Page 1

Biostatistics (2011), 12, 1, pp. 173–191

doi:10.1093/biostatistics/kxq050

Advance Access publication on August 23, 2010

A composite likelihood approach to the analysis of

longitudinal clonal data on multitype cellular systems

under an age-dependent branching process

RUI CHEN, OLLIVIER HYRIEN∗

Department of Biostatistics and Computational Biology,

University of Rochester Medical Center, 601 Elmwood Avenue, Rochester, NY 14642, USA

ollivier−hyrien@urmc.rochester.edu

MARK NOBLE, MARGOT MAYER-PR¨OSCHEL

Department of Biomedical Genetics, University of Rochester Medical Center,

601 Elmwood Avenue, Rochester, NY 14642, USA

SUMMARY

A recurrent statistical problem in cell biology is to draw inference about cell kinetics from observations

collected at discrete time points. We investigate this problem when multiple cell clones are observed

longitudinally over time. The theory of age-dependent branching processes provides an appealing frame-

work for the quantitative analysis of such data. Likelihood inference being difficult in this context, we

propose an alternative composite likelihood approach, where the estimation function is defined from the

marginal or conditional distributions of the number of cells of each observable cell type. These distri-

butions have generally no closed-form expressions but they can be approximated using simulations. We

construct a bias-corrected version of the estimating function, which also offers computational advantages.

Two algorithms are discussed to compute parameter estimates. Large sample properties of the estimator

are presented. The performance of the proposed method in finite samples is investigated in simulation

studies. An application to the analysis of the generation of oligodendrocytes from oligodendrocyte type-2

astrocyte progenitor cells cultured in vitro reveals the effect of neurothrophin-3 on these cells. Our work

demonstrates also that the proposed approach outperforms the existing ones.

Keywords: Bias correction; Cell differentiation; Composite likelihood; Discrete data; Monte Carlo; Neurotrophin-3;

Oligodendrocytes; Precursor cell; Stochastic model.

1. INTRODUCTION

In stem cell biology, there exists considerable interest in studying signals that may modulate or alter the

processes that regulate the formation of tissues during development or repair. Understanding such pro-

cesses has important clinical implications as it may offer potential means to restore or maintain tissue

∗To whom correspondence should be addressed.

c ? The Author 2010. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com

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174R. CHEN AND OTHERS

and organ functions. One of the challenges in conducting detailed analysis of the effects of these sig-

nals, however, is the general lack of quantitative approaches that allow analysis of different aspects of

progenitor or stem cell function.

Thepresentpaperisfocusedonoligodendrocytetype-2astrocyte(O-2A)progenitorcells, whichisone

of the best studied precursor cell populations (e.g. Noble and others, 2004). These progenitor cells may

either divide into 2 new progenitor cells or differentiate into terminally differentiated oligodendrocytes,

which produce the myelin sheaths that enwrap axons in the central nervous system. The differentiation

of progenitor cells into oligodendrocytes needs to be properly regulated to maintain normal function of

the central nervous system. In addition to the function of O-2A progenitor cells in vivo, their ability to

grow and differentiate in vitro has established these cells as an important tool in modern cell biology

that allows the study of the most basic cellular functions (i.e. self-renewal by division, differentiation,

and death) at the clonal level in tissue culture (e.g. Ibarrola and others (1996); Smith and others, 2000).

The development of a typical clone of these cells cultured in vitro started from a single O-2A progenitor

cell is displayed in Figure 1. It may be altered in multiple ways following exposure to agents, such as

cytokines, drugs, or toxicants, and it is of great interest to biologists to determine and quantify these

changes (Dietrich and others, 2006).

Over the past decade, a great deal of attention has been paid to the development of a statistical frame-

work for gaining a quantitative understanding of the biological processes that govern the division of O-2A

progenitor cells and their differentiation into oligodendrocytes (Yakovlev, Boucher, and others, 1998;

Yakovlev, Mayer-Pr¨ oschel, and others, 1998; Yakovlev and others, 2000; von Collani and others, 1999;

Boucher and others, 1999, 2001; Zorin and others, 2000; Hyrien and others, 2005a,b, 2006; Hyrien,

2007). These publications used multitype age-dependent branching processes as a means to model the

temporal development of cell clones. The resulting methods provided quantitative insights into the gen-

eration of oligodendrocytes from cultured O-2A progenitor cells. For instance, Yakovlev, Boucher, and

others (1998) observed that the generation of oligodendrocytes was regulated by a combination of cell-

intrinsic factors and environmental signals that may modulate the probability of differentiation of O-2A

progenitor cells. Other studies based on similar quantitative approaches (Yakovlev, Boucher, and others,

1998; Yakovlev, Mayer-Pr¨ oschel, and others, 1998; Yakovlev and others, 2000; von Collani and others,

Fig. 1. Oligodendrocyte development under in vitro conditions. When cultured under appropriate growing conditions,

any O-2A progenitor cell will either divide into 2 new O-2A progenitor cells or it will differentiate into a single

terminally differentiated oligodendrocyte. O-2A progenitor cells may also die. Oligodendrocytes are terminally dif-

ferentiated cells that arise from O-2A progenitor cells. These cells are not able to divide and they ultimately die (i.e.

disintegrate and disappear from the population). Cell death was not observed in our NT-3 experiment.

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Composite likelihood for branching processes

175

1999; Boucher and others, 1999, 2001; Zorin and others, 2000; Hyrien and others, 2006) confirmed that

thyroid hormone induces the differentiation of O-2A progenitor cells into oligodendrocytes (Barres and

others, 1994; Ibarrola and others (1996); Ahlgren and others, 1997; Rodr´ ıguez-Pe˜ na, 1999), and that the

decision of O-2A progenitor cells to either divide or differentiate is made early in the cell cycle, if not

before, rather than at the end (Hyrien and others, 2005a, 2006, 2010).

Most of these previous studies were concerned either with clonal experiments in which every cellular

clone was observed only once in the course of the experiment thereby yielding independent observations

(Yakovlev, Boucher, and others, 1998; Yakovlev, Mayer-Pr¨ oschel, and others, 1998; Yakovlev and others,

2000; von Collani and others, 1999; Boucher and others, 1999, 2001; Zorin and others, 2000; Hyrien and

others, 2005a,b, 2006, 2010; Hyrien, 2007) or with time-lapse experiments (Hyrien and others, 2006),

where the complete history of each clone was fully recorded. While time-lapse experiments provide com-

plete information about the temporal development of each clone, clonal experiments are far less time-

consuming. This is the reason why clonal analyses remain the experiment of choice to investigate the

ability of agents to impact the regulation of precursor cell functions. There exists a third experimental

setting in which each cell clone is observed longitudinally over time and which offers a compromise be-

tween the above 2 experiments. In this experimental setting, the family trees are not completely observed,

but several snapshots of the temporal development of each individual clone are obtained at multiple time

points thereby providing more information than independent clonal experiments. We shall refer to these

alternate experiments as longitudinal clonal experiments (as opposed to “independent” clonal experiments

where each cell clone is examined only once).

Longitudinal clonal data are not independent across time points, and their analysis presents new chal-

lenges. Some of the estimation methods that have been proposed for independent clonal data could also

apply to the longitudinal setting. Since maximum likelihood estimation remains generally unattainable

with age-dependent branching processes, simulation-based approaches have been adopted by some

authors as a means to construct viable estimators. For instance, Hyrien and others (2005a,b) and Hyrien

(2007) considered the utility of a simulated pseudolikelihood approach. This method relies solely on the

expectation and variance–covariance functions of the process. It is attractive because it provides consistent

and asymptotically Gaussian estimators, while remaining easy to implement, even for longitudinal clonal

data. The disadvantage of this estimator is that it does not enjoy optimality properties, and, for instance,

it is not as efficient as the method of maximum likelihood (Hyrien, 2007). The simulated maximum like-

lihood estimator proposed by Zorin and others, 2000 could be more efficient than the simulated pseudo

maximum likelihood estimator but its implementation faces serious limitations that are amplified when

cell clones are observed repeatedly over time. Specifically, it is very time-consuming, and it suffers from

mismatches, as described by Zorin and others (2000). We shall discuss these limitations in greater details

in Section 4 when motivating the construction of the proposed estimator.

This article proposes an alternative approach that provides a compromise between the statistical effi-

ciency of the maximum likelihood estimator and the computational advantage of the simulated maximum

pseudolikelihood estimator. The proposed method is based on a composite likelihood approach (some-

times also referred to as method of pseudolikelihood). The use of composite likelihood traces back to

Besag (1974). It has been investigated in other settings by several authors, including Azzalini (1983),

Lindsay (1988), Heagerty and Lele (1998), Cox and Reid (1987), Chandler and Bate (2007), Varin and

Vidoni (2005), Hyrien and Zand (2008), to name a few. Composite likelihoods can be defined as estimat-

ing functions formed by combining together likelihood objects that remain tractable for the problem at

hands. Our work shows that composite likelihood estimators provide a viable solution to the analysis of

longitudinal clonal data.

The proposed method relies on a complex stochastic process. Simpler statistical approaches could be

invoked to describe the time course of cell counts and assess whether specific treatment conditions alter

the kinetics of the cell population. The level of sophistication of the proposed method is required for

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176 R. CHEN AND OTHERS

gaining a quantitative insight into the processes of division and differentiation of precursor cells because

the events of interest (division and differentiation) are not observable during clonal experiments.

The remainder of this article is organized as follows. Section 2 introduces the data structure. Section 3

presents a branching process model of the generation of oligodendrocytes from cultured O-2A progenitor

cells. Section 4 describes simulated composite likelihood estimation as applied to longitudinal clonal

data. The finite-sample properties of the proposed method are investigated via simulations in Section 5.

Section 6 presents an application to the analysis of the proliferation of O-2A progenitor cells and their

differentiation into oligodendrocytes exposed to neurotrophin-3 (NT-3) in tissue culture.

2. LONGITUDINAL CLONAL DATA

2.1

Clonal data on the generation of oligodendrocytes in vitro

A typical longitudinal clonal experiment conducted on oligodendrocytes begins by plating initiator O-2A

progenitor cells in culture dishes at a density that is low enough so these cells (and the pools of subsequent

progenies) will not interact with each other. Over time, these progenitor cells divide into O-2A progenitor

cells and/or differentiate into oligodendrocytes to give rise to individual cell clones. These clones may

contain O-2A progenitor cells or oligodendrocytes, or, as is more typically the case, a mixture of both cell

types. O-2A progenitor cells and oligodendrocytes are morphologically distinguishable, which enables

experimentalists to count separately the number of O-2A progenitor cells and the number of oligodendro-

cytes contained in any given clone through visual examination using a microscope. The location of each

clone in the dish is identified for subsequent follow up, and the composition of each clone is examined

repeatedly over time so one can observe how the numbers of O-2A progenitor cells and the number of

oligodendrocytes change over time.

2.2

An example of longitudinal clonal data

We performed a longitudinal clonal experiment, as described above, to investigate the impact of NT-3 on

the processes of division and differentiation of O-2A progenitor cells. We purified O-2A progenitor cells

isolated from the corpus callosum tissue of 7-day old rats as described previously (Ibarrola and others,

1996) and plated cells at a clonal density in defined medium supplemented with NT-3 at 20 ng/ml. No

platelet-derived growth factor (PDGF) was added to the culture medium at any time. In this experiment,

n = 40 clones were followed for up to 6 days. Every clone was generated by a single O-2A progenitor

cell. The composition of each clone was examined daily, so ti = (1,2,3,4,5,6) in this particular ex-

periment (with time being expressed in days). The number of O-2A progenitor cells and the number of

oligodendrocytes were reported at each time point for every clone. Half of these clones were cultured in

the presence of NT-3, and the other half was cultured without NT-3.

Figure 2 shows the histograms for the number of progenitor cells (left panels) and for the number of

oligodendrocytes (right panels) from day 0 to day 6 (from top to bottom). At the start of the experiment

(day 0), all clones contained exactly one O-2A progenitor cell and zero oligodendrocyte. The composi-

tion of each clone evolved over time according to whether O-2A progenitor cells and their subsequent

progenies divided or differentiated. In this particular experiment, all clones growing without NT-3 con-

tained between 0 and 6 O-2A progenitor cells and between 0 and 4 oligodendrocytes. In the presence of

NT-3, the number of O-2A progenitor cells ranged between 0 and 9, and the number of oligodendrocytes

between 0 and 6.

The primary objectives of our experiment were to assess and investigate the effects of NT-3 on the

proliferation of O2-A progenitor cells and their differentiation into oligodendrocytes. A visual comparison

of the histograms of the number of O2-A progenitor cells and of the number of oligodendrocytes clearly

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177

Fig. 2. Histograms for the number of O-2A progenitor cells (PCs) and for the number of oligodendrocytes along with

the fitted model in the absence and in the presence of NT-3.

suggests that the numbers of O-2A progenitor cells were substantially smaller in clones treated with NT-3

starting from day 2. In contrast, the numbers of oligodendrocytes appeared similar among treated and

untreated clones at all time points. The marginal distributions of the number of O-2A progenitor cells

(respectively, oligodendrocytes) at any time point in clones treated with and in clones treated without

NT-3 could be compared using (e.g.) Wilcoxon rank sum statistics. In order to assess the overall effect of

NT-3, irrespective of time, and avoid issues associated with multiple testing, the resulting p-values could

be combined using Fisher’s combination method. Since the individual p-values are dependent across time

points, an overall p-value could be computed using a permutation testing approach, where clones (not

just individual observations) are randomly assigned to one group or the other so the dependencies among

observations are properly accounted for when performing the test.

We assessed the overall effect of NT-3 on the numbers of O-2A progenitor cells and on the numbers of

oligodendrocytes separately using this approach. Our test suggested that the numbers of O-2A progenitor

cells were significantly different depending upon whether they had been cultured with or without NT-

3 (p = 0.02), but it did not detect any significant difference among the numbers of oligodendrocytes

(p = 0.98). Thus, this analysis would suggest that NT-3 had a significant impact on the number of O-2A

progenitor cells but not on the number of oligodendrocytes, at least for the first 6 days of culture.

The conclusion of these analyses is of interest in itself because it reveals the existence of a potential

effect of NT-3 on the regulation of the processes of division and of differentiation of O-2A progenitor

cells. From a biological standpoint, however, it remains limited in scope because it neither offers any

mechanistic interpretation on how these processes might have actually been altered by NT-3 nor quanti-

fies the effects of NT-3 on cellular functions. A number of biological reasons could be invoked to explain

the observed effects of NT-3. For instance, O-2A progenitor cells exposed to NT-3 might have differen-

tiated more frequently in the presence of NT-3, causing the number of O-2A progenitor cells to decrease

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178R. CHEN AND OTHERS

prematurely. An alternative explanation is that O-2A progenitor cells divided more slowly and/or differen-

tiated faster when exposed to NT-3. A combination of these 2 scenarios is also quite plausible. In addition,

experimentalists would also be interested in evaluating the impact of such changes on the number of oligo-

dendrocytes produced by O-2A progenitor cells over an extended period of time. In order to gain such a

quantitative insight into cell proliferation kinetics, we propose to model the dynamics of cell clones using

an age-dependent branching process.

3. A BRANCHING PROCESS MODEL OF OLIGODENDROCYTE GENERATION

We model the temporal development of the population of O-2A progenitor cells and of terminally differ-

entiated oligodendrocytes using a multitype age-dependent branching process defined from the following

set of assumptions:

(A1) The process begins at t = 0 with a single cell of type 1 (an initiator O-2A progenitor cell).

(A2) For every k = 1,2,..., upon completion of its lifespan, every type-k cell (an O-2A progenitor

cell of generation k) either divides into 2 new cells of age 0 and type k + 1 with probability πkor

it differentiates into a single type-0 cell (an oligodendrocyte) with probability 1 − πk. Following

Boucher and others (2001) and Hyrien and others (2005a,b), we assumed that the probability of

division is given by πk= min{1,a+bck}, k ? 1, where a, b, and c are unknown positive constants

with, for instance, a denoting the limiting probability of division of O-2A progenitor cells as the

number of division increases when c < 1.

(A3) Time to division of initiator O-2A progenitor cells: The time to division of any type-1 cell is

represented by a nonnegative random variable (r.v.) T1assumed to follow a gamma distribution

with cumulative distribution function (c.d.f.) G1(t; ξ1), where ξ1= (m1,σ1), and where m1and

σ1denote the expectation and standard deviation of T1.

(A4) Time to division of O-2A progenitor cells of generation greater than or equal to 2: The time to

division of any O-2A progenitor cells of generation k, k ? 2, is described by a nonnegative r.v. T2

assumed to follow a gamma distribution with c.d.f. G2(t; ξ2), where ξ2= (m2,σ2), and where m2

and σ2denote the expectation and standard deviation of T2.

(A5) Time to differentiation of O-2A progenitor cells: The time to division of any O-2A progenitor

cells of generation k, k ? 1, is described by a nonnegative r.v. T0assumed to follow a gamma

distributionwithc.d.f. G0(t; ξ0), whereξ0= (m0,σ0), andwherem0andσ0denotetheexpectation

and standard deviation of T0. The distribution of the time to differentiation is thus independent of

the generation number.

(A6) Independence assumption: Cells evolve independently from each other.

The above process extends a model of the generation of oligodendrocytes previously proposed by

Hyrien and others (2005a) by allowing the distribution of the time to division of O-2A progenitor cells

of first generation to be dissimilar from that of O-2A progenitor cells of subsequent generations. This

extension was motivated by an analysis of time-lapse data on oligodendrocytes (Hyrien and others, 2006),

which indicated that first generation O-2A progenitor cells may have a shorter time to division than cells

of subsequent generations.

Let θ = (m0,σ0,m1,σ1,m2,σ2,a,b,c)?denote the complete set of unknown model parameters. For

every k = 0,1,..., let Zk(t) denote the number of type-k cells at time t ? 0, with Z0(t) corresponding to

the number of oligodendrocytes, and, for every k ? 1, Zk(t) representing the number of O-2A progenitor

cells of generation k. It is not possible to determine the generation of O-2A progenitor cells during clonal

experiments, and our observations consist of the number of O-2A progenitor cells Y1(t) =?∞

k=1Zk(t)

and of the number of oligodendrocytes Y2(t) = Z0(t). The observable process is therefore the vector

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Composite likelihood for branching processes

179

Y(t) = {Y1(t),Y2(t)}. The distribution of the process Y(t) does not generally admit a closed-form ex-

pression, which is why we and others have resorted to simulations to approximate this distribution.

4. STATISTICAL INFERENCE

4.1

A composite likelihood estimator

Let Yij1and Yij2denote the number of O-2A progenitor cells and the number of oligodendrocytes counted

in the ith clone at time tij, where j = 1,...,niand i = 1,...,n. Write Yij = (Yij1,Yij2) and Yi =

(Yi1,...,Yini) for the ordered time points ti= (ti1,...,tini). The log-likelihood function for the sample

Y1,...,Yntakes the form

n

?

where pij(θ) = Pθ{Y(tij) = Yij|Y(ti,j−1) = Yi,j−1,...,Y(ti,1) = Yi,1} denote the conditional distri-

bution function of the r.v. Y(tij), given the trajectory of the process Y(t) is observed at all sampling times

prior to tij.

In the present setting, the pij(θ)s have usually no closed-form expressions, making (exact) maximum

likelihood inference impossible. Zorin and others (2000) proposed a simulated maximum likelihood ap-

proach when a single observation is available per clone (ni = 1). The probabilities pi1(θ)s are approx-

imated by simulating independent clones from the branching process, and by computing the empirical

proportions of simulated clones whose composition at time ti1matches that of the ith experimental clone.

Denoting the corresponding approximation by ˆ pi1(θ), L(θ) is replaced byˆL(θ) =?n

Wolfowitz algorithm (Kiefer and Wolfowitz, 1952; Blum, 1954).

There are 2 limitations to such an approach. The first one is that ˆ pi1(θ) will be identically zero if the

composition of the ith clone as observed during the actual experiment cannot be matched by any of the

simulated clones. Consequently,ˆL(θ) is undefined, making it difficult to compute a simulated maximum

likelihood estimate. This problem was reported by Zorin and others (2000) in their analysis of oligoden-

drocytes generation. The second limitation has to do with the fact that the simulated log-likelihood func-

tionˆL(θ) is a biased estimator of L(θ). The Kiefer–Wolfowitz algorithm considered by these authors is

designed to converge to the parameter vector that maximizes the map θ ?→ g(θ) = E{ˆL(θ)|Y1,...,Yn},

where the expectation is taken with respect to the simulations. Since the maximum of g(θ) will be gen-

erally different than the maximum likelihood estimate, the simulated maximum likelihood estimator will

be biased. The severity of the bias depends, of course, upon the noise of the simulated log-likelihood

that is attributable to the simulations and upon the gradient of L(θ) with respect to θ. One can argue that

the bias will become small if the number of simulations S is large enough. Computing time, however,

may become an issue with branching processes, especially when the process is supercritical (i.e. loosely

speaking, when each cell produces on average more than one cell), which causes cell clones to grow large;

see Hyrien and others (2010) for additional comments on this practical issue. Thus, the strategy that con-

sists of letting S increase is not always applicable in this case. Furthermore, there exists a simple way to

approximate L(θ) that is able to mitigate both limitations simultaneously (see Section 4.2).

The simulated maximum likelihood estimator proposed by Zorin and others (2000) can be extended

in an obvious way to the situation where clones are observed longitudinally over time (ni> 1). However,

the method becomes less appealing than when ni = 1 because now simulated clones need to match the

trajectories of experimental clones observed at several time points, not just one. Since the probability of

mismatching increases quickly with ni, the number of simulations would have to increase with nithereby

making the approach even more time-consuming.

L(θ) =

i=1

ni

?

j=1

log pij(θ),

i=1log ˆ pi1(θ), and

a simulated maximum likelihood estimate is computed using a multidimensional version of the Kiefer–

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180 R. CHEN AND OTHERS

As an alternative to the method of maximum likelihood and simulated maximum likelihood, we pro-

pose to construct estimators of θ using the estimating function

P(θ) =

n

?

i=1

ni

?

j=1

logqij(θ),

where qij(θ) = Pθ{Y(tij) = Yij|Y(ti,j−k) = Yi,j−k,k ∈ Iij}, and where Iij⊂ {1,..., j − 1} for every

i and j. The probability qij(θ) differs from pij(θ) in that the conditioning event is restricted to a subset

of the past observations. By selecting Iijappropriately, we can thus reduce the curse of dimensionality

encountered with the simulated log-likelihood. These probabilities qij(θ) do not generally have explicit

expressions either, but we shall leave aside this aspect of the method until Section 4.2. The estimating

function P(θ) is a composite likelihood, and the maximum composite likelihood estimator maximizes

P(θ). We shall denote it by˜θ.

The conditioning sets Iijneed not be identical for all i and j nor do they have to be collections of

consecutive integers, although they will likely be so in most applications. Thus, Iij= {1,3,5} would be

allowed for instance. They are assumed to be independent of θ, however. Depending upon the choice of

Iij, we obtain different estimating functions, the complexity of which (from a computational standpoint)

increases as Iij“approaches” {1,..., j − 1}. The following cases are the most noteworthy:

• If Iij= {1,..., j − 1}, P(θ) = L(θ), and˜θ is a maximum likelihood estimator.

• If Iij = ∅, no conditioning (other than upon the state of the process at the start of the experiment) is

done. This is the simplest case. Hyrien and Zand (2008) used it in the context of CFSE-labeling data.

• If Iij= {1}, all probabilities are conditioned upon the most recent observation. The method leads to a

maximum likelihood estimator when the process is Markovian and no efficiency is lost in this particular

case. For non-Markovian models, this property does not hold true since the conditional distribution of

Y(tij) given {Y(ti,j−k),k = 1,..., j − 1}, will still depend on Y(ti,1),...,Y(ti,j−2).

• By taking subsets of the form Iij= {1,...,kij}, for some kij∈ {1,..., j − 1}, we can modulate how

far in the past the probability qij(θ) is conditioned. Depending upon the memory of the process, P(θ)

will provide a substitute to L(θ) that will yield more or less efficient estimators. For “near” Markovian

processes, the proposed approach may not sacrifice much efficiency when using kij = 1 or 2 (e.g.)

while greatly simplifying simulation-based inference.

In practice, the choice of Iijwill be driven by practical considerations (such as reducing the number

of mismatches). The maximum composite likelihood estimator possesses desired asymptotic properties

stated below.

PROPOSITION 1 Under classical regularity conditions, the maximum composite likelihood estimator is

consistent as n → ∞. Furthermore, denoting the true value of θ by θ∗, we have that

√n(˜θ − θ∗) → N(0, A(θ∗)−1B(θ∗)A(θ∗)−1),

where

1

n

i=1

and

1

n

i=1

A(θ) = lim

n→∞

n

?

ni

?

j=1

Eθ

?

1

qij(θ)2

∂qij(θ)

∂θ

∂qij(θ)

∂θ?

?

B(θ) = lim

n→∞

n

?

varθ

ni

?

j=1

1

qij(θ)

∂qij(θ)

∂θ

.

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Composite likelihood for branching processes

181

The proof of Proposition 1 rests on standard arguments used to establish asymptotic properties of

M-estimators (for instance, see van der Vaart, 1998) and of multinomial likelihood estimators (Andersen,

1980). In particular, the consistency relies on the fact that the composite likelihood function satisfies the

inequality Eθ∗{P(θ)} ? Eθ∗{P(θ∗)} for every θ, with the equality holding true for θ = θ∗only if a

condition on the identifiability of the model parameters applies. The inequality is easily established by

noticing that each individual terms logqij(θ) is a log-likelihood object and therefore satisfies the classical

inequality Eθ∗{logqij(θ)} ? Eθ∗{logqij(θ∗)}. The matrices A(θ) and B(θ) will generally be different

from each other, and, in most cases, the composite likelihood estimator will not be fully efficient. Lindsay

(1988) commented further on the properties of composite likelihood estimators.

4.2

Practical considerations

The probabilities qij(θ)s will generally have no closed-form expressions and estimation of θ cannot pro-

ceed through direct maximization of P(θ). We describe below 2 algorithms designed to compute a com-

posite likelihood estimate for θ. Both algorithms use bias-corrected simulation-based approximations to

logqij(θ) that are computed in the following way.

Let Cs(θ,Us), s = 1,..., S, be S independent clones simulated under the assumed model for the

parameter value θ, where Usdenotes a set of r.v.’s uniformly distributed on the interval [0,1]. These r.v.’s

are used as seeds to generate the lifespans and the progeny vectors of all cells in the sth simulated clone.

Write U(S)= (U1,...,US). Let Ys

for the D morphologically distinguishable cell types for the sth simulated clone at time tij. To reduce the

notational burden, we shall omit the notation U(S)whenever unnecessary, and, for instance, write Ys

in place of Ys

composition matches that of the ith experimental clone at the time points {ti,j−k,∀k ∈ Iij}. Formally, we

have

S

?

where 1{∙}is the indicator function. The conditional probability qij(θ) can be approximated by

ˆ qS

and a natural substitution in P(θ) produces the simulated composite likelihood function

ij(θ,Us) = {Ys

ij1(θ,Us),...,Ys

ijD(θ,Us)} be a vector of cell counts

ij(θ)

ij(θ,Us). Let Nij(θ; Iij) = Nij(θ,U(S); Iij) denote the number of simulated clones whose

Nij(θ; Iij) =

s=1

1?

Ys

i,j−k(θ)=Yi,j−k,∀k∈Iij

?,

ij(θ) = ˆ qS

ij(θ,U(S)) = Nij(θ; Iij∪ {0})/Nij(θ; Iij),

PS(θ,U(S)) =

n

?

i=1

ni

?

j=1

log{ˆ qS

ij(θ)}.

This simulated composite likelihood, however, suffers from the same limitations as those previously men-

tioned for the simulated log-likelihood function; that is, it is a biased estimator of P(θ), and it is not de-

fined when Nij(θ; Iij) = 0. We mitigate both limitations simultaneously by considering a bias-corrected

version of P(θ) that is built as follows.

Conditional on {Nij(θ; Iij) = sij} and on {Ys

that Nij(θ; Iij∪{0}) ∼ Bin(sij,qij(θ)), whereBin(s,q)denotesthebinomialdistributionwithparameters

s and q. Following Pettigrew and others (1986), a bias-corrected estimator of logqij(θ) takes the form

?Nij(θ; Iij∪ {0}) + 1/2

i,j−k(θ) = Yi,j−k,∀k ∈ Iij,∀s = 1,...,sij}, we have

ˆlS

ij(θ) = log

sij+ 1/2

?

.

Page 10

182 R. CHEN AND OTHERS

The bias ofˆlS

sijqij(θ) → 0, however, the bias of either approximation increases proportionally to s−2

Therefore, even with the biased-corrected estimator of logqij(θ), it will remain important to reduce as

much as possible the number of mismatches so the simulation-based approximation to the objective func-

tion will not be too biased. The performance of the estimator will otherwise be affected. The composite

likelihood approach offers flexibility in choosing an estimating function that reduces the number of mis-

matches or even eliminates them. Note thatˆlS

This motivated us to consider the simulated composite likelihood defined as

ij(θ) is of smaller order—compared to that of log ˆ qS

ij(θ)—when sijqij(θ) → ∞. When

ijqij(θ)−2.

ij(θ) remains defined when there are mismatches.

PS

bc(θ) = PS

bc(θ,U(S)) =

n

?

i=1

ni

?

j=1

ˆlS

ij(θ).

There exist multiple ways to compute an estimator of θ based on PS

below.

bc(θ). Two methods are discussed

Method 1: sample path. In order to estimate θ, a first approach consists in maximizing the simulated

composite likelihood PS

function is not subject to random fluctuations during the optimization. The resultant estimator can be

proven to converge to˜θ as S → ∞ (see Hyrien, 2007, for a proof of a similar result). Due to the discrete

nature of branching processes, the function PS

that do not rest upon differentiability of the objective function (such as the simplex method of Nelder and

Mead) can be used.

bc(θ,U(S)) for a given vector U(S). In doing so, the simulated composite likelihood

bc(θ,U(S)) is discontinuous, and optimizations algorithms

Method 2: stochastic approximation. Alternatively, θ can be estimated using a multidimensional version

of stochastic procedure of Kiefer–Wolfowitz (Kiefer and Wolfowitz, 1952; Blum, 1954). In this alternate

schema, an estimate of θ is computed through the sequence of r.v.’s {θ(k)}k?1defined as θ(k+1)= θ(k)+

ak?k, where ?k= (?k,1,...,?k,q)?, with

?k,l= {PS

where U(S)

with all elements equal to 0, except for a 1 in the lth place, and where (ak)k?1and (bk)k?1are 2 given

sequences ofnonrandom positive numbers such that

bc(θ(k)+ bkel,U(S)

k,+) − PS

bc(θ(k)− bkel,U(S)

k,−)}/2bk,

k,−and U(S)

k,+, k = 1,2,..., are i.i.d. copies of U(S), where el, l = 1,...,q, are q × 1 vectors

lim

k→∞bk= 0,

∞

?

k=1

ak= ∞,

∞

?

k=1

akbk< ∞,

and

∞

?

k=1

(ak/bk)2< ∞.

Usual choices for (ak)k?1and (bk)k?1include the following: ak = a0/kαand bk = b0/kβ, for some

given positive constants a0, b0, α, and β. In our analysis of clonal data on O-2A progenitor cells, we used

a0= 1, b0= 1, α = 1, and β = 1/3. Other choices are possible. If P(S)

P(θ), θ(k)would converge to˜θ as k → ∞. This is not formally the case, but because PS

bias-corrected, θ(k)is expected converge to a value close to˜θ.

The computation of composite likelihood estimates may take up to more than a day, depending on

the number of simulations S, the number of model parameters, and the sample size. The number of

mismatches increases substantially with the number of conditioning observations, and eliminating the

mismatches may require increasing the number of simulations out of practical limits. The numbers of

simulations being equal, the stochastic approximation procedure is expected to provide more accurate

bc(θ) was an unbiased estimator for

bc(θ) is first-order

Page 11

Composite likelihood for branching processes

183

estimates than the sample path approach since the noise arising from the simulations is averaged out as

the sequence {θ(k)}k?1progresses. This procedure is more time-consuming because it requires evaluation

of the objective function at 2q points at every step k, while the sample path approach needs only one

such evaluation at every step. In practice, it is probably best to begin with the sample path approach and

complete estimation using stochastic approximation, with a0and b0appropriately chosen to avoid large

jumps of the sequence {θ(k)}k?1.

5. SIMULATION STUDIES

We performed simulation studies to evaluate the performance of the proposed simulated composite like-

lihood estimator in samples of finite sizes. We also compared it with the simulated pseudo maximum

likelihood estimator proposed in past publications (Hyrien and others, 2005a,b; Hyrien, 2007). In or-

der to keep the computing time reasonable (each estimates required between 1 h of computing time

with S = 10000 simulated clones, and up to 6 h with S = 50000 simulated clones), the study was

conducted using 2 simple branching processes that are special cases of our model of the generation of

oligodendrocytes. The conclusions of these studies are therefore not meant to be exhaustive but they

corroborate what can be intuitively expected. The 2 processes that we considered were defined as

follows:

Model 1 (a binary splitting process) : This model assumes that every cell divides into 2 new cells upon

completion of its lifespan, which follows a gamma distribution with mean m and variance σ2. When

m = σ, the process reduces to the (Markov) linear birth process.

Model 2 (a 2-type process) : This model considers 2 cell types: every type-1 cell divides into 2 new

type-1 cells with probability p or it turns into a single type-0 cell with probability 1 − p. The lifespan

of any type-1 cell follows a gamma distribution with mean m and variance σ2. Type-0 cells have infinite

lifespans.

Longitudinal clonal data were simulated from these 2 models, and cell clones were observed every

day from day 1 to day 6 (same experimental design for both models). We considered various sample

sizes: n = 20, 50, 100; and used S = 10000 and S = 20000 simulated clones to compute the sim-

ulated composite likelihoods for Models 1 and 2, respectively. For each simulated data set, we com-

puted the simulated composite likelihood under various conditioning sets, all of the form Iij = ∅ or

Iij = {1,..., j ∧ j0}, for j0= 1,...,5, where j ∧ j0denotes the smallest value of j and j0. The par-

ticular case Iij = {1,..., j ∧ 5} corresponds to a simulated maximum likelihood estimator. The pseudo

maximum likelihood estimator (Hyrien and others, 2005a,b; Hyrien, 2007), computed for comparison, is

defined as the parameter vector that maximizes the pseudolikelihood function

G(θ) = −

n

?

i=1

[{Yi− mi(θ)}?Vi(θ)−1{Yi− mi(θ)} + logdetVi(θ)],

where mi(θ) = Eθ(Yi) and Vi(θ) = Varθ(Yi), and where detV denotes the determinant of the

matrix V. When these moments have no closed-form explicit expressions, they can be replaced by

approximations computed either from simulations (Hyrien and others, 2005a,b; Hyrien, 2007) or using

the method of saddlepoint (Hyrien and others, 2010). The pseudo maximum likelihood estimator is con-

sistent and asymptotically Gaussian (Hyrien, 2007). A summary of the simulation results is displayed in

Tables 1 and 2.

The biases of the estimators tended to decrease as the sample size increased. The standard errors

depended on the method of estimation being used. Overall the pseudo maximum likelihood estimator ap-

peared to be the least efficient of all tested estimators. This was particularly true for the smallest sample

Page 12

184R. CHEN AND OTHERS

Table 1. Results of the simulation study based on a binary splitting process (see text for explanation).

CLk= simulated composite likelihood with Iij= {1,2..., j ∧ k}; ML = simulated maximum likelihood

PL = simulated pseudolikelihood; s.e. = standard error

Sample

size

Parameter

true value

method

ML = CL5

CL4

CL3

CL2

CL1

Marginal CL

PL

ML = CL5

CL4

CL3

CL2

CL1

Marginal CL

PL

ML = CL5

CL4

CL3

CL2

CL1

Marginal CL

PL

m

σ

40 h20 h

Mean

41.96

42.31

41.99

41.38

40.99

40.97

42.31

40.66

40.52

40.41

40.37

40.03

39.67

41.23

40.14

40.22

40.18

39.86

39.67

39.69

40.31

s.e.

2.05

1.65

2.02

1.93

3.05

4.02

4.70

1.50

1.34

1.37

1.49

1.59

2.30

2.73

1.17

1.15

1.19

1.38

1.61

1.63

2.27

Mean

19.16

19.26

19.31

18.93

18.50

18.50

16.98

20.04

19.94

19.97

19.92

19.59

19.44

18.72

20.12

20.25

20.21

19.73

19.49

19.39

19.27

s. e.

1.24

1.11

1.34

1.67

2.63

3.90

3.51

1.04

1.20

1.11

1.29

1.98

2.74

2.32

1.13

1.37

1.18

1.50

1.84

2.03

2.37

20

50

100

sizes. The simulated maximum likelihood estimator was not particularly more efficient than the simulated

composite likelihood estimators. For instance, in the case of Model 2, the simulations indicated that the

marginal composite likelihood estimator and the composite likelihood estimators defined by conditioning

upon the most recent one or two observations had smaller standard errors. This was particularly true for

the largest sample sizes. The most likely explanation of the fact that some estimators performed worse

when we conditioned upon more observations is that the simulation-based approximations to the condi-

tional probabilities that make up the expression for the estimating functions are more noisy in the latter

case (due to the simulations) than when we condition on fewer observations.

We also found that the bias decreases generally quickly with increasing sample size but not always.

For instance, the estimator of the probability of division (Model 2), say ˆ p, remained substantially biased

even with n = 100 observed clones. When conditioning upon the most recent observation only, the bias

of ˆ p was about 0.06 with a sample size of n = 20 clones and close to 0.05 with a sample size of n = 50

clones. The bias became larger as the composite likelihood estimator was conditioning upon an increasing

number of past observations. We performed additional simulations to assess whether this bias could be

attributed (in part at least) to a value of S that was too small. Thus, we considered S = 50000 simulated

clones (instead of S = 20000). As a result of time constraints, we restricted our investigations to the

above 2 cases and found that the bias of ˆ p decreased to 0.03 both when n = 20 and when n = 50,

and thus was reduced by almost a factor of 2. Also, the standard error of ˆ p decreased to 0.06 and 0.04,

respectively. The estimators of m and σ improved also in this last run of simulations, with standard errors

smaller than those reported in Table 2.

Page 13

Composite likelihood for branching processes

185

Table 2. Results of the simulation study based on a 2-type process (see text for explanation). CLk =

simulated composite likelihood with Iij= {1,2..., J ∧ k}; ML = simulated maximum likelihood; PL =

simulated pseudolikelihood; s.e. = standard error

m

size true value

35 h

method Means.e.

20ML = CL5

CL4

35.971.90

CL3

36.052.07

CL2

36.16 2.04

CL1

35.892.02

Marginal CL 35.50 2.05

PL 36.302.32

50 ML = CL5

CL4

35.591.53

CL3

35.41 1.56

CL2

35.55 1.51

CL1

35.541.67

Marginal CL35.491.24

PL 35.301.63

100ML = CL5

CL4

35.301.20

CL3

35.741.24

CL2

35.601.34

CL1

35.601.12

Marginal CL35.121.03

PL35.52 1.16

SampleParameter

σ

P

10 h

0.35

Mean

10.31

10.24

10.01

10.15

10.34

9.64

12.34

9.73

9.60

9.63

9.60

9.59

9.72

10.56

9.82

9.48

10.07

9.70

9.75

9.90

11.34

s.e.

2.65

2.59

2.57

2.52

2.75

2.42

2.98

1.42

1.36

1.47

1.40

1.23

1.40

1.82

1.30

1.20

1.43

1.48

1.04

1.08

1.45

Mean

0.27

0.28

0.27

0.29

0.29

0.33

0.30

0.29

0.29

0.30

0.30

0.30

0.34

0.33

0.27

0.27

0.28

0.28

0.31

0.34

0.34

s.e.

0.09

0.09

0.09

0.09

0.09

0.07

0.09

0.06

0.07

0.07

0.06

0.06

0.05

0.05

0.06

0.06

0.05

0.05

0.05

0.04

0.04

35.83 2.08

35.611.58

35.78 1.26

6. AN APPLICATION TO OLIGODENDROCYTES GENERATION EXPOSED TO NT-3

Oligodendrocytes produce the myelin sheaths that enwraps axons in the central nervous system. They

are involved in signal propagation along the nerves and are particularly essential for normal impulse

conduction. Some demyelinating diseases, such as multiple sclerosis, have been shown to be associated

with injury or dysfunction of the oligodendrocyte population (Blakemore, 2008; Korn, 2008). The biology

of oligodendrocytes and their lineage is well documented in the literature (e.g. Franklin and Ffrench-

Constant, 2008). Oligodendrocytes are assumed to be nonproliferating, terminally differentiated cells,

and are generated from their immediate precursor cells, the O-2A progenitor cells. These O-2A progenitor

cells have the ability to self-renew by dividing, to differentiate into oligodendrocytes, or to die. Due to

the critical role of oligodendrocytes, the proper regulation of the differentiation of O-2A progenitor cells

into oligodendrocytes is therefore particularly important for the normal functioning of the central nervous

system. O-2A progenitor cells have been shown to respond to the presence of various environmental

signals or factors that may modulate their ability to self-renew and differentiate, and one of the primary

objectives of the research performed on the oligodendrocyte lineage is to identify factors that could restore

the normal balance in the generation of oligodendrocytes.

NT-3 is one member of a family of growth factors known as neurotrophins. Althaus and others (2008)

gave an overview on neurotrophins and their effects on oligodendroglial cells. It has been demonstrated

that, in rodents and humans at least, NT-3 may be synthesized by oligodendrocytes. Although NT-3 has

been shown to be involved in the growth, differentiation, and survival of neurons (Zhou and Rush, 1996;

Page 14

186R. CHEN AND OTHERS

Kalb, 2005), its role (and that of other neurotrophins as well) on the regulation of nonneuronal cells of the

central nervous system is still unclear.

In an attempt to better understand the effect of NT-3 on the regulation of the processes of division

and differentiation of O-2A progenitor cells, we conducted the longitudinal clonal experiments presented

in Section 2.2. In particular, the objectives of this experiment were to assess the hypothesis that the

presence of NT-3 alone (i.e. without coexposure to PDGF) in pure cultures has no effect on the prolif-

eration and differentiation of O-2A progenitor cells, as suggested by Barres and others, 1994, and to

gain some quantitative insight into how NT-3 may alter division and differentiation of O-2A progenitor

cells.

Histograms for the number of O-2A progenitor cells and for the number of oligodendrocytes per clone

with and without NT-3 suggested a striking difference between these 2 experimental conditions in respect

to the number of O-2A progenitor cells. When cells were cultured without NT-3, the number of O-2A

progenitor cells appeared to increase over time, suggesting that these cells maintained some capacity for

self-renewal after 6 days of culture even in the absence of PDGF. By day 6, more than 60% of the clones

still contained at least one O-2A progenitor cell, and the clones contained on average 1.9 O-2A progenitor

cells and 2.1 oligodendrocytes. In contrast, in clones grown in the presence of NT-3, O-2A progenitor cells

underwent division first, just as when they were cultured without NT-3, and then appeared to differentiate

massively. By day 6, less than 20% of the clones still contained at least one O-2A progenitor cell. The

mean numbers of oligodendrocytes per clone (?1.9) was not much different than that without NT-3, but

the number of O-2A progenitor cells was considerably lower (?0.8).

We investigate further the effects of NT-3 on the processes of division and differentiation of O-2A

progenitor cells to determine whether NT-3 affected the decision of O-2A progenitor cells to divide or

differentiate the timing of cellular division or the timing of cellular differentiation, or all of the above.

Changes in cell fate would be seen through either an increase or a decrease of the probability of division

of O-2A progenitor cells, whereas a modification of the time it takes for O-2A progenitor cells to divide or

to differentiate would be reflected by changes in the parameters of the distribution of the time to division

and/or of the distribution of the time to differentiation of these cells.

We fitted our branching process model using the proposed simulated composite likelihood approach.

The ultimate parameter estimates were obtained using 15000 simulated clones to approximate the dis-

tribution of the number of cells of each type per clone under the above model. We started from multiple

initial values and let optimization run for more than a day each time. We used the conditioning sets

Iij = ∅,{1},{1, j ∧ 2},...,{1,2,..., j ∧ 5}). In the last case, we obtain a simulated maximum likeli-

hood estimator. The model was fitted separately to the data obtained with and without NT-3 because none

of the model parameters were common to both experimental conditions.

In our experiments, the follow-up period was long enough so all initiator precursor cells had either

divided or differentiated by the time the experiment ended. By determining the composition of a clone,

it is possible to decipher whether the corresponding initiator cell divided or differentiated (in the latter

case, the clone always contains a single oligodendrocyte). Thus, let Ndivand Ndiff = n − Ndivdenote

the numbers of initiator cells having divided and differentiated by the end of the experiment, respectively.

For every 0 ? j1 < j2, let Ndiv(j1, j2) denote the number of initiator cells having divided between

day j1and j2. Define the vector Ndiv = {Ndiv(t − 1,t); t = 1,...,6}. Conditional on Ndiv = N,

Ndiv follows a multinomial distribution with parameters N and {p1(m1,σ1),..., p6(m1,σ1)}, where

pt(m,σ) = G1(t; m,σ) − G1(t − 1; m,σ) denotes the probability that any initiator cell that will

ultimately divide, completes its cycle between day t − 1 and t. The estimates for m1and σ1can there-

fore be computed by maximizing the multinomial likelihood function associated with the vector of counts

Ndiv. The mean and variance of the time to differentiation of O-2A progenitor cells can be estimated in

a similar fashion for clones starting with an initiator cell that ended up differentiating. We used these

maximum likelihood estimates as starting values in our composite likelihood analyses.

Page 15

Composite likelihood for branching processes

187

Table 3 reports parameter estimates resulting from these analyses. Also reported are the total numbers

of mismatches for the final parameter estimates. In our analyses of cell clones cultured without NT-3, the

number of mismatches remained low (0-1) until we started to condition upon the most recent 3 (or more)

observations. For cell clones exposed to NT-3, the number of mismatches increased substantially when we

started to condition upon the most recent observations. In either case, the largest number of mismatches

we encountered ranged between 6 and 9, which is relatively high compared to the number of cell clones

observed for each experimental condition (n = 20). Alternatively, we could have increased the number

of simulations to reduce the number of mismatches, but each analysis required between 1 and 2 days.

We therefore discuss results obtained with the marginal composite likelihood (no conditioning) for cells

exposed to NT-3, and with the composite likelihood conditioning upon the most recent 2 observations

for control (unexposed) cells. Standard errors for the corresponding parameter estimates were obtained

using a parametric bootstrap approach. We compared parameter estimates across treatment groups using a

Wald test. The fitted marginal distribution for the number of O-2A progenitor cells and for the number of

oligodendrocytes per clone is displayed in Figure 2 as solid lines. The estimated probabilities of division

of O-2A progenitor cells are presented in Figure 3 (panels A and B) as a function of the cell generation

for each culture condition. These analyses pointed out the following conclusions:

(a) In each experiment, the probability of division of O-2A progenitor cells was found to decrease

when the number of divisions increased. This is consistent with the results of previously ana-

lyzeddatasets(Yakovlev, Boucher, andothers,1998;Yakovlev, Mayer-Pr¨ oschel, andothers,1998;

Yakovlev and others, 2000; von Collani and others, 1999; Boucher and others, 1999, 2001; Zorin

and others, 2000; Hyrien and others, 2005a,b, 2006).

(b) Exposure to NT-3 decreased the probability of division of O-2A progenitor cells. For instance,

cells of generation 3 divided with a probability estimated as 0.35 (vs. 0.53) when cells were ex-

posed (vs. unexposed) to NT-3. This may explain that the numbers of O-2A progenitor cells were

Table 3. Estimates of the model parameters (means and standard deviations expressed in hours). See text

for explanations. CLk = simulated composite likelihood with Iij = {1,2..., j ∧ k}; ML = simulated

maximum likelihood; s.e. = standard errors

MethodMismatches

m11

σ11

m12

With NT3

Marginal CL

1 39.917.2 49.3

(s.e.)

(4.1) (3.8) (4.6)

CL1

5 39.417.1 53.8

CL2

7 40.415.4 51.3

CL3

7 37.516.5 52.4

CL4

739.416.454.3

ML = CL5

Without NT3

Marginal CL0 23.722.9 51.0

CL1

0 31.015.9 44.6

CL2

1

31.5 14.244.0

(s.e)

(1.8)(1.9)(1.7)

CL3

7 29.2 15.8 42.5

CL4

6 29.2 15.842.5

ML = CL5

Analysis results were discussed based on the bolded parameter estimates. Bolded values were selected for the discussion.

σ12

m2

σ2

α

bc

46.9

(3.3)

44.2

45.2

42.5

45.4

43.9

36.0

(3.5)

31.0

31.3

31.8

30.1

29.8

29.2

(3.7)

31.8

31.7

30.9

32.6

30.8

0.12

(0.03)

0.09

0.09

0.09

0.08

0.07

0.67

(0.11)

0.70

0.70

0.68

0.68

0.70

0.70

(0.70)

0.59

0.58

0.55

0.62

0.627 38.316.7 53.3

24.2

25.3

27.7

(1.6)

22.6

22.6

26.2

40.2

44.0

43.8

(1.8)

43.8

43.8

44.9

28.7

24.4

24.2

(1.7)

25.7

25.7

23.9

0.18

0.12

0.10

(0.02)

0.11

0.11

0.08

0.90

0.78

0.80

(0.03)

0.73

0.73

0.72

0.74

0.85

0.81

(0.04)

0.80

0.80

0.799 26.8 15.245.9