Open Journal of Statistics, 2013, 3, 1-6 Published Online August 2013 (
A Comparison of Statistical Methods for Analyzing
Discrete Hierarchical Data: A Case Study of
Family Data on Alcohol Abuse
Yuanyuan Liang1, Keumhee Chough Carriere2*
1Department of Epidemiology and Biostatistics, University of Texas Health Science Center at San Antonio,
San Antonio, USA
2Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada
Email:, *
Received June 11, 2013; revised July 11, 2013; accepted July 19, 2013
Copyright © 2013 Yuanyuan Liang, Keumhee Chough Carriere. This is an open access article distributed under the Creative Com-
mons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work
is properly cited.
Although hierarchical correlated data are increasingly available and are being used in evidence-based medical practices
and health policy decision making, there is a lack of information about the strengths and weaknesses of the methods of
analysis with such data. In this paper, we describe the use of hierarchical data in a family study of alcohol abuse con-
ducted in Edmonton, Canada, that attempted to determine whether alcohol abuse in probands is associated with abuse in
their first-degree relatives. We review three methods of analyzing discrete hierarchical data to account for correlations
among the relatives. We conclude that the best analytic choice for typical correlated discrete hierarchical data is by non-
linear mixed effects modeling using a likelihood-based approach or multilevel (hierarchical) modeling using a quasi-
likelihood approach, especially when dealing with heterogeneous patient data.
Keywords: Non-Linear Mixed Effects Model; Multilevel Model; Generalized Estimating Equations; Mantel-Haenszel
Odds Ratio; Specificity; Sensitivity
1. Introduction
The purpose of this paper is to investigate best methodo-
logical approaches that frequently arise in the analysis of
non-independent discrete hierarchical medical data. There
are various methods of handling such types of data. The
most general method is a non-linear mixed effects model,
which uses a likelihood-based approach. Another method
is fitting a multilevel model based on the quasi-likeli-
hood approach proposed by Goldstein (1991) [1]. The
generalized estimating equations (GEE) method propo-
sed by Liang and Zeger (1986) uses the concept of quasi-
likelihood to fit a generalized linear model (GLM) to
clustered data for marginal model building [2]. We com-
pare these methods for their performance, as applied to
hierarchical alcoholism data obtained from Edmonton,
Alberta, Canada. We outline the strengths and weakness-
es of each method.
Data were obtained from a population-based study of
mental disorders conducted in Edmonton, Canada. For
details of the study design, see Newman and Bland (2006)
and the references contained therein [3]. Interviews were
conducted with 924 index subjects, called probands,
randomly sampled from the population, and 2387 of their
first-degree relatives (briefly, relatives). Mental disorders
were diagnosed on a lifetime basis (that is, present at the
time of interview or ever in the past) using a validated
and structured questionnaire. The response variable in
this case is the diagnosis of alcohol abuse in relatives,
which is dichotomous (1 yes, 0 no). Overall, there were
206 (22.3%) and 461 (19.3%) cases of alcohol abuse
among the probands and their relatives, respectively.
We are interested in determining whether (a lifetime
history of) alcohol abuse in probands is associated with
alcohol abuse in relatives, after adjusting for age and sex
of probands and relatives. The data exhibit a hierarchical
(or clustered) structure to the extent that the relatives of a
given proband have more in common than would be ex-
pected in a corresponding random sample from the pop-
ulation. The shared characteristics among relatives from
a given family result in data hierarchies and clustering
*Corresponding author.
opyright © 2013 SciRes. OJS
effects that must be incorporated into the statistical anal-
ysis—the usual assumption of the independent and iden-
tical distribution (i.i.d.) of the variables is not met.
The data consist a large number of families (n = 924).
However, unequal and often small family sizes, ranging
from 1 to 12 individuals, create challenges and possible
complications in statistical data analysis and model
checking. We seek to make general recommendations on
how to best estimate the parameters and test the good-
ness-of-fit of modeling such data.
2. Methods
2.1. Preliminary Analyses
The statistical issue in analyzing hierarchical data, as in
the study of the familial aggregation of mental disorders,
is to sort out and adjust for associations within a cluster/
family. That is, we need to understand whether the pres-
ence of the disorder in a proband increases the risk of the
disorder in a relative. Investigators engaged in this kind
of research typically treat the relatives as if they were a
retrospective cohort followed from the beginning of the
risk period for the mental disorder until either the onset
of the disorder or the time of interview, whichever comes
first. The mental disorder status of the proband (Yes/No)
is the binary independent variable of primary interest. As
a preliminary examination, we compute odds ratios, spe-
cifically, crude and Mantel-Haenszel odds ratio estima-
tors [4]. In the Mantel-Haenszel analyses, we accounted
for the length of time at risk by including the age of the
relative as an independent variable. We considered six
age groups: 18 - 24, 25 - 34, 35 - 44, 45 - 54, 55 - 64, and
2.2. Statistical Modelling
Standard logistic regression that assumes all observations
are independent was performed to examine the effect of
alcohol abuse in a proband on the risk of alcohol abuse in
his/her relatives, adjusting for age and sex of the pro-
bands and the relatives. We treated males and the 18 - 24
age group as reference categories. First-order interaction
terms were also explored.
However, since the observations are not independent,
we also considered regression methods of risk factor
modelling that explicitly address the correlation structure
of the data: 1) non-linear mixed effects (NLME) model,
2) multilevel model, and 3) generalized linear model
(GLM) using generalized estimating equations (GEE).
2.2.1. Non-Linear Mixed Effects Model
Let yij represent the response of the jth relative of the ith
proband (i.e. from the ith family), where ,
with ranging from 1 to 12, and
. Since the response data are binary, we
1,2,, 924i
12 i
1yπij ij
EyP ij
 represents the expected value
of the response variable for the jth relative in the ith fam-
ily, and the non-linear link function to model the odds as:
log π1π
ij ijij
XβZγ. (1)
Model (1) contains a fixed part, X
, and a random
part, . Here,
and are the fixed and random
design matrices, respectively,
is a vector of unknown
fixed effects, is a vector of unknown random effects,
γ and
γΣ, and ij
is the unknown
random error.
When the model (1) has a univariate random effect
with 1
, we have a special case, known as a random
intercept model. Our data have no other available patient
level random effects, and therefore, we consider a ran-
dom intercept model. Then, is assumed to follow a
normal distribution with mean 0 and variance
Non-linear mixed effects models are fitted by maxi-
mizing an approximation to the likelihood integrated
over the random effects. There are several integral ap-
proximations available. We use the default optimization
technique (dual quasi-Newton) and the default integra-
tion method (adaptive Gaussian quadrature) in PROC
NLMIXED to obtain the parameter estimates of the mo-
del [5,6].
2.2.2. Multilevel Model
For the multilevel model, subscript j denotes the level-1
unit (relative) and subscript i denotes the level-2 unit
(family). We define the probability as a function of
an intercept and several explanatory variables similar to
the non-linear mixed effects model considered above.
The full model in terms of can be written as
where ~0,.
ij i
 
Here, ij
is the variance of the level-1 (relative) ran-
dom term with mean 0 and variance 1. The intercept is
being modeled as random at the level of the family (lev-
el-2), that is, the probability of a relative having an alco-
hol abuse disorder at the reference values (all the ex-
planatory variables set to zero) is different across fami-
lies, while all the other parameters
are fixed. We
assume that the random part of the intercept, i
, follows
a normal distribution with mean 0 and variance 2
Note that this model is identical to model (1) under the
random intercept model for two level hierarchical data,
but it is capable of building multilevel models. Also it
uses an algorithm, which is different from that for NLME
to analyze the data when using commercial software.
We use the second-order penalized quasi-likelihood
(PQL2), which has been shown to be least biased, com-
Copyright © 2013 SciRes. OJS
pared to the first-order marginal quasi-likelihood Method
[7]. Bootstrap estimation is an alternative, as it corrects
the bias associated with the quasi-likelihood procedures
[8]; however, the improved accuracy is usually obtained
at the expense of lengthy computational time.
2.2.3. GEE Method
Generalized estimating equations (GEE) method pro-
posed by Liang and Zeger (1986) uses the concept of
quasi-likelihood to fit a generalized linear model (GLM)
to clustered data. The GEE method for estimating
an extension of the independence estimating equation to
the correlated data. The GEE is given by the score func-
 
βVY β
with K = 924 in our case. If is the true correla-
tion matrix of , then the true covariance matrix of
is given as
12 12
iii i
VARαA, where Ai is an ii
diagonal matrix with as the jth diagonal ele-
ment, and
var ij
is the scale parameter. Liang and Zeger
(1986) proposed GEE for
based on Equation (3)
is replaced by an estimator
, which is
-consistent given
, and is replaced by an es-
αβ , which is also
-consistent given
The correlation parameters and the scale parame-
are estimated iteratively using the current value of
the parameter vector ˆr
at the rth iteration. Finally, we
calculate the appropriate functions of the Pearson residu-
Under certain regularity conditions, the solution to
Equation (3) is consistent and asymptotically multivari-
ate normal regardless of whether the working correlation
matrix has been modeled correctly. To account for the
family effect using GEE in marginal model building, we
assume that the correlation among members of the same
family is the same for all 924 families. That is, they all
share the same working correlation matrix. We analyze
the alcoholism data using three different structures for
the working correlation matrices—exchangeable, AR(1)
and unstructured.
2.3. Model Diagnostics
To check the prediction accuracy achieved under each
model, we compare the model prediction results and the
observed presence of the disorder using a classification
table, where a predicted probability of alcohol abuse of
0.5 or more is classified as a “positive” prediction. For
the diagnostic tests, two critical components determine
the model’s accuracy: sensitivity (the probability that a
test is positive given that the person has the disorder) and
specificity (the probability that a test is negative given
that the person does not have the disorder). In a perfect
model, all cases will be on the diagonal of the classifica-
tion table and the overall percent correct will be 100.
The data were analyzed using commercial software:
for the non-linear mixed effects model, we used PROC
NLMIXED in SAS (SAS Institute, Cary, NC), for the
hierarchical model, we used MLwiN (Centre for Multi-
level Modelling, University of Bristol, UK), and for the
GLM/GEE model we used PROC GENMOD in SAS.
The technical details of these three non-linear modeling
approaches and the SAS codes are available upon re-
3. Results
Table 1 provides summary information about the age
and sex distribution of relatives and probands respec-
tively. In a preliminary analysis, we assume that relatives
in a given family are independent; that is, they are un-
correlated. In all models considered, we treat the alcohol
abuse status of probands as the exposure, and the alcohol
abuse status of relatives as the outcome: Exposure is de-
fined as Yes if the proband has had an alcohol abuse
problem, No if the proband has no alcohol abuse problem;
and Outcome is defined as Yes if the relative has had an
alcohol abuse problem, No if the relative has no alcohol
abuse problem.
Table 2 summarizes the contingency table analysis of
Table 1. Age and sex distribution of probands and relatives.
Relatives Probands
Count % Count %
Age 18 - 24298 12.5 61 6.6
25 - 34636 26.6 248 26.8
35 - 44463 19.4 181 19.6
45 - 54333 14.0 129 14.0
55 - 64356 14.9 135 14.6
65+ 301 12.6 170 18.4
Sex Female1358 56.9 604 65.4
Male 1029 43.1 320 34.6
Table 2. Crude and Mantel-Haenszel odds ratio estimates.
Statistic Crudea MHb MHc
Odds ratio 1.525 1.710 2.421
95% CI (1.208, 1.926)(1.325, 2.206) (1.771, 3.309)
Width of CI 0.718 0.881 1.538
aCrude odds ratio based on a standard 2X2 table; bMantel-Haenszel odds
ratio based on a stratification by age and sex of relatives; cMantel-Haenszel
odds ratio based on a stratification by age and sex of relatives and probands.
Copyright © 2013 SciRes. OJS
exposure versus outcome, with and without adjustment.
From this table, we can see that alcohol abuse in a pro-
band is strongly associated with that in a relative. The
second Mantel-Haenszel odds ratio estimate incorporates
stratification by age and sex of both probands and rela-
tives; alcohol abuse in a proband increases the odds of
alcohol abuse in a relative by more than two folds.
In regression modelling, we considered interactions in
model (1) with all methods, but none of the interaction
terms were found to be statistically significant. As men-
tioned in the Methods section, for two level hierarchy,
NLME and hierarchical models were basically identical.
Any differences are more of algorithmic and computa-
tional in nature than anything structural in modeling.
Note that these two models allowed probands to be ran-
dom, accommodating their natural heterogeneity, while
in GEE modeling, we only get the marginal effects.
Table 3 summarizes the estimates, along with their
standard errors in the final model. The results are broadly
similar in all regression approaches qualitatively. As can
be seen, alcohol abuse in a proband significantly in-
creases the odds of alcohol abuse in a relative. The age of
the proband is not statistically significant. For the age of
relatives, there is a decreasing trend in the odds of alco-
hol abuse as they get older. For both probands and rela-
tives, sex is highly statistically significant, but with dif-
fering effects: for probands, being female increases the
odds of alcohol abuse for her relatives, while for relatives
the opposite is true. The GEE estimated the within-clus-
Table 3. Parameter estimates based on the regression ap-
Parameter Non-linear
mixed model
model PQL2 GEE
Intercept 1.079 (0.222) 1.082 (0.217) 0.944 (0.218)
Alcohol abuse 0.842 (0.182) 0.844 (0.179) 0.765 (0.169)
Female proband 0.537 (0.166) 0.539 (0.165) 0.480 (0.158)
Female relative 2.033 (0.147) 2.025 (0.137) 1.826 (0.118)
Relative 25 - 34 0.213 (0.200) 0.212 (0.198) 0.184 (0.186)
Relative 35 - 44 0.026 (0.217) 0.026 (0.215) 0.017 (0.199)
Relative 45 - 54 0.421 (0.242) 0.421 (0.242) 0.391 (0.218)
Relative 55 - 64 0.622 (0.243) 0.622 (0.243) 0.559 (0.215)
Relative 65+ 1.405 (0.293) 1.400 (0.300) 1.280 (0.257)
Random effect/
correlation 0.741 (0.237) 0.726 (0.170) 0.074a
Note: Entries are estimates (standard error). GEE method assumes ex-
changeable working correlation structure. Relative aged 18 - 24 is the refer-
ence category. aParameter in the working correlation matrix, i.e. correlation
coefficient between relatives related to the same proband. Since GEE con-
siders the correlation among clustered observations as a nuisance, no stan-
dard error is calculated.
ter correlation to be rather small at 0.074, indicating that
at the marginal level, the observations were more or less
independent. However, both parametric methods (NLME
and hierarchical models) estimated the random effects to
be quite significant at 0.741 (SE = 0.237) and 0.726 (SE
= 0.170), respectively.
The prediction accuracy using the diagnostic tests for
each approach is given in Table 4. All models performed
well at predicting negative cases correctly, while there
were shortcomings in the prediction of positive cases.
The non-linear mixed model was the best overall, fol-
lowed by the multilevel model. The popular GEE model
did not perform as well in its predictions. It should be
noted that the sensitivities and specificities reported here
are a function of the independent variables included in
the model, and these were chosen purely for illustrative
Finally, Table 5 compares the adjusted odds ratios and
corresponding 95% confidence intervals from the various
analyses. It demonstrates that alcohol abuse in a proband
more than doubles the odds of alcohol abuse in a relative.
It is notable that the three regression methods that take
the intra-familial correlations into account produce nar-
rower confidence intervals than the Mantel-Haenszel
approach and the standard logistic regression, which
treated all observations as independent.
4. Discussion
This paper reviews methods of analyzing hierarchical
data in an effort to highlight their weaknesses and
strengths and draw general guidelines for their use in
Table 4. Model accuracy.
Method Sensitivity Specificity
Non-linear mixed model 27.98 98.65
Multilevel model (PQL2) 25.81 98.65
GEE 9.54 98.18
Note: Entries are in percentages.
Table 5. Summary of odds ratio estimates.
Method Coef. OR 95% CI
of CI
Standard logistic regression0.766 2.152 (1.61, 2.87) 1.26
Non-linear mixed model 0.842 2.322 (1.97, 2.68)0.71
Multilevel model (PQL2)0.844 2.326 (1.97, 2.68)0.71
GEE 0.765 2.150 (1.82, 2.48)0.66
MH Odds Ratioa 2.421 (1.61, 2.86)1.25
aMantel-Haenszel odds ratio is adjusted by age of relatives and sex of both
relatives and probands.
Copyright © 2013 SciRes. OJS
analyzing medical and health data with correlated binary
responses. To illustrate, we made use of alcohol abuse
data from a family study conducted in Edmonton, Can-
The non-linear mixed effects model assumes that the
error distribution is normal, while allowing for the het-
erogeneity of the data in the form of mixed effects of
some covariates. In this situation, multilevel models can
be viewed as a special case of non-linear mixed effects
models, but they are especially useful when the data have
more than two levels of hierarchies. Unlike these two
approaches, which assume a correlated binomial distri-
bution of the data, the GEE method does not require the
data to follow a particular parametric distribution. How-
ever, if the number of clusters is very large, all three me-
thods are expected to perform similarly.
Using the alcohol abuse data, we found that the non-
linear mixed model resulted in the best overall model
prediction with the additional advantage of requiring
only a moderate amount of computing time. However,
only a limited number of researches have been done to
check the model assumptions [9-11]. Further research is
needed to improve this aspect of the non-linear mixed
The multilevel method allows for a model with several
levels, but for two-level hierarchical data such as was
used in this study, it is essentially the same as the usual
non-linear mixed effects model. A disadvantage of this
approach is that the algorithm may not converge, espe-
cially when the cluster size is small with few clusters.
Furthermore, the computation time to convergence is
relatively long.
In the GEE approach, we choose the exchangeable
structure of the working correlation matrix, assuming all
relatives of the proband have the same correlation. We
note that Liang and Zeger (1986) originally considered
the correlation among clustered observations as a nui-
sance, while the regression parameters are the primary
interest [2]. With the GEE approach, regression parame-
ters can be estimated consistently but not necessarily
with complete efficiency, whether the working correla-
tion structure is correct or not. This consistency is based
on the assumption that the regression parameters and the
association parameters are orthogonal to one another,
even when they are not [12]. However, the GEE method
may still be preferable in some cases, when a population
averaged level analysis is suitable for the research ques-
tions and objectives. Further, the computational algo-
rithm is relatively fast and it makes weaker assumptions
about the structure of the variance-covariance matrix of
the response vector.
Regarding the model’s prediction accuracy, in general,
the model’s specificity is quite high, but rather low in
sensitivity. In other words, the model has much more
difficulty predicting cases that have an alcohol abuse
problem, which is a common situation when there are no
strong risk factors for modelling. In the absence of ran-
dom effects, all models have an essentially equal ability
to predict the outcome. However, when there are signifi-
cant random effects, the prediction level improves by
properly accounting for the random effects in the model.
5. Conclusion
Overall, the non-linear mixed effects approach to analy-
sis of these data seems quite competitive with the multi-
level method in terms of convergence properties. The
random family effects were significant, reflecting het-
erogeneity among families, and both the non-linear
mixed effects model and the multilevel model captured
them effectively. The popular marginal modeling via the
GEE method may still be preferable because of its com-
putational ease and relaxed distribution assumptions.
However, caution is advised, as it might underestimate
the odds ratio and its standard error, as indicated in this
case study.
[1] H. Goldstein, “Nonlinear Multilevel Models, with an
Application to Discrete Response Data,” Biometrika, Vol.
78, No. 1, 1991, pp. 45-51. doi:10.1093/biomet/78.1.45
[2] K. Y. Liang and S. L. Zeger, “Longitudinal Data-Analysis
Using Generalized Linear-Models,” Biometrika, Vol. 73,
No. 1, 1986, pp. 13-22. doi:10.1093/biomet/73.1.13
[3] S. C. Newman and R. C. Bland, “A Population-Based
Family Study of DSM-III Generalized Anxiety Disorder,”
Psychological Medicine, Vol. 36, No. 9, 2006, pp. 1275-
1281. doi:10.1017/S0033291706007732
[4] N. Mantel and W. Haenszel, “Statistical Aspects of the
Analysis of Data from Retrospective Studies of Disease,”
Journal of the National Cancer Institute, Vol. 22, No. 4,
1959, pp. 719-748.
[5] J. C. Pinheiro and D. M. Bates, “Approximations to the
Log-Likelihood Function in the Nonlinear Mixed-Effects
Model,” Journal of Computational and Graphical Statis-
tics, Vol. 4, No. 1, 1995, pp. 12-35.
[6] J. Nocedal and S. J. Wright, “Numerical Optimization,”
Springer-Verlag, New York, 1999. doi:10.1007/b98874
[7] N. E. Breslow and D. G. Clayton, “Approximate Infer-
ence in Generalized Linear Mixed Models,” Journal of
the American Statistical Association, Vol. 88, No. 421,
1993, pp. 9-25.
[8] H. Goldstein and J. Rasbash, “Improved Approximations
for Multilevel Models with Binary Responses,” Journal
of the Royal Statistical Society Series A—Statistics in So-
ciety, Vol. 159, 1996, pp. 505-513.
[9] Y. Yano, S. L. Beal and L. B. Sheiner, “Evaluating Phar-
macokinetic/Pharmacodynamic Models Using the Poste-
rior Predictive Check,” Journal of Pharmacokinetics and
Pharmacodynamics, Vol. 28, No. 2, 2001, pp. 171-192.
Copyright © 2013 SciRes. OJS
Copyright © 2013 SciRes. OJS
[10] P. J. Williams and E. I. Ette, “Determination of Model
Appropriateness,” In: H. C. Kimko and S. B. Duffull,
Eds., Simulation for Designing Clinical Trials: A Phar-
macokinetic-Pharmacodynamic Modeling Prospective,
Marcel Dekker, New York, 2003, pp. 68-96.
[11] F. Mentre and S. Escolano, “Prediction Discrepancies for
the Evaluation of Nonlinear Mixed-Effects Models,”
Journal of Pharmacokinetics and Pharmacodynamics,
Vol. 33, No. 3, 2006, pp. 345-367.
[12] K. Y. Liang, S. L. Zeger and B. Qaqish, “Multivariate Re-
gression Analysis for Categorical Data,” Journal of the
Royal Statistical Society, Series B, Vol. 54, No. 1, 1992,
pp. 3-40.