On Properties of Estimators of Testing Homogeneity in r x 2 Contingency Tables
of Small Sample Size
H. Yassaee, Tehran University of Technology, Iran, and
University of North Carolina at Chapel Hill
Abstract
The problem ~f testing homogeneity in r x 2 contingency tables can be converted to the problem of testing homogeneity of r independent binomial distributions.
To estimate the common probability p, of success, we use likelihood
function, statistics ~, x~, I(p; p), and x~, the first of which is to be maximized with respect to p and the rest to be minimized with respect to p.
In
this paper we study the bias and variance of estimators through means of nnalytical and numerical procedures based on approximate rules and computer-obtained
results.
We compare the value of bias and variance for each method and give a
table which presents their order of magnitude.
Key words:
contingency tables; maximum likelihood estimate; minimum modified
x~ estimate; minimum x~ estimate; minimum discrimination information estimate;
2
minimum logit X estimate; bias; mean square error.
page 2
1.
Introduction
Cochran (1952, p.325) has shown that the problem of tesdng homogeneity
in r X 2contingencytables with cell frequencies xij's,
.'
may be converted to the problem stated as follows:
i=l~
2, •.. ,r; j:I,2, •.. ,c
Given r independent binomial
distributions from which samples of sizes xl . ' x . " ..
2
,X
r.
are drawn, respective-
Iy, one is interested in testing the hypothesis that these populations are homogeneous; Le., p.=p, i=l, 2, ... ,r which is the probability of "success".
2
note x. =
1.
L.
J=
1
x .. by n. and x'i by x. throughout the paper.
1J
1
1
1
use the likelihood function, modified
I(p:;), and 10git
xL2 ,
\t 2 ,
cally equivalent.
To estimate p, we
Pearson Xp 2, information statistic
the first of which is to be maximized with respect to p
and the rest to be minimized with respect to p.
__
We de-
1
These estimators are asymptoti-
However, they differ for small sample sizes.
Yassaee (1975,
a, b, c, d) has studied properties of estimators and distribution of various
statistics in rxc and rxcxt cant i ngency tables of small sample sizes.
Some
other researchers have studied the comparison of estimators and statistics in
various models using different approaches.
For example, see Harvey (1977),
Brown and Muentz (1976), PraJhan and Sathe (1976), Chapman (1976), Hommel (1978).
In this paper we study analytically the bias and mean square error of estimators
under consideration, and the criteria, such as biasedness and mean square error
are taken into account for comparing estimators.
Numerical investigations are
restricted to 2x2 contingency tables.
2.
Methods of Estimation
In this section we give estimators of the parameter p, obtained by the
different procedures mentioned in section 1.
page 3
1\
The unique maximum likelihood estimate p for p is given by
/\
P
l
r
i=l
x.
r
1
= - -N- -
N
=L
(2.1)
n.
i=1
1
0'
2
2
The unique minimum modified AM (Neyman X ) estimator for p is given
l. r
p
"Xi
1\
where p. = l-q. =
1
1
n
=
M
'"
'i=l
W.p.
1
1
J\
W
n.
/\
w.1
1
=
A
W=
l
r
i=l
/\
(2.2)
W.
1
We use the normal approximation to the binomial distri-
i
bution to investigate properties of PM'
The minimum Pearson X~ estimate for p is obtained by solving the equation
rl n.(q.-p.)
/\ /\ } p 2.+ 2{rI "2
A 2}
p . n.p - rl n.p.
= 0
{ i=l
1 . 1
1
i=1 1 1
i=1 1 1
(2.3)
2
The only root which is between 0 and 1 and minimizes X is given by
p
r
r
Pp =
l
,,2
n.p
i=l 1 i
r
"2
[i=l
l n.p
1
+ {
l n.
. 1 1
1=
.][
1
r
1\ 2
l n.q.
]
i=l 1 1
}~
(q.1 -p.)
1
Should it happen that the coefficient of p
~-------'H~e~n~c~e'
(2.4)
2
is zero, then we take Pp
there is only one root between 0 and 1.
Let n. (q. -p.)
111
r
= y.,
Y =. l 1
1
1=
1\
Y· •
1
1
= 2'
J?s;ge. 4
Then one deduces the following inequalities
Ir
"
i=1
1\
y.p.
1 1
1\
I
if Y > 0
< Pp < 2
/'I
Y
1\
if y
r
=0
/I
1\
I y.p.
. III
1=- j \ .!.<p < 2
P
1\
if y < 0 .
Y
These inequalities are helpful to check the computed p
p
values.
The minimum
discrimination information estimate p for which information statistic
r
/I
I (p: P)
n.q
n.p
r
Jl,n -1- + L n.q Q,n 1
n. -x. , q
x.1
. 1 1
1=
1 1
= I n.p
i=l
1
= 1-p,
is a minimum, is given by
p*
a
= l+a
(2.5)
where
Q,n a
r
1
=N
1\
( Ln.) logit p.,
. III
1=
and
r
N
= L n.
'1
1=
1
A
Since I(p:p) is a convex function of p, it is a minimum at p*, see Kullback (1959) .
• We define
2
logit XL
r
=
j=l
1
2
1\
/I
Lw.
(L.-L.),
1
1
where
L.1. = logit p
A
1\
1\
w. = n.p.q.
1. 1 1
1
/\
::
log E.
q' L.1
1\
::
logit p,
x.(n.-x.)
=
III
n.
1
\
\
page 5
the minimumlogit
x2
estimator PL is given by
r
logit PL =
,~
A
L w.
logit p.
1
1
i=l
r (\
L w.
i=l
1
A
(2.6)
l+A
In form'P
L
is the same as p* given in (2.5) with~. and
1
r
L~' substituted for
i=l
1
n. and N, respectively.
1
3.
On the Bias and Variance of Estimates
1\
It is well known that p is an unbiased estimate of p for all values of p.
In order to compute different estimates of p simultaneously, we replaced 0 by
f in the analysis of 2x2 contingency tables.
For p
1
= 2'
1
A
E(p) = 2' exactly.
1
A
For fixed N, P is unbiased only at p = 2" in the range of p, 0 < P < 1.
db(p)
is not zero, in general
dp
a2(pAI p = l)
2 ==>
I
dh
[ 1 + (dp)
P =
Since
i]2
(3.1)
4N
For r=2 we can write PM given in (2.2) in the form
(3.2)
which is a function of two independent random variables Xl and x
rational function of independent random variables.
2
and is a
By the use of Laplace,
Mellin, or characteristic function, one can obtain the probability density
function and the moments of such a function.
But the algebra is somewhat
lengthy, and we will not consider the problem as such here.
We refer the
page 6
reader to Prasad (1970, pp 614-625).
We approximate the binomial distribution with probability of "success"
p by a normal distribution with mean and variance the same as that of the
binomial distribution.
Numerically, if
t is not added to xi when it is zero,
1
orx. is taken as n-- when x.=n., and the normality assumption is valid, one
I
2
I
I
should have
n.
A
n.
1
on the condition that W. = ~ is used for W. =
X
1
pq
1
p.q.
A
1
Let
W.
y.
"1
=
w.1
=
1
W
1
W.
, y.
1
= W1
where
n.
n.
/\
w.1
1
pq
1
=
p.q.
1
Then
1
/\/\
PM =
I y.p .
. I 1 1
1=
A
Weights y. 's produce an asymptotically efficient estimate of PM when the ni's
1
are sufficiently large.
1
independent random variables.
;.,
W.
Let Z. =
1
1
w:I
p.
r
Then
I
i=l
W.
1
-Zl
i
r
w.
i=l
Zi
I
"-
/\
Under the assumption of normality, y. and p. are
1
Therefore,
1
page 7
r {
W.
.L
;
Now
'\
I
Z'"i f
.y----_._-
i=l
[L
Wi
i=l
Z.
J2
1
Hence,
To obtain this expectation, we applied a theorem on expectation of a function
of random variables as stated in Welch (1938, pp 330-362).
To get more details
on what follows, one may write to the author for a long preliminary report.
/\2
One may estimate op
M
1·
1
r
~ [1+2.I
by
"'''-
n. Piqi] .
1=1
1
Application of theorem mentioned earlier gives
EJ.) =
w
1. {1-2
W
= 1. {1-2(
W
f
1 p.q. + O( Y. 1.)}
i=l ni 1 1
i=l n i
Y. 1.) pq
i=l n i
+
O(
f
1 )}
i=l n2i
One can see that this estimate has asymptotically a negative bias which is
approximately equal to first term
..... 2
0 p
neglected.
If one wants to obtain an
M
r
estimate with bias of order
1
OC L "2
),
then
1=1 n .
1
a p = -W
M
1
r
{ 1+4 i=l fi
L -
/'..'\ }
p. q.
i
should be estimated by
M
:l
,,\ 2
2
0 p
1
1
page 8
The exact variance of PM under normal distribution is given by
.Zl
But
Zz has an F-distributionwith (nt-I, n 2-1) degrees of freedom,
we get the expectation
0
2
Iz
(P
Consequently,
)with respect to the F-distribution.
M 1
Finally
o2
= --I
+ K
PM W
\'/here
One may refer to (2.4) and expand Pp in Taylor series about values Pi =p
to find approximate bias term for Pp or get the variance of pp'
length of the algebra we will not
A
We expand logit Pi
/'.
= Li
A
A
p.-p.
L.;L = L
+ _1_1_ +
pq
der~ve
Due to the
the bias and variance here,
in a Taylor series about the true value p, to get
A
I (P-q)(Pi- P)
I --. 2 2
P q
2
A
+
1 (Pi- P )
3
3 -3-Y- (p2 - pq
P q
-3
+ q2) + O(n
).
i
p~ge
W~
Consequently, according to (2.5)
9
have
l
E(logit p*) - L + 2N- (p-q) + 1..
3N
rI• 1n. 1 (q-p) n-3pq)
1
1,
because
/\
E(Pi-p)
3
=n
-2
i pq (q-p)
Neglecting the third term, we have
E(logit p*) ,;, L
+
;N (p-q) .
According to this approximation, as p takes small values, i.e., between 0 and .5
the bias value of logit p* is of negative sign.
For p > .50, the value of
1
bias is positive.
For p = 2' logit p* is unbiased.
If the n.1 's are suffi-
ciently large, the value of bias tends to zero.
.e
Following the derivations just presented, we conclude that the biases of
MOl and MLG are of different signs.
Thus one may see that an effective comparison
can not be achieved by using analytical methods.
In the next section we further
study this issue numerically.
4.
Computational Details of
2x2
Contigency Tables
Then all poss~ble tables
Let (n , n2 ) be a set of given raw totals.
l
can be enumerated whenever the set is specified. For each set (n , n 2) there
l
/\
are (n l + I) (n 2+ 1) tables to be generated. For each table, P, PM' Pp' p* and PL
are computed according to formulae (2.1), (2,2), (2,5), (2.6), and (2,7), respect!vely.
S~nce
some estimators are not admissible for
Xi~
following rule is useJ:
I
X.
1
=
{
l
2"
Xi
1
n.- 2
1
if x.1 = 0
if x.1 = 1,2,.oo,n i - I
if x. = n.
1
1
}
the
0 or Xi = n.,
1
page 10
To compare
estim~tors
under the
was applied to all estimator5.
s~e ~xperimental condit~ons,
To get the mean, variance, standard
this rule
deviat~on,
bias, ratio of absolute value of bias to standard deviation, and exact level
of estimators, the following values were used as true values of p.
p
i.e.,
=
.10 (.05), .50
.10, .15, ... , .50
The mean and variance of each estimator, for each p, was computed by using the
definition of mean and variance of a random variable and formula A given in
section·2.
We now study the bias of estimators according to the increasing size of
the sample for different true values of p.
for all possible values of p, p
We computed the bias of estimators
= .10(.05), .50.
easily obtained as the negative of bias at l-p.
The bias for p >.50 is
To make the presentation
of an overall conclusion for the values of biases self-explanatory, we preferred to give details on what we have observed, rather than drawing diagrams
p
If (nl,n ) = (5,5), the bias of
is an increasing function of the
Z
true p for p <.50 and it is zero at p =.375 and p =.50. We note that the bias
for them.
is due to the fact that 0 or "i cells are replaced by
in the estimation procedure, otherwise
B = b(p)
an
= bias.
p is
1
~
and ni -
I
~,
respectively,
For MLE,IBI <.034 where
<
The bias of PM is a decreasing function of p for p =.15 and
increasing function of p for .15
<
unbiased.
<
= P =.50.
It is zero at p =.50.
'
.
part of the b:i,as is due to su h
stltutlng
21 for x.1 =
o and
n
l
- "21 for
Again,
x.=
n ..
;J.
1
Referring to (2,2) and the discussion in section 3, PM would have been unbiased
if the populations were normal.
The bias of Pp is an increasing function of p
for p <.45 and it decreases to zero at p =.50.
Referring to formula (2.3), and
page 11
knowing that , Yi l < 1 1 we see that 181 of p should be smaller than the correp
Y
spond~ng IBI of PM' The maximum 181 of p is equal to ,021, The bias of p*
p
behaves like that of PM except that the bias is almost zero at ,40,
In other
words, p* is an unbiased estimate of p in a small neighborhood ofp =,50,
181
of p* is less than
IBI
of PM'
as Pp which is reasonable due to
Finally, the bias of PL behaves closely
the relationship between
X
2
and XL2 already
given.in Yassaee (1975).
For the case (n ,n 2) = (10,5), (10,10), (10,15), (IS,lS), (20,20), and
1
(20,25) we conclude that one cannot claim that PL is always less biased than
Pp' but other estimators can be arranged in terms of
P - p*-p
As n
~
and n
IBI
in ascending order
M
A
increase, the direction of the bias of P becomes the same as
l
2
those of PM and p*, which are always negative and increase to zero.
-.
It. is interesting to note that the bias of p* and. that of PL are of
different sign for most of the cases except for small values of p, as we have
already concluded approximately in sections (3.4) and (3.5) for bias of logit
p* and logit PL;
5.
Mean Square Error of Estimators (MSE)
We briefly report on the MSE of estimators and compare estimators in
this regard for various true values of P
For p ~,15, the MSEof PM is smaller than for those of others, For
<
>
.15 = P <.30, the MSE of p* is the smallest and for p =.30 the MSE of P L
is the samllest.
'Page 12
For true values of p, the ascending order of estimators
magn~tude
~n
terms of the
of their MSE is given as follows:
-L
Ascending order of MSE
<
P =.15
PM - P* - P - Pp - PL
/\
,r-
p* - PM - P - Pp - PL
.15 < P <.30
<
,.'\
<
PL - Pp - P - p* - PM
.30 = P =.50
for p >.50 the order is the same as that of 1 - P shown here.
We now present a table which summarizes the ascending order of estimators
for the MSE according as (n
'.
l
,1l
2)
= (5,10), (10,10), (15,10), (15,15).
page 13
Table of the Ascending
5,10
Ascending order ofMSE
<
P =.20
/' n PM - p* - p - 'p
PL
p*P
PL
Pp
PM
PL - Pp - P - p* - PM
.25 < P <=.50
<
P =.20
P =.25
.25 < P $.50
'.
e
(15,10)
(15,15)
of Estimators
P
P =.25
10,10
O~der
P
P
P
.20 < P
.30 < P
<
=.10
=.15
=.20
<
=.30
<.50
<
p =.10
< <
. ] 5 = P -.20
P =.20
P =.25
P =.30
< <
.35 = P =.50
A
A
/\
P - p* - p - Pp - PL
'\ M
P - Pp - P* - PL PM
PL - P - P - p* - PM
P
-
1'\
PM - p* - P - Pp - PL
p* - PM - P
" - Pp - PL
A
P - p* - p - pM - PL
I'P
P - P* - PM
Pp - PL - 1\
PL - P - P - p* - PM
P
P - Pp - PL
PM - p* - '"
1\
p* - P - PM - Pp PL
/'\
P - Pp p* - PL - PM
1\
PP - pL - p - p* PM
PP - PL - '"P - p* - PM
1\
P - Pp - P - p* - PM
L
-
-
-
page 14
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2
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.
,
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2
Cochran, W. G. (1952), "The X test of goodness-of-fit", Ann. Math.
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.