3.2. Quadratic inference function (QIF)

The approach of quadratic inference function (QIF) was introduced to improve the efficiency of GEE in longitudinal data analysis under correlation misspecification [19]. The QIF approach expresses the inverse of the working correlation matrix as a linear combination of K basis matrices: R−1(α)=∑k=1KγkMk with Mk as the kth basis matrix and γk the weight. The first basis matrix is usually specified as the identity matrix M1=I. A (Kp)-dimensional score vector is then defined as

Let g‾N(β)=N−1∑i=1Ngi(β) be the list of extended score equations and CN(β)=N−1∑i=1Ngi(β)giT(β) be the empirical covariance matrix of gi(β). The QIF is written as QN(β)=Ng‾NT(β)CN−1(β)g‾N(β). Based on the generalized method of moments (GMM) [20], the estimator βˆ=argminβQN(β) can be more efficient than the GEE estimator in large samples when the working correlation is misspecified. From the first derivative of QN, the QIF estimator obtained by minimizing the QN function is asymptotically equivalent to solving WN(β)=N(∂g‾N/∂βT)CN−1g‾N=0 [19], and the Newton-Raphson algorithm can be used to iteratively update the estimator for β until convergence [9]. A consistent variance estimator of the QIF estimator βˆ then has a sandwich form

Due to the theoretical efficiency improvement of QIF over GEE, there has been increasing efforts in developing the theory of QIF for analyzing correlated data, including the following examples concerning longitudinal data analysis. A QIF likelihood-ratio test statistic, with asymptotic Chi-squared distribution, was proposed and shown useful to test for goodness-of-fit in longitudinal studies [19]. A penalized version of QIF can further accommodate the variable selection [21]. The Godambe Information (TGI) criterion and the trace of the empirical covariance matrix were developed to select the appropriate correlation structure [22,23]. The QIF approach has also been shown to automatically down-weight outlying observations [8], while GEE has unbounded influence function and can be sensitive to outliers. In addition, the QIF approach can also be utilized for meta-analysis with a flexible joint estimation procedure [24]. In small to moderately-sized samples, it was found that the standard errors of parameters can be severely downward-biased and two biased-corrected covariance estimators have shown to provide adequate finite-sample adjustments [9]. Furthermore, it was shown that imbalance of covariate distributions and of cluster sizes can also lead to larger variability of the QIF estimator [25].

In contrast to the longitudinal data setting, in a CRT with a single follow-up time-point, there is no natural ordering or structure of individuals in the same cluster. That is, decay-type structures are not appropriate but the exchangeable working correlation structure with a common ICC parameter is a natural choice. For the exchangeable working correlation structure, we can write R(α)=(1−ρ)I+ρJ, where J is a (m×m)-dimensional matrix of 1's. Following Li et al. [26], we have R−1(α)=(1−ρ)−1I−ρ(1−ρ)−1{1+(m−1)ρ}−1J. As J is not a full-rank matrix, a better use is to specify I and J−I as two full-rank basis matrices for exchangeable correlation structure of QIF [9]. We can also use QIF with an independence correlation structure with only one basis matrix of the identity matrix I. Parallel to Result 1, we provide an additional insight that the results obtained from GEE and QIF are equal under conditions assumed below.

Under the exchangeable working correlation, QIF and GEE produce identical point estimates and robust covariances if two conditions are satisfied: (1) the marginal mean model only includes cluster-level covariates; (2) equal cluster sizes.

The proof of Result 2 is provided in Appendix I.

Furthermore, it has been pointed out previously that QIF and GEE are identical when the independence working correlation is used [8,25]. Combining Result 1 and Result 2, we further summarize an additional result as follows.

The point estimates and robust covariances are identical using either GEE or QIF with either independence or exchangeable working correlation, if the following two conditions are satisfied: (1) the marginal mean model only includes cluster-level covariates; (2) equal cluster sizes.

The proof of Result 3 is also provided in Appendix I.

As previously indicated, these results have implications for sample size procedures assuming equal cluster sizes, in which cases the sample requirements will be equivalent using either GEE or QIF coupled with either independence and exchangeable correlation structures.