A covariance correction that accounts for correlation estimation to improve finite-sample inference with generalized estimating equations: A study on its applicability with structured correlation matrices

Abstract

When generalized estimating equations (GEE) incorporate an unstructured working correlation matrix, the variances of regression parameter estimates can inflate due to the estimation of the correlation parameters. In previous work, an approximation for this inflation that results in a corrected version of the sandwich formula for the covariance matrix of regression parameter estimates was derived. Use of this correction for correlation structure selection also reduces the over-selection of the unstructured working correlation matrix. In this manuscript, we conduct a simulation study to demonstrate that an increase in variances of regression parameter estimates can occur when GEE incorporates structured working correlation matrices as well. Correspondingly, we show the ability of the corrected version of the sandwich formula to improve the validity of inference and correlation structure selection. We also study the relative influences of two popular corrections to a different source of bias in the empirical sandwich covariance estimator.

1. Introduction

Generalized estimating equations (GEE) [1] are commonly utilized for the analysis of correlated data when a marginal model is desired. When GEE incorporates an unstructured working correlation matrix, it has been shown that the covariance matrix of the regression parameter estimates may inflate due to the need to estimate nuisance correlation parameters [2]. We note that although relatively unknown, to our knowledge, in the GEE literature before the work of Westgate [2], this type of small-sample variance inflation is very well known when data arise from a multivariate normal distribution and a linear mixed model is used [3]. In fact, the Kenward and Roger method [4, 5], which accounts for this inflation, has enjoyed great popularity when the working covariance structure is assumed to be correctly specified such that model-based standard error estimates can be utilized.

With respect to GEE, Westgate [2] derived an approximation for this inflation when utilizing an unstructured working correlation matrix, resulting in a corrected version of the well-known sandwich formula for the covariance matrix of the regression parameter estimates. Furthermore, Westgate [6] showed that, in order to improve regression parameter estimation via correlation structure selection, this covariance correction can be used to penalize the estimation of the multiple nuisance correlation parameters within the unstructured matrix in order to reduce its over-selection. In this manuscript, we conduct a simulation study to demonstrate that even when GEE incorporates structured correlation matrices, the variances of the regression parameter estimates can still inflate. Therefore, we also apply and study the use of the covariance inflation correction when GEE incorporates structured correlation matrices. Specifically, we show that use of this correction can improve inference when using structured working correlation matrices, and that this correction should be utilized as a penalty by correlation selection criteria for all structures under consideration. Furthermore, unrelated to the covariance inflation correction, a correction for the bias in the meat of the empirical sandwich covariance matrix estimator is needed in small-sample settings. Therefore, we study the relative influences of two such corrections, proposed by Kauermann and Carroll [7] and Mancl and DeRouen [8], that have found popularity.

Section 2 introduces notation and discusses GEE, the covariance inflation correction, estimation of the sandwich covariance matrix, and correlation structure selection. Our simulation study is presented in Section 3. Finally, concluding remarks are given in Section 4.

2. Notation, GEE, Covariance Correction and Estimation, and Correlation Selection

Assume we have data from N independent clusters. The observed outcome vector for the ith cluster is denoted by Y_i = [Y_i1, …, Y_ini]^T, which has a marginal mean given by E(Y_i) = μ_i that is linked to covariates via a function, f, such that $f({\mu }_{ij})={\mathbf{x}}_{ij}^T\mathrm{\beta }$ for x_ij = [1, x_1ij, …, x_(p−1)ij]^T and β = [β₀, β₁, …, β_p−1]^T. The corresponding working covariance matrix for Y_i is given by ${\mathit{V}}_i={\mathit{A}}_i^{1/2}{\mathit{R}}_i(\mathrm{\alpha }){\mathit{A}}_i^{1/2}$, i = 1, …, N. Here, A_i = diag[ϕν(μ_i1), …, ϕν(μ_ini)] is a diagonal matrix of working marginal variances, ϕ is an assumed common dispersion parameter, ν is a known function, and R_i(α) is a working correlation matrix with 1 along the diagonal and one or more parameters given by α.

Let D_i = ∂μ_i/∂β^T, and denote a consistent working estimate for β by β̃. To obtain the final estimate of the regression parameters, β̂, using GEE [1], we iteratively solve

$$\sum _{i=1}^N{\mathit{D}}_i^T{\mathit{A}}_i^{-1/2}{\mathit{R}}_i^{-1}(\hat{\mathrm{\alpha }}(\tilde{\mathrm{\beta }})){\mathit{A}}_i^{-1/2}({\mathit{Y}}_i-{\mathit{\mu }}_i)=\mathbf{0},$$ (1)

for which β̃ = β̂ at the end of the iterative procedure. The well-known sandwich formula for the covariance matrix of β̂ is given by Cov(β̂) ≈

$$\mathbf{\Sigma }={\left(\sum _{i=1}^N{\mathit{D}}_i^T{\mathit{V}}_i^{-1}{\mathit{D}}_i\right)}^{-1}(\sum _{i=1}^N{\mathit{D}}_i^T{\mathit{V}}_i^{-1}\mathit{\text{Cov}}({\mathit{Y}}_i){\mathit{V}}_i^{-1}{\mathit{D}}_i){\left(\sum _{i=1}^N{\mathit{D}}_i^T{\mathit{V}}_i^{-1}{\mathit{D}}_i\right)}^{-1}.$$ (2)

The sandwich formula of Equation (2) assumes correlation parameters are known, although in practice they must be estimated. Additionally, α̂(β) must be replaced with α̂(β̃) in Equation (1). As a result, because ${\mathit{R}}_i^{-1}(\hat{\mathrm{\alpha }}(\tilde{\mathrm{\beta }}))$ varies about ${\mathit{R}}_i^{-1}(\hat{\mathrm{\alpha }}(\mathrm{\beta }))$, the estimation variability of GEE can increase, thus inflating Cov(β̂) [2]. Specifically, Westgate [2] showed that

$$\mathit{\text{Cov}}(\hat{\mathrm{\beta }})\approx ({\mathit{I}}_p+\mathit{G})\mathbf{\Sigma }{({\mathit{I}}_p+\mathit{G})}^T$$ (3)

after accounting for covariance inflation via a Taylor series expansion. Here, I_p is a p × p identity matrix, and G = (G₀, G₁, …, G_p−1),

$${\mathit{G}}_r=-{\left(\sum _{i=1}^N{\mathit{D}}_i^T{\mathit{V}}_i^{-1}{\mathit{D}}_i\right)}^{-1}\sum _{i=1}^N{\mathit{D}}_i^T{\mathit{A}}_i^{-1/2}{\mathit{R}}_i^{-1}\frac{\partial {\mathit{R}}_i(\hat{\mathrm{\alpha }}(\mathrm{\beta }))}{\partial {\beta }_r}{\mathit{R}}_i^{-1}{\mathit{A}}_i^{-1/2}({\mathit{Y}}_i-{\mathit{\mu }}_i(\mathrm{\beta })).$$

We note that although Westgate [2, 6] only applied the covariance inflation correction in Equation (3) when GEE incorporates an unstructured working correlation matrix, this correction is not restricted and can be applied with GEE regardless of the type of working correlation structure, as will be demonstrated in our simulation study.

In practice, unknown parameters must be estimated within the formula for Cov(β̂). Therefore, an arbitrary estimator can be denoted by (I_p + Ĝ)Σ̂(I_p + Ĝ)^T. Within G, unknown parameters can be estimated using β̂, resulting in Ĝ. However, Σ can be estimated in different manners. If we assume the working covariance structure is correctly specified, then Cov(Y_i) in Equation (2) can be replaced with V_i, i = 1, …, N, resulting in the model-based estimator ${\hat{\mathbf{\Sigma }}}_{MB}={({\sum }_{i=1}^N{\mathit{D}}_i^T{\mathit{V}}_i^{-1}{\mathit{D}}_i)}^{-1}$. However, if the working structure is misspecified, Σ̂_MB will be biased. Therefore, a common form for Σ̂ is the Liang and Zeger [1] empirical estimator, Σ̂_LZ, that replaces Cov(Y_i) in Equation (2) with (Y_i − μ̂_i)(Y_i − μ̂_i)^T, i = 1, …, N. This estimator is routinely used with GEE because it generally is a consistent estimate for Cov(β̂) that does not require the working and true covariance structures to be equivalent [1]. We further note that in small-sample settings, Σ̂_LZ can be biased for Σ because (Y_i − μ̂_i), i = 1, …, N, tends to be too small [8]. Therefore, multiple corrections have been proposed to reduce this bias, such as the popular corrections proposed by Kauermann and Carroll [7] and Mancl and DeRouen [8]. As these two corrections can yield notably different standard error estimates in small-sample settings, in our simulation study we will assess the performances of both corrections in conjunction with the covariance inflation correction.

Accurate modeling of the working correlation structure has the potential to improve estimation efficiency [1, 9]. Therefore, multiple criteria have been proposed to select a working structure, many of which are summarized in and studied by Westgate [6]. For instance, the ‘correlation information criterion’ (CIC) and ‘trace of the empirical covariance matrix’ (TECM) criterion have been shown to work well [6, 10]. When incorporating the covariance inflation correction, as proposed by Westgate [6] to penalize, or account for, the estimation of nuisance correlation parameters, the CIC selects the working structure that gives the smallest value for $\mathit{\text{tr }}({\hat{\mathbf{\Sigma }}}_I^{-1}({\mathit{I}}_p+\mathit{Ĝ})\hat{\mathbf{\Sigma }}{({\mathit{I}}_p+\mathit{Ĝ})}^T)$, where ${\hat{\mathbf{\Sigma }}}_I={({\sum }_{i=1}^N{\mathit{D}}_i^T{\mathit{A}}_i^{-1}{\mathit{D}}_i)}^{-1}$, and the TECM chooses the structure that yields the smallest value for $\mathit{\text{tr }}(({\mathit{I}}_p+\mathit{Ĝ})\hat{\mathbf{\Sigma }}{({\mathit{I}}_p+\mathit{Ĝ})}^T)$. We note that Σ̂ must be Σ̂_LZ or a bias-corrected version of this estimator. Furthermore, Westgate [6] only applied the covariance inflation correction for the unstructured correlation matrix, whereas in our simulation study we show that it should be applied with all working structures that are under consideration for selection. For instance, structures such as independence, exchangeable, AR-1, and less parsimonous Toeplitz forms do not all have the same number of correlation parameters, and therefore each will have a different degree of covariance inflation that must be taken into account.

3. Simulation Study

3.1. Study Description

We now conduct a simulation study to show that variances of regression parameter estimates can inflate when GEE incorporates well-known structured working correlation matrices. Furthermore, we demonstrate the corresponding use, and study the necessity and utility, of the covariance inflation correction in Equation (3). Specifically, we study standard error (SE) estimation and the validity of inference via empirical covarage probabillities (CPs) of 95% confidence intervals (CIs). We further study correlation selection accuracy via the ability of correlation selection criteria to choose the true structure. As our focus in this manuscript is on structured working correlation matrices, we do not present results from an unstructured working matrix. Furthermore, because we focus on small-sample settings, we study the use of the covariance inflation correction in conjunction with the Kauermann and Carroll [7] and Mancl and DeRouen [8] corrections in order to assess the impact these latter two corrections for the bias in Σ̂_LZ has on the necessity and utility of the covariance inflation correction.

Multivariate normal data were generated from

$$Y_{ij}={\beta }_0+{\beta }_1{x}_{1ij}+{\beta }_2{x}_{2ij}+{\epsilon }_{ij};\hspace{1em}j=1,\dots ,n,$$

where β = [0, 0.3, 0.3]^T and Var(ε_ij) = 1, j = 1, …, n; i = 1, …, N, and correlated binary outcomes were generated from the marginal model given by

$$\mathit{\text{logit}}({\mu }_{ij})={\beta }_0+{\beta }_1{x}_{1ij}+{\beta }_2{x}_{2ij},\hspace{1em}j=1,\dots ,n,$$

where β = [0, 0.1, 0.1]^T. In both models, x_1ij and x_2ij, j = 1, …, n, were independently generated from Uniform(0, 1). Models are similar to the ones used in Hin et al. [11], Hin and Wang [10], and Westgate [6].

Simulations were conducted in R version 2.13.1 [12]. Each setting was examined via 1,000 replications. Normal outcomes were generated using rmvnorm of the mvtnorm package [13, 14], whereas binary outcomes were generated using rmvbin of the bindata package [15]. When correlated outcomes are binary, additional constraints are required on the correlation parameters [16, 17]. Therefore, to avoid problems with data generation, and to enhance the stability of working correlation matrices, we utilized α_exch = α_AR−1 = 0.2 in these settings.

In Tables 1 and 2, we focus on results for normal outcomes that only correspond to β̂₁, as results for β̂₂ are similar and the intercept is not of interest. Furthermore, we present two sets of results based on the use of either α̂(β) (no resulting covariance inflation) or α̂(β̃) (results in covariance inflation, as will realistically be the case in practice) within Equation (1), denoted by “Theoretical Analyses” and “Realistic Analyses”, respectively. SE estimates corresponding to the theoretical and realistic analyses are obtained from Σ̂ and (I_p+Ĝ)Σ̂(I_p+Ĝ)^T, respectively, and are denoted by SE_T and SE_R. We note that Σ̂ is Σ̂_LZ with either the Kauermann and Carroll [7] or Mancl and DeRouen [8] correction. For each type of analysis, we present empirical standard deviations (ESDs) of β̂₁, empirical means of SE estimates and corresponding 95% confidence interval (CI) empirical coverage probabilities (CPs). Variance inflation does not occur with the Theoretical Analyses, in which case SE_R is not applicable and is therefore not presented. As in Westgate [2], CIs use critical values based on a t-distribution with N − p degrees of freedom. The working correlation structures for which results are presented are exchangeable, AR-1, and Toeplitz. In Table 1, we present results from when the Kauermann and Carroll [7] correction is used, whereas in Table 2 the Mancl and DeRouen [8] correction is utilized. Results are from settings in which the true structure is exchangeable with α_exch = 0.5. Results for other true structures do not provide additional insight. We therefore include corresponding results from a true AR-1 structure with α_AR−1 = 0.5 in Supplementary Material. For the same reason, results from settings in which correlated outcomes were binary are also included in Supplementary Material.

Table 1.

Empirical standard deviations (ESD) (in bold, underneath β̂₁), empirical mean standard error (SE) estimates (underneath β̂₁) and corresponding empirical 95% confidence interval coverage probabilities (CP) for both Theoretical and Realistic GEE Analyses. The Kauermann and Carroll [7] correction is utilized.

				Theoretical Analyses		Realistic Analyses
Working Correlation	N	n	SE Estimator	β̂1	CP	β̂1	CP
			ESD	0.713		0.892
	10	2	SET	0.675	0.940	0.704	0.901
			SER			0.870	0.942
Any
			ESD	0.305		0.309
	50	2	SET	0.300	0.944	0.300	0.942
			SER			0.307	0.947
			ESD	0.434		0.441
	10	4	SET	0.425	0.952	0.425	0.948
			SER			0.438	0.954
Exchangeable
			ESD	0.191		0.192
	50	4	SET	0.193	0.953	0.193	0.953
			SER			0.194	0.955
			ESD	0.459		0.472
	10	4	SET	0.444	0.951	0.443	0.946
			SER			0.466	0.952
AR-1
			ESD	0.201		0.202
	50	4	SET	0.206	0.955	0.206	0.953
			SER			0.207	0.955
			ESD	—		—
	10	4	SET	—	0.946	—	0.906
			SER			—	0.940
			ESD	0.287		0.303
Toeplitz	25	4	SET	0.274	0.945	0.275	0.925
			SER			0.299	0.942
			ESD	0.190		0.196
	50	4	SET	0.192	0.951	0.192	0.943
			SER			0.198	0.953

N - number of independent subjects; n - number of repeated measurements per subject

SE_T, theoretical SE estimate obtained from Σ̂ that assumes correlation parameters are known

SE_R, realistic SE estimate obtained from 9 (I_p + Ĝ)Σ̂(I_p + Ĝ)^T

GEE-generalized estimating equations

Theoretical Analyses use α̂(β) within Equation (1)

Realistic Analyses use α̂(β̃) within Equation (1)

Table 2.

Empirical standard deviations (ESD) (in bold, underneath β̂₁), empirical mean standard error (SE) estimates (underneath β̂₁) and corresponding empirical 95% confidence interval coverage probabilities (CP) for both Theoretical and Realistic GEE Analyses. The Mancl and DeRouen [8] correction is utilized.

				Theoretical Analyses		Realistic Analyses
Working Correlation	N	n	SE Estimator	β̂1	CP	β̂1	CP
			ESD	0.713		0.892
	10	2	SET	0.796	0.963	0.885	0.938
			SER			1.838*	0.969
Any
			ESD	0.305		0.309
	50	2	SET	0.308	0.952	0.308	0.947
			SER			0.315	0.953
			ESD	0.434		0.441
	10	4	SET	0.465	0.973	0.465	0.970
			SER			0.482	0.975
Exchangeable
			ESD	0.191		0.192
	50	4	SET	0.196	0.959	0.196	0.959
			SER			0.197	0.960
			ESD	0.459		0.472
	10	4	SET	0.491	0.966	0.490	0.961
			SER			0.518	0.968
AR-1
			ESD	0.201		0.202
	50	4	SET	0.210	0.959	0.209	0.958
			SER			0.211	0.961
			ESD	—		—
	10	4	SET	—	0.966	—	0.940
			SER			—	0.957
			ESD	0.287		0.303
Toeplitz	25	4	SET	0.289	0.955	0.311	0.934
			SER			0.333	0.950
			ESD	0.190		0.196
	50	4	SET	0.195	0.956	0.195	0.950
			SER			0.201	0.959

N - number of independent subjects; n - number of repeated measurements per subject

SE_T, theoretical SE estimate obtained from Σ̂ that assumes correlation parameters are known

SE_R, realistic SE estimate obtained from 10 (I_p + Ĝ)Σ̂(I_p + Ĝ)^T

GEE-generalized estimating equations

Theoretical Analyses use α̂(β) within Equation (1)

Realistic Analyses use α̂(β̃) within Equation (1)

^*Empirical mean influenced by outlying estimates

In Tables 3 and 4, we present results from the use of unpenalized, based on Σ̂ and thus unrealistically assuming correlation parameters are known, and penalized, based on (I_p + Ĝ)Σ̂(I_p + Ĝ)^T, versions of the CIC and TECM to select either independence, exchangeable, AR-1, or Toeplitz working correlation matrices for normal outcomes. Corresponding results for binary outcomes are given in Supplementary Material. For each version of each criterion, we present the number of times each structure was selected, with the goal of selecting the true structure as often as possible. For each setting, the true structure is either exchangeable or AR-1 with true parameter value of 0.5, and n = 4. In Table 3, we present results based on the incorporation of the Kauermann and Carroll [7] correction, whereas in Table 4 the Mancl and DeRouen [8] correction is utilized.

Table 3.

Empirical frequencies of selecting each working correlation structure out of 1,000 replications from the use of the given correlation selection criterion. The Kauermann and Carroll [7] correction is utilized.

			Selection Frequencies
True			Ind	Exch	AR-1	Toeplitz
Correlation	N	Criterion
Exchangeable	25	TECMT	18	376	113	493
		TECMR	33	651	150	166
		CICT	16	285	95	604
		CICR	28	690	134	148
Exchangeable	50	TECMT	0	393	62	545
		TECMR	1	699	91	201
		CICT	0	277	52	671
		CICR	1	719	78	202
AR-1	25	TECMT	15	97	315	573
		TECMR	33	167	594	206
		CICT	12	65	277	646
		CICR	36	158	625	181
AR-1	50	TECMT	1	44	331	624
		TECMR	2	83	669	246
		CICT	0	21	256	723
		CICR	1	81	675	243

N - number of independent subjects

Ind - Independence; Exch - Exchangeable

TECM - ‘trace of the empirical covariance matrix’ criterion

CIC - ‘correlation information criterion’

T - No penalty: The criterion is based on Σ̂ that theoretically assumes correlation parameters are known

R - Penalty: The criterion is based on (I_p + Ĝ)Σ̂(I_p + Ĝ)^T that realistically accounts for, or penalizes, covariance inflation due to correlation parameter estimation

Table 4.

Empirical frequencies of selecting each working correlation structure out of 1,000 replications from the use of the given correlation selection criterion. The Mancl and DeRouen [8] correction is utilized.

			Selection Frequencies
True			Ind	Exch	AR-1	Toeplitz
Correlation	N	Criterion
Exchangeable	25	TECMT	16	387	109	488
		TECMR	31	669	141	159
		CICT	15	299	90	596
		CICR	24	705	126	145
Exchangeable	50	TECMT	0	405	57	538
		TECMR	1	706	90	203
		CICT	0	285	48	667
		CICR	1	730	74	195
AR-1	25	TECMT	17	115	328	540
		TECMR	33	184	599	184
		CICT	14	74	287	625
		CICR	34	168	628	170
AR-1	50	TECMT	1	48	343	608
		TECMR	2	86	670	242
		CICT	0	24	274	702
		CICR	1	86	678	235

N - number of independent subjects

Ind - Independence; Exch - Exchangeable

TECM - ‘trace of the empirical covariance matrix’ criterion

CIC - ‘correlation information criterion’

T - No penalty: The criterion is based on Σ̂ that theoretically assumes correlation parameters are known

R - Penalty: The criterion is based on (I_p + Ĝ)Σ̂(I_p + Ĝ)^T that realistically accounts for, or penalizes, covariance inflation due to correlation parameter estimation

3.2. Results

Numerical instability was encountered with the working Toeplitz structure when N = 10. In such instances, we do not present ESDs or empirical means of SE estimates because they were highly influenced. However, because stable results were observed for the analyses of most simulated datasets, empirical CPs are still presented. Furthermore, we include an additional setting with N = 25, as this is a small-sample setting in which results were stable.

For theoretical analyses, the ESD of β̂₁ and corresponding empirical mean of SE_T were typically close in value when utilizing the Kauermann and Carroll [7] correction (Table 1). Furthermore, corresponding CPs were often relatively close to the nominal 0.95 value. Specifically, to determine if empirical CPs are acceptably close to 0.95, we note that empirical CPs between 0.936 and 0.964 have corresponding 95% CIs that cover 0.95. In short, these results suggest that if β did not have to be estimated for use in α̂ within Equation (1), then SE estimates obtained from Σ̂ and utilizing the Kauermann and Carroll [7] correction would result in valid inference. Alternatively, the Mancl and DeRouen [8] correction (Table 2) sometimes yielded positive bias in SE estimates, particularly for small N, and thus resulted in over-coverage of CIs in such settings.

ESDs of β̂₁ from the realistic analyses were greater than the corresponding ESDs from the theoretical analyses, demonstrating that variance inflation does occur when GEE incorporates structured working correlation matrices due to the need for replacing β with β̃ inside α̂ within Equation (1). However, empirical means for SE_T were approximately the same for both theoretical and realistic analyses in most settings. Therefore, when utilizing the Kauermann and Carroll [7] correction, SE_T was negatively biased, or smaller than the ESD, for the realistic analyses in some settings. This bias was also inherently observed via the degree of undercoverage by the CI that is constructed with SE_T in the realistic analyses. In contrast, use of the inflation correction worked very well at approximating the magnitude of variance inflation, and therefore improved inference overall when used in conjunction with the Kauermann and Carroll [7] correction. Specifically, utilizing SE_R notably reduced bias relative to SE_T, and typically resulted in near-nominal empirical CPs (Table 1). Alternatively, when utilizing the Mancl and DeRouen [8] correction (Table 2), use of the covariance inflation correction was often not needed due to the Mancl and DeRouen [8] correction resulting in a positively biased estimate for Σ. However, use of both corrections did perform best when incorporating a Toeplitz working structure.

The magnitude of covariance inflation that occurred, and therefore the need for the inflation correction, depended on n, N, and the number of estimated correlation parameters. For instance, a more notable variance inflation occurred when n = 2 (in which case the three working correlation structures are equivalent), particularly for N = 10, relative to the use of exchangeable or AR-1 when n = 4 because fewer empirical correlations were used to estimate the single correlation parameter. Furthermore, the magnitude of inflation increased as N decreased or the dimension of α̂ increased. For AR-1 and exchangeable, only one correlation parameter was estimated. Therefore, in settings in which n = 4, the covariance inflation was very small, especially when N = 50. Due to this result, SE_T and SE_R were similar, on average, in these settings. More notable inflations occurred with the Toeplitz structure due to the need to estimate three correlation parameters when n = 4. We further note that the number of parameters this structure estimates increases with n. Therefore, the need for the covariance inflation correction will be more apparent for larger values of n when utilizing this structure.

Although it is ideal to select the true structure, either exchangeable or AR-1, in Tables 3 and 4, unpenalized versions of the TECM and CIC using Σ̂ selected Toeplitz more often. However, penalized versions using (I_p + Ĝ)Σ̂(I_p + Ĝ)^T appropriately took into account the degree of covariance inflation that occurs with each of these structures, and therefore the corresponding penalized versions of these criteria correctly chose the true, simpler structure much more frequently, greatly reducing the number of times Toeplitz was selected. We note that the selection accuracy of the penalized criteria enhanced as N increased, because (I_p + Ĝ)Σ̂(I_p + Ĝ)^T is estimated more precisely. Another interesting result is that selection frequencies were similar whether using the Kauermann and Carroll [7] or Mancl and DeRouen [8] correction. In short, the TECM and CIC were not notably influenced by this type of correction for Σ̂_LZ, whereas use of the covariance inflation correction with all working structures under consideration greatly improved the performances of the TECM and CIC.

4. Concluding Remarks

With GEE, correlation parameters are estimated, therefore potentially inflating the covariance matrix of the regression parameter estimates. Westgate [2] derived an approximation for this inflation when utilizing an unstructured working correlation matrix, and Westgate [6] proposed the use of this approximation to penalize the estimation of the unstructured matrix’s parameters. In this manuscript, we showed that the resulting corrected version of the well-known sandwich covariance formula and the use of this correction as a correlation selection penalty are also applicable when GEE incorporates structured working correlation matrices. In our study, use of the corrected formula improved standard error estimation, and thus the validity of inference, when the Kauermann and Carroll [7] correction was used. Alternatively, when the Mancl and DeRouen [8] correction was used, the covariance inflation correction appeared to be useful for attaining valid inference only when GEE incorporated a working Toeplitz structure, as the Mancl and DeRouen [8] correction often over-corrected for the bias in the Liang and Zeger [1] empirical sandwich estimator. Furthermore, irrelevent of which correction is applied to the Liang and Zeger [1] empirical sandwich estimator, use of the covariance inflation correction as a penalty greatly improved correlation structure selection accuracy.

Simulation results showed that, even for small N, the inflation of the variances of regression parameter estimates can be negligible for the AR-1 and exchangeable structures. Therefore, it is no surprise that, to our knowledge, this variance inflation has gone relatively unnoticed in practice with these working structures that require the estimation of only one nuisance parameter. This also implies that the inflation correction is often not needed to penalize these structures when compared against the working independence structure, which is a comparison that is routinely demonstrated in the GEE correlation selection literature. However, the need for the covariance inflation correction can be apparent for structures that require multiple nuisance parameters to be estimated. Another situation in which multiple correlation parameters may be estimated is when different trial arms, for instance, are allowed to have different exchangeable or AR-1 parameter values, in which case the covariance inflation correction can be useful. Furthermore, the need for the covariance inflation correction increases as the number of independent clusters decreases.

An alternative approach to GEE is the quadratic inference function (QIF) method [18]. Theoretically, the QIF approach is equally or more efficient than GEE. However, finite-sample covariance inflation of the regression parameter estimates must be taken into account, as is done in Westgate [19, 20]. Westgate [20] proposed a method that utilizes the TECM to select both a working correlation structure and one of these two methods, analogous to the approach we used in this manuscript. Therefore, our study results imply that the covariance inflation correction can also be used with GEE incorporating structured working correlation matrices in the context of Westgate [20].

An R function that implements GEE and outputs results based on the methods presented in this manuscript can be found in Supplementary Material or obtained by contacting the author.