CHISQUARED
TESTS FOR ANALYSIS
OF C A T E G O R I C A L
J.N.K. Rao
Carleton U n i v e r s i t y
and
and
SUMMARY
SURVEYS
A.J. Scott*
U n i v e r s i t y of A u c k l a n d
Our investigation on the asymptotic d i s t r i b u t i o n
of
X 2 suggests that a simple correction factor
to
X 2 , which requires only the knowledge of
variance estimates (or deffs) for individual
cells in the g o o d n e s s - o f - f i t problem, m i g h t be
satisfactory for many designs.
Empirical results
on the p e r f o r m a n c e of corrected statistic
~2
are given in Section 3.
Results in Section 4,
however, indicate that no such simple correction
for
X~
is possible.
Some empirical results in
Section± 5 indicate that
X 2 is likelv to be less
affected by the survey des~gnT than
X~ .
The effect of stratification and clustering
on the asymptotic distributions of X 2 , the standard g o o d n e s s - o f - f i t chisquared statistic, and X~ ,
the chisquared statistic for testing independence
in a two-way table, is investigated.
It is shown
that both
X 2 and
X~
are a s y m p t o t i c a l l y distributed as w e i g h t e d sums of independent
X 2 random
1
variables.
The weights are then related to the
familiar design effects (deffs) used by survey
samplers. For the special case of stratified
simple random sampling under p r o p o r t i o n a l allocation,
X 2 (or X2) is shown to provide a conservation test w i t h o u ~ any correction.
A simple
correction to
X 2 , which requires only the knowledge of variance estimates (or deffs) for
individual cells in the g o o d n e s s - o f - f i t problem,
is p r o p o s e d and empirical results on the performance of corrected
X 2 provided.
Empirical work
on
X~
indicated that the d i s t o r t i o n of nominal
significance levels is much smaller with
X2
I
2
than
X
.
1.
DATA FROM COMPLEX
Cohen (1976), A l t h a m (1976), and Brier (1978),
among others have p r o p o s e d simple models for
clustering and derived appropriate
X 2 tests for
g o o d n e s s - o f - f i t and independence in the case of
single-stage cluster sampling (with equal cluster
sizes) and two-stage cluster sampling (with equal
subsample sizes).
They hgve shown that a simple
correction to
X2
(or X~) leads to asymptotically correct test statistic, under their models.
Extensions of these m o d e l - b a s e d results to unequal
subsample sizes, three-stage sampling, stratified
two-stage sampling and testing finite p o p u l a t i o n
cell proportions, along with a study on the effect
of model deviations on the corrected
X 2 of
A l t h a m and Brier are given in the Appendix.
2
2.
EFFECT OF SURVEY DESIGN ON X
INTRODUCTION
Methods for analysis of categorical data have
been developed e x t e n s i v e l y under the assumption
2
of m u l t i n o m i a l sampling; in p a r t i c u l a r we have
X
tests for problems involving g o o d n e s s - o f - f i t and
tests of independence and h o m o g e n e i t y in two-way
contingency tables.
Recent extensions of these
methods to m u l t i d i m e n s i o n a l contingency tables
using log linear models (e.g. Fienberg (1977))
have attracted considerable attention due to their
close similarity to analysis of variance in providing systematically tests of various hypotheses.
2.1
X 2 Statistic
Consider the g o o d n e s s - o f - f i t p r o b l e m with
k
cells and a s s o c i a t e d p r o b a b i l i t i e s
Pl'''" 'p
(Pi > 0, 7p. = i).
Let
n ..... ,n~
denote t~e
observed ce~ 1 frequencles
.
~ a sample,
K
in
s , of
n
ultimate units drawn according to a specified
sampling design,
p (s) . The conventional
X2
statistic for testing
H0 : Pi = P0i (i=l,...,k)
is then given by
k
2
X = 7 (n i
nP0i )2/(np0i) .
(2.1)
1
Subject matter research workers have long
been using these classical methods to analyze
sample survey data, but most of the commonly used
survey designs employ stratification or cluster
sampling or both and hence do not satisfy the
assumption of m u l t i n o m i a l sampling.
Operational
and cost considerations often dictate the use of
a complex clustered design and, therefore, it
becomes important to assess the impact of survey
design on the classical
X 2 tests.
The statistic (2.1) is appropriate when E(n.) =
np.
w h i c h , for instance, is satisfied for ~elfl
w e l g h t i n g designs or under models for cluster
sampling i n v e s t i g a t e d in Section 4.
For other
cases we e n c o u n t e r difficulties w i t h noncentral
distributions, so it is natural to consider a
more general statistic
in this paper we propose to investigate the
effect of stratification and clustering on the
asymptotic distributions of
X 2 , the goodnessof-fit
X 2 statistic, and
X 2 , the
X 2 statistic
I
for testing independence in a two-wa~ contingency
table.
In Section 2 we show that
X~
is distributed a s y m p t o t i c a l l y as a w e i g h t e d sum of independent
X~. random variables, and then relate the
weights to the familiar "design effects (deffs)"
used by survey samplers.
For the special case of
stratified simple random sampling (srs) under
p r o p o r t i o n a l allocation, we show that
X2
provides a conservative test w i t h o u t any correction.
Similar results for
X~
are obtained in
Section 4.
k
X 2 = n 7 (Pi - P0i) 2/p 0 i '
1
(2.2)
where
~.
is an unbiased (or a s y m p t o t i c a l l y
unbiased~ estimate of p. . If n~. = n i , (2.2)
reduces to the usual statistic (2.1~.
2.2
Asymptotic
Distribution
We now look at
X 2 . To this end,
is a s y m p t o t i c a l l y
0 and covariance
P =
58
the asymptotic d i s t { i b u t i o n of
we suppose that
n2(p - p)
(k-l)-variate normal with mean
matrix
V
say, where
(Pl ..... Pk_l ) '.
If
i~'p~ for any
2
X2/A- ~ Xk_l
!
=
(£i ..... £k-i )
asymptotically
can be w r i t t e n
Corollary
(at least
as
~ w t ( s ) y t , an e s t i m a t o r of
tss
the p o p u l a t i o n m e a n for some
y
and w e i g h t s
wt(s) , then a s u i t a b l e c e n t r a l t l i m i t t h e o r e m for
means ensures that
n½(~ - p )
is a s y m p t o t i c a l l y
normal.
~
~
if and only if
IP0i(l-
V = AP 0
^
= n(~ - p0 ) ,~-i(~ _ P0 )
= n(~ - p0 )
Poi)/n
and
Cov
(i.e. Var
or
A
(~i) =
(Pi' Pj)
.... IP0iP0j/n)"
(1979)
A 1 = ... = lk_ 1 = I .
These general results can also be framed in
more familiar s a m p l i n g terms.
In the case of
k = 2 categories,
D reduces to the o r d i n a r y
design e f f e c t (deff) , n Var (pl)/[P01 (l-p^.) ]
For general
k , D can be thought of as ~ e
natural m u l t i v a r i a t e e x t e n s i o n of the d e s i g n
effect.
In particular,
(2.3)
2
w h i c h will be d i s t r i b u t e d a s y m p t o t i c a l l y as
Xk i
under
H0..
A good i l l u s t r a t i o n of this a p p r o a c h
is given in K o c h et al (1975), N a t h a n (1975) and
S h u s t e r and D o w n i n g (1976).
Note that
X 2 can
be e x p r e s s e d in the form
X
V = IP 0
for some c o n s t a n t
Cohen (1976), A l t h a m (1976) and B r i e r
have all p r o p o s e d models w i t h the p r o p e r t y
If a c o n s i s t e n t e s t i m a t o r of V , say V ,
is available, the a n a l y s i s is fairly s t r a i g h T forward.
The n a t u r a l test is b a s e d on the corresponding Wald statistic
4
2.
11 - sup [c' v c/c'P c]
C
k-i
= sup [Vex
c
(~ - p0 ) ,
k-i
( 7. ciPi)/Vsrs(
1
F. ciPi^ )],
1
where
V
denotes the v a r i a n c e under
srs.
srs
Thus
I_
r e p r e s e n t s the largest p o s s i b l e d e s i g n
e f f e c t itaken over all i n d i v i d u a l cell p r o p o r t i o n s
and over all p o s s i b l e linear c o m b i n a t i o n s of the
cell p r o p o r t i o n s .
In the same way
lk_ 1 can be
where
P_ = diag. (p^) - p p'
is the c o v a r i a n c e
u
~u
~0~0
m a t r i x ~or m u l t i n o m i a l s a m p l i n g under
H_ , so
that
X 2 is e s s e n t i a l l y a special case oOf the
W a l d statistic.
The exact a n a l o g u e is the modk
2 ^
ified
X 2 statistic
n ? ( ~ i - P 0 i ) /Pi w h i c h
1
2
is a s y m p t o t i c a l l y e q u i v a l e n t to
X
under
H0
and under local alternatives.
shown to be the s m a l l e s t p o s s i b l e design e f f e c t
taken over all p o s s i b l e linear c o m b i n a t i o n s of the
cell p r o p o r t i o n s .
The other
l.'s
also r e p r e s e n t
1
design effects for special linear c o m b i n a t i o n s of
the ~. 's . Thus the
I. 's may be termed
1
1
" g e n e r a l i s e d design effects"•
Routine c a l c u l a t i o n of V
is a very desirable practice, b u t u n f o r t u n a t e l y this is still
the e x c e p t i o n rather than the rule.
W h e n no
estimator
V
is available, p r a c t i t i o n e r s very
often use the s t a n d a r d
X 2 s t a t i s t i c (2.1) or
(2.2) even w h e n they realize that m u l t i n o m i a l
a s s u m p t i o n s are i n a p p r o p r i a t e 2
The c o r r e c t
a s u m p t o t i c d i s t r i b u t i o n of
X
follows d i r e c t l y
from the a s y m p t o t i c n o r m a l i t y of ~
and s t a n d a r d
results on q u a d r a t i c forms (Johnson and Kotz
(1970, p. 150))•
C o r o l l a r y 1 e n a b l e s us to c o n s t r u c t a cons e r v a t i v e test, if we can find an upper b o u n d for
the d e s i g n effects of linear c o m b i n a t i o n s of cell
proportions.
C o r o l l a r y 2 says that to have
2
X 2 ~ IXk_l
requires not only that all the
i n d i v i d u a l cells have the same design e f f e c t
but that the deff for each of the c o v a r i a n c e
terms must also be equal to
A . This
c o n d i t i o n is rather s t r i n g e n t in practice.
T h e o r e m i.
Under the h y p o t h e s i s
H0 : ~ = ~0 '
k-i
2
2
X = E liZ.l where
Z 1 ..... Zk_ 1 are a s y m p t o t 1
ically i n d e p e n d e n t
N(0,1)
r a n d o m v a r i a b l e s and
11 ~ 12 ~ ... ~ Ik_ 1 are the e i g e n v a l u e s of
-i
D = P
V .
2.3
I
Special Cases
We now c o n s i d e r two special cases of the
general r e s u l t of S e c t i o n 2.2.
(i)
Stratified
srs
(Proportional
Allocation)
t0
In general, then,
X 2 is d i s t r i b u t e d
a s y m p t o t i c a l l y as a w e i g h t e d sum of i n d e p e n d e n t
X~
r a n d o m variables.
The f o l l o w i n g results are
i m m e d i a t e c o n s e q u e n c e s of T h e o r e m i.
Corollary
n(~ically
i.
ortion of units in s t r a t u m
h and the p r o p o r t i o n
of units from s t r a t u m
h b e l o n g i n g to c a t e g o r y i.
Then
pi = hE % P i h
and m h = nW h
for p r o p o r t i o n a l
k-i
k-i
E Z~i where
E Z 21. =
1
1
_ ~0 ) is d i s t r i b u t e d a s y m p t o t -
as
2
Xk_l
under
Consider
L strata and a s t r a t i f i e d sample
(s I ..... s L)
where
sh
is a r a n d o m sample of
size
m h drawn with r e p l a c e m e n t from s t r a t u m
h .
Let W h and
Pih
r e s p e c t i v e l y denote the prop-
X2/11 _<
p0 )' - i ( ~
(~)
s =
allocation.
H0 .
With this design we have
L
!
v:~-
z w h(~h- ~)(~h- ~)
1
If the largest e i g e n v a l u e can be s p e c i f i e d
(or a r e a s o n a b l e upper b o u n d can be set) we can
obtain, throug~ this result, a c o n s e r v a t i v e test
by t r e a t i n g
X~/A 1 as Xk_l .
!
where
Ph =
(Plh ..... P k - l , h )
0<[c'Vc/c'Pc]
Therefore
= 1 - Z Wh[C' (ph-p) ]2/ (c '~~
h
59
"
< 1
or
11 < 1
2
as a
I = 7~I./(k-l)
l
2
Obviously
X 2 has the same expected value as Xk_l
but its asymptotic variance is
k-i
V(X 2) = 2(k-l) + 2 7. (I. - ~ ) 2 / ~ 2
1
1
and
0 < X 2 < 7, Z 2 ~ ¥ 2
--
--
Thus the ordinary
(ii)
i
X
2
'~k-I
test is always conservative.
.Xk_l random variable,
which is larger than
1.'s
are equal.
Two Stage Sampling
where
V(X2_I
)~
g
= 2(k-l)
unless all
l
Suppose we have
R p r i m a r y sampling units
(psu's) with
M t secondary units in the t th
p.s.u.
(t = I,...,R; ~ML = M^) . Consider the
following two-stage design whUch is commonly
used:
r psu's selected with p r o b a b i l i t y
p r o p o r t i o n a l to size (pps) of M t with replacement and subsamples each of size m drawn srs
with replacement i n d e p e n d e n t l y from each selected
psu
(n = mr) . In this case
E(n.) = np.
where
Pi = Pi = 7tMtPit/M
0_
and
iPit
The importan_t p o i n t to note about the modification is that
I only depends on the cell
variances
v..
(or e q u i v a l e n t l y the cell design
effects
dl,ll...,dk ) since
k
= tr(p-iv) / (k-l) = 7, vii/[Pi (k-l) ]
1
k
-i
= (k-l)
? (l-Pi)d'l "
1
i~ the
p r o p o r t i o n in the
t th psu b e l o n g i n g to
category i. We also have
V=P
Some knowledge about design effects for individual
cells in a g o o d n e s s - o f - f i t p r o b l e m is often available (Kish et al (1976) for example), whereas
information about effects for the covariance terms
is much less common.
Note that
I is not in
general the same as ~ , the average cell design
effect, except when
Pi = i/k or d. = d
(i =
1 ..... k).
1
+ (m-l)Y~tW t(pt-p) (pt-p) '=P + (m-l)A 1 (say),
where
Let
W t = Mt/M 0
Pl >-'" ">- Qk-i
p-IAI
' then
and
Pt =
(Plt ..... Pk-l,t ) '"
denote the eigenvalues of
Ii = l + (m-l)Pi
and
k-i
2
2
X = Y [i + (m-l) Q i ] Z . .
1
l
Also
(c'A_c)/(c'Pc)~
~ .
.
. < 1 . so .that
and we get 1
Empirical Results.
Table i, taken from Ewings
(1979), shows the significance levels of the
ordinary
X 2 test and of the m o d i f i e d test,
~2,
for a nominal level of 0.05 for various items
taken from the 1971 General H o u s e h o l d Survey
(GHS) of the U.K.
This is a stratified threestage sample of a p p r o x i m a t e l y 13,000 households.
Here
b denotes the average cluster size and d.f.
is an a b b r e v i a t i o n for degrees of freedom.
The
m o d i f i e d test gives very good results in all cases
of Table I.
In contrast, if we naively use the
ordinary c h i - s q u a r e d test w i t h o u t m o d i f i c a t i o n the
true significance is as high as 0.41 which is
o b v i o u s l y unacceptable.
(2.4)
Pl < 1
k-i 2
k-i
2
2
~. Z.l -< [i + (m-l)Qk_l ] Y. Z.1 -< X
1
1
k-I
2
< [ i + (m-i)Pl ]_ k~l- Z.2 < m 7. Z.
-1 -1
1
1
since
p. > 0 . Thus
X2/m
gives a conservative
test w h a % e ~ e r the
0. 's , but we can get a b e t t e r
1
conservative test if a value for
p.
can be
,
I
specified.
We call the
0. s "generalised
1
measures of homogeneity" analogous to the measure
of h o m o g e n e i t y based on the intraclass correlation
O
(Kish et al, 1976 and Kalton, 1977).
The measure
p is "portable" in the sense that
it is relatively insensitive to cluster sizes
unlike the deff.
3.
M O D I F I C A T I O N TO
X
Table i: Significance Levels
(SL) for Ordinary X
~2
and M o d i f i e d Test,
X
Variable
2
1
b
~
Var(1.)
SL(X 2)
SL(X 2)
1
If the
I. 's were known we could get
l
accurate a p p r o x i m a t i o n for the p e r c e n t a g e 2 P o i n t s
of the true asymptotic distribution of X
,
using the methods in Solomon and Stephens (1977),
for example.
However, knowledge of the
I. 's
1
is essentially e q u i v a l e n t to knowing the
full covariance matrix
V = (v0 .) and if we have
this we can always construct ~ 3 a s y m p t o t i c a l l y
correct Wald statistic.
In practice we w o u l d
like ~ simple a p p r o x i m a t i o n to the d i s t r i b u t i o n
of X
that requires only very limited information about V . One very simple approach is
to treat the m o d i f i e d statistic
k-i
~2 = X 2 / ~ = Y (I./~)Z 2,
1
d.f.
2
1
6O
Age of Bldg
2
33.1
3.42
1.65
0.41
0.05
Home
(own/rent)
3
34.4
2.54
1.82
0.37
0.06
No. of
bedrooms
6
34.7
i. 29
0.20
0.15
0.06
Bedroom
Standard
4
34.5
1.01
0.18
0.06
0.055
No. of rooms
9
34.6
1.19
0.25
0.14
0.06
No.of cars
3
34.6
1.48
0.85
0.16
0.06
Household
gross weekly
income
3
26.6
1.40
0.41
0.14
0.055
4.
2
XI
E F F E C T OF S U R V E Y D E S I G N ON
where
~ = H(~) , w h i c h is a s y m p t o t i c a l l y
X2
m
~ ~under
H0
or on a m o d i f i e d v e r s i o n
(r-l) (c-l)
of the W a l d s t a t i s t i c in w h i c h
V (or H)
is
estimated assuming
H 0 is true. ~ W i t h m u l t i n o m i a l
sampling
H V H'
reduces to P ~ ~
when
= ~r ~ ~c ~so that the o r d i n a~r
r y X I2c is equiva-
C o n s i d e r the p r o b l e m of t e s t i n g for i n d e p e n d ence in a t w o - w a y table.
Suppose the
k = rc
c a t e g o r i e s are set out in a table w i t h
r rows
and
c
columns.
Let
P~ =
(PlI' PI2 ..... Prc )'
be the v e c t o r of e s t i m a t e d cell p r o p o r t i o n s and
assume as b e f o r e that
~n(~ - p)
is a s y m p t o t i c ally n o r m a l w i t h mean v e c t o r and c o v a r i a n c e
matrix
V . (Note that
V will be s i n g u l a r here
since it is c o n v e n i e n t to m o d i f y the d e f i n i t i o n
of p
to i n c l u d e all
rc
cells in this section.)
Let Pr = (Pl+ ..... Pr+ )' denote the v e c t o r of
m
~
marginal
lent to the c o r r e s p o n d i n g
and
Pc =
"
(P+I ..... P+c ) '
in the null h y p o t h e s i s of i n d e p e n d e n c e
rows and columns, i.e.
: Pij = Pi+P+j
(p-l~r O
.
~ C
the
is the d i r e c t p r o d u c t of
statistic
A
Pr
and
Pc
n
h
(p)
and
S
-r
~c
~
rc-i
where
~_
is the l a r g e s t p o s s i b l e deff a m o n g all
1
linear c o m b i n a t i o n s of the cell p r o p o r t i o n s and
is the s m a l l e s t
(More p r e c i s e l y
it
rc-i
"
'
follows from results in A n d e r s o n and Das Gupta
A
(1953)
can be w r i t t e n
in
that
I i ~ 6 i ~ li+r+c_2.)
A
(mr
p)
(PI+ ..... P ( r - l ) + )'
for
P
~c
and
p
~c
- p~r P~'
r
with
X~/6
< ~ Z~
so that we can get a c o n s e r v a t i v e
1 -1
test w h e n e v e r we can specify
61 (or a r e a s o n a b l e
upper b o u n d for
61).
A
The m o s t i m p o r t a n t a p p l i c a t i o n of this result
is to s t r a t i f i e d srs w i t h p r o p o r t i o n a l allocation.
Since we m u s t have
0 < 6. < 1 for all
i ,
A
equivalent
to
where
~ =
Pr
--
h..
13
is a s y m p t o t i c a l l y
h*lj = Pi j - Pi+P+j
--
1
1
X(r-l) (c-l)
--
so that the o r d i n a r y
c h i - s q u a r e d test is conservative, just as in the
g o o d n e s s - o f - f i t case.
The gains from s t r a t i f i c a tion are often rather small and the usual test
should w o r k very well in these cases.
similar d e f i n i t i o n
.
It can be shown that
Clearly
~
~
P~r = diag.
ZI,Z 2 .... 'Z(r-l) (c-l)
P^ i j 's ' we have
--
and
of
,
11 ->- 61 ->-
where
h(~)
d e n o t e s the (r-l) ( c - l ) - d i m e n s i o n a l
vector with components
hij = Pij - P i + P+j
~
are the e i g e n v a l u e s
p--l)mc(H~ V~ H')~
for t e s t i n g
which, after some m a n i p u l a t i o n ,
the form
=
--
"
r c (Pij - Pi+ P+j ) 2
2
XI = n y y.
1 1
Pi+ P+j
X I
--
,
~
The usual c h i - s q u a r e d
i n d e p e n d e n c e is
1
The
Z.'s
are now linear c o m b i n a t i o n s of the
h* '
w h i c h are t h e m s e l v e s linear c o m b i n a t i o n s
l]
of the s
's
and
~
is the deff for
Z
~ij
'
i
i "
Since
Z i is a p a r t i c u l a r linear c o m b i n a t i o n of
or
p
i
are a s y m p t o t i c a l l y i n d e p e n d e n t N(0,1) r a n d o m
v a r i a b l e s under the null h y p o t h e s i s of independence.
between
•
i.e.,
i
"'" > ~(r-1)(c-l)
(i=l ..... r; j=l ..... c)
H 0 : P=Pr@P
I
I
the v e c t o r of column p r o p o r t i o n s , w i t h
Pi+ =
c
r
Z P
and
=
7.
We are i n t e r e s t e d
j=l ij
P+J
i=l Pij "
H0
statistic.
As in the case of
X 2 , the true a s y m p t o t i c
d i s t r i b u t i o n of
X2T follows d i r e c t l y from standard results on q u a d r a t i c forms:
(r-l) (c-l)
Theorem 2
X2 7
6 Z.~ w h e r e
~ > 62 >
~
row p r o p o r t i o n s ,
modified Wald
~
We could follow the a p p r o a c h of S e c t i o n 2.2
to o b t a i n an a p p r o x i m a t e c h i - s q u a r e d s t a t i s t i c by
dividing
X~ by ~.
U n f o r t u n a t e l y this requires
i n f o r m a t i o n on the d e s i g n effects of the h..'s
(or h$.'s) and such i n f o r m a t i o n is rarely 13
- Pi+P+j + P i + P + j
u n d e r the c o n d i t i o n s above, so that
h has the
same a s y m p t o t i c d i s t r i b u t i o n as h*= H~ (~ - p r ( g P c )
say, w h e r e
H
is the (r-l)(c-l) x rc m a t r l x
±
1
m
where
=
mr
r
J~r =
S
J
~c
r
(I
r~ _i) I0).~ and
~rl
®
a v a i l a b l e at p r e s e n t e x c e p t in the 2 × 2 case
(Kish and Hess (1959)).
We could p e r h a p s divide
by
I , in the hope that
I is close to
6 . As
before
I can be c a l c u l a t e d from the value of the
d e s i g n effects for the cells (i.e. from the
d i a g o n a l e l e m e n t s of V) but i n f o r m a t i o n on
d e s i g n effects for i n d i v i d u a l cells is not likely
to be a v a i l a b l e either.
(Fellegi (1978) suggests
d i v i d i n g by the averag__e cell deff
d w h i c h will
u s u a l l y be close to
I.)
Perhaps the best that
we can r e a s o n a b l y hope for in m o s t cases is to
have some i n f o r m a t i o n about design effects for the
m a r g i n a l row and column totals, (the sort of
~ ~c
is an
r × 1
v e c t o r of ones.
This means that
h
is a s y m p t o t ically normal w i t h m e a n
H(p - p_ ~ pc ) and
covariance matrix
HVH'/n ~
As~~efore, if we
have a c o n s i s t e n t estimator,
V , of V a v a i l a b l e
a n a t u r a l test for H 0 w o u l d be b a s e d ~ o n the
Wald statistic
2
= nh
.
m
(p)(H. ~. ~,)
. . 1. h(p)
61
i n f o r m a t i o n given in Kish, Groves and Krotki
(1976), for example), and it w o u l d be d e s i r a b l e
to have an a p p r o x i m a t i o n b a s e d on the m a r g i n a l
design effects.
A simple special case w h e r e
this is p o s s i b l e is w h e n
~I = ~2 = "'" = ~rc-l= ~
Table
E M P I R I C A L RESULTS F O R
2
XI
under
H0 .
E x a m p l e i.
The first set of v a r i a b l e s comes from
the B r i t i s h E l e c t i o n Study (BES), w h i c h is a
s t r a t i f i e d t h r e e - s t a g e sample, like
GHS,
(voters w i t h i n wards w i t h o u t constituencies) of
about 2500 B r i t i s h voters.
The sample is
a p p r o x i m a t e l y self-weighting.
We are g r a t e f u l
to the SSRC Survey A r c h i v e at the U n i v e r s i t y of
E s s e x for m a k i n g the data available.
We give
results for c r o s s - c l a s s i f i c a t i o n s of the four
v a r i a b l e s set out in Table 2, along w i t h the
m a r g i n a l d e s i g n effects.
Deffs are r e a s o n a b l y
small for all v a r i a b l e s in this study.
Table
2:
D e s c r i p t i o n of V a r i a b l e s
Variable
No. of
Categories
2
1.06
B2:
"Election campaign
gave p e o p l e facts"
2
1.29
"USA looks at W o r l d
P o l i t i c s as we do"
3
Social Grade
B5: L e n g t h of residence
~
~
4
B1 x B2
2 x 2
1.14
1.02
B1 x B3
2 x 3
1.15
1.02
18.68
B1 x B4
2 x 3
1.30
1.12
146.75
B1 x B5
2 x 4
1.25
1.02
2.52
B2 x B3
2 x 3
1.13
0.97
4.94
B2 x B4
2 x 3
1.21
0.84
2.24
B2 x B5
2 x 4
1.26
1.00
0.58
B3 x B4
3 x 3
1.24
1.07
19.74
B3 x B5
3 x 4
1.19
1.03
17.89
B4 x B5
3 x 4
1.19
0.97
20.59
2.39
Description
of V a r i a b l e s
Variable
for GHS
No. of
Categories
Deff
2
3.28
G2: Age of d w e l l i n g
2
4.28
G3: Age of h e a d
3
1.26
G4: Type of A c c o m m o d a t i o n
3
2.83
GI: Rented,
owned
V a l u e s of
classifications
~ and
~
for the 6 crossare given in Table 5.
Table
for C r o s s - c l a s s i f i c a t i o n s
Deff
Sex
B4:
(BES)
r x c
Table 4.
for BES
BI:
B3:
Cross-classifications
E x a m p l e 2.
The second set of v a r i a b l e s comes from
the General H o u s e h o l d Survey (GHS) of 1971.
As
can be seen from Table 4, the deffs are r a t h e r
h i g h e r for the v a r i a b l e s c o n s i d e r e d here.
!
random variable
for
The o u t s t a n d i n g feature of the results (and
of all the results for other v a r i a b l e s in the study)
is the small value of
~ ; the o r d i n a r y c h i - s q u a r e d
needs no m o d i f i c a t i o n in any of these examples.
M o d i f i e d s t a t i s t i c s b a s e d on
l or the average
of the m a r g i n a l deffs w i l l be c o n s e r v a t i v e in
every case (and in e v e r y case in the larger study),
w i t h the one b a s e d on ~ p e r f o r m i n g s l i g h t l y
better.
The u n m o d i f i e d c h i - s q u a r e d statistic,
however, is c l e a r l y the b e s t choice in all cases.
An e x t e n s i v e e m p i r i c a l i n v e s t i g a t i o n b a s e d
on the results of S e c t i o n 4 has b e e n c a r r i e d out
by Holt, Scott and Ewings (1979) u s i n g data from
two l a r g e - s c a l e U.K. surveys.
Some results of
that study are s u m m a r i z e d here for illustration.
In all the e~amples the v a r i a n c e of the 6 i s is
s~all and
X~/~
can be r e g a r d e d as a
X(r_l) (c_l)
Deffs
Variables
and hence
6. = ~ for
i = 1 ..... (r-l) (c-l).
1
All m a r g i n a l design effects are also equal to
in this case and we can find the a p p r o p r i a t e
m u l t i p l i e r from either margin.
C o h e n (1976),
A l t h a m ( 1 9 7 6 ) a n d B r i e r (1979) have all p r o p o s e d
models w i t h this property.
However, the
a s s u m p t i o n of a common d e s i g n e f f e c t t h r o u g h o u t
the table seems less r e a l i s t i c here than in the
g o o d n e s s - o f - f i t case, e s p e c i a l l y if the row and
column v a r i a b l e s are of quite d i f f e r e n t types.
For example, average d e s i g n effects for s o c i o e c o n o m i c c l a s s i f i c a t i o n s r e p o r t e d in Kish, Groves
and Krotki (1976) range from 4 to 8 w h i l e average
d e s i g n effects for d e m o g r a p h i c v a r i a b l e s range
from 1.0 to 1.6.
Furthermore, the limited a m o u n t
of e m p i r i c a l e v i d e n c e a v a i l a b l e suggests that
will tend to be less than the average design
effect of either m a r g i n in general (Kish and Hess
(1959), Kish and Frankel (1974), N a t h a n (1975)).
5.
3.
5.
Deffs
Variables
r × c
.
.
.
.
.
2
l
.
.
(GHS)
P
22.87
G1 x G2
2 x 2
3.18
1.99
G1 x G3
2 x 3
1.61
1.13
20.17
G1 x G4
2 x 3
2.50
1.94
592.36
G2 x G3
2 x 3
1.75
0.97
156.39
G2 x G4
2 x 3
2.36
1.97
651.33
G3 x G4
3 x 3
1.51
0.96
50.83
1.32
3
1.59
4
1.55
In Table 3 we give values of
~ and
6
for the c r o s s - c l a s s i f i c a t i o n s , along w i t h the
value of the W a l d statistic to give some feeling
for the relative degree of dependence.
62
Again
~ is always much smaller than
~ , which
is slightly smaller in turn than the average of
the two marginal deffs.
Thus modifications based
on
~ or the average marginal deffs will be too
conservative for the GHS data too, although
is too large for some of the cross-classifications
for the unmodified chi-squared test to be satisfactory.
Kalton, G. (1977),
"Practical Methods for
Estimating Sampling Errors", Paper Presented
at the Meetings of the Int. Statist. Inst.,
New Delhi.
Kish, L., Groves, R.M., and Krotki, K.P. (1976),
Sampling Errors for Fertility Survey,
Occasional Paper No. 17.
London: World
Fertility Survey.
These empirical results give extra support
to the hypothesis that the ordinary chi-squared
statistic for independence is very much less
affected in a complex survey than the corresponding statistics for goodness-of-fit, although both
these surveys are of very similar structure
(approximately self-weighting).
The results
also indicate that modifications based on the
average deff or the average marginal deff may
be too severe.
However, the values of
~ in the
GHS results mean that blind application of the
naive test still has dangers.
We need models
which explain the difference between
~ and
and which leads to a smaller divisor than
~ .
A simple model for two-stage sampling will be
explored elsewhere.
Kish, L. and Frankel, M.R. (1974),
"Inference from
Complex Samples", J. R. Statist. Soc. B, 36,
1-37.
Kish, L. and Hess, I. (1959),
"On Variances of
Ratios and their Differences in Multi-stage
samples", J. Amer. Statist. Assoc., 54,
416-46.
Koch, G.C., Freeman, D.H. Jr. and Freeman, J.L.
(1975),
"Strategies in the Multivariate
Analysis of Data from Complex Surveys,"
Int. Stat. Rev., 43, 59-78.
Nathan, G. (1975),
"Tests for Independence in
Contingency Tables from Stratified Samples,"
Sankhya C, 37, 77-87.
REFERENCES
Shuster, J.J. and Downing, D.J. (1976),
"Two-way
Contingency Tables for Complex Sampling
Schemes,"
Biometrika 63, 271-6.
Altham, P.A.E. (1976), "Discrete Variable Analysis
for Individuals Grouped Into Families",
Biometrika, 63, 263-9.
Solomon, H., and Stephens, M.A. (1977),
"Distribution of a Sum of Weighted Chi-square Variables,
J. Amer. Statist. A s s o c , 72, 881-5.
Anderson, T.W. and Das Gupta, S. (1963). "Some
Inequalities on Characteristic Roots of
Matrices", Biometrika, 50, 522-4.
FOOTNOTE
Brier, S.S. (1978),
Categorical Data Models for
Complex Data Structures, Unpublished Ph.D.
Dissertation, School of Statistics,
University of Minnesota.
The names of the authors are listed in alphabetical order.
Most of this work was carried out at
the University of Southampton under Grant No.
HR4973/I from the U.K. Social Science Research
Counc±l.
The authors are grateful for the
stimulus and help given by D. Holt, G. Nathan,
T.M.F. Smith and G. Kalton.
Cohen, J.E. (1976), "The Distribution of the Chisquared Statistic under Cluster Sampling
from Contingency Tables", J. Amer. Statist.
Assoc., 71, 665-70.
Ewings, P.D. (1979),
The Effect of Complex
Sample Designs on the Chi-square d Goodness
of Fit Statistic, Unpublished M.Sc.
DiSsertation, University of Southampton.
APPENDIX
X 2 UNDER MODELS FOR CLUSTER SAMPLING
A.i.
Fellegi, I.P. (1978)
"Approximate Tests of
Independence and Goodness of Fit based on
Stratified and Multi-stage Samples", Survey
Methodology, 4, No. 2, 29-56.
Two-stage Sampling
(Altham' s Model)
The population consists of R psu's with M t
secondaries in the t th psu.
A two stage sample
s is denoted b y
(s 1 . . . . . S r ) w h e r e
st
is a
r
subsample of size mt( ~ m t = n and m t ~ M t) and
Fienberg, S.E. (1977).
The Analysis of Crossclassified Categorical Data, New York: MIT
Press.
r
is the number of sampled psu's.
1 if
Holt, D., Scott, A.J. and Ewings, P.D. (1979),
"The Behaviour of Chi-squared Tests with
Complex Survey Data", Unpublished Manuscript.
Zt~i_ =
)th
population
element
Let
of
0 otherwise;
l=l,...,Mt;
i=l,
,k.
t=l,...,R;
Then
Johnson, N.L., and Kotz, S. (1970),
Continuous
Univariate Distributions - 2, Boston:
Houghton Mifflin Co.
ni =
63
i th
in category i
r
m
r
~
~t
~ = ~ mti
t=l ~=i Zt"i
t=l
' say.
psu
is
In the m o d e l approach, we r e g a r d the
Ztl: 's
as r a n d o m v a r i a b l e s w i t h a s s u m e d d i s t r i b u t i o n a ~
p r o p e r t i e s and e x p e c t a t i o n s etc. are t a k e n w i t h
r e s p e c t to the a s s u m e d model.
S u p p o s e (Altham,
1976)
E(Zt~i)
= Pi
and
C o v ( g t ~ i, Zt~,j)
= b..1]
and the a s y m p t o t i c d i s t r i b u t i o n of the m o d i f i e d
statistic
X~ = X 2 / ( I + ( m 0 - 1 ) Q) is e x a c t l y
X2
k-1
u n d e r H O.
The p a r a m e t e r
p is like a g e n e r a l ~ z e d
measure-of homogeneity.
This is an e x a m p l e of the
s i t u a t i o n in C o r o l l a r y 2 of S e c t i o n 2.2 and, as we
n o t e d there, the m o d e l is f a i r l y r e s t r i c t i v e since
it i m p l i e s the same d e f f for all i n d i v i d u a l cells
and for all l i n e a r c o m b i n a t i o n s of the cell totals.
This m o d e l was i n t r o d u c e d for s i n g l e - s t a g e
s a m p l i n g by C o h e n (1976) for M t = M = 2 and A l t h a m
(1976) for g e n e r a l
M , and they s u g g e s t s e v e r a l
w a y s in w h i c h it m i g h t arise n a t u r a l l y .
(A.I)
for
~ ~ ~' . Note that these v a l u e s do n o t
d e p e n d on the c l u s t e r label, so we are a s s u m i n g a
form of w e a k e x c h a n g e a b i l i t y b e t w e e n clusters.
This m o d e l will n o t be s u i t a b l e if the c l u s t e r s
are o b v i o u s l y d i f f e r e n t (as in the case of d i f f e r ent s t r a t a for example).
If we also a s s u m e that
v a l u e s in d i f f e r e n t c l u s t e r s are u n c o r r e l a t e d then
it is s t r a i g h t f o r w a r d to show that, c o n d i t i o n a l on
the r e a l i z e d sample sizes
(m I ..... mr ) ' the cov a r i a n c e m a t r i x of
n = (n I ..... nk_l~' can be
B r i e r (1978) c o n s i d e r e d a m o d e l for t w o - s t a g e
s a m p l i n g (for the case
m =m) with multinomial
s a m p l i n g w i t h i n e a c h s a m p l e d cluster.
Thus
is m u l t i n o m i a l w i t h
~m t = (mtl ' " " " 'mt,k-I )'
-%
probabilities
p
= (p _ ,
,p
_ ) ' and the
,
~t
tl "" " t k - i
Pt s are a s s u m e d to be s a m p l e d i n d e p e n d e n t l y f r o m
Dirichlet distribution with density function
e x p r e s s e d in the form
n V = n P + (7.m.
~-n)B
,
w h e r e P = d i a g (p) - p p'
as b e f o r e and B = (b..),
~
~ ~
~
i]
or as
V = P + (m 0 - I)B ,
~
(A.2)
r
where
m 0 = 7 m~/n
1
N o t i c e the close s i m i l a r i t y to the e x p r e s s i o n
for pps s a m p l i n g given in (2.4).
The d i f f e r e n c e
is that (2.4) a p p l i e s o n l y to a s p e c i f i c d e s i g n
but r e q u i r e s no a s s u m p t i o n s a b o u t the p o p u l a t i o n
elements, w h i l e (A.2) is v a l i d for any t w o - s t a g e
d e s i g n b u t r e q u i r e s the w e a k e x c h a n g e a b i l i t y
a s s u m p t i o n s of the model.
Of course, the c h o i c e
of d e s i g n and m o d e l are u s u a l l y i n t i m a t e l y
r e l a t e d and we w o u l d n o r m a l l y feel more secure
a b o u t r e s u l t s b a s e d on an e x c h a n g e a b l e m o d e l if
the sample was c h o s e n w i t h the s e l f - w e i g h t i n g
d e s i g n of S e c t i o n 2.3.
k
~Pi -I
f(~t I~'~) = [ F ( ~ ) / F ( ~ p i )] ~i mti
' ~>0, mi>0,
.%
k
7.p.=l.
1 1
g
If we make the s t r o n g e r
dependence between different
is a p p r o x i m a t e l y
Theorem
1,
V : [(m+~)/(l+~)]P
,
given
~
r
and,
P-
from
is n o n - n e g a t i v e
Let
B* = P -
B
noting
that
Pi=Pii+j~i
since
k-i
2
2
7 P i k C i + 7 7. P. 0 (c.-c.)
> 0
1
i<j
13
1
]
-
-
Pij > 0 .
It follows
f r o m the l e m m a that p. < 1
for
1
k-i 2
7, Z. . H e n c e we
i=l,...,k-I
so that
X2/ m 0 - -<
1
1
get a c o n s e r v a t i v e test if we t r e a t
X2/m0
as a
2
random variable.
Xk- 1
A.2.
Special
Case:
In the special
Qi=p
(i=l, .... k-l)
Constant
case w h e n
k Mti
Mt
= ~ (
)/(m )
1 mti
t
Mt =
Mti
.
If we r e g a r d
the
,
M t s as
~2
To use the m o d i f i e d s t a t i s t i c
X
we n e e d a
v a l u e for
p or a l t e r n a t i v e l y for
C =
[(m + ~ ) / ( i + ~ ) ] = 1 + (m -I)Q . If the cell
frequencies
m t in e a c ~ c l u s t e r are o b s e r v e d , it
is p o s s i b l e to c o n s t r u c t e s t i m a t o r s of
C (or Q)
w h i c h are u n b i a s e d .
Since
E ( mt)_ = m t P~ and
r
nV = Cov(n) = 7. Cov(mt) , an u n b i a s e d e s t i m a t o r
1
Deffs
B = QP
(k-1)with
independent random variables having compound multinomial distributions with
E ( M t) = M ~
and
C o v (M t) = [Mt(M + ~ ) / ( i + ~ ) ] P , then ~he m a r g i n a l
d i s t r i b u t i o n of Zm t is a g a i n c o m p o u n d m u l t i n o m i a l
with
E(m ) = m p
and
Cov(m~) = [ m ( m . + ~ ) / ( l + ~ ) ] P
~t,
~~.
~~
~ ~
~
a n d the m t s are i n d e p e n d e n t .
Weohave
E(9) = nD
as a b o v e and
V = [(Tm~/n+~)/(l+~)]P
=
[i + (m^-l) Q]P
where
m^ = ~. m t / n
and Q=I/(I+~)
u
u a tconstant design effect
as before.
A~g a i n we have
m o d e l and the m o d i f i e d s t a t i s t i c
~2 = [ (I+~) / (m0+~) ]X 2
~
2
under
H_
T h e s e r e s u l t s h o l d for any
Xk-i
0"
a e s l g n for p s u s e l e c t i o n p r o v i d e d the
m's
are
u n c o r r e l a t e d and the c o m p o u n d m u l t i n o m i a ~ m o d e l
is valid.
P i j ' we can write
k-I
c'B*c =
~ ~ ~
distribution
.
k
Then,
~t
is n o w a
k
definite.
and let P i j = E ( Z t l i Z t l , j
(Mtk ..... M t , k _ I)
f(mtlMt)
where
Proof.
Mt =
dimensional hypergeometric
density function
~
7
(i + (m0-l) pi ) Z 2. w h e r e
1
i
are the e i g e n v a l u e s of
p-IB .
B
(m-1) p]P
We can e x t e n d these r e s u l t s e a s i l y to the m o r e
g e n e r a l case of u n e q u a l s u b s a m p l e sizes and finite
M t , a s s u m i n g srs w i t h o u t r e p l a c e m e n t in e a c h
s a m p l e d cluster.
The d i s t r i b u t i o n of
m
for
a s s u m p t i o n of inclusters, then
n
for large
= [i +
where
p = I/(i+~).
Thus this m o d e l is an e x a m p l e
of the c o n s t a n t d e f f m o d e l and
~2 = [ ( l + ~ ) / ( m + V ) ] X 2
~
2
under
H0
Xk_ 1
X2
Pl ..... Pk-i
Lemma.
normal
k-I
The m a r g i n a l d i s t r i b u t i o n of
m
is c o m p o u n d
multinomial with
E ( m t) = mp
atd
Cov (m) =
[m(~+m)/(~+l) ]P . U s i n g these m o m e n t s ,
is
r e a d i l y seen that
E(n) = np
and
we have
k-i
so that X 2 ~ (l+ (m0-1) p) ~ Z 2.
1
l
64
of
V
is given by
V = [ l + ( m 0 - 1 ) p ] P + ( m 0 - 1 ) ( l - p ) Y W h ( P h - p) (ph-p)',
^
r [r
_
r
-- 2
,]
n VI = r~_--[7.1 (mt-m) (mt-m) ' - ~'I (mt-m)
P0P0
]
r
m = 7~ mt/r
1
where
vii/Pi(l-Pi)
and
= vij/piPj
_
r
m = 7 mt/r.
1
so that deffs
all
h .
Noting
(k-l)-i
1
or
~
and
I is the i d e n t i t y matrix•
Since the rank of
p-i (~i-~2) (pl-P2) , is one,
k-2
of its e i g e n v a l u e s
are zero
and the n o n z e r o e i g e n v a l u e is
k
given by its trace:
1~' (Pli - P2i ) 2/pi . C o n s e q u e n t l y
k
)
for
P -I V = [i + (m0-1)p]I + (m0-1)(l-p) W I W 2 P - I (~l-P2) (pl-P2)'
,
C2 = k-I ? v i i / P 0 i ( l - P 0 i
1
k
k
^
^
2
C 3 = 7. vii/(l - 7. P0i )
1
1
_Ph = p~
In the special case of L = 2, we can evaluate
the eigenvalues,
I. , of p-Iv
explicitly, where
= C , we can c o n s t r u c t
7 vii/P0i
1
unless
that
several u n b i a s e d e s t i m a^ t o r s of C.
Using only the
diagonal e l e m e n t s of V we could use
k
C1 =
are n o n c o n s t a n t
(A.3)
(k > 2)
the e i g e n v a l u e s
1 + (m0-1)kQ;
li
are given by
%1 = 1 + (m0-1) p +
~2 =. " "=lk-i =
(m0-1) (l-Q) 6
with
= WIW2
Y~ (Pli - P2i ) 2/pi " Since the model (A.6)
1
is a special case of (A.I) it follows from Section
A.I that
Ii -< m0
or
0 _< ~ _< 1
w h i c h are all u n b i a s e d for C under
H : D _ = p 0 i.
Approximately unbiased estimators
of
C0
are valid for all
~ , are o b t a i n e d from (A.3)
if P0i is r e p l a c e d by its e s t i m a t o r Pi = ni/nThe m o d i f i e d statistic,
X 2, d e r i v e d above
does not involve the finite p o p u l a t i o n sizes
M
t
and
R since we are d e a l i n g w i t h model
parameters
p..
If our i n t e r e s t is in the finite
1
population proportions
P. = 7. Y. Z. x i / N = N i / N ,
•
1
then zt can be shown, after some a ~ g e b r a we omit,
that
~2 = X2/A i s a s y m p t o t i c a l l y
Xk_l when
B = p P , where
r
n
n
A = ( I - ~ ) (i + (m0-1) p)-2 -Qn 7.mt(Mt-mt )+ N ( M 0 - m 0 )Q' (A.4)
1
r
R
m 0 = 7. m2/n
as before and M 0 = 7. M2/N . In the
1
1
special case of m t = m, M t = M
for all
t ,
(A. 4)
reduces to
A =
mr
(i - ~ ) ( i
If
p = I/(I+~)
reduces to
as in Brier's
r
- -~') +
A = C[(l
A. 3
mr
+ (m-l) p) - ~ ( M - m )
~r
(m.{~O)R
(I
p.
model,
m
- ~-)]
(A.5)
then
~ WhPhi
h=l
= pi' Wh > 0 '
Zt~'A j ) =
where
b
-I
2
+ b
2
~ (a + k-l)
(a Xk_l
XI)
a = 1 +
(m0-1)p,
Three-stage
ssu's
from
The sample has
r
psu's
from the t th sampled p s u and
ktl
m
'
t
elements
in the I th ssu of the t th p s u
Let
k t = { ktl
(7 7~ k
= n).
t I tl
be the n u m b e r of s a m p l e d elements
in the t th psu.
As in the case of t w o - s t a g e
sampling, we let
Ztlsi = 1 if the s th e l e m e n t
of the (t,l) th ssu is in c a t e g o r y
o t h e r w i s e and we assume that
E(Ztlsi) = P i '
7.Wh = 1
= b..,
z3
(m0-1)(l-p)~.
Sampling
i and
C°v(Ztlsi'Ztls'j)=bij
Cov(Zt~si,Zt~,s,j)
Z Whbhij
h= 1
b =
(A.7)
The c o e f f i c i e n t of v a r i a t i o n of the ~. 's is
given by
c = b(k-2)½/(b+(k-l)a)
w h i c h telnds to
(k-2) ½(m^-l)/(m0+k-2)
as
~ ÷ 1 and
p ÷ 0 .
As
p ÷ ~ or
~÷0
, (A.7) a p p r o a c h e s
X 2~_I •
Using (A. 7) and S a t t e r t h w a i t e ' s a p p r o x i m a t ~ o ~
(or the m e t h o d s of S o l o m o n and Stephens (1977))
we can compute the true s i g n i f i c a n c e level of
X2/C l for any d e s i r e d c o m b i n a t i o n (m0,P,k,~).
(7. ?
= N)
t I Mtl
"
L
Cov(Zt~i,A
C1
Suppose the p o p u l a t i o n consists of
R psu's,
M t s e c o n d - s t a g e units (ssu's) in the t th psu, and
Ktl e l e m e n t s in the I th ssu of the t th psu
(A.5)
Since the a s s u m p t i o n of c o n s t a n t deffs is
rather restrictive, it is of i n t e r e s t to study
the a s y m p t o t i c d i s t r i b u t i o n of
~2
under model
deviations•
A simple model deviation, w h i c h
leads to n o n c o n s t a n t deffs, is given by the
f o l l o w i n g "mixture" a s s u m p t i o n s :
L
E(Ztli ) =
1
2
A.4
.
D i s t r i b u t i o n of ~2 Under a D e v i a t i o n
C o n s t a n t Deffs Model
2 The a s y m p t o t i c d i s t r i b u t i o n s of
X2/CI
and
X / ~ (= ~2)
are the same under (A.6) so using
the above
~. we have
=dij
Ztlsi = 0
(s ~ s'),
(lfl ~'),
~ ~ ~'
Cov(Ztlsi,
Zt,l,s, j) = 0
(t ~ t')
.
(A.6)
Bh =
where
(bhij)
It is easily shown that the c o v a r i a n c e
n is given by
= P Ph' h = 1 ..... L
Ph = diag (~h) - p h P h
and
Ph
(Phl .....
n V = n P + (~.
t Y~ k t2 l - n ) B
~
~
p. ~ .)'.
The model (A.6) is a special case of
K--±
t~6 general model (A.I) of Altham. It follows that
where
B =
~
65
+
(b0 .) and D =
l]
~
m a t r i x of
(Y. 2 - Y 7. 2 ) D
t kt
t I ktl
~
(d. 0).
i]
We can show that
-~
and
P -D
are n o n - n e g a t i v e d e f i n i t e as in
S e c t i o n A.I, so that the eigenvalues, of
p-iv
are
all b o u n d e d above by k 0 = ~ k~/n . Thus t r e a t i n g
(k-2) of its e i g e n v a l u e s roots are zero and the nonk
2
zero e i g e n v a l u e is
7 (Pli - P2i ) /Pi"
Hence
1
X2/k0
as a ~
r a n d o m variablet p r o v i d e s a
conservative
s~ for H . In the special case
with
m = m, k . = k
t~is reduces to X2/mk.
Thus th~ m o d i f i ctA
a t i o n to X2
that is r e q u i r e d
gets w o r s e as we switch from t w o - s t a g e s a m p l i n g
to t h r e e - s t a g e sampling.
i=l ..... k - 2 ; = [i + ( m - l ) p ] ( l - 6 ) ,
k
where
6 = W I W 2 X (Pli-P2 i ) 2/p i and
1
0 < 6 < 1 . Therefore
I.
2
X
If B = Q1 P and
D = p ~ , V reduces to
C P w h e r e ~C = 1 +~QI (7'27~ k2?/n~A - i) +
V
~
~
t I
P2(~t k2t/n - ~t ~ k2tl/n) ' s° we have a n o t h e r
where
where
kti = ~ ktl i ,
A. 5
Stratified
r
_n_)2
2
~= (k t
P0i t 1
r
'
is an u n b i a s e d estimator.
Two-stage
Sampling
We have
R h psu's in s t r a t u m
h with
M
h
ssu's in the tth p s u (h=l ..... L).
A stratifie~ t
t w o - s t a g e sample consists of
r h psu's and
mh
ssu's in the t th sampled p s u of s t r a t u m h.
Let
Zhtli = 1 if the
(h,t,1) th e l e m e n t is in
category
i and assume the m o d e l of S e c t i o n 4.1
holds in each stratum.
Then, if n~
~ = (n~fl.,...,
n. k 1 )' is the v e c t o r of c a t e g o r y ~ o t a l s ~or
s~ga[um
h , we have
E(nh) = nh Ph ' Coy (nh) =
nhPh + n h ( ~ - l )
Bh where
nh = r h ~
"
L
L
p = Y W h Ph
and p = Y W h n h / n h w h e r e
~
1
~
1
Let
W 1 ..... W L
are the strata weights.
We have
E(p) = p~
L
Cov (~)
7. Wh2 (Ph + (mh-l) B h ) / n~h = V/n
~ = 1
~
"
and
In the case of p r o p o r t i o n a l a l l o c a t i o n with
/n = W
V
reduces to V = 7 W. (P~ +(nk-l)B~ ).
'
~
n
~
~n
Nhte tha~
P~-~B h
is n o n - n e g a t i v e d e f i n i t e as
in S e c t i o n Ahl, so that V < 7. W h m h P h < m*Y.WhP h
m*[P - 7. W h(ph-p) (ph-p)' ] < m*P
max
(mh)
11 <__ m ~
.
,
where
Thus the largest e i g e n v a l u e
and
X2/m *
provides
m* =
of P IV,
a conservative
test.
An i m p o r t a n t special case occurs w h e n
Bh = Q Ph"
Assuming
m h = m , we get
L
Z W h ( P h - p) (ph-p)'].
1
For the special case
L = 2 , we can o b t a i n
e i g e n v a l u e s explicitly.
We have
V = [I
+
(m-l)p][P-
p-Iv = [i + ( m - 1 ) p ] [ I - W i W 2 P - l ( p l - P 2 )
Since the rank of P l ( p l - P 2 ) --(DI-D2)'
~
~
~ =
(1-6)
+ 1-6/(k-l)
2
X1
k
(k-l)-i ~ vii/Pi . As
6 ÷ 1 , X 2/ ~
1
[ (k-l) / (k-2) ] Xk_2
2
, (k > 2).
The
If a value for
p can be obtained, an
a p p e a l i n g a l t e r n a t i v e test s t a t i s t i c is
X .2 =
X2/[l + (m-1) p] ~
2
+ (i 6*)
2
This
Xk_ 2
X1 •
p r o v i d e s a c o n s e r v a t i v e test and should p e r f o r m
b e t t e r than
X2/I .
(k-l)-i ~ ~ i i / P 0 i
1
with
2
Xk-2
1-6/(k-i
c o e f f i c i e n t of v a r i a t i o n of the
I. 's is given
by
c = (k-2) ½6/(k-l-6)
w h i c h ten~s to
i/(k-2) ½
as
6+1
.
E s t i m a t o r s of
C can be o b t a i n e d from
c l u s t e r totals in the same way as in S e c t i o n A.2.
For example,
k
r
n.
^
r t7~
_ _~l) 2
r
V.ll. = r-i
=l(kti
r
- ~
~
approaches
example of a model w i t h c o n s t a n t d e s i g n e f f e c t and
2
under
H0.
~2 = X2/C ~ Xk_l
C1 =
= 1 + (m-l) p,
I
i =k-l,
the
(pl-P2)'].
is one,
66