Marginal analysis of multiple outcomes with informative cluster size

Abstract

In surveillance studies of periodontal disease, the relationship between disease and other health and socioeconomic conditions is of key interest. To determine whether a patient has periodontal disease, multiple clinical measurements (eg, clinical attachment loss, alveolar bone loss, and tooth mobility) are taken at the tooth-level. Researchers often create a composite outcome from these measurements or analyze each outcome separately. Moreover, patients have varying number of teeth, with those who are more prone to the disease having fewer teeth compared to those with good oral health. Such dependence between the outcome of interest and cluster size (number of teeth) is called informative cluster size and results obtained from fitting conventional marginal models can be biased. We propose a novel method to jointly analyze multiple correlated binary outcomes for clustered data with informative cluster size using the class of generalized estimating equations (GEE) with cluster-specific weights. We compare our proposed multivariate outcome cluster-weighted GEE results to those from the convectional GEE using the baseline data from Veterans Affairs Dental Longitudinal Study. In an extensive simulation study, we show that our proposed method yields estimates with minimal relative biases and excellent coverage probabilities.

1 |. INTRODUCTION

Periodontal disease is a serious gum and bone infection that affects almost half of the adult population in the United States (Eke et al., 2012). Many physical and socioeconomic risk factors such as metabolic syndrome (MetS), smoking, and level of education have shown to be associated with periodontal disease (Kaye et al., 2016). The progression of disease is assessed by multiple clinical measurements made on each tooth. Increasing probing pocket depth (PPD), clinical attachment loss (CAL), radiographic alveolar bone loss (ABL), and tooth mobility (Mobil) scores are all indicators of worse prognosis of periodontal disease. Because a universal definition for periodontal disease does not exist, researchers who study the prognosis of periodontal disease use a combination of the four outcomes to indicate the status of the disease, or build four separate models to describe the association between each outcome and the exposures of interest. The latter method is less efficient than analyzing all outcomes simultaneously because it ignores the extra information given by the correlation between the outcomes (Teixeira-Pinto and Normand, 2009). Indeed, PPD, CAL, ABL, and Mobil are all correlated with each other. Marginal models using generalized estimating equations (GEEs; Liang and Zeger, 1986) provide an attractive alternative to simultaneously analyze multiple clustered outcomes, and have been studied by several authors.

Lefkopoulou et al. (1989) used GEE to analyze multiple correlated binary outcomes in the cross-sectional setting. Following Liang and Zeger (1986), they used the moment estimators approach to obtain estimates of the working correlation matrix that consists of correlations between the outcomes. Lefkopoulou et al. (1989) argued that when the effects of the predictor on each outcome are similar, assuming a common coefficient estimate for the predictor across all outcomes is reasonable. Also within the GEE context, Lipsitz et al. (1991) used the odds ratio parameterization over the correlation coefficient to describe the association between the pairwise longitudinally correlated binary outcomes. They argue that the odds ratio is a more intuitive choice to describe the pairwise association between binary variables, but demonstrate that the two types of models (odds ratio vs correlation coefficient parameterization) yield similar efficiency in estimating the common marginal predictor. The marginal models of each of the above studies are based on the logit link function. Lu and Tilley (2001) used GEE to estimate the common relative risk for multiple correlated binary outcomes based on the log link function for clinical trials. Finally, several authors (Catalano and Ryan, 1992; Lin et al., 2010) have studied the marginal analysis of multiple correlated outcomes of varying distributions (eg, one binary outcome and one continuous outcome) using GEE.

The method of GEE is useful in analyzing clustered data because it is robust to the choice of the working correlation matrix (Liang and Zeger, 1986) and, as mentioned above, can accommodate combined analysis of multiple outcomes. In periodontal disease studies, data are subject to the issue of informative cluster size (ICS), which describes a situation when the outcome of interest (CAL, for example) is related to cluster size (number of teeth per patient). However, to date, few have described the use of GEEs to model multiple clustered outcomes while taking into consideration the presence of ICS. For subject-specific inference, Hwang and Pennell (2018) introduced a Bayesian joint modeling approach for binary and continuous outcomes with ICS. They account for ICS by modeling cluster size using a Poisson regression model with a cluster-specific random effect that is shared with the outcome variables. However, the focus of our paper is on modeling marginal (or population-average) inferences through estimation via GEE rather than conditional (or subject-specific) inferences using random-effects models.

Because one of the assumptions in GEE is the independence between the outcome and cluster size, when ICS is present in the data, the method of conventional GEE has been shown to produce biased parameter estimates (Hoffman et al., 2001). One approach to overcome this challenge is to include weights that are the inverse of cluster size in the GEE score function (Williamson et al., 2003; Wang et al., 2011). For the cross-sectional setting, Williamson et al. (2003) proposed cluster-weighted GEE (CWGEE) for binary outcomes that involves taking the weighted average, where the weight is the inverse of cluster size, of the GEE during the estimation process while using an independence working correlation matrix. Williamson et al. (2003) showed that the inclusion of the cluster-size weights in the GEE corrects the bias introduced by ICS. Wang et al. (2011) extended CWGEE to the longitudinal setting for continuous outcomes. When the outcome is repeatedly measured over time, the association parameter between the visits within each unit (tooth) needs to be estimated. They used the method of quasi-least squares (Chaganty, 1997; Chaganty and Shults, 1999) to estimate the association parameter and the model parameter estimates.

In this paper, we build on the CWGEE method by Wang et al. (2011) to analyze multiple binary outcomes measured on each tooth simultaneously in a cross-sectional framework that has not been described in the statistical literature to date. The pairwise correlations between the outcomes will be estimated using the method of quasi-least squares. Our goal is to simultaneously describe the relationship between multiple binary outcomes measured on each tooth per patient and predictors measured at the patient level. The rest of the paper is organized as follows. In Section 2, we introduce our proposed multivariate CWGEE approach in more detail. In Section 3, we illustrate how the results from our proposed method are different from the conventional GEE approach when applied to a real data set. An extensive simulation is presented in Section 4. We conclude with a discussion in Section 5.

2 |. METHOD

Suppose the data consists of i = 1, …, N patients, each with j = 1, …, n_i teeth. For each tooth j of the ith patient, k = 1, …, K binary outcomes are recorded. Let Y_ijk be the kth outcome on the jth tooth of the ith patient, for k = 1, …, K, j = 1, …, n_i, and i = 1, …, N. Similarly, let X_i be the 1 × P row vector of patient-specific covariates for the ith patient. Here, it is assumed that the covariates are consistent across all n_i teeth within each patient i; however, each of the K outcomes are allowed to vary between each tooth j of patient i. Our goal is to describe the relationship between μ_ijk = (Y_ijk) = Pr(Y_ijk = 1) and X_i. The most general approach presented by Lefkopoulou et al. (1989) is to use a logistic regression model with a unique intercept, α_k, and P × 1 vector of regression coefficients, β_k, for each outcome, k = 1, …, K, that is,

$$\text{logit}\left({\mu }_{ijk}\right)=a_k+X_i{\beta }_k.$$ (1)

For the pth covariate, the outcome-unique fixed regression coefficient can also be expressed as ${\beta }_k^p={\beta }_1^p+{\beta }_{1k}^p$ for k = 2, …, K, and ${\beta }_1^p$ as the pth coefficient for outcome k = 1. If all ${\beta }_1^p,\dots ,{\beta }_K^p$ are similar, then assuming a common regression coefficient, β^p, for all K outcomes for the pth covariate may be reasonable. The decision to use a common regression coefficient across the K outcomes for the pth covariate can be based on the hypothesis test $H_0:{\beta }_{12}^p=\cdots ={\beta }_{1K}^p=0$ versus H₁: at least one of ${\beta }_{1k}^p\ne 0$, using the multivariate Wald statistic that has a χ² distribution with K − 1 degrees of freedom (Fitzmaurice et al., 2004). As a result, a more parsimonious model can be used:

$$\text{logit}\left({\mu }_{ijk}\right)=a_k+X_i^0{\beta }^0+X_i^1{\beta }_k^1,$$ (2)

where $X_i^0$ is the 1 × Q row vector of covariates with a common set of coefficients β⁰ across the K outcomes, and $X_i^1$ is the 1 × (P − Q) row vector of covariates with unique sets of coefficients ${\beta }_k^1$ across the K outcomes. Using matrix notation, for each tooth j of patient i, we have a response vector Y _ij = (Y_ij1, …, Y_ijK) of length K. Similarly μ_ij = (μ_ij1, …, μ_ijK). Let V _ij be the diagonal matrix containing the variance of elements of Y _ij, where var(Y_ijk) = μ_ijk(1 − μ_ijk) for the binary case, and let R_ij be the matrix of correlations, which is assumed to be a function of α, between the pairwise elements of Y _ij. Further details on matrix R_ij are provided in Section 2.2. Let $Z_{ij}=V_{ij}^{-1/2}\left(Y_{ij}-{\mu }_{ij}\right)$. The method of quasi-least squares (QLS) introduced by Chaganty (1997) is an alternative approach to estimating parameters via two sets of estimating equations, by partially minimizing the generalized sum of squares for error with respect to β and α. The estimating equation for β is equivalent to the familiar GEE introduced by Liang and Zeger (1986). The advantage of QLS is the ability to more easily estimate complex correlation structures than GEE (Shults and Ardythe, 2002). Here, we construct cluster-weighted generalized sum of squares for error as follows:

$$Q_W(\beta ,\alpha )=\sum _{i=1}^N\frac{1}{n_i}\sum _{j=1}^{n_i}{Z}_{ij}^{\prime }{R}_{ij}^{-1}{Z}_{ij}.$$ (3)

Our key objective is to find estimates of β and α that minimize this estimating equation. By weighting the estimating equation with the inverse of cluster size, each cluster contributes to the inference equally regardless of cluster size.

2.1 |. Estimation of β

Taking the partial derivative of Equation (3) with respect to β and setting it equal to 0, we obtain the typical CWGEE estimating equation originally proposed by Williamson et al. (2003):

$$\frac{\partial Q_W(\beta ,\alpha )}{\partial \beta }=\sum _{i=1}^N\frac{1}{n_i}\sum _{j=1}^{n_i}{D}_{ij}^{\prime }{W}_{ij}^{-1}\left(Y_{ij}-{\mu }_{ij}\right)=0,$$ (4)

where D_ij = ∂μ_ij/∂β is the gradient matrix and $W_{ij}=V_{ij}^{1/2}{R}_{ij}{V}_{ij}^{1/2}$ is the working variance-covariance matrix. Williamson et al. (2003) have shown that estimates obtained from CWGEE are consistent and asymptotically normal (we present the robust sandwich estimator of the variance of the multivariate outcome CWGEE in Section 2.3). Importantly, however, interpretations of the parameter estimates obtained from unweighted GEE and CWGEE differ when ICS is present. Inference of estimates obtained from unweighted GEE will be based on the population of all units (all teeth) ignoring cluster memberships, whereas those from CWGEE will be based on a typical unit (tooth) of a typical cluster (patient). With our proposed multivariate CWGEE, the inference will be based on the average periodontal disease experience of a typical tooth of a typical patient.

2.2 |. Specification and estimation of correlation matrix R_ij

The matrix R_ij contains the correlation between each pair of the K outcomes. Below we describe two correlation structures commonly used for multiple correlated outcomes: the unstructured and exchangeable correlation structures. Note that we can also use the independence correlation structure, which is represented by a K × K identity matrix. This assumes no correlation between any pairs of the K outcomes.

2.2.1 |. Unstructured correlation structure

Let α_kl be the correlation between outcomes Y_ijk and Y_ijl for k ≠ l, k, l = 1, …, K, where α_kl = α_lk. The most general form of R_ij has the unstructured working correlation structure with K(K − 1)/2 unknown correlation parameters:

$$R_{ij}=\left(\begin{array}{cccc}1& {\alpha }_{12}& \cdots & {\alpha }_{1K}\\ {\alpha }_{21}& 1& & ⋮\\ ⋮& & \ddots & {\alpha }_{(K-1)K}\\ {\alpha }_{K1}& \cdots & {\alpha }_{(K-1)K}& 1\end{array}\right).$$

To estimate R_ij, we want to minimize Equation (3) with respect to R_ij. Following Example 4.4 in Chaganty (1997) and section 3.1 in Chaganty and Shults (1999):

$$\begin{gathered} \underset{R}{\text{min}}\sum _{i=1}^N\frac{1}{n_i}\sum _{j=1}^{n_i}{Z}_{ij}^{\prime }{R}_{ij}^{-1}{Z}_{ij}=\underset{R}{\text{min}}\text{ tr}\left(\sum _{i=1}^N\frac{1}{n_i}\sum _{j=1}^{n_i}{Z}_{ij}^{\prime }{R}_{ij}^{-1}{Z}_{ij}\right) \\ =\underset{R}{\text{min}}\text{ tr}\left(\sum _{i=1}^N\frac{1}{n_i}\sum _{j=1}^{n_i}{Z}_{ij}{Z}_{ij}^{\prime }{R}_{ij}^{-1}\right) \\ =\underset{R}{\text{min}}\text{ tr}\left(Z{R}_{ij}^{-1}\right), \end{gathered}$$ (5)

where $Z={\sum }_{i=1}^N\frac{1}{n_i}{\sum }_{j=1}^{n_i}{Z}_{ij}{Z}_{ij}^{\prime }$ is a K × K positive-definite matrix. Chaganty (1997) describes the following iterative procedure to estimate ${\widehat{R}}_{ij}$ that minimizes (5):

1. Select any diagonal positive-definite matrix Λ₀ as the initial matrix.

2. Iterate ${\Lambda }_k=\text{diag}\left({\Lambda }_{k-1}^{1/2}Z{\Lambda }_{k-1}^{1/2}\right)$ until ${\Lambda }_{k-1}\approx {\Lambda }_k=\tilde{\Lambda }$.

3. Compute the initial estimate for R_ij denoted by ${\tilde{R}}_{ij}$, where

   $${\tilde{R}}_{ij}={\tilde{\Lambda }}^{-1/2}{\left({\tilde{\Lambda }}^{1/2}Z{\tilde{\Lambda }}^{1/2}\right)}^{1/2}{\tilde{\Lambda }}^{-1/2}.$$

4. Compute $\Delta =\text{diag}\left\{{\left({\tilde{R}}_{ij}\circ {\tilde{R}}_{ij}\right)}^{-1}J\right\}$, where J is a K × 1 vector of 1’s and ◦ denotes the Hadamard product of two matrices.

5. Obtain the final bias-corrected estimate for R_ij, given by

   $${\widehat{R}}_{ij}=\{\begin{array}{ll}{\tilde{R}}_{ij}\Delta {\tilde{R}}_{ij}\hfill & \text{ if }\Delta >0,\hfill \\ {(\text{diag}(Z))}^{-1/2}Z{(\text{diag}(Z))}^{-1/2}\hfill & \text{ otherwise.}\hfill \end{array}$$ (6)

2.2.2 |. Exchangeable correlation structure

If the correlation coefficients are similar between each pair of outcomes, the exchangeable correlation structure that assumes α_kl = α for all k, l can be a good alternative and requires the estimation of only one correlation coefficient:

$$R_{ij}={\left(\begin{array}{cccc}1& \alpha & \cdots & \alpha \\ \alpha & 1& & ⋮\\ ⋮& & \ddots & \alpha \\ \alpha & \cdots & \alpha & 1\end{array}\right)}_{K\times K}.$$ (7)

To minimize Equation (3) with respect to α, we take the partial derivative of Q_w(β, α) with respect to α and set it equal to 0. Because R_ij is the only matrix that is a function of α, the minimizing equation is simply,

$$\sum _{i=1}^N\frac{1}{n_i}\sum _{j=1}^{n_i}{Z}_{ij}^{\prime }\frac{\partial R_{ij}^{-1}}{\partial \mathit{\alpha }}{Z}_{ij}=0,$$ (8)

which has a closed form solution for the exchangeable correlation structure (Chaganty, 1997). The derivative of $R_{ij}^{-1}$ with an exchangeable correlation structure (Equation (7)) with respect to α is

$$\frac{\partial R_{ij}^{-1}}{\partial \alpha }={\left(\begin{array}{cccc}{d}_a& d_b& \cdots & d_b\\ d_b& d_a& & ⋮\\ ⋮& & \ddots & d_b\\ d_b& \cdots & d_b& d_a\end{array}\right)}_{K\times K},$$

where $d_a=\frac{{\alpha }^2(K-1)(K-2)+2\alpha (K-1)}{{(1-\alpha )}^2{[1+(K+1)\alpha ]}^2}$ and $d_b=\frac{-\left[1+{\alpha }^2(K-1)\right]}{{(1-\alpha )}^2{[1+(K+1)\alpha ]}^2}$.

Let scalars $G_1={\sum }_{i=1}^N\frac{1}{n_i}{\sum }_{j=1}^{n_i}{Z}_{ij}^{\prime }{Z}_{ij}$ and $G_2={\sum }_{i=1}^N\frac{1}{n_i}{\sum }_{j=1}^{n_i-1}{\sum }_{j^{\prime }=j+1}^{n_i}{Z}_{ij}^{\prime }{Z}_{ij}$. Then, Equation (8) reduces to

$${\alpha }^2(K-1)(K-2)G_1+2\alpha (K-1)G_1-s\left[1+{\alpha }^2(K-1)\right]G_2=0.$$ (9)

To solve for the stage one estimator of α in Equation (9), we apply the quadratic formula:

$${\widehat{\alpha }}_0=\frac{-(K-1)G_1+\sqrt{(K-1)\left(G_1+2G_2\right)\left[G_1(K-1)-s{G}_2\right]}}{(K-1)\left[(K-2)G_1-2G_2\right]}.$$ (10)

The bias-corrected stage-two estimator can be obtained from solving the following equation:

$$\sum _{i=1}^N\frac{1}{n_i}\sum _{j=1}^{n_i}\frac{1}{t_{ij}}\text{trace}\left(\frac{\partial R_{ij}^{-1}\left({\widehat{\alpha }}_0\right)}{\partial {\widehat{\alpha }}_0}{R}_{ij}(\alpha )\right)=0,$$ (11)

where ${\widehat{\alpha }}_0$ is the solution to Equation (10). Solving Equation (11), the final bias-corrected estimator of α for the exchangeable correlation structure is

$$\widehat{\alpha }=\frac{{\widehat{\alpha }}_0\left[{\widehat{\alpha }}_0(K-2)+2\right]}{{\widehat{\alpha }}_0(K-1)+1}.$$ (12)

2.3 |. Variance-covariance matrix for $\widehat{\beta }$

The robust variance-covariance matrix, Ψ, for $\widehat{\beta }$ is constructed in a similar way as done by Wang et al. (2011) using the familiar sandwich estimator (Williamson et al., 2003):

$$\widehat{\Psi }={\widehat{B}}^{-1}\widehat{M}{\widehat{B}}^{-1}$$ (13)

where

$$\widehat{B}=\sum _{i=1}^N\frac{1}{n_i}\sum _{j=1}^{n_i}{\widehat{D}}^{\prime }{\widehat{W}}_{ij}^{-1}\widehat{D}$$

and

$$\begin{gathered} \widehat{M}=\sum _{i=1}^N\left[\frac{1}{n_i}\sum _{j=1}^{n_i}{\widehat{D}}^{\prime }{\widehat{W}}_{ij}^{-1}\left(Y_{ij}-{\widehat{\mu }}_{ij}\right)\right] \\ {\left[\frac{1}{n_i}\sum _{j=1}^{n_i}{\widehat{D}}^{\prime }{\widehat{W}}_{ij}^{-1}\left(Y_{ij}-{\widehat{\mu }}_{ij}\right)\right]}^{\prime }. \end{gathered}$$

Finally, the entire parameters estimation procedure can be summarized as follows:

1. Select a starting value for β₀ (typically from fitting a generalized linear models assuming independence between observations).

2. Compute ${\widehat{Z}}_{ij}={\widehat{V}}_{ij}^{-1/2}\left(Y_{ij}-{\widehat{\mu }}_{ij}\right)$.

3. Compute ${\widehat{R}}_{ij}$ applying the procedures described in Sections 2.2.1 or 2.2.2 if assuming unstructured or exchangeable correlation structure, respectively. If assuming independent working correlation structure, let ${\widehat{R}}_{ij}=I$ where I is the K × K identity matrix.

4. Obtain $\widehat{\beta }$ by solving Equation (4).

5. Repeat steps 2–4 until convergence for $\widehat{\beta }$ is reached.

6. Obtain bias-corrected ${\widehat{R}}_{ij}$ using Equations (6) and (12) if assuming unstructured or exchangeable correlation structure, respectively.

7. Compute $\text{Cov}(\widehat{\beta })$ using Equation (13).

An R package named CWGEE contains the function mvoCWGEE that computes the procedure described above (see the URL provided in the Supporting Information).

In the next section, we apply this multivariate CWGEE procedure as well as the standard GEE procedure to our motivating data set.

3 |. EXAMPLE: VETERANS AFFAIRS DENTAL LONGITUDINAL STUDY

Our motivating data comes from the Veterans Affairs (VA) Dental Longitudinal Study (Kaye et al., 2016; Kapur et al., 1972). In this paper, we focus on baseline data, which consists of 760 men who have at least one tooth. Baseline health and oral examinations occurred between 1981 and 2011. The number of teeth (excluding third molars) ranged from 2 to 28 for each patient. The periodontal measures recorded for each tooth per patient included: PPD, CAL, ABL, and Mobil scores. Kaye et al. (2016) provides a clinically meaningful dichotomized version of each of these measurements as: PPD ≥5 mm, CAL ≥5 mm, ABL ≥40%, and Mobil ≥0.5 mm. Patient-specific health measurements were also recorded including age, level of education (college degree or higher/none), smoking status (yes/no), and metabolic syndrome (MetS) status (yes/no) as defined by the National Cholesterol Education Program Adult Treatment Panel III criteria. Note that our approach can flexibly handle covariates of all types: continuous, binary, and ordinal. We analyzed the population-average effects of age, level of education, smoking status, and MetS on the probabilities of three binary outcomes: CAL ≥5 mm, ABL ≥40%, and Mobil ≥0.5 mm. We decided to not include PPD ≥5 mm as one of the outcomes because PPD and CAL measurements are extremely similar.

Figure 1 shows the relationship between the number of teeth for each patient and the proportion of each patient having teeth with CAL ≥5 mm, ABL ≥40%, and Mobil ≥0.5 mm. Each periodontal disease outcome is inversely associated with number of teeth, that is, patients with few number of teeth tend to have high proportions of teeth with CAL ≥5 mm, ABL ≥40%, and Mobil ≥0.5 mm and patients with a full set of teeth have low proportions. The inverse relationship between these proportions (periodontal disease outcomes) and cluster size (number of teeth) implies that ICS is present in this data set.

FIGURE 1.

Relationship between number of teeth and proportions of teeth with clinical attachment loss (CAL) ≥5 mm, radiographic alveolar bone loss (ABL) ≥ 40%, and Mobil ≥ 0.5 mm at baseline per patient from the Department of Veterans Affairs Dental Longitudinal Study

We first conducted a combined (multivariate) analysis with separate regression coefficients for each outcome, based on Equation (1). We will refer to this as the general model:

$$\begin{gathered} \text{logit}\left({\mu }_{ij\text{CAL}}\right)=a^{\text{CAL}}+X_i\beta \\ \text{logit}\left({\mu }_{ij\text{ABL}}\right)=a^{\text{ABL}}+X_i\left(\beta +{\beta }^{\text{ABL}}\right) \\ \text{logit}\left({\mu }_{ij\text{Mobil}}\right)=a^{\text{Mobil}}+X_i\left(\beta +{\beta }^{\text{Mobil}}\right), \end{gathered}$$ (14)

where $X_i=\left(X_{\text{Age}}^{\prime },X_{\text{Smoking}}^{\prime },X_{\text{Education}}^{\prime },X_{\text{MetS}}^{\prime }\right)$ is the complete data matrix of the four predictors for each patient i, and β = (β _Age, β _Smoking, β _Education, β _MetS). The parameters β^ABL and β^Mobil are additive effects that each of the predictors have on the outcomes ABL and Mobil, respectively, above and beyond the reference outcome, CAL. For each predictor, if we reject the null hypothesis ${\text{H}}_{0p}:{\beta }_p^{\text{ABL}}={\beta }_p^{\text{Mobil}}=0$ (where p = 1, …, 4 for each predictor), then we keep separate regression coefficients for each outcome. Otherwise, we use a common regression coefficient across the three outcomes. The dependencies among the three outcomes were modeled using each of the three working correlation structures for comparison: unstructured, exchangeable and independent. Note that under the independent working correlation assumption, Equation (14) is essentially equivalent to fitting three separate logistic regression models for each of the three outcomes.

Parameter estimates, standard errors, and P-values for each coefficient from fitting the general model in Equation (14) using the methods of GEE and CWGEE with unstructured correlation structure are shown in Table 1. The results with exchangeable and independent correlation structures were similar. However, we observe some notable discrepancies in both estimates and inference between the two methods. For example, based on the results from the GEE method, we will use a common slope for age because the P-value from multivariate Wald statistic for ${\text{H}}_0:{\beta }_{\text{Age}}^{\text{ABL}}={\beta }_{\text{Age}}^{\text{Mobil}}=0$ is not significant at the .05 level (p = .231 when assumed an unstructured correlation structure, with similar results for other correlation structures). However, based on the results from the CWGEE method, we will keep separate slopes for age because the P-value for the same test is significant (p = .018 when assumed an unstructured correlation structure, with similar results for other correlation structures). The remaining results from both GEE and CWGEE methods are in agreement regarding the use of common slopes for smoking, education, and MetS. Table 2 shows the results from parsimonious models with unique slopes for age and common slopes for smoking, education and MetS. The results from GEE indicate that MetS is a significant predictor (p = .018) while those from CWGEE indicate that MetS is not a significant predictor (p = .089).

TABLE 1.

Multivariate analyses: Parameter estimates with robust standard errors and P-values when analyzing CAL ≥ 5 mm, ABL ≥ 40%, and Mobil ≥ 5 mm with separate effects for Age, Education, Smoking, and MetS using GEE and CWGEE with unstructured correlation structure. P-values are from multivariate Wald test for H₀ : β^ABL = β^Mobil = 0

	GEE		CWGEE
	Estimate (SE)	P-value	Estimate (SE)	P-value
Int (CAL)	−4.500 (0.772)		−4.810(0.789)
Int (ABL)	−4.042 (0.710)		−3.750 (0.786)
Int (Mobil)	−4.821 (0.788)		−4.174 (0.917)
Age	0.041 (0.011)		0.051 (0.011)
Age (ABL)	−0.017 (0.010)	0.231	−0.024 (0.009)	0.018
Age (Mobil)	−0.010 (0.010)		−0.023 (0.011)
Smoking	0.710 (0.198)		0.657 (0.221)
Smoking (ABL)	0.253 (0.177)	0.360	0.132(0.178)	0.726
Smoking (Mobil)	0.078 (0.182)		−0.018 (0.207)
Edu	−0.401 (0.111)		−0.424 (0.123)
Edu (ABL)	0.002 (0.100)	0.683	−0.041 (0.102)	0.454
Edu (Mobil)	−0.083 (0.104)		−0.157 (0.125)
MetS	0.403 (0.176)		0.336 (0.185)
MetS (ABL)	−0.197 (0.160)	0.288	−0.267 (0.165)	0.197
MetS (Mobil)	0.096 (0.178)		0.067 (0.188)

TABLE 2.

Multivariate analyses: Parameter estimates with robust standard errors and P-values when analyzing CAL ≥ 5 mm, ABL ≥ 40%, and Mobil ≥ 5 mm with separate effects for Age and common slopes for Education, Smoking, and MetS using GEE and CWGEE with unstructured working correlation structure

	GEE		CWGEE
	Estimate (SE)	P-value	Estimate (SE)	P-value
Int (CAL)	−4.520 (0.767)	<0.001	−4.774 (0.787)	<0.001
Int (ABL)	−3.928 (0.700)	<0.001	−3.751 (0.762)	<0.001
Int (Mobil)	−4.851 (0.760)	<0.001	−4.295 (0.857)	<0.001
Age	0.042 (0.011)	<0.001	0.052 (0.011)	<0.001
Age (ABL)	−0.019 (0.010)	0.059	−0.025 (0.009)	0.007
Age (Mobil)	−0.012 (0.010)	0.264	−0.024(0.011)	0.028
Smoking	0.793 (0.171)	<0.001	0.695 (0.195)	<0.001
Edu	−0.415 (0.097)	<0.001	−0.458 (0.108)	<0.001
MetS	0.365 (0.154)	0.018	0.277 (0.163)	0.089

Table 3 shows the estimates of the unstructured and exchangeable working correlation structure, R_ij(α), using GEE (upper half of the matrices) and CWGEE (lower half of the matrices). The estimates between the two methods are similar, although the GEE method produced slightly lower correlation estimates between each pair of outcomes when assuming an exchangeable correlation structure. Each pair of outcomes has a moderately strong positive correlation. Based on the unstructured correlation estimates from the CWGEE method, CAL and ABL have the highest correlation (0.40), whereas Mobil and ABL have the lowest correlation (0.29) among the three outcomes.

TABLE 3.

Estimates of the working correlation matrices (unstructured and exchangeable): GEE estimates are shown in the upper half of the matrices and CWGEE estimates are shown in the lower half of the matrices

	Unstructured				Exchangeable
	CAL	ABL	Mobil		CAL	ABL	Mobil
CAL	–	0.40	0.33	CAL	–	0.31	0.31
ABL	0.40	–	0.29	ABL	0.34	–	0.34
Mobil	0.32	0.29	–	Mobil	0.34	0.34	–

4 |. SIMULATION STUDY

4.1 |. Design

To demonstrate the performance of our proposed method compared to the conventional GEE method under various scenarios with ICS, we conducted an extensive simulation study. For each simulation scenario, we simulated 1000 data sets including three correlated binary outcomes with a common regression coefficient estimate for one patient-specific binary covariate that is common across all teeth within a patient, based on Equation (2).

$$\text{logit}\left\{\text{Pr}\left(Y_{ijk}=1\right)\right\}=a_k+X_i\beta ,k=1,2,3$$ (15)

where (α₁, α₂, α₃) = (−1, −1.5, −2) and β = 0.5.

To simulate K = 3 correlated binary outcomes for each tooth j of patient i, we generated random effects b_ijk for each outcome k of tooth j of patient i that follow the bridge distribution (Parzen et al., 2011). A large b_ijk value indicates an increased risk of the kth periodontal outcome occurring at the jth tooth of the ith patient. Details on how the data were simulated are provided in the Supporting Information. Briefly, we employed the exchangeable correlation structure with parameter τ to generate the correlation between teeth within each patient and the unstructured correlation structure with parameters (α₁₂, α₁₃, α₂₃) to generate the correlations between each pair of the three outcomes. To induce informative cluster size, we computed λ_i for each patient as a function of the random effects such that ${\lambda }_i=1-\text{logit}\left({\overline{b}}_i\right)$, where ${\overline{b}}_i$ is the mean of the set of random effects generated for the ith patient that follow a bridge distribution. Here, λ_i is bounded between 0 and 1, and is inversely related to the magnitude of the patient-averaged random effects. Then, the number of teeth for each patient (n_i) was generated from a truncated binomial distribution with size 28 and probability λ_i with the restriction n_i > 1. Finally, the outcomes were generated with the probability Pr(Y_ijk = 1|b_ijk, α_k, X_i, β) = logit⁻¹{b_ijk + (α_k + X_iβ)ϕ⁻¹} with the scale parameter ϕ = 0.5. Because of the properties of the bridge distribution, the marginal probability of Y_ijk is Pr(Y_ijk = 1|α_k, X_i, β) = logit⁻¹(α_k + X_iβ).

Table 4 summarizes the parameters used in the simulation study. The maximum number of teeth per patient (m) and the number of outcomes (K) per tooth were fixed at 28 and 3, respectively. We varied the number of patients (N), correlation between teeth (τ), and correlations between each pair of outcomes (α₁₂, α₁₃, α₂₃). Four levels of correlations between outcomes were considered: None (α₁₂, α₁₃, α₂₃) = (0, 0, 0); Low (0.4,0.35,0.3); Medium (0.6,0.55,0.5); and High (0.8,0.75,0.7).

TABLE 4.

Simulation scenarios

Parameter	Values
Number of patients (N)	20, 100, 750
Maximum number of teeth per patient (m)	28
Number of outcomes (K)	3
Correlation between teeth (τ)	0, 0.25, 0.5, 0.75
Correlation between outcomes 1 and 2 (α12)	0, 0.4, 0.6, 0.8
Correlation between outcomes 1 and 3 (α13)	0, 0.35, 0.55, 0.75
Correlation between outcomes 2 and 3 (α23)	0, 0.3, 0.5, 0.7

For each simulated data set, we applied multivariate CWGEE and unweighted multivariate GEE methods with unstructured (Unstr), exchangeable (Exch), and independent (Ind) working correlation structures to model the correlation between each pair of the three outcomes. For each scenario, we obtained the parameter estimates (${\widehat{a}}_1^s$, ${\widehat{a}}_2^S$, ${\widehat{a}}_3^s$, ${\widehat{\beta }}^s$) from each simulated data set, s = 1, … , 1000 and computed: (a) the mean parameter estimates (Mean Est), $\frac{1}{1000}{\sum }_{s=1}^{1000}{\widehat{\beta }}^s$; (b) the 1000 mean robust standard error (Mean SE), $\frac{1}{1000}{\sum }_{s=1}^{1000}\widehat{\text{s.e.}}\left({\widehat{\beta }}^s\right)$; (c) the empirical standard error (SD Est), ${\left[\frac{1}{1000-1}{\sum }_{s=1}^{1000}{\left({\widehat{\beta }}^s-\beta \right)}^2\right]}^{1/2}$; (d) the mean relative bias (Rel Bias), $\frac{1}{1000}{\sum }_{s=1}^{1000}\left({\widehat{\beta }}^s-\beta \right)/\beta$; and (e) the 95% coverage probability (CovP) defined as the percentage of time the 95% confidence interval for ${\widehat{\beta }}^s$ from each simulated data set includes the true value β.

We also fit a general model with separate regression coefficients for each outcome:

$$\begin{gathered} \text{logit}\left\{\text{Pr}\left(Y_{ij1}=1\right)\right\}=a_1+X_i\beta , \\ \text{logit}\left\{\text{Pr}\left(Y_{ij2}=1\right)\right\}=a_2+X_i\left(\beta +{\beta }_{12}\right), \\ \text{logit}\left\{\text{Pr}\left(Y_{ij3}=1\right)\right\}=a_3+X_i\left(\beta +{\beta }_{13}\right). \end{gathered}$$ (16)

We then computed the type I error rates for the null hypothesis H₀ : β₁₂ = β₁₃ = 0 from both GEE and CWGEE methods for each simulation scenario. The type I error rate is the percentage of time H₀ is rejected at a 0.05 level of significance out of the 1000 simulated data sets.

4.2 |. Results

4.2.1 |. Type I error from general model

In Figure 2, we depict the relationship between the type I error rate for H₀ : β₁₂ = β₁₃ = 0 from fitting the general model from Equation (16) by the correlation level between teeth (τ) and the correlation level among the three simulated outcomes (α) for each simulation scenarios with N = 750. We present the results for GEE and CWGEE with unstructured correlation structures. Results obtained from using other correlation structures were similar. With lower correlations between teeth (τ = 0, 0.25), we observe that the type I error rates are similar between GEE and CWGEE. However, with higher correlations between teeth (τ = 0.5, 0.75), the type I error rates are noticeably larger for GEE, while the type I error rates for CWGEE remained close to 0.05. Higher type I error rates from GEE indicate that GEE models tend to falsely reject H₀ more often and incorrectly conclude to use a common slope. This pattern was observed in our example data set in Section 3 where results from GEE indicated to use a common slope for age while results from CWGEE indicated otherwise.

FIGURE 2.

Type I error rate for H₀ : β₁₂ = β₁₃ = 0 based on the multivariate Wald test by correlation level between teeth (τ) and correlation level among outcomes (α) from GEE and CWGEE using unstructured correlation structures for α (N = 750)

4.2.2 |. Parameter estimates from parsimonious model

In Table 5, we present the simulation results for GEE and CWGEE unstructured correlation structure from the scenarios with N = 750, τ = 0.5 and low (α₁₂, α₁₃, α₂₃) = (0.4, 0.35, 0.3), medium (0.6, 0.55, 0.5), and high (0.8, 0.75, 0.7) correlations among the three outcomes. The simulation results from exchangeable and independent working correlation structures are comparable and presented in the Supporting Information.

TABLE 5.

Simulation results for unstructured correlation structure, N = 750, τ = 0.5, Number of simulations = 1000

Parameter	Truth	Results	Correlation levels among outcomes
			Low		Medium		High
			GEE	CWGEE	GEE	CWGEE	GEE	CWGEE
a1	−1	Mean Est	−1.393	−0.999	−1.850	−1.023	−1.951	−1.039
		Mean SE	0.047	0.052	0.061	0.072	0.065	0.076
		SD Est	0.046	0.051	0.061	0.074	0.067	0.075
		Rel Bias	0.393	−0.001	0.850	0.023	0.951	0.039
		Covp (%)	0	95.4	0	93.0	0	92.3
a2	−1.5	Mean Est	−1.931	−1.500	−2.451	−1.531	−2.567	−1.552
		Mean SE	0.050	0.056	0.065	0.078	0.069	0.082
		SD Est	0.049	0.055	0.064	0.080	0.069	0.081
		Rel Bias	0.287	0	0.634	0.021	0.712	0.035
		Covp (%)	0	94.9	0	91.8	0	90.8
a3	−2	Mean Est	−2.480	−1.999	−3.071	−2.045	−3.204	−2.069
		Mean SE	0.055	0.063	0.071	0.086	0.075	0.091
		SD Est	0.056	0.063	0.071	0.088	0.076	0.090
		Rel Bias	0.240	−0.001	0.536	0.023	0.602	0.035
		Covp (%)	0	95.3	0	91.7	0	89.0
β	0.5	Mean Est	0.537	0.496	0.604	0.506	0.623	0.514
		Mean SE	0.061	0.065	0.081	0.095	0.086	0.101
		SD Est	0.063	0.066	0.079	0.095	0.087	0.097
		Rel Bias	0.074	−0.008	0.207	0.013	0.246	0.027
		Covp (%)	90.4	95.2	73.7	95.2	68.6	95.2
α12		Mean Est	0.198	0.147	0.278	0.219	0.332	0.270
α13		Mean Est	0.179	0.133	0.253	0.200	0.302	0.247
α23		Mean Est	0.175	0.132	0.245	0.196	0.297	0.247

First, we observe the increasingly large relative biases and consistently 0 coverage probabilities in the GEE intercept parameter estimates (${\widehat{a}}_1$, ${\widehat{a}}_2$, ${\widehat{a}}_3$) with increasing correlation levels among the outcomes. The extremely large biases in the GEE intercept estimate when ICS is present have been observed in previous studies (Wang et al., 2011; Bible et al., 2016; Mitani et al., 2019). The relative biases also increase for the CWGEE estimates too but their magnitudes are consistently much lower compared to the GEE estimates. CWGEE coverage probabilities were close to 95% with low correlation levels among the outcomes and decreased slightly to between 89% and 92.3% with high correlation levels. Second, the relative biases of the GEE main effect estimates, $\widehat{\beta }$, are also larger compared to the CWGEE estimates (0.074, 0.207, 0.246 for GEE and −0.008, 0.013, 0.027 for CWGEE with low, medium, high correlation levels, respectively). The coverage probabilities from GEE are all lower than 95% (90.4%, 73.7%, 68.6% for low, medium, and high correlation levels, respectively), whereas those from CWGEE are all 95.2% regardless of the correlation levels, reflecting the robustness of the multivariate CWGEE method to ICS. The mean model standard errors (Mean SE) and empirical standard errors (SD Est) are comparable within the same parameter, method, and levels of correlation. The simulation results from scenarios with smaller sample size (N = 20, 100) yielded similar mean relative biases but larger model standard errors and empirical standard errors (see the Supporting Information).

The mean of the 1000 correlation coefficient estimates among the outcomes (${\widehat{\alpha }}_{12}$, ${\widehat{\alpha }}_{13}$, ${\widehat{\alpha }}_{23}$) for each method and scenario are shown at the bottom of Table 5. In our simulation, we use the (α₁₂, α₁₃, α₂₃) values as correlations among random variables that are multivariate normally distributed, and then transfer these variables using the bridge distribution to generate binary outcomes. Therefore, the true values for (α₁₂, α₁₃, α₂₃) listed above cannot be estimated exactly from GEE or CWGEE methods using QLS. In most marginal analyses, the primary interest lies in estimating the main effects and estimating the correlation coefficients are considered a “nuisance” (Fitzmaurice et al., 2004). Nonetheless, we observe that the correlation coefficient estimates from both GEE and CWGEE methods do increase as the true levels of correlation increase. Within each scenario, the estimates from GEE are slightly larger compared to those from CWGEE.

The distributions of the relatives biases for $\widehat{\beta }$ by correlation between teeth (τ) and correlation among outcomes, α = (α₁₂, α₁₃, α₂₃), from CWGEE and GEE methods are depicted in Figure 3. Correlation levels among the outcomes along the x-axis are as follows: None (α₁₂, α₁₃, α₂₃) = (0, 0, 0); Low (0.4, 0.35, 0.3); Medium (0.6, 0.55, 0.5); and High (0.8, 0.75, 0.7). We present the results using the unstructured correlation structure among the outcomes from the scenario with N = 750. Similar results were obtained when using the other correlation structures. For both GEE and CWGEE estimates, the relative biases have means close to 0 with small variability when there is no correlation between teeth (τ = 0). The relative biases are generally unaffected by levels of α. The variability starts to increase as we increase the values of τ and α for both methods. The variability for both methods is largest when τ = 0.75 and (α₁₂, α₁₃, α₂₃) = (0.8, 0.75, 0.7). The mean relative biases for CWGEE remain close to 0 with increasing τ and α, whereas the mean relative biases for GEE shift upward with increasing τ and α.

FIGURE 3.

Distribution of relative bias for β by correlation level between teeth (τ) and correlation level among outcomes (α) from GEE and CWGEE using unstructured correlation structures for α (N = 750)

The coverage probabilities for $\widehat{\beta }$, also by τ and α, from CWGEE and GEE methods are depicted in a figure in the Supporting Information. The coverage probabilities for both methods are close to 95% when τ = 0 regardless of the levels of α. Similar to the relative bias pattern, the CWGEE coverage probabilities are robust to the varying τ and α, whereas GEE coverage probabilities decrease as τ and α increase.

There were also discrepancies in parameter estimates between GEE and CWGEE from the general model. Similar to the results from the parsimonious models, the relative biases for the intercepts and β from CWGEE were close to 0 and coverage probabilities ranged from 92.3% to 95.1% regardless of the working correlation structure for the scenario with τ = 0 and medium levels of correlation among outcomes. On the other hand, the relative biases from GEE ranged from 0.141 and 0707, and coverage probabilities were 0% for the intercepts and 86.1% for β. Results from one of the scenarios are presented in Table S1 in the Supporting Information. Simulation results from scenarios with smaller samples sizes, N = 20 and 100, are presented in Tables S4 and S5, respectively. The improvement in relative biases and coverage probabilities using CWGEE compared with conventional GEE were apparent with smaller sample sizes also. However, in general, when N = 20, both methods yielded higher relative biases and lower coverage probabilities.

Overall, our proposed multivariate CWGEE approach produced parameter estimates with extremely low relative biases and excellent coverage probabilities compared to those from the standard GEE approach. The performance of the multivariate CWGEE is robust to varying levels of correlations between teeth and correlation among the outcomes.

5 |. DISCUSSION

In this paper, we propose a novel extension to CWGEE for analyzing multiple binary outcomes measured on each tooth simultaneously for use in cross-sectional studies. Previous marginal methods for ICS in the current literature have only focused on one outcome variable, either measured at a single time point or longitudinally. In our proposed approach, we allow the assessment of more than one outcome variable by implementing the method of QLS to model the pairwise correlations between the outcomes. Our proposed method makes more efficient use of the available data by simultaneously modeling two or more correlated outcomes where an alternative might be to model each outcome separately. To our knowledge, no research has previously been conducted on marginal models for multiple outcomes in the presence of ICS, which commonly arises in many settings where cluster size is influenced by the outcome or outcomes of interest.

Besides dental studies, ICS is present in many other clinical settings including pregnancy studies where women who have experienced an adverse pregnancy outcome may have fewer subsequent pregnancies (Chaurasia et al., 2018), and psychological studies where the frequency of depressive episodes by a participant may also be related to the severity level of each event (Iosif and Sampson, 2014). In some of these studies, usually more than one outcome will be of interest, especially in psychological studies where questionnaires contain multiple items (Iosif and Sampson, 2014).

In our motivating data set of periodontal disease, we found that the methods of GEE and CWGEE can lead to discrepancies in inference on some of the predictors. In particular, in the most general model, results from GEE indicated including a common slope among the three outcomes for age, whereas results from CWGEE indicated otherwise. In addition, in the parsimonious model, MetS was a significant predictor in the GEE model, whereas it was not a significant predictor in the CWGEE model. In our extensive simulation study, we observed that CWGEE consistently produces coefficient estimates with low relative biases and coverage probabilities close to 95% regardless of the levels of correlation between teeth and levels of correlation among the outcomes, demonstrating a robustness to the issues raised due to ICS. On the other hand, coefficient estimates produced by GEE suffered large relative biases and low coverage probabilities especially when the levels of correlation were high. Another strength in our proposed CWGEE is its performance reliability in data with no ICS. As a sensitivity analysis, we also simulated data with no ICS and found that CWGEE yields low relative bias and excellent coverage probabilities for all estimated coefficients.

In the conventional GEE approach, correlation between units within a cluster can be modeled by choosing a working correlation structure in order to gain efficiency in the parameter estimates. In CWGEE, however, assuming a nonindependence working correlation structure in attempt to model the correlation between units within a cluster will alter the weights given to each cluster. Thus, the cluster-specific weights would then need to be adjusted. However, no efficiency will be gained from choosing a nonindependence correlation structure because the new adjusted cluster-weights will cancel out with the working correlation matrix (Williamson et al., 2003).

One limitation of this paper is in the application of multivariate outcome CWGEE to cross-sectional data only. Many studies that exhibit data in ICS are longitudinal, including the VA Dental Longitudinal Study. However, we anticipate that extending our proposed method to longitudinal studies is not difficult. In a longitudinal study with multiple outcomes, in addition to modeling the correlation among the outcomes, we also would need to model the correlation among the repeated visits. One way is to use the Kronecker product to join the two correlation structures and then use QLS, which can be applied to complex correlation structures, to estimate the correlation coefficients (Shults and Ardythe, 2002). Another interesting extension to this research is to allow the model to analyze multiple outcomes of different distributions, such as a binary outcome and a continuous outcome. In the VA Dental Longitudinal Study, all periodontal disease measures were recorded using an ordinal scoring system, which were dichotomized for the current study. After 1987, CAL and PPD were recorded in millimeters, rather than the previously used ordinal scale, and the normality assumption may be more suitable for these measurements.

We recommend the use of CWGEE to produce valid parameter estimates if data are potentially subject to ICS. Employing the standard GEE method to analyze dental data, without considering the impact of ICS, may result in incorrect assessment of the relationship between periodontal disease and individual health characteristics.