SCHO OL OF STATISTIC S
UNIVERSITY OF THE PHILIPPINES DILIMAN
WORKING PAPER SERIES
SEMIPARAMETRIC PRINCIPAL
COMPONENTS POISSON REGRESSION ON
CLUSTERED DATA
by
Erniel B. Barrios
and
Kristina Celene M. Manalaysay
UPSS Working Paper No. 2011-06
November 2011
School of Statistics
Ramon Magsaysay Avenue
U.P. Diliman, Quezon City
Telefax: 928-08-81
Email: updstat@yahoo.com
SEMIPARAMETRIC PRINCIPAL COMPONENTS POISSON
REGRESSION ON CLUSTERED DATA
Erniel B. Barrios
School of Statistics, University of the Philippines Diliman
Kristina Celene M. Manalaysay
School of Statistics, University of the Philippines Diliman
ABSTRACT
Clustering of observations and interdependency of predictors are two common
problems in modeling count data with multivariate predictors. We propose to use principal
components of predictors to mitigate the multicollinearity problem. To abate information
losses due to dimension reduction, a semiparametric link between the count dependent
variable and the principal components is postulated. Clustering of observations is accounted
into the model as a random component and the model is estimated via the backfitting
algorithm. A simulation study illustrates the advantages of the proposed model over standard
poisson regression in a wide range of simulation scenarios.
Keywords: semiparametricpoissonregression, clustered data, multicollinearity, principal
components analysis
1. Introduction
In many diverse fields, outcomes of certain phenomena are measured using indicators
that possess the characteristics of poisson events, e.g., prevalence of a disease, number of
customers patronizing products/services, number of student enrollees. Poisson regression is
used to characterizesuch dataand in predicting the average number of instances an event
occurs, conditional on one or more factors. Ruru and Barrios (2003) demonstrated using
malaria data that poisson regression is advantageous over classical regression in modelling
count data. Classical regression analysis requires more predictors to achieve as much
predictive ability as poisson regression.
Spatial aggregation causes certain poisson events to manifest clustering.The spread of
AH1N1 is influenced by determinants leadings towards vulnerability of individuals in the
same community, this may be different from those causing vulnerability of other individuals
from a different community. Clusters may still be independent but members of the same
cluster (or neighborhood) are necessarily dependent since there is some spatial endowment
commonly shared among units that formed the cluster. Classical statistical inference assumes
independence of observations, i.e., data are independently collected on similar, homogenous
units. This assumption is not necessarily true for clustered data. Thus, in analyzing clustered
data with methods that implicitly consider independence of observations may yield incorrect
analyseson the dynamics of the events/phenomena being characterized.
Predictors that explain occurrence of poisson events within the cluster can also be
naturally correlated. The interdependence among predictors usually causes problems in
statistical inference involving linear models. Themulticollinearityproblem exists when two or
more explanatory variables in a regression model are highly correlatedimplicating the
inefficiency of ordinary least squares estimates of the regression coefficients. As an
illustration, consider income and educational attainment as predictors of political preference.
Income and educational attainment are structurally correlated since income varies according
to the level of educational attainment of an individual. The presence of multicollinearity in a
statistical model inflates the standard error of the estimated coefficients, resulting in
unreliablecharacterization of the coefficients (Curto and Pinto, 2007). It further weakens in
sensitivity of the dependent variable on changes in independent variables and makes it
difficult to assess the relative importance of the independent variables in the model.
There are several solutions to the multicollinearity problem. For example, instead of
individual predictors, some important principal componentsare used in the model. In the
presence of multicollinearity, the design matrix becomes ill-conditioned if not singular and
hence, principal components analysis transforms correlated variables into fewer independent
components.
Since Principal Components Regression (PCR) uses only a subset of the
principal components, there is a loss of information resulting to thedeterioration of the
predictive ability of the estimated regression function compared to the model that usesall the
individual predictors (Dunteman, 1989). It is also possible that the use of a subset of principal
components can result to bias in the assessment of the relative importance of a predictor in
explaining the dependent variable.
The lost information in principal components regression can be recovered by allowing
flexibility on the functional relationship between the dependent variable and the principal
components, (Barrios and Umali, 2011). In nonparametric regression, the functional form of
the link between the dependent and independent variables is allowed to be flexible with only
the requirement of smoothness of the function incorporated into the objective function of the
estimation. With a flexible functional form, the principal components can have a more
accurate characterization of the variation of the dependent variable, hence improving its
predictive ability.
We postulate an additive combination of nonparametric functions on principal
componentsand random effects in a regression model with measurements of poisson events as
the dependent variable.
This semiparametricpoisson regression model can be used in
characterizing high dimensional clustered data. Clustering effect is accounted into the model
through a random intercept term. Dimension reduction is achieved through principal
components and due to the inherent deterioration in model fit due to dimension reduction, the
covariate effect summarized in terms of the principal components will be postulated as
nonparametric functions.
2. Some Modeling Strategies
Classical linear regression assumes continuous dependent variable and will lead to
inefficient, inconsistent and biased estimates when used in count dependent variable. Poisson
regression is appropriate in modeling with count dependent variable data.Even if poisson
regression can be approximated by classical linear regression, e.g., large sample size, poisson
regression is advantageous over classical linear regression since it usually requires fewer
predictorsto achieve a good fit, as demonstrated in the study of malaria incidence by Ruru and
Barrios (2003).
Nelder and Wedderburn(1972) introduced the generalized linear models (GLM) to
relax some of the classical assumptions of a linear model. The model is given
by
where
exponential family and
systematic component
for every i, Yibelongs to the
is a function that links the random component Yi to the
.
These are developed for regression
models with non-normal dependent variables; special cases include poisson regression where
Y is a count variable and logistic regression where Y is a binary outcome.
Demidenko (2007) compared the following models for clustered data: (1) ordinary
poisson regression, which ignores intracluster correlation, (2) poisson regression with fixed
cluster-specific intercepts, (3) a generalized estimating equations approach with an equicorrelation matrix, (4) an exact generalized estimating equations approach with an exact
covariance matrix, and (5) maximum likelihood. All five methods lead to consistent estimates
of slopes but have yield varying efficiency levels speciallyfor unbalanced data.
Poisson regression assumes heterogeneous mean that is expressed as a linear
combination of explanatory variables. Since the parameter λ is positive, it is convenient to
express this parameter through an exponential function,
, where β is an
mx1 vector of regression parameters and xi is an mx1 vector of explanatory variables or
covariates, i = 1,2,…,n. In the context of generalized linear models, poisson regression has log
link because
, (Demidenko, 2004).Poisson model is useful for clustered data
since the cluster-specific intercepts may be eliminated and can be viewed as a limiting
maximum likelihood estimates when the variance of the intercepts approaches infinity
(Demidenko, 2007).
Principal components analysis (PCA) transforms a set of p correlated variables into
uncorrelated linear combinations called principal components (PC’s). PCA rotates the original
variable space to a point where the variance of the new variate is maximized. Since the PC’s
are ranked by order of explained variance, the last PC’s have the smallest variance but it is
through the last PC’s that the relationship of the independent variables to the dependent
variable are determined, that is, the variables with high loadings on the last PC’s are proven to
be highly correlated (Jolliffe, 2002). In modeling where only a subset of the PC’s areused,
there is substantial loss of information. Typically, only a subset of the PC’s is included in
regression modeling, though there is no universally acceptable procedure yet to determine the
PC’s to retain (Draper and Smith, 1981). Filzmoser and Croux (2002) proposed a procedure
that simultaneously chooses the components while model fit is optimized.
Montgomery and Peck (1982) and Dunteman (1989) justified why the principal
components with low eigenvalues are not included in the model. Since the variance of the
estimator
(of ) is a linear combination of the reciprocal of the eigenvalues,
inclusion of one or more components with small eigenvalues in the model yield high variance
of .
Nevertheless, Marx and Smith (1990)noted that given specific theoretical models
oriented towards parameter estimation, principal component regression can yield desirable
(maximum) variance property with minimal bias.
The bias and the lost information in principal component regression should be
addressed. For instance, nonparametric smoothing techniques which aim to provide a strategy
in modeling the relationships between variables without specifying any particular form for the
underlying regression function may be considered.When several covariates are present, Hastie
and Tibshirani (1986) proposed to extend the idea of linear regression into a flexible form
known as generalized additive model (GAM). The regression model is given by
where
are nonparametric components. Additive models assume nonparametric smoothing
splines for each predictor in regression models. Hastie and Tibshiranie (1990) suggested that
additive models are used as initial procedure to locate the patterns and behavior of the
predictors relative to the response, suggesting a possible parametric form for which to model
Y at a later stage.
Alvarez and Pardinas (2009) formulated a censored regression model with additive
effects of the covariates. The additive model sped up computation in the inference process and
yieldmore promising results over a class of linear models especially where there is violation
in the linearity assumption.
3. Methodology
Clustered data is characterized by homogeneity of elements within the cluster and
heterogeneity between clusters. This case would require a model structure that accounts for
variation across the clusters, while accounting for similarity of subjects within the cluster.The
choice of covariates or predictors that can sufficiently account for between-cluster variability
and within-cluster homogeneity is often complicated resulting to very long list. This would
necessarily lead to a high dimensional data type and often, this can invite other confounding
problems. High dimensional data becomes prone to multicollinearity and suffers from the
curse of dimensionality.
Multicollinearitycan be a crucial issue in modeling.
However, there are various
approaches to mitigate its ill-effects, e.g., transform the individual predictors into linear
combinations, the combinations are chosen so that they are independent, yet it contains the
maximum amount of variance that the original predictorscontain. With linear combinations as
predictors instead of the individual variables in a linear model, the predictive ability could
suffer.
On the other hand, high dimensionality of predictors, multicollinearity, discreteness of
the dependent variable (count data), and the clustering usually associated with poisson process
that generates the count data can posed many complications in many model structures. Thus,
we offer a solution that can potentially alleviate the predictive ability of the model through a
nonparametric postulated link function in poisson regression for count data.
3.1 Postulated Model
De Vera (2010) proposed asemiparametricpoisson regression model for spatially
clustered count data given npredetermined clusters with n k observationsin eachcluster given
by
(1)
Equivalently, Model 1 can also be written as:
(2)
where
are the explanatory variables
are smoothfunctions of
are clusterrandom intercepts
isthe error term
isthe response variable
As the number of predictorsincreases, the likelihood of themulticollinearity problem
also increases and either can cause potential problems in estimating Model 2. We propose to
use principal components of the predictors instead of the individual variables in the model.
This however leads to bias since some information on the predictors can be lost in the process.
Thus, the link function is postulated as a nonparametric function resulting to the following
semiparametric model:
(3)
where
are cluster random intercepts
is the score for jth principal component on the ith observation
is the error term
is the response variable
The flexibility in form of the nonparametric function of the principal components will
compensate for the bias in the estimation of the regression coefficients due to the information
lost by selecting only the most important principal components to be included as predictors in
the model. The random intercepts will account for cluster differences (homogeneous within a
cluster) and the possible peculiarities in the model caused by clustering.
3.2 Estimation Procedures
The principle of backfitting is used to estimate the parametric and nonparametric
components of the model.Assuming additivity of Model (3), the parametric and
nonparametric frameworks are imbedded into the backfittingg algorithm to estimate the
parameters and the nonparametric components of the model.
Two approaches are presented in this section.In Method 1, the backfitting algorithm
estimates the nonparametric part first, and then the parametric part is estimated from the
residuals. In Method 2, ordinary poisson regression is usedbut with principal components as
predictors.
3.2.1 Method 1: Backfitting of the Semiparametric Model
After the extraction of principal components, the parametric and nonparametric parts
of the model are estimated iteratively in the context of backfitting. The nonparametric
functions of the principal components are estimated through spline smoothing.
Smoothing splines are used to fit the nonparametric part of the model. Consider a
simple additive regression model Y = f(X) + ε, where E(ε) = 0 and Var(ε) = σ2. We want to
estimate the function fas a solution to the penalized least squares problem given by
given
a
value
of
the
smoothing
parameter  >0. The first term measures the goodness of fit and the second term served as the
penalty for lack of smoothness in f due to the interpolation in the first term.The smoothing
parameter  controls the tradeoff between smoothness and goodness of fit. Largevalues of
 emphasizes smoothness of f over model fit, while small  values put higher leverage on
model fit rather than on smoothness of f. As
, the solution to f is an interpolation of the
data points.The choice of the value of smoothing parameter  is optimized through the
generalized cross validation (GCV).  is chosen to minimize the generalized cross validation
mean squared error given by
, see (Golub, et. al., 1979), (Hardle, et.
al., 2004) for further details.
The partial residual e i is computed and used to estimate the random intercepts (with a
priori information on clustering of observations) through methods like maximum likelihood
methods, EM algorithm, see for example (Seidel, 2011).
Spline smoothing and mixed model estimation in the backfitting framework are then
iterated until convergence, see (Opsomer, 2000) for some optimal properties of the backfitting
estimators.
3.2.2 Method 2: Ordinary Poisson Regression on the Principal Components
With poisson link function, a general linear model (GLM) with the principal
components as predictors was also estimated. In GLM, the outcome Y is assumed to be
generated from a particular distribution in the exponential family, in this case, from a poisson
distribution. The heterogeneous mean, μ, of the distribution depends on the independent
variables, in this case, the PC’s, through:
(4)
These two methods are compared with Ordinary Poisson Regression on the original
predictors in terms of their predictive ability.
3.3 Simulation Studies
We conduct a simulation study that covers the features of typical data to be analyzed
using Model (3), i.e., either it includes high dimensional predictors, or that the
multicollinearity problem is present, or both. We then compared the predictive ability of
Methods 1 and 2 with ordinary poisson regression with the individual variables as predictors.
The simulation scenarios are summarized in Table 1:
Table 1 Summary of Simulation Scenarios
Number of Variables/
Expected Number of
PC’s
Few (5)
Many (30) - Single Pattern
Many (30) - Three Patterns
Multicollinearity
Absence
Strong
Sample Size
50
100
Number of Clusters
5
10
Model Fit
Good
Poor
Starting with five X’s, the rest were simulated as an additive combination of a function
of another X and the random error term. The initial set of predictors are simulated from:
(5)
The
error
term
is
distributed
asN(0,1),
the
multiplier
of
the
error
term
inducesmulticollinearityamong the predictors, i.e., higher multiplier implies absence of
multicollinearitywhile lower multiplier indicates presence of multicollinearity.The twenty five
other predictors were generated to fulfil various multicollinearity structures.
The response variable was computed as the linear combination of the X’s and added
with a cluster mean (from the normal distribution) and an error term. The means of the normal
distribution from where the cluster means were simulated were spread thoroughly to
differentiate the clusters. A multiplier to the error term is included to altermodel fit (large
multiplier implies poor model fit, small multiplier implies good model fit). A model usually
fits the data well when the functional form used is correct and that thecorrect predictors are
aptly accounted into the model. When the functional form of the model is incorrect or that
there are missed out predictors, variation of the error term will dominate. Hence, to simulate
misspecification, we magnified the error by multiplying it with a constant.
Scenarios for varying sample sizes (50, 100) as well as varyingnumber of clusters
(5,10) were generated to assess robustness of the proposed model. Furthermore, correlations
among predictors that can result to absence of multicollinearity or severe multicollinearity
were included. The number of variables can be 5 or 30, with the 30-variable scenarios further
divided into single and three-patternmulticollinearitystucture. Single-pattern scenarios
represent data with the variables being correlated with one variable, i.e., they are computed as
a function of a particular variable only. Three-pattern scenarios, on the other hand, have a
number of these “base” variables, from whom the rest of the variables are computed. These
variations in pattern were incorporated to simulate varying predicted number of principal
components. Each scenario is replicated 100 times.
4. Results and Discussion
The predictive ability of the proposed model is assessed by comparing the mean
absolute prediction error (MAPE)using estimation methods 1 and 2 as well as the MAPE
obtained using Ordinary Poisson Regression (OPR) based on the original predictors,
4.1 Effect of Misspecification Error
In empirical modeling, misspecification is very common especially when variables
were measured in an ad hoc manner, i.e., not based on some theoretical foundation.
Misspecification error is introduced by magnifying the error term, i.e., multiplying with a
constant to inflate the variance. As the magnitude of the constant increases (equivalently,
model fit worsens or that the extent of misspecification error increases), the mean absolute
prediction error (MAPE) also increases.
In the simulation study, “good” model fit is represented by linear equations where
coefficient of determination is at least 60%, while “poor” model fit is associated with
coefficient of determination lower than 60%. Simulation shows that whether misspecification
error is present or not, the proposed semiparametric model always outperformed the other
models in terms of predictive ability. In the absence of misspecification error, the proposed
model is advantageous to ordinary poisson regression by about 10% in MAPE. Shifting from
ordinary poisson regression to the parametric principal components regression, on the other
hand, results in 65% increase in MAPE. This illustrates the effect of lost information due to
the summarization of the predictors into principal components instead of using the individual
predictors.
The advantages of the proposed method are also observed even with the presence of
misspecification errors. MAPE in the semiparametric model improved 9% over ordinary
poisson regression. Furthermore, there is advantage of 48% in terms of MAPE in using
ordinary poisson regression relative to the parametric model principal component model. The
MAPE of the three models by nature of model fit are summarized in Table 2.
Table 2. Comparison of MAPE for Varying Model Fit
MAPE (%)
Model Fit
Semiparametric Model
Parametric Model
Good
Poor
24.26
141.54
44.28
230.64
OPR on Original
Predictors
26.88
156.23
Since substantial number of cases with “poor” model fit was included, subsequent
results yield higher MAPE levels since predictive ability of those models with “good” fit were
contaminated. The discussions then will focus on the comparison of the three models, rather
than on the magnitude of MAPE.
4.2 Effect ofMulticollinearity
In the absence of multicollinearity, the proposed model yields better prediction over
ordinary poisson regression on the original predictors (OPR) where the MAPE is lower by
5%. The parametric principal component regression on the other hand, yields48% higher
MAPE than OPR. This is explained as the effect of lost information due to selection only of
the more important principal components. With multicollinearity present, similar ranges of
MAPE can be observed from the proposed model with improvementby 14% over ordinary
poisson regression. Again, the parametric model yield 52% higher MAPE relative to ordinary
poisson regression on original variables.Principal Components Analysis is a procedure aimed
at addressing multicollinearity, thus giving the proposed model advantage over ordinary
poisson regression with original correlated variables as predictors. Thesemiparametric nature
of the proposed model makes a more flexible regression and endowed advantages over the
parametric model.The proposed model performs best among the three methods whether
multicollinearityis present or not. The MAPE values are summarized in Table 3.
Table 3. Comparison of MAPE with or withoutMulticollinearity
MAPE (%)
Multicollinearity
Semiparametric Model
Parametric Model
OPR on Original
Predictors
Absence
Presence
108.84
65.61
170.28
115.58
114.79
76.06
4.3 Effect of Sample Size
We simulated two sample size values: 50 observations (small) and 100 observations
(large).In small sample size,while ordinary poisson regressionyields better predictive ability
among the three models, the proposed model is still within a comparable range. For large
sample size, however, the proposed model is most advantageous, yielding 19% lower MAPE
than ordinary poisson regression. This is consistent with the observation ofOsborne and
Costello (2004) that in small samples, errors can easily occur particularly among multivariate
techniques such as principal components analysis (eg extraction of erroneous principal
components). For small samples, estimates of correlation among the predictors are relatively
unstable, hence, the estimates of component loadings may not be accurate resulting to losses
in information.
This lost information is somehow retrieved by relaxing the parametric
structure of the model and employ the more flexible nature of nonparametric regression.
In Table 4, the parametric model is the most robust to sample size, with its MAPE
changing only by 3% as sample size changes from 50 to 100. MAPE of the proposed model
increased 23% as sample size increased, whereas MAPE of ordinary poisson regression
increased 63%. In the parametric model, a sample of size 50 is already fairly large, hence no
significant changes in MAPE is observed as the sample size is increased further to 100. While
MAPE for the proposed model increases with the sample size, it still performed better than
ordinary poisson regression and the parametric model.
Table 4. Comparison of MAPE for Varying Sample Size
MAPE (%)
Sample Size
Semiparametric Model
Parametric Model
50
100
74.19
91.61
135.26
139.66
OPR on Original
Predictors
69.59
113.51
4.4 Effect of Number of Clusters
A priori information on clustering of observations can inform the modeler of possible
sources of variation of the response variable. More clusters mean that there aremore basis for
the attribution of heterogeneity of the observations.
Furthermore, the proposed model
included a random intercept term that basically explains the cluster differences.
The
simulation study illustrates the influence of the number of clusters in the proposed model.
Increasing the number of clusters yield improvement on the predictive ability of the proposed
model. In fewer clusters, MAPE of the semiparametric model improved by 7% compared
toordinary poisson regression. On the other hand, MAPE increased by 49% in the parametric
model from ordinary poisson regression. Similar trend can be observed for cases with more
clusters, but the greater advantage of the semiparametric model over ordinary poisson
regression (about 12% decline in MAPE) is observed in these cases.
The above result is consistent with Arceneaux and Nickerson (2009) who
recommended adding more clusters, not the individual observations, for increased efficiency
in the analysis of clustered data. Table 5 provides details related to the effect of number of
clusters on predictive ability of models in clustered data. Table 6 illustrates the implication of
the number of clusters and number of observations per cluster in the analysis of clustered
data.
Table 5 Comparison of MAPE for Varying Cluster Count
MAPE (%)
Number of
Clusters
5
10
Semiparametric Model
Parametric Model
87.49
78.31
140.57
134.36
OPR on Original
Predictors
94.19
88.91
Table 6. Comparison of MAPE for Varying Cluster Count and Cluster Size
MAPE (5)
Number of
Clusters (Cluster
Size)
5 (10)
5 (20)
10 (5)
10 (10)
Semiparametric Model
79.40
95.58
68.98
87.64
Parametric Model
131.69
149.45
138.84
129.87
OPR on Original
Predictors
64.20
124.18
74.98
102.84
4.5 Effect of Number of Variables
While multicollinearity may appear even with few predictors, the chances of
observing it in high dimensional data are higher. Scenarios with 5 (few) and 30 (many)
variables were simulated in this study.In high dimensional data, principal component analysis
is knownas an effective data reduction technique. Whether there are few or many variables,
the semiparametric model still exhibitsbetter predictive ability.MAPE values are even lower
in cases where there are 30variables. With fewer variables, the proposed model yields 17%
lower MAPE compared toordinary poisson regression, whereas the parametric model yields
8% higher MAPE than that of ordinary poisson regression. Table 7indicates similar results for
scenarios involving many variables.
Table 7. Comparison of MAPE for Varying Number of Variables
MAPE (%)
Number of
Variables
5
30
Semiparametric Model
81.53
78.19
Parametric Model
106.84
146.52
OPR on Original
Predictors
98.36
83.37
For scenarios with thirty variables, presence of multicollinearityis further examined
for the effect of only a single pattern of interdependence among the predictors and cases
where there are three patterns of such interdependencies. The purpose of heterogeneity in
interdependencies is to account for the number of principal components that need to be
included in the model. The simulation study illustrates that predictive ability of the
semiparametric model improves as the interdependencies become more complicated, i.e.,
more patterns of interdependencies. Compared to the ordinary poisson regression and the
parametric model, the proposed model is still advantageous in terms of predictive ability.
Table 8. Comparison of MAPE for Single- and Three-Pattern Scenarios
MAPE (%)
Number of
Correlations Pattern
30 Single-Pattern
30 Three-Pattern
Semiparametric Model
95.05
61.33
Parametric Model
180.60
112.43
OPR on Original
Predictors
94.31
72.43
5.Conclusions
The semiparametric principal component poisson regression model is aimed to
characterize clustered data with high dimensional or correlated predictors.The proposed
model provides solution to the modelling issues associated with high dimensional or
correlated predictors while it mitigates the bias caused by lost information in dimension
reduction.
The predictive ability of the semiparametric model is superior in the presence of
multicollinearity. While it is superior to parametric poisson principal component regression
and ordinary poisson regression in any sample size, it is best in cases of large sample sizes.
Although ordinary poissonregression performed better than the proposedmodel for small
sample size, the performance of the proposed model is still at par with the other methods, and
it is the more robust than ordinary poisson regression to changes in sample size.
The proposed model and the corresponding estimation procedure is capable of
mitigating the problem of multicollinearity by regressing on the principal components instead
of on the original predictors. Furthermore, the nonparametric specification of the effect of
principal components abate the potential reduction in the predictive ability of the model that is
usually observed in principal components regression caused by loss in information from
dimension reduction.
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