THE INFORMATION IN CONTINGENCY TABLES - AN APPLICATION OF INFORMATION-THEORETIC CONCEPTS TO THE ANALYSIS OF CONTINGENCY TABLES Fkz _SI_ tar by C. T. Ireland and S. Kullback -......... J;+ + ' : ........ .. ......... ... .... ....... ........ ......................... IT . ...... ....... .... ... - Technical Report No. 235 August 4, 1976 Prepared under Contract N00014-76-C-0475 (NR-042-267) Office of Naval Research Herbert Solomon, Project Director Reproduction in Whole or in Part is Permitted for any Purpose of the United States Government Approved for public release; distribution unlimited DDC DEPARTMENT OF STATISTICS -F STANFORD UNIVERSITY 1OCT 1 1976 STANFORD, CALIFORNIA n a D
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THE INFORMATION IN CONTINGENCY TABLES -
AN APPLICATION OF INFORMATION-THEORETIC CONCEPTS
TO THE ANALYSIS OF CONTINGENCY TABLES
Fkz _SI_ tar by
C. T. Ireland and S. Kullback
-.........
J;+ + ' : ........ .. .........
... .... ....... ........ .........................
IT . ...... ....... .... . . .
- Technical Report No. 235
August 4, 1976
Prepared under Contract N00014-76-C-0475(NR-042-267)
Office of Naval Research
Herbert Solomon, Project Director
Reproduction in Whole or in Part is Permitted forany Purpose of the United States Government
Approved for public release; distribution unlimited
DDC
DEPARTMENT OF STATISTICS -F
STANFORD UNIVERSITY 1OCT 1 1976
STANFORD, CALIFORNIA n a
D
The Information in Contingency Tables -
An Application of Information-Theoretic Concepts
to the Analysis of Contingency Tables
by
C. T. Ireland(1) and S. Kullback(2)
1. Introduction
The primary purpose of this paper is to present an exposition of the
methodology underlying the analysis of the information in contingency
tables. We shall stress the concepts, techniques, analyses and inferences
without entering into extensive technical statistical proofs or detailed
references to the bibliography at the end.
It is useful to note that we are concerned with an aspect of multi-
variate (multiple variates) analysis with particular application to
qualitative or categorical as well as quantitative variables. The basic
data we deal with are counts in multiway cross-classifications or multivay
contingency tables. Multiway contingency tables, or cross-classifications
of vectors of discrete random variables provide a useful approach to the
analysis of multivariate discrete data.
As we shall see, the analytic procedures serve to bring out various
interrelationships among the classificatory variables in a multivay
cross-classification or contingency table in many dimensions. Classical
problems in the historical development of the analysis of contingency
(1)C. T. Ireland is Professor of Statistics, George Washington University.
(2)S. Kullback is Visiting Professor of Statatics.
tablej concerned themselves with such questions as the independence or
conditional independence of the classificatory variables, or homogeneity
or conditional homogeneity of the claasificatory variables over time or
space, for example. Such classical problems turn out to be special cases
of the techniques we shall discuss. These techniques result in analyses
which are essentially regression type analyses. As such they enable us
to determine the relationship of one or more "dependent" qualitative or
categorical variables of interest on a set of "independent" classificatory
variables as well as the relative effects of changes in the "independent"
variables on the "dependent" variables. In particular such problems as
the determination of possible factors and measures of their effect and
interactions in the representation of the logits of one or more
dichotomous variables lend themselves to-the analysis we shall
examine.The methodology is based on the Principle of Minimum Discrimination
Information Estimation, associated statistics and Analyses of Information.
General computer programs are available to provide the data for the
inferences.
2. Contingency Tables
We assume that the reader has some familiarity with cross-
classificationsin the form of contingency tables. We use a
slightly modified conventional notation. For example, for a
four-way contingency table, that is, one with four classifications
or variables, each of several categories, not necessarily the
same in number, we represent the observed number of occurrences
in the (ijkl) cell of the contingency table by x(ijkl), where
the indices i,j,k,l, range over the respective categories of
tne variables. The corresponding probabilities are represented
ty p(ijkl). Summation over one or more indices, resulting in
various marginal distributions or marginals, is indicated by a
dot or dots, thus
Z x(ijkl) - x(.jkl), EZ x(ijkl) = x(i-k-), etc.,i ji
witn a similar notation for the probabilities.
2
We shall denote estimates under various hypotheses or
models by x*(ijkl) , where values of the subscript u will4Urange over the hypotheses or models.
AIn example of a 2x2 two-way contingency table is shown
in Taole 2.1.
Table 2.1
x(ij)j = 1j = 2
i = 1 x(ll) x(12) x(l.)
i = 2 x(21) x(22) x(2-)
x(1l) x(#2) x = n
The estimated two-way table under the hypothesis or model
of independence is shown in Table 2.2.
Table 2.2
x*(ij)
i = 1 x(l.)x(l/n xlx(2)/n l)
i = 2 x(2.)x(-l)/n x(2-)x(-2)/n x(2-)
x(l) x(.2) n
3
... .. . . . .. . . . .. > -
- - - - - - - - - - - - -•-
A co..irnon statistical ueasure of the assocJatLon or interactioa
1etween the variabls of a tw.o-w4ay 2x2 contingenc.y table i th2 cruis-
product ratio, or its logurith-a. The cross-product ratio is defined by
(2. )x (11) x (22)x(12)x(21)
though we shall be more concerned with its logarithm
(2.2) fn% x(ll)x(22)
x(12)x(21)
We shall use natural logarithms, that is, logarithms to the base e
rather than common logarithms to the base 10, because of the nature of the
underlying mathematical statistical theory. Note that with the estimate
for independence, or no association, the logarithm of the cross-product
ratio is zero., . x(l')x('l) x(2")x("2)
(2.3) Zn x (ll)x (22) _ 4n n n 1 = 0x*(12)x* (21) x(l-)x(2) x(2-)x(-l)
n n
The logarithm of the cross-product ratio is positive if the odds satisfy
the inequalities
> Ll(12) or LL) >x(21) x(22) x(12) x(22)
since then we get for the log-odds
en x(lZ)n(22) -x(ll) n >r x2) > ox(12)x(21) (2.I) x(,22)
4in x~L!) - n x(21)> , 0
x(12) :.-(22)
The logarLhAn of the cross-product r-.tio it nug-ttive if the o.dds satisfy
the : - ua I ties
-- or _ _-x(2 1) x(22) x(12) x(22)
since tza-2. .;e ';2t for the log--oddr
4
Lo i ,: . 7., t I aitr m-
1) l u n
For tae th cc way 2x2x2 contingency table in addition to the
clasic, types ot ui.dpendetnce, interaction or association, there
arises ani additio'nal unet impirtLant historically and practically.
This is known as no tnree-factor
or no second-ordlar interactiarn. 11o three-factor or nio _,;econd-ordar
interaction imp]iies that the logarithm. of the associatrion measured by the
cross-product raL1o for aay twio of the variables is, thia sa-me for all the
valuds of the third variable, that is, there is no second-order interaction
if
x(121)x(211) x(122)x(212)'
(2.4) _ _ -- , 1~~~~~~ xt)(11- (12 2) x(1) -
(1 ~(l~ 2 2) _ ~_x( 211)-.(222)
On'e is coac.2rrI*2d .- it'! thle po03&rL~le 1yo1' or :liodal of no
ca~-or~ rintera 2tion uhen Pnone of the other tyicas o- inctenenca are
f o, md. yvri~n thii,3 Cjle, the correspondinj esv1eCannot 1-) ex-
pr3'4 0C! e:2-.iiitly in Lolr2:s or olbservedl maririnalz. IUCct estimate
13 un!ct-,'im 1 to liav i:. sac t,''3-!.ay mar-Ln:i L3 a,; t* ose cz~
oiird ttoriti; :j'oc.:dur-_±s ex:isttoet!,i,2-h
U:d~ ~'~hyY~~vc~o~r :Dclof no sec'0:hI-order Pr±nio.For thn-
;-rmnitcal ;iree--,iay r ;:< : t contingency t;!ble Lher-1 nvr. of C-earS--~. r-1-e
r,- L:iL. orb aron - Ltue 1,), cross-product ratios like Xi -)wich r-.u:-t bie
ju K i tC ~ :ie p rocedlurous to 'ltrIn.:t!c~ I xtC ~~e
5
For four-way and higher order contingency tables the problem of
presentation of the data increases, as do the variety and number of rues-
tions about relationships of possible interest and varieties of interaction.
The basic ideas, concepts, notation and terminology te have mentioned for
the two- and three-way contingency tables extend to the more general
cases as tie consider the methodology. For some additional prefatory
remarks see Au et al (1971).
3. Discrimination Information
To make the discussion more specific and with no essential restric-
tion on the generality, we shall present it in terms of the analysis of
four-way contingency tables. Let us consider the collection of four-way
contingency tables RxSxTxU of dimension rxsxtxu . For convenience let
us denote the aggregate of all cell identifications by P. with individual
cells identified by W so that the generic variable is W - (i,j,k,t)
i - l,...,r, j - 1,...,s, k - l,...,t, £ - l,...,u . Suppose there are
two probability distributions or contingency tables (we shall use these
terms interchangeably) defined over the space Q , say p(u), ir(w),
E p(w) - 1, E r(w) - 1 . The discrimination information is defined by
(3.1) I(p:w) E p() 9.n PM
The basis for this definition, its properties, and relation to other
definition3 of information measures will not be considered in detail in
this exposition. For the particular types of application to which we
shall restrict this exposition the 7-distribution, .T(w) , in the definition
(3.1) according to the problem of interest may either be specified, or it
may be an estimated distribution. The p-distribution, p(w) , in t'
definition (3.1) ranges over or is a member of a family of distributions
of Interest.
Of the various properties of T(p:!w) we rantion in particular the
fact that 1(p:w) > 0 and - 0 if and only if p( -(
6
*:r1' obI IQ.ll: ysi ;ot crn ' , , y ua charac-
0? .9 :,t i ~iLJ. i (I L,; t i ;' : L ~ a i t -bl s je t Lo
L: C. L J e. 2 r i i nt s and t h en-, coj7 yari .1 at. tc1 L1 e; witi an obse!rved
tableDl to Itrrn :ethcr LhK ob-ser\vcd ttblc sat is_-ies a nail hypothesis
or mod"1. iri Led by, Cae reaitraiir.ts. In accordnca rwitih the principle of
riinun, Oi cririinatli I afornatioa en t .uta.;e tLnn hat '.itaaa of
the collection or faT.ily of p-distributions sac isfyOL1; the r2straints
-hich nilizst~ciciniain[irain I( 7)over all members
of ti:,,fniyo pertinent p-distributions. W dranote thie rnininum di.s-
crirnin:.!tiDn inforrmation estimate by p GO) so that
(4.1) TO :) p~(W) R~) Xnin l n I)Tr (W)
Unless otherwise stated, "he summation is over iw~.hich will ba omitted.
In a wide class of problems which can be characterized as '1 sroothiag"
or fitting an obsetvecl contingency table the restraints specify that the
estimated distribution or- contin-ency table have some -,?t of marginals
-.hich arc rhz! same au; those of an obsorved continge-ncy table. In such
cases rT L.) is taken to be eiliher the uniform diatcihution -.'(jkZ)
1/rstu or a distributioni already estitnated subject to restraints contained
in and irplied by the restraints under ey:arinatioai. T--- latt-er case
includes,- rhe Classical 1hypoth.eS.s Of iridope ndenra, conu'itional ladepaidance,homoc ,enco.ry, condlitloaal. hoimooneity aiid initeraction, all of which cart be
considored 1-; [Ilst3.aces Of 1-11ralized inecdn~ and wiLl he conaidared
in somta &deaLl in Ci i; paper. BY L.rol!ralhized icd t~neis ;-'sant the
f:rcttlat Ltha estDiatL~s maiy be expressed asa ncetor z actorq which arc.
Iun-ccioir; of apocropri.:re :narginals. See Ku et al (1971).
j. L~r.i~:. usrrlri.iinction Inftormaition Statistic
fo .-. t wlictther a-n cbsc-rve~d contingency tam3 ii. coosis1'tei'C Wuith
tre iralI an LS 3 i.~ o d~l 13 rpvir.3nai'ed by LbsinietUicr ai-
a ~ ~ ~ ~ f':eLvnL1t2> 'r m-aeOf Ch .av2 . on ,t;-nc:
o s i t. aad C a niorn:-r 7 7 re a'-n' is,
12 I '19a a-: i~onst2tiotic. Foc n-oi* [on-il .:?c3:n I
7
r t.'J o - 1. a rt ii Z.atilsI
I: d; LoK i j ~9 r.io s)ot jti utis sritneesaiy o
othier kinds; of appi ia oris of th, general theory.
6. 2.nrunDiscrimtir-!tji IULorrmatio.; Lhoorem
Uie HJWe'es'it atheorem tzKicn is the basis for the principle of
Lii mL:'Ji~, rim Lfdt- it). fl Aorrnto sati tion anld its applications. 14
>;'~11~)eQ~tit il Z.! :7, re- Latcd to the cotntei:t of tis discuss Lon on
-n,.ra ly;, a; of c(Jllinq :v=5 <cyables.
Let us con.;, ide; We .m;a.; A2 rntioned in S-u tion 3 and !:he i-
cr iAl wi.nnf Wiirmn.r i at rodue ed in (3.1) . Suppeo .: ijw, for U*.-vnaple,
crht Irir aret> o. :ariy independont stat is~ics of inzerest define-d
7.2) i(P:) Z pGO r.
t'i ly (if P-ti Ci".V.atio Wh.' *- sai"LsO reitraitL:s
-Y TI(,p< =
(6.3) T
where e 0 2 , 0 3 are specified values, and n (L:) is a fixed distri-
bution.
If -,(T ) s;Iti.;iics the restraints (6.3), then of course the
r,Linimum value of i(p:.) is zero and the mininizing distribution 13
p (G) (.) More geLarally, the ainimum discrimiination information
theorem states that the minimizing distribution is given by
( e).p (T 1 TI(W) + T 2 T2 (u) + T3 T3 (W))i "4)(6.4) p (T)3 )
where
(6.5) M(t 1,, 2 , 3 ) exp (T1 TI( ) + t2T 2 (w) + 13T3(w)),v(w)
is a normalizing fauLor so that E p*(4 1 , and th2 T's are para-
meters which technically are in essence undetermined Lagrange multipliers
whose value.- are defined in terms of 1, 02 , a3 by
*01 ;A-I9 M(TiI 2T3
(g exp ( 1lTI(,w) + - 2 r2(O) + T3T3( Tl(W),T( ))/M(Tl9r2, 3)
E T1 (w)p ( )
* Z
2 9r n M(-rt , I )
= (: Cxp (rTI () + T 2 T 2 (w) + T3T3(w))T2(W)r(w)J V ' ,3
(6.6) ,= . "2 (G ) p ( W )
*
03 na3 n 3)
(Z exp(TrT 1.(, + T 2 T2 (w) + T3T3(w))T3((J))/A('1l, 2,T3 )
= T. (M)p (w)
We can now state a number of consequences of the preceding.
AWe n fe fit; L tl 'r 1 ) is a ::ut- l ;In (a\!;oiinL i1I I-filV o I
dit;t i-i u i. 1) : . r, -, by :( ) and A; tii ha ,; , ,irab .1 1.b a :;t tiF;t ici
prop ,i ic,-; of 01k[.U ;, o Li an O.po;&' tia! i.I ly ,iih in : ,,." all the
commuu and 'las;ic di,t rib';tions. We nay ,Iso writ (6.4) as
(6. i) n pT, (u ,-) - __ - n2 1. , "2' T3) T T 1 j(a + T2 T2 (W0) + c 3 T 3 ('.
n I T + T G) + T (M + r. TT(.1: , + L T ( ) + 'Tr (U)) + -3T3(
ii 2 3 3
with L 2 n V( 23) The regression or Log-linear expres.sion in
(6.7) for inp p(n)/T(c )) ith T!(w) , T2 (b) , T3((,) as the explanatory
variables nd[ 1 , t,, , 3 as the regression coefficients plays an L-mpor-
tant role ill the ansiy. -is we shall consider.
We note next that the minimum value of the di.acrimination inform.ation
(6.2) is
01 .o) i(p : C. + = e +2 " M( T21 22 3 3 1' 2' 3
.iere the 's are defined in (6.3) and the - 's are determined to
cs't isfy (6.6) Us ing the value in (6./) it may be shown that if p(Uw) is
any ;ic:her of the fari ly of distributions satisfyin: (6.3), then
(6.9) I(p:7) = I(p :7) + l(p:p
T1he pytlagoreAn typo property (6.9) plays an important role in the analysisSr ira .rc..atiao tabl .sq
Am "exp!rirent" has been designed and observations made resulting
in a mui.ti--dive;!;ional contingency table with the desired classifications
aad rat.dori,.;. All the information the analyst hopes to obtain from the
.exp,."r~innt is contained in the contingency table. In the proce.-, of
anrily:is, the i! i; L to fit the observed table by a minimal or parsimonious
number of par,,i etCr,; t c eerifng ou sone or all of the marg inals , that- is,
10
U
to 1j!k Gl. t, 1!'u, 11.ch o ti . tota j i i o( r2 i is cor:.. o :;'I in a summary
CoIO1: i ',i ;i" ot i of :2 I., Is . Lindeed, the re I at, iou-,h i p bcetueen th-e
con ept 1 j 'II ief I i I C0 )I ;S S C' i -It L f k) an d i c nt I- t- kA n- i cunt i:ngency
tabi cs; :.rrd Lhc tlU1e hem ~inasplay is evidlemet. inl the historrical
ceve1op w-'t.; ill thl ex t eisive lit rat u rc on the aiimily's is of contingency
tabl 10;.rhu:, ,the s in the prccld i g disc us.sion .. ill be the mar-
g'inals (fA intert-Lt. Se.e KuI et al (1b)71)
7.1 . he'() :nt i ()- n:. 'fIre T(w) fuzictions fr the_ R*':SxTxU
tab he turn out. tim be a bas;ic_ set 0f Sirmple fuol-Ct iDns and their various
products . Tfhu,; , for exiplc, the T(w) tunct ion aessociated WiUf the
onie-way marginal p (2 . . ) is
(7.1) T <(ijkZ) 1. for i =2 , any j,k,2
0 otherwise
since
(7.2) Ipijk .) 2~ijZ
Similarly theT 1(w) function associated with the one-way miarginal p(. .3.)
for exnmple, is
(73T (ji I for k =3 , any i. jZ3
since
ihus for the r:ssxtxi Ltble the2re art?
Y T R
(5 ~ ~~~~~ La f 1 ual ~c~n uixenf i S ijk ~ 1 = ,.s-1
(7.5)t-1) ri ely i nd.,pcndc~nt funictin T ( iJk) , -y 1
(u-i) lii'r"I y I ird 'eedent funcetions 'Y( ij kzh), 6 ,...,-
since, to:r examrpie ,
r R
(t: I , jk ) r
cc--]
C hiv hj t r1eriv :c t.J , the tut .0 t KOrt:..,1i to U ( - r,
Qa s , t, , a natiur k'". c l-'-.nic c (i r( 'd have selected
S , 8 - 1, y ., 1 or any ot!,cr ce'vt of valll.s.
Thu T(Ue) filt 0ion assoc i at d ,, Lh the two-way narginal p(Z.
Say, i' T ) (i j.'-) , ( ) since from :h,(. definition of T (jjk) andSI
T2(1 ii.k) it m.v be !en that
K S(7.6) a ijk) T(ijkL) - I for i = I, j 2, any k,Z
= 0 ctnrlise
and
(7.7) p(ijk;.) T R(ijkk) T S~ik, p(12..)
R S R
For convenience we s!lal write TR(ijkZ) TS(ijkx) = TRS(ijk) , etc. Thus
the T(ws) function associated with any two-way marginal is a product of
two appropriate functions of the set (7.5).
Similarly the T(to) function associated with any three-way marginal
will be a product of ttiree of the appropriate functions of the set (7.5),
for example,
R ~jk) I T U(7.8) Z p(ijk,) T2(ijk),) T1 (ijkt) T3 (ijkZ.) = p(2.13)
For cvovnience we shrill write T R(ijk.) if (i k.) T (i .t:.) = " ,e. t '
2 i-ilarly the T( .,) iunction assoia ted with any four-way mar-ginal
,:i1l be a product of fouir of the appropriate functions oC te sot (7.5),
for ex4Apie,
RS T U,(7.9) E p(ijkL,) i' ( i.,j .) TI (iJkZ) TI(ijkZ) T2 p(2112)
For corivenienct!y shall. write T R (ijk).) T S(ljkZ) T T (ijkk) TU (jl"Y = 6(2112)
RST ,racy:(ijk;')
12
.'e lu t that they'd. ilre a total of
( = (r-) '- (s-I) + (ti) + (u-I)N2 - (r 1 ( -) + (r-1)(t-1.) +- (r--3)(u--1) +- (s,-1)(t-1)
(7.10) + (s-1)(u-) + (t-.)(u--):5 = (r-L)( -l(t-1) 4 (r-1) (s-1) (u-]) 4 (r-1) (t-1) (u-1)
+ (S-) (t-l) (u-)
respectively, of the simple linearly independent functions and their
products tvo, three, four at a time. It may he verified that
(7.11) rstu - I = N = N1 + N2 + N3 + N4
These values of the number of T(w) functions (or associated tau para-
meters) appear as appropriate degrees of freedom in the analysis of
information tables.
7.2. The Estimated p (u) Values. In the usual least squares
regression analysis procedure, one first computes the regression coeffi-
cients and then gets the values of the estimates. In the methodology we
use we reverse the procedure. Instead of trying to obtain the values of
the 7's from (6.6) (which is possible) we shall first obtain the values
of the t!stiunates p (t) by a straig'htfonnrd convergent iterative
procedure and then dorive the values of the ,'s from (6.7). We shall
not discuss the details of the iteration here, as they are in the computer
program and have been described elsewhere. The iteration may be described
as successively cycling through adjustments of the marginals of interest
starting with the ,;(o) distribution until a desired accuracy of agree-
ment between the set of observed marginals of interest and the computed
inarginals hani been attained. See Ku et al (1971.).
7.3. The T Values or Interaction Parameters. From the definitions
of the T(:j) functin; in ection 7.1 it is clear that they take on only
the values 0 or 1. for each value of w. From the nature of the T(w)
13
- - - -
lunc t ion tI. st o ! rc :' cI cr lo',- Iincar :-li ion'; ( .7) will htve
witn a ingl, value whii(ll can be dteii:id ien there i l bit
O', With one aditir unknown vallno and so on the I's alrcady
d, t er iT. Thu..se n,,i Iunnown I va lues c-n b( Litn dot', ri ned T Pis
p-1cess ef su,ocessiVe eviluation is carried on unt,1 all the values of
ace deer-ined. 'ivy :ire also available as output of a general coin-
titer prOLa-n.
. r Ipic heo resen Lar ion
A useful graphic representation of the log-linear regression (6.7)
is given in Figure 8.1 for a 2x:2x2x2 contingency table. This is the
analogue of the design r.atrix in normal regression theory. The blank
spaces in Figure 8.1 represent zero values. The (ijkZ)-columns are the
cell identifications in the same lexographic order as the cell entries
for the estimates in the computer output. Column I corresponds to L
which is essentially a normalizing factor. Each of the columns 2 to 16
........ ti.le.....g v4alues of the T(') functions, coluln~s 2
to 5 those for the one-way marginals, columns 6 to 11 those for the two-
•ay marginals, columns 12 to 15 those for the three-way marginals, and
column 16 that for the four-way marginal. For convenience the colut.ns
are also arranged in lexoqraphir order. The tau parameter associated
S-ita t'ie T(.) function is given at the head of the coluan. The full
~r ,is .at ion wiI all the columns of Fig'ure 5.1 gen-.races the o .sscved
values. Thui- the rows represent
__i____ _ __ _i ij,,ij.(8.l) ;.n p(ijk2,)
= Zi j = L + ti T(ijk; ) + ... + TllI'l(ijk1' 1Ki Jk , ijkZ j k
+ ... + T (ijkZ + ... + T T iikZ)
we!re ,r(ijkZ) in the 2x2x2x2 case is l/2x2x2x2 and the numerical
values of 1, and the tans depend on the observed values x(ijkZ) . The
,esign matrix corr,;punding to an estimate uses only those columns asso-
ciated with the marginal:, explicit and implied in the fitting process.
'Tis i; a r&t4 Lction of the fact. that hLgher order marginals Jmply certain
14
p -L
W 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
i k ij Tik it jk II k. ijk ijT ik. Jk. Tijk1
1111 1i I 1 1 1 11 1 1
1 II I 1 1 1 1 1
1121 1 1 1 1 1 1
1122 1 11 1
1211 1 1 1 1 1 1
12 2 1 1 1 1
1221 1 1 1
1222 1 1
2111 1 1 1 1 1 1
2112 1 1i 1
2121 1 1 1
2122 1 1
2211 1 1 1
2212 1 1
2221 1 1
2222 1
Figure 8.1. Graphic representation.
lower order marginals, for example, the tw:o-way marginal x(ij..) implies,
by sumnation over i and j , the one-way margiuals x(.j..) , x(i...) ,
and the total n = x( .... . Thus the estimate based on fitting the one-way
marginals will use only columns 1-5. The values of L and the taus for this
estimate will be different from those for x(ijkZ) and depend on the esti-
mate xl(ijkZ) . Thus if we denote the estimate based on fitting the one-way
marginals as x (ijkt) , the representation in Figure 8.1 implies
x (11)1 1 k 1
In - L + -rl + TJ
+ TI k
nn£n L+ 1 1 J T1
(8.2)
xi(2222)Zn - L.
nir
15
la" "D b -oD ma
111, -.l~k io k l t Lt i i o I -4u io o e-4d o l m ,i y
No II1 t i[ Id 1 v itII I a I I i
NII
N 6i4 4-I I
N I I + I /I (co ll icm. I ? t ci )
I + ~'1i i
.41 ~ m. .'cdI Jig d i .j.4 f41 (,it :1tt r r I. i*' W i fri I o IF-~a
-t. .it t~ ill .ibI I it i ior rel.AL IVo rc-'iI4C-I ch , 44 1 Jrai,
I t i i 11 (1 fmtil :;,ri, It4~il ' )[41 I i I v f det -v Iv i I iI III by 11 , 014. f (it :
41 , -, (I Ur'A - " l ., 1 tit .1 I I i 'i , %vd Ir t It Lii:tt t!i olcc t rriricc-e:,, ct~t-
i I -JI L'* c . 'fliit, .1 [4.11Iy!; I. -; tit tll orm4. it I olI 1.~4 bi.:,d till Lhe JId.L4iI It ii
k. it d11 1w G.) (w 1 !i ti ho4i111 I~' llui hat I ml n !( I -i -tI 1.41 i hif o 1 A 1:.1 i
1!1,~ 11i. u cur r ta: p mi Ili', I (Ia . :ot, It (i I of ivon 44f).44 ii i fl 1 d xbL (W~) is the
16
sycub esIah OVDt doe. notISGfA9 - r.ducin
dji.- LI 2. J i 10, 1. T Ut --j t LJi us:. to I. L i Zof 1-1.i::-O c ~ I 1. LA 0 y Y ; re (I LI
It. , th. L , I LIEC
(x ~ ,., - .: (-< :r. ) + f: ( ::-
(9.1).L ;n-.T) 2[(-< :niT) + 2i1(x" )
2L( Xa 21 % :&a) + 2 ( b
with a correspondiag dditive relaLion for the asaociated degraes of
freedom.
In terms of the representation in (6.4) or (5.7) or Figure 8.1 as
an exoonential family, for our discussion, the two extreme cases are the
uniform distribution for which all T's are zero, and the observed con-
tiagency table or discribution for which all N - rstu - 1 t's art
needed.
Nleasures of the form 21(x:x ) , that is, tha comparison of ana
observed contingency table rith an estimated contin;ency table, are called
measures of interaction or goodness-ot-fit. ALeasur3s of the form
2-(.::) , comaaring -i'.o estima .ad contir-nc7 i ' l - are called mea-
sues of -:act, tIat is th2 ei:act o -: e air-r a in -2 sac E bu
no: in the ,: H zaui.±: in jUz oor x a X noU e tshat
2T(:::1.:) L7ts a nIul u "0 t '1,3 r1 t t . ... >z 1 Z! : .. a:ameters in
the repr2sa~.-.n o.'- the 'Thsa~r;ad cornv:4-".vc- :! :z su- no:: the
.r_ ........ r o o w .- _ st-it -. - _ ,- C .) :ra :;ciro -'d t ' = c'" be_-of
e :'"s is t2 ::. o. J o- oa S . n. 5 -1cly 2- : a
tests a null nypotdesis that tae values of the set of T parameters
in tne representation of tile estimateA table x*(,) but not in
tae representation of the estimatcd table x*() are zero and thea
numoner of tnese taus is the nu.'ber of degrees of freedom.
We sum-marize taie additive relationships of the m.d.i.
statistics and the associated degrees of freedom in the Analysis
of Information Table 9.1.
17
Copy avabl* to Mel)C doda iroIPVMA~ NiUT 1oaible mipredutiofl
TABLE 9.1
ANALYSIS OF INFORMATION TABLE
CompoM'nt dLh to Information D.F.
Ha : Interaction 21(x:x*) Na -
lib : Effect 2 1(x :xa Na -N
Interaction 21(x:x ) Nb
Since measures of the form 21(x:x a ) may also be interpreted as measuresa
of the "variation une~xplained" by the estimate x a the additive rela-
tionship leads to the interpretation of the ratio
21(x:Xa - 21(x:xb) 21( :xa)
(9.2) a = - _
21(x:x ) 21(x:x )
as the percentage of the unexplained variation due to x accounted for* a
by the additional constraints defining xb "The ratio (9.2) is thus
similar to the squared correlation coefficients associated vith -.0-al
distributions.
We remark that the marginals explicit and implicit of the estimated
table x a() :hich form the set of restraints H used to generate xa(w)a a
are the sanie as the curresponding marginals of tht observed x() table
and all. lower order i,:p1ied mar:ia1..;. TI may be shorw, that 21(:x is
a~ppoxiat-y ,a quadraric in the differenrces betweea the remaining mar-
4inals of the x(w) table and the corresponding ones as calculated from
the x ( ) table.a
Similarly 21( :x*) is also approximately a quadratic in the
differences between those additional marginal restraints in H but not
in Ha and the corresponding marginal values as computed from the xa("G)aa
table.
As may be seen, because of the nature of the T(w) functions
described in Section 7.1 or indicated in Figure 8.1, the t's are deter-
mined from the log-linear regression Equations (6.7) (see (8.2) and (10.3))
18
'3 ',.;. . ; F . .; ; ". (; , A variLy o0 stiati-+
ic;l A n ii 1trAA ~ riy , CUL LL tin- ae!y
Lt i,., . ' .::- qu ' ia dZ.t ,(,Li L 't r 1 . jr:I or quadratic&
-j' c! E: C ii' Iciil t .I,; of lo-, iL~ Wi. the ob ; rv,2d or
C'3L ti-; d v'i ,_2, "i'.~' rOF-ri l_ u L[ ,iT i d.i;,cri:zLn:!,t.iot, i-tor:antiort
citV.ato'' d its pr-)' -,!'ITas thus prvi L!es a uni f y i4 rclatioas!ip sirn::e
"u:iC !3L1J>I rwry s, ;za us qtf.'tdr, ri2 ,uppro:zi .- ionj of. the nini.um
isCrii i ct- i0IrM'r 0,1 statistic. I7e remarh that the corresponding
appco:c.ata X2 ar ' ;It g:mterally aldicive.
fUe rrltioa thr .2Dro:irations ia terms of quadratic forms in the
marzinals ,r the C'S "a.; a possible bridge conn :cting the familiar pro-
cedures of classical regression analysis and the procedures proposed here
to assist in understanding and interpreting the analysis of informatioa
tabl. The covariance matrix of the T(w) functions or the taus can
be obtained for either the observed table or aRy of the estimated tables,
as well as the inverse matrices as part of the output of the general
computer program. See (10.4) to (10.9)
10 - ne 2%2 Tab I
It way be useful tc rc xaaine the 2x2 table Lrom the point of
view, of the preceding discussion. The algebraic details are simple in this
ca3e and exhibit the unification of the information theoretic development.
Suppose we have the observed 2x2 table in Figure 10.1
x(tU) x(12) x (l.
Figure 1U.l
19
If we obtai:i the m.d.L . e tiriate fittiug the: l () .-. ,y marginais, the
generalizt-d hirdependen. c hypoLhti,;is isi the c Lv;;icil indepeondence hy-
pothes i,; an1 the n.inimu:' discrimination inforation estill te is x (ij) =
x(i.)x(.j)/n . The representation of the log-linear regression (6.7) as
in Figure 8.1 for the full model is given in lI gure 10.2. The entries in
the colunnns -1 9 2 ' I3
i j L I, 2 ! 3
1 2 1 1
2 1 1 1
2 2 1
Figure 10.2
are, respectively, the values of the functions Tl(ij) , T2 (ij) , T3 (ij)
associated with the marginals 61 = x(l.) , 0 2 = x(.1) , 83 = x(11)
and the cc1u'.n hcadcd L corresponds to the normalizing factor (the
negative of the; logarithn of the moment-generating function as in (6.7)).
We recall the interpretation of Figure 10. 2as the log-linear
relations=x(I ) L + T + - 1 3
nTr
Z __ n L +
(I0.i)Z, 1 ) L + -
rUT
n 2L(22)
From (10.1) we find
1, - ;.I, (x(22)/n/4) ,
T I= Zn (x(12)/x(22)) ,
(10.2) 2 = ,n (x(21)/x(22)) ,
:- 9.n (x(1L).x(22)/x(12)x(21))
20
or
I Z11 x~ 2-- ilX(22)
(10.3) 2 x(21) Zn x(22)
, Zin x(tl) I in x(22) - 9n x(12) - ,n :.-(21)
If wC call T thte 1.,aLri: with colulins the Column;: of LiiC desigLt matrix
of Figure 10.2 t!,L' is,
(10.4 ) 1 1 0 0
0 01 0 0 0,
and define a diagonal matrix D with main diagonal the elements x(ij)
that is,x(I1) 0 0 0
(10(5) D 0 x(12) 0 00 0 x(21) 0
0 0 0 x(22)
then the estimate 4,t the covariance matrix of 01 x(l.) , 2 = x(.1)
x(ll) for the observed contingency table is Z A2.1 where
-. 1 2 .
2 . 1 11-12
atnd A11 is I x I ,-&:" is 3 x 3 , -1=A..is I x 3 .it is
found tha't
/" x~l)x(2.)x(] ~.I ) x(. ) :(Il) x(2.)nn n
A\
x(l)x(2.) x( 2)
(L fli 1111n2
21
/ -. ) -I ,_I_ 1
x(2 x(22)2-_ + _+-
-'- xQ.) : (2l) x(22.(21) x(22)1 1 1 1 1 1 1 1- ] i
v:e ,r:rrk that thL Intri x i(l 1.9) Li- the ccvariance iatrix of tsLj -E'
in (i0.3). Similuh results hold in general and for -stimated
taoles.Note that the value of the logarithm of the croas-product ratio, a
measure of association or interaction, appears in the coure of the analysis
as the value of T for the observed values x(ij) , and that 7 = 0 for
x (ij) , the estimatc onder tile hypothesis of independeuce, for which the
representation as in Figure 10.2 'ces not involve the last column since it
is obtained by fitting the one-way marginals.
The log-linear relations for the estimate x (ij) are
*n x(1) + +I nit 1 2
( . x (1.2) L +
___ 1
Un" 2) L
n7[
Witer, tt., nierical vaiues of L , I ,2 in( 10 .10) depend on x and
di Ifer ft on the valuos in (10 .1)
The minimum discrimination information statistic to test the null
hypothes is or model of indeipendence is 21(x:x ) with one degree of free-
dom. In this case the quadratic approximation is
(10.11) 21(X:Kx) (A() ) 2_ + +
2 \x(1.)x(1?) )
22
Remembering that x (ij) x(i.)x(,j)/n , the rig ' t-ha d si-le of (10.11)
may also be shown to be
2 2/ x(i .)x.(10.12) X ((ij) -x).
2
the classical X -test for independence: uith one degee of freedom. Another
test which has been pro!rsed for the null hypothe,;is of no association or
no interaction in the 2x2 table is
- i xJ 2 -Zn2 11 1- .j(10.13) (Zn x(ll) + n x(22) -x(21)) (I- x(1) +(21) x(22))
which may be shown to be a quadratic approximation for 21(x:x ) in terms
of T3 with the covariance matrix estimated using the observed values and
not the estimated values. We remark that if the observed values are used2
to estimate the covariance matrix then inst id of the classical X -test in
(10.12) there is derived the modified Neyman chi-square
(10.14) ×2= r (x(ij) - x(i.)x(.j)/n)-/x(ij)
11. An Analysis
In ordhr to coordinate and relate the variou.i delinitions, concepts,
parameters, ccnoutatiunai f _I ur&:s, L.to. discussei n the preceding sec-
ticns w tha- Onder u detai- .e _nalys Eu of -ipeclfic cctiiretz':
table.
Table 11.1 is a four-way continzeucy table A 1-,,053 men in a
training program , cross-cla:Lified on the \'ari_.Dles ho:e region
level )f education, racu: and program co; 1recion. ;e denote the
occurrencei in the four-way cross-classification or contingency Yable 11.1
by x(ijk2o) with the notation
Variable Inde x 1 2 3 4
Home Itegion i East North West South
Level of Education j Below V.S. H.S. Above II.S.
Race k White Non-white
Program 9. Failed Passed
23
r-I
H1
Kr,
,f 00
LC4
24
A 0
n i ) 0
v. ... I *if !1) 1 i* Ln 2, i' l ;I, i .. p
'7'] 1i[> it I v .Z1 it A )n tit. ii ii ';4nif i-
C it * tA C. -I. iO -. 1. L 0 i '~:'!(! of 1i'i.Ij InM
di~" WA it ixi~ not iA wl ''o i nIWw SUS 01t MUCCIt
'~~' i L~~ 0~tt 0~!. . f mIIr, Li,11
p*'i t ly w mi- u .: nsrQ 2Y nJi w2 the pia-c.ding
CSL i 1W nI z: : ii Nut W' do.rmiin. It- cff..wt of th:c dddi-
ii:~ .> r I- ' :,, 2i 'r ioni I iii piiramters. The
auillyVslO' o1 ,2 I I 2. :!:Li;In J "-r:I i Z 0 t1 o L he signifciie or
non-sit.':t ic igc,: tK ao ef M L .or interac ioen tiLl parameters.
Pi.. itt iu, viL~ NEtS WV MA0gina0S. SUMr we are interested in
the c.n h~I <. . lcn:in Lrdaining c n region ,level
Of Ciicitib, iod racc, .c lirzt: fit thV ii.!tar-its x(ijk.) , x( ...
o:,,Cd ti S< Lc K:01 a osyt P x (i x ( xi jt- .) . ) /n is that
un..cr t.h, W I f vpoLW.. I or n~udolit i. ric-oen e of s uccess and the
jointi voni Wo 2-,* c Jiof , IcvI ()I tduc it iOn, rIce-) or no inter-
aCt. ioO- bUcIdt i .. )i Iec- IOU, 7~t olt v:ib 1-. In otwr words w'- first
want to WIt. -0 r t-ie 2-4 col -. a of TAW 1. 11, i.wrC homogeneous
oi rA r i t111 under2 prial)i i a. pas,,io g - r f a iling.
flie~ ~ :~.cti d~ i atistc IC S
with [I Ia " n Z f urnf~. .. FJ- Cic hixicn;ctindependence or
I'. i~111t.,i. , I riulr. Sh'il l tok tot t .lintr ffec(-ts.
liiVA ai: 1.re i. L'; he iok)Mpt'tc citiit or toae log-
liioa r The.ct t n- n iprnaa o fur tLe ti etinjffL~CI of join 1t
26 - 3 1. co9 1i r i i i t,' I I t.Iic rn.ir,- i ia ls exp I iQ i t anid inqp Lici t in the
f i ttcd !i r-,, i i I Con,,iL r iitz; . We Call a1so iliL.2 prot 21 (x:x )as testing
a null ipdi.Usor t 1 it.i tit,' 23 ti piuiu o: in the ropresenta-
thon of 1). 1 ni:ot in: -Ire li i ktit. L;, t p.ir.ieters; corresponding
to ... (1 ' , 32-,6.
25
I . 1~ t ~
I I - __
...
I I-'
I - __ ___ ___
I I-. -~ I a
I i a
- I-. - .. -. -. - -.
p 77K-, -. - -.
I *-~- K -4-
* I
* I * -I
- 1.-. . .. I . I I
-~ I.-.-- . ~.. i I
I I...-. -- . I
- - -. I _
-.-....- -. . -. .. ..>t+-. L>; I.. . - * -. - - - -
I -'-*-'- '--4 -4---
....I - - ... . - -. .. -. -
~1___
26
A
1" ' V ,. ,f i'I .:.. ) I:; ) l.argt. that o,, rej.'ct the riodel of
joint ijtA,, , . t r f , • l , cr ,i tI) fi, :TC0, .1 , of nested
riarg inat. I i t' 11 , i X( ijk.) ,a,, v..io .; ci)"-, ato s; of two- and
thcue-way "',2 j i n l:a ; , otit inin.g so, t , witi ot li,. vr I ;11) 1 s. W1e summa-
rizo Iome iLe l!i ; ill i tuct ; '..mty:i:; of [Liformaton Table 11.2.
W:e have not aiiiludcd al the intrn, i te fitting soqnences for concise-
nIss. V, r' . , tlhit ilthough tli: c .o ure of the effect of additional
rIarja ina I t., ;Ir L:v;<,iatd par;imeLt:rs may vary according to the
scquencc in ci oh thty hive been addod, significa:nt effects tend to
remain s ig.n ii iaut .ian non-si nlfcarL effects tend to -,Lay non-
significant ,) tiit the tirst overal I survey should determine the
estimates ,id interactian parameters which warrant further investigation.
For example, the effect of adding x(..ki) t c x(ijk.) , x(i..Z) ,
x(.j.Z) is given in Analysis of Information Table 11-3as 21(xf:x) -
1.410 with One degree of freedom, but the effect of adding x(..ki) to
x(ijk.) , x(ij.:) is given in Analysis of Information Table ll.2 as
21(x*:x) 1.239 with one dugree of freedom. In neither case is thee m kP
cffcct or tle corespoodng cau parameter -i1 significant.
The calumns of Figure 11.1 which occur in the log-linear repre-
sentations of the estimates retained in Analysis of Information Table
i. 2 are
MIr.,inals Fitted EiLimate Columns of Figure 11.1
x..; 1-17, 21-22, 26-31
x(ijk.) , :(,i. .), x , !., ') 1-24, 26-31
x(ijk.), x(ij.,) x* 1-24, 26-37x (ijk .) , > i . ) ( . )x * 1- 3 7
FroM the anal ytic form of the log-linear representation or by
taking difftriIces 01 op,),opriate rows of Figure 11.'within the columns
used for thy es timate, th log-odds of fail to pass for each of the
estimates ar' jwiven by the respective parametric representations in (11.1)
where tie sup,!rscripts relate to the variables and the subscripts range
over the po;:Jible indices. The values of the parameters depend of course
on the corresponding estimate.
27
TABLE 11.2
ANALYSIS OF INFORINATION TABLE
Component Due to Information D.F.
x(ijk.), x(...Z) 21(x:x*) 160.551 23
a) x(ijk.), x(i..L), x(.j.z) 21(x* :x) = 138.732 5a
21(x:x*) 21.819 18a
m) x(ijk.), x(ij.X) 21(x*:x*) = 7.384 6
21(x:x*) = 14.435 12m
e) x(ijk.), x(ij.Z), x(..ki) 21(x*:x*) = 1.239 1e m
21(x:x*) = 13.196 11
21(x:x ) - 21(x:x a) 138.732 = 0.86
2 I x x2 1 x x1 4 . 1 = 0 . 9 121(x:x*) 160.551
21(x:x ) - 21(x:x ) -146.116
=*=-0.91
21(x:x 160.55121(x:x )2(x)
21(x__x_)_-_21(_:_e) 147.355
*• 160.551-092L(x:x )
TABLE 11.3
ANALYSIS OF INFOR.tATION TABLE
Counponent Due to Information D.F.
a) x(ijk.), x(i..Z), x(.jil) 21(x:x*) - 21.819 18a
f) x(ijk.), x(i..Z), x(.j.Z), x(..ki) 21(xf:xa) = 1.410 1
21(x:x*) - 20.409 17
28
I -A .. ...
x (ijkl)
x ij ,2)j
x (ijkl) i 9,. rj ijz
(11. 1) Ln t + +x*(ijk2) I ii . Tiijl
LU
*e(ijizl) + + +j r +t
T + =__ + + + ij+
x (ijk2) 1 1I ji 11 ijl
We recall that parameters with indices i = 4 and/or j 3
and/or k = 2 and/or Z 2 are by convention set equal to zero.
We remark that x m(ijk£) , determined by fitting the marginals
x(ijk.) , x(ij.Z) , is expressible explicitly as
(11.2) x (ijki) = x(ijk.)x(ij.Z)/x(ij..)
and is the estimate under a null hypothesis that race and success are
conditionally independent given home region and level of education.
In Analysis of Information Table 11.2 the value 21(x:x ) 14.43512 degrees of freedom, indicates an acceptable fit of this model. Fur-
thermore, 2(x* :x*) = 1.239 , one degree of freedom, implies that thee i
additional eofect of the marginal x(..kZ) is not significant or that
in the parametric representation of the log-odds in (11 .1) the parameterki
T l measuring tao effect of race on the dependent variable success isnre
not s ignif icant. 1t'e therefore investigate the estimate x min greater
detail. T"he values of (ijkZ) are given in Table 11.4.
* £In the expression for the log-odds under x in(l 1.1) T I is
an overall average, -ii and TIj are the effects of home region and
level of education on program completion and T ij is the interaction
effect of home region x level of education on program completion.
The numerical valuLs of the tau parameters are given in Table ll.5.We
recall that by convention parameters with an index corresponding to
i = 4 and/or j 3 and/or L = 2 are equal to zero.
29
u
c
U)
-o 0 1- 1
0'
- 4 C,1 '0 -4
.~ '0 CN -40 00
cn 0.0 - CD Q .4 (z -4-4 -
C14 - 0
co 1-1 co C4
Co 0
-4 cL 0 '
~ ' -4
-4 ODan -4 '0T4
fn 1 004
0 -4 C)
o '0-4 c '-4 co1 0 C14 CI
A44 V4 -1a
=C-4
300
TABLE 11. 5
VALU!-'S OV ',ARALETERS IN LOG-uODS FOR 'JN (11.1)m
= 44347 i z -0.292!78iIn
Til= 0.728653 ti 1 = -0.689433
i9. ij 2T = 0.041549 i21 = -0.602435
Ti -1.632427 • = -L.00304531 221
T 1 .312903 T ij,= 1..13793211 311
j z ijkT21 0.648130 '321 = 0.360697
From the pararmetric representation of the log-odds in (11.1)snd
the values In Tablu 11.5 one can determine differences in the log-odds
associated with changes in various categories. Thus the differences in
the log-odd' (fail to pass) as one changes the home region , for fixed
level of education,are given by
E-N E-W E-S
Below H.S. 0.9970 0.7287 0.4362
H.S. 1.0007 1.3110 0.0392
Above H.S. 0.6871 2.3611 0.7287
The diffe-ences in the log-odds as one-changes the level of education for
fixed home region arc, given by
Below H.S.-H.S. H.S.-Above H.S.
East 1.0617 -0.0413
North 1.0654 -0.3549
West 1.4420 1.0088
South 0.6648 0.6481
For eatiret interpretation, we convert the log-odds values to ratios
of the odds of failure.
31
F/N F/ [/4d'
Below uH.S. 2.7 2.1 1.6
H.S. 2.7 3./ 1.0
Above i.S. 2.(0 10.6 2.1
h lUwVt{.S./it.S. H.S./Above H.S.
East 2.9 0.96
North 2.9 0.70
West 4.2 2.7
South 1.9 1.9
Note that the odd, of failure in training of a man with home
region East and Above H.S. level of education are 10.6 times the odds
of a man with the ,arie level of education but home region West.
Men with home region East or North but with level of education
H.S. do botter than men with same home region but Above H.S. level
of education.
We have also computed the odds of failure x m(ijkl.)/x (ijk2) and
listed the results iii increasing values. The odds are expressed to 1,000,
that is, 5 to 1,000, 6 t' 1,000, etc.
Home region Level of Education Odds
Wes t Above H.S. 2
West H.S. 6
North H.S. 9
South Above H.S. 12
North Above H.S. 12
South H.S. 22
East H.S. 23
East Above H.S. 24
North Below H.S. 25
West Below H.S. 26
South Below H.S. 43
East Below H.S. 67
32
Note that tile O%OraV l ,'[- of 'li)uII ( O' t i' . rr& 311 I37L =
0.0226~ or 23.
For ease of compi ri-,;on and inf er'ict,, wo, aI.;, lis t t he f'aregoing
re';ult,; by home region and ltewl Of edu, itioii.
}West North South East
Above H.S. 2 12 12 24
H.S. 6 9 22 23
Below H.S. 26 25 43 67
12. Ou i [ zr-.s
'e d?'ine outliers as observations in one or more cells of a con-
ingecy tobl,,~ which appazrently deviate signficantly from a fitted model.
I.e- e ctlicr3 ray L,.,ad one to reject a model which fits the other.
ur_,vra,.tics. l- ex- c, in multi-dieznsional contingency tablas in
.lica tieor a.e i o. of the classificaitons there ,.iay occur an age
It cL Kuit a r -: nay be rejected for the entire tasle ut a model
'ai.in- Lhe pLhL ; tbJI - J.o 10. fet inito account nay liod to an acce.Prable
P !3-t ij Lil. t i.T" ( t( e i oi: , I
lit ol,_r casc .; ,n though a model s cs to f t, the outlieri con-
t ri bute t"(az r.-ire t L - .,-a+,njb Ie to the r'e":ure of deviat ion between the
I aroL!'.d t. r t 'i -1 v i 1: ,:;S of the mI-L. I.. in nt l,_-r ,.ord tle outliers
Ike Up a , rye perconta;e or the "une:-.pl ined v.'riat iou' 2 (:x )
A c,. to po,. ib - outlier s is p rov Lded by tIw ut p,,.,t of the con-
'tscur [-ogr I. ri th,- wic'te" otput for each ,tifate five entries are
33
t.
f . ...n:.
t)L.~ L t1 .1 C I) it. .L1 1 ""! r o r Mac
V ~ ~ ~ ~ -ft th ILK12 : : t I 7 i I t t.,rp r , t it. t) l ,l~ . La
ini rg,,iti ilt ti I t nL t' Lte ir. i:riiiaiuin info--
na t- li ,:I cPa i'Le C-L; t i t Maie ;ia rg inral res tra i its a s x except
ti-ati thct tic ;'e' x( ) s'! Ia ill ziuden, so 'A'ar >b (w I X(w 1 )
Pic b,;.;L, !:tL vvL' jy erry ol HLi :'iniinui discr~iiiaLiun infirratiofl
2 1Q:X 21(v :x) + 2 1(-)
0or
2 1 (x x a -1(~b 21(Q% :x)
liiero iI i :.i :.:,i t:i z ed i n th e Analys is o f lif 0rmatLion Tab le 12.1
TABIL 12.1
z".XLY c I S OF' I,~g v T&J; LE
COFIpontnc. du-2 to Ill foC)ri' t ionl D. F.
2[( 1 b Nh~
a it! 1? um l-ti~ X(, X )
it * itt' .r: r'';i:
34
/ (' / ,1%),... .)
(12.2) "
, n- "(, ,'L)-- (. - % <,,} Z. - ,,•n
n - a 1)
we 'Ct 1v ; (12.1)t L: i.( ( < )) EXb(
(12.3) x -0 1 X (w
2(.( ) . - - + (n - x(u 1 )),n n-/x a(I 1 ) n Xa (i)
The last vaue can be computed and is listed as the OUTLIER entry for eachCell 01 tI' corputer Output for thle estinate x a We remark that a
separate outlier computation for each cell is time consuming.Thc rat io
21(x:x - 21(x:x 21(x. :xa b nD I(12.4) 2I(x:x ) 2(x:x. )
a cl
then indicates the ,<:r:>1nta , of the "anexplained variatrin" due to the
outlier ;l1."
We shall iilus;trate tie outlier procedures and analysis using
datd originally (jtven by H.F. I)urn (The relationsiiip of cancer of
the lung ana tile use ot tobacco. Airs,-rican Sttistician, 8(1954),
7-13) and ,IJalyzed Luy J. Cornfield (A statistical problem arising
from rutrospectivc studies, Proc. 3rd Berkeley Symposium 4(1956),13b-14 ).
35
tu I it 111k ), w. iu r. a r I i ( I Ii t I (4 n X L: v
La(3X d t, u a s t r (,- 1 idua L wI I hk Ii i I i1 i: JV,( i prC r m tr i I y
k ~ ~ dh i >1 ia Is tIor r ardim fwj remn Ll( ii j, ~L flu 1. 1
di L tri bi Lion . fi' i I LISLLatU.; tL:J 1 CUpI12,1 11, an I y'; i S Wit 11 ti I
Citj a ori(ji-_1 IlIy q.i\7t:fl by. Lorui Aiid in clIyzu2d Ir: Co )f t ield'2 as
In Tablel2.lare listed the obse-rvatLions from 14
,:ci-rosp~ctiva studies on tho possible associa tion between
5[Oiol4._ and lungj cancer. lie denote the nccurrences in
the_ t'hirea-.iay I 'I>2x2 continci-3acy table hb x (Jjk) wi t
tho2 notation
VariIbe Incex 1 2 3 ... 14
S uLI-Iy i "o. 1Io a i..N.1
P a L f:,'Ita Con tro I lui cn~cr
Smoking I ,l~oiismoker Soo
D,-)--; Lh is3 cata li.i,-,sOcci aII-JOn hbe *:-n :;rokiniq and lung
canrcer, and if sro, i:.- the da;s-rujatLion ho;7cvrcOus ove:r
th, 1 -' zLu,.ics? Ucethe am !0 a;sia o is iho
1 L -c;h of Lh2 croz73-prcOIuct ratio.
36
'J',<Llo_ ]2. .
-c n t s,
(- t-- 1 -)--t-.ic nts
S72 8 3
2 4i 227 90
3 ) 41 i i/ 129
4 125 397 1) 70
5 3.1 299 U2 412
6 114 666 8 597
7 12 174 5 988 61 1296 7 1350
9 27 106 3 60
I0 0.1 53". 12 459
-2 7211 54 2, , 72412 56 462 9
13 4536 1725 -
14"2-9 " 260
37
i e tiy pt ll i .', (, c1U . . LI,]C_)t i ut' .- *; i 'e.' ti'.
L Ltul y
(12.5) l : p L- p(i-k)
p( ") p(i. -) p(i" )
imposes the restraiits on the estimate x*(ijk) thaLa
(12.6) x*(ij.) = x(ij-)and x*(ik) x(iK)d I
in fact x*(ijk) may be exlicitly represented bya
(12.7) x*(ijk) = x(ij-)x(i'k)/x(i--)a
Similarly the hypothesis of no second-order interaction
(12.8) H2 : p(ijk) = a(ij)b(ik)c(jk)
imposes the restraints on Lhe estimate x*(ijk) that
(12.9) x*(ij-) - x(ij. , x (i-k) x(i-k) , x* -jk) = x(-jk)
The estimate x*(ijk) cannot be reprt:sented as an explicit
function of the observed ;argqnals.
In this s tId-, th: ri -isrAt i-1 ) t -Ion info 7'a ttionsatizt:Lcz are o.-]iVce] i. ratio c. , .r,,; anl the
a',,;['.t ic c:)
t:;O a ,~ ;:nll'; is or n olcot i: '.oi tao! :r.Ltsta :.
te:', t":2:' )o< n! .-o -f t- ! L , o t 1. It , ,.ct
38
).( p) (. >) :. )~iC ~ O
(12.10) X 1 .. .. .2
,' a , . .1 tI VdI t il , r , I [( lu ., i c a n c e r
'Ind -' ar_ C v,.n c), Lh loq-lincar'" 2
x(il " ijtill I + I
(12.11)
+j kn x ( 2 -) - T + T i + 'L .I
,. cr -7 L e ~V a .II I of hc tau pa rarw tL rs demn d of c ,ucse
On
2n 1
I :.( i -.) ,:<(i .' ) 21 ( .: :: ) S c9.74 ],1
39
hile ""I II- of 2 [ Cx :x* ) when1 cu".'.jA0
~rc ni a: t- l Is i 1)tr It 1 *1 .. r IL
11 1) t 1" .!s is of 1 ci ' Io I)~I~td Le ter~ 1~ Uf
c'1nCer1 con()Pitiun2d ,'11 tihe t Iy s fi ae I i con-I itl tonaIl
I y po tne tSi - al I I c t InI a ( -ulr U 1, tjf o-u1 )f i i o., 1yn't io1 n f rof-
(litteret. StUdiezi ,.ttho)ut ilnposixiq the rejlulirement that
toeL popl±-L-ion characteristics of each stul' be similar.
The rejuction of tni conditional independence- hypothesis
-IS Of coulrse exp(-cted:. The degree of departure from
indepe2ndence is func-tionally dependent on the study. Is
this dependence the result of a small sub)set of the studies
whichi are suL)b!"ti~aliy different- from the remainder, or
does the dup'ArtAlre Vcry along a continuum?
Tiie, value, ut z i x. :x*) suggests that in accordance
with (12.10) the vaiLue of ijk 1 .687 is significantly
diffe-rent froin zero-. Moieover, 21(x:x*) is alzso significant2
when coitpared to thc -th percentile of -i d istribution.
Tevi Lue of 21 (x :,..) 5-juqqests thlat we rceect ther null
iiypotno-sis Of no second-oruer interacin tItis h
mo11)Iti with a conumon value of. the interactio-w param-eter t jk
Js not a good fit. The values of x* are clivt.n in Table 12.3.2
We now proceed to determine the outliers.
40
,,ntro1 P i t,: ; : 3 Canc,.r Patients
;t LnSl n Sl ,1: C,,-S: ck, rf Smokers
].0 ( 71 .99 99 83.01
2 4?.86 227.14 3.14 89.86
3 19.99 80.11 6.01 129.99
4 122.16 389.83 4.84 77.16
5 1_0.03 300.00 32.97 410.99
6 105.06 G74.94 16.94 588.06
7 15.47 170. 53 1.54 91.47
8 57.06 1299.93 10.94 1346.08
9 27. 15 105.85 2.85 60.15
10 85.21 529.79 13.79 463.21
i1 3.62 261.39 19.38 708.62
12 52 23 4J5.77 12.77 505.23
13 W15. 32 1721.66 31.69 458.33
27 , 3 259.16 5.17 259.84
41
'.x-minaLiun of the co:ututer oltput rcor x i usinq
1 2
all 14 studies showed a largest OUTLTi value of 18.14
for the cell (11,2,1). A new estimate fitting the mar-
qin: 1, x(ij.) , x(i.k) , x(.jk) and. omitting the cell
(11,2,1) was obtained. In fact Study 11 -aas o:mitted
because with the constraints for the new estimate
x.(ll,j,k) = x(ll,j,k). Since this estimate yielded
(12.12) 2I (x:x*) = 28.40, 12d. f.
the deletion procedure was continued. We suin'arize the
results in Table 12.4 and Analysis of Information Table 12.5.
Table 12.4
Fittinq x Lj.) x (i.k) , Y(. jk) wJi: siun i
d_i<i on of stucdes
-Cell V_1Ue I 1 ozration D.F.
I-l (11,2,1) 18]. 1 2(.:>: f 55. 19 13
I-10,12--1 (6,2,1) 7.89 20(:::. -2,.40o 12
1 ,7- LO,]2- 1.4 (4,2,1-) 4 .07 21(.::c 18 .03 1
1- 7, 1. ,1:!-]"4 (7,2,1) 3.9] 2 )--11.9,1 10
1-3, :J, --i , I 2'' .::j-) 7.03 9
42
TabL- 12.)
ALoalsins c t ior.-. t i,, r
C•zowiit dutonOi ion D. F.
All 14 studiei 2i(x:x2)255.19 13
Loss 11 2I(x*.)=2C.79 1~b 2__
2 I (x- *) =28. 40 12Less 11,6 21(x*:x*)=10.37
Less11,6c b2I (x:x c) :18.03 I
Less 11,6,4 21(x*:x*)= 6.08 1
2I(x:x*)=1.94 10
Less 11,6,4,7 2I(.*:x*)= 4.92 1e d21(x:x*)= 7.03 9e
Since (21(x:x*) - 21(x:x*))/2I(x:x*)2 e 2xx~
21(x*.x)/2I(x:x*) = 48.16/55.19=0.37 we see that the
four studies numbered 4,6,7,11 contributed 87% of the"unexplained variation" 21(x:x*) Tho values of the" 2
estii:ite x: are given in Tablel 2.b- The value of thee
log :ros;-procuct ratio is
-.x* (ill) x* (i22)(12.13) : n 1.53, i-3,58-0,12- 4x* (i12) x* (i21)
43
'ibLj 12.b
Ci i
tL'-c) V - -tr n t . 1 IuN; Cance r Patien-s
St y on . rk.. .. ... i .r rs Smo:rs-T. o,... r..--.g- ,o , s i o]- s
1 13.9 7:.32 3. 32 82.69
2 42. 6 227.53 3. 54 89. 47
3 19.40 80.60 6.60 129.40
5 126.85 303.18 36.15 407.83
8 55.79 1301.21 12.21 1344.80
9 26.00 S 106.20 3.20 59.80
10 83.61 531.39 15.39 461.62
12 60.81 I 457.20 14.19 503.81
1 639.35 1725.64 35.66 454.36
27.- 2 5.76 259.2"
44
We note thlIt Cox (1970) in analyzing the data of
Tablel2.1concluded that studies 8,6, aid 11 were outliers.
For the 14 studies he found a residual sum of squares 47.7
with 13 degrees of freedom. With studies 8, 6, and 11
omitted he found a residual sum of squares 15.1 with 10
degree3 of freedom. (Cox (1970) p. 83 gives the degrees
of freedom as 11, a misprint).
Following the procedure described when Studies 6,
8, and 11 were omitted the results led to the Analysis of
Information Table 12.7. Note that omitting Studies 11, 6,
4 as per Table 12.5 accounts for more of the unexplained
variation. Table 12.7
Analysis of Information
Comp-onent due to Information D.F.
All 14 stuies 2I(x:Y*)=55.l9 13
Loss 6,P,11 2I(xf:x)=41.62 32 1 (x: X*) =1 3. 57 10
Th-c sequun'ial procedure discussed herein was also
applied to data relating Iathar and son professions
45
F
pubLi ,LeCd by Karl Pear son (1904) , "un the th,,,o ty of
,x)lntii19elc da1 i 1 Lion Lo ass ItOS ,and normal
coirelLion," r,3printed in Karl Pear,.on ' s Early Papers,
Caibridqe University Press, 1948, and considered by
i.'ieniurq (1969) and Good (1956). Using the Pearson data
2Ficnoerg obtained an X = 184.9 with 44 out of 196 cells
2deleted whereas the sequential procedure led to an X = 155.3
with 25 cells deleted.
13. Zero Marginals
As may be noted from the analysis in Section 11, zero
occurrences in cells of the observed contingency table
present no special problem provided that no marginal entering
into the fitting specification is zero. When the latter is
the case, however, the interpretation may be distorted
because of inflated degrees of freedom. A procedure to
circumvent this problem is similar to that used for getting
revised estimates when outliers are indicated. We shall
present the procedure in terms of a specific example.
The following data resulted from a study of Christmas
tree consumption. We are indebted to Dipl. Forstwirt Dietrich
V. Staden, Institut f. Forstbenutzung, Universitaet Goettingen
for the data and permission to use it. In Table 13.1 are
listed responses to the question "Did you have a Christmas
tree in your apartment/house last year or not?" according to
size of household and size of city. We denote the occurrences
46
in the t1hree-waY -'x9x5 cOntililency tit I I,(, i k.) w i t h the(
notaLion
Var Lable Lndex 1 2 3 4 5 6 7
Tree 1. YcL No
l1ouzolld size J 1 2 3 4 1 b 7
City size k '<2000 2000 20000 10O000 500000to Lu to or
20000 100000 500000 more
For a 2x9x5 wxCxD contingency t-ible wa compute ,nn
estimate under a Iypothesit of no second-order interaction
by fitting all tho two-way marginals. C.ill this estimate
x*(ijk). A tost for the null hypothesis ot no uecond-order
interaction is given by
2LU. x(ijk) (n (ijk) 21(x:x*) 32 o.V.
If there is no second-order interaction thun the associations
between H and C, R ind D, C and D are the L;ame for alL values
of the third variable, that is,
:2(ljk) x* (29k)In C depends only on j
x* (ljk) x, (2 j 5)tn 2jk)x2(lj5) depends only on k,
X*(ijk) x*(05,)In (i2 _x ( is independent of i.
x(i j 5) X*T T is
47
t III t c; s 1vodtz . I 'Iat~ A w~ t hki I. I~ a
~t i bJ LC: k*CJk) L ti Lji beL~t tp~~..~ .
\(Ij -. X( jk) /X(1 "I WOJ X*(~ix.i L.i)I Llw c.,,( iwa~kt
Nf~IMUO anlylpOU~h2s Of CO\kif~ fdfe~h. ll d1)
qivoln C. We fluvW 1Livo Allaliz IS t ElitorWIt (km T'ibloI I 1._1
TaIble- 13.2
Com~ponent Jue to lntocnildLioli 1). F.
Condi tiondl ilopewndenceof R( anid D) qmim' C k(x: 3b
[El fc":t of X(i-k) .venX(Lj-) alid x(jkJ 2 L(x*:.~ 4
No s,;cond-OrderintltCd:tivf 2L1(x:.\ J
Por Lteo pcrt tc.kav dUlt -t i n queti~on h'w\,- ))WQ 'Q1* ,
x ( ijk) U for j -t , /, , i- -2 anct a1l.io Lor :;()inQ ~t i 4 L,
j 7 , 8 , 9 , thc! es Lji..i Luvi for iw h en t ries ct Iclipond iiy
x* ( ij Q~ f or j -O, 7 , , b(,Lh or x*and x w i J L n ro, di LlI -I'
from' tue obLjLv rveLd vol ui.. Accordiri(I 1y 1,!L w; mu Lc aw
e( btiwLatc :* ( ijk) wlr ich is obtai i ticd by I i LL i ml tho two-waiyt
Lflar(ina~ls 01 the 2:<',x5 Lab I j:.l., 2, 3 , irld A
j-fb,7,t3,9. 5ivitiliiy lut x* (ijk)-x (ij-)x ( -jk)/.()Iu
the! 2x,-x) table L>-12,3,4, and x,*(ijk) *.;c( i jk) , O j< 7,,
48
We now find
Table 13.3
Component due to Information D.F.
Conditional independenceof R and D given C 2I(x:x*)=25.532 20
Effect of x(i-k) givenx(ij-) and x(-jk) 2 (X*:x*)=5.821
No second-orderinteraction 21(x:x*)=19.711 16f
Note the reduction in the degrees of freedom between Table 13.2
and Table 13.3. It is also interesting to note that when
actually carrying out the procedures for Table 13.2, the same
estimates and statistics were obtained as for Table 13.3. See
Table 13.4 and 13.5, Table 13.6 and 13.7.
It seems reasonable to conclude that the purchase of a
Christmas tree is independent of the size of the city given
the size of the household (j=1,2,3,4,5) i;ii lioiseholds of
size 6,7,8,9 seem almost sure to buy C:,rist',is trees.
The log-odds for the purchase ot a (Tiistrnas tree as a
function of household size is given in Toble 13.8. The
probability estimate for a purchase as a lunction of household
size is given in Table 13.9.
Table 13.8
fn (x*(ljk)/x*(2jk))= tn (x(lj-)/x(2j-))e e
j=1 -0.25862 0.86623 2.17024 3.40125 2.3716
49
'Q0 111 3 )x C j- ,'x ( j'
.[ (1/140 u.43L_72 214/304 u 103-13 219,244 = 891'4 180/18b 0.96775 7/3/82 0 .9146
I(-r more copLex situations there is also the Log-linear
canalysis, which is ot course avoilable for this problem too,
but it would not add anything to the analysis o- this
p'Irticular data.
50
L3 ,: 134
: , I ifi . I ,':4
L722 II_11
", - -.'-,I ~ . , '*'' " ,/ ,'I , 1 2 . 3 292 24 865 63. 44
C. : 0 "- 9 U "L .. j) 22. / ;. . C: ,-I 49 9532 62.5%' -* 2o 4: 30 4:. 6 .. LS,
2 ,; .; 'D56 32.098 45.213.. 2 , 974 12 036 8 . 5
S7 3 Oi 2 < * "C , 2 0C1 1.000 1.000i~~~~ I 2 0, 0 0 ; ;!O5 0; L00 0.000 1.000
: 0 J 0 .-0000 0 000 000tu
2 ' 4 13 9 13 40 79 .?2 12 1 7.184 12.953 42.76322 5 18 19 i3 19 . '6 17.795 15.134 32.157
2 3 3 o 4 10 26 .877' 61, 4.050 4.046 8.4102 4 0 3 G 0 2 6 0.636 1 7 0.9'5 0.903 1.78623! 1 1 2 22 7 0 807 1.026 0.964 0.8342 6 0 o o 0 0 ol 0. o000 0.000 0.0002 7 0 0 0 0 0 0 0.00G 0 0' 0 0.000 0.000 0.0002 1 0 0 0 0 .000 0. OO) 0.000 0.000 0.0002i 9 0 01 0 0_0 O 0 0 003 0.000 0.000 0.000
,able 13.5
1 3 2 1 2 2 3 J 4 9 5 S-,*ud* . * ,~n 2 . . = --
I 2i 17.233! 3%.S"$ 56.2 4 o13 03.8,733, 2l.123~ , )I41 91o " 50
I 4- 24.364 4I.26 33 1 0 :213115 11. 193 2 60 8 .97,11 12 036 8 .151 4 12 1 617 3 C 21L 8, 2 2 L o c
2 1 3.93 12. 16/ 7 1 Z i2 953 ,2.7632 2' 5.747 1 .I, 7.79 15. 32.1572 3 1..77 ,.61 i 4. . 8.4A 02 4 U 7,' . .1) 3 1 . 7362 2) 0.8G-7 3 1 .026 0.9 4 0.8342/ 0 0 0 0
02 3< 0 0 0 G2' 0 0 o
51
3 C . r . .
C ..
--
• ° • • , ':-. , -, •
C* C CD-
C, ) ...
Cm) C) 'u' -C".; , C <' C.
*) -(C *() 9 ~ .. .. . ..C* C C ,' "
r , u , ( , .:.... " c ' to i.11i
(70 ,
c.. ') CD) C7 ) C ll t Wi11 7
(C C - CC C). -. ) - ' r \ '
. .. . ,-"
C.) 2 ' ' -.9. I , .CAC ( I A C > l
C:..
" C) 0 C7 ) . ) ,I-- C'',, ,' , , 52
C" i C -- ', CC ' * .
52-. C-
' t "V ~ i - i L F ,I -
rcf -utliu ed :.Av VOndei Grant
31 3.. ueluni and 'li qIp icedtto Lof -the
t CCtl Ld I ;yur pai ticl 2 U 2O 1t ) y the
IN a I. Na' :.e.Iaai-e unv r; Contr -, ct .'dlmoer IhUu0 i4-67-A-
cz u, <t 147' 0 24 and! 1) tli.: :,rr Psearch
.~xi ~nc~~handAir Furee2 Office of
-1 'L)t iC t~e~ i~ Contrt No . n 0 0O 14 -b 7-,%-0 214-0O01 5
CcOi,,utation3 were carried out at the ComTputer Center,
The ~r~ aeLaUnUniversi ty. Thu2 support and encourage-
ii,2nt o~f P~roiessor Henry Solom-ion and Profesisor Herbert Solomon
are yr- __ utIly ici-xw1k2dkyid.
TILhis paper. wais tiie basis of lucturos presented by
Pruic-suu t ,. Ku~ik15 iuno 1973 at O.SakaL Univrsity,
Osa~.,<ei ;20J ,JUn~ J973 at The Institute of Statistical
.*itnua i C , 'e~ys ,Japan ; ind 6 JulIy 197 3 at. ii oshim-'a
U11 IV, YS t-Y , III C:. 1'. -I , 'J -Pn hu oppertsriity ind iiiv itaticons
~w pPeri Laiis ntr I we-re matde possibl, a ky thie partial
5upL)(IrL oIn.- inLet en t ()f th(u Jipan octyLu;- thu Prormotion
of ;uLcnce , Pr nrNiznor u ;c kL1(JLC11i, k'rufu-s sor Ki yonori
KuuisaaDr. Yukdv-k'QSi Kawadla, Dr. Kamiuo Moltu-sita, Processor
U. .~n~ii nd PLIol ssor Suiiiya.su Yamamoto. Their support
andj C(MIllItr aLien urk -ralt eLui~ly acknow le qtid.
53
'i'nu bibliography is essentially a compilation of
those propared Dy Dr. Harvin A. Kastenbaum and Dr. H.H. Ku
covering the period through 197L and permission to use their
results is jratefully acknowledged.
54
lxt i lb ii c, ii t icV yA tab 10 ei LLC Ii anay ;ss
-,(); ''a'l iAt~ ity to iP
55
REF ERENCES
1972
AA.V , J. and ENKE, ii. (1972). Analyse iw Icrdiiuen;i (,ana Lr kont i n!La v!In
:,Lt iitlfe d(jc nutr.jt on ina::;es von Kuilback. Biometri: ,t-
Zeitschrift 14, 5, pp. 305-323.
BE.KKS'N, J. (1972). >Linimun discrimination infor-'iation, the "n. inter-
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I:t ,1 . AriA. r"art. t i t , yp . 190-- 1 u
19 Y 4
tLV', 2',. A. (1i i4) . 01 , I - andi Io y 2 ;' d ' ., -; ,, V i:i rli
. i l t .rti A, j[i,: t. A t' . I A :. (II V' r ,inJ tIOcy.! IA . , 3<cc 1 icci 2 t.22i.
2Y A. 'F' , f. . ( I') .',) .:I(t : y' II t 1 i ! v : I v i 1 , + :. I I tu ' U .-, a .i , tJ ' , 2
t, t y I ;< . P: i 'L._ K.nc_. - ,~ ,i '.,; ! ,. 2 . IS,.
SO+
FIS HER, H. A. (1924). 'ide cono a it. ion s i in cn: whr: I L i_ 1 -e e -!u ures the
discrepancy betwcc [erv: ion ao hyp , !es i . J. hoy. Stati.St. Soc.
37, pp. 442-450.
1922
FISHER, R. A. (1922). On the lnfterr-cetatioa of cl:i-sqna-e from contingency
tables, and the calculation of P. J. Roy. ?tatist. SoCe. 85, pp.
87-94.
1900
PEARSON, K. (1900). On the criterion that a given systLem of deviations
from the probablc in the case of a correlated system of variables is
such th:t it can be reasonably suppcsed to have arisen from random
Ph' .. eris 5 5,, pp. 157-172.
81
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20.-- rhe analysis of the intu-mation in contingency tables is an as- ctof maltivariate (mulviple variates) analysis wlth particular applicationto qualitative or c:cg.rical as well as quantitative variables.
"The analysis is e'rieerned with couwts in muiltiway cross-cla.sificationsor multiway contingen/. tables. Multiway contir.gency tables, or cross-classifications of' vet rs of discrete rm.ndom vmriables, provide a u-efulapproach to the analysis of multivariaie discrete3 data.
'The method of analysis presented will bring out the various inter-relationships among the classificatory variables in a multivay cross-classification or conting-ency table in many dimensions.
The proc-2dure is basei on the Principle of Minimum DiscriminationInformation Estimaticni, associated statistics and Analysis of Informa-tion. Genei-ul cxmnputer programs are available to provide the necessaryresults for :Inference. An analysis of a four-way contingency table ispr.?sented for tilustraticn of these techniques.
#235
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