American Journal of Epidemiology
Copyright © 2005 by the Johns Hopkins Bloomberg School of Public Health
All rights reserved
Vol. 161, No. 1
Printed in U.S.A.
DOI: 10.1093/aje/kwi017
Appropriate Assessment of Neighborhood Effects on Individual Health: Integrating
Random and Fixed Effects in Multilevel Logistic Regression
Klaus Larsen1 and Juan Merlo2
1
2
Clinical Research Unit, Hvidovre University Hospital, University of Copenhagen, Hvidovre, Denmark.
Department of Community Medicine, Malmö University Hospital, Lund University, Malmö, Sweden.
Received for publication October 3, 2003; accepted for publication August 30, 2004.
The logistic regression model is frequently used in epidemiologic studies, yielding odds ratio or relative risk
interpretations. Inspired by the theory of linear normal models, the logistic regression model has been extended
to allow for correlated responses by introducing random effects. However, the model does not inherit the
interpretational features of the normal model. In this paper, the authors argue that the existing measures are
unsatisfactory (and some of them are even improper) when quantifying results from multilevel logistic regression
analyses. The authors suggest a measure of heterogeneity, the median odds ratio, that quantifies cluster
heterogeneity and facilitates a direct comparison between covariate effects and the magnitude of heterogeneity
in terms of well-known odds ratios. Quantifying cluster-level covariates in a meaningful way is a challenge in
multilevel logistic regression. For this purpose, the authors propose an odds ratio measure, the interval odds ratio,
that takes these difficulties into account. The authors demonstrate the two measures by investigating
heterogeneity between neighborhoods and effects of neighborhood-level covariates in two examples—public
physician visits and ischemic heart disease hospitalizations—using 1999 data on 11,312 men aged 45–85 years
in Malmö, Sweden.
data interpretation, statistical; epidemiologic methods; hierarchical model; logistic models; odds ratio; residence
characteristics
Abbreviations: IOR, interval odds ratio; MOR, median odds ratio.
A growing number of epidemiologic studies apply multilevel regression analysis for the investigation of associations
between area of residence (e.g., neighborhood) and individual health (1, 2). A majority of studies have focused on
traditional measures of association such as fixed effects
using regression models for the relation between neighborhood (or cluster) characteristics and individual health. Other
multilevel regression analysis approaches have concentrated
on determining the components of health variation (3). On
some occasions, difficult questions arise, such as “Is it
worthwhile to study the association between neighborhood
characteristics and health when the neighborhood variation
is very small?” (4). Neither pure fixed-effects regression
analysis nor calculation of intraclass correlations can
provide sufficient insight to answer such questions in a satisfactory way.
The partition of variance at different levels (e.g., neighborhood and individual) is the sine qua non of multilevel regression analysis, and its consideration is relevant for both
statistical reasons (improved estimation) and substantive
epidemiologic reasons (quantification of the importance of
the neighborhoods for understanding individual health) (5).
However, contrary to normally distributed continuous variables, components of variance are tricky to investigate when
it comes to dichotomous response variables. Forcing classical interpretative schemes wastes information and may be
inappropriate. New measures are needed in order to quantify
effects and ultimately provide a better understanding of the
data.
In this paper, our aim is to highlight two measures previously described (6): the median odds ratio (MOR) and the
interval odds ratio (IOR). These measures facilitate the inte-
Correspondence to Dr. Klaus Larsen, Clinical Research Unit, Section 136, Hvidovre University Hospital, Kettegård Allé 30, DK-2650 Hvidovre,
Denmark (e-mail: [email protected]).
81
Am J Epidemiol 2005;161:81–88
82 Larsen and Merlo
gration and presentation of both fixed and random effects in
logistic regression.
MOTIVATION FOR NEW MEASURES OF EFFECT SIZE IN
TWO-LEVEL MODELS WITH DICHOTOMOUS
RESPONSES
The multivariate normal distribution is an attractive framework for statistical modeling, when data on the response
variable are continuous and normal. However, when the
response variable is dichotomous, the distribution is incorrect, and therefore conclusions based on the normal distribution are likely to be flawed.
When formulating a two-level model, it is common to
assume normality for the cluster-level (level 2) variation and
to assume independence of units within-cluster (level 1)
conditional on the cluster variable, thus generating a model
in which individuals are marginally correlated within clusters. In the multivariate normal case, the interpretation of
both fixed effects and random effects (individual residual
error and cluster variation) is simple. This is not true in the
case of dichotomous response variables, because of the
nonlinear relation between the covariates and the response
variable (typically a logit relation).
Choosing a logistic regression model leads to the choice
between two different odds ratio interpretations of the fixedeffects parameters, the subject-specific interpretation and the
population-averaged interpretation (7, 8). Taking a population-averaged approach, heterogeneity is considered a
nuisance, whereas the subject-specific approach opens up
the possibility of quantification of heterogeneity based on
the two-level model (6).
Quantifying the variance component is a different matter.
Usually, researchers either calculate a so-called intraclass
correlation coefficient based on an estimate of the variance
components and the residual variance or report the variance
component, neither of which is very useful. Many different
measures of intraclass correlation have been suggested (9).
Nevertheless, intraclass correlations have serious interpretational drawbacks (10) for binary responses. First, the intraclass correlation may be interpreted as the proportion of the
total variation attributable to variation between clusters, or
as correlation between persons within the same cluster. That
is, the intraclass correlation does not convey information
regarding variation between clusters. Therefore, it is not a
very useful measure when determining whether or not clustering is an important factor. Second, the intraclass correlation is not comparable with the fixed effects, which have
odds ratio interpretations. This is unfortunate, because
random effects are not very different from fixed effects in
nature; random effects are fixed effects with an additional
distributional assumption. Therefore, it seems natural to
quantify variation between the random effects using odds
ratios. Alternatively, one may use the variance component
itself, but it is quite difficult to interpret, since it is on the log
odds ratio scale.
In this paper, we unify interpretations of fixed and random
effects in a subject-specific approach, explaining the use of
the MOR and IOR measures in the case of a two-level
logistic regression model.
The model
Consider a population of N individuals. Each individual
has a vector of covariates, x, and each individual belongs to
one of K clusters. The parameters corresponding to the covariates are in the vector β. The K mutually independent cluster
variables, u1, u2, ..., uK, are not to be estimated, since they are
not of interest per se; rather, it is the variation between clusters that needs be quantified. Therefore, a normal distribution is assumed for the u’s, and parameters characterizing
this distribution can then be used to characterize the heterogeneity induced by the random effects.
The response variable, Y, is a dichotomous variable. That
is, for each individual, it is observed whether Y = 0 or 1. The
model has two levels.
Level 1: For a person with covariate vector x, corresponding
to the kth cluster, the probability of observing Y = 1 is
exp ( η ( x ,u k ) )
-.
P ( Y = 1 u k ;x ) = ----------------------------------------1 + exp ( η ( x ,u k ) )
Level 2: For that individual, the second-level equation is
η(x, uk) = βx + uk,
where uk ∼ N(0, σ2). The covariate vector, x, contains individual-level (level 1) and cluster-level (level 2) covariates.
Although there is no difference in terms of formulating the
model, there are differences when interpreting the effects of
variables varying within clusters and cluster-level variables.
For variables varying within a cluster, the usual odds ratio
interpretations apply for comparisons of persons belonging
to the same cluster; for example, a gender effect may be
interpreted as an odds ratio between a woman and a man
belonging to the same cluster and with the same covariates,
except for gender.
For variables varying on the cluster level, the quantification is more difficult; the usual odds ratio interpretation is
incorrect, because it is necessary to compare persons with
different random effects, since the variable of interest does
not vary between individuals within-cluster. A measure for
this situation is described below in the section “The IOR.”
It is of interest to quantify the clustering as something
other than variation on the underlying linear scale, because
this is difficult to interpret and relate to the fixed effects,
which are quantified in terms of odds ratios. Therefore, it
would be useful to have an odds ratio interpretation of the
cluster variation as well. A measure is described below.
The MOR
The MOR quantifies the variation between clusters (the
second-level variation) by comparing two persons from two
randomly chosen, different clusters. Consider two persons
with the same covariates, chosen randomly from two different
clusters. The MOR is the median odds ratio between the
person of higher propensity and the person of lower propensity.
The MOR is very easy to calculate, because it is a simple
function of the cluster variance, σ2:
Am J Epidemiol 2005;161:81–88
Quantification of Heterogeneity due to Clustering 83
2
–1
MOR = exp ( 2 × σ × Φ ( 0.75 ) ),
where Φ(·) is the cumulative distribution function of the
normal distribution with mean 0 and variance 1, Φ–1(0.75) is
the 75th percentile, and exp(·) is the exponential function. A
theoretical derivation of the formula is provided in the
Appendix.
The measure is always greater than or equal to 1. If the
MOR is 1, there is no variation between clusters (no secondlevel variation). If there is considerable between-cluster
variation, the MOR will be large. The measure is directly
comparable with fixed-effects odds ratios.
The IOR
The IOR is a fixed-effects measure for quantification of
the effect of cluster-level variables. Consider two persons
with different cluster-level covariates, x1 and x2. The IOR is
an interval for odds ratios between two persons with covariate patterns x1 and x2, covering the middle 80 percent of the
odds ratios.
The IOR is only slightly more difficult to calculate than
the MOR. The lower and upper bounds of the interval are
2
–1
2
–1
IOR lower = exp ( β × ( x 1 – x 2 ) + 2 × σ × Φ ( 0.10 ) )
and
IOR upper = exp ( β × ( x 1 – x 2 ) + 2 × σ × Φ ( 0.90 ) ),
where Φ–1(0.10) = –1.2816 and Φ–1(0.90) = 1.2816 are the
10th and 90th percentiles of the normal distribution with
mean 0 and variance 1. A theoretical derivation of the
formula is provided in the Appendix.
The interval is narrow if the between-cluster variation is
small, and it is wide if the between-cluster variation is large.
If the interval contains 1, the cluster variability is large in
comparison with the effect of the cluster-level variable. If the
interval does not contain 1, the effect of the cluster-level
variable is large in comparison with the unexplained
between-cluster variation.
According to the model, the odds ratios can take any value
between zero and infinity. However, to put meaning into the
IOR, it is natural to report an interval that is likely to contain
the odds ratio when randomly choosing two persons with
two specific sets of covariates. Reporting an 80 percent
interval has been suggested (6), and that is reasonable,
because it covers a large fraction of the odds ratios. Note that
the IOR is not a confidence interval.
EXAMPLES
It is known that different geographic boundaries may have
different effects on the same response, a problem that in
medical geography is known as the “modifiable area unit
problem” (11). Analogously, the same neighborhood definition may affect different individual outcomes differently.
Moreover, one also needs to consider a temporal or longitudinal perspective to define individual exposure to the neighAm J Epidemiol 2005;161:81–88
borhood environment, because individuals move between
different areas during the course of their lives. Therefore, in
cross-sectional analysis of neighborhood effects on health,
one should expect that atherosclerotic disorders with a long
natural history (e.g., ischemic heart disease) are influenced
only slightly by the actual neighborhood territorial limits.
Following the same reasoning, behavior-related outcomes
(e.g., the choice of a specific kind of physician) may be
much more susceptible to neighborhood influences.
Multilevel analysis has provided evidence that the socioeconomic characteristics of the neighborhood environment
affect individual risk of ischemic heart disease (12). However,
neighborhood variation seems to be low (13).
In Sweden, cost is not a specific determinant of people’s
choice of a private versus public health-care practitioner, since
the county councils support both economically. However,
private physicians are outside of the public health-care system
and therefore are less susceptible to health-care strategies
directed by the county councils. Preferring a public practitioner versus a private practitioner might suggest dysfunction
in some parts of the public health-care system, but it might
also be explained by individual preferences, demands, and
expectations related to socioeconomic position. The greater
confidentiality of medical records in the private sector is
another factor that could explain individual preference. Moreover, area of residence might influence individual decisions
over and above individual characteristics.
Using a multilevel approach, we investigated the probability of being hospitalized for ischemic heart disease and the
probability of visiting a public practitioner versus a private
practitioner among Swedish men. We give two examples,
followed by some comments on issues that have not already
been covered.
Study population and assessment of variables
The study population consisted of 11,312 men aged 45–85
years residing in 98 of the 110 neighborhoods in the city of
Malmö, Sweden. All of the men had visited a physician
during the year 1999. Information was obtained from the
Register on Health Care Utilization in Skåne, Sweden (14).
For the analysis, the variable “neighborhood” was used as
a cluster (level 2) variable. Two individual-level (level 1)
explanatory variables were considered: “age,” which was a
four-category variable with the categories 65–69, 70–74,
75–79, and 80–85 years, and “education,” which was a
dichotomous indicator of having 9 or fewer years of
schooling. A cluster-level (level 2) explanatory variable,
“neighborhood education,” was also used in the analyses.
This variable was a dichotomous indicator of the neighborhood’s educational level being below the median for all
neighborhoods. In example 1, the response variable was an
indicator of whether a person had been hospitalized for
ischemic heart disease (coded as 1) or not (coded as 0).
Ischemic heart disease was defined by hospital discharge
diagnosis in 1999. The relevant codes according to the International Classification of Diseases, Tenth Revision, are I20
(angina pectoris), I21 (acute myocardial infarction), I22
(subsequent myocardial infarction), and I50 (heart failure).
In example 2, the response variable was whether a person
84 Larsen and Merlo
TABLE 1. Estimates and standard errors from analyses of hospitalization for ischemic heart disease (models 1 and 2)
and visiting a public physician (models 3 and 4), Malmö, Sweden, 1999
Hospitalization
Model 1
Visits to a public physician
Model 2
Model 3
Model 4
Estimate
SE*
Estimate
SE
Estimate
SE
Estimate
SE
70–74 vs. 65–69
0.458
0.132
0.458
0.132
0.116
0.066
0.116
0.066
75–79 vs. 65–69
0.498
0.137
0.506
0.137
0.262
0.073
0.264
0.074
80–85 vs. 65–69
0.733
0.142
0.750
0.142
0.437
0.083
0.442
0.083
Education (low vs. high)
0.223
0.116
0.200
0.117
0.243
0.062
0.239
0.062
Neighborhood education
(low vs. high)
—†
0.236
0.088
—
1.018
0.266
0.009
0.020
1.815
1.593
0.246
Fixed-effects variables ( β̂ )
Age group (years)
2
Random effects ( σ̂ )
Neighborhood
0.028
0.025
0.278
* SE, standard error.
† Not included.
had visited a public physician (coded as 1) or not (coded as
0). We restricted our analysis to persons who had used the
health-care system and visited a general practitioner at least
once during the year 1999.
Parameters in the random-effects model were estimated
using restricted iterative generalized least squares. The
MLwiN software package (15), version 1.1, was used to
perform the analyses. Extrabinomial variation was explored
systematically in all of the models, and there was no indication of either under- or overdispersion.
Population-averaged parameters were calculated on the
basis of the approximate formula (7)
tion versus a high level were 1.25 in model 1 and 1.22 in
model 2. These odds ratios are conditional on age and
neighborhood.
Note that the above odds ratios are correct only for
comparisons of persons belonging to the same cluster. When
comparing persons from different clusters, it is necessary to
calculate an IOR or leave the subject-specific interpretation
and consider population-averaged odds ratios. The population-averaged odds ratio is the odds ratio between two
persons from different clusters. In models 1 and 2, the estimates on the linear predictor scale are only shrunk by factors
of
2
2
2
2
1 ⁄ 1 + ( 16 × 3 ⁄ ( 15 × π ) ) × 0.028 = 0.995
2
2
2
β̃ = β̂ ⁄ 1 + ( 16 × 3 ⁄ ( 15 × π ) ) × σˆ ,
where β̃ is the population-averaged parameter and β̂ is the
2
subject-specific parameter. The cluster variance is σ̂ .
and
Example 1: hospitalization for ischemic heart disease
respectively. This indicates an apparently small amount of
heterogeneity of the clusters. Thus, in effect, the attenuation
is hardly visible in this analysis.
In model 1, the cluster heterogeneity is interpreted in the
following way. Consider two randomly chosen persons with
the same covariates from two different clusters (e.g., two
more-educated persons aged 65–69 years) and conduct the
hypothetical experiment of calculating the odds ratio for the
person with the higher propensity to be hospitalized versus
the person with the lower propensity. Repeating this comparison of two randomly chosen persons will lead to a series of
odds ratios. This distribution is estimated when estimating
the parameters, and it is shown in figure 1 (on the log scale).
The median of these odds ratios between the person with a
higher propensity and the person with a lower propensity is
estimated to be 1.17 in model 1. This is a low odds ratio, and
it suggests that the clustering effect is small even without
inclusion of any cluster-level covariates. When neighbor-
In example 1, two models are considered. In model 1, age
and individual education were included together with a
random neighborhood effect. Model 2 was an extension of
model 1 that also included the cluster-level covariate neighborhood education.
In both analyses, it appeared that older people were more
likely to be hospitalized than younger people and that lesseducated persons were more likely to be hospitalized than
the more educated. The estimates shown in table 1 were very
similar in the two models, but there were some differences
(discussed in detail below). The parameter estimates were
transformed into odds ratios, which are shown in table 2.
The individual-specific fixed effects are conditional on the
random effects. That is, they may be interpreted as odds
ratios for within-cluster comparisons. For the individual
education variable, the odds ratios for a low level of educa-
1 ⁄ 1 + ( 16 × 3 ⁄ ( 15 × π ) ) × 0.009 = 0.998,
Am J Epidemiol 2005;161:81–88
Quantification of Heterogeneity due to Clustering 85
TABLE 2. Odds ratios for being hospitalized for ischemic heart disease (models 1 and 2) and visiting a public
physician (models 3 and 4), Malmö, Sweden, 1999
Hospitalization
Model 1
Visits to a public physician
Model 2
Fixed-effects variables, varying within cluster
Model 3
Model 4
Odds ratio
Age group (years)
70–74 vs. 65–69
1.58
1.58
1.12
1.12
75–79 vs. 65–69
1.65
1.66
1.30
1.30
80–85 vs. 65–69
2.08
2.12
1.55
1.56
1.25
1.22
1.28
1.27
Education (low vs. high)
Random effects
Neighborhood
Median odds ratio
1.17
Fixed-effects variables, constant within
cluster
Neighborhood education (low vs. high)
1.09
3.61
3.33
Interval odds ratio
—*
[1.07; 1.50]
—
[0.28; 27.3]
* Not included.
hood education is included (model 2), the unexplained
cluster heterogeneity (comparing persons from neighborhoods of the same kind—for example, both neighborhoods
with a high level of education) decreases, yielding an MOR
of 1.09, which is a very low odds ratio. Thus, there is very
little variation between neighborhoods in the propensity for
hospitalization for ischemic heart disease.
Although the cluster heterogeneity is small in both models,
it is interesting that neighborhood education accounts for
quite a large part of the variation between clusters. This can
best be seen from the IOR in the following way. The IOR is
[1.07; 1.50]. This means that when randomly choosing two
persons with identical individual covariates (e.g., 80- to 85year-olds with a low individual level of education) from a
low-education neighborhood and a high-education neighborhood, the odds ratio between the two lies within the interval
in 80 percent of the cases. The estimated distribution of the
odds ratios is shown in figure 2 (on the log scale). Two
things are worth noticing regarding this interval. First, the
interval does not contain 1. This suggests that the effect of
neighborhood education is large relative to the cluster effect,
because apparently there is little chance that the person from
FIGURE 1. Model 1: density of the distribution of log odds ratios
between a person with a higher propensity for being hospitalized for
ischemic heart disease and someone with a lower propensity. The
median value in the distribution of the log odds ratios is 0.16. MOR,
median odds ratio.
FIGURE 2. Model 2: density of the distribution of log odds ratios
between a person from a less-educated neighborhood and a person
from a more-educated neighborhood, when the persons belong to the
same age group and have the same personal educational level. The
10th and 90th percentiles in the distribution of the log odds ratios are
0.07 and 0.41, respectively. IOR, interval odds ratio.
Am J Epidemiol 2005;161:81–88
86 Larsen and Merlo
the less-educated neighborhood has a lower propensity for
hospitalization than the person from the more-educated
neighborhood. Second, the interval appears to be relatively
narrow, which suggests that there is little cluster heterogeneity (as does the MOR). Therefore, a fairly large proportion
of the (small) variation between neighborhoods in the
propensity to be hospitalized may be explained by neighborhood educational level.
A population-averaged approach cannot convey the ramifications of the effect of neighborhood education on the
propensity to be hospitalized in an equally satisfactory way.
The results of such an analysis would be quantified in terms
of a (slightly attenuated) odds ratio of
2
2
exp ( 0.236 ⁄ 1 + ( 16 × 3 ⁄ ( 15 × π ) ) × 0.009 ) = 1.27
and some measure of within-cluster association. Contrary to
the MOR and the IOR, these two measures do not convey
much information regarding the effect of clustering on the
likelihood of being hospitalized.
Example 2: visits to a public or private physician
In example 2, two models are considered. In model 3, age
and individual education were included with a random
neighborhood effect. Model 4 was an extension of model 3
that also included the cluster-level covariate neighborhood
education.
The parameter estimates are shown in table 1, and the odds
ratios are shown in table 2. The odds ratios for individual
education were 1.28 and 1.27 in models 3 and 4, respectively; this suggests a higher propensity to visit a public
physician for the less educated, conditional on age and
neighborhood.
However, the results regarding clustering are quite
different from those in example 1, and this implies substantial differences between the subject-specific and populationaveraged estimates in this example. In models 3 and 4, the
linear predictor parameters are attenuated by factors of
2
2
1 ⁄ 1 + ( 16 × 3 ⁄ ( 15 × π ) ) × 1.815 = 0.784
and
2
2
1 ⁄ 1 + ( 16 × 3 ⁄ ( 15 × π ) ) × 1.593 = 0.803,
respectively. This yields population-averaged effects of 1.21
for individual education in both models. These effects are for
comparisons of persons of the same age belonging to
different clusters.
In model 3, for two persons with the same individual-level
covariates, the MOR between the person living in the neighborhood with the higher propensity to visit a public physician and the person living in the neighborhood with the
lower propensity is 3.61. This is a high odds ratio, suggesting
that the heterogeneity is substantial. Including neighborhood
education as a covariate reduces the unexplained heterogeneity between neighborhoods to an MOR of 3.33, which is
still high. Thus, the propensity to visit a public physician
varies a great deal between neighborhoods. This is also
reflected in the IOR, which is very broad: [0.28; 27.3]. The
interpretation is that if one is randomly selecting two
persons, one from a low-education neighborhood and one
from a high-education neighborhood, and comparing their
odds of having visited a public physician, the middle 80
percent of the odds ratios will lie within this interval. The
interval contains 1, which implies that neighborhood education does not account for a substantial amount of the neighborhood heterogeneity. In addition, the interval is quite
broad, reflecting a large amount of unexplained variation
between neighborhoods in the propensity to visit a public
physician. Other cluster-level variables are needed to explain
the cluster heterogeneity.
A population-averaged approach might lead to a different
conclusion. The population-averaged effect of neighborhood
education is
2
2
exp ( 1.018 ⁄ 1 + ( 16 × 3 ⁄ ( 15 × π ) ) × 1.593 ) = 2.26.
This is by far the most important of the fixed effects, and it
suggests that neighborhood education is indeed an important variable when analyzing the propensity to visit a public
physician. However, this is not completely in agreement
with the previous interpretations based on the randomeffects approach. The reason for this apparent discrepancy
is the large unexplained cluster heterogeneity, which in the
population-averaged approach is only considered a
nuisance.
Other issues regarding the MOR and IOR
In a previous study (6), the IOR for being prescribed
morphine when calling two different doctors and giving
“back pain” versus “other” as the reason for the call was
[2.08; 22.9]. This is an example of a large cluster effect
(variation between physicians) along with a strong fixed
effect (cause for the call).
Another, perhaps less interesting situation is also
possible—the situation where there is relatively little variation between clusters and there is little or no effect of the
cluster-level covariate. Although this may be a common situation, it is also the one that is most likely to be overlooked
because of type II error.
It is also possible to include continuous cluster-level covariates using the formulas given above in the section “The
IOR,” and the interpretation is straightforward.
DISCUSSION
In this paper, we discuss interpretational aspects of the
multilevel logistic regression model. Two measures, the
MOR and the IOR, are proposed and applied to data
concerning neighborhood effects on people’s propensity to
visit public physicians and their likelihood of being hospitalized because of ischemic heart disease.
Regarding interpretation of fixed effects, the choice
between generalized estimating equations and the randomeffects model is the choice between population-averaged and
subject-specific interpretations. Which one to choose
depends entirely on the substantive research question. In
Am J Epidemiol 2005;161:81–88
Quantification of Heterogeneity due to Clustering 87
examples 1 and 2, the effect of individual education is conditional on neighborhood. A subject-specific approach is
proper, because it facilitates measures of effect on the individual level: What is the direct effect of having a high individual level of education versus a low level for a person
under his/her specific conditions? In this sense, the subjectspecific parameters are more “clean” measures of the effect
of individual education, because they are not attenuated by
unmeasured heterogeneity.
A general argument favoring the random-effects model
is the possibility of quantification of heterogeneity in an
intuitively attractive way. As is shown in the four models
in the examples, the heterogeneity may be quantified by
characteristics from the distribution of the odds ratios
between pairs of randomly chosen persons from different
neighborhoods. This provides a measure of heterogeneity
on a scale that is familiar to researchers who have worked
with the logistic regression model, namely odds ratios.
The usual odds ratio interpretations (conditioning on all
other covariates and random effects) are proper for covariates that vary within-cluster, whereas they are improper for
cluster-level covariates, because it is impossible to make
comparisons within-cluster. That is, all comparisons must be
made between persons belonging to two different clusters,
and thus the odds ratio is no longer a fixed quantity but a
random variable. In this way, the heterogeneity becomes
relevant when quantifying the effect of a cluster-level
covariate. The IOR incorporates both the fixed effect and the
cluster heterogeneity in an interval, allowing for a more
detailed description of the covariate effect.
The MOR and the IOR have been applied in a two-level
model for the quantification of between-rater variability in
a study of mammographic screening (16). Other studies
have used the pairwise odds ratio (17, 18), which is similar
to the MOR and IOR in the sense that it is also based on
comparisons between pairs of individuals. However, where
the MOR and IOR quantify between-cluster heterogeneity,
the pairwise odds ratio quantifies association (concordance/discordance) between pairs of individuals withincluster.
Both the MOR and the IOR are very easy to calculate,
since they are simple functions of the parameters in the
model. Thus, no additional analyses are necessary. A pocket
calculator with exponential and square root functions is
sufficient for calculating the measures from the parameters.
The MOR and the IOR may be extended to higher-order
multilevel models, models with a nonhierarchical structure,
and models with random slopes. They can also be extended
to log-linear (Poisson) and linear (normal) responses, as well
as to models with nonnormal random effects.
ACKNOWLEDGMENTS
This study was supported by grants 2002-054 and
2003-0580 (Principal Investigator, Dr. Juan Merlo) from the
Swedish Council for Working Life and Social Research.
Am J Epidemiol 2005;161:81–88
The authors express their gratitude to Prof. Niels Keiding
of the Department of Biostatistics, University of Copenhagen, for his comments on an early draft of this paper. The
authors also thank Dr. Basile Chaix of the Research Team on
the Social Determinants of Health and Health Care, French
National Institute of Health and Medical Research, for his
comments on the final manuscript.
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88 Larsen and Merlo
APPENDIX
2
–1
z = exp ( 2 × σ × Φ ( 0.75 ) ).
Mathematical Derivations
The interval odds ratio (IOR)
The median odds ratio (MOR)
The odds ratio between two persons with identical covariates from two different clusters is exp(u1 – u2), where u1 and u2
are the two random cluster variables. Consequently, the odds
ratio for the person with the higher propensity versus the
person with the lower propensity is exp ( u 1 – u 2 ) . Since data
for the two cluster variables are assumed to be independent
and normally distributed with mean zero and variance σ2, the
distribution of exp ( u 1 – u 2 ) may be characterized by the
cumulative distribution, F, in the following way. For z > 0,
F ( z ) = P ( exp ( u 1 – u 2 ) ≤ z )
⎛ u1 – u2
log ( z ) ⎞
- ≤ -------------------⎟
= P ⎜ -----------------2
2
⎝ 2×σ
2×σ ⎠
⎛ log ( z ) ⎞
= 2Φ ⎜ -------------------⎟ – 1,
⎝ 2 × σ 2⎠
where Φ(·) is the cumulative distribution function for the
standard normal distribution—that is, a normal distribution
with mean 0 and variance 1. Thus, the density function, f (·),
for the distribution of exp ( u 1 – u 2 ) becomes
∂
f ( z ) = ----- F ( z )
∂z
log ( z ) ⎞
1 2 ⎛ ------------------⎟ .
= --- -----ϕ
⎜
2
z σ ⎝
2
2×σ ⎠
The MOR is the median of this distribution, so it can be
calculated as the solution to the equation, F(z) = 0.5, which
leads to
The odds ratio between two persons from two different clusters with covariates x1 and x2 is exp(β × (x1 – x2) + (u1 – u2)).
Since data for the two cluster variables are independent and
normally distributed, the odds ratio is lognormally distributed
with cumulative distribution function G, where, for z > 0,
G ( z ) = P ( exp ( β × ( x 1 – x 2 ) + ( u 1 – u 2 ) ) ≤ z )
⎛ u 1 – u 2 log ( z ) – β × ( x 1 – x 2 )⎞
= P ⎜ ------------------- ≤ ----------------------------------------------------⎟
2
⎝ 2 × σ2
⎠
2×σ
⎛ log ( z ) – β × ( x 1 – x 2 )⎞
-⎟ .
= Φ ⎜ --------------------------------------------------2
⎝
⎠
2×σ
Consequently, the density, g(·), becomes
∂
g ( z ) = ----- G ( z )
∂z
⎛ log ( z ) – β × ( x 1 – x 2 )⎞
1 1
= --- ------------------- ϕ ⎜ ----------------------------------------------------⎟ .
z
2 ⎝
2
⎠
2×σ
2×σ
The a-percentile in the distribution of the odds ratio is the
solution to G(z) = a, which leads to
–1
2
z = exp ( β × ( x 1 – x 2 ) + 2 × σ × Φ ( a ) ).
In particular, the 10th percentile is
2
–1
2
–1
IOR lower = exp ( β × ( x 1 – x 2 ) + 2 × σ × Φ ( 0.1 ) ),
and the 90th percentile is
IOR upper = exp ( β × ( x 1 – x 2 ) + 2 × σ × Φ ( 0.9 ) ).
Am J Epidemiol 2005;161:81–88