a formal derivation of the conditional likelihood for ...osius/download/papers/map55osius.pdf · 4...
Post on 30-Jun-2018
217 Views
Preview:
TRANSCRIPT
MATHEMATIK-ARBEITSPAPIERE
M A T H E M A T I K - A R B E I T S P A P I E R E
A: MATHEMATISCHE FORSCHUNGSPAPIERE
FACHBEREICH MATHEMATIK UND INFORMATIK
UNIVERSITAT BREMEN
BibliothekstraBe D-28359 Bremen
Germany
A Formal Derivation Of The Conditional Likelihood For Matched Case-Control Studies
Gerhard Osius
1 Introduction and Notation
2 Odds Ratios
3 Odds Ratio Models
4 Conditional Sampling Distribution and Likelihood
5 Case-Control Studies With l:M Matching
A Multivariate Noncentral Hypergeometric Distributions
B Rank and Order Statistics
Abstract
Likelihood analysis for logistic regression models in matched case-control studies is based on a conditional likehood. We provide a formal derivation for this conditional likelihood in a general setting by conditioning the sampling distribution of the risk factors (in each matching group) on its observed order statistic with respect to lexicographical ordering. The required joint distribution of the rank and order statistics for vectors is derived in a more general context. The conditional likelihood depends only upon the odds ratios of interest, i.e. for different values of the risk factor and same value of the matching variable. Hence the conditional analysis may be used in parametric (eg.. logistic repesssion) as well as in nonparametric models for the odds ratios. In the presence of only two disease categories (case and control) and 1M matching the conditional likelihood is a product of multinomial probabilities (reducing to binomial probabilities for M = 1). In this case the conditional analysis can be done within the framework of generalized linear models. In the presence of more than two disease categories the conditional distribution of the covariate in each matching group is a multivariate noncentral hypergeometric distribution (whose basic properties are also given) and the conditional analysis requires a higher computational effort.
1 This work was supported by the German National Science Foundation (DFG), grant 0s-14411 and is a revised version of the German report (Oct. 1997) for the DFG.
2 Institut fiir Statistik, Fachbereich 3, Universitat Bremen, Postfach 330440, 28334 Bremen, Germany. E-mail: osius@math.uni-bremen.de
Conditional likelihood for matched case-control studies
1. Introduction and Notation
Let the status of a specific disease (eg. lung cancer) be given by a discrete variable
YE{O, 1, ..., K} where 0 represents no disease and 1, ..., K are the different disease
categories of interest. In the case K= 1 the variable Y is binary and hence an
indicator. In epidemiology one investigates the dependence of the disease status Y
upon a vector X€lRR of (suspected) risk factors (eg. consumtion of tobacco,
exposition to asbestos) taking into account an additional vector Z E lRT of
confounders (eg. age and gender) which are not of primary interest. An important
(often retrospective) sampling design for this situation is a matched case-control
study in which risk variable is sampled from conditional distributions given the
disease status Y = k and the confounder Z = z which is also referred to as the
matching variable.
The association between X and Y conditional upon Z = z is completely determined
through the family of odds ratios (cf. Osius 2000)
for all disesase categories k = 1, ..., K and all pairs u, v of values of the risk
variable X. In the logistic regression model thes odds ratios depend on unknown R parameter vectors PI, ..., PK E lR through
Although the likelihood of a matched case-control study involves the above odds
ratios of interest (and hence the parameters Pk of the logistic model) it also
depends on additional nuisance parameters. The number of the nuisance
parameters typically increases with the number of matching groups. This may
result in severely biased and inconsistent maximum likelihood estimates bk if the
matching variable Z ranges over infinitely many values (eg. if one component of Z is continuous), c.f. Breslow und Day (1980), Sec. 7.1. Therefore a conditional
likelihood has been proposed (Liddell et. a1 1977, Breslow und Day 1980), which
depends only on the parameters Pk of interest and leads to consistent estimates.
Our aim is to provide a formal derivation of this conditional likelihood in a general
context (not restricted to parametric logistic models) by conditioning the sample of
a the matched case-control study upon a suitable family of random variables,
namely the order statistics of the risk factors in each matching group. Besides a
clarification of the term conditional likelihood this derivation also allows a baysian
Conditional likelihood for matched case-control studies 3
approach by considering sampling from conditional distribution as a (pseudo)
experiment (cf. van der Linde und Osius 2001). Our derivation of the conditional
sampling distribution is not limited to the logistic regression model (2) but allows a
general type of models (including nonparametric ones) for the odds ratios (1). For
binary Y (i.e. K = 1) we have a closer look at 1M matching where the conditional
likelihood is a product of multiniomial likelihoods (and hence a product of
binomials for M = 1). This allows a conditional likelihood analysis. using standard
techniques and software.
The formal derivations require some assumptions and notations which will be given
prior to the more substantial considerations. Although the discrete variable Y was
introduced as a disease status this is not essential and more generally Y may
represent any kind of status and its value 0 is considered a normal or reference
status. The distribution of the risk factor XER' and the matching variable R Z E R will not be restricted and in particular the components of each vector may
be dicrete, continuous or mixed. We only assume that the joint distribution
2(Y, X, Z ) of (Y, X, Z ) has a density with respect to some product measure S u = u x u x u on the product space R x R xRT. Here u#
# X Z is the counting measure
on R and typically (but not necesassarily)
"x = uxlX. . . X u and
XR "z = '121X. . . X u ZT
are product measures, too, with each factor being Lebesgue's resp. the counting
measure on R provided the corresponding component of X or Z is continuous resp.
discrete distributed.
R Y has support Oy={O, 1, ..., K} and let O X c R resp.. O2clRT denote the
support of X resp. Z, i.e.
In analogy to the generic notation P for the probability we will use p for the density,
e.g. p(Y=y, X=x, Z=z) denotes the densitity of 2(Y, X , Z ) in (y, x , z ) and
p ( X = x, Z = z I Y= y ) the (conditonal) densitity of 2 ( X ,Z I Y = y ) in (x ,z).
Furthermore we assume that the joint densitiy of (Y, X , Z ) is positive on its
support Oyx Oxx Oz
(4) p ( ~ = y , X=X, Z = Z ) > o for all (y ,x , z ) E O ~ X O ~ X O ~
The lexicografical ordering on O x c R R needed later may be defined recursively
over the length R of the vectors by
(Y 5 (v, vR) * u < v or ( u = v und uR<vR)
Conditional likelihood for matched case-control studies 4
for all u, v E E~- ' and uR, vR E E.
R Any two vectors u, v E E are lexicografically comparable, i.e. u 5 v or v 5 u holds,
and hence the lexicografic minimum min(u,v) and maximum max(u,v) of (u,v) are
defined. As usual, u < v abbreviates (u 5 v and u s v).
2. Odds Ratios
A statistical analysis typically focusses on certain parameters of interest and not
on the whole distribution of (Y, X ,Z) itself. Of primary interest in applications is
the conditional distribution of the status Y given the values X E O ~ and Z E O ~ of
the risk factor X and the matching variable Z, i.e. the conditional probability
for status k E Oy Instead of the conditional probabilities the odds with respect to
the reference status 0 may be used, i.e.
The probability vector ~ ( x , z) E (0,l) ltK is uniquely determined by the K K-dimensional vector Odds (~(x , z)) E (0, co) .
The odds ratio of a status k = 1, ..., K for two values u, v E Ox of the risk factor and
fixed value z E OZ of the matching variable is given by
(3) Odds T ~ ( u , Z)
ORk(u,v I Z) = Odds T~(v , Z)
From ORk(u,v 1 z) = ORk(v,u 1 z)-I and ORk(u,u 1 z) = 0 we conclude, that the
family of all odds ratios (3) - which characterizes the association of (X,Y') for a
given Z = z (cf. Osius 2000) - is already determined by its subfamiy with u < v.
Conditional likelihood for matched case-control studies 5
The logarithm of the odds ratio will be denoted by
(4> $k(u,v I Z) = log O R k ( ~ , v I Z )
= log Odds T ~ ( u , z)) - log Odds T~(v , z)
= logitk(?r(u, z)) - logitk(?r(v, z)),
where the multivariate logit transform of a probability vector ?r E (0,l) 1+K
is defined as the K-dimensional vector logit(?r) with components
(5) Iogitk(?r) = log T~ - log 7r0 . k = I, ..., K.
To arrive at further representations of the odds ratios we consider other
conditional distributions and start with conditioning (X,Z) upon Y. The density of
the conditional distribution %(X, Z I Y = k) for k E Oy is given by
with the marginal probability
(7> P(Y=k) = Jp(Y=k, X = x , Z = z ) ux(dx) uZ(dz)
The densitiy ratio for two values u, v E Ox is
(8) D R ( U , V ~ z,k) = ~ ( x = u , Z = Z 1 ~ = k ) / ~ ( x = v , Z = Z 1 Y=k)
= p(Y=k, X = u , Z = Z ) / p ( ~ = k , X=V, Z = Z ) .
Hence the odds ratio also appears as a ratio of density ratios
Consider now the conditional distribution of X given (Y,Z). The density of
%(XIY=k,Z=z) for any k ~ O ~ a n d z€Ozis given by
(10) p ( X = x 1 Y=k, Z = z ) = p(Y=k, X = x , Z = z ) / p ( ~ = k , Z = Z )
with
(11) p(Y=k, Z = z ) = Jp(Y=k, X = x , Z = z ) ux (dx) .
The density ratio (8) may also be written as a ratio for the density (10)
Conditional likelihood for matched case-control studies 6
Hence the odds ratio (9) is also a parameter of the conditional distributions
.d(X I Y=k, Z = z )
From the representations (3), (9) and (13) we conclude that the odds ratio
OR(u,v I z ) are common parameter of the three conditional distributions .d(Y I X, Z),
.d(X,ZIY) and .d(X IY,Z). This is important since the major sampling schemes
in epidemiology are obtained by drawing samples from these conditional
distributions: cohort studies, case-control studies and matched case-control studies
3. Odds Ratio Models
The logistic regression model specifies the conditional probability sk(x,z) for
status k = 1, ..., K through K
(x, Z) = ex^ { Q ~ ( Z ) + x T ~ k } / ex^ {QI(z) + x T ~ l } k resp.
T I =o logitk a(x, z) = ak(z) + x Pk
R with unknown parameter vectors pkcIR and an unknown function ak:Oz+ IR.
Since the model (1) has no interaction between the risk factor X and the matching
variable Z the corresponding log-odds ratio
does not dependent upon the value z of the matching variable. Replacing the linear T function x pk in (1) by an arbitrary (eg. sufficiently smooth) function hk leads to
the model
with log-odds ratios
(4> $,(u, I Z ) = hk(u) - hk(v)
independent of z. Our model approach will based on (4) which may alternatively
be specified by two conditions. The first is a matching condition
Conditional likelihood for matched case-control studies 7
(MC) All odds ratio functions ORk(-, - I z): Oxx Ox+ (0, co) for k = 1, ..., K do
not depend upon Z E O ~ i.e. there are functions ORk(-, -) such that
ORk(u, v 1 z) = ORk(u, v ) resp..
$,(u, I Z ) = $ k ( ~ , V) : = log O R k ( ~ , V) for all k, u, v, z.
And the second condition spezifies the actual odds ratio model by restriction the
structure of the functions ORk resp. $k = log ORk in (MC) through
(ORM) dk(u , V) = hk(u) - hk(v) for all k = 1, ..., K and u , v E Ox,
with arbitrary functions hk: Ox+ IR. Both conditions (MC) and (ORM) together
are equivalent to (4).
The log-linear odds ratio model is given by linear functions
T (LLM) h k ( u ) = u P k' (log-linear OR-modell),
with unknown parameters pl, ..., PK€IRR and in this case the model (ORM)
reduces to the logistic regression model (2).
4. Conditional Sampling Distribution and Likelihood
Before turning to sampling with matching in its general form let us look at
case-control studies with binary Y and l : M matching. Here sampling is typical
divided into two steps. First, the cases are sampled, i.e. for all i = 1, ..., I the pairs
(Xil,Zi) of the risk factor and matching variable are independently drawn from
the conditional distribution %(X,Z I Y=l) . Second, for each case i = 1, ..., I a fixed
number M of controls is collected each having the same value zi forthe matching
variable as the case, i.e. the risk factors Xoim are drawn independently from the
conditional distribution %(X I Y = 0, Z = zi) for m = 1 ,..., M. Since the distribution
%(Z I Y = 1) of the matching variable among the cases contains no information
about the odds ratios of interest we may equally well pass among the cases to the
conditional distribution given the observed matching values zi, i.e. we may assume
that the risk factors Xil among the cases are sampled from the conditional
distribution %(X I Y = 1, Z = z .) with given values zi of the matching variable. 2
For general Y with arbitrary K 2 1 and Mo : Ml : ... : MK matching we now assume
that the risk factor is sampled conditional upon the status Y and the matching
variable Z. More precisely, for each matching group i =1, ..., I specified by a fixed
value Z . E O ~ and each status k = 0, ..., K an independent sample Xikm for 2
Conditional likelihood for matched case-control studies 8
m = 1, ..., Mk is drawn from the conditional distribution %(X I Y = 5, Z = zi), i.e. from
the corresponding subpopulation { Y= 5, Z=z .}, 2
(1) %(Xikm) = %(X I Y=k, Z=zi) for all i, 5, m.
Since all Xikm are independent, the joint distribution of the sample is given by the
product probability measure
From 2 (13) we conclude that the odds ratios of interest
for 5 = 1, ... ,K are parameters of the sampling distribution (2).
Now we want to pass to a conditional distribution of the sample which depends only
on the odds ratios of interest. To begin with we look at a single matching group
(Xkm, Z ) - the index i is dropped for convenience - satisfying
The basic idea is to condition on the observed distribution of the risk factor X resp.
on the empirical distribution of the observed values (xkm) viewed as a vector with
M : = Mo + ... + MK components in Ox The empirical distribution is determined by M the order statistic (ukm) = ord(xkm) E Ox of (xkm) (cf. appendix B1 where the M
components are indexed here by a pair ,,5mU of indices). More precisely, if
(5) < u < . . . U(l) (2) < U(4
represent the J>1 distinct components of (ukm) - and hence of (xkm) - then
the empirical distribution of (xkm) is given by the frequencies
of uO1 among (ukm) resp. (xkm) for j = 1, ..., J. Hence conditioning on the observed
distribution of X is the same as conditioning upon the observed order statistic.
Now consider the random JxM table A = ( A 3 given by the indicators (cf. 3 k
appendix B1)
Conditional likelihood for matched case-control studies 9
and let a = (a. ) be the corresponding observed table (Table 1). Then by appendix 3 km
B3 the conditional distribution of A given ord(Xkm) = u is hypergeometric
with row sums n(u) = (n,(u)), constant column sums 1 = (1) and noncentrality
parameters
= p(X=uO)IY=k, Z = z ) for all j, 5, m.
The corresponding family 8(u) = OR(p(u)) of odds ratios is given by
p ( X = u g ) I Y=k, z = z ) . ~ ( X = U ( ~ ) I Y=O, z = z ) (10) e . 3 km (u) = ~ ( X = U ( ~ ) I Y=k, z = z ) . p ( X = u g ) 1 Y=O, z = z )
= oRk(u@'u(l) I Z ) cf. 2 (13)
and depends only on the odds ratios (3) of interest. Since the hypergeometric
distribution depends only through 8(u) upon p(u) the conditional distribution
4(xkm) I ord(Xkm) = (ukm) ) is obtained as (cf. appendix B3 (8))
(I1) (Xkm) = (xkm) I Ord(Xkm) = (ukm) = A = a 1 ord(Xkm) = (ukm)
= h(aIe(u) ,n ,I)
for (xkm) E ordC1{(ukm)}.
Table 1: The JxM table a = (a. ) using double indices for columns. 3 km
(1)
U(4 C
0 1 . . . . km . . . . K " ~
. . . . a l k m ""
a . . . . . a . 3 01
. . . . a . IKMK 3 km
. . . . a J k m ""
1 . . . . 1 . . . . 1
C
n 1
n . 3
n J
M = n +
Conditional likelihood for matched case-control studies 10
For all m € M k the variables Xkm are independent and identically distributed
according to .d(X I Y = 5, Z=z) and hence (by appendix B4) the hypergeometric
probabilities depend on the table a only through the collapsed table at E IN Jx(l+K)
(Table 2) with
(12) a t = a 3k 3k+ = #{rnlxkm=uO} for all j and 5.
Table 2: The collapsed Jx(1 +K)-Tafel at = ( a t ) 3k '
By appendix B 4 (6), conditioning the collapsed table A+ on ord(Xkm) = (ukm)
yields a hypergeometric distribution
(I3) dA+ I ord(Xkm) = (u~,)) = JX @+I) (p("> I n,") with column sums M = (Mk) and noncentrality parameters
(14) j . 3k (u) = J? (X=U @ I Y = k , Z = z ) for all j and 5.
Again, this hypergeometric distribution depends on p(u) only through the odds
ratios B(u) = OR(~(U) ) given by
(15) s. 3k = oRk(uO,u(l) I "I for all j and 5.
In the case J= 1 the observed values u k m € Ox coincide for all 5 and all m and
hence the conditional distribution of (Xkm) resp. A given ord(Xkm) = (ukm) has a
unit mass and thus contains no information about the odds ratios (3) since
Qj(km)(u) = 1.
Conditional likelihood for matched case-control studies 11
After the principal investigations for a single matching group we now return to all I
matching groups. All terms defined for a single group now receive an additional
index i = 1 . . I referring to the corresponding matching group, i.e. M (uikm) = ord(xikm) m)EOf denotes the observed order statistic of ( x i k m ) k m ~ O x
with Ji distinct components
(16) U i ( l ) < ~ i ( 2 ) < . . . < U . 2 (J; .
Furthermore A. = ( A . . 3 resp. A: = (A?. ) are J x M resp. J . x(l+K) tables z 211% 2 1 1% 2 2
(17) Ai j km = I { X . zkm = u . 20 } resp.
(18) A+ 2 1 1% = A . 2 1 k+ = # { m I X i k m = u i O } , +- + with observed tables a . = (a . ) resp. ai - (a . ) having row sums
2 zkm zkm
(19) 2 1 = # { ( k , m ) I xikm=uiO)} for j = I , ..., J.. 2
The Likelihood for the observed sample (xikm) is given by
using the notation ba from appendix A.1 (7) and
(21) p . . 211% (u.) z = p ( X = u . 2 0) . l Y = k , Z = z . ) 2 for all i , j, 5, m.
Since the likelihood depends on the sample only through the collapsed tables at 2
the familiy of tables (A+) is a sufficient statistic. Conditioning all tables A: on 2 2
the observed order statistic of (X ikm)km yields the corresponding conditional
likelihood which - by (13) - turns out as a product of hypergeometric probabilities
with noncentrality parameters
P3) s. 211% . (u) = ORk(uio, u 2 . (1) 1 z i ) for all i , j and 5.
Denoting the set of all J . x (l+K) tables with row sums n . = (n . .) and column sums 2 2 2 1
M = (Mk) by 2. = q n ., M ) the hypergeometric probabilities may be written as 2 2
(cf. appendix Al)
Conditional likelihood for matched case-control studies 12
As already mentioned a matching group with constant values for the risk
factor, i.e. Ji= 1, does not contribute to the conditional likelihood because A: is
concentrated on a single value. For this reason we exclude these matching groups
when discussing the conditional likelihood which amounts to the assumption J. > 1 2
for all i.
Since conditional likelihood becomes more complex if either the number K of
categories or the row columns Mk increase we only look at the binary case K = 1 in
some detail.
Following usual practice we have assumed that the sample sizes Mk are the same
in all matching groups. However the same arguments apply for group specific sizes
Mik and lead to the conditional likelihood (22) with M. = (Mik) instead of M. 2
5. Case-Control Studies With l:M Matching
Let us now assume that Y E {O,l} is an indicator (eg. for a disease) and look at
case-control studies with 1M matching, which are typically used in epidemiology if
the disease is rare. The index 5=1 for the status will now be omitted in notations
like T ~ ( x , z), ORk(u,v I Z) etc.
For each matching group i = 1, ..., I with given value zi of the matching variable we
now have only one risk factor Xill for the corresponding case and M risk factors
XiOl, ..., XiOM drawn as controls. Using the notation in section 4 we have M = 1, 1
Mo = M, and that the risk variables Xil, Xiom are independently distributed for all
m = 1, ..., M and all i = 1, ..., I such that
for all i, m.
The observed values (x. ) = (x. ..., xioM, x . ) for machting group i can zkm 2 01' 2 1 1 eqivalently be described in terms of their order statistic (uikm) = ord(xikm) with J.
2
distinct components
Conditional likelihood for matched case-control studies 13
and the collapsed Jix2 contingency table a t = ( a t ) (cf. Table 3 ) with entries 2 zkm
(3) a? . 2 1 1% = # { m l x i k m = u . 2 0) j .
The jth row sum n.. in this table is the frequency of u . . in (xikm) %' 20
(4) a? . 2 1 + = n . . 2 1 = # { ( k , m ) I x i k m = u i O }
+- + Table 3: The Jix2 table a . - (aikm) for a matching group i. 2
Ui (1)
Ui
U . z (J;)
C
Given the row and column sums the table a t is uniquely determined by its second
column which will now be abbreviated by ri = ( r . ) E 10, u J i , i.e. 2 1
(5) + - r . . = a , . - a , . = I { x .
2 1 211 2 1 1 z l ~ = ~ i ~ ) for j = 1, ..., J . 2
control case Y=O Y = l
+ a . + r . = a . 2 10 2 1 2 1 1
+ a . . + r . . = a . . 2 1 0 %' 211
+ a . + 2 Ji0 2 J i l r~ = a .
M 1
indicates which component of (u. . ) belongs to the case x i l l . 2 0)
C
n . 2 1
n . . 2 1
n . 2 Ji
M + l = n
If Ri = (R . ) E 10, llJi denotes the corresponding random vector with components 2 1
for j = 1, ..., Ji ,
then by 4 (13) and appendix A3 the conditional distribution of R. given 2
ord(Xi km) = (U . ) is multinomial zkm
The probability vector r ( u 1 zJ is given by its logit transform - cf. 4 (15) 2
Conditional likelihood for matched case-control studies 14
(8) Iogit. 3 r (u i 1 ZJ = log s 3 2 ( u . I zi) - log s1(ui I zi)
= logOR(u. . , u . Iz.) 2 0 2 (1) 2
= d ( u i O l , ~ i ( l ) I ~ i ) .
The expectation of Ri is r(uil ZJ and may be written as
(9) log E(R..) 2 3 = log s ( u . 1 1 2 2 z.) = ai + $ ( u . , u . I z . ) 2 0 2 (1) 2
with
(lo) ai = log E(R. 2 1 ) = log s 1 2 2 (u.1 z.).
The conditional likelihood 4 (22) simplifies in the present situation to a product
of multinomial probabilities
Note that the numbers Ji of classes for the multinomials may vary with i = 1, ..., I. If however at least one component of the risk factor is continuously distributed then
all 1+M components of (X . . ) are distinct (almost surely) and hence Ji = 1+M for 2 3 m
all i. Then R = (R. .) is a IxJ table with J= 1+M whose expected table is given by 2 3
(9).
Using (8) and (9) the log-linear odds ratio model
reduces to a multivariate logistic regression model
(13) T
logi t . r (u. lz . ) = (u . . - u . ) P 3 2 2 2 0 2 (1)
or equivalently to a log-linear model for the expected table E(R)
log E(R..) = ai + (u . . - u . lTP. 2 3 2 0 2 (1)
The (conditional) maximum likelihood estimate for P can be obtained with
standard software by maximizing the conditional likelihood LC. The computation
of the estimate B may be performed as if all entries R . . are independent and 2 3
Poisson-distributed (cf. e.g. Habermann 1974) with expectations given by the
log-linear model (14). In this case the additional nuisance parameters ai (for each
matching group) need to be estimated too. The estimated (asymptotic)
covariance matrix 2 of the estimate B may also be obtained assuming a P
Poisson distribution, at least if the supports OX and OZ of the risk factor and
matching variable are both finite (c.f. Haberman 1974).
Conditional likelihood for matched case-control studies 15
6. Case-Control Studies with 1:1 Matching
For case-control studies with 1:l-matching, i.e. M = 1, further simplifications of the
the results in section 5 are availbale. Now each matching group consists of a pair
(Xi ,, Xi J : = (Xi 01, Xi 11) of independent random vectors
(1) %(Xi,) = L ( X I Y = k , Z = z i ) fork=0,1.
The order statistic of the observation (xio, xiJ is given by
(2) ui = min(x 2 . 0' Xiill , U . 2 2 = max(x. 2 0' Xi 1).
If all "uninformative pairs" with xio = xil are ignored, we have
(3) -
Ui(l) - Uil < Ui2 = Ui(2)
and hence Ji = 2 and nil = ni2 = 1. The pair r . = (r. r . ) is determined by the rank 2 2 1 ' 22
indicator
(4) r . 2 := ril = I{X. 2 0 - < x . 2 1 }
and may be represented as a 2x2 table (cf. Table 4) in which all row and column
totals equal 1
Table 4: The 2x2 table for a matching pair (xio, xil)
The conditional distribution of the rank indicator
(5> R . 2 := Ril = I{Xio<Xil}
is binomial
(6) -qRi l ord(Xi 0, XiJ = (Ui 1, Ui2)) = B(1, +i I zi))
whith probability given by
(7) logit s(ui 1 zi) = +(ui 2, ui 1 z 2 .) .
The two possible values of ri correspond to the following 2x2 tabels
C
1
1
2
risk factor
o, xi l)
max(x 2 0 ' ~ i l ) .
C
control case Y = O Y = l
r . 2 I - r . 2
1 - r 1 r . 2
1 1
Conditional likelihood for matched case-control studies 16
control
The log-linear odds ratio model 5 (12) here reduces to the (univariate) logistic
regression model1
(8) T logit ~ ( u . 1 z . ) = (u. - u . ) p .
2 2 22 2 1
Hence standard software for logistic regression is available for estimating and S testing the paramter vector IR .
A detailed analysis of a real and simulated 1:l matched case-control data using
nonparametric as well as logistic regression models may be found in van der Linde
and Osius (2001).
Appendix A: Multivariate noncentral hypergeometric distributions 17
Appendix A Multivariate Noncentral Hypergeometric Distributions
This section summarized the relevant features (needed in appendix B) of the
multivariate noncentral hypergeometric distribution which will be defined as a
conditional (multivariate) Poisson-distribution.
1. Definition
Let Y = (Y. ) be a JxK contingency table where all components Y. are 3k 3k
independent and Poisson-distributed with positive expectation ,LL. = E(Y. ) so that 3k 3k
Y has a (multivariate) product Poisson-distribution
For a given vector n = (n.) E INJ of row sums we denote the set of all JxK tables 3
having these row sums by
where No = {0) U IN is the set of nonnegative integers and the index ,,+" indicates
summation over the corresponding index. Similar for given column sums K m = (mk) E W let
(3) JXK
g = $ m ) = { y = ( y ) E W 0 3k I y+k=mkfo ra l l k= l , . . . ,K) ,
be the set of all JxK tables with these column sums. Assuming n = m the + +' intersection
(4) X= %n,m) = %(n) n q m ) contains all JxK tables with these margin totals (Table A.l).
The conditional distribution of Y under the condition Y E Wn,m) is a multivariate
noncentral hypergeometric distribution and will be denoted by
(5> HJxK(~ I "w) = dY I Y E %n,m)) .
The hypergeometric probabilities are
for y E 2
Appendix A: Multivariate noncentral hypergeometric distributions 18
Table A.1: The general JxK table y = ( y . ) E Wn,m) with row sums 3k
n = (n .) and column sums m = (m ). 3 k
Using the notation
we get
(8) Yjk . ,Pjk 1 yjk!
for
= P Y . e-'++ . c(y) with
and hence
for y E 2
Note that for constant n = (1) or m = (1) we get yjk E {O, 1) and hence c(y) = 1 for all
y E 2
The hypergeometric probabilities depend on the vector of expectations p = E(Y)
only through its odds ratio familie 8= OR(,u) defined by
(10) 8. = pll ' y j k
3k P1k ' P j l
Indeed, from
p . = a : b . 8 with a . = p . 3k 3 k jk 3 31'
for all j, 5.
Appendix A: Multivariate noncentral hypergeometric distributions 19
we conclude for y E 2
pY = n n ( a . b 8,)YJ"e = eY.na>+. Y+k = g Y . a n . b m j k 3 k 3k j n b k k
and thus
(11) . C(Y)
h(y I P, n, m) = = h(y I 8, n, m) for Y E 2 c eZ.+) z€ 2
Hence the hypergeometric distribution depends on p only through 8
(12) HJxK(p I = H JXK (8 I n,m),
and with no loss in generality we may asssume that 8 = p holds, i.e. yk = p . = I for 31
all j and 5. The parameter 8 is called the noncentrality of HJxK(81 n,m) and is
already determined by its (J-l)x(K-1) subtable (8jk)j,k>l. If 8 . = 1 holds for all j and 3k
5 we get a central hypergeometric distribution.
T For the tranpose Y of Y we get
Hence any result on hypergeometric distributions entails a "dual" result for the
tranposed table.
The hypergeometric distribution may also be derived as a conditional
product multinomial distribution if conditioning for Y is broken up in two steps:
first on the column sums and then on the row sums. Under the condition Y E q m ) the columns of Y are independent with multinomial distributions, i.e.
J with probability vectors T. = ( T ~ ~ ) E (0,l) given by
(15) Tjk = P$ /P+~ for all j, 5.
Suppose Z is a JxK contingency table with product multinomial distribution
then the distribution of Z under the condition Z E %(n) is hypergeometric
= HJxK(" I n ~ ) , since 8 = OR(p) = OR(.lr).
Appendix A: Multivariate noncentral hypergeometric distributions 20
Using (13) we get the corresponding result, if we condition Y upon the row sums
Further properties of hypergeometric distributions can be found in Haberman
(1974, Chapter 1) within the more general framework of conditional Poisson
distributions.
2. Collapsing Over Groups of Co lumns
For a JxL contingency table Y with hypergeometric distribution
the summation of columns with the same noncentrality yields again a
hypergeometric distributed table. More formally, we consider a decomposition of
the L columns into K > 1 groups. Let each group k = 1, ..., K contain (say) Lk > 1
columns and hence L = L For notational convenience we replace the column +' index 1 = 1, ..., L by a pair (k, l) with k = 1, ..., K and 1 = 1, ..., Lk thus writing the JxL
table as Y = (Y. ). $1
We assume that the columns of the noncentrality table p= (,L. ) are constant $1
within each group k and put
for all j, 5, 1.
Collapsing over groups, i.e. summing within all groups, may be described by the
matrix operator
+- JxK Y = (yjk1) lRJxL y - (Yjk+) E ,
and y+ will be called the collapsed table.
We now show that the collapsed JxK contingency table Y+ has a hypergeometric
distribution
(4> qy+) = HJrK(b I nim+)
+ where m is the collapsed vector of column sums
(5) m+ = (rnk+).
To prove (4) we view the distribution of Y according to 1 (17) as a conditional
product multinomial distribution
(6) q Y ) = HJd(* I n,m) = 4~ I Z E 3) with
Appendix A: Multivariate noncentral hypergeometric distributions 21
and
(8) T j k ~ = pjk~lp+k~ for all j, 5, 1.
To establish (4) it suffides to shown
(9) 4 ~ ' I Z E 3) = HJxK(,ii I n,mt).
By (2) the probability vectors kl coincide within each group k
(lo) T,, := t. 1k = b. 1k //I +I for all jand I .
Hence the kth column Z; =(Zlk+, ..., Z ) of the collapsed table Z+ has a Jk+
multinomial distribution
(11) 4 2 ; ) = M&mk+>" k ) .
Thus the collapsed table Z+ is a product multinomial
and using 1 (17) yields
(13) + + 4 Z 1 Z E 3) = HJxK(i 1 n,mt)
= HJxK(b I nn,m+) , since OR(i) = OR@).
+ Because Z and Z have the same row sums we get
(15) Z + E % o Z E % ,
and hence (9) follows from (13).
3. Special Case: T h e Mul t inomia l Distr ibut ion
In the case with only K= 2 columns a Jx2 contingency table Y = (Y. ) (Table A2) ~k
with a hypergeometric distribution
(I) B Y ) = HJx2(e 1 n,m)
is already determined by its second column Y. = (Yj2), because Y E %(n) implies
(2) Y. = n . - Y . 11 1 12 for all j.
Appendix A: Multivariate noncentral hypergeometric distributions 22
Table A.2: The general J x 2 table y = (y . ) E Wn,m) with row sums 3k
n = (n .) and column sums m = (m m ). 3 1' 2
Hence the hypergeometic distribution is determined by the (marginal) distribution
2(Y. 2) of the second column. Let us additonally assume, that all row totals are not
smaller that the second column total
(3) n . > m 2 3 - for all j,
which always holds for example in the case m2 = 1. Then the second column has a
multinomial distribution
(4) J(Y. 2) = MJ(m27. 2) provided (3) holds, J with probability vector T . ~ = ( ~ ~ 2 ) E (0,l) given by
(5> T . 32 = 19. 32 / e +2 for all j.
With no loss in generality we may assume i3= OR(i3), i.e.. elk = 19. = 1 for all j and 31
5. Using the representation 1 (17)
@I dz I %) = HJx2(e I sm) with
2
(7) g z ) = n ~ ~ ( m ~ , r . ~ ) , k = l
the result (4) will follow from
(8) d Z . 2 1 Z ~ % ) = MJ(m2i?r2).
In view of (7) it suffices to prove
Appendix A: Multivariate noncentral hypergeometric distributions 23
for any a E IN; with a = m2. First, for any z E 3 we get + (10) P { Z . , = z . , l Z ~ 3 1
= P { Z . ~ = Z . ~ I Z . ~ = n-z . 2 ' Z E INF~} = P { Z . ~ = Z . ~ ~ Z . , = n - z . 2 ' z E i q K }
= P{Z.2=z.21Z.l= z . I ' Z E INF~}, since z E 3
= P{Z.2=z.21Z.1= z . l , Z E IN;*}
- = P{Z.2=~ .21Z .1 - z . J , since z E 3
= p{z.2=z.2},
where the last step exploits the independence of the columns Z. and Z. 2.
To establish (9) define for any a~ IN; with a = m a Jx2 table z having the + 2 columnsz . 2 = a a n d z l = n - a . F r o m a . < m and(3) weget
3- 2
z. = n . - a , > n.-m > 0 for all j, 31 3 3 - 3 2 -
and hence z E 3. Thus (9) follows from (10).
The probability vector is uniquely determined by its J- 1 logits
(11) logit 3 (T . 2 ) = log T . 32 - log s12 for j = 2, ..., J,
and hence we get the representation
(12) logit = log 0. , 1.e. logit 3 .(T . 2 ) = log 19. 32 for j = 2, ..., J.
Thus ?r2 is uniquely determined by B and vice versa.
Appendix B: Joint distribution of the rank and order statistic 24
Appendix B Rank and Order Statistics
The purpose of the section is to derive the joint distribution of the rank and the
order statistic for a sample of random vectors with respect to lexicografical S S ordering 5 in R . For simplicity we put R = R , let d denote the a-algebra of
S Borel-sets in R and "forget" the special euclidean nature of the totally ordered set
R except that all intervals with respect to the ordering should be measurable, (i.e.
members of 4. In fact, the following consideration hold for any such totally
ordered structure (0, @ 5).
For fixed K E IN we start with a the definition of the rank and order statistic for
K-dimensional samples x E fl which allows tied observations. Then we derive the
joint (and conditional) distribution of the rank and order statistic of a random
vector X having an arbitrary distribution (dominated by a product measure).
Furthermore we look at typical situations with independent and partially
identically distributed components of X.
1. Definitions
The rank statistic of a vector x = (xl, ..., xK) E fl is a permutation rk(x) = Q on X
(1, ..., K} given by
(1) eX(k) = #{i lxi<xk} + #{i l i < k , xi=. L } for k = 1, ..., K.
As usual a < b stands for (a 5 b and a s b) which for a total ordering is equivalent to
not b 5 a. The fundamental properties are
PI xk < x1 * ex(k] < ex(L) ,
(3) xk = x~ + ( ~ , ( k ) < eX(l) * 6 < 1 ) for all 5, 1.
Let YK denote the set of all permutations on {I, ..., K}. For any D E YK the index
permutation is a map IIo : fl+ fl defined by
o o II (x) = II (xl, ..., xK) = (x o(k) )k = 1, ..., K .
The order statistic of x is defined as
where a is the inverse rank statistic of x. The components of ord(x) are X X
ordered
x < x < . . . . < x res p. x < x < . . . . < x ox(1) - ox(2) - - ox(K) PI- PI- - [Kl
Appendix B: Joint distribution of the rank and order statistic 25
using the common notation [ k ] := ox(k). If x has no ties (i.e. all components of x
are distinct) then (5) uniquely defines the (inverse) rank statistic and may serve as a
definition of the rank statistic. However in the presence of ties the rank statistic ex is no longer determined by (5). Furthermore by (3), the indices of tied components
in x have the same ordering in ord(x) as in x itself.
Example: For x = (2, 3, 1, 2) we get ord(x) = (1, 2, 2, 3) and QX(l) = 2, QX(2) = 4,
Q (3) = 1, Q (4) = 3. Here we have x = x = 2 and Q preserves the ordering of the x x 1 4 x
indices 1 and 4, i.e. 1 < 4 implies QX(l) < ~ ~ ( 4 ) .
Any x E fl is uniquely determined by its rank ex E YK and order statistic
(7) ord(x) E 4 - = { x ~ f l l xl<x2< . . .< xK}
since
Hence the mapping (rk, ord): fl+ YKx fl is injective (one-to-one) but not I surjective (onto) because its range
(9) 93 := {(rk(x), ord(x)) I x E fl} = { ( Q , U ) E Y ~ X ~ ~ for all k , lwi thuk=ul : k < 1 u e(k)<e(l)}
-
is not the whole space YK x fl. In particular, if all components of u coincide, we I have ( Q , u) E 93 if and only if Q is the identity. But if u has no ties, then ( Q , u) E 93 holds for any permutation Q.
The inverse mapping H : 9 3 1 fl of (rk, ord) is given by
(10) H(Q, 4 = lIe(u) for (Q, u) E 93.
For a fixed ordered vector u = (uk) E 4 any vector x E fl with ord(x) = u will
now be characterized using a binary contingency table. First let
(11) < u < . . . U(l) (2) < denote the J> 1 distinct components of u, and
(12) n . 3 = n(u) 3 = # { k I uk = uD1} for j = 1, ..., J
the frequency of u in u. For any x E fl with ord(x) = u we defined the J x K Ci> table a(x) = (a. (x)) of indicators
3k
Appendix B: Joint distribution of the rank and order statistic 26
Now ord(x) = u implies for the row resp. column sums of a(x)
(14) a . (x) = n . a (x) = 1 for all j, 5 3+ 3 +k
and hence a(x) lies in the following set of contingency tables
(15) JxK = { ( a k ) ~ { O , 1 } la . = n . f o r a l l j , a = l f o r a l l k } .
3+ 3 +k
Thus the assignment x H a(x) defines a mapping
ordC1{u) = { x ~ f l l ord(x) = u ) + 2
which is injective. Conversely, for any table a E 2 the vector x(a) = (xk(a))E fl is
defined
Now a = 1 implies that there is a unique j such that a . = 1 (the other entries in +k 3k
the kth row are 0). Hence the product in (16) reduces to a single factor which gives
(17) x ( a ) = u L u a . = I . O 3k
The frequency of u in x(a) is n . because a . = n . and hence Ci> 3 3+ 3
By (17) the assignment a H x(a) is inverse to x H a(x) and thus both mappings
are bijections. Hence any vector x E ordC1{u) may also be viewed as a JxK
contingency table a(x) E 2
2. Jo in t And Condit ional Distribution
A "random vector" X = (XI, ..., XK) taking values in fl is uniquely determined by
its rank and order statistic rk(X) and ord(X). To derive the joint distribution of the
rank and order statistic we assume that X has a density p(x) =p(X = x) with
respect to some product measure uK arising from a 0-finite measure u on d We
will provide a joint densitiy of (rk(X), ord(X)) on its support BC Y K x f l with
respect the product measure u xuK, u being the counting measure on YK # #
We first observe that for any permutation Q E the measure induced by u K
under the mapping lIe : fl+ fl concides with u .
Appendix B: Joint distribution of the rank and order statistic 27
Indeed, for arbitrary All ..., AK E d we have
with -1 ( ne ) - l [n~k] = I-I a = @ k k
which proves (1) in view of
The joint distribution of rk(X) and ord(X) is determined by the following
probabilities for any Q E YK and A E with {Q)XA c B
K = J p ( X = x ) u (dx)
lie [A1 K Q-I -1
= J P(X = lIe(x)) (u (n 1 )(dx) A
= J p(X = lIe(x)) uK(dx) by (1). A
Let C c YKx fl be any (measurable) set such that C(Q) = { u 1 (Q, u) E C) E dK for
all Q E YK Then
This implies that the distribution of (rk(X),ord(X)) with support B has the
following density with respect u K # x u
(4) for (Q,u) E B
p(rk(X) = Q, ord(X) = u) = for (e,u) Sf B.
The conditional distribution of the rank rk(X) given ord(X) = U E fl has I support
and is given by
Appendix B: Joint distribution of the rank and order statistic 28
(6) P{rk(X) = Q 1 ord(X) = u ) = P(X = fle(4)
for Q E T(u). C P(X = f l T ( ~ ) )
Given the order statistic ord(X) the random vector X is uniquely determined by
its rank rk(X). Hence the conditional distribution of X given ord(X) = u has (finite)
support
and (6) implies
p ( ~ = nrk(x)(u)) (8) P { X = x l o r d ( X ) = u ) = for ord(x) = u.
C p(X=flrk(v)(u)) v E ord-l{u}
Using the bijection a(-): ordC1{u) + %'we finally obtain from (8) the conditional
distribution of the table a(X)
(9) P{a(X) = a(x) I ord(X) = ord(x) ) = P{X = x I ord(X) = ord(x) ) .
3. Independen t Componen t s
We now look at the special case where all components X1, ..., XK of X are
independent. If p(Xk=x) denotes the density (with respect to u) of Xk the joint
density of X is
for x = (x,) E fl.
For fixed u E 4 and arbitrary Q E T(u) we get -
K
PI P ( ~ = '@(,)) = n P ( ~ ~ = 1' k=l
For x E ordC1{u) with rk(x) = Q we consider the table a(x) =(a. (x)) E %' from 1 3k
(12) and the table p(u) = (p. (u)) given by 3k
(3) p . 3k (u) = p ( X - u . ) k - b) where u are the distinct componente of u. Then (2) may be expressed as
Ci> K J
(4) p ( ~ = ILQ(U)) = n n p ( ~ k - - 0) . )$(x) = P(U) a(x) , ~ f . A1(7), k=l j=1
and the conditional probabilities 2 (8) and (9) reduce to
Appendix B: Joint distribution of the rank and order statistic 29
Using appendix A1 we conclude that the conditional distribution of the
contingency table a(X) given the condition ord(X) = u is hypergeometric
(7> .d(a(X)lord(X)=u) = H JxK (p(u)~n(u) , l ) .
The row sums n(u) = (nj(u)) are the frequencies from 1 (12) and the column sums
are constant =l. Thus the conditional distribution .d(X I ord(X) = u) corresponds
(up to a bijection of its support) to the hypergeometric distribution (7), i.e.
= h(a(x) I ~ ( 4 , n(u),l) for x E ordC1{u).
4. Ident ical Repl icat ions
Suppose now that we have b = 1, ..., K independent samples each consisting of Mk
random elements X k m ~ X which are independent and identically distributed
replications of a random element, say Xk, SO that
(1) .d(xkm) = .d(xk) for all m = 1, ..., M ~ ,
with all Xkm being independent. Viewing X = (Xkm) as a random vector of length
M = Ml + ... + MK with independent components the results from section 3 apply. In
particular the JxM tables a(x) =(a. (x)) and p(u) are defined by 3 km
(1) a . 3 km ( X ) = I { X ~ ~ = U , ) , pj km(u) = dXkm = u(j) )
Using the matrix operator from appendix A2 (4) we get
with p(u) = (jjk(u)) and the collapsed JxM table a+(x) = ( a t (x)) given by 3 k
Appendix B: Joint distribution of the rank and order statistic 30
(3) P 3 k (4 = p(Xk = uO) ) ,
(4) + a . (x) = a . (x) = #I{mIxk,=u,}. 3 k 3 k+
Hence the hypergeometric probabilities 3 (6) depend on a(x) only through the
collapsed table at(x). Instead of the conditional distribution of a(X) given
ord(X) = u
(5) 44x1 I ord(X) = u) = HJXM(p(~) I 4 ~ ) ,I)
we might equally well consider the conditional distribution of at(x) which is
hypergeometric by appendix A2
@I qat(~) I ord(X) = U) = HJxK(P(u) I "(u) W) .
The connection between the hypergeometric distributions (5) and (6) is given by
(7) P{~+(x) = a+(x) I ord(X) = u)} = d(x). P{a(X) = a(x) I ord(X) = u)}
for any x E X w h e r e the number
@I d(x) = # { z E XI ord(z) = u , at(z) = a+(x) }.
does not depend on the noncentralities.
References
Breslow, N.E. and Day, N.E. (1980). Statistical Methods i n Cancer Research, Volume I: The Analysis of Case-Control Studies. International Agency for Research on Cancer, Lyon.
Haberman, S.J. (1974). The analysis of frequency data. The University of Chicago Press, Chicago and London.
Liddel, F.D.K., McDonald, J.C. and Thomas D.C. (1977). Methods of Cohort Analysis: Appraisal by Application to Asbestos Mining. J.R.Statist.Soc. A, 140, 469-491.
Osius, G. (2000). The association between two random elements: A complete characterization i n terms of odds ratios. Mathematik-Arbeitspapiere No. 53, Universitat Bremen (http://www.math.uni-bremen.de/"osius/download).
van der Linde, A. and Osius, G. (2001). Estimation of nonparametric multivariate risk functions i n matched case-control studies: with application to the asessment of interactions of risk factors i n the study of cancer. To appear in Statistics i n Medicine.
Date: 29-Jan-2001 (printed edition) 7-Feb-2000 (PDF-File, with minor corrections)
Universität Bremen Mathematik-ArbeitspapiereStand Januar 2001 ISSN 0173 - 685 X
Vertrieb der Hefte 4, 14, 23, 26 durch Universitätsbuchhandlung, Bibliothekstr. 3, D-28359 Bre-men. Vertrieb der übrigen Hefte (soweit nicht vergriffen) durch die Autoren oder FB 3 Mathema-tik/Informatik Universität Bremen, Postfach 330440, D-28334 Bremen.
1. Ulrich Krause (1976): Strukturen in unendlichdimensionalen konvexen Mengen, 74 S.
2. Fritz Colonius, Diederich Hinrichsen (1976): Optimal control of heriditary differential systems.Part I, 66 S.
3. Günter Matthiessen (1976): Theorie der heterogenen Algebren, 88 S.
4. H. Wolfgang Fischer, Jens Gamst, Klaus Horneffer (1976): Skript zur Analysis, Band 1 (11.Auflage 2000), 286 S.
5. Wolfgang Schröder (1977): Operator-algebraische Ergodentheorie für Quantensysteme, 59S.
6. Rolf Röhrig, Michael Unterstein (1977): Analyse multivariabler Systeme mit Hilfe komplexerMatrixfunktionen, 216 S.
7. Horst Herrlich, Hans-Eberhard Porst, Rudolf-Eberhard Hoffmann, Manfred Bernd Wisch-newsky (1976): Nordwestdeutsches Kategorienseminar, 193 S.
8. Fritz Colonius, Diederich Hinrichsen (1977): Optimal Control of Hereditary Differential Sy-stems. Part II: Differential State Space Description, 36 S.
9. Ludwig Arnold (1977): Differentialgleichungen und Regelungstheorie, 185 S.
10. Rudolf Lorenz (1977): Iterative Verfahren zur Lösung großer, dünnbesetzter symmetrischerEigenwertprobleme, 104 S.
11. Konrad Behnen, Hans-Peter Kinder, Gerhard Osius, Rüdiger Schäfer, Jürgen Timm (1977):Dose-Response-Analysis, 206 S.
12. Hans-Friedrich Münzner, Dieter Prätzel-Wolters (1978): Minimalbasen polynomialer Moduln,Strukturindizes und BRUNOVSKY-Transformationen, 53 S.
13. Konrad Behnen (1978): Vorzeichen-Rangtests mit Nullen und Bindungen, 53 S.
14. H. Wolfgang Fischer, Jens Gamst, Klaus Horneffer, Eberhard Oeljeklaus (1978): Skript zurLinearen Algebra, Band 1 (13. Auflage 2000), 249 S.
15. Günter Ludyk (1978): Abtastregelung zeitvarianter Einfach- und Mehrfachsysteme, 54 S.
16. Momme Johs Thomsen (1977): Zur Theorie der Fastalgebren, 146 S.
17. Klaus Horneffer, Horst Diehl (1978): Modellrechnungen zur anaeroben Reduktionskinetik desCytochroms P-450, 34 S.
18. Horst Herrlich, Rudolf-Eberhard Hoffmann, Hans-Eberhard Porst, Manfred BerndWischnewsky (1979): Structure of Topological Categories, 252 S.
19. Hans-Friedrich Münzner, Dieter Prätzel-Wolters (1979): Geometric and moduletheoretic ap-proach to linear systems. Part I: Basic categories and functors, 28 S.
20. Hans-Friedrich Münzner, Dieter Prätzel-Wolters (1979): Geometric and moduletheoretic ap-proach to linear systems. Part II: Moduletheoretic characterization and reachability, 28 S.
21. Eckart Beutler, Hans Kaiser, Günter Matthiessen, Jürgen Timm (1979): Biduale Algebren,165 S.
22. Horst Diehl, Detlef Harbach, Jürgen Timm (1980): Planung und Auswertung von Atomabsorp-tions-Spektrometrie-Untersuchungen mit der Additionsmethode, 44 S.
23. H. Wolfgang Fischer, Jens Gamst, Klaus Horneffer (1981): Skript zur Analysis, Band 2 (7.Auflage 2001), 299 S.
24. Horst Herrlich (1981): Categorical Topology 1971-1981, 105 S.
25. Horst Herrlich, Rudolf-Eberhard Hoffmann, Hans-Eberhard Porst, Manfred BerndWischnewsky (1981): Special Topics in Topology and Category Theory, 108 S.
26. H. Wolfgang Fischer, Jens Gamst, Klaus Horneffer (1984): Skript zur Linearen Algebra, Band2 (7. Auflage 1999), 257 S.
27. Rudolf-Eberhard Hoffmann (1982): Continuous Lattices and Related Topics, 314 S.
28. Horst Herrlich, Rudolf-Eberhard Hoffmann, Hans-Eberhard Porst (1987): Workshop on Cate-gory Theory, 169 S.
29. Harald Boehme (1987): Zur Berufspraxis des Diplommathematikers, 16 S.
30. Jürgen Timm (1986): Mathematische Modelle der Dosis-Wirkungsanalyse bei den experimen-tellen Untersuchungen der Arbeitsgruppe zur karzinogenen Belastung des Menschen durchLuftverunreinigung, 65 S.
31. Dieter Denneberg (1988): Mathematik für Wirtschaftswissenschaftler. I. Lineare Algebra, 97S.
32. Peter E. Crouch, Diederich Hinrichsen, Anthony J. Pritchard, Dietmar Salamon (1988, previ-ous edition University of Warwick 1981): Introduction to Mathematical Systems Theory, 244S.
33. Gerhard Osius (1989): Some Results on Convergence of Moments and Convergence in Dis-tribution with Applications in Statistics, 27 S.
34. Dieter Denneberg (1989): Verzerrte Wahrscheinlichkeiten in der Versicherungsmathematik,Quantilsabhängige Prämienprinzipien, 24 S.
35. Eberhard Oeljeklaus (1989): Birational splitting of homogeneous Albanese bundles, 30 S.
36. Gerhard Osius, Dieter Rojek (1989): Normal Goodness-of-Fit Tests for Parametric MultinomialModels with Large Degrees of Freedom, 38 S.
37. Dieter Denneberg (1990): Mathematik zur Wirtschaftswissenschaft. II. Analysis, 59 S.
38. Ulrich Krause, Cornelia Zahlten (1990): Arithmetik in Krull monoids and the cross number ofdivisor class groups, 29 S.
39. Dieter Denneberg (1990): Subadditive Measure and Integral, 39 S.
40. Ulrich Krause, Peter Ranft (1991): A limit set trichotomy for monotone nonlinear dynamicalsystems, 31 S.
41. Angelika van der Linde (1992): Statistical analyses with splines: are they well defined? 22 S.
42. Dieter Denneberg (1992): Lectures on non-additive measure and integral (new edition: Non-additive measure and integral. TDLB 27, Kluwer Academic, Dordrecht (1994)), 114 S.
43. Gerhard Osius (1993): Separating Agreement from Association in Log-linear Models forSquare Contingency Tables With Applications, 23 S.
44. Hans-Peter Kinder, Friedrich Liese (1995): Bremen-Rostock Statistik Seminar, 5. - 7. März1992, 110 S.
45. Dieter Denneberg (1995): Extension of a measurable space and linear representation of theChoquet Integral, 30 S.
46. Dieter Denneberg, Michael Grabisch (1996): Shapley value and interaction index, 20 S.
47. Angelika Bunse-Gerstner, Heike Faßbender (1996): A Jacobi-like method for solving alge-braic Riccati equations on parallel computers, 24 S.
48. Hans-Eberhard Porst editor (1997): Categorical methods in algebra and topology - a collecti-on of papers in honour of Horst Herrlich, 498 S.
49. Angelika van der Linde, Gerhard Osius (1997): Estimation of nonparametric risk functions Inmatched case-control studies, 28 S.
50. Angelika van der Linde (1997): Estimating the smoothing parameter in generalized spline-based regression, 46 S.
51. Ursula Müller, Gerhard Osius (1998): Asymptotic normality of goodness-of-fit statistics forsparse Poisson data, 15 S.
52. Ursula Müller (1999): Nonparametric regression for threshold data, 18 S.
53. Gerhard Osius (2000): The association between two random elements – A complete charac-terization in terms of odds ratios, 32 S.
54. Horst Herrlich, Hans-E. Porst (2000): CatMAT 2000, Proceedings of the Conference: Catego-rical Methods in Algebra and Topology, 490 S.
55. Gerhard Osius (2001): A formal derivation of the conditional likelihood for matched case-control studies, 30 S.
top related