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In longitudinal data analysis, our primary interest is in the estimation of regression parameters for the marginal expectations of the longitudinal responses, and the longitudinal correlation parameters are of secondary interest. The joint likelihood function for longitudinal data is challenging, particularly due to correlated responses. Marginal models, such as generalized estimating equations (GEEs), have received much attention based on the assumption of the first two moments of the data and a working correlation structure. The confidence regions and hypothesis tests are constructed based on the asymptotic normality. This approach is sensitive to the misspecification of the variance function and the working correlation structure which may yield inefficient and inconsistent estimates leading to wrong conclusions. To overcome this problem, we propose an empirical likelihood (EL) procedure based on a set of estimating equations for the parameter of interest and discuss its characteristics and asymptotic properties. We also provide an algorithm base d on EL principles for the estimation of the regression parameters and the construction of its confidence region. We have applied the proposed method in two case examples.

Longitudinal studies are common in areas such as epidemiology, clinical trials, economics, agriculture, and survey sampling. In longitudinal studies, we are interested in the changes in the variables over time as a function of the covariates, generally under the assumption that observations from different individuals are independent. For example, longitudinal studies are used to characterize growth and ageing, to assess the effect of risk factors on human health, and to evaluate the effectiveness of treatments. To obtain an unbiased, efficient, and reliable estimate, we must properly model the correlation between the repeated responses for each individual. However, the modelling of correlation, especially when the responses are discrete, is a challenging task even if the responses are collected over equispaced time points.

The approaches used for the analysis of longitudinal data can be classified as mixed effects models, transitional models, and marginal regression models. A potential disadvantage of mixed effects models is that they rely on parametric assumptions, which may lead to biased parameter estimates when a model is misspecified. Moreover, the estimation of the parameters is challenging when the random effects have a high dimension; it typically involves integrals that do not have an explicit form. Transitional models are more difficult to apply when there are missing data and the repeated measurements are not equally spaced in time. In addition, the interpretation of the regression parameters varies with the order of the serial correlation, and the regression parameter estimates are sensitive to the assumption of time dependence. Because of the aforementioned difficulties in modelling and performing inference, we focus on marginal models in this paper.

We start with brief review on existing methods for longitudinal data under the framework of generalized linear models (GLMs). The longitudinal observations consist of an outcome random variable y i t and a p-dimensional vector of covariates x i t , observed for subjects i = 1, ⋯ , k at a time point t, t = 1, ⋯ , m i . For the ith subject, let y i = ( y i 1 , ⋯ , y i m i ) T be the response vector, and let X i = ( x i 1 , x i 2 , ⋯ , x i t , ⋯ , x i m i ) T be the m i × p matrix of covariates. Marginal models for longitudinal data can be extended to the GLM framework. The marginal density of y i t is assumed to follow an exponential family [

f ( y i t ) = exp [ ( y i t θ i t − a ( θ i t ) ) ϕ + b ( y i t , ϕ ) ] , (1)

where θ i t = h ( η i t ) , h is a known injective function with η i t = x i t β , β is a p × 1 vector of regression effects of x i t on y i t , and a ( ∗ ) and b ( ∗ ) are functions that are assumed to be known. The mean and variance of y i t can be written

E ( y i t | x i t ) = a ′ ( θ i t ) = μ i t and Var ( y i t ) = a ″ ( θ i t ) = v ( μ i t ) ϕ ,

where ϕ is the unknown over-dispersion parameter and v ( ∗ ) is a known variance function. For simplicity, we set the nuisance scale parameter ϕ to 1 for the rest of this paper.

The generalized estimating equation (GEE) approach [

g ( β , α ^ ( β ) ) = ∑ i = 1 k X i T A i 1 / 2 R i − 1 ( α ^ ) A i − 1 / 2 ( y i − μ i ) = 0, (2)

where A i is an m i × m i diagonal matrix with Var ( μ i t ) as the tth diagonal element and R i ( α ^ ) is the m i × m i working correlation matrix of the m i repeated measurements. For j = 1, ⋯ , m i and j ′ = 1, ⋯ , m i , the ( j , j ′ ) th element of R i is the known, hypothesized, or estimated correlation. The working correlation may depend on an unknown s × 1 correlation parameter vector α . The observation times and correlation matrix may differ from subject to subject, but the correlation matrix R i ( α ) for the ith subject is fully specified by α . Some common working correlation structures are independent, autoregressive of order 1 (AR(1)), equally correlated (EQC), moving average of order 1 (MA(1)), or unstructured.

It has been demonstrated that in some situations the use of an arbitrary working correlation structure may lead to no solution for α ^ , which may break down the entire GEE methodology (see [

The estimate for β is obtained by solving the following estimating equations:

g ( β , ρ ) = ∑ i = 1 k X i T A i Σ i − 1 ( ρ ^ ) ( y i − μ i ) = 0, (3)

where Σ i ( ρ ^ ) = A i 1 / 2 C i * ( ρ ) A i 1 / 2 , with C i * ( ρ ) the stationary lag correlation structure for the AR(1), MA(1), or EQC models. The stationary lag correlations can be estimated via the method of moments introduced by [

The GEE approach requires only the assumption of the existence of the first two marginal moments and a correlation structure. GEE estimators are consistent and asymptotically normal as long as the mean, variance, and correlation structure are correctly specified. Marginal models have satisfactory performance when the assumptions are satisfied. Misspecification can cause estimates based on marginal models to be inefficient and inconsistent, and inference in this situation can be completely inappropriate. Confidence regions and hypothesis tests are based on asymptotic normality, which may not hold since the finite-sample distribution may not be symmetric. These problems motivate us to investigate the applicability of empirical likelihood (EL), a nonparametric likelihood method, based on a set of GEEs for the parameter of interest.

The EL introduced by [

The remaining part of the paper is organized as follows. In Section 2, we develop the subject-wise EL via a set of GEEs of the parameter of interest and discuss its characteristics. We then introduce an adjusted EL (AEL) inference to longitudinal data. We discuss its characteristics and asymptotic properties in Section 3. In Section 4, we developed an algorithm based on EL principles for the estimation of the regression parameters and the construction of its confidence region. In Section 5, the performance of the proposed method is assessed based on Monte Carlo simulations. The implementation of the proposed method in two real case examples is discussed in Section 6 and the conclusions are given in Section 7.

EL is a nonparametric-likelihood-based approach, introduced by [

In a seminal paper, [

In longitudinal data analysis framework, [

Following [

l ( β ) = sup [ ∑ i = 1 k log ( p i ) : p i ≥ 0 , i = 1 , 2 , ⋯ , k ; ∑ i = 1 k p i = 1 , ∑ i = 1 k p i g i ( β , ρ ) = 0 ] .

The EL is maximized when

p ^ i = 1 k { 1 + λ ^ T g i ( β , ρ ) } , i = 1,2, ⋯ , k , (4)

where the Lagrange multiplier λ ^ = λ ^ ( β ) is the solution of

∑ i = 1 k g i ( β , ρ ) 1 + λ T g i ( β , ρ ) = 0. (5)

This result leads to the profile empirical log-likelihood function

l ( β ) = − k l o g ( k ) − ∑ i = 1 k l o g ( 1 + λ ^ T ( β ) g i ( β , ρ ) )

and the profile empirical log-likelihood ratio function

W l ( β ) = − ∑ i = 1 k l o g ( k p ^ i ) = ∑ i = 1 k l o g [ 1 + λ ^ T ( β ) g i ( β , ρ ) ] . (6)

Under some regularity conditions, we have 2 W l ( β 0 ) → D χ p 2 as k → ∞ if

E [ g ( β 0 , ρ ^ ( β 0 ) ) g T ( β 0 , ρ ^ ( β 0 ) ) ]

is full rank where β 0 is the true parameter value. This conclusion is similar to that for the parametric likelihood ratio function. The vector β can be estimated by minimizing

W l ( β ) = ∑ i = 1 k log ( 1 + λ ^ T ( β ) g ( β , ρ ) ) (7)

with respect to β . Note that the profile log-likelihood ratio function can be minimized with respect to β when ρ is known. In practice, ρ is unknown, but can be consistently estimated using the method of moments by [

The computation of the profile EL function is a key step in EL applications, and it involves constrained maximization. In some situations, the algorithm may fail because of poor initial values of the parameters. Moreover, the poor accuracy of EL confidence regions has been reported by several authors, including [

The computation of the profile EL ratio function W l ( β ) given in (7) is a key step in EL applications. The solution for λ must satisfy { 1 + λ ^ T ( β ) g i ( β , ρ ^ ( β ) ) } > 0 for all i = 1, ⋯ , k . A necessary and sufficient condition for its existence is that the vector 0 is an interior point of the convex hull of { g i ( β , ρ ^ ( β ) ) , i = 1, ⋯ , k } . Under some moment conditions on g ( β , ρ ^ ( β ) ) [

Let g i ( β ) = g i ( β , ρ ^ ( β ) ) and g ¯ k ( β ) = 1 k ∑ i = 1 k g i ( β ) for any given β . For some positive constant b k , by the addition of an artificial observation

g k + 1 ( β ) = − b k k ∑ i = 1 k g i ( β ) = − b k g ¯ k ( β )

with b k = l o g ( k ) / 2 . The adjusted profile empirical log-likelihood ratio function is

W l * ( β ) = inf [ − ∑ i = 1 k + 1 log [ ( k + 1 ) p i ] : p i ≥ 0 , i = 1 , 2 , ⋯ , k + 1 ; ∑ i = 1 k + 1 p i = 1 , ∑ i = 1 k + 1 p i g i ( β ) = 0 ] = ∑ i = 1 k + 1 l o g [ 1 + λ ^ T ( β ) g i ( β ) ]

with λ ^ = λ ^ ( β ) being the solution of ∑ i = 1 k + 1 g i ( β ) 1 + λ T g i ( β ) = 0 . Note that 0 always lies inside the convex hull of { g i ( β , ρ ^ ( β ) ) , i = 1, ⋯ , k + 1 } . The adjusted profile empirical log-likelihood ratio function is well defined after adding a pseudo value g k + 1 ( β ) . For a wide range of b k , following [

W l * ( β ) = ∑ i = 1 k + 1 [ l o g ( 1 + λ ^ T ( β ) g i ( β , ρ ^ ( β ) ) ) ] (8)

with respect to β .

The adjustment is particularly useful because, even for some undesirable values of β , the algorithm guarantees a solution. The confidence regions constructed via the AEL are found to have better coverage probabilities than those for the regular EL and the algorithm provides a promising solution for λ , particularly when the sample size is small. The improved coverage probability is achieved without resorting to more complex procedures such as Bartlett correction or bootstrap calibration.

In the next section, following [

In this section, we present the first-order asymptotic properties of β ^ and the adjusted profile empirical log-likelihood ratio statistics. We first introduce some notation and regularity conditions that are used in the theorems and lemma.

Regularity Conditions:

A1. E { g ( β 0 , ρ ^ ( β 0 ) ) } = 0 , where β 0 is the true value of β , g ( β , ρ ^ ( β ) ) = ∑ i = 1 k D i T Σ i − 1 ( ρ ^ ) ( y i − μ i ) be the estimating function for β ∈ R p (defined in (3)), D i = ∂ { a ′ i ( θ ) } / ∂ β , Σ i ( ρ ^ ) = A i 1 / 2 C i * ( ρ ^ ) A i 1 / 2 , and A i = d i a g { a ″ i ( θ ) } for i = 1,2, ⋯ , k . Let g ¯ k ( β , ρ ^ ( β ) ) = 1 k ∑ i = 1 k g i ( β , ρ ^ ( β ) ) and g k + 1 ( β , ρ ^ ( β ) ) = − b k g ¯ k ( β , ρ ^ ( β ) ) , where b k is a positive constant.

A2. { a ′ ( θ ) } is three times continuously differentiable and { a ″ ( θ ) } > 0 in Θ ∘ , where Θ be the natural parameter space of the exponential family distributions presented in (1) and Θ ∘ the interior of Θ . Also, h ( η ) is three times continuously differentiable and h ′ ( η ) > 0 .

A3. E β 0 { ∂ g k ( β , ρ ) ∂ β } and V k ( β 0 , ρ ^ ( β 0 ) ) = E β 0 { g k ( β , ρ ^ ( β ) ) g k T ( β , ρ ^ ( β ) ) } are positive definite.

A4. The rank of E { ∂ g k ( β , ρ ) ∂ β } is p in a neighbourhood of β 0 .

A5. There exist functions G ( y , X ) such that in a neighbourhood of β 0 .

| ∂ g k ( β , ρ ) ∂ β | < G ( y , X ) , ‖ g k ( y , X , β , ρ ^ ( β ) ) ‖ 3 < G ( y , X )

with E [ G ( y , X ) ] < ∞ .

Theorem 3.1. Under regularity conditions A1-A5, suppose ( y i , X i ) , i = 1,2, ⋯ , k is a set of independent and identically distributed random vectors. Let

2 W l * ( β ) = 2 ∑ i = 1 k + 1 l o g [ 1 + λ ^ T ( β ) g i ( β , ρ ^ ( β ) ) ] (9)

be the adjusted profile empirical log-likelihood ratio function. Then, as k → ∞ , ρ ^ ( β ) is a consistent estimator in the neighbourhood of β ; the correlation matrix of y i is C i * ( ρ ) , defined in (3) and W l * ( β ) attains its minimum value at some point β ^ in the interior of ‖ β ^ − β 0 ‖ < k − 1 / 3 in probability.

This result corresponds to Lemma 1 in [

Theorem 3.2. In addition to the regularity conditions A1-A5, suppose that ∂ 2 g ( β , ρ ) ∂ β ∂ β T is bounded by some integrable function G ( y , X ) in the neighbourhood. Then, there exists a sequence of adjusted profile EL estimates β ^ of β such that

k ( β ^ − β 0 ) → D N ( 0 , Δ ) ,

where

Δ = [ E β 0 { ∂ g ( β , ρ ^ ( β ) ) ∂ β } T { E β 0 { g ( β , ρ ^ ( β ) ) g T ( β , ρ ^ ( β ) ) } − 1 } E β 0 { ∂ g ( β , ρ ^ ( β ) ) ∂ β } ] − 1

It is noted that the proof of Theorem 3.2 is similar to the proof of Theorem 1 in [

Theorem 3.3. Under regularity conditions A1-A5, the adjusted profile empirical log-likelihood ratio function 2 W l * ( β 0 ) , where β 0 is the true value of β , is asymptotically chi-squared distributed with degrees of freedom p.

The proof of Theorem 3.3 can be achieved by using similar arguments as those used in the proof of Theorem 2 in [

To implement our method, we need an efficient algorithm. We minimize the profile EL ratio function W l ( β ) with respect to β using a Newton-Raphson algorithm. At each Newton-Raphson iteration, we compute the Lagrange multiplier for updated values of β and ρ ^ ( β ) . We used the modified Newton-Raphson algorithm proposed by [

The Lagrange multiplier λ is estimated by solving the equation

∑ i = 1 k g i ( β , ρ ^ ( β ) ) 1 + λ T g i ( β , ρ ^ ( β ) ) = 0

for a given set of vectors g i ( β , ρ ^ ( β ) ) , i = 1 , 2 , ⋯ , k . Note that the above equation is the derivative of R with respect to λ for a given β , where

R = ∑ i = 1 k log { 1 + λ T g i ( β , ρ ^ ( β ) ) } . (10)

In the EL problem, the solution must satisfy

1 + λ T g i ( β , ρ ^ ( β ) ) > 0 , i = 1 , 2 , ⋯ , k .

The modified Newton-Raphson algorithm for estimating λ for a given value of β and ρ ^ ( β ) is as follows:

1. Set λ c = 0 , c = 0 , γ c = 1 , ϵ = 1 e − 08 , ρ = ρ 0 , and β = β 0 .

2. Let R λ and R λ λ be the first and second partial derivatives of R (given in (10)) with respect to λ :

R λ = ∑ i = 1 k [ g i ( β , ρ ^ ( β ) ) { 1 + λ T g i ( β , ρ ^ ( β ) ) } ] ,

R λ λ = − ∑ i = 1 k [ g i ( β , ρ ^ ( β ) ) g i T ( β , ρ ^ ( β ) ) { 1 + λ T g i ( β , ρ ^ ( β ) ) } 2 ] .

Compute R λ and R λ λ for λ = λ c and let Δ ( λ c ) = − [ R λ λ ] − 1 R λ .

If ‖ Δ ( λ c ) ‖ < ϵ stop the algorithm and report λ c ; otherwise continue.

3. Calculate δ c = γ c Δ ( λ c ) . If 1 + ( λ c − δ c ) g i ( β , ρ ^ ( β ) ) ≤ 0 for some i, set γ c = γ c 2 and go to Step 2.

4. Set λ c + 1 = λ c − δ c , c = c + 1 , and γ c + 1 = ( c + 1 ) − 1 2 and go to Step 2. Step 2 will guarantee that p i > 0 and the optimization is carried out in the right direction.

Let λ ^ ( β ) be the estimated value of λ for a given β . We minimize the profile EL ratio function defined in (7) over β . The Newton-Raphson algorithm is as follows:

1. Set β = β 0 , h = 0 , and ϵ = 1 e − 08 .

2. Let λ ^ = λ ( β ) and ρ ^ ( β ) be the estimated values of λ and ρ .

3. Compute the new estimate of β via

β ( h + 1 ) = β ( h ) − { W l β β ( β h ) } − 1 { W l β ( β h ) } (11)

where W l ( β ) is the profile empirical log-likelihood ratio function defined in (7), with

W l β = ∂ W l ( β ) ∂ β , W l β β = ∂ 2 W l ( β ) ∂ β ∂ β T .

Note that to compute W l β and W l β β , we need to estimate the Lagrange multiplier λ ^ ( β ) as in Section 4.1. In practice, ρ is unknown, and the correlations can be consistently estimated by [

4. If m i n | β ( h + 1 ) − β ( h ) | < ϵ stop the algorithm and report β ( h + 1 ) ; otherwise set h = h + 1 and go to Step 3.

The simplified expressions for W l β and W l β β are as follows. Let R β , R β β , and R β λ be the first and second partial derivatives of (10) with respect to β and λ

R β = ∑ i = 1 k [ g ′ i ( β , ρ ^ ( β ) ) λ { 1 + λ T g i ( β , ρ ^ ( β ) ) } ] ,

R β β = ∑ i = 1 k { [ g ″ i ( β , ρ ^ ( β ) ) λ T { 1 + λ T g i ( β ) } ] − [ g ′ i ( β , ρ ^ ( β ) ) λ λ T [ g ′ i ( β , ρ ^ ( β ) ) ] T { 1 + λ T g i ( β , ρ ^ ( β ) ) } 2 ] } ,

and

R β λ = ∑ i = 1 k [ { 1 + λ T g i ( β ) } g ′ i ( β , ρ ^ ( β ) ) − g ′ i ( β , ρ ^ ( β ) ) λ [ g i ( β ) ] T { 1 + λ T g i ( β ) } 2 ] .

The first derivative of W l ( β ) with respect to β is

W l β = ∑ i = 1 k [ [ ∂ λ ( β ) ∂ β ] T g i ( β , ρ ^ ( β ) ) + g ′ i ( β , ρ ^ ( β ) ) λ ( β ) { 1 + λ T ( β ) g i ( β , ρ ^ ( β ) ) } ] = [ ∂ λ ( β ) ∂ β ] T R λ + R β .

Note that for λ = λ ^ ( β ) , R λ = 0 . Therefore,

W l β = R β . (12)

Similarly, the second derivative of W l ( β ) with respect to β is

W l β β = ∑ i = 1 k [ { 1 + λ T ( β ) g i ( β , ρ ^ ( β ) ) } { [ ∂ 2 λ ( β ) ∂ β ∂ β T ] g i ( β , ρ ^ ( β ) ) + 2 g ′ i ( β ) [ ∂ λ ( β ) ∂ β ] T + g ″ i ( β , ρ ^ ( β ) ) λ ( β ) } { 1 + λ T ( β ) g i ( β , ρ ^ ( β ) ) } 2 ] − ∑ i = 1 k [ { [ ∂ λ ( β ) ∂ β ] T g i ( β , ρ ^ ( β ) ) + g ′ i ( β , ρ ^ ( β ) ) λ ( β ) } { [ ∂ λ ( β ) ∂ β ] T g i ( β , ρ ^ ( β ) ) + g ′ i ( β , ρ ^ ( β ) ) λ ( β ) } T { 1 + λ T ( β ) g i ( β , ρ ^ ( β ) ) } 2 ] = [ ∂ λ ( β ) ∂ β ] T R λ λ [ ∂ λ ( β ) ∂ β ] + 2 [ ∂ λ ( β ) ∂ β ] T R β λ + R β β .

Following [

[ ∂ λ ( β ) ∂ β ] = ( R λ λ ) − 1 R β λ ,

so

W l β β = R β β − R β λ ( R λ λ ) − 1 R λ β . (13)

We use the bisection method to construct the lower and upper confidence limits based on the profile EL ratio for β . Let β ^ = ( β ^ 1 , β ^ 2 ) T be the estimated value of β from Section 4.2, where β ^ 1 is a scalar and β ^ 1 is the 1 × p − 1 vector of parameters and we are interested to construct confidence interval for β 1 .

1. Compute a reasonable lower confidence limit β 1, L for β 1 . Set L 1 = β ^ 1 , L 2 = β ^ 1 − a × SE ( β ^ 1 ) , and ϵ = 1 e − 05 , where SE ( β ^ 1 ) is the standard error of β ^ 1 using any existing method. We can choose a such that W l ( L 2 , β ^ 2 ) > [ χ 1,1 − α 2 ] / 2 > W l ( L 1 , β ^ 2 ) , where χ 1,1 − α 2 is the ( 1 − α ) th quantile from a χ 2 distribution with one degree of freedom.

2. Compute the profile empirical log-likelihood ratio values W 1 = 2 W l ( L 1 , β ^ 2 ) and W 2 = 2 W l ( L 2 , β ^ 2 ) .

3. Minimize the profile EL ratio function defined in (7) over β 2 for a given L n e w = ( L 1 + L 2 ) / 2 . Let β ^ 2 n e w be the new estimate of β 2 and W n e w = 2 W l ( L n e w , β ^ 2 n e w ) .

4. If W n e w < χ 1 , 1 − α 2 , set L 1 = L n e w and W 1 = W n e w ; else set L 2 = L n e w and W 2 = W n e w .

5. If | W 1 − W 2 | < ϵ stop the algorithm and report β 1, L ; otherwise go to Step 3.

We can use this approach to construct the upper confidence limit by setting U 1 = β ^ 1 and U 2 = β ^ 1 + a × SE ( β ^ 1 ) .

In this section, we conduct simulation studies to investigate the performance of our EL-based approach. We compute the coverage probabilities based on the ordinary EL, AEL and compare them with those of the GEE approach, which is based on a normal approximation. We also compute the coverage probabilities based on the extended empirical likelihood (EEL) by [

We consider the stationary correlation models for count data discussed by [

(i) Poisson Autoregressive Order 1 (AR(1)) Model

Let y i 1 ~ Poi ( μ ˜ i ) , where μ ˜ i = exp ( x ˜ i β ) . The repeated responses follow the AR lag 1 dynamic model given by

y i t = ρ ∗ y i , t − 1 + d i t , t = 2, ⋯ , m i . (14)

Given y i , t − 1 , ρ ∗ y i , t − 1 is the binomial thinning operation. That is,

ρ ∗ y i , t − 1 = ∑ j = 1 y i , t − 1 b j ( ρ ) = z i , t − 1 ,

where the b j ( ρ ) are independent and identically distributed Bernoulli ( ρ ) random variables. We assume that d i t ~ Poi ( μ ˜ i ( 1 − ρ ) ) and it is independent of z i , t − 1 . Let x ˜ i = ( x ˜ i 1 , ⋯ , x ˜ i p ) be the time-independent covariate for the ith individual.

(ii) Poisson Moving Average Order 1 (MA(1)) Model

The repeated responses follow the MA lag 1 dynamic model given by

y i t = ρ ∗ d i , t − 1 + d i t , t = 2 , ⋯ , m i , (15)

where ρ ∗ d i , t − 1 = ∑ j = 1 d i , t − 1 b j ( ρ ) is a binomial thinning operation and d i t ~ Poi [ μ ˜ i 1 + ρ ] , t = 0, ⋯ , m i , with μ ˜ i = exp ( x ˜ i β ) . Here t = 0 is the initial time.

(iii) Poisson Equally Correlated Model

Let y i 0 ~ Poi ( μ ˜ i ) and d i t ~ Poi [ μ ˜ i ( 1 − ρ ) ] for all t = 1 , ⋯ , m i . The repeated responses follow the dynamic equicorrelation model given by

y i t = ρ ∗ y i 0 + d i t , for t = 1 , ⋯ , m i . (16)

We simulated 1000 data sets from each of these models follow the AR(1), EQC, or MA(1) structure, and used EL-based methods to estimate the parameters using different working correlation such as AR(1), EQC, and MA(1) as well as lag correlation. In each simulation we use the parameters β = ( β 1 , β 2 ) T = ( 0.3,0.2 ) T and ρ = 0.5 . We consider k = 100 subjects and m = 4 time points. For the ith subject, we generate the covariates x ˜ i = ( x ˜ i 1 , x ˜ i 2 ) from a normal distribution with mean 0 and standard deviation 1. For the analysis, we consider the working correlation to be either a true correlation or a lag correlation. We did not consider other possible values for ρ since the working correlation structure may lead to no solution for α ^ in some situations.

In the simulation studies discussed in Section 5.1 we considered the correlation structure used to generate the data as the working correlation in the GEE-based modelling. However, in practice, we do not know the correlation structure of the data. As discussed before, if the working correlation is misspecified, we may lose the efficiency of the parameter estimates [

We conducted a simulation study to assess the loss of efficiency. We generated repeated counts with the AR(1) correlation structure given in Section 5.1(i) with ρ = 0.49 and 0.70 and m = 5 time points. We used three working correlation structures: EQC, MA(1), and lag correlation.

In this section, we consider the performance of our approach when the variance function is misspecified, in the context of stationary count data. We generate over-dispersed stationary count data y i t using μ ˜ i = u i exp ( x ˜ i β ) for the three models discussed in Section 5.1, where u i is a random sample such that E ( u i ) = 1 and Var ( u i ) = ω . Marginally, we have E ( y i t ) = μ ˜ i and Var ( y i t ) = μ ˜ i ( 1 + μ ˜ i ω ) . The distribution of u is chosen to be gamma with shape parameter ω and scale parameter 1 / ω , where ω is the over-dispersion parameter. We choose over-dispersion parameter ω = 1 / 4 . However, the GEE, EL, EEL, and AEL CIs are constructed under the assumption that there is no over-dispersion.

In this section, we investigate the performance of our EL approach on a class of stationary and nonstationary correlation models for longitudinal continuous data. The random errors ( ϵ 1 , ϵ 2 , ϵ 3 , ϵ 4 ) T are generated from the multivariate normal distribution with marginal mean 0, marginal variance 1, and an auto-correlation coefficient ρ = 0.5 . In this performance analysis, we consider three correlation models: exchangeable, AR(1), and MA(1).

(i) AR(1) Structure

For t = 1 , ⋯ , m i , for the ith individual

y i t = x i t β + ϵ i t , (17)

and we assume that

ϵ i t = ρ ϵ i t + a i t ,

with | ρ | < 1 and a i t ~ N ( 0,1 ) .

(ii) MA(1) Structure The ϵ i t in (17) follow the model

ϵ i t = ρ a i , t − 1 + a i t

where ρ is a suitable scale parameter that does not necessarily satisfy | ρ | < 1 , and a i t ~ N ( 0,1 ) .

(iii) Equicorrelation (EQC) Structure The ϵ i t in (17) follow the model

ϵ i t = ρ a i 0 + a i t ,

where a i 0 is an error value at the initial time, and ρ is a suitable correlation parameter. We assume that

a i t ~ N ( 0,1 ) and a i 0 ~ N ( 0,1 ) ,

and a i t and a i 0 are independent for all t.

We simulated 1000 data sets from the above models under stationary and nonstationary covariates, using the parameters β = ( β 1 , β 2 ) T = ( 0.4 , 0.5 ) T , ρ = 0.5 , and m = 4 . For the ith subject, we generate the covariates x ˜ i = ( x ˜ i 1 , x ˜ i 2 ) from a normal distribution with mean 0 and standard deviation 1.

The coverage probabilities of the intervals based on the EL, EEL, and AEL are similar to those of the GEE. For instance, in the MA(1)/MA(1) case in

In this section, we compare the performances of the methods when the correlation model for continuous data is misspecified. The stationary and nonstationary correlation models for longitudinal continuous data are generated from (17) for the parameter set in Section 5.4, and the correlated random errors ( ϵ 1 , ϵ 2 , ϵ 3 , ϵ 4 ) T are generated from the χ 2 ( 1 ) − 1 distribution instead of the normal distribution for the three correlation models:

True model/ Working correlation | Estimate | Statistic | Coverage Probabilities | |||||||
---|---|---|---|---|---|---|---|---|---|---|

95% level | 99% level | |||||||||

GEE | EL | EEL | AEL | GEE | EL | EEL | AEL | |||

IND/IND ρ = 0.50 | β ^ 1 | 0.2975 | 0.953 | 0.929 | 0.936 | 0.937 | 0.986 | 0.985 | 0.987 | 0.987 |

(0.046) | (0.184) | (0.175) | (0.178) | (0.179) | (0.242) | (0.232) | (0.240) | (0.238) | ||

β ^ 2 | 0.1978 | 0.944 | 0.927 | 0.934 | 0.936 | 0.990 | 0.980 | 0.984 | 0.984 | |

(0.048) | (0.188) | (0.179) | (0.183) | (0.184) | (0.248) | (0.238) | (0.246) | (0.244) | ||

AR(1)/AR(1) ρ = 0.50 | β ^ 1 | 0.3006 | 0.947 | 0.928 | 0.937 | 0.937 | 0.985 | 0.980 | 0.981 | 0.981 |

(0.065) | (0.258) | (0.246) | (0.251) | (0.252) | (0.339) | (0.326) | (0.338) | (0.335) | ||

β ^ 2 | 0.1978 | 0.954 | 0.934 | 0.940 | 0.942 | 0.990 | 0.980 | 0.984 | 0.983 | |

(0.067) | (0.264) | (0.251) | (0.256) | (0.257) | (0.347) | (0.332) | (0.343) | (0.340) | ||

AR(1)/Lag ρ = 0.50 | β ^ 1 | 0.3006 | 0.946 | 0.930 | 0.936 | 0.939 | 0.985 | 0.978 | 0.981 | 0.981 |

(0.066) | (0.256) | (0.246) | (0.250) | (0.252) | (0.337) | (0.325) | (0.337) | (0.334) | ||

β ^ 2 | 0.1978 | 0.952 | 0.931 | 0.938 | 0.940 | 0.990 | 0.980 | 0.985 | 0.985 | |

(0.067) | (0.263) | (0.250) | (0.255) | (0.256) | (0.345) | (0.331) | (0.342) | (0.339) | ||

EQC/EQC ρ = 0.50 | β ^ 1 | 0.2984 | 0.955 | 0.942 | 0.946 | 0.948 | 0.988 | 0.986 | 0.988 | 0.987 |

(0.074) | (0.288) | (0.274) | (0.280) | (0.281) | (0.379) | (0.364) | (0.376) | (0.373) | ||

β ^ 2 | 0.1936 | 0.950 | 0.939 | 0.941 | 0.941 | 0.990 | 0.986 | 0.986 | 0.986 | |

(0.074) | (0.295) | (0.282) | (0.288) | (0.289) | (0.387) | (0.372) | (0.386) | (0.383) | ||

EQC/Lag ρ = 0.50 | β ^ 1 | 0.2985 | 0.954 | 0.940 | 0.948 | 0.948 | 0.987 | 0.984 | 0.987 | 0.986 |

(0.074) | (0.288) | (0.274) | (0.280) | (0.281) | (0.379) | (0.363) | (0.376) | (0.373) | ||

β ^ 2 | 0.1933 | 0.952 | 0.937 | 0.940 | 0.943 | 0.990 | 0.986 | 0.986 | 0.986 | |

(0.075) | (0.294) | (0.281) | (0.287) | (0.288) | (0.387) | (0.372) | (0.385) | (0.382) | ||

MA(1)/MA(1) ρ = 0.50 | β ^ 1 | 0.2989 | 0.943 | 0.926 | 0.929 | 0.931 | 0.989 | 0.982 | 0.985 | 0.983 |

(0.058) | (0.222) | (0.211) | (0.215) | (0.216) | (0.291) | (0.280) | (0.289) | (0.287) | ||

β ^ 2 | 0.2022 | 0.952 | 0.932 | 0.935 | 0.936 | 0.994 | 0.984 | 0.991 | 0.989 | |

(0.056) | (0.227) | (0.216) | (0.220) | (0.221) | (0.298) | (0.285) | (0.296) | (0.293) | ||

MA(1)/Lag ρ = 0.50 | β ^ 1 | 0.2990 | 0.944 | 0.928 | 0.929 | 0.929 | 0.986 | 0.980 | 0.985 | 0.985 |

(0.058) | (0.220) | (0.210) | (0.215) | (0.216) | (0.290) | (0.279) | (0.289) | (0.287) | ||

β ^ 2 | 0.2024 | 0.946 | 0.931 | 0.934 | 0.937 | 0.992 | 0.983 | 0.989 | 0.988 | |

(0.056) | (0.225) | (0.215) | (0.220) | (0.221) | (0.296) | (0.285) | (0.295) | (0.293) |

True model/ Working correlation | Estimate | Statistic | Coverage Probabilities | |||||||
---|---|---|---|---|---|---|---|---|---|---|

95% level | 99% level | |||||||||

GEE | EL | EEL | AEL | GEE | EL | EEL | AEL | |||

AR(1)/AR(1) ρ = 0.70 | β ^ 1 | 0.3001 | 0.952 | 0.938 | 0.944 | 0.946 | 0.987 | 0.983 | 0.984 | 0.984 |

(0.070) | (0.279) | (0.265) | (0.270) | (0.272) | (0.367) | (0.351) | (0.364) | (0.361) | ||

β ^ 2 | 0.2009 | 0.950 | 0.928 | 0.938 | 0.941 | 0.988 | 0.986 | 0.989 | 0.988 | |

(0.073) | (0.286) | (0.270) | (0.275) | (0.276) | (0.375) | (0.356) | (0.369) | (0.366) | ||

AR(1)/EQC ρ = 0.70 | β ^ 1 | 0.2997 | 0.911 | 0.932 | 0.935 | 0.936 | 0.973 | 0.975 | 0.978 | 0.978 |

(0.073) | (0.247) | (0.273) | (0.278) | (0.280) | (0.325) | (0.361) | (0.375) | (0.371) | ||

β ^ 2 | 0.1956 | 0.902 | 0.934 | 0.936 | 0.936 | 0.963 | 0.977 | 0.982 | 0.980 | |

(0.076) | (0.252) | (0.278) | (0.284) | (0.286) | (0.331) | (0.368) | (0.381) | (0.378) | ||

AR(1)/Lag ρ = 0.70 | β ^ 1 | 0.3002 | 0.952 | 0.932 | 0.940 | 0.940 | 0.986 | 0.982 | 0.982 | 0.982 |

(0.070) | (0.278) | (0.264) | (0.268) | (0.270) | (0.366) | (0.349) | (0.361) | (0.359) | ||

β ^ 2 | 0.2007 | 0.950 | 0.926 | 0.938 | 0.940 | 0.988 | 0.985 | 0.988 | 0.987 | |

(0.073) | (0.284) | (0.268) | (0.273) | (0.275) | (0.373) | (0.354) | (0.367) | (0.364) | ||

AR(1)/AR(1) ρ = 0.49 | β ^ 1 | 0.2989 | 0.938 | 0.918 | 0.922 | 0.923 | 0.988 | 0.977 | 0.982 | 0.978 |

(0.062) | (0.237) | (0.225) | (0.229) | (0.231) | (0.311) | (0.298) | (0.309) | (0.306) | ||

β ^ 2 | 0.1956 | 0.940 | 0.920 | 0.928 | 0.928 | 0.993 | 0.985 | 0.988 | 0.987 | |

(0.062) | (0.243) | (0.231) | (0.235) | (0.236) | (0.0.319) | (0.305) | (0.316) | (0.313) | ||

AR(1)/EQC ρ = 0.49 | β ^ 1 | 0.2992 | 0.899 | 0.931 | 0.936 | 0.936 | 0.968 | 0.978 | 0.985 | 0.984 |

(0.061) | (0.207) | (0.231) | (0.236) | (0.237) | (0.272) | (0.306) | (0.317) | (0.314) | ||

β ^ 2 | 0.1987 | 0.908 | 0.945 | 0.948 | 0.950 | 0.980 | 0.987 | 0.992 | 0.990 | |

(0.062) | (0.212) | (0.236) | (0.241) | (0.242) | (0.278) | (0.313) | (0.324) | (0.321) | ||

AR(1)/MA(1) ρ = 0.49 | β ^ 1 | 0.2991 | 0.897 | 0.931 | 0.934 | 0.936 | 0.968 | 0.979 | 0.985 | 0.984 |

(0.061) | (0.205) | (0.228) | (0.232) | (0.233) | (0.270) | (0.302) | (0.313) | (0.310) | ||

β ^ 2 | 0.1985 | 0.905 | 0.936 | 0.944 | 0.944 | 0.981 | 0.990 | 0.993 | 0.991 | |

(0.061) | (0.210) | (0.233) | (0.238) | (0.239) | (0.276) | (0.309) | (0.319) | (0.317) | ||

AR(1)/Lag ρ = 0.49 | β ^ 1 | 0.3006 | 0.946 | 0.930 | 0.936 | 0.939 | 0.985 | 0.978 | 0.981 | 0.981 |

(0.066) | (0.256) | (0.246) | (0.250) | (0.252) | (0.337) | (0.325) | (0.337) | (0.334) | ||

β ^ 2 | 0.1978 | 0.952 | 0.931 | 0.938 | 0.940 | 0.990 | 0.980 | 0.985 | 0.985 | |

(0.067) | (0.263) | (0.250) | (0.255) | (0.256) | (0.345) | (0.331) | (0.342) | (0.339) |

True model/ Working correlation | Estimate | Statistic | Coverage Probabilities | |||||||
---|---|---|---|---|---|---|---|---|---|---|

95% level | 99% level | |||||||||

GEE | EL | EEL | AEL | GEE | EL | EEL | AEL | |||

IND/IND ρ = 0.50 | β ^ 1 | 0.2978 | 0.808 | 0.937 | 0.941 | 0.941 | 0.902 | 0.985 | 0.989 | 0.988 |

(0.074) | (0.191) | (0.273) | (0.278) | (0.280) | (0.252) | (0.361) | (0.374) | (0.370) | ||

β ^ 2 | 0.1980 | 0.813 | 0.934 | 0.937 | 0.937 | 0.919 | 0.979 | 0.985 | 0.983 | |

(0.071) | (0.188) | (0.265) | (0.270) | (0.271) | (0.247) | (0.349) | (0.361) | (0.358) | ||

AR(1)/AR(1) ρ = 0.50 | β ^ 1 | 0.2974 | 0.876 | 0.916 | 0.926 | 0.931 | 0.971 | 0.978 | 0.982 | 0.982 |

(0.080) | (0.276) | (0.314) | (0.317) | (0.319) | (0.363) | (0.410) | (0.425) | (0.421) | ||

β ^ 2 | 0.2016 | 0.898 | 0.924 | 0.929 | 0.932 | 0.973 | 0.983 | 0.987 | 0.986 | |

(0.086) | (0.282) | (0.316) | (0.323) | (0.324) | (0.370) | (0.417) | (0.432) | (0.428) | ||

AR(1)/Lag ρ = 0.50 | β ^ 1 | 0.2916 | 0.899 | 0.928 | 0.930 | 0.931 | 0.973 | 0.978 | 0.983 | 0.983 |

(0.085) | (0.282) | (0.310) | (0.316) | (0.318) | (0.371) | (0.408) | (0.423) | (0.419) | ||

β ^ 2 | 0.1978 | 0.952 | 0.931 | 0.938 | 0.940 | 0.990 | 0.980 | 0.985 | 0.985 | |

(0.088) | (0.288) | (0.315) | (0.321) | (0.323) | (0.378) | (0.415) | (0.430) | (0.426) | ||

EQC/EQC ρ = 0.50 | β ^ 1 | 0.2960 | 0.903 | 0.917 | 0.922 | 0.926 | 0.967 | 0.975 | 0.980 | 0.977 |

(0.091) | (0.305) | (0.336) | (0.342) | (0.344) | (0.401) | (0.443) | (0.459) | (0.455) | ||

β ^ 2 | 0.2000 | 0.892 | 0.905 | 0.912 | 0.913 | 0.963 | 0.970 | 0.979 | 0.977 | |

(0.094) | (0.311) | (0.339) | (0.346) | (0.347) | (0.409) | (0.447) | (0.463) | (0.458) | ||

EQC/Lag ρ = 0.50 | β ^ 1 | 0.2957 | 0.901 | 0.917 | 0.923 | 0.924 | 0.967 | 0.974 | 0.978 | 0.976 |

(0.091) | (0.305) | (0.335) | (0.341) | (0.343) | (0.400) | (0.441) | (0.457) | (0.453) | ||

β ^ 2 | 0.2001 | 0.894 | 0.905 | 0.912 | 0.912 | 0.961 | 0.974 | 0.978 | 0.977 | |

(0.095) | (0.311) | (0.338) | (0.344) | (0.346) | (0.409) | (0.445) | (0.461) | (0.457) | ||

MA(1)/MA(1) ρ = 0.50 | β ^ 1 | 0.2980 | 0.859 | 0.928 | 0.933 | 0.935 | 0.949 | 0.980 | 0.983 | 0.983 |

(0.080) | (0.234) | (0.289) | (0.295) | (0.297) | (0.307) | (0.382) | (0.395) | (0.392) | ||

β ^ 2 | 0.1987 | 0.861 | 0.915 | 0.917 | 0.917 | 0.938 | 0.975 | 0.982 | 0.981 | |

(0.080) | (0.239) | (0.289) | (0.295) | (0.296) | (0.314) | (0.381) | (0.395) | (0.391) | ||

MA(1)/Lag ρ = 0.50 | β ^ 1 | 0.2975 | 0.897 | 0.922 | 0.935 | 0.936 | 0.967 | 0.979 | 0.982 | 0.981 |

(0.080) | (0.257) | (0.288) | (0.294) | (0.296) | (0.337) | (0.380) | (0.394) | (0.391) | ||

β ^ 2 | 0.1986 | 0.891 | 0.915 | 0.920 | 0.920 | 0.962 | 0.973 | 0.984 | 0.981 | |

(0.080) | (0.262) | (0.288) | (0.294) | (0.295) | (0.344) | (0.380) | (0.393) | (0.390) |

True model/ Working correlation | Estimate | Statistic | Coverage Probabilities | |||||||
---|---|---|---|---|---|---|---|---|---|---|

95% level | 99% level | |||||||||

GEE | EL | EEL | AEL | GEE | EL | EEL | AEL | |||

IND/IND ρ = 0.50 | β ^ 1 | 0.4010 | 0.955 | 0.947 | 0.950 | 0.951 | 0.989 | 0.988 | 0.988 | 0.988 |

(0.050) | (0.197) | (0.191) | (0.195) | (0.196) | (0.259) | (0.254) | (0.262) | (0.260) | ||

β ^ 2 | 0.5015 | 0.953 | 0.936 | 0.942 | 0.942 | 0.993 | 0.985 | 0.990 | 0.989 | |

(0.051) | (0.197) | (0.191) | (0.195) | (0.196) | (0.259) | (0.253) | (0.262) | (0.260) | ||

AR(1)/AR(1) ρ = 0.50 | β ^ 1 | 0.4019 | 0.949 | 0.944 | 0.950 | 0.952 | 0.990 | 0.989 | 0.990 | 0.990 |

(0.071) | (0.278) | (0.270) | (0.276) | (0.277) | (0.365) | (0.358) | (0.371) | (0.368) | ||

β ^ 2 | 0.5027 | 0.959 | 0.948 | 0.951 | 0.952 | 0.994 | 0.993 | 0.993 | 0.993 | |

(0.068) | (0.278) | (0.270) | (0.275) | (0.277) | (0.365) | (0.358) | (0.371) | (0.367) | ||

AR(1)/Lag ρ = 0.50 | β ^ 1 | 0.4018 | 0.946 | 0.944 | 0.949 | 0.950 | 0.990 | 0.988 | 0.989 | 0.989 |

(0.071) | (0.277) | (0.270) | (0.275) | (0.277) | (0.364) | (0.357) | (0.370) | (0.367) | ||

β ^ 2 | 0.5026 | 0.959 | 0.948 | 0.950 | 0.953 | 0.994 | 0.992 | 0.994 | 0.994 | |

(0.077) | (0.310) | (0.302) | (0.308) | (0.310) | (0.408) | (0.401) | (0.415) | (0.411) | ||

EQC/EQC ρ = 0.50 | β ^ 1 | 0.4016 | 0.949 | 0.942 | 0.949 | 0.951 | 0.993 | 0.986 | 0.988 | 0.987 |

(0.079) | (0.310) | (0.302) | (0.308) | (0.310) | (0.408) | (0.400) | (0.415) | (0.411) | ||

β ^ 2 | 0.5023 | 0.963 | 0.947 | 0.953 | 0.954 | 0.995 | 0.989 | 0.994 | 0.992 | |

(0.091) | (0.307) | (0.332) | (0.338) | (0.340) | (0.404) | (0.437) | (0.453) | (0.448) | ||

EQC/Lag ρ = 0.50 | β ^ 1 | 0.4015 | 0.948 | 0.945 | 0.948 | 0.949 | 0.992 | 0.986 | 0.988 | 0.987 |

(0.079) | (0.310) | (0.301) | (0.307) | (0.309) | (0.407) | (0.399) | (0.414) | (0.410) | ||

β ^ 2 | 0.5024 | 0.961 | 0.947 | 0.953 | 0.954 | 0.996 | 0.989 | 0.994 | 0.994 | |

(0.077) | (0.310) | (0.301) | (0.308) | (0.309) | (0.407) | (0.400) | (0.414) | (0.410) | ||

MA(1)/MA(1) ρ = 0.50 | β ^ 1 | 0.3981 | 0.955 | 0.945 | 0.955 | 0.954 | 0.992 | 0.986 | 0.991 | 0.991 |

(0.065) | (0.254) | (0.247) | (0.252) | (0.253) | (0.334) | (0.327) | (0.338) | (0.335) | ||

β ^ 2 | 0.4974 | 0.958 | 0.944 | 0.948 | 0.951 | 0.995 | 0.989 | 0.992 | 0.990 | |

(0.062) | (0.254) | (0.246) | (0.251) | (0.252) | (0.334) | (0.327) | (0.338) | (0.335) | ||

MA(1)/Lag ρ = 0.50 | β ^ 1 | 0.3981 | 0.956 | 0.942 | 0.951 | 0.953 | 0.994 | 0.986 | 0.989 | 0.987 |

(0.065) | (0.252) | (0.246) | (0.251) | (0.253) | (0.332) | (0.326) | (0.338) | (0.335) | ||

β ^ 2 | 0.4974 | 0.957 | 0.943 | 0.948 | 0.950 | 0.994 | 0.989 | 0.991 | 0.991 | |

(0.062) | (0.252) | (0.246) | (0.251) | (0.252) | (0.332) | (0.326) | (0.338) | (0.335) |

· AR(1): ϵ i t = ρ ϵ i , t − 1 + a i t , t = 1,2 , 3,4 ,

· EQC: ϵ i t = ρ a i ,0 + a i t , t = 1,2 , 3,4 ,

· MA(1): ϵ i t = ρ a i , t − 1 + a i t , t = 1,2 , 3,4 .

However, the confidence regions for the GEE are constructed under the normality assumption.

When the model is misspecified, the EL, EEL, and AEL outperform the GEE. For example, in the AR(1)/Lag case in

In this section, we illustrate the applicability of our proposed method to two real-world examples.

We consider longitudinal health care utilization data [

var ( y ) = μ + α μ

and

var ( y ) = μ + α μ 2 .

Thus, the variance function is different from that of the Poisson model, var ( y ) = μ . To confirm the over-dispersion, we test H 0 : α = 0 against

True model/ Working correlation | Estimate | Statistic | Coverage Probabilities | |||||||
---|---|---|---|---|---|---|---|---|---|---|

95% level | 99% level | |||||||||

GEE | EL | EEL | AEL | GEE | EL | EEL | AEL | |||

IND/IND ρ = 0.50 | β ^ 1 | 0.4003 | 0.947 | 0.940 | 0.945 | 0.945 | 0.986 | 0.986 | 0.988 | 0.988 |

(0.051) | (0.196) | (0.194) | (0.198) | (0.199) | (0.258) | (0.257) | (0.266) | (0.263) | ||

β ^ 2 | 0.4996 | 0.948 | 0.946 | 0.949 | 0.949 | 0.985 | 0.986 | 0.988 | 0.988 | |

(0.051) | (0.196) | (0.195) | (0.199) | (0.199) | (0.258) | (0.258) | (0.267) | (0.264) | ||

AR(1)/AR(1) ρ = 0.50 | β ^ 1 | 0.3998 | 0.939 | 0.926 | 0.936 | 0.937 | 0.989 | 0.987 | 0.989 | 0.989 |

(0.043) | (0.160) | (0.158) | (0.161) | (0.162) | (0.211) | (0.209) | (0.216) | (0.214) | ||

β ^ 2 | 0.4980 | 0.959 | 0.949 | 0.954 | 0.955 | 0.992 | 0.992 | 0.993 | 0.993 | |

(0.040) | (0.160) | (0.159) | (0.162) | (0.163) | (0.211) | (0.210) | (0.218) | (0.215) | ||

AR(1)/Lag ρ = 0.50 | β ^ 1 | 0.3998 | 0.935 | 0.923 | 0.932 | 0.934 | 0.988 | 0.986 | 0.989 | 0.988 |

(0.043) | (0.159) | (0.157) | (0.160) | (0.161) | (0.209) | (0.208) | (0.216) | (0.214) | ||

β ^ 2 | 0.4979 | 0.958 | 0.946 | 0.955 | 0.955 | 0.991 | 0.993 | 0.994 | 0.993 | |

(0.040) | (0.159) | (0.158) | (0.161) | (0.162) | (0.209) | (0.209) | (0.217) | (0.215) | ||

EQC/EQC ρ = 0.50 | β ^ 1 | 0.4018 | 0.954 | 0.947 | 0.955 | 0.956 | 0.992 | 0.988 | 0.989 | 0.989 |

(0.040) | (0.155) | (0.153) | (0.156) | (0.157) | (0.204) | (0.203) | (0.210) | (0.208) | ||

β ^ 2 | 0.5001 | 0.953 | 0.947 | 0.952 | 0.954 | 0.989 | 0.989 | 0.991 | 0.990 | |

(0.040) | (0.155) | (0.153) | (0.156) | (0.157) | (0.204) | (0.203) | (0.210) | (0.208) | ||

EQC/Lag ρ = 0.50 | β ^ 1 | 0.4019 | 0.947 | 0.945 | 0.950 | 0.951 | 0.988 | 0.985 | 0.990 | 0.990 |

(0.040) | (0.154) | (0.153) | (0.156) | (0.156) | (0.202) | (0.202) | (0.209) | (0.207) | ||

β ^ 2 | 0.5011 | 0.946 | 0.938 | 0.944 | 0.945 | 0.988 | 0.987 | 0.989 | 0.989 | |

(0.040) | (0.154) | (0.153) | (0.156) | (0.157) | (0.202) | (0.202) | (0.209) | (0.207) | ||

MA(1)/MA(1) ρ = 0.50 | β ^ 1 | 0.4000 | 0.942 | 0.936 | 0.946 | 0.947 | 0.992 | 0.993 | 0.995 | 0.993 |

(0.043) | (0.165) | (0.166) | (0.169) | (0.170) | (0.217) | (0.220) | (0.227) | (0.225) | ||

β ^ 2 | 0.5027 | 0.938 | 0.931 | 0.939 | 0.939 | 0.987 | 0.979 | 0.986 | 0.983 | |

(0.044) | (0.165) | (0.165) | (0.168) | (0.169) | (0.217) | (0.218) | (0.226) | (0.223) | ||

MA(1)/Lag ρ = 0.50 | β ^ 1 | 0.4004 | 0.926 | 0.926 | 0.931 | 0.932 | 0.982 | 0.976 | 0.983 | 0.981 |

(0.038) | (0.138) | (0.136) | (0.139) | (0.140) | (0.181) | (0.180) | (0.187) | (0.185) | ||

β ^ 2 | 0.5001 | 0.943 | 0.950 | 0.957 | 0.957 | 0.992 | 0.988 | 0.990 | 0.990 | |

(0.035) | (0.138) | (0.137) | (0.140) | (0.141) | (0.181) | (0.182) | (0.188) | (0.187) |

True model/ Working correlation | Estimate | Statistic | Coverage Probabilities | |||||||
---|---|---|---|---|---|---|---|---|---|---|

95% level | 99% level | |||||||||

GEE | EL | EEL | AEL | GEE | EL | EEL | AEL | |||

IND/IND ρ = 0.50 | β ^ 1 | 0.3971 | 0.836 | 0.906 | 0.909 | 0.913 | 0.930 | 0.963 | 0.975 | 0.971 |

(0.103) | (0.281) | (0.357) | (0.367) | (0.369) | (0.369) | (0.479) | (0.512) | (0.501) | ||

β ^ 2 | 0.4988 | 0.838 | 0.904 | 0.926 | 0.926 | 0.928 | 0.966 | 0.978 | 0.976 | |

(0.101) | (0.281) | (0.350) | (0.370) | (0.372) | (0.369) | (0.473) | (0.513) | (0.502) | ||

AR(1)/AR(1) ρ = 0.50 | β ^ 1 | 0.3954 | 0.805 | 0.923 | 0.934 | 0.934 | 0.910 | 0.971 | 0.973 | 0.977 |

(0.157) | (0.397) | (0.587) | (0.610) | (0.613) | (0.522) | (0.783) | (0.827) | (0.814) | ||

β ^ 2 | 0.4986 | 0.807 | 0.935 | 0.942 | 0.944 | 0.913 | 0.975 | 0.983 | 0.980 | |

(0.154) | (0.397) | (0.593) | (0.610) | (0.613) | (0.522) | (0.784) | (0.827) | (0.815) | ||

AR(1)/Lag ρ = 0.50 | β ^ 1 | 0.3949 | 0.790 | 0.918 | 0.931 | 0.932 | 0.902 | 0.971 | 0.980 | 0.977 |

(0.156) | (0.391) | (0.581) | (0.601) | (0.604) | (0.513) | (0.772) | (0.816) | (0.803) | ||

β ^ 2 | 0.4983 | 0.801 | 0.924 | 0.937 | 0.937 | 0.911 | 0.977 | 0.986 | 0.983 | |

(0.152) | (0.391) | (0.581) | (0.602) | (0.604) | (0.513) | (0.714) | (0.818) | (0.805) | ||

EQC/EQC ρ = 0.50 | β ^ 1 | 0.3970 | 0.885 | 0.927 | 0.940 | 0.943 | 0.953 | 0.981 | 0.989 | 0.987 |

(0.143) | (0.444) | (0.570) | (0.590) | (0.593) | (0.584) | (0.759) | (0.803) | (0.790) | ||

β ^ 2 | 0.4935 | 0.893 | 0.943 | 0.955 | 0.955 | 0.960 | 0.984 | 0.991 | 0.990 | |

(0.140) | (0.444) | (0.565) | (0.585) | (0.588) | (0.584) | (0.753) | (0.798) | (0.785) | ||

EQC/Lag ρ = 0.50 | β ^ 1 | 0.3973 | 0.811 | 0.928 | 0.935 | 0.941 | 0.902 | 0.981 | 0.989 | 0.987 |

(0.143) | (0.371) | (0.568) | (0.588) | (0.591) | (0.487) | (0.756) | (0.800) | (0.787) | ||

β ^ 2 | 0.4940 | 0.812 | 0.941 | 0.953 | 0.956 | 0.928 | 0.986 | 0.990 | 0.989 | |

(0.140) | (0.371) | (0.562) | (0.583) | (0.586) | (0.487) | (0.750) | (0.795) | (0.782) | ||

MA(1)/MA(1) ρ = 0.50 | β ^ 1 | 0.3996 | 0.802 | 0.935 | 0.941 | 0.942 | 0.897 | 0.976 | 0.984 | 0.981 |

(0.144) | (0.363) | (0.563) | (0.584) | (0.587) | (0.477) | (0.750) | (0.796) | (0.782) | ||

β ^ 2 | 0.4974 | 0.808 | 0.949 | 0.955 | 0.955 | 0.917 | 0.985 | 0.991 | 0.989 | |

(0.139) | (0.363) | (0.560) | (0.580) | (0.583) | (0.477) | (0.747) | (0.792) | (0.779) | ||

MA(1)/Lag ρ = 0.50 | β ^ 1 | 0.3997 | 0.802 | 0.932 | 0.943 | 0.943 | 0.906 | 0.977 | 0.983 | 0.982 |

(0.144) | (0.366) | (0.561) | (0.582) | (0.585) | (0.481) | (0.748) | (0.794) | (0.780) | ||

β ^ 2 | 0.4980 | 0.808 | 0.948 | 0.960 | 0.960 | 0.921 | 0.985 | 0.991 | 0.989 | |

(0.138) | (0.366) | (0.558) | (0.578) | (0.581) | (0.481) | (0.745) | (0.791) | (0.777) |

True model/ Working correlation | Estimate | Statistic | Coverage Probabilities | |||||||
---|---|---|---|---|---|---|---|---|---|---|

95% level | 99% level | |||||||||

GEE | EL | EEL | AEL | GEE | EL | EEL | AEL | |||

IND/IND ρ = 0.50 | β ^ 1 | 0.3992 | 0.843 | 0.930 | 0.943 | 0.943 | 0.945 | 0.983 | 0.990 | 0.989 |

(0.098) | (0.279) | (0.377) | (0.392) | (0.393) | (0.366) | (0.491) | (0.544) | (0.530) | ||

β ^ 2 | 0.4936 | 0.846 | 0.926 | 0.934 | 0.935 | 0.936 | 0.979 | 0.987 | 0.985 | |

(0.098) | (0.279) | (0.377) | (0.392) | (0.394) | (0.366) | (0.489) | (0.544) | (0.531) | ||

AR(1)/AR(1) ρ = 0.50 | β ^ 1 | 0.3949 | 0.776 | 0.925 | 0.930 | 0.932 | 0.890 | 0.972 | 0.983 | 0.979 |

(0.094) | (0.221) | (0.354) | (0.368) | (0.369) | (0.291) | (0.477) | (0.512) | (0.500) | ||

β ^ 2 | 0.4982 | 0.777 | 0.933 | 0.942 | 0.945 | 0.887 | 0.984 | 0.988 | 0.986 | |

(0.092) | (0.221) | (0.353) | (0.368) | (0.369) | (0.291) | (0.476) | (0.511) | (0.499) | ||

AR(1)/Lag ρ = 0.50 | β ^ 1 | 0.3990 | 0.750 | 0.919 | 0.931 | 0.933 | 0.870 | 0.981 | 0.988 | 0.985 |

(0.094) | (0.214) | (0.355) | (0.369) | (0.371) | (0.281) | (0.477) | (0.513) | (0.500) | ||

β ^ 2 | 0.4952 | 0.762 | 0.917 | 0.927 | 0.928 | 0.864 | 0.972 | 0.982 | 0.979 | |

(0.096) | (0.214) | (0.355) | (0.369) | (0.371) | (0.281) | (0.478) | (0.514) | (0.501) | ||

EQC/EQC ρ = 0.50 | β ^ 1 | 0.3980 | 0.772 | 0.928 | 0.938 | 0.938 | 0.896 | 0.979 | 0.986 | 0.985 |

(0.107) | (0.256) | (0.408) | (0.424) | (0.426) | (0.336) | (0.548) | (0.588) | (0.574) | ||

β ^ 2 | 0.5018 | 0.778 | 0.923 | 0.934 | 0.936 | 0.890 | 0.980 | 0.988 | 0.984 | |

(0.106) | (0.256) | (0.404) | (0.421) | (0.422) | (0.336) | (0.543) | (0.583) | (0.569) | ||

EQC/Lag ρ = 0.50 | β ^ 1 | 0.3981 | 0.751 | 0.921 | 0.932 | 0.934 | 0.888 | 0.979 | 0.983 | 0.983 |

(0.109) | (0.253) | (0.405) | (0.421) | (0.423) | (0.332) | (0.544) | (0.585) | (0.570) | ||

β ^ 2 | 0.5026 | 0.761 | 0.918 | 0.937 | 0.938 | 0.880 | 0.978 | 0.986 | 0.983 | |

(0.107) | (0.253) | (0.402) | (0.419) | (0.420) | (0.332) | (0.541) | (0.581) | (0.566) | ||

MA(1)/MA(1) ρ = 0.50 | β ^ 1 | 0.3961 | 0.709 | 0.920 | 0.930 | 0.931 | 0.827 | 0.980 | 0.989 | 0.987 |

(0.100) | (0.213) | (0.369) | (0.385) | (0.386) | (0.274) | (0.497) | (0.534) | (0.521) | ||

β ^ 2 | 0.5009 | 0.728 | 0.926 | 0.941 | 0.941 | 0.834 | 0.983 | 0.991 | 0.989 | |

(0.100) | (0.213) | (0.369) | (0.385) | (0.386) | (0.274) | (0.497) | (0.534) | (0.522) | ||

MA(1)/Lag ρ = 0.50 | β ^ 1 | 0.3972 | 0.749 | 0.913 | 0.929 | 0.930 | 0.866 | 0.981 | 0.989 | 0.987 |

(0.096) | (0.222) | (0.356) | (0.370) | (0.372) | (0.292) | (0.478) | (0.514) | (0.502) | ||

β ^ 2 | 0.5002 | 0.762 | 0.929 | 0.938 | 0.939 | 0.872 | 0.981 | 0.991 | 0.989 | |

(0.096) | (0.222) | (0.356) | (0.370) | (0.372) | (0.292) | (0.478) | (0.515) | (0.502) |

H a : α > 0 using the likelihood ratio test. The result confirms the presence of over-dispersion in both variance function models.

Our analysis used the GEE with a working correlation matrix (AR(1), EQC, MA(1), or lag correlation) and our EL approach.

Covariates | Estimate | 95% Confidence Interval | |
---|---|---|---|

GEE | EL | ||

Working Correlation: AR(1) Gender effect ( β ^ 1 ) Chronic effect ( β ^ 2 ) Education effect ( β ^ 3 ) Age effect ( β ^ 4 ) | −0.1929 0.1668 −0.4738 0.0308 | (−0.313, −0.073) (0.177, 0.216) (−0.624, −0.324) (0.029, 0.033) | (−0.421, 0.020) (0.094, 0.241) (−0.768, −0.180) (0.028, 0.033) |

Working Correlation: EQC Gender effect ( β ^ 1 ) Chronic effect ( β ^ 1 ) Education effect ( β ^ 3 ) Age effect ( β ^ 4 ) | −0.1772 0.1681 −0.4354 0.0302 | (−0.306, −0.048) (0.115, 0.222) (−0.597, −0.274) (0.028, 0.033) | (−0.407, 0.034) (0.095, 0.237) (−0.726, −0.146) (0.027, 0.033) |

Working Correlation: MA(1) Gender effect ( β ^ 1 ) Chronic effect ( β ^ 2 ) Education effect ( β ^ 3 ) Age effect ( β ^ 4 ) | −0.1922 0.1669 −0.4720 0.0308 | (−0.299, −0.086) (0.123, 0.211) (−0.605, −0.339) (0.029, 0.033) | (−0.421, 0.021) (0.094, 0.241) (−0.766, −0.179) (0.028, 0.033) |

Working Correlation: Lag Gender effect ( β ^ 1 ) Chronic effect ( β ^ 2 ) Education effect ( β ^ 3 ) Age effect ( β ^ 4 ) | −0.1819 0.1677 0.4469 0.0304 | (−0.311, −0.053) (0.114, 0.221) (−0.608, −0.286) (0.028, 0.033) | (−0.411, 0.029) (0.095, 0.238) (−0.738, −0.156) (0.027, 0.033) |

in our performance analysis. We conclude that the EL approach is more appropriate for this data set, and the significant variables identified by this approach are more reliable.

This data set contains 2376 observations of the CD4 cell counts of k = 369 men infected with the HIV virus [

This data set has the subject-specific evolution over time of the CD4 cell counts with and without drug use. The cell counts are right-skewed, so the analysis was conducted on square-root transformed CD4 cell counts whose distribution is more nearly Gaussian. Tables 9-11 summarize the analysis for the AR(1), EQC, and lag working correlations. The GEE indicates that SMOKE, DRUG, SEXP, AGE × SEXP, SMOKE × DRUG, SMOKE × SEXP, and DRUG × SEXP are significant. Under EQC, AGE × SMOKE and AGE × DRUG are also significant. The EL selects SMOKE, DRUG, SEXP, and DRUG × SEX. Under EQC and lag AGE × SEXP is also significant. The GEE approach is sensitive to the choice of correlation structure. In this real data set, the true correlation structure is unknown, so the lag correlation approach is appropriate since it can accommodate all three correlation structures. The Shapiro-Wilk test shows that the square-root transformed CD4 cell counts are not normally distributed. The GEE-based method is, therefore, not appropriate. We, therefore, conclude that the EL is a better choice.

Variable | Method | |
---|---|---|

GEE | EL | |

INTERCEPT AGE SMOKE DRUG SEXP AGE × SMOKE AGE × DRUG AGE × SEXP SMOKE × DRUG SMOKE × SEXP DRUG × SEXP | 25.37 (25.25, 25.49) −0.001 (−0.014, 0.012) 0.938 (0.864, 1.012) 0.716 (0.597, 0.834) 0.390 (0.365, 0.414) −0.001 (−0.007, 0.004) 0.001 (−0.013, 0.013) 0.008 (0.006, 0.009) −0.242 (−0.315, −0.169) 0.041 (0.033, 0.049) −0.270 (−0.295, −0.245) | 25.37 (24.97, 25.77) −0.001 (−0.060, 0.060) 0.938 (0.669, 1.211) 0.716 (0.316, 1.115) 0.390 (0.306, 0.471) −0.001 (−0.032, 0.031) 0.001 (−0.069, 0.071) 0.008 (−0.003, 0.016) −0.242 (−0.500, 0.023) 0.041 (−0.005, 0.089) −0.270 (−0.358, −0.183) |

Variable | Method | |
---|---|---|

GEE | EL | |

INTERCEPT AGE SMOKE DRUG SEXP AGE × SMOKE AGE × DRUG AGE × SEXP SMOKE × DRUG SMOKE × SEXP DRUG × SEXP | 25.10 (24.96, 25.25) −0.023 (−0.037, 0.008) 1.241 (1.161, 1.322) 1.132 (1.006, 1.257) 0.545 (0.521, 0.569) −0.011 (−0.016, −0.006) 0.036 (0.022, 0.050) 0.017 (0.015, 0.018) −0.398 (−0.477, −0.319) 0.038 (0.030, 0.045) −0.184 (−0.209, −0.159) | 25.10 (24.70, 25.50) −0.023 (−0.085, 0.043) 1.241 (0.972, 1.515) 1.132 (0.732, 1.532) 0.545 (0.459, 0.633) −0.011 (−0.044, 0.023) 0.036 (−0.031, 0.106) 0.017 (0.007, 0.027) −0.398 (−0.650, 0.131) 0.038 (−0.010, 0.091) −0.184 (−0.274, −0.091) |

Variable | Method | |
---|---|---|

GEE | EL | |

INTERCEPT AGE SMOKE DRUG SEXP AGE × SMOKE AGE × DRUG AGE × SEXP SMOKE × DRUG SMOKE × SEXP DRUG × SEXP | 25.35 (25.22, 25.46) −0.001 (−0.015, 0.012) 0.942 (0.867, 1.016) 0.727 (0.608, 0.845) 0.389 (0.364, 0.414) −0.002 (−0.007, 0.003) 0.002 (−0.011, 0.015) 0.007 (0.005, 0.009) −0.233 (−0.306,−0.160) 0.042 (0.035, 0.050) −0.268 (−0.356,−0.182) | 25.35 (25.34, 25.36) −0.001 (−0.061, 0.059) 0.942 (0.673, 1.215) 0.727 (0.327, 1.127) 0.389 (0.305, 0.470) −0.002 (−0.032, 0.030) 0.002 (−0.067, 0.072) 0.007 (−0.003, 0.016) −0.233 (−0.490, 0.032) 0.042 (−0.004, 0.090) −0.268 (−0.356,−0.182) |

Longitudinal data modelling using the GEE approach assumes a working correlation model for the within-subject correlation of the responses. When the working correlation is incorrectly specified, the GEE based estimates are not necessarily consistent and may lose efficiency. Any misspecification can cause estimates based on marginal models to be inefficient and misleading conclusions. Also, the construction of a confidence region and hypothesis testing are based on asymptotic normality, which may not hold since the finite-sample distribution may not be symmetric.

Taking these issues into account, we have proposed an EL-based longitudinal modelling based on a data-driven likelihood ratio approach sharing many of the properties of the parametric likelihood. We do not need to specify the complete parametric distribution to perform the inference. We can, therefore, use likelihood methods without assuming that the data come from a known family of distributions. We defined the subject-wise profile EL based on a set of GEEs. The estimation and confidence region construction using the EL approach are proposed, which has advantages over other methods such as those based on normal approximations. We introduced the adjusted EL to avoid any computational issues, which improve the coverage probabilities. A major advantage of EI is that involves no prior assumptions about the shape of an EL-based confidence region, which is data-driven. The construction of the confidence region based on the EL method does not involve any variance estimation.

The proposed approach yields more efficient estimators than the conventional GEE approach and achieves the same asymptotic properties as [

The authors’ research was supported by grants from Natural Sciences & Engineering Research Council of Canada and Canadian Institute of Health Research.

The authors declare no conflicts of interest regarding the publication of this paper.

Nadarajah, T., Variyath, A.M. and Loredo-Osti, J.C. (2020) Empirical Likelihood Based Longitudinal Data Analysis. Open Journal of Statistics, 10, 611-639. https://doi.org/10.4236/ojs.2020.104037