UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
ON TESTING FOR INDEPENDENCE OF UNBIASED COIN TOSSES
2
WMPED IN GROUPS TOO SMALL TO USE X
by
Richard F. Potthoff and Maurice Whittinghill
January 1963
Contract No. AF 49(638)-213
National Institutes of Health Research Grant No. RG-9358
In genetics research and perhaps in other fields also,
if we encounter data classified into two categories
know to be equally likely, we may suspect that the
individual observations are not independent and hence
do not observe the binomial distribution, and we may
wish to find out whether this suspicion is correct.
If the groups (families, e.g.) into which the observations are lumped are sufficiently large, we may use the
~2 test to test for independence; but if the group sizes
are too small, an alternative test is needed. Such an
alternative test is developed in this paper.
This research was supported in part by National Institutes of Health
and in part by the Mathematics Division of the Air Force Office of
Scientific Research.
"
Institute of Statistics
Mimeo Series No. 347
....~-:t.)~.,'"
..
~<.,."
ON TESTING FOR INDEPENDENCE OF UBBIASED COIN TOSSES
LUMPED IN GROUPS TOO SMALL TO USE
2
X
by
1
2
Richard F. Potthoff and Maurice Whittinghill
Department of Statistics and De~artment of Zoology
UNIVERSITY OF NORTH CAROLINA
Cha~el Hill, N. C.
= = = = = = = == = = == = =
~
SUMMARY;
The problem treated in this
= = = = = = = = = = == = = =
- - =====
~
ORIGIN OF THE PROBLEM
~a~er
manifested itself in genetics research, al-
though it is of a general enough nature that it could arise in many other areas
also.
In genetics one sometimes encounters a situation where two
~henotypes
are
known to occur in a 1:1 ratio; where data are available from (say) N families, the
= 1, 2, ••• , N) producing
i-th family (i
type and
Xi offs~ring observed to be of one pheno-
(n
- Xi) offspring of the other phenotype; and where it is suspected
i
that some factor (SUCh as gonial crossing-over, e.g.) may be operating to cause
the random variables x.
~
model of n
i
independent tosses of an unbiased coin.
cr~ssing-over,
e.g.) is not present, then each
variate with parameters
sumed that the two
~i
If the factor (gonial
will behave as a binomial
n~... and ~2 ; if the factor is present, then it is pre-
phenotypes will still occur in a 1:1 expected ratio and that
the distribution of x..
~
x.~
to have greater dispersion than they would under the
will still be symmetric, but that the distribution of
will be heavier in the tails and lighter in the middle than the binomial
~esearch supported in part by National Institutes of Health Research Grant
RG~9358
and in part by the Mathematics Division of the Air Force Office of Scientific Research.
2Research supported by National Institutes of Health Research Grant RG-9358.
e
2
distribution.
To test the null hypothesis that the factor is absent against the
alternative hypothesis that the factor is present, it is appropriate to use
with N degrees of freedom (and reject the null hypothesis for large
provided that the
for the
2
X
test.
values),
ni's (the family or group sizes) are of sufficient magnitude
If the
n.'s
are too small to allow us to use
~
2
X •
we need to have a test available in place of
this paper,
x2
2
X
and~umerical example
2
X,
however,
Such a test is developed in
based on some genetics data is presented.
Essentially the same basic (statistical) problem which gave rise to this paper
has been recognized and considered before, by GrUneberg and Haldane (1937).
The
test employed by those authors, however, is eVidently usable only when N is of
a magnitude somewhat greater than that required for our test.
Although our test was developed specifically for the purpose of handling a
situation where the
n. 's
2
X , we should mention that there
are too small to use
~
is no reason why our test cannot also be used, if desired, even in the case where
the n.' s
~
are large enough for
2
X •
1.
INTRODUCTION
We suppose that we have random variables
Zij (j
N) such that the marginal distribution for each
Zij
= 1,
2, ••• , n ; i
i
= 1,
2'0'"
corresponds to the model for
the toss of an unbiased coin, i.e.,
for all (i,
1
1
(1.1)
= "2
'2 '
j). The Zij'S, however, are unobservable: we assume that it is pos-
sible to observe only the quantities
xl
=
x.!
=
n
i
~
j=l
Zij
(1.2) and
~
= n.
~
3
4It
~or
i
= 1,
2, ••• , N.
Under either the null hypothesis
hypothesis, we assume that the vector random variables
zi2' ••• , z.
J.n.
J.
)'
-
7 are
or the alternative
z.J . -;-where
z.(n.
x 1)
J.
J.
=
mutually independent (thereby implying that the
xi 's are mutually independent), and that (1.1) holds.
We consider a null hypothesis
elements of
of X.
J.
zi
H which specifies that, for each i, the
o
are mutuallY independent. Hence, under H, the density function
o
is
~'
(1.3)
n
(x) = 2
i
i
-no
J.
We will consider alternative hypotheses
of
z. are not
J.
-
mutually independent.
H which specify that the elements
l
More specifically, we will be primarily
concerned with alternative hypotheses under which (roughly speaking) either (a)
the larger values of IXi - ~ nil generally occur with greater frequency than
under Ho (1,3), and the smaller values with lesser frequency; or (b) the larger
values of
IXi - ~
nil
generally occur with lesser frequency than under Ho
(1.3), and the smaller values with greater frequency. We will refer to these two
situations as alternative hypotheses of the forms
H
and Hlb respectively,
la
Thus, under an alternative hypothesis of the form Hla , the distribution of Xi
spreads out more into the tails than does (1.3) (roughly speaking); under an
alternative hypothesis of the form H , the distribution of Xi
lb
is bunched in
the center more than (1.3).
In the particular problem which motivated this paper, the alternative hypothesis was (as already indicated) of the form
H • Because of this, our emphasis
la
in this paper, particularly in Section 3, will be slightly more on the Hla situa-
e
tion tran the H ,
lb
However, the general approach we develop in Section 2 will be
just as applicable to Hlb as to Hla •
We will assume that, for ar~ alternative hypothesis we consider, the distribution of
zi
will be determined solely by nit
In other words, for any
(1, I)
4
= nI ,
such that n.
~
the distribution of z.
~
zI (under any permissible alternative h~~othesis).
of
If the
n. 's (the group sizes) are sufficiently large, the statistic
~
2
(1.4)
X
=
N
121
(x.; - -2 n.) !(T. n.)
i=l·
~
~
~
L:
will follow approximately the
Ho
will be the same as the distribution
f
see (1.317 is true.
X2 distribution with N degrees of freedom if
H against an alternative of the form Hla ,
o
we reject Ho at the a-level if x2 (1.4) exceeds the (l-a) fractile of the ~
distribution; to test H against an alternative of the form H , we reject H
lb
o
o
2
.
2
at the ~level if X (1.4) falls short of the a fractile of the XN distribution.
This
2
X
To test
test cannot be used, however, if the n.'s
do not satisfy certain
A traditional rUle-of-thumb for using x2 is that the ex~
minimum reQ.uirements.
pected freQ.uency in every cell should be
valent to reQ.uiring that every n.
~
~
5 (which, in our case, would be eQ.ui-;;;
be > 10).
-
More recently, some authors _;-see
Cochran (1952), e.g.:.7 have maintained that this rule-of-thumb can be liberalized
to a certain extent, and that some of the expected cell freQ.uencies can be permitted to drop below 5 under certain conditions.
Regardless of what rule is
accepted, though, experimental situations can clearly arise in which the rule says
that the
ni 's
are not large enough for the
X2 test to be satisfactorily
accurate.
2
In such situations, where X cannot be used, it is desirable to have available another test which can be used instead.
This paper presents a method of
constructing such an alternative test.
The test statistic we present will be approximately normally distributed
.
2
under H rather than approximately X - distributed. Furthermore, as N ->
o
00
5
~
(with an upper bound assumed for the
n 's)1 the distribution of our test statisi
tic under Ho will tend exactly to normality in the limit; note that an analogous property is ~ generally true for X2 (1.4) ~i.e., if we increase N, this
need not improve the approximation of the distribution of X2 (1.4) to the
distribution if the
n. t s remain sma117•
-
~
The test used by Gruneberg and Haldane (1937) employs
the usual way.
X2
x
2
(1.4), but not in
The exact value of var(x2IHo) (which will be < 2N) is calCUlated,
and is required for the denominator of the statistic
(1.5)
Under Ho ' (1.5) will be approximately N(O,l) for sufficiently large N ~Cochran
"
(1952) seems to feel that N should be ~ 69.,7. Thus, in the test of Gruneberg
2
and Haldane (1937) which uses (1.5), X
is taken to be approximately normal under
Ho rather than approximately { - distributed (which it is not for small l1 1 S ).
i
2
However, the distribution of X (1.4), like that of ~, is evidently rather slow
to approach normality with increasing N; for this reason, a somewhat larger N
seems to be required for the test using (1.5) than for our test.
When the
ni's
are too small to use
2
X,
it is evident that a possible test
of H could be obtained by pooling certain of the groups so that the sizes of
o
2
the pooled group~ would be sufficiently large to calculate a X based on the
pooled groups.
However, such a
X2
test, in addition to forcing arbitrary ,
decisions as to how the pooling would be done, would apparently have the more
serious disadvantage of adversely affecting the power of the test at least against
certain alternatives.
kind of test of H.
o
Thus it looks as though we should, if possible, avoid this
6
Our proposed technique of testing
H will be introduced in Section 2; also
o
:i.n Section 2 it will be demonstrated that our test statistic exhibits the desired
Section 3 deals with the role of the alter-
behavior under the null hypothesis.
native hypothesis.
Section
4, which is intended to suit the needs of the non-
mathematical reader, treats a numerical example based on data from a genetics
experiment.
2.
THE TEST STATISTIC, AND ITS BEHAVIOR UNDER
H
o
We will start by constructing some new random variables which are functions
of the
Xi's (1.2).
the subscript
i
In this section and the next section, we will sometimes drop
on Xi
and n , and write simply x and n
i
respectively.
Suppose we let
(2.1)
We
r
the null hypothesis:
using
we can write, if n
is odd
f
and if
n
nr
=
21-n
is even
f'
(2.2b)
nr
(n
=
f nr
(n = 2m+ 1, say) ,
n
(r)
(for
= 2M,
say)
n
21 - n ( )
r
(2.3a)
n
odd (n
= 2m
+ 1), and
r(2.1)
to denote the density function of
r
= 0,
1, 2, ••• , m)
,
,
(for
r
= 0,
(for
r
= M)
We will define random variables
for
n-x)
that we can obtain from (1.3) the distribution of
remark
(2.2a)
= min(x,
1, 2, ••• , M-1)
r
under
under H '
o
7
2-
=
n
r2 r-l (n) + (n) 7
E
k=O
-
r-l
= 2-n~2 to (~)
k=
for
n even (n ~
member that
r-
k
(for
r
= 0,
+ ~ (~)_7 (for r=M), i.e., YtiM
2M). We also define y.
n is a constant and r
u
=
= 1_2-n-l(~)
= y
Our proposed technique for testing the null hypothesis
(2.4)
... , M-l)
• In the notation (2.3), reniri
is a rano.om variable.
J.
native of the form H
la
1, 2,
H
o
against an alter-
or H
will be based on a test statistic of the form
lb
N
E
i=l
wi
,
Yi
which is a weighted linear combination of the y 1 s (2.3).
property possessed by these
The special desirable
y1s (2.3) has to do with their expectations:
we
will show that
1
= '2
(for all n)
whereas it appears that E(Ynr)
under Ho
'
should generally be < ~
or > ~ under the
respective (as yet somewhat imprecisely defined) alternative hypotheses
H
la
or
Hlb ~in fact, see (3,1 - 3.2)_7.
At the end of this section we will prove that, under H, the distribution of
o
(2.6)
z =
u - E(u
fvar(u
approaches the
I Ho )
/HJ7
1
~
N( 0,1) distribution as
N ->
00.
Thus we will be ab Ie to con-
clude that, in order to test H against H
at (approximately) the a-level,
o
la
we can use a test with critical region
z <
-za
,
8
~
where
za is the (I-a) fractile of the N(O,l) distribution; and that, in order
to test Ho against
z
>
H , we can use the critical region
lb
Z
a:
The consistency of the tests (2.7) will be discussed in Section ;.
The weights
w. in
J.
(2.4) are constants whose selection must of course not be
influenced by knowledge of the yif S (or r i ' s ) • It seems appropriate to try to
choose the w. 's
J.
in such a way as to maximize the power of the test against the
alternative hypothesis, and this line of approach will be explored more fully in
Section;.
Beacuse of our assumption, indicated in Section 1, to the effect that
the distribution of z.
J.
is determined solely by n. (for any alternative hypotheJ.
sis), it would seem logical to stipulate that Wi
should also be determined sole-
ly by n :
i
(2.8)
Except for this restriction
(2.8), however, our development in this section will
be in terms of general Wi's) and will be valid for any appropriate weight function w(n)
which might be chosen.
under Ho • We now prove (2.5).
is odd (n = 2m + I), then
Expectation of y
=1-r
and if n
is even (n = 2M), then
First note that, if n
m
... (n)
-'72 =
r _,
r=O
LI
=
2
2n-2
,
(2.10)
From (2. 2a), (2 .3a), and (2.9) we obtain
(2.11)
E(Y /H)
nr 0
=
m
f
Y
r='O nr nr
~
=
1 n
n
2n-2
2 - . 2- . 2
,
1
= "2- (for odd n);
from (2.2b), (2.3b), and (2.10) we obtain
M-l
(2.12)
E(YnrlHo )
=
~
f nr Ynr + f nM Y.nM
x= O
(for even n) .
.e
Thus (2.11) and (2.12) together prove (2.5).
Note that (2.4) and (2.5) give us
(2.13)
E(u
I H)
=
o
1
-2
N
~
. 1
~=
Variance of Y under Ho •
w~
...
Observe from (2.4) that
(2.14)
where we define
(2.15)
Thus, in order to be able to calculate the denominator of (2.6), we will need to
have available the values of v(n) (2.15) .
We proceed to develop formulas for these variances.
combinatorial identities.
For odd n, we get
First we establish two
10
= (4/3)r6
z
- o~<f<~ m
+ 3
z
o<k<r<m
-
-
(~)(~)
2
3
m
3
+ Z (~) _7-(1/3) z (~)
r=O
r=O
m
1
= (1/3) 23n- - (1/6) Sn
,
where we define
(2.17)
In a similar manner, we obtain for even n the identity
(2.18)
M-1
r-1
M-1
M-1
Z (~) £2 Z (~) + (~72 = (4/3)£ z (~L73 - (1/3) z (~)3
r= 0
k=O
r=O
r="O
Sn again being defined by (2.17) •
11
tit
We now use (2.2a), (2.3a), and (2.16) to get
(2.l9a)
v(n)
-
=
1
~
= (1/12) - (1/3) 2- 3n Sn
(for odd n)
From (2.2b), (2.3b) and (2.18) we obtain
(2.l9b)
v(n)
M-l
= E
r= 0
f nr Y2 + f nM Y2
nr
nM
-
1
1j:
The values of v(n), as calculated by the formulas (2.19), are exhibited in Table
2.1 for
n
= 1, 2, •.• , 20. Table 2.1 also presents (as a matter of information)
the values of
(2.20)
S (2.17), which were obtained via the recursion formula
n
2 2 2
n Sn = (7n - 7n + 2) Sn- 1 + 8(n-l) Sn- 2
This formula (2.20) was derived by Franel (1894), and is listed by Gould (1959)
fsee p. 69, formula (X.13)_7.
12
e
T/IBLE
2.1. Values of Sn (2.17) and v(n) (2.19) •
n
.-
v(n)
S
n
1
2
.000000
2
10
.062500
3
56
.046875
4
346
.068359
5
2,252
.060425
6
15,J.84
.071655
7
104,960
.066650
8
739,162
.073759
9
5,280,932
.070218
10
38,165,260
.075211
11
278,415,920
.072529
12
2,046,924,400
.076274
13
15,148,345,760
.074148
14
112,738,423,360
.077087
15
843,126,957,056
.075346
16
6,332,299,624,282
.077728
17
47,737,325,577,620
.076267
18
361,077,477,684,436
.078247
19
2,739,270,870,994,736
.076997
20
20,836,827,035,351,596
.078677
13
Observe that, if we substitute (2.4), (2.13), and (2.14) into (2.6), we obtain a computing formula for Z:
N
I:
(2.21)
=
Z
w. y.
i=l J. J.
1
N
- -2
w.
I:
i=l
J.
1
v(n.)
J.
-
7
'2
We now show that Z has the limiting distribution which was previously indicated.
Asymptotic normal!ty of Z
under Ho '
To prove that Z (2.6, 2.21) is
asymptotically N(O,l) ur.der H , we will utilize Liapounoff's version of the ceno
tral limit theorem. In order to simplify our argument, we will assume an upper
bound on the
ni's:
we specify that there exists an integer no
(2.22)
for all
i
We also assume that there are no groups with
n
=1
i
bute nothing to testing the hypothesis anyway).
weight function wen)
£ see
such that
(such groups could contri-
Our proof will be valid for any
(2.8t7, provided that wen) > 0 for all n > 2 .
Let us define
Kl
=
min £W(2), w(3),
K
=
max
=
min LT V (2),
2
f
w( 2 ),
w(:~),
... , wen
)
7
0-
... , wen ) 7
0-
and
(2.24)
Now Kl
that
is
> 0 by assumption.
v(3), .•. , v(n0)-7
Hence it follows from (2.8), (2.22), and (2.23)
(2.25)
for all
i
It is easily shown fVia an induction proof based on formula (2.20), e.g~7 that
v(n) > 0 for all
that
n > 2.
Hence K is > 0, and it follows from (2.22) and (2.24)
3
14
(2.26)
vle
for all i
are now in a position to show that the independent but non-identically
distributed random variables
u
w i
y'
i= i
satisfy the Liapounoff condition under H . Let P~ be the third absolute ceno
J.
tral moment of ui (2.27) under Ho ' From (2.3) we know that 0 ~ y ~ 1. Hence
3
( 2 •28)
If we let
.-
7j
Pi
2
~i
Wi
=
E(
fy i
-
13
"2 J ) ~
7j
Wi
be the variance of u i (2.27) under
2
2
~i = wi v(ni )
Ho ' then
Using (2.28), (2.29), (2.26), and (2.25) , we find that
1
"3
.rr,i=l Pi3 _7
1
.r:J.=. 1 ~~ 7 2
N
N
<
-
J.-
as
.ri=l ~_7
"3
r,
r,
which -> 0
1
1
N
N ->
N
.rN
<
1
-2
K2
=
1
2
-rK 3 N K1-j2
2
r, w 7
fK 3 i=l
ico.
K~7
"3
1
K
l
1
-5
N
K3'2
Hence the Liapounoff condition is established for the
N
u. f S (2.27) under H, and (since u = r, u.) it follows immediately that
:J.
0
i=l J.
z(2.6, 2.21) has the N(O,l) limiting distribution.
We have not made any very detailed attempts to investigate how rapidly the
distribution of Z approaches normality, or how large
N needs to be in order
for the normal approximation to be sufficiently accurate.
y
(2.3) is essentially a binomial variate;
nr
n becomes large, it looks as though the distribution of ynr tend.s
ply point out that, for
and that, as
=2
However, we might sim-
n
and 3,
to the uniform (rectangular) distribution on the interval
.r0, J:.7.
Since the dis-
tribution of a sum of binomial variates and the distribution of a sum of rectangular variates both approach normality rather rapidly, we might be justified in sus-
15
~
pecting that the distribution of Z converges to nOl~ality with roughly the same
The numerical example which we give in Section 4 has
order of rapidity.
N
= 11.
3 • THE ROLE OF THE ALTERNATIVE HYPOTHESIS
In Section 1, alternative hypotheses of the forms
RIa
and H
lb
were "de_
fined!! heuristically in language which was mathematically somewhat vague.
that we have defined the
fide definitions for
Now
y's (2.3), however, we are in a position to set up bona
RIa
and
RIb' definitions which may seem to be less meaning-
ful but which will be mathematically precise:
We say that an alternative hypothesis is of the form RIa
<
2:2
for all
n > 2
Similarly, we say that an alternative hypothesis is of the form
for all
(3.2)
Our "definitions" of RIa
tive notions.
and RIb
if
Hlb if
n > 2 .
given in Section 1 were based on intui-
The definitions (3.1 - 3.2) represent an attempt to translate these
intuitive notions into precise
tel~s,
the
yls (2.3) having been constructed ex-
plicitly for the purpose of discriminating between Ro and RIa
Consistency of the tests (2.7).
In establishing consistency, we will make
the same assumptions as in the asymptotic normality proof:
wen) > 0
or RIb •
for all n ~ 2, and the assumption (2.22).
no
n. 's equal to
J.
Under these assumptions, the
test (2.7a) is consistent against all alternatives of the form
RIa (3.1), and
the test (2.7b) is consistent against all alternatives of the form
RIb (3.2).
We need prove only the first of these two statements, since the proof for
analogous to that for
Let us define
RIa'
1,
RIb
is
16
and
=
min (d2 , ~, ... , dn )
o
From (3.1) it follows that D > O. vie also define
(3.4)
D
N
W.
].
= w./{ E wI )
].
I=l
Applying (2.25) to (3.5); we find that
(3.6)
Y ~ 1, var (Yi ) will be
i
vTe conclude from (3.6) that
Now, since
0
~
< 1 under either Ho or Hla .
(under either Ho
. ~
Hence
or HI)
a
Having established these preliminaries, we use (2.7a), (2.2l), (3.5), (3.3),
and (3.4) to write
(3.8)
p [rejecting Ho } = p,{z < - z
-5)
ex
= pILr: wi (yi - 1:)
<
2
- za fvar{ r: WiYi/Hot71!2 }
1
= p{-r:, vli Yi - E{r: WiyiIHla )< - zcl"var(r: itliYi!Hot7"2
+ E Vl.]. dn. }
].
1
1 }.
(L: H.y. - E(r: W.Y./H )< D - z rvar(r: W.y.jH )
-> p L
J.].
J. J.
Cif]. J.
0-'
la
From (3.7) it follows that the expression on the right of the < sign in the last
line of (3.B) will becoffie positive for sufficiently large N.
Hence,. for large
enough N, we may apply Tchebycheff' s inequality to (3. 8) to obtain
17
p {rejecting Ho IHla
1>
1
Again utilizing (3.7), we see that the right-hand side of (3.9) -> 1 as N ->
co
•
This completes the consistency proof.
§election of the weight function wen).
In Section 2 we indicated that it
would be sensible to try to choose the weight function wen) in such a way as to
maximize the power of the test against the alternative hypothesis.
of Section 3 will be concerned with this problem.
find some sort of rational basis for choosing wen)
The remainder
Our approach will be to try to
rather than to attempt a more
rigorous (and more complicated) development aimed at finding a more exact solution
to a problem of maximizing the power.
In other 'Words, our line of argument will
be heuristic to a certain extent, since there will be a few steps which will represent approxinations or which we will not justify rigorously.
This lack of rigor
at this stage of the game will not be particularly disadvantageous, however, in
the sense that all of the results which we have already proved are valid for any
choice of wen) ~so long as
does not matter how wen)
wen) > 0 for all
is obtained.
n ~ 2_7; thus, in this sense, it
In the remainder of the paper, we will
consider in detail only alternative hypotheses of the form H , since (as indicated
la
previously) the problem motivating this paper was concerned with an alternative
hypothesis of the form H
rather than R .
lb
la
For the test (2.7a), we can use (2.6, 2.21) and (3.3) to write
(3.10)
P [rejecting
Ho
J
18
~
If u (2.4) is approxi~ntely normally distributed under an alternative hypothesis
of the form Hla , then we can conclude from (3.10) that the power of the test
(2.7a) against H
is approximately
la
1
Lvar(u/H,.lj'2
v-
t·
I
IP ( - - - - - . . . - 1 ~ - z +
(3.11)
'2 )
Lvar(u/Hl <327
....where
~(.)
is the
~
N(O,l) c.d.f.
a
N
Z w d
i=l i n.
't
1)
J.
.ri=l
z w.
N
2
2j
,
v(n.) 7
J.
J. -
To maximize (3.11) (with respect to the Wi's),
it of course suffices to maximize the arg1.l24ent of
~
• Instead of
~.aximizing
the
entire argument in (3.11), however, we will attack the much more tractable task of
maximizing only that portion of the argument between the curly brackets, and will
hope that the
w.J. 's thus obtained will result in producing approxiwately the maxi-
mum possible ve.lue of the entire argument. It would seem plausible that the ex\"71 / 2
.
pressJ.on
Lvar(u/HJ71/2 / Lvar(uIHla"L1
may not vary too much under different
choices of wen); if such be the case, then our approximation (which ignores this
expression in front of the curly brackets in order to circumvent certain other
problems) should not hurt us noticeably.
Maximizing the expression within the curly brackets in (3.11) is equivalent
to maximizing the quantity
(3.12)
where vi
)2/(Z
d
n
denotes
i
v(n.).
J.
w~J. v.)
J.
,
First we use Cauchy's inequality to write
(3.13 )
Since (3.13) tells
2
that (3.12) cannot exceed ~ dn /v
for any choice of the
i
2
w. 's, it follows that (3.12) achieves its maximum value (= ~ d /v.) with respect
US
i
J.
~
J.
19
~
•
= dn Iv.~ ,
to the wits if we put w.
~
(3.14)
wen)
i.e., if we use
i
= dn Iv(n)
as the weight function.
Thus, it appears that the weight function (3.14) approxi-
mately maximizes the power of the test (2.7a) against an alternative of the form
H1e. (3.1).
Although the denominator of (3.14) will clearly be the same for all alternative hypotheses, the numera.tor will be different.
The imagination can concoct a
number of alterT'ativc hypotheses seeming to be of the for:'1l RIa' each with its
own d is (3.3).
n
Here, however, it will suffice for us to explore two such hypo-
theses, to be designated as
the
d I~
n "
H
and H • For both of these, we will evaluate
laA
JaB
is "close" to H0' and the two sets of dn 's will turn
when the
H
la
out to be markec.ly different from each other.
Alter.nat~Te
hypothesis
we suppose that the
z.. I S
according to (1.1).
For
].J
In the first of our two alternative hypotheses,
H .
laA
are determined as follows. Let zil be determined
j > 1, we suppose that, with probability (1-0),
is determined by a coin toss
~as
in
(1.117,
but that the Temaining
time
automatically assumes the same value as
(3.15)
P [Zij -- Zi,j-l.}
I
Zi,j_l}
where the statement
"
.1
zi'
r
P [ Zij
J
0 of the
Zi,j_l' in other words,
1
1+&
= "2 (1-&) + 0 = ""2
(j > 1)
1-0
(j > 1) ,
= 2
zi ,J. 1"
Zij
-
= 1 if z. j 1 = 0 and
means that
~,
vice versa.
If
*
~n.(xi)
denotes the density function of
~
duce from (3.~)
that
Xi
under Hla , then we can de-
20
,
where
R represents the number of runs in a sequence of x l's and (n-x) O's,
and where
;-(n) g
(R) 7 is the total possible number of such sequences conx
nx
taining exactly R runs. That is, gnx(R) can be thought of as the probability
-
of obtaining R runs in a randomly arranged sequence of x lIs and (n-x) O's,
n
assuming all (x) possible sequences to be equally likely; the formula for gnx(R)
is given (e.g.) by Feller (1950) ~see p. 57, formulas (4.1 - 4.217.
(3.16) ~and remembering that
gnx(R)
Re-writing
can be thought of as a density functioE7,
we obtain
<l> *(x)
n
where
<l>n(x)
is defined by (1.3).
Now we may interpret E R gnx(R)
simply the expectation of R, the formula for which is well-kno'WIl
as being
~ see
Dixon and
Massey (1957), p. 289 or 422, or utilize Feller (1950), Chapter 9, Problem 12,
pp. 188 and 40§l:
n
(3.18)
ERg vCR)
R=l
nA
=
2x(n - x)
n
+ 1
Substituting (3.18) ~ (3.17), we end up with
If
f*
nr
also for H
1aB
(3.20a)
r under H , then for
1a
, it will turn out) we can write the formulas
denotes the density function for
f*
nr
=
n*
2 <l> (r)
(for r
= 0,
H
(and
laA
1, 2, ••• , m)
e
21
for
odd n(n = 2m + 1), and
*
f nr
(3.20b)
= 2 ~:(r)
=
<I>
for even n(n=2M).
(for r= 0, 1, 2, .•• , M-l)
*n (r)
(for r = M)
Before we proceed to use (3.19 - 3.20) to evaluate dn (3.3),
however, we will first need to establish some combinatorial identities.
First note that
6 (n)
(n)
(3 •21) 2 k:O
k + 6+1
If n is odd (n
= 2m
s r(n-2)
(n-2)
(n-2) "7+r(n-2)+2(n-2)+(n-2) 7
k-2 + 2 k-l + k - - s-l
6
6+1 O
= 2k:
+ 1), we can use (3.21) and (2.9) to write
/
Similarly, for even n(n
M-l
(3.23)
= 2M),
r-l
r~O (~:i)f2k:() (~)
+
we U6e (3.21) and (2.10) to obtain
(~17
M-2
s
= s:o (n;2) £2 :{)
k
(~)
From (2.9) and (3.22) we obtain for odd n the formula
+
(s~1)_7
22
For even n, we can apply (2.10) and (}.2,) and write the formula
M-l
r2k=r-lZ 0 (n)+(n)7+
2: f(n)_4(n-2)7 r2 Z (n)+ 2:(n) 7
k r- 2
M M-l- - k=O k 2 M
M-l
Z r(n)_4(n-2) 7
r-l - r=o - r
-~
•
Now that we have (,.24) and (,.25), we can evaluate
Using (,.,), (3.20a),
d •
n
(,.19), (2.11), (2.,a), and (,.24), we get
(,.26)
(for odd n, n
= 2m
+ 1)
By using (,.,), (,.20b), (,.19), (2.12), (2.,b), and (,.25), we obtain
.
(3.27)
r
.
2
d = 2 e- n (n_l)·2- n 1 .-E;... (n) 7
n
- 1j: n-l M = 2- 4M
M(~)2
2
5 + 0(0 )
0 + 0(0 2 )
(for even n, n
= 2M)
•
Thus, if 0
is near O(i.e., if HlaA is ,"close II to H) , it would seem approo
priate to use the weight function
=
2
2- 4m m(2m) /v(n)
m
= 2- 4M M(~)2/v(n)
(for n
+ 1)
(for n = 2M)
fThere of course is no effect on Z (2.21) if w(n)
such as
= 2m
is multiplied by a constant,
1/0._7 If we utilize Stirling's formula, we find that
l/rr .
,
23
which indicates that
d is about the same for all n.
n
might consider using the weight function
Because of (3.29), one
(3.30)
in place of (3.28) •
Our second alternative hypothesis will result in a weight
r~nction
which is
radically different from (3.28) or (3.30).
Alternative hypothesis
H
• Under H
we suppose that, just before
laB
laB
the Zij'S are determined, a number
c
is obtained by a coin toss (so that
i
this ci has a 500/0 chance of being either 0 or 1). Then for any j we
suppose that, with probability
is determined by a coin toss ~as in
(1-5), Zij
(1.117, but that the remaining 5 of the time
value
Cit
with P tZij
Thus, given
= l}
c
=~
i
Zij
automatically assumes the
= 0, the Zij'S will be independent binomial variates
(1-5); conditional on
f
= 1, the Zij'S will be inde-
ci
pendent binomial variates with P Zij = 11 = ~ (1 +0).
for
Hence
HlaB " Re-writing (3.31), we have
n
(~) ~-x
= <l>n(x)
+ 2-
= <l>n(X)
+ 2- n (n)
2
+ (n;
2X
t7 02 + 0(0 4 )
r(n)
_ 4(n-2)
7 02
x
x-l-
L
+ 0(0 4 )
Except for tbe important matter of a factor of (n/2), (3.32) is virtually
identical with (3.19).
d for
n
Thanks to this similarity, all of our work in evaluating
H
has essentially already been done.
laB
We obtain
24
= 2-4M-1
(for n
= 2m
(for n
= 2M)
2
Mn
(2M)
M
+ 1)
corresponding to which we have the weight function
=
=
if we consider
52
t"'l_ 2
2- 4m mn (~) /v(n)
m
2
2-1.Jl.1 Mn (~) / v(n)
to be near
(for
n
= 2m
(for
n
= 2M)
+ 1)
o( 54).
0 and disregard the term
If the numera-
tors in (3.34) are replaced by their approximate values based on Stirling's
formula, we can write the weight function
Looking at (3.30) and (3.35), it appears possible that
H
laA
and H
may
1aB
represent two rather extreme (and opposite) types of alternative hypotheses of the
H1a " If this is so, then it might be reasonable to consider some sort of
form
"minimax weight function", such as
1
(3.36) wAB(n)
=
=
2- 4m
"2
1
2","4M Mn"2
or
2
) /v(n)
m
(for
n
= 2m +
(~)2 /v(n)
(for
n
= 2M)
(
ron
2m
1)
1
(3.37)
wA'B' (n)
"Hhich lies
'~etween"
=
2
n /v(n)
the weight functions for
H
1aA
and H
•
1aB
Such weight func-
tions as (3.36) or (3.37) might be appropriate particularly when the exact nature
of H
1a
is only haZily envisioned.
'We now make some final retne,rks relevant to our rather limited and somewhat
~
heuristic investigation of the optimum weight function:
25
The sharp difference between (3.30) and (3.35) might suggest toot the
(i )
use of a non-optimal weight function could have an emphatic adverse effect on
the power.
However, some rough calculations indicate that the power lost by
using (3.30) when (3.35) should be used, or vice versa, may not be as great as
one might suspect.
Moreover, it appears that the maximum possible power loss
which could result
from use of a non-optimal weight function can be cut sub-
stantially by employing a ''minimax'' weight function such as (3.37) for (3.3617.
(ii)
Sometimes it might be most suitable to base the choice of wen)
experimental data.
That is, the weight function to be used for a given experiment
E (say) could be selected on the basis of data from a different
experiment
--
mate
d
n
E (say).
o
function
(iii)
and H
laB
For example, the data from
E
0
(independent)
could be utilized to esti-
by fitting a linear regression of the form (0: + t3n), and this
estimate of
d
A
n
A
(Le., a + (3n)
E.
A number of possible specific alternative hypotheses besides
can evidently be conceived of.
to be a function of
n
f5
Eo
could then be used as the numerator of the weight
wen) (3.14) emplqyed for experiment
= 5(n>-7
weighted combinations of different
new patterns for
on
For example, just by allowing
in either H
laA
H
la
or
H , and/or
laB
H
laA
0
by taking
density functions, we can obtain several
H •
la
(iv)
For both H
and H , the formulas obtained for wen) were
laA
laB
arrived at by assuming 5 to be near o. No attempt vas made to determine
whether larger values of
formulas for
w(n).
either
or
H
laA
5 would result in any important change in the suggested
fIn fact, strictly speaking, we did not even prove (for
H )
laB
formally proved only for
that (3.1) holds for larger
0
near
0~7
5; this relation (3.1)
'WaS
26
(v)
Although our discussion of the choice of wen)
was directed only
toward H , part of our development is evidently applicable to Hlb also. Cerla
tainly a formula analogous to (3.14) can be obtained for Hlb , via an argument
analogous to the one we used in getting
(3.14). If, in HlaA , we assume that
(3.15) ff. (which amounts to assuming
is negative rather than positive in equations
that,
0
of the time,
automatically takes the value opposite from
Zij
&
z . . l'
J.,J-
rather than the same value as
zi . 1)' then (for 0 near 0) we will have an
,Jalternative hypothesis of the form H
and will end up with the weight functions
lb
wA(n) (3.28) and wAt(n) (3.30). There seems to be no way, however, of obtaining
an alternative hypothesis of the form H
which is an analogue of
lb
HlaB
(3.31).
4. NUMERICAL ILLUSTRATION
For illustration we will use some data obtained by one of the authors (M.v1.)
in a pilot experiment which was concerned, among other things) with trying to
detect gonial crossing-over ~see Vlhittinghill
(1950) for a more detailed discus-
sion of this phenomen0,E7 is non-irradiated Drosophila melanogaster females.
The
offspring of 12 females in this test-cross experiment were classified according
to the characteristics purple eye color (pr) and straw bristle (stw), both of
which are on the second chromosome.
four categories
Thus each offspring belonged to one of the
+ +, pr stw, pr +, or +stw.
For purposes of the illustration
here, we are not concerned with the data for the two non-cross-over classes
(+ + and pr stw).
The results of the classification for the two cross-over
classes (pr + and + stw) are presented in the first four columns of Table
(E.g., the 6th
a total of n
female had x=l pr +
=8
offspring and x i =7
cross-over offspring altogether.)
overs and + stw cross-overs will occur equally often.
~
+stw
4.1.
offspring, or
On the average, pr + crossFurthermore, if classical
genetic theory holds strictly, the cross-overs for each of the 12 families should
27
tit
observe the binomial distribution (1.3).
If some gonial crossing-over (which
is not allowed for in classical theory) is present, however, the distribution
~~ll
not be binomial, since the cross-overs will no longer all be independent events.
Rather, in the presence of gonial crossing-over, it is expected that pr + and
+ stw cross-overs will still occur
~qually
. often in the long run, but that the
more lop-sided ratios (lop-sided in either direction) of x to xt(pr + to + stw)
will occur more frequently than in the binomial distribution (1.3).
In order to
determine whether the experimental data in Table 4.1 support the null hy:pothesis
of no gonial crossing-over, we will calculate the statistic Z (2.21); we will
use the one-tail test (2.7a) and reject the null hypothesis of no gonial crossing-
.05 level if Z < - 1.645 •
over at the
.-
We may write the formula (2.21) more explicitly in the form
N
I:
(l~.l )
Z
=
v(n.)
Yn..
r
. 1 w(n.)
~
~=
~ ~
J1~1
..
1
'2
N
I:
i=l
w(n )
i
,
~w(ni)_72 v(ni )
is specified by (2.3), and
niri
depends on our choice of weight function based on the suspected alternative
~
is obtained from Table 2.1,
hypothesis (see Section 3).
y.
Since it seems to be rather difficult in this case to
formulate the alternative hypothesis (i.e., the hypothesis that there does exist
some gonial crossing-over) in explicit mathematical terms, we will make what seems
to be the relatively "safe" choice and use the "minimaX" weight function (3.37)
for purposes of our illustration:
(4.2)
•
28
4It
N in
(4.1) is the number of families (females), and, in effect, is equal to 11
rather than 12 in our example, since we must discard the data for the 3rd family;
= 1) can contribute nothing
i
toward distinguishing between the null hypothesis and the alternative hypothesis,
any family having only one cross-over offspring (n
and is therefore discarded.
The last four columns of Table 4.1 represent the intermediate steps in t'he
col~s,
r. (2.1), is just the
calculation of Z (4.1).
The first of these four
lesser of Xi and xi.
The second column is obtained from Table 2.1, and the
third column from formula (4.2).
(2.3), which is
nir i
a certain linear function of some binomial coefficients and is easily evaluated.
For example, for
The final column is for
~
i = 1 we have n
l
= 5 and r
l
y
= 2, so that fusing (2.3aL7
we get
Alternatively, we note that (2.3) may be
(4.4)
Ynr
=f
2-
n r-l
Z,
k=O
(~L7 + f2-
:t'~written
n r
Z (~L7
k=O
in the form
(if n ~ 2r)
(if n
which tells us that
= 2r
) ,
may be obtained simply by using tables of the cumulative
nr
binomial distribution fsee Owen (1962), p. 265, e.g._7; thus, if we use (4.4) to
y
evaluate Y52 (e.g.), we have
.1875
+
.5000
= .6875
,
29
e
which is the same as (4.3).
for
ynr
We should emphasize the difference in the formula
in the special case where n
second line_7:
for i
yn r
12
12
=
= 12,
= 2r Lr
see (2.3b), second line, or (4.4),
e.g., we have
n
= 2 and r
12
12
-2
2
1 2
Y2l = 2 ["2 (0) + -2 (1)_7 = .75
=1
or
- 2 -2-1 (2)
1
=
= 1,
so that
.75
or
•
Using Table 4.1, we now calculate
and
.e
Plugging (4.5), (4.6), and the sum of the w(ni)'s (see Table 4.1) into (4.1),
we obtain
(4.7)
z =
171.3969 -
21
(393.597)
=
25.4016
31.5220
=
- .806
•
)993.6335
Since our computed Z-value (4.7) is not
< ... 1.645,
'We
donot have sufficient
evidence to reject the null hypothesis (i.e., the hypothesis that there is no
gonial crossing-over and that the data are binomially distributed).
might note that the sign of Z (4.7)
However, we
is negative (as we would anticipate it to be
if gonial crossing-over were present), and that the significance level corresponding to Z
= - .806
is
.21.
Thus the results of the eA'1>eriment are not con-
elusive and seem to suggest that additional investigation is needed in order to
establish something definite about gonial crossing-over.
30
In order to get a rough idea of the sensitivity of our test for detecting
gonial crossing-over, we used the artifieial device of arbitrarily adding 1 to
the larger of x.
2
for i
= 8,
and x!
2
the data became
for each value of
x
= 1,
ted using this artificial data.
Xl
= 6,
n
i (each family) in Table 4.1 (e.g.,
= 7),
and then Z (4.1) was re-compu-
The resulting (artificial) Z turned out to be
-2.80, which is highly significant (significance level
= .0026).
Thus this might
seem to suggest that the test is rather sensitive for detecting certain departures
from the null hypothesis.
In the example discussed in this section, there are both a priori and experimental reasons for believing that the two classes (pr + and + stw in this case)
occur with equal likelihood (50 0/0
for each).
Oftentimes, however, it may be
desired to test the null hypothesis of independence within each set of n.
2
vations when it is known or suspected that the two classes do not
equal likelihood.
obser-
occur with
In such situations, our test is of course not applicable; this
problem provides a possible area for further investigation.
32
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2
The X test of goodness of fit.
COCHRAN, W. G. (1952).
Ann. Math. Statist.
23, 315-345.
DIXON,
vI.
J., and MASSEY, F. J., JR. (1957).
Analysis, Second Edition. New York:
FELLER, WILLIAM (1950).
Applications.
New York:
FRANEL, J. (1894).
GOULD, H.
w.
(1959).
Introduction to Statistical
McGraw-Hill Book Co., Inc.
An Introduction to Probability Theory and Its
John Wiley and Sons., Inc.
A note in l'Interm~diaire des Math~maticiens 1,
Combinatorial Identities, a Standardized Set of Tables
Listing 500 Binomial Coefficient SUIJml8.tions (mimeographed).
Virginia:
45-47.
West Virginia University.
Morgantown, west
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NSA 14, Contract
DA l8-110-sc-294, Assignment R4.013.1).
"
GRUNEBERG,
HANS, and HALDANE, J .B.S. (1937).
to records of Mendelian segregation in mice.
OWEN, D. B.
(1962).
Tests of goodness of fit applied
Biometrika 29, 144-153.
Handbook of Statistical Tables.
Reading, Massachusetts:
Addison-Wesley Publishing Co., Inc.
vlHITTINGHILL, MAURICE (1950).
Genetics 35, 38-43.
Consequences of crossing over in o3g onia1 cells.