (3) is more general as seen in the examples to follow.
well.
Before exemplifying the conditioning method, we briefly consider the
properties of CSPRT's when the
distributed.
XIS
are nmtually independent and identically
Let the c.l.r. of the stage n data be denoted by r
. n
ratio of conditional densities of X given Un under H and H ).
n
l
o
Rn
=~
~=
1 log r .•
~
(the
Then log
Now the r.~ tS,considered as random variables, are uncon-
ditionally independent and identically distributed; conditionally, they
are independent but not identically distributed.
ThUS, considered uncon-
ditionally, the CSPRT is based on a cunmlative sum of a random number of
independent and identically distributed terms.
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Therefore, Wald1s method
[31] of proving the certainty of termination is applicable.
equation (namely, E log ~
=E
log r
l
Also, Wald1s
• EN), on which his ASN approximation
21
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is based, is applicable.
However, r.). is not (unconditionally) a 1.r., and
hence Wald's methods of investigating and approximating the OC function are
not generally applicable.
Thus, the OC is only known for
where it is (approximately) l-a and
~,respective1y.
only known (approximately) in HO and
where
H:t'
e in Ho and
in
I1.
Therefore, the ASN is
being given by
H denotes the (random) number of stages required for terminatioo.,
and where
a a + {l-a)b for
e in Ho
(l-~)a + ~ b for
e in I1.;
Ee log ~ = (approximately)
a
= log
(l-~)/a
and b
= log~/(l-a).
Of course, if (for each n) the conditioning
statistic U is Fraser-sufficient, then the CSPRT is a valid SPRT for arbitrary
n
'I)
and Wald' s methods of approximating the OC and ASH for any e-value are
applicable.
One further property of CSPRT' s may be noted.
Lehmann's method ([22], pp.
101-2) of proving I1xmotonicity of the OC is applicable for CSPRT's.
His
condition that X. have a monotone l.r. in r and in t{Xi ) may be replaced'by the
condition that ti have a monotone c.l.r. in r and t(Xi) for fixed Ui; identical
distributions are not required. Then one can conclude that the conditional OC
is monotone in r, whatever the values of the U-sequence, and hence unconditionally
monotone as well. Examples of this will be noted below.
We now turn to examples, first reviewing the double dichotoID¥ example of
Wald, then extending it, and then introducing a permutation test.
Example:
double
dichotomy~.
of the previous section, with r
We return to the double dichotoID¥ example
= Pl/P2 = P1' ~ '/q1 'pq' • Denote the data from
a single stage by (X,Y) where X and Y each
= 1 (success) or 0 (failure) in the
~
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r
respective trials.
For fixed 7, the density is
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where
Hence, U is sufficient for (pi, P2) for eacll fixed 7-va1ue,
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u
=
x + y.
and the conditioning on U is possible.
In contrast,
non-se~uentially,
after
n stages, one vrould condition the distribution of accumulated successes in each
se~uence
on tlle total accumulated number of successes.
bution of (X,Y) given U
=u
is:
with probability 1 when u =
(0,0)
(x,Y) =
pairs with
°,
(1,0) or (0,1) with probabilities 7/(1+7) and 1/(1+7),
respectively, when u = 1,
(1,1) with probability 1 when u = 2 •
= 1,
The c.l.r. is thus unity except ifllen u
when (X,Y)
T11e conditional distri-
= (1,0) and
u = 1 --the
and then is (7 /7 )(1+7 )/(1+7 )
1 0
0
1
(1+70 )/(1+71 ) when (X,Y)
= (0,1).
In effect, only those
untied pairs--contribute to the cumulative sum log R
n
on which the test is based.
This test has been described this i'lay (by "lald
(31] and Armitage [3]) as being a conditional one confined to the se~uence of
untied pairs.
This latter
usual SPRT for testing p
se~uence
= 701(1+70 )
is simply a Bernoulli
vs.
p
= 71/(1+71 )
se~uence,
and the
is applied.
Its
properties are described by Wald, utilizing the fact that it is simply a test
based on a Bernoulli
se~uence.
TIle ASN has to be adjusted by dividing by the
probability of an untied pair.
As noted above, its properties could also be derived from the fact that
it is a SPRT of (70' ~') vs. (7 , ~'), for arbitrary ~', after ties have been
1
merged. Not at all apparent from the method (3) approach, but obvious from the
method (1) approach, is that this test has Wald's optimal property uniformly
in ~--that is, the (unconditional) ASN is minimized at (7.,
~') among
l.
23
all tests of these simple hypotheses with strength
uniformly in ,,'.
(a,M, and this is true
From Wald' s approach (method (3», it is only apparent that
the conditional ASN (which depends only on r) is minimized at r. among all
I
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1
tests of the same strength Which ignore tied pairs.
This optimality of Wald's
double dichotomy test has not been pointed out before.
ExamPle:
extended double
dichotomy~.
We consider the same model as
in the double dichotomy example, namely, that of two (independent) Bernoulli
sequences; but we now no longer confine attention to sampling in pairs.
Instead, ,re permit sampling in groups of arbitrary and varying sizes, with
the sole requirement that at least one observation from each Bernoulli sequence
be included in each group (or otherwise the data from some groups may have to
be ignored).
There are of course possible advantages of pairing in that
certain kinds of variation may be eliminated by matching the pairs [3].
On
the other hand, there may be advantages to permitting larger group sizes;
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for example, a group may be constituted by all observations reported during a
particular m:mth to a center supervising clinical trials in a number of hospitals.
Or
it may be desirable to place a larger proportion of patients on the
new treatment, in the context of comparing a new treatment with a standard
one in clinical trials, than on the standard, or vice-versa.
We temporarily confine attention to a single stage of the experiment,
Suppose n and ~ (positive
l
integeIJs) observations are taken from the two Bernoulli sequences, and let
and suppress its index n from the notation.
U denote the total number of successes among the
~
+
~
trials.
Let X and Y
represent the numbers of successes from the respective Bernoulli sequences
(U = X + y); (X, Y) is sufficient for this stage of the eJq>eriment.
density (probability) of (X,Y) may be expressed in the form
24
The
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where Y is the odds ratio Pl~/P2ql. The sufficiency of U when y is specified is thus apparent; that is, y is partially separable.
fixed U, say U
= u, the distribution of (X,Y) is effectively that of X
(given U); this conditional density (probability) is readily found to be
An alternative form is obtained by replacing the coefficient of
u u'
denominator by (j)(j') where u'
and j'
= nl
+ ~ - u
in the
= total number of failures,
= n1-j (failures in first sequence), and likewise in the numerator.
u
t
" J
u
u u'
J'
'V
/ t
( , ) ( j' ) Yl
( u)
, (u. ") ' 0
j=l J J
j=l J
•
The CSPRT for testing YO vs. Y may be carried out by computing log R
n
l
=
n
t, 1 log r, and comparing it ~rlth Wald's boundaries a and b in the usual way.
~=
~
We now briefly examine this test in the case of sampling in double pairs--
= ~ = 2 at every stage. T1le c.l.r. for a single stage is seen to assume
l
one of six possible values, given in the table below; the unconditional
n
probabilities of these six types of data (indexed 1 to 6 for convenience)
are also given.
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.yi
The c.l.r. r n for this one stage is thus
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Conditioning on
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=rn
type
(x,y)
1
(o,o) or (2,2)
1
2
(0,1) or (1,2)
(1+10 )1 (1+1 )
1
3
(1,0) or (2,1)
1 {1+10 )110 {1+1 )
1
1
4
(0,2)
{1+h10 +1~)1 (1 +411 +1i)
5
(1,1)
11 {1+410+1~)110{1+411 +ri.>
6
(2,0)
Letting c.
J.l1
unconditional probability
1i (1 +1-~10+10)110 (1+411 +ri)
denote the accunn.lJ.ated number (through stage n) of stages with
data of type i, and letting di be the logarithm of the c.1.r. for a
single stage with data of type i, vre thus have
log Rn
6
=~.
1 C.~n d.~
~=
and the test may be carried out conveniently by plotting this cumulative
sum until it reaches one of the I1Orizontal termination boundaries with
ordinates a and b.
= 1/10 , it may be shown that this test is
1
identical "nth the double dichotomy test (except that termination is
In tIle special case when 1
permitted here only after even numbers of pairs of observations).
Otherwise,
it is readily seen that the extended test is not the same as Wald's test.
For, ~he extended test would treat two successive pairs (mW,ing up the nth
stage, say) vnth outcomes SS and FF identically with two pairs "\'r.i.th outcomes
SF and FS (each of these double pairs is of type
5), whereas Wald's test
ivould ignore the tied pairs SS and FF but not the untied pairs SF and FS.
(If 1 = 1/1 , the contribution of SF to the cunn.lJ.ative sum would cancel
1
0
the contribution of FS, but not othervnse.) It is thus seen tlmt the
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extended test utilizes zoore of the data.
Now except in the case
Fraser-sufficient.
ASN at 'YO and "1
1
ll. = n2 = 1
already considered, Un is not
The CSPRT for the extended problem is not a SPRT.
is approximable (see Appendix 1).
to its OC is available,
a.lthO~l
Its
But no approximation
it is zoonotone in "I for
fixed~.
The
monotonicity property holds since, for fixed U , the stage n c.l. r. is
n
monotone in xn ' and Leh!lm.nn t s method of proof cited above th.e~.efore leads
to the conclusion that the conditional OC for fixed U , U , ••• is monotone
l
2
in
r;
since this is so for every fixed U-sequence, zoonotonicity also holds
unconditionaJJ.y.
on
It is apparent that the OC and the ASN do indeed depend
~.
The ASN I s of the double 0.ichotomy test and the extended test (under
H and
O
1\)
are compared in Appendix 1.
the table above.)
(E e log r 1 is readily con;mted from
It is found that a slight reduction may be possible--but
typically only a very slight one--by using the extended test.
On the other
hand, the increased group size in the extended test will, if sufficiently
increased, have an adverse effect on the ASN.
Some numerical studies are
planned for later publication.
This extended test, for the case "10
=1
(i.e. ,pi
= P2)'
appears in Part
II as Test 5b; Wald's double dichotomy test is Test 5&.
ExaEIPle:
tyro-samPle permutation test.
The non-sequential Pitman permu-
tation test for the two-sa.n:q;>le problem, as derived by Lehmann and Stein [23]
(or see [10] or [22]), is a test of the nonparametric null hypothesis that
the tvro independent samples com.e from identical populations (''lith some unspecified continuous distribution), against a parametric alternative that the tvro
populations are normaJ., with a COIlllOOn variance, but with differing meau•
Under the null hypothesis, the two samples constitute a single random sa.n:q;>le
27
and the order statistic of this single sample is a sufficient statistic.
Thus,
under H ' the conditional distribution of the observations, given the order
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statistic, is completely known--it is in fact an equal-probability distribution over the possible permutations of the values making up
th~
order statistic.
The parametric family of alternatives serves to define a critical region which
is conditionally most powerful, whatever the order statistic; in fact, it
is uniformly most powerful, among conditional tests, against
all such normal alternatives with positive (say) difference in means.
We now develop an analogous sequential test by using a similar line of
reasoning within
!!£!! sta.te of the sequential experiment, each state consisting
of at least one observation from each population.
It will be found that the
order statistic based on the combined data from a single stage is not only
sufficient for the null hypothesis but also sufficient for certain families
of parametric alternatives, and hence a CSPRT may be cCXlstructed by conditioning
on the sequence of stage-wise order statistics.
We sl1all nOli describe the simple special case in Which sampling is done
in pairs; each stage consists of a single observation from the X-population
and a single observation from the Y-population.
Part II as Test 6.
The general case appears in
The null hypothesis is that the X- and Y-poPUlations
are identical, both '\-lith continuous but unspecified distributions.
this connnon distribution could vary from stage to stage.).
(Actually,
The alternative
hypothE!sis, for the present, is that the X-population is normal 'torith mean A
2
and variance
0"
2
and the Y-population is normal with mean A+O and variance
0" ,
represent the unordered pair (X , Y },
all parameters being specified.
n
n
X and Y being the respective observations at stage n. (U is 1-1 in the
n
n
n
.
order statistic.) Since the alternative hypothesis is simple, it is trivially
true that U is sufficient for the stage n data--in fact, a constant is a
n
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sufficient statistic for the stage n data under this alternative hypothesis.
We shall explore presently to 1-That extent the alternative hypothesis can be
enlarged. vlithout destroying this sufficiency of U •
n
Vie nOl-T fix attention on a single stage, and omit the subscript n.
fixed unordered observations are denoted u = {x,y}.
probability of (x,y) is
The
Under H ' the conditional
o
1/2, as is the conditional probability of (y,x). Ties
may be ic;nored since we are confining attention to continuous distributions.
Let ~(x,y) denote the density of (X,Y) under the alternative hypothesis (the
product of two normal densities).
The conditional probability of (x,y) is
then seen to be ~(x,y)/[~(x,y) + cp(y,x)] (see [10] or [22]), and the conditional
probability of (y,x) is the same "lith x and y interchanged.
Algebraic reduc-
tion leads to the value l/{l + e-~) for the conditional probability of (x,y),
2
c/a.
I_
where z = y-x and A =
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normal alternatives with a fixed value of A, say A'--but otherwise arbitrary
,I
Hence, the conditional distribution under the
normal alternative depends solely on the parameter A, and we thus see that
2
values of A, 0, and a --may be substituted for the simple alternative above,
and U
= {X,Y} remains sufficient (the eonditional distribution being known).
Further enlargement will be noted later.
~1US, for the stage n data, the c.l.r. r
n
is
2/(1
-A' z .
+ en).
A CSPRT
1
- 0 z.
-~=l log 2"{1 + e
J.).
At
thus may be based on the cumulative sum log Rn =
Sampling is continued until this cumulative sum reaches one of the boundaries
a and b.
Since, for each fixed sequence of values of the unordered pairs
{x ,Y }, the test is a valid SPRT i'Tith conditional strength (a,I3), it is also
n n
a test with unconditional strength (a, 13) (ignoring overshoot).
tion is (approximately) constant vTithin HO and within
.
unknown.
6
=M
11.'
The OC func-
but is otherwise
It is, however, lOOnotone in A for fixed values of " and a
2
(and
2
) 1vhenever the two populations are normal with COllIlJX)n variance and
29
means A and A+o.
(See earlier remarks on proof of monotonicity.)
The test
terminates with certainty, being based on a cumula.tive sum of independent
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and identically distributed random variables, namely the swmnands in the
expression for log R •
n
(This assumes identical distributions at each
but this can be relaxed.)
The ASN under H and under
O
11. may be
stage,
approximated;
an asymptotic fQrm of this approximation, for small 6', is given in Appendix 2.
It is also shown there that, asymptotically, these ASN approximations coincide
with the
~
as under
11..
of the fully parametric SPRT assuming normality under H as well
O
Hence, this test, which has a broader validity than the normal
theory test, is asymptotically efficient in this sense.
Some numerical
studies will be reported in a subsequent paper.
~~
final comments about this test will now be made.
far can the class of alternatives be extended?
models will the test have power
First, just how
That is, for "mat other
It may be shown that if the density
l-~?
functions of the two POPULations differ only in an exponential factor ePX and
e P 'y where p' - p = 6', then U remains sufficient and the above CSPRT remains
n
valid.
For example, if the X-population and Y-population both have gamma
distributions with densities proportional to e
-~
x
A-l and e -~'y y
A-l, where
IJ--Il' = 6' but A is arbitrary, then the above test has power l-~.
Comparison
of this test with other sequential tests appropriate for exponential and gamma
distributions is planned for a subsequent paper.
The· final conunent is that, although this test is not a SPRT for any simple
hypotheses included within HO and
Hl--Un
is not Fraser-sufficient--it is an
SPRT for testing the simple hypothesis HOthat the differences Zl' Z2' •••
have logistic distributions with distribution function F(z)
= 1/(1+e-6 'z),
against the simple alternative Hi that the Z's have the distribution function
[F(z)]2--that is, that each Z may be regarded as the larger of two independent
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random variables each with the logistic distribution function F(z).
a curiosity.
the form:
It does permit approximation of the OC against alternatives of
k
Z has distribution function F for any k (6' fixed), but any real
use of this fact is not apparent.
other composite hypothesis testing problems are certainly amenable to
method
(3).
Specifically, CSPRT's for the two-sample problem
,~1en
testing
against discrete parametric alternatives may be derived in complete analogy
with the permutation test above; in fact, the double
dichtor~
tests may be
viewed as tests of this type, but Poisson or negative binomial alternatives
could lil:e,nse be considered.
Also, CSPRT's for tests of independence against
correlated normal alternatives llave been derived and will be reported elsewhere.
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This is
31
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PART II:
SEQUB:NTIAL PROBABILITY RATIO TlSTS FOR THE TWO-SAMPIE PROBIEM
SUMMARY OF PART II
A number of SPRT's for the two-sample problem are listed and briefly
described.
The two-sample problem considered is that of testing whether two
populations, from which samples are sequentially taken, are identical or
whether observations from the second population tend to be larger than those
from the first.
Thus, the tests would be appropriate for comparing a treat-
ment with a control, or a new treatment with a standard treatment.
Analogous
tests are available, but not presented, for testing against two-sided alternatives; these would be appropriate when attempting to determine which of two
treatments is superior or whether there is no real difference.
Such two-sided
tests can be constructed by running two one-sided tests simultaneously, as
proposed by Armitage ([2] and [3]), or by use of invariance theory (one sets
up a model for the two samples "lith the "labels lost"--ignoring which population is which); these will not be considered further here.
The tests given here all presume that each population has a continuous
probability distribution (although the sign tests are more
gener~
and the only parametric tests given presume normal populations.
valid),
All of the
nonparametric tests given are also valid if the populations are actually normal, with the exception of the rank tests; these presume Lehmann alternatives
and such alternatives imply that at least one of the populations is non-normal.
All tests are SPRT's (or CSPRT's--lee Part I)--that is, sampling is continued until a joint likelihood ratio falls outside a fixed interval (B,A);
actually, the normal tests with variance unknown are not quite of this type,
but are sufficiently similar
(and relevant) to be included.
of sequential tests, nor any non-sequential or two-stage tests,
32
No other types
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are included.
In every case, the hypotheses are composite; and in every case
bounds on the two types of error are guaranteed (except for inaccuracies due
to overshoot).
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Two basic types of sequential sampling are considered:
(1) sampling one
at a time from the two populations in an arbitrary order, or in batches of
arbitrary and varying compositions, and (2) sampling in pairs from the two populations; the stages of the sampling scheme are always (tacitly) assumed to
be independent.
Sometimes there is a restriction in (1) that there be at
least one observation from each population at each stage.
When sampling scheme
(2) is used, tests can be based on the sequence of differences between the
observations
mal~ing
up the pairs; such tests do not require the assumption of
independence between the two populations, and also permit the elimination of
certain factors by matching the pairs--that is, the two populations from which
one samples may vary from stage to stage as long as the distributions of the
observed differences remain constant.
Some of the tests using sampling scheme
(1) also permit some variation from stage to stage in the two populations.
Four general categories of tests are presented:
sign tests, perrautation tests, and rank tests.
normal theory tests,
Some empirical comparisons
of the tests are planned for later publication (with Peter NemenJ~).
INTRODUCTION TO THE TESTS
We introduce some notation, conventions, and overall comments relevant
to the tests that follow.
We refer to an X-population and a Y-population.
\fuen sampling in pairs from the X- and Y-populations, the difference between
the
Y-membe~l'
of a pair and the X-member of a pair is denoted by 3.
The null
hypothesis :is always that the t,'lO populations are identical or that the distribution of Z is centered around zero (together with additional assumptions
33
varying from test to test), and the alternative is always that Y-values tend
to be larger than X-values--at least in the same stage of sampling--or that
Z-values tend to be positive (again with additional
ass~}-:~,:tons
He:
'Y
=e
vs. Hl : 'Y = 'Y' (>0) for some numerical parameter 'Y. Whenever the OC function
is known to be l1xmotone in 'Y, the hypotheses could be enlarged to He: 'Y 5 e
~:
'Y ~ 'Y'.
The follo't'J1.ng sampling scheme notations will be used:
81:
*
81:
Sl**
32:
32:
*
sampling in batches of arbitrary composition and size. Let m' and
mlf (which may vary with n--if they do not, we say the scheme is regular)
denote the numbers of X- and Y-observations in stage n, and let n' and
n" denote the accumulated numbers of X- and Y-observations through
stage n. Both n' and nlf are assumed to tend to infinity with n (batch
size and composition may be random, but not dependent on previous
observations; see relevant discussion in [17]).
~
Sl "lith m'
1
~
mlf 2: 1 for every n.
31 with m' ~ 2 .2t mlf 2: 2 (or both) for every n.
sampling in pairs (regular Sl with m'
= mlf = 1
sampling in groups of pairs (31 with mt
= mlf).
n
out as follows:
Unless otherwise indicated, the test is carried
for each n
= 1,
2, ••• , in turn, calculate Ln and
He
if Ln 5 b ,
terminate and reject H if L > a ,
nO
proceed to stage n+l otherwise
,
terminate and accept
where
b
= log[~/(l~)]
the type I error and
~
and a
= log[(l-~)/a];
a is the prescribed bound on
is the prescribed bound on the type II error.
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for every n).
The tests are described by defining a statistic L (typically a log likelihood ratio of some sort).
,
in each test).
In every instance, the hypotheses could be expressed in the form:
vs.
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Some of
the group tests are carried out by calculating a group-specific statistic L*
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for each Group (stage), and lettinG
Ln = ~=l L: .
The following five properties are frequently investigated for SPRT's of
specified strength:
(i)
certainty of termination;
(ii)
monotonicity of the OC in some numerical parameter
(iii)
approximability of the OC (by Wald' s method);
(iv)
approximability of the ASN (by Wald's method);
(v)
minimality of ASN under H and H SUbject to strength requirements.
o
l
r of interest;
All tests considered here can be shown to terminate with certainty under
broad
assur~tions--always broader
tllan the hypotheses specifically tested.
Relevant references are [31], [6], [18], [17] and [30].
No further comments
about the certainty of termination 'f.lll be made.
Cor~nts
are [22]
on monotonicity have been made in Part I.
Relevant references
and [11].
Only ,then the test is a simple
~
(SPRT of two simple hypotheses based
on a sequence of independent and identically distributed observations) may
the operating characteristic function (OC) be approximated (I-lith the exception
of Tests 2a and 2b); Wald's method is described in [31].
vThenever Wald's equation is valid, the ASN may be related to the OC.
The
I1.
OC is always knovffi (approximately) under H and
and thereby tile ASN may
O
frequently be approximated under H and H ; otherwise, the approximability of
l
O
the ASN depends on the approximability of the OC and is essentially confined to
the case of simple SPRT's.
When H ld's equation is not valid, some conjectured
methods of approximation of the ASH under H and
O
I1.
have been connnonly used;
see Johnson [19](and [16]) for discussion.
Wald's optimal property
(v) is known to hold only when the test is in fact
a simple SPRT.
35
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NORM\L TESTS
One of the following two assumptions will be made throughout this
section.
The X-population is N{t-., 02), the Y-population is N{A+o, 02),
ASSUMPTION Al:
and the X- and Y-populations are independent (_eo < A < co, -eo < 8 < eo,
O<cr<co).
Z = Y-X is N{o, 202) (-co < 0 < eo, 0 < cr < eo).
ASSUMPTION A2:
Assum;ption A2 does not require independence of X and Y, nor equal variances
of X and Y, nor constant joint distributions of X and Y; it i·Till only be invoked
when Sampllllg in pairs; the successive Z's are then assumed independent and
identically distributed.
The null hypothesis will ali'layS be that 0
be some specific form of 0
1.
___________L
Normal mean
= 0,
and the alternative will
> O.
_
variance knovm:
In this section we shall always assume either Al or A2, and in addition,
assume tl1at
02
is known (specified); 0 is the parameter of interest and A the
nuisance parameter, and 0' is a specified positive number.
TEST la:
Assum;ptions:
Al, Sl, and 02 known.
Hypotheses:
H :
O
Metll0d:
0
= 0,
Hl:
0
= 0'.
Method (2)--invariance under conunon location shifts in all
observations from all stages; test based on l.r. of difference between
sam;ple means of accumulated data.
Test statistic:
Properties:
anel m'
L
n
= ?0' ·
1
1
n' + n ti (n'~ - n"Ex - 2" n'n"o')
ASN and OC are only Imown when the sampling scheme is regular
= m";
then the test is a simple SPRT, method (l) being applicable.
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TEST Ib:
As surgpt ions :
HyPotheses:
A2., S2, and r? lmovrn.
H :
O
0
= 0,
H :
l
0
= 0'
<
Method: Methods (1) and (2); data reduction to differences Zl':2"'"
and then separation of 0 (see Part I); test based on l.r. of Z (sample
mean of available Z's), and hence coincides with l.r. of Test lao
0'
Properties:
1
1
Ln = r;z • 2" (~z - 2' no')
properties of simple SPRT, uniformly in A.
Test statistic:
T&lT8 lc and Id:
=
Group tests, analogous to Tests la and Ib, respectively,
based on a statistic L computed from the data from each stage and then
n
summed over stages.
Permits variation in the nuisance parameter A from
stage to stage, and uses group sampling Sl* (Test la) or S2* (Test 2a).
Wald's methods for approximating the OC and ASN are applicable in the case
of rec;ular sampling.
2.
___________
Normal
meanL variance unknovm:_
We continue to assume Al or A2, with 0 as the parameter of interest, but
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now A and r? are nuisance parameters; 0' is as before.
are not really SPRT's; they are modifications of SPRT's developed in analogy
with Stein's two-stage test for such problems ([4] and [14]).
d2
An estimate of
is based on the data in the first stage of sampling, and the tests may be
considered conditionally (on this variance estimate) as SPRT's ,dth termination
boundaries depending on the variance estimate.
Alternatively, unconditionally
they may be described by a test statistic L which is a 1.1'. except that
n
a2
is replaced by its estimate and the termination boundaries a and b are
replaced by
~ =
I.
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The tests given here
where
~
2'1
~
(-2/~)
ex
- 1 and
b~ = -
2'1 ~ (-2/~
f3
- 1)
denotes the degrees of freedom of the variance estimate.
details.
37
See [14] for
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TEST 2a:
Assunwtions: Al and Sl with the restriction that the first stage has
at least two observations from at least one of the populations.
H : 8 = 0, H : 8 = 8' .
O
l
Method: See above; ~
= nwnber of observations in first stage less k
where k = 2 if both POPUlations are sampled in the first stage and k
otherwise.
I
HYPotheses:
=1
Test statistic: L same as in Test la with oF replaced by S2 = pooled
n
within-sample mean square from first stage data (~ degrees of freedom)
= ~1
a
~
- 2
- 2
+ E(Yi- Y) ] (first stage data only); termination boundaries
[~(xi-x)
and b are used.
~
Properties:
ASN and OC may be approximated when m'
= mil = m,
free of
n; see [14].
TEST 2b:
Assunwtions: A2 and S2 with the nx>dification that the first stage consists
of ~+l pairs, rather than a single pair, of observations (~>o).
~~otheses:
Method:
Ho:
8
= 0,
Hl:
8
= 8'
,
See above; based solely on differences.
Test statistic: L same as in Test lb with a2 replaced by
1
2 n
S2 = -E (z. -Z)
(first stage data only);
~
~
termination boundaries a and b are used.
~
Properties:
3.
2'1
~
ASN and OC may be approximated; see [14].
Normal mean in standard deviation units--t-tests:
We continue to assume Al or A2 but re-parameterize, replacing (8,"A, 02)
by (r, A,if!) ..There r
= 8/(j; r is the parameter of interest, and A and r? are
nuisance parameters; r' is a specified positive number.
analogous to tests la, lb, lc and ld, respectively.
Four tests are given,
The first two are based
on the invariance method (or Cox's method) and are analogs both of RUShton's
[28] one-sided sequential t-test, with the degrees of freedom and non-centrality
38
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parameter modified to adapt it to the two-sample problem, and Hajnal's [17]
two-sample sequential t-test except that we consider one-sided rather than
two-sided alternatives.
The exact test statistics involve confluent hypergeometric functions
M(.,.,.);
suitable tables are given by Rushton and Lang [29].
Approximation
formulas are also given, using Rushton's third approximation [2E)]; such
approximations are probably satisfactory for large degrees of freedom, which
are not available in Tests 3c and 3d.
The ASN's and OC' s are not
Im01'ffi,
but the ASN is presumably close to
that for the corresponding tests 'I'Tith variance known (for small 'i"), and
approximations are discussed by Johnson [19] for analogous sequential t- and
F-tests.
Hald's methods can be applied to approximate the OC and ASN in
Tests 3c and 3d when sampling is regular.
TEST 3a:
AssUl1]?tions:
H,y:potheses:
Metllod.:
Al and Sl .
H :
O
'i' = 0,
HI:
'i' = 'i".
Method (2)--invariance under corrunon location and scale changes;
test based on l.r. of t-statistic based on difference between sample
means of accumulated data and pooled within-sample variance estimate
based on accumulated data.
Test statistic:
\'There k
=
L
n
= - ~ k!2.,i
.[ n'n'n"
+n '
,
\1
+ 10gD.ft;1 ,
= n'+n"-2,
~ , ~ w2,.?) + J2
,,1)'
'H( \1;2
,~,~ if/F)
and
,,1 =
(L
11
-- 0 if n' or nil
approximate Ln
=
= 0) '~
1( ll'/' )2 + ( \1+1 )1/2 (w'i'
' ) [1 - 11-{\1+1)
1
2"1 k2 'i' i2 + 4'
+ ~]
~ •
39
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Properties:
••I
See above.
TEST 3b:
AssUll1Ptions:
HYPotheses:
~thod:
A2 and S2.
r
HO:
= 0,
Hl: r = r' ·
Method (2)--data reductiop to differences and invariance under
scale cha.nges
0" the differences.
Test is Rushton's test applied to
the differences.
Test statistic:
As in Test 3a with k =..[(n/2), v = n-l, and w = Ez(2tz2,-1./2
(approximations as in Test 3a).
PrOJ?erties:
TESTS 3c and 3d:
See above.
Group tests, analogous to Tests 3a and 3b, respectively,
based on a statistic L computed from the data from each stage and then
n
summ.ed over stages.
stage to stage.
Permits va;riation in the nuisance parameters from
(See section 1.7 of [16] for co:mments on analogous two-
sided tests.)
SIGN TESTS
In this section, the actual observations are replaced by dichotomous
observations--either the XiS and Y's are separately dichotomized, or the
differences, the Z's, are dichotomized.
required.
Pl
Continuous distributions are not
We introduce the following notation
= Prob(X <
c), P2
= Prob(Y <
c), p = Prob(Y > X) (c fixed).
Under Assumption Al, and with l > representing the standard normal distribution
function, Pl
= ,ll>[(C-A)/cr]
p = ll>[(8/ .f2cr)].
and P2
= ll>[(C-A-8)/cr];
While under Assu.mpt10n A2,
Thus, tests about the parameters Pl and P2' or p, may be
related 'to tests about underlying n::,:"'18.l models, but of course these tests
do not require a normality assu.mption.
40
We also let ~
= 1-Pi (i=1,2)
and
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We introduce the assumptions:
ASSUMPTIOn Aj:
The X- and Y-populations are independent and the odds ratio
p
is constant from stage to stace.
ASSUMPTION A4:
I'Then sampling in pairs, p is constant from staGe to stage.
As always, we assume independence among stages, but the distributions may vary
from stac;e to stage subject to either A3 or
A4.
The tests presented here are based on dichotomous classifications, but
they could all be extended to per:cti.t more than two classifications; such an
extension of Test 4 belovl is treated elsewhere [15].
other double-dichotomy
tests arc r;i ven by Cox [8].
TEST
4:
JIeTe, the parameter of interest is p and all other specification of
tIle Joint Q:Lstributions of successive pairs is ascribed to nuisance
para::lctcrs; p' is a specified number betvreen 1/2 and 1.
A4
ASsU1mtions:
H :
HnJ otl1cses:
Method:
O
and S2.
p=1/2,
l~:
p=p'.
I,rethod (2)--data red.uction to the signs of the differences.
be justified by invariance under monotone transformations c;
each of the observations in stage n; see Section I.O of
n
(May
applied to
[16].) The
reduced. data constitute a Bernoulli sequence with success (positive sign)
probability p.
The simple SPRT for Bernoulli data (C11apter 5 of [31])
is applicable, the 1. r. at s,tage n being that of a binonial random variable I'd tl1 parameters n and p.
,laS
originally described by Armitage [2].
Test statistic:
s
n
In the context of normal data, this test
= nluooer
Propertics:
L
n
= sn loc;(p' / q') + n 10g(2q') where q' = l=p' and
of positive differences in the first n pairs.
Approximations to the OC and ASN, as functions of p, are
,,,'cll-len01m (e.g., [31]).
I
double-dichotolny tests:
The parameter of interest is now P, which equals unity if and only
if Pl = P2 and exceeds unity if and only if Pl > P2. All other specification of the distributions of X and Y (assumed independent) is ascribed
to nuisance parameters; pi is a specified constant exceeding unity. This
test is different from all others so far in that it is not location-free;
adding a constant to all observations would rot leave this fornmlation
of the two-sample problem invariant.
Test 5a is Wald's double-dichotomy test (Chapter 6 of [31]; see also
Armitage [3]), and is a special case of Test 5b which follows.
Test t:a:
sSw!!ptions: A3 and S2.
HYpotheses:
He:
p = 1,
lil:
p =p' •
Method: Methods (2) and (1), or methods (2) and (:3). Data reduction
to the trichotonw: "success" if X < c and Y 2: c, "failure" if X 2: c and
Y < c, and "tie" if X-c and Y-c have the same signs; and then separation
of the Parameter p (see Part I). Alternatively, data reduction to the
double dichotonw: "success" in the X-observation if X < c and "failure"
otherwise, and likewise for the Y-observation; then condition on U =
number of successes (0, 1 or 2) in the data for each stage. The first
leads to a simple SPRT about (p, TJ) where TJ = Pl ~ + P2ql and TJ is
assigned an arbitrary but common value under both He and lil. The second
leads to a CSPRT which is in effect a SPRT about the Bernoulli sequence
of "successes" and "failures" as defined in the trichotomy with all "ties"
ignored.
l'
Test statistic:
L = s log pi - t log -2(1+p')
n
n
n
"There s = number of pairs among the first n stages with X-successes
n
and Y-failures
and t = number of untied pairs among the first n stages (X-success
n
and Y-failure or X-failure and Y-success).
Properties: Approximations to the OC and ASN are well-lmOi'1l1 [31]; since
the test is a simple SPRT about the trichotomous data, it has minimaJ.
ASN in He and in
among a..ll tests of this strength based on the
reduced (trichotomous) data.
1\
42
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~lis
TEST 5b:
is the extended double-dichotomy test, described in Part I; the
assumptions and notation 1rill be similar to that in Test 5a. The sampling
sc:ileIle is n0'l'1 '1* (see accoIrila.rWing notation in the Introctc.1.ction of Part II).
*
AssUtIl?tions:
Hypotheses:
Method:
A3 and Sl •
HO:
p
11.:
= 1,
p
•
Methods (2) and (3)--data reduction to the stage-i·lise double-
dichotomy, letting s' and sit denote the nwnbers of X-successes and Yn
n
successes at stage n, and rA' -s' and rrl'-s" the respective nwnbers of
n
failures; then conditioning on
the cur,mlative
sum of
1.1
n
stage-~Tise
lil:cli:ilood ratio of s' given u
= s'n
n
+ s".
n
The test sta.tistic is
statistics, each being a conditional
(parameterized by p).
See Part 1.
All
n
n
staGcs llitIl s' or s" = 0 are in effect ignored sincc their log c.lor. 's
n
n
arc zero.
Test statistic:
Properties:
Ln
...n
= ~.
~=
-l(.
1 L. and
~
1.1
*
L = s'
n
n
log p' - log
n
OC is not knm-ffi; ASH under H and
G
p' ;j
L:
j=O
11. may be
approximated (see
Appendix 1).
mm·mATION
\'1e
nOir
TESTS
describe a test which is distribution-free i·;ith reGard to the
null hypotllesis, but is directed against parametric alternatives.
He here
describe tIle alternatives as a certain family of normal hypotheses about the
X- and Y-po}nllations, but the alternatives may be enlarged to include certain
other exponential family distributions; see Part I (p. 30).
We introduce the assumption:
ASSUMPTION A5:
The X- and Y-populations are independent idt11 continuous
distribution functions, denoted F and G, respectively.
Actually, in this section, F and G may
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=p'
We denote ~
= o/~,
0 and
dF
vro.~y
from stage to stage.
having been defined in tile section on NORMAL
TESTS; A and c;2 are nuisance par8..l:wters under the alternative (and they may
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vary from stage to stage), and 6' is a specified number.
-TEST 6:
Assumptions:
A5 and Sl*
Hypotheses:
H : G = F (unspecified),
O
Method:
as for H •
O
Al and 6 = 6' •
A CSPRT is constructed by conditioning on the stage-wise
Method(3).
order statistics,
H :
l
may be shown to be sufficient for H as well
l
At a particular stage n, an unordered collection u of
~n1ich
n
the values of the n' X-observations and n" Y-observations is fixed
n'+n"
(u is 1 to 1 in the order statistic). Let j index the k = (n" )
n
n
ways of selecting n" numbers out of the collection u
of the Y-observations.
n
to play the role
The conditional probability that any particular
n" of them, say Yl' ••• , Y II, are the y-observations is 11k under HO'
6f~ n kn 6,~(j)
n (j)
and is found to be e
I Ej=l e
under ~ where Yl ' ••• ,
yn,,(j)
are the (unordered) numbers making up the jth selection.
Hence, the c.l.r. for the stage n data is this latter ratio with the
sum in the denominator replacedby the average--that is, the sum is
pre-multiplied by 1/1\.;L* is the log of this c.l.r.
n
Test statistic:
i'rhere
snl
L
n
is the
=
n
~~=1 L~~ where
Sl
L
* =6
n
f
s
nl
k
6's.
l
- logE -k E.nle
nJ ]
n
J=
m of the stage nY-observations,
is the sum of the numbers making up the j th selection
of n" numbers from among the X- and Y-observations at
stage n (the k
~TaY
kn =
n
selections are ordered in an arbitrary
except that the actual Y-observations make up the
first selection)
nf+n"
( n" ) •
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(An alternative form is:
Properties:
1
*
k
log [k
L ::::
n
6'(s .-s
~ n e
~
llj
)
nl]
.)
n j::::l
Tl1e OC is unlmovm, but may be shmm to be monotone in 6
uncier Al (normal populations).
The ASN is approxi:mable under H and
O
HI' at least for sDall 6', where it approximates the ASN for Test la
under Al--that is, Test 6 is asymptotically efficient (as 6' ~ 0).
See Appendix 2.
RANK TESTS
He no\'1 consider nonparametric rank tests, continuing to assume continuous
clistributions of X and Y (or of Z :::: Y-X).
distribution-free,H
1
But nO'iT both H and HI will be
O
being a "power" (Lehmann) alternative.
Instead of such
pouer alternatives, other distribution-free families could be considered ilitl1
1
G :::: 11(F) (rather than G :::: F :') for other specified monotone h(·).
These are
the first tests Which are not completely valid if X and Y are indeed normally
distributed; H and HI are not both compatible with normality (see discussion
O
in [32] ).
~1e
first of these tests is described (and derived) in
[16], and also--
by a different approach--in [26]and some of its properties derived in [30].
~1e second test (whiCh may seem somewhat artificial in its formulation,
594.
especially 1Vith power alternatives) is new, but mentioned in [16], page
Group rank tests were introduced in [32].
F and (} are the di:::tribution fUnctions of X and Y; H denotes the distribution function of Z :::: Y-X, p :::: Prob(Z
> 0), H+ denotes the conditional
distribution function of Z Given that Z
> 0, and H- denotes the conditional
distrib11tion function of -Z given that Z < 0; k' is a specified number
exceeding unity, and p' is a specified number in the interval
(t ,1).
He
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shall use Assumption A5 or:
ASS1n1PTION A6:
Z has a continuous distribution function.
This assumption does not require the independence of X and Y; the joint
distribution of (X, Y) need not remain constant from pair to pair, but the
differences are assumed to be independent and identically distributed.
TEST 7a:
Assumptions:
A5 and Sl.
Hypotheses: H :
O
Method:
G
=F
(unspecified),
G
= Fk
f
•
Method (2)--invariance under a common monotone continuous
transformation applied to every observation.
The test is based on
the l.r. of the ranks of the accumulated Y-observations in a combined
ranking of all (accumulated) X- and Y-observations.
The relevant
distributions are given in [10], p. 192 (and in [22], p. 256, where
Sj+l is twice misprinted as
s .+1); an alternative expression appears
J
in [30].
n"
Test statistic:
L = n" log k' + 1: log [r(s. +jk'-j)/r(sj +jk'-j-k'+l)]
n
j=l
In
n
- log[r(n'+n"k'+l)/r(n'+n"+l)]
where
sln' ... , sn"n
are the ranks (integers betw'een 1 and n'+n"),
in increasing order, of
am::mg
(Yl'''. 'Yn " }(the accUlllU18.ted Y-observations)
(XJ.' ... ,xn"Yl, ... ,Yn " ) at stage n.
Properties:
Sure termination is proved in [30], but the OC and ASN are
unknown, nor is any monotonicity of the OC known.
TEST 7b:
Assumptions:
A6 and S2 •
Hypotheses:
HO·• P -- 1.
2
and H+
46
= H-
(Z is symmetrically distributed
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about the origin),
~~thod:
H : p
l
= p'
k'
and
H+ = H-
•
Method (2)--data reduction to the differences Zl,Z2' ••• ' and
invariance under a common monotone continuous transformation applied
to the absolute value of every difference.
Tne test is based on
the 1. r. of the number of positive Z-values and the ranks of the
positive Z-values in a combined ranking of the absolute values of
the accumulated Z-values.
Tcst statistic:
+
L
n
= n +log
See [32], p.204.
p' + n - log ( l-p' ) + n + 1013 1<;:'
n
+
~
j=l
'\'T!1Cre
n
=
log [r(s. +.:j;:'-;;)!r(s. +.jk'-j-l<;:'+lJ - 10G[r(n-+n+k'+1)!nn
In
In .
n
+
= accu.r:mlated
n - n
+
nUl1ber of positive Z I S at stage n,
, and
sJ..n -' ••• ' s n+n
are the ranks in increasing order of the positive Z' s
among the absolute values of all the accumulated Z's.
Pl~oJ?erties:
Unkno'm (but presumably termination can be proved as in [50J).
TESTS 7c and 70.:
Group tests, analogous to Tests 7a and '7b, respcctivel;'h based on
a statistic L
n
computed from the data from each stage separately anel
,
then summed over stages.
*
(Test 7d uses sampling scheme 82.)
Test 7c
is der.cribed, rold its properties investigated, in [32]; see also the
co~~ents in
[16].
Test 70. could be investigated in an analogous "my.
TESTS 7e and 7f:
Group tests, again analogous to Tests 7a and To, respectively, and
to
Ta~ts
7c and 70..
But these tests are based on the stage-wise l.r.
of the' rank sum L: s. (the sum of the ranks of the stage nY's, or
. - -In
posit::ve Z's, in a conbined ranking of the staGe n (lata only).
47
Test '/e
I
is described, and its properties investigated, in [32]; Test 7f could
lD:e~dse
be investigaged.
These group rank sum tests are feasible
since the exact distribution of the rank sum can be vrorked out under
H and under ~ --at least in small samples (small group sizes); non-
o
group versions are not feasible since invariance and sufficiency
theory is not applicable, and the required joint distributions quickly
become unmanageable.
The test statistics for carrying out these
tests are not presented here; see [32].
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A p pen d i x
1
ASN OF THE EX'IENDED DOUBLE-DICHarOlrl' TEST
In this appendix we silall sketch an investigation of the ASN of the ex-
A
tended double-dichotomy test (Test 5b, p.43; see also Part I, pp.24-27).
more thorough investigation seems warranted, however.
We shall assume a regular
s8.l-:!pline; scheme, with m X-Observations and mil Y-observations per stage, and
l
assume the basic observations are independent and identically distributed.
It was noted in Part I (pp.2l-22) that, under the above assumptions, Wald's
method of approximating the ASN is applicable to CSPRT's when H or HI is true.
For, ignoring overshoot, and ,rlth
E~
= {-c(a,13)
c(13,a)
where
L:,
log ~ = ~, log r n =
= -cO
ifHO is true
= cl
o
if HI is true
c(a,13) = (l~) log[(l~)/13] - a log[(l-13)/a]; also, Wald's equation is
L;'S
applicable and hence E LN = EN· E L~ since ~ = t;=l L; and the
are
independent and identically distributed. Here N represents the number of stap,es.
If I're let M = m' + mil, the number of observations per stage, then the expected
number of observations is
(i
= 0,1)
(1)
Hhere E. denotes expectation vr11en some specific distribution in H. is true, and
~
~
the subscript has been deleted from L*.
the
~xpected
It thus remains to evaluate
increment (per observation) in the cumulative
~
*
E. L /r,1 -l
defining the test.
He shall evaluate this when pI, the hypothesized value of the odds ratio under
HI' is close to the null hypothesis value of unity.
Using the notation of pp. 40-43, we find the stage-I'rise log c.l. r. to be
L*
=
- log { pI
-s
u
E
( ~) (M-u .)
J
j=O
('lIe have replaced s I by s, u
n
n
m'-J
M
(m' )
by u, and mI + m" by M).
We now set € = pl-l (> 0), expand (l+€)j in (2) as a binomial sum, intercllanc;e the order of summation, expand
th~
49
log(l+€) term, and thus find
I
=
L*
.-I
-(a -s)€
1
l
where
aj
u m
M
= (j)(j
)/(j).
Now S is (marginally) a binomial random variable with parameters (m',Pl)'
and U is (marginally) the sum of two independent binomial random variables with
parameters (m I ,Pl) and (mil, P2 ), so that moment s of S and U are readily available.
Ta.}dng expectations in (3) term-by-term, we find (after some labor)
~E
L*
=
rc(1-rt)(P l -P 2 ) €
_ -lM-l
where rt
= m'/M,
+ ~(l-rt) e.2 ( M~l [rtpl+(1-rc)P2 ]2
[rtp 2+(1_rt)p 2] _ P _ rt(p _p»)
1
2
1
1 2
(4 )
+ 0(€3)
the proportion of the observations in any stage Which are from
the X-population.
.-
The modulus of the RHS of (4) is seen to be maximized (and hence E.N
~
minimized) when rt
for simplicity.
Ho~.
Case 1:
= t.
Hereafter, we shall confine attention to this case
We now examine E L* separately for H and for H •
O
l
ml
= mil = M/2,
Pl
= P2 = P
(say), and q
= l-p
•
Then (4) becomes, with some additional terms in (3) also evaluated,
.2
1M E0 L* = - 1
8' pq
~
[ 1 -
€
+
M-2)
8'1.2
~ (7 + M-l pq ]
This expression is increasing in M (for
€
+ O(eS).
small), but very slowly, and hence
the approximate ASN under H is seen to decrease (slowly) with increasing batch
o
size M; on the other hand, if M is too large, the overshoot cannot safely be
ignored and the approximation to the ASN would not be valid.
that there is an optimal batch size (for small
€,
It would appear
i.e., pI close to unity),
presumably greater than M = 2, but that not much improvement over the case H
is typically possible.
eM
denote the ratio of ASN approximations (equation (1» when
,p
the batch size is 2 (numerator) and when the batch size is M (denominator),
when H is true and the common Pi value is p. (When the batch size is 2, the
O
test is Wald's double-dichotomy test.) Then eM,p is a measure of efficiency
Let
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=
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1
+
1
"8
'l'rl1ere it is to be recalled that
-2
pq
E
t
relative to Hald's test.
He find
M-2
M-l
= pI -1
and pI (> 1) is the value of the odds
ratio under H • Thus, the efficiency under H is slightly improved with M > 2,
l
o
uniformly in p; but it never exceeds 1 + pq E?-/8 + O(ES ), which is at most
+ O(ES ).
1 + €2/32
This supports the conclusion drawn in the previous
paragraph.
Case 2:
Hl~.
ml
= TI" = M/2,
PI ~/P2ql
= pI
•
Tl1en ().j.) becomes
(6)
\';l1ich is also increasing in i'1 (for
Letting
E
small) •
be the relative efficiency under H of the extended
;Pl,P2
1
= mil ana Pl~/P2ql = pI) relative to Vlald's test, "le find frora
eM
test ("lith ro l
(1) and (6)
eM;Pl,P2
=
Again, the efficiency is sliehtly increased by choosing M > 2.
The least
upper bound on the second order term in (7) may be shown to be € 2/32, the
Sal"1e
value as obtained in case 1 above.
Hence, only a small percentage re-
d.uction in the ASN, under both H and Hl' can be achieved by increasing the
O
'IJatch size.
The ASN can be evaluated from (1), (5) and (6); a fe"l values of the
.~onstant c., when ex
l
l~
c.
J.
I.
I
I
= mil),
under H of the extended test (with ml
O
=
= c ex~~) =
=13,
are r;iven beloH:
.1
.01
2. 6~-
51
4.50
.001
I
Appendix
.-I
2
ASH OF THE PERMUTATION TEST
In this appendix we
sl~l
briefly investigate the
~r
of the sequential
I
I
I
I
I
I
permutation test (Test 6) '~1en sampling in pairs and When the observations are
actually normal.
A more thorough investigation, under broader assumptions,
is warranted, however.
Noting the remarks in Appendix 1 on the ASN' s of CSPRT' s, we only need
evaluate the expected increrrent, E L* , for the permutation test in order to
approximate the ASN under
He
and H • When sampling in pairs (S2), and with
l
Z = Y-X, a typical increment L* is given by
L*
6
= 10g[2/(1
+ e- 'Z)] •
We shaJ.l approximate its expectation under the assumption that Z is normal
and 6' is small.
(Recall that 6' is the hypothesized value for the mean of Z
divided by one-half the variance of Z under H , and is positive.)
l
as follows:
Let
= -
g(w)
w
log ~l + e- )
so that
L*
= g(6'Z).
.-
We proceed
Expanding g in a
Maclaurin series, we find
= ~
g(w)
-> o. Let
-> O. Taking
\T
~ w2
-
+
l~
4
w
-
= O(h)
2§-80
"1
+
O(W
= O(h)
S
)
as v1
W be N(J.l., 2v) and assume that J.l.
h
expectation of g(H) term-by-term, He find (after some labor)
=
E g(W)
1
'2J.l.
-
1
J+v
1
2
E~
-
and v
6
1
+ Ibv~
Nmv let He be the hypothesis that Z is N(O, 202) and
where 0'
= 6'02;
= g(W).
Hi
that Z is N(o', 202)
Letting 11 = (6' 0)2
'Ive llave equivalently:
a.nd L*
(8)
these are compatible with the hypotheses Ho and Hl' respectively,
v1hich are being tested by the permutation test.
He:
for some
W is N(O, 2h)
,
Hi:
Hence, from (0), we have
52
W is N(h, 2h)
and VI
= 6' Z,
I
I
I
I
I
I
.-I
I
I
I.
I
I
I
I
I
I
I
••I
I
I
I
I
I
I.
I
I
For the normal theory test of H~ vs. ~ for specified (j (Test la or lb),
"le find the corresponding increment, say L
, is given by
t ~'Z
L** =
so that
E L**
o
=
,
E L**
1
=
(10)
Note that, to the first order of approximation, (9) and (10) agree.
se~1.'.ently,
Con-
to the first order of approximation, the ASN's of the two tests--
tIle rermutation test (test 6) and the normal SPRT (Test lb)--agree when one of
the normal test hypotheses is true.
=
h
(~' (j)2
= ('0' / (j) 2
The approximation is valid for small
•
The approximate ASN' s (actually, expected number of stages, 'in thL,\'io
= L~ci/h,
observations per stage) for the normal test are EiN
in Appendix 1.
For the pel~utation test, E.N
J.
=
is defined
i
2
(4c./h).(1 + ~ + 0(h )). T11e
J.
ratio of these ASN's provides a measure of efficiency:
of the permutation test relative to the normal test.
than unity, but is asymptotically (h
->0)
where c
namely,
1 -
-&
h + 0(h2 ),
T11e efficiency is less
unity.
SUbtracting the normal test ASN from the correspondine permutation test
ASH, lIe obtain c. + O(h).
J.
Hence, the permutation test requires approximately
c. ac..C.itional pairs of observations, compared with the nonnal test, when sampJ.
linG in pairs, '-Then h is
SL~J~,
and vmen H! is true.
J.
[';ivcn at the end of Appendi:c 1. (p. 51).
53
S01~
values of c. are
J.
I
.1
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a
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~---.;...;.;..~~
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55