Over-and Underdispersion Modelslmb.univ-fcomte.fr/IMG/pdf/ch30_kokonendji2014.pdfOver-and Underdispersion Models 509 mixed Poisson model, taking into account the heterogeneity of the

Over- and Underdispersion Models

Celestin C. Kokonendji

30.1 Introduction

This chapter explores some count statis-tical models, which are tied to the phe-nomenon of over-/equi-/underdispersion. This concerns many fields in applied statis-tics, including public health, medicine, and epidemiology.

In general, one says o v e r d i s p e r s i o n if the observed variability exceeds the ex-pected variability and u n d e r d i s p e r s i o n if it is lower than expected; sometimes, e q u i d i s p e r s i o n means there is no dis-crepancy between both variabilities. For example, when applying generalized lin-ear models (e.g., [69,73]) with a known scale or dispersion parameter, there are two possible scenarios to examine the qual-ity of the fitting model through the simi-larity of residual deviance and residual de-grees of freedom: standard model diagnos-tics (e.g., outliers, omitted terms or vari-ables in the linear predictor, incorrect re-lationship between mean and explanatory variables) and the phenomenon of over-underdispersion (i.e., the empirical vari-ance or variation may be greater/smaller than that predicted by model). The sec-ond scenario is of interest to us and it was observed a long time ago in the particu-lar case of the Poisson model [103; see also

32]. Overdispersion considered as a bur-den by statisticians is most frequent than underdispersion hardly noticed, in several domains of statistics for which we can be inspired by it for clinical trials; see, for ex-ample, [2-23,28-32,36-40,65-122].

The origin of such phenomena of over-and under-dispersion can be interpreted as a failure of some basic assumptions of the model. It may arise because of one or more possible causes from the hy-pothesized structure of the population. Some possibilities are variability of ex-perimental material, correlation or lack of independence between individual item responses, cluster sampling or clustered structure of the population, heterogene-ity or lack of assumption of homogeneous population, small sample size, omitted un-observed variables, contagion, aggregation and repulsion; see, for example, [113] for some detailed explanations. The cause of over-underdispersion may also be appar-ent from the mechanism of the data col-lection process. Ignoring these phenom-ena, there are some possible problems such as incorrect estimations of parameters and standard error, incorrect interpretation of the considered model and any predictions will be too precise, although Milanzi et al.

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Methods and Applications of Statistics in Clinical Trials: Planning,Analysis, and Inferential Methods. Edited by N. Balakrishnan.

© 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

Over- and Underdispersion Models 507

[74] recently pointed out some solutions for particular situations. One of the popular measures to detect such failures, for exam-ple, from the Poisson distribution is the so-called Fisher index, which is the ratio of variance to the mean of the count dis-tribution. There are many different statis-tical tests, which are introduced with re-spect to the models and situations (see, e.g., [30,73,76]).

The basis of the main over- and un-derdispersion models is centered around a special class of distributions, the ex-ponential families of distributions (e.g., [64,Chap.54]), which are widely used as flexible models in different contexts. An example is in the theory of generalized lin-ear models (see, e.g., [16] with many refer-ences therein). For single parameter densi-ties the actual choice of distribution affects the assumptions on the variances, since the relation between the mean and the vari-ance is known and also characterizes the distribution (e.g., [60,61]). The Poisson distribution, for instance, has the follow-ing count equidispersion property, "mean = variance." It is useful to distinguish be-tween the exponential family (or the natu-ral exponential family) and the exponential dispersion family; see, for example, [29,43-46,68] for more details and possible exten-sions of exponential dispersion models. In the exponential dispersion family an extra parameter, the so-called dispersion param-eter, is introduced. This allows for a de-scription of distributions without a unique relation between the mean and the vari-ance. The Gaussian or normal distribu-tion is such an example. The exponen-tial dispersion family enables the possibil-ity of separating the parameters in a struc-tural parameter describing the mean value and another parameter, which typically is a scale, index, or precision parameter.

The key to adequate over- and underdis-persion models is to improve or modify the previous failing models considering the de-

tected causes. There are many ways, such as mixing and stopped-summing, which are often for overdispersion situations. An interesting class of distributions for both phenomena of over- and underdispersion is the weighted distribution (e.g., [82,83] and also [6,7,58,62]). The weight function in-troduced in the initial distribution brings the necessary corrections that the appro-priate model or process needs. Combin-ing weighted distributions and exponential dispersion models, we obtain a more flexi-ble class of distributions to remedy certain causes of over-underdispersion.

We focus here on some over- and under-dispersion models for count data, which oc-cur in many fields and need specific treat-ments. Clinical trials also produce, in quantity, this type of discrete data be-longing in the nonnegative integers set N = {0,1,2,---}. This could be done for proportion and polytomous data. The rest of the chapter is organized as fol-lows. Section 30.2 introduces basic count dispersion models. Five families of para-metric and nonparametric models are re-visited: mixed Poisson, compound Pois-son, weighted Poisson, Lagrangian Poisson and semiparametric Poisson models. Some probabilistic interpretations are given, if possible, for parametrical models. A non-parametric approach to over- and under-dispersion models, using the recent no-tion of discrete associated kernels, is briefly presented. Section 30.3 is devoted to three types of count explanatory mod-els. First, we present parametric regression models through the framework of the gen-eralized linear models. Then, we browse some integer-valued time series for over-underdispersion phenomena. Finally, we describe some nonparametric models for count explanatory variables. The last part, Section 30.4, concludes the chapter with a summary and some final remarks. A list of references is supplied.

508 Over- and Underdispersion Models

30.2 Count Dispersion Models

Following the structure of count data it is natural, even "normal," to begin with the standard Poisson random variable X ~ V{n) with probability mass function (p.m.f.):

Pr(X = X\JJL) = ^ ? V * e N , x\

a ) for a given mean parameter \i > 0. How-ever, it has only one parameter and its variance is equal to the mean: Equidis-persion is very restrictive for practitioners. For many observed count data, it is com-mon to have the sample variance greater or smaller than the sample mean, which is referred to as over- and underdispersion, respectively, relative to the Poisson distri-bution. This raises the question of whether an appropriate two-parameter distribution or more, such as the negative binomial and binomial, should be used routinely for analyzing over- and underdispersed count data. Note that the excess of zeros (e.g., [36]) is a special case of overdispersion but then it is a very particular kind, measured by the zero-inflation index (see, e.g., [88-90]) and admitting various treatments (see, e.g., [51,85]). Zero-truncation and general truncation also produce the phenomenon of over-underdispersion (see, e.g., [105]), and we omit them here.

Fixing the same mean /i as in (1), it is necessary to "generalize the Poisson" by introducing a second parameter 0 > 0 of the type "dispersion" for obtaining the so-called count d ispers ion m o d e l Y with p.m.f.

P r ( y = y|/x,0) = /(2/;/x,0), Vy € N. (2)

Its variance function tjy(/x, 0) must be greater or smaller than the variance /i = cr2

x{lj) of the Poisson (1):

(3)

Hence, the over- and underdispersion (3) relative to Poisson can be extended to an-other referential distribution with mean fji — /x(0); see, for example, [58]. The dis-persion parameter <\> could be fixed equal to 1 and, then, the variance function

1) = tfy(^) becomes the unit vari-ance function (see, e.g., [43]), which makes easy some comparisons and constructions of count dispersion models.

There are many ways for getting count dispersion models satisfying (3) relative to a given referential count model. For exam-ple, using framework of exponential disper-sion models one can build count dispersion models satisfying conditions (3) through classifications of variance functions (e.g., [60]). In particular cases, with respect to Poisson (1) we have some classical and single over- and under-dispersion mod-els such as binomial and Poisson-binomial for underdispersion; negative binomial or Poisson-gamma, generalized Poisson or Consul's Lagrangian Poisson, gener-alized negative binomial or Lagrangian negative binomial, Poisson-inverse Gaus-sian or Sichel, strict arcsine, Hermite, and Neyman Type A for overdispersion; Conway-Maxwell-Poisson or COM-Poisson for both over- and underdispersion phe-nomena; see, for example, [12,42,54-58,88-90,93,101,112]. In addition, we have some different classes of univariate count dis-persion models that are described below. These families have two or three parame-ters and unify some groups of single count models. Some of the parametric count models that we describe below can be found in dictionaries of discrete distribu-tions with more comments (e.g., [41,42]); however, the nonparametric count models are more recent (see, e.g., [59,63,123,124]).

30.2.1 Mixed Poisson A popular class of overdispersion models, for instance, relative to Poisson (1) is a


mixed Poisson model, taking into account the heterogeneity of the population. In-deed, a count random variable Y follows a m i x e d P o i s s o n d i s t r i b u t i o n with mean E(Y) = p > 0 if, and only if, there ex-ists a nonnegative random variable T with cumulative distribution function FT such that

Pr(Y = » ) = r (4) Jo W-

for all y € N; that is, the distribution of Y given T = t is the Poisson distribution with mean pt, or we randomize the Pois-son parameter by some probability mea-sure on (0, oo). More actually, since the variance of the mixed distribution is the sum of the variance of its conditional mean and the mean of its conditional variance, the variance of the mixture model is al-ways greater than that of the simple com-ponent model and this explains the use of the term "over-dispersion models" used for mixture models. See [50] for a detailed re-view. Particular cases of mixed Poisson distributions consist of the negative bino-mial, the Poisson-Lomax, the Dellaporte, the Poisson-uniform, the Polya-Aeppli, the Poisson-lognormal, the Yule, the Poisson-Lindley and, more globally, the Poisson-Tweedie distributions; see [18,37,60] for de-tails.

Let us mention that the Tweedie model is a large class of a-stable distributions with power unit variance function pP for p = p(a) G (—oo, 0] U [1, oo), including nor-mal for p = 0, Poisson for p = 1, gamma for p = 2, noncentral gamma for p = 3/2 and inverse-Gaussian for p = 3; see, e.g., [43,Chap.4]. Denoting with Tp(p,(j>) any Tweedie distribution, the correspond-ing Poisson-Tweedie distribution, denoted by VTp(p, </>), has the p.m.f.

f°° e~* ty

fp{y;p,(f))= / — — T p f a f l d t (5) Jo y-

for all y e N from (4) and only for p £ {0}U [1, oo). Some recent properties, including a

recursive version of the p.m.f. (5), can be found in [18,Th.2.1]; see also [60,Prop.3]. The unit variance function of the Poisson-Tweedie model, proving its overdispersion from (3), is given by

a2p(p) = p + pp exp{(2 - p)%(p)h (6)

for p > 0 and where $p(p), implicit most of the time, denotes the inverse of the in-creasing function s d[logEexp(sY)]/ds with Y ~ VTP(p, 1); see [60,Prop.2]. The well-known particular cases are Hermite for p = 0, Neyman type A for p = 1, neg-ative binomial for p = 2, Polya-Aeppli for p = 3/2, Poisson-inverse Gaussian for p = 3, and Poisson for p —» oo. For a more general use of the Poisson-Tweedie model, for example, in generalized linear models, it is necessary to reparametrize one of two expressions of p.m.f. (5) and unit variance function (6); see, [46,47] for a simplification of (6) through a new notion of (discrete) exponential dispersion models, called fac-torial or discrete dispersion models.

For a multivariate mixed Poisson model, one can refer to [21]; see also [46,50]. The mixed Poisson processes can be here envis-aged. Instead of Poisson (1) in (4), it is also possible to consider another mixture to provide over-dispersion models (e.g., [42]).

30.2.2 Compound Poisson In order to introduce also an over-dispersion model relative to Poisson (1), one can consider a (count) c o m p o u n d P o i s s o n distribution, which is the count probability distribution of the sum Y of a "Poisson-distributed number" X ~ V(p) of independent identically distributed (i.i.d.) count random variables Zi , Z2, • • •:

x y = = + + (7)

One can refer to [117] for some details on count compound Poisson, also called "stut-tering Poisson" distribution, because there


are continuous compound Poisson distribu-tions if Z j s are continuous random vari-ables in (7). This compound Poisson repre-sentation (7) can be found under the name of the Poisson stopped-sum. Particular cases are for randomizing the Poisson pa-rameter such as mixed Poisson.

From (3) and to detect departures from the Poisson random variable, one of the common variability measures of a count random variable Y is the so-called Fisher index of dispersion (see [22] and also [76] for tests), defined by

(8 ) where 1 2 ( Y ) = E[Y(Y - 1)]/[E(Y)}2 is the factorial moment of second order and vary-ing as Fi(y); see [75]. Thus, the compound Poisson (7) is an overdispersed model be-cause, by conditioning one successively has E ( Y ) = E(X)E(Zi ) = / iE(Zi) and

Var{Y) = M(X)Var(Zi)+Var(X)[E(Zi))2,

with E(X) = Var(X) = [x and Var(Zx) = E(ZX) + [E(^i)]2[I2(ZI) - 1].

Particular cases can be found among the mixed Poisson models in Section 30.2.1 since there is connexion between these two count models; see, for exam-ple, [50,Prop.2] and [106]. Another global class of compound Poisson is the Hinde-Demetrio model, belonging to the expo-nential dispersion models; see [56,60,61] for more details. Briefly, if 7iDq(fi,<l>) de-notes an Hinde-Demetrio distribution then q £ {0} U [l,oo), the additive semigroup N -f qN is its support and

+ (9)

is the form of its unit variance function, compared to (6). Special cases are positive-translated Poisson for p = 0, scaled Pois-son for p = 1, negative binomial for p = 2, strict arcsine for p = 3 and standard Pois-son for p —> oo.

From representation (7), the compound Poisson processes are easily deduced by re-placing X to the Poisson process (Xt)t>o-Also, it is always possible to extend this compound Poisson (7) to "compound com-pound ..." (i.e., change the Poisson ran-dom variable X ~ ^(aO to any count random variable such as compound mixed Poisson) and investigate their over- and underdispersion properties. However, it is also important to look at the flexibility of the resulting model. Let us note here that "compound weighted Poisson distri-butions" of Minkova and Balakrishnan [75] produce both over- and under-dispersion models. In Section 30.2.3, we introduce the "weightening operation" on standard Poisson distribution.

30.2.3 Weighted Poisson

The following weighted Poisson distribu-tion is, in my mind, the most complete and flexible of the generalizations of the Poisson distribution, which provides a uni-fied approach to handle both over- and underdispersion. Originally introduced by Fisher [22] through the method of ascer-tainment, which was just a method of adjustment applicable to many situations (see, e.g., [62,82,91]), the weighted Pois-son distributions are now used widely as a tool in the selection of appropriate models for observed data drawn without a proper frame. More practical, for example, in clin-ical trials, several mechanisms on the count data collections are included in the weight function that we define as follows.

Let X be a standard Poisson random variable with p.m.f. f ( x ; /J,) = Pr(X = x\/i) given in (1). Suppose that when the event X = x occurs, the probability of ascertaining it is w(x). The recorded x is thus a realization of the count random variable Y = Xw, which is said to be the w e i g h t e d v e r s i o n of X. Its p.m.f. is

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given by

™{v)f(v\m) E > ( X ) ]

= Pp[Xw = »|M,ti;(.)]> (10)

for all y G N and where E^[w(X)] = exp(-/u)/x! is the normaliz-

ing constant belonging in (0, oo). The fact that different count distributions can be formulated as a weighted Poisson distribu-tion (see, e.g., [58]) does not mean that their generating process is necessarily, im-perfect recording of classical Poisson dis-tribution. The weight (or recording) func-tion w(-) is a nonnegative function, that is, w : N —> [0,oo). The weight func-tion w(-) = can depend on free/orthogonal parameters and p of the initial Poisson model.

Particular cases are the standard Pois-son with w(y) = constant for all ye N, the size-biased Poisson with w(y) = y for all y e N, the popular COM-Poisson with w(y;u) = (y\)1~~u for all y G N (see, e.g., [27,49,96-101,120]), and, more globally, the exponentially weighted Pois-son distributions with

w(y\6,v(')) =exp[8v(y)]J Vy G N, (11)

where <5 G R and x V(X) is the so-called "exponent weight" function (may or may not depend on the original Poisson param-eter p and another free parameter such as 0); see [10] and also [62]. As examples of v(-) in (11), one has: v{y\<t>) = log(y + 0) of Castillo and Perez-Casany [9], v(y\ p) = \y — p\ of Ridout and Besbeas [93], and v(y;p) = ( y / p ) v which is the Kullback-Leibler distance between y and p\ see, for example, [62]. Note also an illustration of the fact that "any count distribution is a weighted Poisson (or another count) distri-bution" [58]. Let $ > 0 and 0 G © C R. Then, the p.mj. of count (additive) expo-nential dispersion models of the form

f(y; (j>) = a(y; <t>) exp[6y - «(0; 4>)]

is a weighted Poisson distribution with mean p = exp(0) and the corresponding Poisson weight function is

w(y9</>) = y\a(xi<f>), Wye N.

In order to characterize the type of dispersions in the exponentially weighted Poisson models (11), the convexity of v(-) is required and then 5 > 0 (<5 = 0 and 8 < 0, respectively) implies the over (equi-and under-, respectively)-dispersion. More generally, the weighted Poisson random variable Y = Xw is overdispersed (under-dispersed) if and only if the mean weight function p Etl[w(X)] is log-convex (log-concave), that is,

E fl[w(X)}Efl[w(X + 2)) -{Ep[w(X + 1)]}2^0.

An important property connecting over-and underdispersion count models is the (pointwise) duality between two weighted Poisson random variables Y\ = XWl and y2 = Xw2 such that

WI(x) W2{x) = 1, Mx e (12)

Thus, it enables us to build an opposite dispersion model; see [62] for mire de-tails. For practitioners, it is of interest to use a family of count distributions closed by duality, that is, possessing both over-and under-dispersion properties with re-spect to the parameters. Such a process would lead automatically to an appropri-ate model depending on the type of ob-served count data. The property of clo-sure by duality contributes to the success of the flexible COM-Poisson and some ex-ponentially weighted Poisson distributions. One can refer also to J0rgensen and Koko-nendji [46] for a new approach to this dual-ity through a transformation on a kind of variance function of the mean.

According to the weighted Poisson rep-resentation of any count distribution, it is


shown in [58] that the weightening oper-ation commutes with several useful trans-formations applied to count distributions such as left-truncation, zero-modification, and size-biasing. The weightening repre-sentation is also possible with respect to another count distribution than the stan-dard Poisson distribution [58]. See [75] for compound weighted Poisson distributions and, for example [75,Sect.3.10], for weight-ening a mixed Poisson distribution. Fi-nally, one can refer to [6,7] for stochastic process versions of weighted Poisson dis-tributions.

30.2.4 Lagrangian Poisson A large class of parametrical count mod-els is the family of general Lagrangian dis-tributions (see, e.g., [42,Sect.7.2] and also [53,Sect.4]), which is yet another promi-nent extension of the Poisson distribution providing over- and underdispersion mod-els. This family of count distributions con-tains the "modified power series distribu-tions" [34]. The well-known example, the title of this section, is the generalized Pois-son or Consul's Lagrangian Poisson dis-tribution [12]. After defining, we present some examples and global expresions of mean and variance before giving two useful probabilistic interpretations.

Let g(z) and h(z) be simply two probabil-ity generating functions (p.g.f.s) such that 2 = £(u) is the smallest root of z = ug(z), or £(u) = ug(£(u)), and

- [ I ^ H ^ L -for all j e N + := { 1 , 2 , . . . } . Then, the general Lagrangian distribution or model Y of the pair (g, ft) can be defined from its p.g.f. of the form

^ qiV h(z) = h(£(u)) = ft(0) + £ — My; g, f )

y=l V'

and, therefore, its p.m.f. Pr[Y = • |g, f ] = /(•;#> h) is given by

*t m _ / h ( ° ) i f y = 0

( y , g j ) / y l if y > 0. (13)

As examples of the general Lagrangian models Y , generating by pairs (<7, ft), we can combine various expressions of p.g.f. g with those of ft. Without normalizing con-stant, here are five basic generating func-tions of g: gi(z) = exp ( z ) for Poisson; g2{z) = (1 + z)T, r € N + , for binomial; 9s(z) = (1 — z)~r-> t > 0, for negative bi-nomial; #4(2) = —rlog(l — z), r > 0, for logarithmic; and, <75(2) = exp[r sin-1(z)], r > 0, for strict arcsine. Note that the corresponding Lagrangian function £ of a given g does not have always a closed form. As for ft, we also have several possibilities for each g given above. An outline of ft is given by

h(z) = [b(z)g(z)]+, (14)

with cf) > 0 and &(•) corresponds to a gener-ating function of positive measure. In the same spirit of Consul [12] the generalized Poisson or Abel, generalized negative bino-mial or Takacs and generalized strict arc-sine or large arcsine are, respectively, ob-tained with b(z) = constant and g = gj in (14) for j = 1,3,5. See, for example, [42,53] for extensions and other combina-tions of ft in (13) with b equals or not to gi in (14) and g equals or not to one of gj.

For any general Lagrangian model Y with p.m.f. given in (13), its mean and variance can be expressed, respectively, as

and

Var(Y) = (l-eg'/g)-2{fi[l-ig'/g

+tg-2(gg'+egg"-t(g')2)} +e2(hh" - {h'f)h-2}.

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One can also refer to [53] for some (unit) variance functions satisfying the over-dispersion phenomenon and for the two fol-lowing interpretations.

The first and global probabilistic inter-pretation of any general Lagrangian model Y of (13) is in terms of branching pro-cesses. Let g(z) = a P-S-f-on N. Let (Wk)kL0 be a Markov chain on the integers with transition function de-fined by

E (zw^\Wk) = [g(z)]Wk.

If E(Wi) = XVX < 1, it is routine to show that there exists fc € N such that Wk = 0 almost surely. In this case, we write the count random variable

CO r = £ w f e , (15)

0

the so-called "total progeny" in the theory of branching processes. Indeed, if Wo = 1 then t(z) = E (zY) = zg(£(z)). Fur-thermore, if Wo is random and if h(z) = E ( z W o ) then Y of (15) is also characterized by the p.m.f. (13) or its p.g.f. See, for ex-ample, [114] for other elementary interpre-tations in particular cases of Poisson, bi-nomial, and negative binomial, which rep-resent basical models of equi-, under-, and overdispersion.

The second probabilistic interpretation of general Lagrangian model Y of (13) is given for the following particular situation of

h(z) = [g(z)}\ fc€ N+ In fact, for g a p.g.f. on N with g(0) > 0, one considers X i , X 2 , • • • a sequence of independent count random variables such that

E {zXi)=g(z), V i = 1 , 2 , . . . .

Let So = 0, Sn = n-(Xi + • • - + Xn,) and, for all k = 1,2, • •«, let

Tk = inf{n > 0; Sn = fc},

with Tk = oo if this set is empty. Then, the image of Tk (restricted to finite part) under the transformation x \ x — k has the same distribution as Y following the general La-grangian distribution with h(z) = [g(z)]k.

30.2.5 Semiparamet r ic Poisson

For a given i.i.d. sample Yi, Y2, • • • ,Yn of a count random variable Y , it may satisfy the empirical version of conditions (3) of over- or underdispersion using the Fisher index (8) such as

a*(Y)%Ynt (16)

where Y n is the sample mean and the sample variance. Without assumption or choice of (parametric) count dispersion distribution on Y , that is, with unknown p.m.f. / on the support T C N, we are go-ing to "let talk count data" for estimating this discrete function / through a nonpara-metric procedure that is suitable for count data and inspired by the continuous case.

Following the discrete structure of count distribution, it is necessary to consider the discrete associated kernel estimator fn of /, which is defined as

U y ) = lY,Ky,hn{Yi), ye T, (17) 1=1

where hn > 0 is an arbitrary sequence of smoothing parameters (or bandwidths) that fulfills lim hn = 0, and Ky,hn{-) is

n—•oo '

the "discrete associated kernel" with the target y and the bandwidth hn. More pre-cisely, a p.m.f. Ky,hn(') on its support Sy

(not depending on hn) is said to be a d i s -c r e t e a s s o c i a t e d k e r n e l if it satisfies the following three conditions:

lim E(Zythn) = y,

lim Var{Zy,hn) = 0, hn~+ 0


where Zy,hn is the discrete random variable whose p.m.f. is Ky,hn. See [59,123,124] for more details such as bias and variance and for choosing both bandwidth and type of discrete kernel, even for small sample size. Note that, apart from a discrete tri-angular kernel, an underdispersed kernel, says binomial, is more appropriate as a discrete smoother than equi- and overdis-persed ones. Up to^the normalizing con-stant Cn = J2yeT fn(y)> w e assume that

y l~> fn(y) is a p.m.f. In the particular situation of over- and

underdispersion with respect to Poisson model (1), for instance, one can consider a semiparametric approach for estimating the unknown p.m.f. / instead of a purely nonparametric approach (17). Moreover, according to the global representation (10) of any count distribution may be viewed as a weighted Poisson, we can henceforth consider this s e m i p a r a m e t r i c a p p r o a c h by decomposing

f(y) = u{y,p)p(y',p) (18)

for y G T with p( •; p) being the parametric part well specified and depending on /i, say Poisson or binomial, and •; p) being the unknown discrete weight function or non-parametric part for fixed p. Work in these directions has been recently done for p.m.f. [63] and for count regression function [1]. In the previous papers, the semiparamet-ric estimation of /(•) = •; m) following the decomposition (18) is represented as

where first the estimation of parameter p, pp, is obtained by maximum likelihood or by least squared methods, and second, for the value of pp obtained, one gets the dis-crete kernel estimation of a;(-) = ;pp), namely, £(•) = u>(-;p,p). From (18) and (17) one can express

f ( y ) i n Ky,hn(Yi)

p(Yt; ftp)

= P { V ' r " P )

Its . 1 1=1 p(Xi,V>pY i / G T ,

and, therefore,

n piXi'^p)

This procedure is under progress for inves-tigating the convergence of the estimator, with some new materials related to the dis-crete associated kernel method for smooth-ing discrete functions.

Note that another way to estimate

f(')=fw(",p) = ^2w(y)p(y; p)

of (18) represented in the sense of (10) is to consider an empirical weighted parametric estimation

f ( y ) = fw(y,fiw) = ™yv(y\ V>w)

fiw)

for all y, where wy is the frequency of count y in a sample of size n and pw is an estimation of p. Work in this direc-tion must be done for investigating the con-vergence of /(•) to something that makes sense, method and consistency of p w > and comparison of /(•) with respect to /(•) and their perfomance through simulations and real count data.

While the parametric count model is ap-propriate to handle over- and underdis-persion, semiparametric Poisson must be preferable to purely nonparametric (17) without consideration of any known count model; see [63] for model diagnostics.

30.3 Count Explanatory Models

Count data are encountered in a wide range of applications, including medical

176


and biomedical research. In order to model this type of data with covariates, one ap-peals to appropriate and adapted frame-works such as the class of generalized linear models [73]. However, the same problem of over- or underdispersion (16) also oc-curs in the framework of explanatory mod-els for count data, that is, when the Pois-son distribution does not fit well, and the observed dispersion is greater or smaller than that predicted by the standard dis-tribution; see, for example, [30].

Various contexts offer themselves to practitioners and we briefly describe be-low three families of count explanatory models: generalized linear models with re-peated measures, random effects, and its zero-inflated version; an introduction to count time series models; and nonpara-metric count models using discrete or con-tinuous kernels for expected (or observed) value of count response variable.

30.3.1 Generalized Linear Models

The well-known g e n e r a l i z e d l inear m o d e l s have the following basic setup. From any parametric count dispersion model Y of Section 30.2, which is here called a response variable, one needs some covariates x i , • • • , x^ for explaning Y with x i = 1.

The standard assumptions are that Y\, • • • , Yn are independent count response variables, with mean and variance function

E(Yi)=iMenidVar(Yi) = (f>a2(ni), (19)

respectively, and p.m.f. of the (reproduc-tive) exponential dispersion form

fi(yi\0u<t>) = a(yi] <£)exp{^[Oiyi - /c(0<)]};

(20)

and, finally but most important, a linear function of covariates m called link func-

tion

= V i = P i x n + • • • + (3kXik, (21)

for a l H = 1, • • • , n and where (3\, • • • , fa are unknown regression coefficients and <f)> 0 is the dispersion parameter for which cf)"1 can be weighted as (j)~lWi in (20) by prespecified weights. In (20), the canoni-cal or natural parameter 6i is directly con-nected to the mean parameter /i* > 0, that is, fii = fii(0i) = m~l(r)i) and «(•) and a(-, •) are known functions specifying a particular member of the exponential fam-ily. From the unit variance function cr2(-) in (19), the canonical link function is ob-tained by

rjj, m0(fi) = I

J V 0 <T*{Z) dz

and, in practice we use the common log-linear link function of count data m(fi) = log/i or /i = exp(xT/3) from (21). The unit deviance function is also given from the unit variance function as follows:

d(y; /i) - 2 f O U L dz.

Thus, one can express the p.m.f. (20) of exponential dispersion models, as

f(y\ n, (j)) = a(y; <t>) exp

See, [8,16,27,43-47,57,61,68-74,78-80] for more details, variants, extensions, and applications. One can also find in the above references some parameter estima-tions of /3 and (j) [e.g., (nonparametric) maximum likelihood, maximum and ex-tended quasi-likelihood, pseudo-likelihood, moment methods and Bayesian approach], analysis of deviance and software (e.g., SAS, R, S-plus, STATA, Minitab, GEN-STAT, and SPSS). Here are two ways for extending the above standard generalized linear models for clinical trials.


Within the framework of the repeated measures and random effects, one considers Y^ as the j th outcome measured for sub-ject i with i = 1, • • • , n and j = 1, • • • , n .̂ Thus, the p.m.f. (20) stays of the same expression but denoted by

(22) where the conditional mean is modeled as

E(li j|bi, Z, Oij) = dij^ij

with lij — M (xjjlt + zfjbi)

for a known link function M, x -̂ and zij two vectors of known covariate values, re-spectively, Z a vector of unknown fixed regression coefficients. The measurement-specific parameters 0^ follow a conjugate distribution (e.g., gamma if Y^ is a Poisson for overdispersion) and the subject-specific parameters 7^ follow a centered normal distribution b̂ with a given variance. In general, the measurement-specific random effect Oij is considered for accommodating over- or underdispersion, while the subject-specific normal random effect bi is used to model extra variation, outliers, correlation coming from an hierarchy in the data, and other unexplained sources of variation.

In additioning zero-inflation phenomena, a general form of its mixed p.m.f. can be deduced from (22) as follows:

Pr(Y^ = yij\hli, Z, Oij, <fi,Pij)

' Pij + (1 - Pij)fi(0; bli, Z, </>,pi:i) = < if Vij = 0

(1 ~ Pij)fi(Vij\ b l i , Z, 4),Pij) if y^ > 0.

Indeed, in zero-inflated count models, it is well-known that there are two scenar-ios that can generate zeros: For obser-vation i at time j , zeros come both a point mass with probability p^ as well as from the count component with prob-ability 1 — p^. The zero-inflation compo-nent p^ — pij(x2fjT + z2fjb2i) is mod-eled by a Bernoulli: In the simplest case

with only an intercept, but potentially con-taining known regressors x2^- and a vector of zero-inflation coefficients T to be estimated, as well as random effects b2^. Common link functions can be used. Note that x^j, z^, and b̂ of (22) are now re-placed by xl^-, z l i j , and b2^, respec-tively, for the nonzero count part. The regressors in the count and zero-inflation component can be overlapping, a subset of the regressors can be used for the zero-inflation, or entirely different regressors for the two parts can be used. Sometimes, a simple random-intercept model is ade-quate. In practice, we need more flexibility on assumptions to treat this kind of model and also for finite mixture models. Also, one can refer to [11,31,38,51,77,85,115].

See, for example, [8] for censored count response models and [78] for longitudinal count data. A few directions can be new dispersion models for other types of count response variables (e.g., [44-47]), general-ized nonlinear regression models, and mul-tivariate generalized linear models for mul-tivariate response Y (e.g., [3]).

30.3.2 Count Time Series Models

The same problems of over-, equi- and underdispersion (3) or (16) appear, at various levels, in large classes of count time series models. For example, one can consider a count variable measured on a patient's regular/irregular visits to a clinic. With the use of generalized lin-ear models (Section 30.3.1) and of branch-ing processes (e.g., at the end of Sec-tion 30.2.4) with immigration, one can model time series of counts following one of two categories: observed driven mod-els and parameter driven models. Refer to [14,24,25,48,84,108-111,119-122] for some details and references therein.

One of the main facts in modeling time series of counts is that the conditional dis-

176


tribution of observed counts given past outcomes or latent process is commonly as-sumed to come from some count dispersion distributions, described above in Section 30.2 and, from here, we generally denote by 0,5), which extends the standard Poisson with <f> = 0 and 5 = 0 for con-ventional notation. Let us poset that the following modeling approach is based on the observation that the total number of count data can be thought as a collec-tion of individual count data that corre-spond to different clinical strategies for ex-ample. Of course, a kind of flexibility on the considered CD distribution is required for modeling time series of counts. In the context of over- and underdispersion from CD distribution, we briefly present below two classes of integer-valued time series models: integer-valued generalized autore-gressive conditional heteroscedastic (IN-GARCH) models and integer-valued au-toregressive moving average (INARMA) models.

Assume that (Yt)tez is a count time se-ries, also called the response process. Let (At)tez be a sequence of the unobserved mean process of Yt given its past. De-note by & t = the past of the pro-cess up to and including time that is, the a-field generated by {Ys, s < t\ Ao}. Then, the count process (Yt)tez is a CD-INGARCH(p, <?) of integer orders p > 1 and q > 0, respectively, when random vari-ables Yil&o, Y2\&u ' ' • > Ynl&n-x are con-ditionally independent and the conditional distribution of Yt generally satisfies both conditions:

Yt\&t-i~CD()it;<t>95) (23)

and

p Q

At = a 0 4- ] T caYt-i + J T fyXt_j , (24) *=1

where t > max(p, q), ao > 0, ai > 0 for all i = 1, • • • p j > 0 for all j = 1, • • • <j)

and 5 are other suitable parameters of the CD distribution. When q = 0 the above model is denoted by CD-INARCH(p); see, for example, [111] for p = 1.

The At in (23) and (24) can be trans-formed to A£ = A£(</>, S) for suitability and flexibility of the models (e.g., [120]). For example, with the help of some in-trinsic properties of the CD distribution, we can exactly calculate E{Yt\^t-i) and Var(Yt\^t-i). Thus, many aspects of the model can be investigated correctly, such as existence or stationarity, ergod-icity, estimating methods, testing, auto-correlation functions, unconditional dis-tributions checking the over- and under-dispersion with /i = E ( Y t ) , higher-order moments, diagnostic, forecasting, general-izations and real applications (e.g., [48]). Particular cases of CD distribution (e.g., Poisson, negative binomial and, recently, COM-Poisson) have been considered for this model and some other of its de-riviations. Noting in passing ulk}1 = ([/*_!, • • • , Ut~k), the linearity (24) of the

mean process At = r ( y ^ ; A c a n be changed to nonlinearity on the known form of r( ), like an additive form

r Al«y = n ( y W ) + r 2 ( a ^ , ) and a purely functional INGARCH model.

The second type of count time series models (Yt)tez is the class of INARMA models. In contrario of the scalar multi-plication in usual real-valued ARMA mod-els, these are built through the probabilis-tic operation of "binomial thinning," de-noted below by o and defined as follows: If Y is a count random variable and let a € [0,1], then the random variable

Y

ao y : = ] T B 4 ( a ) i=l

is said to arise from Y by binomial thin-ning, where the counting series Bi(a) are i.i.d. Bernoulli random variables with suc-cess probability a and independent of Y;

518 Over- and Underdispersion Models

see, for example, [109] for a review. This random variable a o Y counts the number of successes in a random number of Bernoulli trials with a fixed; that is, for given Y = y then a oY follows the binomial distribu-tion with parameters y and a. Thus, the count process (Yt)tez is said to be a CD-INARMA(p, q) of integer orders p > 1 and q > 0, respectively, if it has the following representation:

p q

Yt = J2ai°Yt-i + Yl Pj ° > (25) 1 0

with t > m a x ( p , q ) and where (et)tez is a sequence of i.i.d. count dispersion random variables following CD(p; <5) distribution with mean p and variance a2 = cr2(/i; </>, <5), and all p + q -f 1 thinning operations with respect to a* £ [0,1] and (3j £ [0,1] are per-formed independently of each other, and independently of (et)t£z and — s < t\eo} and where (et)tez is independent of J^. One can choose /3o = 1 and simplify the assumption of variance a2 = a2(p; as a2 = p in the Poisson case. Two particular cases are CD-INAR(p) := CD-INARMA(p,0) and CD-INMA(g) := CD-INARMA(0,g). Similar comments after defining CD-INGARCH in (23) and (24) can be made here for this class of CP-INARM A. However, once again, the flex-ibility of the models (25) always remains crucial for developing all procedures of the count time series analysis. As for a direc-tion of a multivariate extension, one can re-fer to [84] for the bivariate Poisson-INAR.

30.3.3 Nonparametric Models From Equation(17), the discrete associated kernel estimator, we also have its classical continuous version, which is defined by

(26)

where /C is the continuous kernel function, generally a symmetric probability density function, not depending both on the tar-get y and the bandwidth /in, and admit-ting zero mean and finite variance; see, for example, [59] and references therein. Both discrete and continuous versions of smoothers Ky,hn(-) lead to consideration of a nonparametric part in treatments of over- and under-dispersion models. For ex-ample, the inverse link function m~l of the generalized linear models (21) such that p = m _ 1 ( x i , • • • , Xfc) could be an unknown function and need to be estimated. In the same spirit, one can introduce a kernel es-timator of r in the functional I N G A R C H such that the mean process \ t of (24) is

written by At = r ( V ^ ; A ^ ] ^ Another

way is to use the appropriate kernel, dis-crete or continuous, for analyzing count models with or without covariates; see, for example, [35].

In general, suppose that X i , • • • ,Xk are k regression variables observed jointly with a count response variable Y satisfying the empirical over-/equi-/underdispersion con-dition (16) or that follows a count disper-sion distribution, denoted by CD(p,]<f>,5) with p = E ( Y ) and Var(Y) = a2(//;<M) such that the over-/equi-/underdispersion condition

cr2(p](f),5) ^ p

holds as in (3). Two possible nonpara-metric models for association between the regressors and the expected (or observed) value of Y given X\, • • • , Xk are

p(ovY) = R ( X l r . . ,Xk), (27)

where R is an unknown function to be es-timated. According to the nature of each regressor Xj and then to the form of R in (27), there exists a solution to esti-mate R at least by using the recent and more global notion of "associated kernel estimator" (26) and for given any support TkCRk. See, for example, the Ph.D. the-


sis manuscript of Tristan Senga Kiesse1 for some solutions and tracks of both univari-ate and multivariate associated kernels, for which current works are well begun.23

In fact, from a probability density esti-mator as (17) for p.m.f. or a regression es-timator of the Nadaraya-Watson type (see, e.g., [1]), one can need a m u l t i v a r i a t e as-s o c i a t e d kerne l KX)Hn,fc (') o n its support §x,Hn,fe (can here depend on Hn,fc) that satisfies the corresponding simple condi-tions:

x € Sx,Hn)fc,

lim H n & = Ofc, n—• oo

lim E(-Zx,H n,J = x , n—•oo

lim Cov(Z^Hn,fc) = n—+ oo

where 2X )Hn fc is the multivariate ran-dom variable whose density distribution is K X j h* . ( • )• The above ingredients are defined by analogy to those of univariate cases of Section 30.2.5: the vector x = (a?i, • • • ,£fc)T € Tk is the target, the size sample is n, the fc x fc matrix = (hij(n))i,j*=i,— ,k is for bandwidths and is symmetric and positively defined, and Ok denotes the fc x fc null matrix. Under as-sumption of diagonal bandwidths matrix Hn,fc = Diag(f t i (n) , • • • , hk(n)) and with respect to the nature of each regressor Xj, continuous, discrete, or even mixed, one can use the following polyunivariate asso-ciated kernel

k Kx,Hntfc(-) =

J=1 1 Nonparametric Approach by Discrete

Associated-Kernel for Count Data (in French), http:/ / tel .archives-ouvertes.fr / tel-00372180/fr/

2Francial G. Libengu£, Ph.D. thesis on Non-parametric Methods by Mixing Associated Kernels and Applications, defended on June 13, 2013, Uni-versity de Franche-Comt£.

3 Sobom M. Som£, Ph.D. thesis on Multivariate Associated Kernel Estimators, in progress, Uni-versity de Franche-Comty.

where is an appropriate uni-variate associated kernel for the j t h com-ponent or regressor. This flexibility allows us to handle many situations now. See, for example, [81] who had to transform the count covariates before smoothing by clas-sical continuous kernel (26) in a binomial regression model. See also [35] for extend-ing the multivariate treatment by continu-ous kernels to "associated discriminant ker-nels." Finally, refer to [1] for a direct semi-parametric regression on count covariates.

30.4 Summary and Pinal Remarks

This chapter began with a discussion of general concept of the phenomenon of over-/equi-/underdispersion. We pointed out several origins of such phenomena that are based on different kinds of variabil-ity from the hypothesized structure of the population. The exponential dispersion models were presented as special classes of distributions for treatment by separat-ing the mean parameter from the dis-persion one. In addition, it was recom-mended to start with the initial model and then to improve it according to the possi-ble causes of the phenomenon. The rest of the chapter focused on definitions of some models of count data sets for which over-/equi-/underdispersion occurs gener-ally and these were applied in diverse do-mains of the statistics among clinical trials.

From the equidispersed Poisson model, Section 30.2 presented several modifica-tions of this standard Poisson distribution that we denoted count dispersion models in the sense of over- and underdispersion with respect to Poisson, and we briefly dis-cussed some extended models. In fact, the situation of count data sets is easy to un-derstand before generalizing to other types of data sets. The well-known mixed Pois-son distribution was first introduced; it is exclusively for the over-dispersion by het-


erogeneity. Afterward, it was also the case of the compound Poisson models that we produced connection, combination and ex-amples with respect to the mixed Poisson models. The complete case of weighted Poisson models that one can apply to both over- and underdispersion models was then defined. Many families of examples illus-trated this large count model. The prop-erty of duality which could be changed over- and underdispersed models to its op-posite (if one exists) was shown in a point-wise way. It is encouraged to produce some multivariate versions and the most flexible as possible as COM-Poisson. However, the multivariate notion of over- and underdis-persion must be well defined. Among other classes of univariates and parametric count models, a part of the general Lagrangian distributions was investigated with prob-abilistic interpretations such as the total progeny in the theory of branching pro-cesses. Finally, we included a semipara-metric count models for over- and under-dispersion that are oriented by the stan-dard Poisson model. The nonparametric part used the notion of discrete associated kernels.

Section 30.3 was devoted to three types of count explanatory models always based on over-/equi/-underdispersion. The first concerned some families of the popular generalized linear models. An introduc-tion to the repeated measures and ran-dom effects was provided and completed with its zero-inflated version. Then, in a similar spirit, we evoked two classes of count time series models that connected to parametric count distributions: integer-valued GARCH and integer-valued ARMA for which the binomial thinning operation replaced the scalar multiplication in stan-dard real-valued ARMA models. The as-sumption of i.i.d. in the INARMA can be relaxed, such as the continuous one, for some particular studies of count data sets. Also, the multivariate or vector IN-

ARMA could be useful. The last part of this section, and of this chapter, consid-ered the problem of nonparametric regres-sion models under various angles. More-over, we used the complete notion of associ-ated kernels both continuous and discrete, and in its multivariate version, to deal with multiple kinds of covariates. The link be-tween over- and underdispersion models and treatments of clinical trials is insepa-rable from progress in statistics and prob-ability, with even computer science provid-ing additional tools and incentives for the implementation of the models. The list of references supplied below is for more de-tails and some practical illustrations.

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