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Taylor & Francis, Ltd.
American Statistical Association
Hierarchical Models for the Probabilities of a Survey
Classification and Nonresponse: An
Example from the National Crime Survey
Author(s): Elizabeth A. Stasny
Source: Journal of the American Statistical Association, Vol.
86, No. 414 (Jun., 1991), pp. 296-
303
Published by: on behalf of the Taylor & Francis, Ltd.
American Statistical Association
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Hierarchical Models for the Probabilities of a Survey
Classification and Nonresponse: An Example
From the National Crime Survey
ELIZABETH A. STASNY*
A goal in many survey sampling problems is to estimate the
probability that elements of the population within various small
areas or domains have some characteristic or fall in some
particular survey classification. The estimation problem is
typically
complicated by nonrandom nonresponse in that the probability
that a unit responds to the survey may be related to the char-
acteristic of interest. This article presents a random parameter
or hierarchical model approach to modeling the small-domain
probabilities of the characteristic of interest and the
probabilities of nonresponse. The general model allows
nonresponse prob-

abilities to depend on a unit's survey classification. A special
case of the model treats nonresponse as occurring at random.
Empirical Bayes methods are used to obtain parameter estimates
under the hierarchical models. The method is illustrated using
data from the National Crime Survey.
KEY WORDS: Categorical data; Empirical Bayes; Nonrandom
nonresponse; Small-domain estimation.
1. INTRODUCTION
Data from large-scale sample surveys are often used to
estimate the probability that an individual falls into a par-
ticular survey classification or has a certain characteristic.
For example, data from the National Crime Survey (NCS)
are used to estimate the probability of being victimized, and
data from the Current Population Survey are used to esti-
mate the probability of being unemployed. It is often of
interest to obtain such estimates for subgroups of the pop-
ulation or small domains, such as neighborhoods or age/
sex/race groups, as well as for the entire population. If few
data are available for a particular domain, then it may be
difficult to obtain an accurate estimate of the desired prob-
ability within that domain. The problem of small-domain
estimation has become an important concern for survey or-
ganizations, as evidenced by recent publications such as that
by Platek, Rao, Sarndal, and Singh (1987). In addition to
the problem of small sample sizes, a further complication
is that not all sampled units respond to a survey and the
probability that a sampled unit responds may be related to
the survey classification of that unit. The work presented
here addresses this problem of estimating probabilities in
population subgroups in the presence of possibly nonran-
dom nonresponse.
This article presents hierarchical models for the proba-

bilities of the classification of interest and the probabilities
of response within subgroups of the population. Under these
hierarchical models, we think of the probabilities that in-
dividuals within subgroups have the characteristic of inter-
* Elizabeth A. Stasny is Assistant Professor, Department of
Statistics,
The Ohio State University, Columbus, OH 43210. The National
Crime
Survey data used in this article were made available by the
Inter-Uni-
versity Consortium for Political and Social Research. The data
were orig-
inally collected by the United States' Law Enforcement
Assistance
Administration. The longitudinal data set used here was created
by the
Bureau of Justice Statistics, using the quarterly public-use data
files. This
research was supported in part by a grant from the Bureau of
Justice
Statistics, U.S. Department of Justice, and the Committee on
Law and
Justice Statistics, American Statistical Association, which
permitted the
author to attend two workshops on the design and use of the
National
Crime Survey. The author takes sole responsibility for the work
presented
in this article.
est or respond to the survey as belonging to distributions
of such probabilities. The advantage to using such models
is that information from the entire sample may be used to
estimate parameters of the distributions of probabilities and
hence information from the entire sample is used to esti-

mate the probabilities for a single subgroup (see, for ex-
ample, Morris 1983). Such hierarchical models have been
proposed in the survey sampling context by Lehoczky and
Schervish (1987), who worked with data from the NCS and
modeled the distribution of the probabilities of victimiza-
tion as a beta distribution.
In this article, we extend the model for victimization
probabilities proposed by Lehoczky and Schervish to allow
for nonresponse. The models presented here allow the non-
response probabilities to come from a single distribution,
which corresponds to random nonresponse, or from two
distributions depending on the presence or absence of the
characteristic of interest, which corresponds to informative
or nonrandom nonresponse. Although the models are de-
veloped in the context of estimating probabilities of victim-
ization and are fit to data from the NCS, they are applicable
to surveys other than the NCS.
Section 2 of this article presents a brief description of the
NCS. The general hierarchical model for probabilities of a
survey classification and nonrandom nonresponse is pre-
sented in Section 3. The special case of the model corre-
sponding to random nonresponse is presented in Section 4.
In Section 5 the models are fit to simulated data, which
were generated based on probabilities obtained from NCS
data, and to actual NCS data. Conclusions and areas for
future research are presented in Section 6.
2. THE NATIONAL CRIME SURVEY
2.1 The Survey Design
The NCS is a large-scale, household survey conducted
by the U.S. Bureau of the Census for the Bureau of Justice

? 1991 American Statistical Association
Joumal of the American Statistical Association
June 1991, Vol. 86, No. 414, Applications and Case Studies
296
2015 20:08:52 UTC
Stasny: Hierarchical Models for Survey and Nonresponse
Probabilities 297
Statistics. Data from the NCS are used to produce quarterly
estimates of victimization rates and yearly estimates of the
prevalence of crime. The survey uses a rotating panel of
housing units (HU's) under which members of households
(HH's) living in sampled HU's are interviewed up to seven
times at six-month intervals. Individuals interviewed for the
NCS are asked about crimes committed against them or
against their property in the previous six months. The sur-
vey covers the following crimes and attempted crimes: as-
sault, auto or motor vehicle theft, burglary, larceny, rape,
and robbery. Crimes not covered by the NCS include kid-
napping, murder, shoplifting, and crimes that occur at places
of business. Additional information on the design and his-
tory of the NCS is provided, for example, by the U.S. De-
partment of Justice and Bureau of Justice Statistics (1981).
In this article we consider models for estimating the
probability that anyone within an HH reports at least one
victimization of any type for the previous six-month period.

The models also allow for the nonresponse in the NCS data.
Previous work with NCS data suggests that nonresponse
does not occur at random with respect to victimization sta-
tus [see, for example, Saphire (1984) and Stasny (in press)].
Thus the hierarchical model described in Section 3 allows
the probability that an HH responds to depend on the vic-
timization status of the HH. A random nonresponse model
described in Section 4 will be fit for the purpose of com-
parison.
2.2 The Data
The data used in this work are from a large, longitudinal
data set that includes all of the regular NCS interview in-
formation collected from January 1975 to June 1979, ex-
cept for the HU's that rotated into the sample in 1979. To
make it easier to handle the data, this article uses only a
subset of this large data set. The subset was created by tak-
ing a random start at the record for the eighth HU in the
full data set and then every fifteenth record after that. Be-
cause the HU's on the original longitudinal file are ordered
in such a way that units from the same cluster appear to-
gether, the 1-in-15 systematic sample should not include
two or more HU's from a single cluster. Thus this article
does not consider the problem of correlations among HU's
within clusters.
It should be noted that during the time when the data
were collected, a reference-period experiment was being
conducted using a sample of NCS HU's. Since individuals
in HU's included in the experiment were asked to report
victimizations for reference periods other than the usual six-
month period, those HU's were not used in the analyses
presented here.

The models of Sections 3 and 4 are fit to the NCS data
collected in the first half of 1975. The data are poststratified
into domains according to three neighborhood characteris-
tics: (a) urban and rural, (b) central city, other incorporated
place, and unincorporated or not a place, and (c) low pov-
erty level (9% or fewer of families below poverty level)
and high poverty level (10% or more of families below pov-
erty level). Since it is practically impossible for a rural area
to be a central city, this poststratification results in 10 do-
mains. The NCS data summarized according to these 10
domains are shown in Table 1.
Note that the sample sizes within 8 of these 10 domains
are fairly large. For those domains, reasonable estimates of
the probability of victimization within the domain might be
obtained using only the data from that domain. In practice,
domains of interest would most likely be much smaller than
those defined here and the corresponding sample sizes within
each domain would also be smaller. In cases where sample
sizes within domains are small, the empirical Bayes pro-
cedure, which allows us to borrow information from the
entire sample to estimate probabilities in small domains,
may provide more accurate estimates within domains than
do standard procedures. We use the larger domains here for
illustrative purposes.
3. THE GENERAL HIERARCHICAL MODEL
This section presents a general form of the hierarchical
model for the probabilities of having a particular survey
Table 1. National Crime Survey Data: From January 1975 to
June 1975
Naive Random Nonrandom
NCS data estimator nonresponse nonresponse

Domain* Y,+ - Z+ Zj+ ni - Y,+ pi 7Ti Pi ri A iij, rjo
U/C/L 555 156 104 .219 .872 .217 .873 .272 .689 .937
U/C/H 364 95 73 .207 .863 .205 .869 .265 .684 .937
U/I/L 557 162 101 .225 .877 .222 .876 .276 .692 .937
U/I/H 262 72 36 .216 .903 .212 .885 .254 .694 .937
U/N/L 297 92 79 .237 .831 .230 .855 .305 .679 .937
U/N/H 40 15 9 .273 .859 .228 .872 .287 .687 .937
R/l/L 36 1 1 7 .234 .870 .210 .873 .265 .687 .937
R/l/H 105 10 20 .087 .852 .130 .870 .185 .682 .937
R/N/L 274 35 32 .113 .906 .129 .886 .166 .687 .937
R/N/H 413 79 64 .161 .885 .165 .879 .213 .686 .937
NOTE: Yj+ number responding in domain i, Yj+ - Zi+ number
reporting crime free in domain i, Zj+ = number reporting
victim-
izations in domain i, and ni - Yj+ = number of nonrespondents
in domain i.
*Poststratified into domains by neighborhood characteristics
urban or rural (U or R), central city, other incorporated place,
or un-
incorporated or not a place (C, 1, or N), and low or high poverty
level (L or H).
2015 20:08:52 UTC
298 Journal of the American Statistical Association, June 1991
classification and of responding to the survey. The model
is an extension of the hierarchical model for victimization

probabilities proposed by Lehoczky and Schervish (1987).
We will refer to the sampled units as "individuals," al-
though, for our example, an HH is the responding unit. We
will assume that the sample is chosen using a stratified ran-
dom sampling plan and that the goal is to estimate the prob-
abilities that individuals within each domain have the char-
acteristic of interest. In the development of the hierarchical
model, we use the context of the NCS and let the charac-
teristic or survey classification of interest be whether or not
the individual reported being victimized. The model, of
course, is applicable to surveys other than the NCS.
3.1 Model for the Observed Data
Suppose that the population of interest has been divided
into K domains. We assume that individuals within a single
domain have a common probability of being victimized but
individuals in different domains may have different prob-
abilities. Let pi be the probability that an individual in the
ith domain is victimized. We will model the distribution of
the pi as a beta distribution with parameters a and b. That
is, we will assume that
iid
Pi llid beta(a , b) for i = I1, 2, . . .,9 Kg
so that the pi have density function
((pi I a, b) = [B(a, b)]-pa1( -
where B(a, b) = F(a)F(b)/F(a + b) is the complete beta
function.
Within each domain, suppose that we take a random
sample of ni individuals and observe the survey classifi-
cation of each individual. Thus we observe, say,

Xij = 1 if the jth individual in the ith domain is victimized
= 0 otherwise.
We assume that the victimizations within a domain are con-
ditionally independent of each other given the probability
of victimization within the domain. Thus
Xil, Xi2, . . ., Xini I Pi
d
Bernoulli (pi)
fori=1, 2, ...,K.
Naturally, we observe the Xij only for individuals who
respond to the survey. Let us denote the response status for
a sampled individual by Yij, where
Yij = 1 if the jth individual in the ith domain responds
= 0 otherwise.
We denote summary counts for the observed data as fol-
lows: Y1+ is the number of respondents in domain i, ni-
Y1+ is the number of nonrespondents in domain i, Zi+ -
Xjn'> XijYij is the number of responding victims in domain
i, and Y1+ - Z1+ is the number of crime-free respondents
in d omain i.
We now add the hierarchical model for the response
probabilities to the hierarchical model for the victimization
probabilities. We will allow the probability of nonresponse
to differ by domain and by victimization status. For i = 1,
2, . .., K, let 7rij be the probability a victimized individual

in domain i is a respondent (i.e., Xij = 1) and let 7TiO be the
probability a crime-free individual in domain i is a respon-
dent (i.e., Xij = 0). We also model the distributions of the
7rij and 1Tjo probabilities as beta distributions. In particular,
we assume that, given Xij, the 7nij and 7Tio for i = 1, 2, ...,
K are random samples from beta distributions with param-
eters a1 and 131, and a0 and (30, respectively. That is,
7i I X = v i beta(av, Pv) for v = 1,0 ,
so the 7Tiv have density functions
&((7Tiv = Xij V, vr,a Iv) = [B(av, v3)] 1 7T.avv1(1 - r i)3v-1
We assume that the response statuses of individuals within
a domain are conditionally independent of each other given
the probability of responding within the domain and the
victimization status of the individual. Thus
id
Yij I Xij = v, TiV id Bernoulli(71iv)
forv= 1,0 andi=1,2,...,K.
We will take an empirical Bayes approach to estimating
the parameters of the model described previously. (Note
that a full Bayes approach could also be taken by placing
prior distributions on the a, b, a,, ,81, ao, and (30 parame-
ters. The full Bayes approach leads to more difficult com-
putations, however, and we will not consider it here.) Un-
der the empirical Bayes approach, data from the entire sample
are used to estimate the parameters of the beta distributions
for pi, 7riI, and 7Tio. These estimated distributions are then
used as priors in a Bayesian analysis. Through these priors,
information from the entire data set is used to provide es-
timates for small domains.

To carry out the empirical Bayes procedure, we will first
integrate the likelihood for the observed data over the unob-
servable pi, 7rij, and 7rio parameters to obtain the marginal
distribution of the data given the a, b, a,, ,81, ao, and (30
parameters. Then we will obtain maximum likelihood es-
timators (MLE's) of the a, b, all, 38I, a0o, and (30 parameters
from this marginal distribution. The pi, 7Tij, and 7rio param-
eters may then be treated as a random sample from a dis-
tribution with parameters equal to the MLE's, a, b, c ,
I(31, ao, and (30. This distribution will be used as a prior
distribution for the pi, 7Tij, and 7rio parameters, and a pos-
terior distribution given the data, {Xij, Yi0}, will be com-
puted. The means of the posterior distribution will be used
as estimates of the pi, 7TiI, and 7rio parameters.
In the following sections, we derive the empirical Bayes
estimators for the parameters of our hierarchical model for
victimizations and nonresponse.
3.2 Estimates of the Parameters of the
Beta Distributions
Under the hierarchical model described previously, the
probabilities for each possible type of observation in the
survey data for individual]j within domain i are as follows:
2015 20:08:52 UTC
Probabilities 299

Pr(Xij = 1, Yij = 1) = Pr(responding and victimized)
= Pr(Xij = I)Pr(Yj = 1 I Xij = 1)
= Pi ITil-
Pr(Xij = 0, Yij = 1) = Pr(responding and crime free)
= Pr(X0j = O)Pr(Y1j = 1 I Xj = 0)
= (1 - pi)Tfio.
Pr(Y1j = 0)
= Pr(nonresponding and either victimized or crime free)
= Pr(Xij = I)Pr(Yj = O | Xij = 1)
+ Pr(Xm = O)Pr(Y11 = 0 I Xij = 0)
= PiO 1- 7ril) + ( -PY)(, - 7rio) -
Thus the likelihood function for the observed data is
i= (Yi+) (z+) [Piil ]Z [(1 Pi) 7Tio] '+
x [pi(l - Til) + (1 pi)(I - YT0)]i,. (1)
We obtain the marginal distribution of the data given the
a, b, a,, 831, a0, and (30 parameters by integrating (1) with
respect to the beta densities of the pi, 7Til, and 1TiO param-
eters. Details of this integration are provided in the Ap-
pendix. The result of the integration is
{B(a, b)B(al, f31)B(ao, 130)}1Y
x {K tEY (i~2 (Yi: (ni - Yi+)
i= {I r= (Yi+ (Z,+)( r )
x B(Z,+ + a + r, ni - Z+ + b - r)

x B(Zi+ + a,, 831 + r)
x B(Y1+ - Zi+ + ao, ni - Y, + Io - r)} (2)
Note that the summation in Equation (2) is over all possible
combinations of victimized and crime free for the nonre-
spondents in each domain.
The expression in Equation (2) must be maximized using
numerical methods to obtain the MLE's of the a, b, a(x, I3 ,
a0, and ,30 parameters. The methods used to obtain the MLE's
for the examples of Section 5 will be discussed in that
section.
3.3 Estimates of the Probabilities of Victimization
and Response
The MLE's described in the previous section are now
used to obtain the joint posterior distribution for the pi, Til ,
and 1rio parameters. In the ith domain, the desired posterior
distribution is
f(p1, Ti , 7TiOl {Xij, Yij1})
= {({Xij, Ye>} I, P ig , m7Til(7io)T(Piq , i7TO)}
X {(Pi, 11, Xioj) dpi d1n d7rio} , (-1
where ~(pi, 1Til, lTio) = d(pi I a, b)&(irlI = 1, c1,
I3i)4 Oo I xj = 0, &o fo). Using the MLE's obtained from
maximizing Equation (2) we have that the numerator of
Equation (3) is
f ({Xij, Yij} I Pi, '7Til, '7io)f(Pi, '7Til, '7TiO)
(ni )Yi ++){B(a b)B(a^1, f31)B(a^ o, 8o)}1

x Pia++-I (1 -Pi)
x 7Ti Z++6l-(1 -7il))'I
X Y7TiY+ -Z,++dO-1(1 - '7TO)i3o-
I
x [pi(l - 7Til) + (1 -pi)(l - 7Tio)]ni *+
= (i') (Y'+){B(a b)B(a&1, f1)B(a& Po)}1
X { + (ni -Yi+)pZ++d+r1
r=O
X (I _ -pi)n,-Z,++b-r-I ,7TZ,+ +di-I
X _ 7TiJI+r-1,7Y,+-Zi++do-i
X ( 1 - mil)p 7Ti
X (1 - '7T )Ol-Y,++P30-r-1 }
where the summation is obtained by rewriting [pi(I 1 ril)
+ (1 - pi)(l - lTio)]ni-YI+ as a binomial expansion. The
integration of the foregoing function with respect to pi, 1Til,
and 1Tio, needed in the denominator of Equation (3), is com-
pleted using methods similar to those described in the Ap-
pendix for the integration of Equation (1). Canceling the
common terms in the numerator and denominator, we find
that the desired joint posterior distribution is
f(Pi, 7Til, 7TiO {Xij, Yij})
n,-Yi+ ={ nr=O (ni Yi+) pZ,++a+r-1
X (1 -
X (1 - 7Til)PI+r-i7 '+-Z?++o-
X (1 - '7Ti)n-Y,++Po-r-1 }

rn,-Y,+
x E ni Yi+) B(Zi+ + a + r, ni - Zi+ + b - r)
r=O r
x B(Zi+ + a^, fl1 + r)
x B(Yi+ - Zi+ + &^0, ni - Yi+ + 1% - r)}
The expected values of the pi, 1Til, and iTio parameters
under the posterior distribution may be used as the corre-
sponding parameter estimates within each domain. For any
domain i, the integration is easily completed to show that
the expected value of p, is
{ EKi[Zi+ + a + rln, + a + b]} t /- il
2015 20:08:52 UTC
where
Ki= (ni +)B(Zi+ + a + r, ni - Z,+ + b - r)
x B(Zi+ + d1, f1 + r)
x B(Yi+ - Zi+ + ao, ni - Yi+ + go - r)
Similarly, the expected value of 7rij may be shown to be
n,-Yt rni-Y1+A

I Ki[Zi+ + al/Zi+ + a^ + f31 + r]} 2 Ki
r=O r-O
and the expected value of rio may be shown to be
E K[Yj+ - Zi+ + O/ni -Zi+
+ 60 + d o+P-r]}{> Ki}.
4. SPECIAL CASE OF RANDOM NONRESPONSE
In this section we consider the special case of the model
described previously in which individuals within different
domains may have different probabilities of responding but
those probabilities do not depend directly on victimization
status. Thus, in the terminology of Little and Rubin (1987),
victimization status is missing at random (MAR) within each
domain. The distributional assumptions concerning the pi
are as in Section 3. But we now assume that 7il = vio-
ir, with
.id
1ti iid beta(a, ,8) for i = 1, 2, ... ., K.
The likelihood function for the observed data using this model
is
J.?I nh)( l).[p^T]Z,+[(l p)ir]Yi+Zi+[l - ,r]n-Yi+}
To find the marginal distribution of the data given the a,
b, a, and ,X parameters we must complete the following
integration:
f ({i, Yij} | a, b, a, f)

Jo Joij l {(ni)(Yi+)[piT]zI+
X [(1 - pi)ir,]Yi+-Z+ [1 - _]ni-Yi+
x [B(a, b)]- Pi(-i
x [B (a, 1]-1ira-'(1 - li)1 1}dp dirX
This integration can be solved simply by rewriting the in-
tegrands as beta probability density functions. The result of
the integration is.
fB(a, b)B(a, 8)1} x {H nYi+)(+)
x B(Zi+ + a, 1j+ - Z1 i+ b)B(Y1+ + a, n1 - Y4 + 13)}.
(4)
The expression in Equation (4) must be maximized using
numerical methods to obtain the MLE's of the a, b, a, and
,[ parameters. The maximization is made easier, in this case,
by the fact that Equation (4) may be factored into two parts-
one a function of the a and b parameters alone and the other
a function of the a and f8 parameters alone-which may be
maximized separately. This factorization is expected under
the MAR assumption since, for likelihood-based infer-
ences, the estimates of probabilities of victimization cannot
be affected by the nonresponse mechanism (see Little and
Rubin 1987). The part of the likelihood function involving
only the a and b parameters for the distribution of the prob-
abilities of victimization is the same as that given by Le-
hoczky and Schervish (1987).
The MLE's of the a, b, a, and [3 parameters are used to
obtain the posterior distribution for the pi and iri parame-
ters. Then the expected values of these parameters under

the posterior distribution are used as the parameter esti-
mates within each domain.
For any domain, i, the expected value of pi is easily shown
to be
(Zi+ + a)/(Yi+ + a + b).
Note that this is the usual mean of a posterior distribution
for the binomial parameter, p, with a beta(a, b) prior. For
any domain, i, the expected value of 7i is easily shown to
be
(Yi+ + a)/(ni + & + ,B).
This is also the usual mean of a posterior distribution for
the binomial parameter with a beta prior.
5. FITS OF THE MODELS TO NCS DATA
In this section, we discuss the computer algorithm used
to fit the two hierarchical models described in Sections 3
and 4. Then we present the results of fitting those models
to NCS data and some randomly generated data.
5.1 Algorithm for Fitting the Model
Numerical algorithms for obtaining the MLE's of the a,
b, a, and [3 parameters under both the nonrandom and ran-
dom nonresponse models must be carefully written to avoid
overflow, underflow, and rounding error problems on the
computer. The computer programs for the analyses de-
scribed here were written in double precision FORTRAN
using IMSL subroutines to perform the required maximi-
zations, to evaluate the complete beta functions, and to
compute combinations. Since the complete beta function,
B(a, b) = F(a)I(b)/r(a + b), is large even for moderate

values of a and b, the calculations were carried out using
logarithms wherever possible.
In the case of the random nonresponse model, the log-
arithm of Equation (4) was maximized using IMSL func-
tions to evaluate the logarithm of the complete beta func-
tion. For the nonrandom nonresponse model, since
I( I . I I [I - Z)!n - -
2015 20:08:52 UTC
Probabilities 301
Equation (2) may be rewritten as follows to facilitate
maximization:
K ni-Y,+
HI E exp{ln[F(ni + 1)] - ln[F(Zi+ + 1)]
i=lI r=O
- ln[F(Yi+ - Zi+ + 1)] - ln[r(r + 1)]
- ln[r(ni - Yi+ - r + 1)]
+ ln[B(Zi+ + a + r, ni - Z1+ + b -r)]
- ln[B(a, b)] + B(Zi+ + a1, I31 + r) - ln[B(al, I31)]
+ ln[B(Yi+ - Z1+ + ao, ni -Yi+

+ 30 - r)] - ln[B(a0, 130)]}.
Again, the logarithm of the foregoing equation was used in
the maximization.
An additional problem encountered in the data analyses
described here was that, because the probability of respond-
ing was rather large, the maximization routine tended to
converge toward an estimate of 1.0 for the probability of
responding, skipping possible estimates that gave larger
values of the likelihood function. To avoid this problem,
each beta(a, b) distribution in both the random and non-
random nonresponse cases was reparameterized in terms of
a/(a + b) and (a + b). In addition, in the nonrandom non-
response case a partial grid search was used to locate a rea-
sonable starting point for '-io in the iterative procedure.
5.2 The Simulated Data
We fit the hierarchical models described in Sections 3
and 4 to two sets of data. The first set of data was randomly
generated from distributions based on summaries of the NCS
data. Inspection of the raw NCS data suggests that the prob-
ability that an HH is touched by crime in a six-month period
is about .2 and the overall probability that an HH responds
to the survey is about .9. The data set, therefore, was gen-
erated to agree with these probabilities. The data were gen-
erated for K = 10 domains with a sample of ni = 100 HH's
sampled from each domain. The probability of victimiza-
tion within the ith domain, pi, was randomly chosen from
a beta(15, 60) distribution so that E[pi] = .2. For those
HH's that were victimized, the probability of responding
to the survey for HH's within domain i, vil, was randomly
chosen from a beta(7, 3) distribution so that E[vnil] = .7.
For HH's that were not victimized, the probability of re-

sponding to the survey, i-io, was randomly chosen from a
beta( 19, 1) distribution so that E[ 7rio] = .95. IMSL sub-
routines were used to generate the values of the pi, 7-il, and
l0io parameters and the resulting data. The randomly gen-
erated data set is shown in Table 2.
5.3 Results
The random and nonrandom nonresponse models were
fit to obtain parameter estimates from both the simulated
data described previously and the actual NCS data de-
scribed in Section 2. These parameter estimates are given
in Table 2 for the randomly generated data and in Table 1
for the NCS data. In addition to the parameter estimates
obtained under the two hierarchical models, "naive"
parameter estimates are provided in both tables. These es-
timates are obtained using only the information in an in-
dividual domain to compute the estimates for that domain.
Thus the naive estimators in domain i are Pi = Zi+/Yi, and
'ri = Yi+/ni.
Consider the effects of the hierarchical estimation schemes
on the parameter estimates in the case of the randomly gen-
erated data in Table 2. The naive estimates of pi are simply
the observed proportions of victimized HH's in the 10 do-
mains ignoring nonrespondents. The estimates of pi in Ta-
ble 2 under the random nonresponse model are pulled from
the naive estimates toward the overall proportion of victim-
ized HH's, Z++/Y++ = 145/903 = .161. Similarly, the
estimates of 7ni under the random nonresponse model are
pulled from the naive estimates, the observed proportions
of respondents in each domain, toward the overall propor-
tion of respondents, Y++/n+ = 903/1,000 = .903. In this
way the information from all domains is used to estimate
the probabilities of victimization and nonresponse in each
individual domain.

Under the nonrandom nonresponse model, the estimates
of pi shown in Table 2 are again pulled toward an overall
probability of victimization, but in this case that overall
probability is somewhat larger than the naive overall esti-
mate because it has been adjusted for the fact that victim-
ized HH's are less likely to respond than are crime-free
Table 2. Data Randomly Generated With p, beta(15, 60), 1ir -
beta(7, 3), and 7jO - beta(19, 1) So That Efpj] = .2, E[7ril] = .7,
and
Erio] = .95, Using Sample Sizes of ni = 100 for i = 1, 2, ..., 10
Randomly generated Naive Random
Randomly generated data probabilities estimator nonresponse
Nonrandom nonresponse
i Yj+ - Z,+ Z+ ni- Yj+ Pi n-1 jT1o A r1 p; A p 1 AT11 7rjo
1 72 19 9 .267 .765 .920 .209 .91 .168 .906 .176 .861 .914
2 73 21 6 .222 .813 .966 .223 .94 .170 .921 .179 .861 .931
3 76 17 7 .243 .768 .977 .183 .93 .164 .916 .172 .861 .927
4 72 17 1 1 .226 .541 .977 .191 .89 .165 .897 .173 .861 .903
5 68 15 17 .210 .796 .860 .181 .83 .163 .867 .171 .861 .866
6 80 10 10 .211 .629 .986 .111 .90 .153 .901 .161 .861 .913
7 83 12 5 .189 .738 .976 .126 .95 .155 .926 .163 .861 .940
8 72 14 14 .215 .803 .879 .163 .86 .161 .882 .169 .861 .886
9 76 10 14 .175 .527 .954 .116 .86 .154 .882 .161 .861 .889
10 86 10 4 .171 .565 .986 .104 .96 .152 .931 .159 .861 .946
See Table 1 Note.
2015 20:08:52 UTC

Table 3. Errors in Estimation for Randomly Generated Data
Random Nonrandom
Naive estimator nonresponse nonresponse
P iT11 IT0 P~~ 1T11 ITl IT11 i 1 pj ri, 7Tjo pj lki, Pij pjiT,io
Mean absolute error .052 .209 .045 .052 .208 .047 .045 .167
.039
Root mean squared error .058 .240 .054 .057 .237 .055 .050
.200 .047
HH's. Thus the estimates of the probabilities of victimiza-
tion are all larger under the nonrandom nonresponse model
than under the random nonresponse model. The estimates
of the probabilities of responding for crime-free HH's are
generally larger under the nonrandom nonresponse model
than are the single response probabilities under the random
nonresponse model (the only exception occurs in the fifth
domain). The estimates of the probabilities of responding
for victimized HH's are all smaller under the nonrandom
nonresponse model than are the single response probabili-
ties under the random nonresponse model. The values of
Vril shown in Table 2 are all identical to three decimal places
because, in this example, the estimated value of a1 + ,31
is very large. Since this term appears in the denominator
of the variance of the prior beta distribution for the 77il. the
prior variance is quite small. Thus the information from the
sample does not greatly affect the estimates of ril's.
Using the results presented in Table 2 for the simulated

data, we may compare the naive, random nonresponse, and
nonrandom nonresponse estimates to the actual parameter
values to determine how effective the hierarchical models
are. The mean absolute errors and root mean squared errors
for the Pi, Vril, and fio are given in Table 3. Note that for
the naive and random nonresponse estimators, the single
estimator of the probability of responding, ire, is compared
with both 7Til and 7Ti0 since in those cases the probability of
responding is taken to be the same for both victimized and
crime-free HH's. The errors shown in Table 3 indicate that
the naive and random nonresponse estimators are approx-
imately the same in terms of mean absolute errors and root
mean squared errors, whereas the errors associated with the
nonrandom nonresponse model are somewhat smaller.
Now consider the results for the actual NCS data pre-
sented in Table 1. Again, the estimates of the pi under the
random nonresponse model are pulled from the naive es-
timates toward the overall proportion of victimized HH's,
Z++/Y++ = 727/3,630 = .2003. Similarly, the estimates
of ri under the random nonresponse model are pulled from
the naive estimates toward the overall proportion of re-
spondents, Y++/n+ = 3,630/4,155 = .8736. In this way
information from all domains is used to estimate the prob-
abilities of victimization and nonresponse in each individ-
ual domain. Note, of course, that in domains where the
sample size is particularly large the estimate is not pulled
toward the overall proportion as much as it is in cases where
the sample size is smaller.
In the case of the nonrandom nonresponse model, the
estimates of the pi are again pulled toward a larger overall
probability of victimization that has been adjusted for the
fact that victimized HH's appear to be less likely to respond
than are crime-free HH's. Thus the estimates of the prob-

abilities of victimization are all larger under the nonrandom
nonresponse model than under the random nonresponse
model. Under the nonrandom nonresponse model, the es-
timate within each domain of the probability of responding
for crime-free HH's is larger than the single response prob-
ability obtained under the random nonresponse model,
whereas the estimate of the probability of responding for
crime-free HH's is smaller. The values of frj0 are all iden-
tical to three decimal places, because the estimated value
of a0 + X30 is very large and hence the variance of the prior
distribution of -ri0 is quite small. Thus the information from
the sample does not greatly affect the estimates of 77i0.
6. CONCLUSIONS AND FUTURE WORK
We have developed hierarchical models for the proba-
bilities of victimizations and nonresponse and fit those models
to randomly generated data and actual data from the NCS.
The hierarchical models allow for either random or non-
random nonresponse. The nonrandom nonresponse model
fit to the simulated data succeeded in capturing the differ-
ence in response probabilities for victims and nonvictims
that was present in the distributions from which the data
were generated. Since the parameter estimates obtained when
the nonrandom nonresponse model was fit to the actual NCS
data show similar differences for victims and nonvictims,
it seems reasonable to conclude that nonresponse in the NCS
is informative nonresponse. The values of the parameter
estimates suggest that victims of crime are less likely to
respond to the survey than are nonvictims. Any estimation
procedures that do not allow for this difference will result
in estimates of probabilities of victimizations that are biased
downward.
The empirical Bayes approach taken here has the advan-
tage of allowing information from all domains to be used

to provide estimates of probabilities within each domain.
The disadvantage is that the computations are more difficult
than for the standard, nonhierarchical approach. It will be
important to improve the computation procedure in the fu-
ture, particularly if more complex models are developed.
Gelfand and Smith (1990), for example, described adaptive
sampling techniques for calculating marginal densities that
may be useful for fitting the models described in this ar-
ticle. Obtaining variance estimates under these hierarchical
models is an additional problem. One must be wary of vari-
ance estimates based on using the MLE's as the parameters
in the beta priors for these hierarchical models because such
variance estimates would not include the uncertainty in the
MLE's themselves. A possible remedy for this problem was
suggested by Morris (1983).
2015 20:08:52 UTC
Probabilities 303
Although the subsample of NCS data used here allowed
us to avoid dealing with clusters, we did ignore other as-
pects of the NCS design in our models. Future work should
consider the complex design of surveys like the NCS that
use multistage stratified cluster samples. Other future work
would involve extending these hierarchical models to allow
the probabilities of victimizations to be influenced by co-
variates in the data. Saphire (1989) developed hierarchical
models for estimating the number of victimizations expe-
rienced by an HH that make use of covariates but do not

address the nonresponse problem. Another extension of the
models would be to allow them to handle the longitudinal
nature of NCS data. Lehoczky and Schervish (1987) sug-
gested a hierarchical Markov-chain model for victimiza-
tions, and Stasny (1987) presented Markov-chain models
that are not hierarchical but that do allow for random or
nonrandom nonresponse. A combination of these ideas could
be used to develop hierarchical Markov-chain models for
both victimizations and nonresponse.
APPENDIX: MARGINAL DISTRIBUTION OF THE DATA
GIVEN a, b, a4, PBi, a0, and 13o
To find the marginal distribution of the data given the a, b, a(,
,f1, ao, and 13o parameters of the nonrandom nonresponse
model
of Section 3, we must complete the following integration:
f({X0j, Y11} I a, b, a,, 131, ao, PO)
- A J J rI {(ni)(Yj+) X1' ]+(i - p)1T ]Y'+-Zi+
x [pi(l _ il) + (1 pi)(I - rio)]
x [B(a, b)]V'Pi(1-Pi)
x [B(al, /31)]1 siTll1 (1 -7Til)
x [B(ao o)] -)1a'?
x (1 -Tio) 4} dpi dTil dirio
fi (ni)Q'+){B(a, b)B(al, /3,)B(ao, Po)}'
fl Yi+ +
x f fJf {pZI++a1(1 - pi)Y,+ Zi++b 1 TZ,++ajI (1 Til)
X ITYi+-Z+++a0-1 ( 1 -r.0)P1

x [pi(l- iril) + (1-Pi)
X ( 1- Ti0)]nj-Yi+} dpi d'Til diriO,
where B(a, b) = F(a)F(b)/F(a + b) is the complete beta function.
Using only the terms involving the pi in the innermost integral,
we can use a binomial expansion to complete that integration as
follows:
Z++ (1 - P)Yi+-Zi++b-1
0
X [pi(l-
_
il) + (1- pi)(l - Tio)]ni-Y+ dpi
Il nljY,+/
= J Fi++a-(l -pi)yi+-Zi++b-1 (fni -Yi+1
? [pi(Il-1Til)]r [(I-pi)(I-_ TO)]niYi+ rdpi
_I
nii
(ni -Yi+) 1-Xlrl-XOn-i-
JO r-O 'I
= nrY+ ( ri il)r(l (- Tio)ni-Yi+-r
r=O
X r(Zi+ + a + r)r(ni - Z1+ + b - r)/r(ni + a + b),

where the final step is obtained by rewriting the integrand as a
beta probability density function.
The remaining two integrals can be solved simply by rewriting
the integrands as beta probability density functions. The result
of
the integration is given in Equation (2) of Section 3.
[Received September 1989. Revised September 1990.]
REFERENCES
Gelfand, A. E., and Smith, A. F. M. (1990), "Sampling Based
Ap-
proaches to Calculating Marginal Densities," Journal of the
American
Statistical Association, 85, 398-409.
Lehoczky, J. P., and Schervish, M. J. (1987), "Hierarchical
Modelling
and Multi-level Analysis Applied to the National Crime
Survey," un-
published paper presented at the Workshop on the National
Crime Sur-
vey, July 6-17, 1987.
Little, R. J. A., and Rubin, D. B. (1987), Statistical Analysis
With Miss-
ing Data, New York: John Wiley.
Morris, C. N. (1983), "Parametric Empirical Bayes Inference:
Theory
and Applications,' Journal of the American Statistical
Association, 78,
47-65.

Platek, R., Rao, J. N. K., Sirndal, C. E., and Singh, M. P.
(1987),
Small Area Statistics: An International Symposium, New York:
John
Wiley.
Saphire, D. G. (1984), Estimation of Victimization Prevalence
Using Data
From the National Crime Survey, (Lecture Notes in Statistics,
Vol.
23), New York: Springer-Verlag.
(1989), "An Empirical Bayes Model With Covariates Applied to
Victimization," unpublished paper presented at the Joint
Statistical
Meetings, Washington, D.C., August 6-10, 1989.
Stasny, E. A. (1987), "Some Markov-Chain Models for
Nonresponse in
Estimating Gross Labor Force Flows," Journal of Official
Statistics,
3, 359-373.
(in press), "Symmetry in Flows Among Reported Victimization
Classifications with Nonrandom Nonresponse," Survey
Methodology.
U.S. Department of Justice and Bureau of Justice Statistics
(1981), The
National Crime Survey: Working Papers, Volume I: Current and
His-
torical Perspectives, Washington, D.C.: Author.
2015 20:08:52 UTC

http://www.jstor.org/page/info/about/policies/terms.jspArticle
Contentsp. 296p. 297p. 298p. 299p. 300p. 301p. 302p. 303Issue
Table of ContentsJournal of the American Statistical
Association, Vol. 86, No. 414 (Jun., 1991) pp. 257-555Front
Matter [pp. ]Editors' Report for 1990 [pp. 257]1990 R. A.
Fisher Memorial LectureStatistical Inference: Likelihood to
Significance [pp. 258-265]Applications and Case
StudiesNondetects, Detection Limits, and the Probability of
Detection [pp. 266-277]Evaluation of Procedures for Improving
Population Estimates for Small Areas [pp. 278-284]Using
Medical Malpractice Data to Predict the Frequency of Claims: A
Study of Poisson Process Models With Random Effects [pp.
285-295]Hierarchical Models for the Probabilities of a Survey
Classification and Nonresponse: An Example from the National
Crime Survey [pp. 296-303]Shrinkage Estimation of Price and
Promotional Elasticities: Seemingly Unrelated Equations [pp.
304-315]Theory and MethodsSliced Inverse Regression for
Dimension Reduction [pp. 316-327]Sliced Inverse Regression
for Dimension Reduction: Comment [pp. 328-332]Sliced Inverse
Regression for Dimension Reduction: Comment [pp. 333]Sliced
Inverse Regression for Dimension Reduction: Comment [pp.
333-335]Sliced Inverse Regression for Dimension Reduction:
Comment [pp. 336-337]Sliced Inverse Regression for
Dimension Reduction: Rejoinder [pp. 337-342]Transformations
in Density Estimation [pp. 343-353]Transformations in Density
Estimation: Comment [pp. 353-354]Transformations in Density
Estimation: Comment [pp. 359]Transformations in Density
Estimation: Rejoinder [pp. 360-361]On the Problem of
Interactions in the Analysis of Variance [pp. 362-367]On the
Problem of Interactions in the Analysis of Variance: Comment
[pp. 367-369]On the Problem of Interactions in the Analysis of
Variance: Comment [pp. 369-372]On the Problem of

Interactions in the Analysis of Variance: Comment [pp. 372-
373]On the Problem of Interactions in the Analysis of Variance:
Rejoinder [pp. 374-375]Structural Image Restoration Through
Deformable Templates [pp. 376-387]Least Squares Estimation
of Covariance Matrices in Balanced Multivariate Variance
Components Models [pp. 388-395]Sensitivity in Bayesian
Statistics: The Prior and the Likelihood [pp. 396-399]An
Approach to Robust Bayesian Analysis for Multidimensional
Parameter Spaces [pp. 400-403]Fast Computation of Exact
Confidence Limits for the Common Odds Ratio in a Series of 2
× 2 Tables [pp. 404-409]A Unified Approach to Rank Tests for
Multivariate and Repeated Measures Designs [pp. 410-419]A
Lack-of-Fit Test for the Mean Function in a Generalized Linear
Model [pp. 420-426]Smooth Goodness-of-Fit Tests: A Quantile
Function Approach [pp. 427-431]Optimal Sample Allocation for
Normal Discrimination and Logistic Regression Under Stratified
Sampling [pp. 432-436]A Large Deviation-Type Approximation
for the "Box Class" of Likelihood Ratio Criteria [pp. 437-
440]An Unbiased Estimator of the Covariance Matrix of the
Mixed Regression Estimator [pp. 441-444]Estimating a
Population Total Using an Area Frame [pp. 445-449]An
Approach to the Construction of Asymmetrical Orthogonal
Arrays [pp. 450-456]Limitations of the Rank Transform
Procedure: A Study of Repeated Measures Designs, Part I [pp.
457-460]Walsh-Fourier Analysis and Its Statistical Applications
[pp. 461-479]Walsh-Fourier Analysis and Its Statistical
Applications: Comment [pp. 480]Walsh-Fourier Analysis and
Its Statistical Applications: Comment [pp. 481-482]Walsh-
Fourier Analysis and Its Statistical Applications: Comment [pp.
482-483]Walsh-Fourier Analysis and Its Statistical
Applications: Rejoinder [pp. 483-485]Social Statistics and
Public Policy for the 1990sSocial Statistics and Public Policy
for the 1990s: [Introduction] [pp. 486]Contextually Specific
Effects and Other Generalizations of the Hierarchical Linear
Model for Comparative Analysis [pp. 487-503]Social Statistics
and an American Urban Underclass: Improving the Knowledge

Base for Social Policy in the 1990s [pp. 504-512]National
Surveys and the Health and Functioning of the Elderly: The
Effects of Design and Content [pp. 513-525]The Effects of
Census Undercount Adjustment on Congressional
Apportionment [pp. 526-541]Book Reviews[List of Book
Reviews] [pp. 542]Review: untitled [pp. 543]Review: untitled
[pp. 543-544]Review: untitled [pp. 544-546]Review: untitled
[pp. 546]Review: untitled [pp. 546-547]Review: untitled [pp.
547-548]Review: untitled [pp. 548]Review: untitled [pp. 548-
549]Review: untitled [pp. 549]Review: untitled [pp.
549]Review: untitled [pp. 550]Review: untitled [pp. 550-
551]Review: untitled [pp. 551-552]Review: untitled [pp.
552]Review: untitled [pp. 552-553]Review: untitled [pp.
553]Review: untitled [pp. 553-554]Publications Received [pp.
554-555]Back Matter [pp. ]
940 Am J Epidemiol 2003;157:940–943
American Journal of Epidemiology
Copyright © 2003 by the Johns Hopkins Bloomberg School of
Public Health
All rights reserved
Vol. 157, No. 10
Printed in U.S.A.
DOI: 10.1093/aje/kwg074
Estimating the Relative Risk in Cohort Studies and Clinical
Trials of Common
Outcomes
Louise-Anne McNutt1, Chuntao Wu1, Xiaonan Xue2, and Jean
Paul Hafner3

1 Department of Epidemiology, School of Public Health,
University at Albany, State University of New York,
Rensselaer, NY.
2 Department of Environmental Medicine, Division of
Biostatistics, New York University School of Medicine, New
York, NY.
3 Departments of Pulmonary and General Internal Medicine,
Samuel S. Stratton Department of Veterans Affairs Medical
Center,
Albany, NY.
Received for publication June 14, 2001; accepted for
publication March 14, 2003.
Logistic regression yields an adjusted odds ratio that
approximates the adjusted relative risk when disease
incidence is rare (<10%), while adjusting for potential
confounders. For more common outcomes, the odds ratio
always overstates the relative risk, sometimes dramatically. The
purpose of this paper is to discuss the incorrect
application of a proposed method to estimate an adjusted
relative risk from an adjusted odds ratio, which has
quickly gained popularity in medical and public health research,
and to describe alternative statistical methods
for estimating an adjusted relative risk when the outcome is
common. Hypothetical data are used to illustrate
statistical methods with readily accessible computer software.
clinical trials; cohort studies; odds ratio; relative risk
The study of common outcomes is becoming more frequent
in medicine and public health. Studies of symptoms, health
behaviors, health care utilization, and even rare diseases in
high-risk populations all have the potential to occur frequently
(>10 percent) in a study population. This fact becomes an

important consideration in deciding on the appropriate statis-
tical analysis for a study. Typically, researchers use statistical
methods designed for studies of rare diseases, sometimes
incorrectly applied to studies of common outcomes. An
example of this problem is the use of logistic regression to
compute an estimated adjusted odds ratio and the subsequent
interpretation of this estimate as a relative risk. This relation is
approximately true when the incidence of outcome is less than
10 percent but usually not true when the outcome is more
common. Although logistic regression may be correctly
applied to studies of common outcomes, in public health we
are often interested in estimating a relative risk (e.g., the prob-
ability of the outcome for one exposure group divided by the
probability of the outcome for another exposure group
(referent)), not the odds ratio, and it is this inference that
becomes troublesome. In studies of common outcomes, the
estimated odds ratio can, and often does, substantially overes-
timate the relative risk.
A method proposed by Zhang and Yu (1) to correct the
adjusted odds ratio in cohort studies of common outcomes was
proposed in 1998 and has gained popularity in medical and
public health research. A review of the Journal Citation
Reports (accessed on May 15, 2001) identified 74 citations of
this paper, and 56 reported studies utilized Zhang and Yu’s
method in the data analysis. Unfortunately, in most cases the
method was incorrectly applied. By March 28, 2003, 214
scientific publications had cited Zhang and Yu’s paper.
The purpose of this paper is to discuss the drawbacks of the
Zhang and Yu method as applied by many researchers and
briefly review alternative methods for estimating an adjusted
relative risk and its confidence interval when the incidence of
disease is common and confounding exists. The study designs
we focus on include cohort studies and clinical trials with

equal follow-up times for study subjects, and the cumulative
incidence in at least one exposure or treatment group is greater
than 10 percent.
We focus on methods that are compatible with statistical
programs widely used in medical and public health research,
including stratified analysis, Poisson regression, and the log-
binomial model. Other methods to estimate confidence inter-
vals of adjusted relative risks (e.g., delta method, bootstrap)
have attractive properties (2, 3); however, user-friendly soft-
ware is still developmental for these methods and not yet
widely available to researchers. We focus here on the situation
where effect modification (interaction with other factors) of
the relative risk does not exist.
Correspondence to Dr. Louise-Anne McNutt, Department of
Epidemiology, School of Public Health, University at Albany, 1
University Place,
Room 125, Rensselaer, NY 12144 (e-mail: [email protected]).
by guest on O
ctober 30, 2015
http://aje.oxfordjournals.org/
D
ow
nloaded from
Relative Risk Estimation 941

Am J Epidemiol 2003;157:940–943
COMPARISON OF AVAILABLE METHODS
For the purpose of illustration, we created several hypothet-
ical studies; each focuses on the association between a specific
risk factor (E) and disease (D) and needs to be adjusted for a
confounder (C). The data and calculated adjusted and crude
measures of the relative risk for the method reviewed are
shown in table 1. Additionally, we provide results from a
simulation study that highlights the potential bias that may
occur with the Zhang and Yu correction method (table 2).
MODEL SELECTION: STUDYING ASSOCIATION
VERSUS PREDICTION
Rarely is there only one statistical model that adequately fits
a set of data. Rather, researchers find themselves choosing
among a few models that fairly summarize the information.
The choice between models that adequately fit the data is
based on various criteria, one of which is the research ques-
tion. Relative risks are computed for studies that focus on
measuring an association(s) between an exposure(s)/risk
factor(s) and an outcome. Unlike predictive models where
parsimony is revered, regression models for studies of associ-
ation often keep several factors that may not explain large
amounts of the variance in the outcome; however, these vari-
ables confound the association between exposure(s) and
outcome sufficiently to warrant adjusting for them in the anal-
ysis (4, 5). Other criteria considered in model selection
include the existence of influential individuals, extreme
outliers, and other factors related to model fit (4).
ZHANG AND YU’S PROPOSED METHOD
Zhang and Yu proposed an intriguing, simple formula to

convert an odds ratio provided by logistic regression to a rela-
tive risk (1):
In this formula, P0 is the incidence of the outcome in the
nonexposed group, “OR” is an odds ratio from a logistic
regression equation, and “RR” is an estimated relative risk.
Most researchers apply this formula to the adjusted odds ratio
to estimate an adjusted relative risk. Using the formula in this
manner is incorrect and will produce a biased estimate when
confounding is present. If no confounding exists, then regres-
sion analysis is not needed and simple calculations can be
used to compute an estimated relative risk (6).
With logistic regression, an estimated relative risk can be
computed for each covariate pattern (i):
where Y is the outcome factor of interest (dependent vari-
able), E is the exposure of interest, and x2, …, xk are
RR OR
1 P0–( ) P0 OR×( )+
---------------------------------------------------=
RRi
P Y E x, 2 i
… xki,,( )
P Y E x, 2 i … xki,,( )
-----------------------------------------------=
1 e
β0 β1 E β2 x2 i … βk xk i+ + + +( )–+
1 e

β0 β1 E β2 x2 i … βk xk i+ + + +( )–
+
--------------------------------------------------------------------=
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942 McNutt et al.
Am J Epidemiol 2003;157:940–943

confounders. Although the formula looks complicated, these
probabilities are just the predicted values that statistical
programs provide routinely. It should be noted that this
formula cannot be used for classical case-control studies, as
the intercept cannot be validly estimated.
In data from our studies on the health effects of violence,
the Zhang and Yu correction, applied to the adjusted odds
ratio and using the incidence among the unexposed for the
entire sample, usually tends to be biased away from the null,
suggesting that the strength of association is greater than is
true. This bias occurs because the formula, used as one
summary value, fails to take into consideration the more
complex relation in the incidence of disease related to expo-
sure for each covariate pattern. This finding also occurred in
Zhang and Yu’s simulation studies (1). Although the
formula can be applied to specific covariate patterns, taking
the ratio of the predicted probabilities is a simpler method to
obtain covariate pattern-specific relative risks.
It is also important to note that, in general, if an outcome is
common, then homogeneity of the odds ratio cannot coexist
with homogeneity of the relative risk. It is useful to note that
more than one statistical model may adequately fit the data;
however, allowance for effect modification will depend on
which model is selected.
The most difficult problem in estimating an adjusted rela-
tive risk for studies of common outcomes is not the point
estimate (which we discuss below), but rather the confidence
interval. Zhang and Yu’s proposed confidence interval for
the adjusted relative risk, computed by applying the above
formula to the bounds on the adjusted odds ratio’s confi-
dence interval, also can be biased, leading one to believe that
the relative risk estimate is more precise than is true (7). This

bias occurs because the proposed calculation does not take
into consideration the covariance between the estimated
incidence and estimated odds ratio. Yu and Zhang note that
a “trade-off between simplicity and precision” (8, p. 529) is
at issue with their method; however, we believe that it is
important, particularly when there are policy implications,
not to overstate precision. In the simulation study results
presented in table 2, the computed 95 percent confidence
interval coverage is only 63 percent (it should be 95 percent),
suggesting that in some typical situations substantial misrep-
resentation of precision is possible.
STRATIFIED ANALYSIS
One of the simplest and best-known techniques for calcu-
lating an adjusted relative risk is stratified analysis (9). Using
stratified analysis, the relative risk between the risk factor of
interest (E) and disease (D) is computed for each level of the
confounder. These stratum-specific relative risks can be
pooled together to create one adjusted relative risk, usually
by taking a weighted average of the stratum-specific relative
risks. Typically, the weights are chosen so that they are
larger for strata with the most individuals and smaller for
strata with fewer individuals (4).
LOG-BINOMIAL MODEL
The log-binomial model has been proposed as a useful
approach to compute an adjusted relative risk. Like logistic
regression, the log-binomial model is used for the analysis of
a dichotomous outcome. Both model the probability of the
outcome (e.g., probability of disease given the exposure and
confounders), and both assume that the error terms have a
binomial distribution. The difference between the logistic
model and the log-binomial model is the link between the

independent variables and the probability of the outcome: In
logistic regression, the logit function is used and, for the log-
binomial model, the log function is used. In general, the log-
binomial model produces an unbiased estimate of the adjusted
relative risk. Although it has a couple of drawbacks, these
appear to pose minimal restriction on its usefulness unless
adjustment for many confounders is needed. First, the confi-
dence interval for the adjusted relative risk computed may be
narrower than is true (10, 11). As seen in table 2, our simula-
TABLE 2. Simulation study comparing methods of estimating
adjusted relative risk and coverage of
confidence interval*
* Sample size equals 500 with 50% of subjects exposed.
Specified (true) adjusted relative risk is 2.00.
Exposed group: prevalence of confounder, 60%; cumulative
incidence of disease, 0.80 when confounder
present and 0.40 when confounder absent. Nonexposed group:
prevalence of confounder, 40%; cumulative
incidence of disease, 0.40 when confounder present and 0.20
when confounder absent. One thousand
random data sets were created, and each statistical method was
applied to every data set to estimate the
adjusted relative risk and its confidence interval.
† % of relative bias = [(median of adjusted relative risk
estimated from 1,000 random data sets – true
adjusted relative risk) / true adjusted relative risk ] × 100.
Method
Median of adjusted
relative risk
% of relative bias†

95% confidence interval
coverage
Stratified, Mantel-Haenszel 1.9989 –0.05 93.80
Stratified, logit 1.9972 –0.14 93.90
Log-binomial 1.9999 –0.01 93.80
Poisson 2.0029 0.15 98.80
Zhang and Yu 2.3053 15.27 63.00
Logistic 4.3282 116.4 8.00
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Relative Risk Estimation 943
Am J Epidemiol 2003;157:940–943
tion study results suggest that this bias is minor and similar to
that found in stratified analysis. Similar coverage was seen in
simulations of a range of relative risks and confounding

patterns (data not shown). Second, in some situations, the log-
binomial model does not converge to provide parameter esti-
mates (10, 12). The lack of convergence may simply be due to
software programs that have a default convergence criterion
that is insufficient. This problem can be remedied by requiring
additional iterations in the modeling fitting process. Another
reason the model fits may not converge to the maximum like-
lihood estimate(s) is that the maximum likelihood estimates
may lie near a boundary of the parameter space. When this
occurs, the iteration can become stuck at the boundary, and a
small adjustment of the interim fit away from the boundary
may be needed to keep the iterations moving toward the
value(s) that maximizes the likelihood.
POISSON REGRESSION AND THE CONCEPT OF
PLACING BOUNDS ON THE CONFIDENCE INTERVAL
Poisson regression is generally reserved for studies of rare
diseases where patients may be followed for different lengths
of time, such as cohort studies of rare outcomes conducted over
many years with some patients being lost to follow-up. In
contrast, unconditional logistic regression is typically utilized
when every patient is followed for the same length of time or
for a defined period with equal follow-up for subjects. For
cohort studies where all patients have equal follow-up times,
Poisson regression can be used in a similar manner as logistic
regression, with a time-at-risk value specified as 1 for each
subject. If the model adequately fits the data, this approach
provides a correct estimate of the adjusted relative risk(s). For
studies of common outcomes, Poisson regression is likely to
compute a confidence interval(s) that is conservative,
suggesting less precision than is true (tables 1 and 2). The
reason Poisson regression produces wider confidence intervals
compared with a log-binomial model and stratified analysis is
that the Poisson errors are overestimates of binomial errors
when the outcome is common (Poisson errors approximately

equal binomial errors when the outcome (disease) is rare). As
the examples in table 1 illustrate, although the confidence
interval is more conservative, the actual difference compared
with a stratified analysis is moderate. Conceptually, this
interval can be thought of as bounding the true confidence
interval.
Computer programs for the log-binomial and Poisson
regression are widely available. For example, many general-
ized linear models’ programs (e.g., PROC GENMOD in
SAS; SAS Institute, Cary, North Carolina) can be used for
both log-binomial and Poisson regression analysis.
Checking the fit of the model can be done using standard
methods.
CROSS-SECTIONAL STUDIES
For cross-sectional studies, two common measures of asso-
ciation are the prevalence ratio and the prevalence odds ratio
(13). The mathematical computations for these measures are
identical to the relative risk and the odds ratio, respectively.
Thus, the methods presented in this paper can be utilized for
cross-sectional studies; however, a temporal association
between risk factors and “outcome” cannot be assessed.
CONCLUSIONS
The use of an adjusted odds ratio to estimate an adjusted rela-
tive risk appropriate for studies of rare outcomes, however,
may be misleading when the outcome is common. The over-
estimation may inappropriately affect clinical decision-making
or policy development. Additionally, overestimation of the
importance of a risk factor may lead to unintentional errors in
the economic analysis of potential intervention programs or
treatments. Options exist to obtain unbiased estimates of rela-

tive risks in studies of common outcomes. Two methods that
have widely available user-friendly software and often are
statistically appropriate (e.g., fit the data) include stratified
analysis and log-binomial modeling.
REFERENCES
1. Zhang J, Yu KF. What’s relative risk? A method of correcting
the odds ratio in cohort studies of common outcomes. JAMA
1998;280:1690–1.
2. Rao CR. Linear statistical inference and its applications. 2nd
ed. New York, NY: John Wiley & Sons, Inc, 1965.
3. Efron B, Tibshirani R. An introduction to the bootstrap. New
York, NY: Chapman & Hall, 1993.
4. Kleinbaum DG, Kupper LL, Muller KE, et al. Applied regres-
sion analysis and other multivariable methods. Pacific Grove,
CA: Brooks/Cole, 1998.
5. Rothman KJ, Greenland S, eds. Modern epidemiology. 2nd
ed.
Philadelphia, PA: Lippincott Williams & Wilkins, 1998.
6. Kleinbaum DG, Kupper LL, Morgenstern H. Epidemiologic
research: principles and quantitative methods. New York, NY:
Van Nostrand Reinhold, 1982.
7. McNutt LA, Hafner JP, Xue X. Correcting the odds ratio in
cohort studies of common outcomes. (Letter). JAMA 1999;282:
529.
8. Yu KF, Zhang J. Correcting the odds ratio in cohort studies
of
common outcomes (in reply). (Letter). JAMA 1999;282:529.

9. Mantel N, Haenszel W. Statistical aspects of the analysis of
data from retrospective studies of disease. J Natl Cancer Inst
1959;22:719–48.
10. Skov T, Deddens J, Petersen MR, et al. Prevalence
proportion
ratios: estimation and hypothesis testing. Int J Epidemiol 1998;
27:91–5.
11. Ma S, Wong CM. Estimation of prevalence proportion rates.
(Letter). Int J Epidemiol 1999;28:175.
12. Wacholder S. Binomial regression in GLIM: estimating risk
ratios and risk differences. Am J Epidemiol 1986;123:174–84.
13. Lee J. Odds ratio or relative risk for cross-sectional data?
Int J
Epidemiol 1994;23:201–3.
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