Estimation of intra class correlation parameter for correlated binary data in common correlated models

Some robust estimators which are independent of the distributions of S i have beenintroduced, such as the moment estimator Klienman, 1973, analysis of variance es-timator Eslton, 1977, q

Estimation of Intra-class Correlation Parameter for Correlated Binary Data In Common Correlated

Models

Zhang Hao(B.Sc Peking University)

A THESIS SUBMITTEDFOR THE DEGREE OF MASTER OF SCIENCEDEPARTMENT OF STATISTICS AND APPLIED PROBABILITY

NATIONAL UNIVERSITY OF SINGAPORE

2005

For the completion of this thesis, I would like very much to express my heartfeltgratitude to my supervisor Associate Professor Yougan Wang for all his invaluableadvice and guidance, endless patience, kindness and encouragement during the pasttwo years I have learned many things from him regarding academic research andcharacter building.

I also wish to express my sincere gratitude and appreciation to my other lecturers,namely Professors Zhidong Bai, Zehua Chen and Loh Wei Liem, etc., for impartingknowledge and techniques to me and their precious advice and help in my study

It is a great pleasure to record my thanks to my dear friends: to Ms Zhu Min,

Mr Zhao Yudong, Mr Ng Wee Teck, and Mr Li Jianwei for their advice and help

in my study; to Mr and Mrs Rong, Mr and Mrs Guan, Mr and Mrs Xiao,

Ms Zou Huixiao, Ms Peng Qiao and Ms Qin Xuan for their kind help and warmencouragement in my life during the past two years

Finally, I would like to attribute the completion of this thesis to other members andstaff of the department for their help in various ways and providing such a pleasantworking environment, especially to Jerrica Chua for administrative matters and Mrs.Yvonne Chow for advice in computing

Zhang HaoJuly, 2005

1.1 Common Correlated Model 2

1.2 Two Specifications of the Common Correlated Model 5

1.2.1 Beta-Binomial Model 5

1.2.2 Generalized Binomial Model 6

1.3 Application Areas 7

1.3.1 Teratology Study 7

1.3.2 Other Uses 9

1.4 The Review of the Past Work 10

1.5 The Organizations of the Thesis 11

2 Estimating Equations 12 2.1 Estimation for the mean parameter π 12

2.2 Estimation for the ICC ρ 14

2.2.1 Likelihood based Estimators 14

2.2.2 Non-Likelihood Based Estimators 16

i

2.4 The Estimators We Compare 27

2.5 The Properties of the Estimators 28

2.5.1 The Asymptotic Variances of the Estimators 28

2.5.2 The Relationship of the Asymptotic Variances 39

3 Simulation Study 41 3.1 Setup 41

3.2 Results 45

3.2.1 The Overall Performance 45

3.2.2 The Effect of the Various Factors 48

3.2.3 Comparison Between Different Estimators 49

3.3 Conclusion 52

4 Real Examples 62 4.1 The Teratological Data Used in Paul 1982 62

4.2 The COPD Data Used in Liang 1992 62

4.3 Results 63

ii

In common correlation models, the intra-class correlation parameter (ICC) vides a quantitative measure of the similarity between individuals within the samecluster The estimation for ICC parameter is of increasing interest and important use

pro-in biological and toxicological studies, such as the disease aggression study and theTeratology study

The thesis mainly compares the following four estimators for the ICC parameter

Gaussian likelihood estimator (ρ G ) and a new estimator (ρ U J) that is based on theCholesky Decomposition The new estimator is a specification of the UJ methodproposed by Wang and Carey (2004) and has not been considered before

Analytic expressions of the asymptotic variances of the four estimators are obtainedand extensive simulation studies are carried out The bias, standard deviation, themean square error and the relative efficiency for the estimators are compared Theresults show that the new estimator performs well when the mean and correlation aresmall

Two real examples are used to investigate and compare the performance of theseestimators in practice

keyword: binary clustered data analysis, common correlation model, intra-class

corre-lation parameter/coefficient, Cholesky Decomposition, Teratology study

iii

1.1 A Typical Data in Teratological Study (Weil, 1970) 8

3.1 Distributions of the Cluster Size 43

3.2 The effect of various factors on the bias of the estimator ρ U J in 1000 simulations from a beta binomial distribution 53

3.3 The effect of various factors on the mean square error of ρ U J in 1000 simulations from a beta binomial distribution 54

3.4 The MSE of ρ F C and ρ U J when the cluster size distribution is Kupper 55 3.5 The MSE of ρ F C and ρ U J when the cluster size distribution is Brass 55

3.6 The ”turning point” of ρ when π = 0.05 55

4.1 Shell Toxicology Laboratory, Teratology Data 63

4.2 COPD familial disease aggregation data 63

4.3 Estimating Results for the Real Data Sets 64

4.4 The Estimated value of the Asymptotic Variance of ˆρ (By plugging the estimates of (π, ρ)) into formulas: (2.29), (2.28), (2.26) and (2.21) 65

iv

4.5 The Estimated value of the Asymptotic Variance of ˆρ (by using the

Robust Method) 65

v

3.1 The two distributions of the cluster size n i 44

3.2 The overall performances of the four estimators when k = 10 46

3.3 The overall performances of the four estimators when k = 25 47

3.4 The overall performances of the four estimators when k = 50 48

3.5 The Legend for Figure (3.8), (3.7), (3.6), (3.9) and (3.10) 56

3.6 The Relative Efficiencies when k = 25 and π = 0.5 57

3.7 The Relative Efficiencies when k = 25 and π = 0.2 58

3.8 The Relative Efficiencies when k = 25 and π = 0.05 59

3.9 The Relative Efficiencies when k = 10 and π = 0.05 60

3.10 The Relative Efficiencies when k = 50 and π = 0.05 61

1

Chapter 1

Introduction

Data in the form of clustered binary response arise in the toxicological and biologicalstudies in the recent decades Such kind of data are in the form like this: thereare several identical individuals in one cluster and the response for each individual

is dichotomous For ease of the presentation, we name the binary responses here as

”alive” or ”dead”, and the metric (0,1) is imposed with 0 for ”alive” and 1 for ”dead”

Suppose there are n i individuals in the i th cluster and there are k clusters in total The binary response for the j th individual in the i th cluster is denoted as

y ij = 1/0 (i = 1, 2, , k; j = 1, 2, , n i ) So S i =Pn i

j=1 y ij is the total number of the

individuals observed to respond 1 in the i th cluster It is postulated that the ”death”

rate of all the individuals in the i th cluster are the same, which is P (y ij = 1) = π.

The correlation between any two individuals in the same cluster are assumed to be the

2

same We denote this Intra-Class Correlation parameter as ρ = Corr(y il , y ik) for any

l 6= k For individuals from different clusters, they are assumed to be independent,

which means y ij is independent of y mn for any i 6= m.

The variance of S i often exhibit greater value than the predicted value if a simplebinomial model is used This phenomenon is called the over-dispersion, which is due

to the tendency that the individuals in the same cluster would respond more likelythan individuals from different clusters

According to the above assumptions, we can see that:

Ey ij = π and Vary ij = π(1 − π) i = 1, 2, k j = 1, 2, n i

And for the sum variable S i =Pn i

j=1 y ij , which is the sufficient statistics for π:

The second moment of S i is determined by ρ but the third, forth and the higher order moment of S i may depend on the other parameters Only when we know the

likelihood of S i (such as the Beta-binomial model or the generalized binomial model),

we can get the closed forms of these higher order moment of S i

Define a series of parameters:

Qj=s

j=1 (y ij − π)

For the common correlated model, we can show that φ2 = ρ and the s th moment

m si = E(S i − n i π) s of S i only depends on {π, φ2, , φ s }

When π is fixed, ρ can not take all the values between (−1, 1) Prentice( 1986) has

where n max = max{n1, n2, , n k }, ω = n max π − int(n max π) and int(.) means the

integer part of any real number For the different specifications of the model, theconstraints might be different

The model described above was first formally suggested as the Common lated Model by Landis and Koch (1977a) It includes various specifications, such asBeta-Binomial and Extended Beta-Binomial model (BB) of Crowder (1986), Corre-lated Beta-Binomial model (CB) of Kupper and Haseman (1978) and the GeneralizedBinomial model (GB) of Madsen (1993)

Corre-Kupper and Haseman (1978) has given an alternative specification of the common

correlated model when ρ is positive It is assumed that the probability of alive (success)

varies from group to group (but keep the same between individuals in the same group)

according to a distribution with mean π and variance ρπ(1 − π) All the individuals

(both within the same group and different groups) are independent conditional on thisprobability If this probability is distributed according to Beta distribution, it willlead to the well-known Beta-Binomial model

1.2 Two Specifications of the Common Correlated

Model

Of the specifications of the common correlated model, Beta-Binomial model is themost popular Paul (1982) and Pack (1986) has shown the superiority of the beta-binomial model for the analysis of proportions However, Feng and Grizzle (1992)

found that the BB model is too restrictive to be relied on for inference when n i arevariable

The beta-binomial distribution is derived as a mixture distribution in which theprobability of alive varies from group to group according to a beta distribution with

parameters α and β S i is binomially distributed conditional on this probability

In terms of the parameterizations of α and β, the marginal probability of alive for any individual is: π = α/(α + β) and the intra-class correlation parameter is: ρ = 1/(1 + α + β) Denote θ = 1/(α + β), we can get the probability function for the

j=0 (1 − π + jθ)

Qy−1

If the intra-class correlation ρ > 0, it is called over-dispersion, otherwise it is called

under-dispersion Over-dispersion is much more common than under-dispersion in

Chapter 1: Introduction 6

practice since the litter effect suggests that any two individuals are tended to respondmore likely and therefore they are positively correlated But this does not mean that

ρ must be positive For BB model, it is required that ρ > 0 However, Crowder (1986)

showed that to ensure (1.1) to be a probability function, ρ only needs to satisfy

1 − π

In this case, ρ can take negative values, which makes the BB model also suitable for

under-dispersion data This is called extended beta-binomial model

The generalized binomial model is proposed by Madsen (1993) It can be treated asthe mixture of two binomial distributions:

An advantage of the generalized binomial model is that ρ contains information for the higher(≥ 3) order moment As we know, the correlation for any pair

Corr(y ij , y ik) = E(y ij − π)(y ik − π)

E(y ij − π)2 = ρ = φ2

For the GB model, it can be shown that:

E(y ij − π)3 = φ3 = ρ

E(y ij − π)4 = φ4 = ρ That means ρ also determines the third and forth moment of S i

Of the various applied areas of the common correlated model, we mainly focus on theTeratology studies In a typical Teratology study, female rats are exposed to differ-ent dose of drugs when they are pregnant Each fetus is examined and a dichotomousresponse variable indicating the presence or absence of a particular response (e.g., mal-formation) is recorded For ease of the presentation, we often denote the dichotomousresponse as alive or dead Apply the common correlation model and the notations

above to the teratology study, it can be described as: k female rats were exposed

to certain dose of drug during their pregnancy For the i th rat, she gave birth to n i fetuses Of the n i fetuses, y ij denotes the survival status for the j th fetus y ij = 1

means the fetus is observed dead or it is alive Then S i =Pn i

j=1 y ij is the total number

Table 1.1: A Typical Data in Teratological Study (Weil, 1970)

be accounted for by binomial variation (Liang and Hanfelt, 1994) This is a typicalover-dispersion clustered binary response data and the ICC parameter ought to bepositive.

Besides the Teratological studies, the estimation for the intra-class correlation ficient are also widely used in the other fields of toxicological and biological studies.For example, Donovan, Ridout and James (1994) used the ICC to quantify the extent

coef-of variation in rooting ability among somaclones coef-of the apple cultivar Greensleeves;Gibson and Austin (1996) used an estimator of ICC to characterize the spatial pat-tern of disease incidence in an orchard; Barto (1966), Fleiss and Cuzick (1979) andKraemer et al.(2002) used ICC as an index measuring the level of interobserver agree-ment; Gang et al (1996) used ICC to measure the efficiency of hospital staff in thehealth delivery research; Cornfield (1978) used ICC for estimating the required size of

a cluster randomization trial

In some clustered binary situation, the ICC parameter can be interpreted as the

”heritability of a dichotomous trait” (Crowder 192, Elston, 1977) It is also frequentlyused to quantify the familial aggregation of disease in the genetic epidemiologicalstudies (Cohen, 1980; Liang, Quaqish and Zeger, 1992)

Chapter 1: Introduction 10

Donner (1986) has given a summarized review for the estimators of ICC in the casethat the responses are continuous He also remarked that the application of continuoustheory for the binary response has severe limitations In addition, the moment method

to estimate the correlation, which is used in the GEE approach proposed by Liang andZegger (1986) is also not appropriate for the estimation of ICC when the response isbinary

A commonly used method to estimate ICC is the Maximum likelihood methodbased on the Beta-Binomial model (Williams 1975) or the extended beta binomialmodel (Prentice 1986) However the estimator based on the parametric model mayyield inefficient or biased results when the true model was wrongly specified

Some robust estimators which are independent of the distributions of S i have beenintroduced, such as the moment estimator (Klienman, 1973), analysis of variance es-timator (Eslton, 1977), quasi-likelihood estimator (Breslow, 1990; Moore and Tsi-atis, 1991), extended quasi-likelihood estimator (Nelder and Pregibon, 1987), pseudo-likelihood estimator (Davidian and Carroll, 1987) and the estimators based on thequadratic estimating equations (Crowder 1987; Godambe and Thompson 1989).Ridout et al (1999) had given an excellent review of the earlier works and con-ducted a simulation study to compare the bias, standard deviation, mean square errorand the relative efficiencies of 20 estimators The reviewing work is based on the datasimulated from beta binomial and mixture binomial distributions and the simulationresults showed that seven estimators performed well as far as these properties were

concerned Paul (2003) introduced 6 new estimators based on the quadratic estimatingequations and compare these estimators along with the 20 estimator used by Ridout

et al (1999) Paul’s work shows that an estimator based on the quadratic estimating

equations also perform well for the joint estimation of (π, ρ).

Chapter 1(this chapter) gives an introduction to the clustered binary data, common

correlated model and reviews the past works on the estimation of the ICC ρ

Chap-ter 2 will introduce the commonly used estimators and the new estimators that weare going to investigate Then we will obtain the asymptotic variances of the four

estimators that we are going to compare: κ-type (FC) estimator, ANOVA estimator,

Gaussian likelihood estimator and the new estimator based on Cholesky tion Chapter 3 will carry the simulation studies to compare the performances of thesefour estimators We will compare the bias, standard deviation, mean square error andthe relative efficiency of these four estimators To investigate the performance of theestimators in practice, chapter 4 will apply these four estimators on two real exampledata sets Chapter 5 will give general conclusions and describe the future work

decomoposi-Chapter 2

Estimating Equations

Since S i is the sufficient statistics for π, modelling on the vector response y ij does

not give more information for π than modelling on S i = Pn i

j=1 y ij On the otherhand, the estimating equation should not dependent on the order of the fetuses in

the developmental studies Denote the residual g i = S i − n i π and the variance V i =

Var(S i − n i π) = σ2

we can get the estimating equation for π:

Simplify (2.1), we get the Quasi-likelihood estimating equation for π:

From another point of view,we may also use the GEE approach, which is modelled

on the vector response y i = {y i1 , y i2 , , y in i }.

Note that (2.3) also does not depend on the order of y ij even though it is modelled

on the vector response It has the same form with the Quasi-likelihood estimatingequation (2.1)

Consider a general set of estimators for π:

Chapter 2: Estimating Equations 14

When w i = [1 + (n i − 1)ρ] −1 = ν −1

i , we can get (2.2) The weight factor ω i can also

take other values For example, when ω i = 1, the estimator for π is ˆ π =Pi S i /Pi n i

and when ω = 1/n i , the estimator for π is (Pi S i /n i )/k

The maximum likelihood estimators are based on the parametric models However,

when the parametric model does not fit the data well, these estimators may be highly

biased or inefficient

• MLE Estimator Based on Beta Binomial Model

As mentioned in (1.2.1), the likelihood of the beta binomial distribution is:

j=0 (1 − π + jθ)

Qy−1

j=0 (1 + jθ) Denote the log-likelihood as l(π, ρ), so the jointly estimating equations for (π, ρ)

n i −SXi −1 r=0

1 − ρ (1 − ρ)(1 − π) + rρ

r − (1 − π)

(1 − ρ)(1 − π) + rρ −

nXi −1 r=0

r − 1

(1 − ρ) + rρ

)

= 0

Denote the solution for the above estimating equations as the maximum

likeli-hood estimator ρ M L

• Gaussian Likelihood Estimator

The Gaussian likelihood estimator was introduced by Whittle (1961) when ing with the continuous response and Crowder(1985) introduced it to the analysis

deal-of binary data As shown in Chapter 1, we know that the Gaussian likelihoodmodel only needs to assume the first two moments and are very easy to calculate

of all the moment based methods Paul (2003) also showed that the Gaussianestimator for the binary data performance well, compared with the other knownestimators for ICC

Assume the vector response y i = {y i1 , y i2 , , y in i } is distributed according to

the multivariate Gaussian distribution, with the mean and variance:

Here A i = diag{π(1 − π), π(1 − π), , π(1 − π)} is the diagonal variance matrix.

Denote the residual

Chapter 2: Estimating Equations 16

the standardized residual

y i2 −π

√ π(1−π)

y ini −π

√ π(1−π)

Denote the solution for (2.5) as the Gaussian likelihood estimator ρ G

The non likelihood based estimators are supposed to be more robust than the

maxi-mum likelihood estimators since they are independent of the distributions of S i We

will introduce the new estimator ρ U J which based on the Cholesky decomposition, aswell as some other commonly used estimators

• New Estimator Based on Cholesky Decomposition

The new estimator is a specification of the U-J method proposed by Wang andCarey (2004), which is based on the Cholesky Decomposition:

Chapter 2: Estimating Equations 18

1 + (j − 2)ρ (1 − ρ)[1 + (j − 1)ρ]

Let’s consider all the permutations of ε ij We use ε i[l]represent one permutation.

Since there are n i ! permutations for the i th cluster, we shall use 1/n i! as the weight

for the i th cluster

Chapter 2: Estimating Equations 20

Denote the solution for (2.6) as the new estimator ρ UJ

• The Analysis of Variance Estimator

The analysis of variance estimator is by defination:

where MSB and MSW are the between and within group mean squares from a

one-way analysis of variance of the response y i And

• Direct Probability Interpretation Estimators

Assume the probability that two individuals have the same response to be α if they are from the same cluster or β if they are from the different clusters The

assumptions of the common correlation model shows that:

and hence that

Similarly, we can get other estimators with the different estimators of α and β.

Mak (1988) has proposed the Mak’s estimator:

• Direct Calculation of Correlation Estimator

Donner (1986) suggested to estimate ρ by calculating the Pearson correlation

coefficient over all possible pairs within one group Karlin et al (1981) proposedthe general form of such kind of estimators Ridout et al (1999) extended this

Chapter 2: Estimating Equations 22

method to the binary data and proposed the Pearson correlation estimator as:

Denote the estimator that use the constant weight ω i = 1/Pi n i (n i − 1) as the

Pearson estimator ρ P earson

P

P

i n i (n i − 1) (2.12)

• Pseudo Likelihood Estimator

Davidian and Carroll (1987) and Carroll and Ruppert (1988) introduced the

pseudo likelihood estimator Treat the count number S i =Pi y ij as a Gaussian

distribution random variable So the likelihood for S i is:

Denote the solution for (2.13) as the pseudo likelihood estimator ρ P L Note that,

ρ P L is different with the Gaussian likelihood estimator ρ G ρ G is got by treating

the vector response y i = {y i1 , y i2 , , y in i } as a multivariate normal distribution

while ρ P L is got by maxmizing the pseudo likelihood of S i =Pj y ij

• Extended Quasi Likelihood Estimator

Nelder and Pregibon (1987) extended the quasi likelihood estimating equation for

the common correlation model to estimate the ICC ρ Note that the traditional quasi likelihood approach can not be used here, since the residuals ε i does not

likelihood estimator ρ P Another way is to replace D i (S i , π) with k − 1D k i (S i , π),

this will yield the unbiased version of the quasi likelihood estimator ρ EQ

• Moment Estimator

Chapter 2: Estimating Equations 24

Kleinman (1973) proposed a set of moment estimators in the form of:

Two specifications of the moment estimators are used in Ridout et al (1999),

one with weights (ω i = 1/k) and the other with (ω i = n i /N) They are labeled

ρ KEQ and ρ KP R If S ω is replaced by S ∗

ω = k − 1 k S ω, we can get two slightly

different moment estimators ρ ∗

KEQ and ρ ∗

KP R

A more general moment estimator proposed by Whittle (1982) is by using the

iterative algorithms Take ω i = n i

1 + (n i − 1)ˆ ρ, where ˆρ is the current estimate

of ρ, we can get a new moment estimator ρ W and ρ ∗

W (by replacing S ω with S ∗

ω

mentioned above)

• Estimators Based on Quadratic Estimating Equations

The quadratic estimating equations was first proposed by Crowder (1987) It is

a set of estimating equations with the quadratic form of S i − n i π:

He also proposed that the optimal estimating equations is obtained by setting:

a iπ= −(γ 2iλ + 2) + γ 1iλ (1 − 2π)σ iλ /π(1 − π)

Here γ 1j and γ 2j are the skewness and kurtosis of S i − n i π

n i and σ iλis the variance

of S i − n i π

n i However we do not know the exact form of γ 1i and γ 2i for the nonlikelihood estimators Paul (2001) suggested to use the 3rd and 4th momentsderived from the beta-binomial distribution instead:

esti-Chapter 2: Estimating Equations 26

The Gaussian likelihood estimator and pseudo likelihood estimator are specialcases of the optimal quadratic estimating equations For the Gaussian likelihoodestimator, the parameters are:

a iρ = n i (1 − 2π) and b iρ = n

2

i [1 + (n i − 1)ρ2]

[1 + (n i − 1)ρ]2

For the pseudo likelihood estimator, the parameters are:

a iρ = 0 and b iρ= n

2

i (n i − 1)

[1 + (n i − 1)ρ]m 2i

It also coincides with the optimal estimating equations when we set γ 1i = γ 2i= 0

Ridout et al (1999) compared 20 estimators of the intra-class coefficient for theirbias, standard deviation, mean square error and relative efficiency He suggested that

the analysis of variance estimator (ρ A ), the κ-type estimator (ρ F C) and the moment

estimator (ρ KP R and ρ W) performed well as far as the median of the mean square

error were concerned He also found that the Pearson estimator (ρ P earson) performed

well when the true value of the intra-class correlation parameter ρ was small But the overall performance of ρ P earson depends on the true value of ρ The conclusion of Rid-

out et al (1999) were based on the simulation results on the data generated from thebeta binomial distribution and the mixed distribution of two binomial distributions.Paul (2003) introduced 6 other estimators based on the quadratic estimating equa-tions and compare these 6 estimators along with the 20 estimators used by Ridout

et al (1999) With similar setup of the simulation, Paul (2003) showed that the

estimator based on the optimal quadratic estimating equations ρ QB, which used the

3rd and 4th moment from beta binomial distribution, also performs well in the jointlyestimation of (ˆπ, ˆ ρ) For the data generated from the beta binomial distribution, it

even has higher efficiency than that of ρ A He also found that the performance of

ρ P earson depends on the true value of ρ, which is consistent with the findings of Ridout

et al.(1999)

Zou and Donner (2004) introduced the coverage rate as a new index to comparethe performance of the estimators They obtained the closed form of the asymptotic

variances of the analysis of variance estimator ρ A , the κ-type estimator ρ F C and the

Pearson estimator ρ P earson, under the distribution of the generalized binomial models

(Madsen, 1993) The simulation results indicated that the κ type estimator ρ F C formed best among these three estimators as far as the coverage rate of the confidenceinterval was concerned

We are going to compare four estimators The κ-type estimator ρ F C, the analysis of

variance estimator ρ A , the Gaussian estimator ρ G and the UJ estimator based on theCholesky decomposition

The κ-type estimator ρ F C and the ANOVA estimator ρ A are widely used tors for ICC and performs well in many situations (Ridout et al 1999) Gaussianlikelihood method is the most general form of all the moment based methods And it

estima-Chapter 2: Estimating Equations 28

also only relies on the first two moments, that is what we know in the common lated model Besides, the Gaussian likelihood method is also the most convenient tocalculate method of all the pseudo likelihood methods (Crowder 1985)

corre-We are going to compare these three estimators with the new estimator ρ U J based

on the Cholesky decomposition, which is the specification of the UJ method proposed

by Wang and Carey (2004)

The asymptotic variance quantifies the limit properties of the estimators As shown

above, we have two types of estimators for ρ One type of the estimator is the solution

of some estimating equation, such as the new estimator ρ U J and the Gaussian

Like-lihood estimator ρ G Another type of the estimator has the closed form, such as the

obtain the asymptotic variances of these two types of estimators

• Estimators Without Closed Forms

This type of the estimator is the solution of some estimating equation and has

no closed forms The typical example is the NEW (UJ) estimator

the estimating equations for (π, ρ) jointly So the choice of the estimators of ˆ π

may affect the asymptotic variance of ˆρ Here we will use (2.2):

as the estimating equation for ˆπ The advantage of this estimator is that it would

maximize the efficiency of ˆπ.

Of all the estimators mentioned above, the MLE estimator(ρ M L), the Gaussian

estimator (ρ G ), the Pseudo likelihood estimator (ρ P L), the extended quasi-likelihood

estimator (ρ EQ), the estimator based on the quadratic estimating equations

(ρ QB ) and the New (UJ) estimator ρ U J based on Cholesky decomposition are

Chapter 2: Estimating Equations 30

So the asymptotic variance-covariance matrix is

Here Var(ˆρ) = V22 Simply plugging in the estimates of (ˆπ, ˆ ρ) can not ensure the

positiveness of matrix M and sometimes we will get the negative values of the

asymptotic variances of ˆρ G and ˆρ U J Here we define:

M ] is a positive matrix So, use M ] instead of M if necessary, then the

asymp-totic variance of ˆρ G and ˆρ U J will always be positive

For our choice of the estimating equation for π:

Var(ˆπ) = V11 =

µ1

where m 2i = E(S i − n i π)2 = n i π(1 − π)[1 + (n i − 1)ρ] is the 2 nd order centralized

moment of S i m 3i and m 4i are the 3rd and 4th order centralized moment of S i

Apply the sandwich method on the NEW(UJ) estimator and the Gaussian

like-lihood estimator, with m 3i and m 4i to denote the 3rd and 4th order centralized