Marginal models are often fitted using the GEE methodology, whereby the relationship between the response and covariates is modeled separately from the correlation between repeated measurements on the same individual 2.

The correlation between successive measurements is modeled explicitly by assuming a "correlation structure" or "working correlation matrix". The assumption of a correlation structure facilitates the estimation of model parameters 2. Examples of working correlation matrices include: exchangeable, auto-regressive of order 1 (AR(1)), unstructured, and independent correlation structures2. For binary data, correlation is often measured in terms of odds ratios 11. A plausible working correlation matrix can be chosen using a visual tool known as the lorelogram 11.

Details of the correlation structure and response-covariate relationship are included in an expression known as the quasi-likelihood function2, which is iteratively solved to obtain parameter estimates. Estimates obtained from the quasi-likelihood function are efficient when the true correlation matrix is closely approximated '[see Additional file 1]'. In other words, the large-sample variance of the estimator reaches a Cramer-Rao type lower bound3 '[see Additional file 2]'.

The pros and cons of using GEE are summarized in Table 1.

The QIF methodology overcomes some of the disadvantages of GEE highlighted in Table 13. It is largely based on observing that the inverse of many commonly used working correlation matrices can be expressed as a linear combination of unknown constants and known matrices '[see Additional file 2]'. This linear expression is substituted back into the quasi-likelihood function from which an extended score vector 3 is obtained. Qu et al 3 used the generalized method of moments 12 to obtain an objective function consisting of the extended score vector and its inverse variance matrix. This function is termed the "Quadratic Inference Function", which is minimized through a numerical algorithm to obtain parameter estimates '[see Additional file 2]'.

The estimates obtained from QIF are as efficient as those from the quasi-likelihood function provided the true correlation structure is specified. Further, the estimates obtained from QIF are still efficient, even if the correlation structure is misspecified 3. This is confirmed from simulation results obtained by Qu et al 3 comparing the simulated relative efficiency (SRE) of parameter estimators from GEE and QIF:

SRE = mean squared error of GEE estimator mean squared error of QIF estimator . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@8A94@

Given a true correlation structure of AR(1) and a correlation of 0.7 between repeated observations, Qu et al 3 obtained an SRE of 1.34 (QIF more efficient) if the working correlation structure is misspecified as "equicorrelated". An SRE of 2.07 (QIF more efficient) was obtained if a true equicorrelated structure is misspecified as AR(1). SRE is in the range 0.97–0.99 if correlation structure is correctly specified, meaning GEE and QIF are similarly efficient 3. The reliability of these simulation results is assessed in this paper.

The pros and cons of using QIF are listed in Table 1.

The NLSCY dataset consists of long-term, population-based survey data collected since 1994, and is designed to evaluate the determinants of developmental outcomes of Canadian children and youth. Each two-year period from 1994 constitutes a cycle 13.

For this paper, we selected a sub-sample of children meeting the inclusion criteria outlined below.

Child must be four or five years old in Cycle 1 of the survey. Child must also have complete data (Cycles 1 to 4) on the following variables: hyperactivity-inattention, age, gender, family functioning, maternal (or person most knowledgeable) depression, household income adequacy, maternal immigration status and maternal educational level. The "person most knowledgeable" (PMK) is usually the child's mother 13.

From a total of 2,090 (weighted sample of 795,856) four to five year olds in Cycle 1, a sub-sample of 1,052 (weighted sample of 384,306) children met the inclusion criteria outlined above. A flowchart of this process is shown in Figure 1.

Summary statistics are expressed as count (percent). Hyperactivity-inattention is expressed as a function of time, gender, family functioning, maternal depression, maternal immigration status, household income adequacy and maternal educational level using marginal logistic regression models in GEE and QIF (Equations 8 and 9). The "adjusted Cycle 4 longitudinal weight" is included as a weight variable in the GEE and QIF models to account for study design.

Logit(μ_ij) = α + β₁t_j + β₂gender + β₃FF + β₄MD + β₅MIS + β₆ME + β₇IA

logit( μ i j ) = α + β 1 t j + β 1 ∗ t j 2 + β 2 g e n d e r + β 3 F F + β 4 M D + β 5 M I S + β 6 M E + β 7 I A MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@75A2@

The goodness-of-fit (GOF) test in QIF is used for model assessment. We compared the fit of different models using the Q statistic 3 and its extensions such as AIC (Akaike Information Criterion) and BIC (Bayes Information Criterion). Smaller Qs, AICs and BICs indicate better fits 13.

QIF and GEE are compared with respect to relative efficiency of parameter estimates. We also illustrate how to use the GOF statistic from QIF in selecting an optimal working correlation matrix between AR(1) and exchangeable correlation structures. All statistical tests were conducted at 5% level of significance.

Graphs and analyses results were obtained using SAS^© (Version 9.1), SPSS^© (Version 14.0) and R (Version 2.5.1).

Table 2 represents the weighted frequencies of the baseline and follow-up characteristics of the study population. Figure 2 shows the estimated proportion of hyperactive-inattention among the selected cohort between 1994 and 2000. The graph is not linear. Hyperactivity-inattention appeared to diminish as children in this cohort grew older.

Figure 3 is a lorelogram which measures the correlation between repeated binary outcomes using odds ratios 11. The x-axis (index) is the time-lag between two measurements. From Figure 3, correlation appears to decrease with increasing lag between repeated responses, thus an AR(1) correlation structure may be appropriate for describing the relationship between hyperactivity-inattention scores at different cycles.

Results in Table 3 were obtained from fitting Model (8) in GEE and QIF, and assuming an AR(1) correlation structure. GEE and QIF produce different conclusions for maternal education level and family functioning, although the odds ratios are of similar magnitudes. Figure 2 shows that a quadratic term may be required to improve model fit, but GEE does not provide a goodness-of-fit test with, for instance, the SAS^© implementation.

In addition to the results obtained in Table 3, QIF – in contrast to GEE – provides direct measures of goodness-of-fit (GOF) with SAS^© software output to assess model adequacy 3. QIF facilitates comparison among different plausible models using the Q statistic 3. The Q statistic is obtained from the asymptotic limiting distribution of the quadratic inference function (QIF). Just like the likelihood ratio test, it enables one to test the null hypothesis that a simpler model is just as predictive as a saturated model. The difference between QIF (for saturated model) and QIF (for simpler model) is asymptotically chi-squared under the null hypothesis irrespective of the underlying true correlation structure. This difference is asymptotically non-central chi-squared under the local alternative hypothesis 3. The mathematical proof and simulation results are found in Qu et al 3. The Q statistic has properties similar to the likelihood ratio test used for generalized linear models 3. Thus, extensions of the Q statistic such as AIC (Akaike Information Criterion) and BIC (Bayes Information Criterion) can also be used to compare the fit of different models. In comparison to a saturated model, a fitted model is considered inadequate if the p-value for the goodness-of-fit test is less than 0.05 3.

From Table 4, GOF tests show that Model (8) is inadequate to describe the observed data (GOF statistic Q = 22.82, p = 0.0066; AIC = 40.82, BIC = 85.45). Considering the non-linearity of Figure 2, a quadratic term was added to Model (8) to obtain Model (9). Model (9) appears to provide a better fit (GOF statistic Q = 11.74; p = 0.3027; AIC = 31.74, 81.33).

Table 5 shows parameter estimates for GEE and QIF using Model (9). The quadratic term (t²) is statistically significant in both GEE and QIF (p < 0.05). Also, the results from GEE and QIF appear to be in agreement in Model 9, suggesting that the results are robust. Next we provide the clinical implication of the results.

✔ Male children have significantly higher odds of developing hyperactivity-inattention than their female counterparts (OR = 1.73, 95% CI of 1.10 to 2.71).

✔ Children from dysfunctional families have significantly higher odds of developing hyperactivity-inattention than those from non-dysfunctional families (OR = 2.84, 95 CI of 1.58 to 5.11).

✔ Children of moderate to severely depressed mothers have significantly higher odds of developing hyperactivity-inattention than those whose mothers are not depressed (OR = 2.49, 95% CI of 1.60 to 2.60).

✔ Children of immigrants, children with mothers having university/college degree and children in the high income adequacy group have lower estimated odds of developing hyperactivity-inattention (OR = 0.69, 95% CI of 0.35 to 1.37; OR = 0.59, 95% CI of 0.34 to 1.02; and OR = 0.95, 95% CI of 0.58 to 1.57 respectively). The odds ratios in these three situations are not statistically significant.

QIF facilitates an optimal choice among the available correlation structures. Assuming Model (9), Table 6 shows the results for AR(1) and exchangeable correlation structures. The results for both correlation structures are similar, but AR(1) is the more appropriate working correlation matrix from the goodness-of-fit tests in Table 7. This is also supported by the lorelogram in Figure 3.

We compared the efficiency of parameter estimates from QIF and GEE using:

Relative Efficiency (RE) = mean square error of estimate from GEE  mean square error of estimate from QIF = trace of covariance matrix of parameter estimates from GEE trace of covariance matrix of parameter estimates from QIF = sum of squares of SEs from GEE estimates sum of squares of SEs from QIF estimates MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeWabmqaaaqaaiabbkfasjabbwgaLjabbYgaSjabbggaHjabbsha0jabbMgaPjabbAha2jabbwgaLjabbccaGiabbweafjabbAgaMjabbAgaMjabbMgaPjabbogaJjabbMgaPjabbwgaLjabb6gaUjabbogaJjabbMha5jabbccaGiabbIcaOiabbkfasjabbweafjabbMcaPGGaaiab=1da9KqbaoaalaaabaGaeeyBa0MaeeyzauMaeeyyaeMaeeOBa4MaeeiiaaIaee4CamNaeeyCaeNaeeyDauNaeeyyaeMaeeOCaiNaeeyzauMaeeiiaaIaeeyzauMaeeOCaiNaeeOCaiNaee4Ba8MaeeOCaiNaeeiiaaIaee4Ba8MaeeOzayMaeeiiaaIaeeyzauMaee4CamNaeeiDaqNaeeyAaKMaeeyBa0MaeeyyaeMaeeiDaqNaeeyzauMaeeiiaaIaeeOzayMaeeOCaiNaee4Ba8MaeeyBa0MaeeiiaaIaee4raCKaeeyrauKaeeyrauKaeeiiaacabaGaeeyBa0MaeeyzauMaeeyyaeMaeeOBa4MaeeiiaaIaee4CamNaeeyCaeNaeeyDauNaeeyyaeMaeeOCaiNaeeyzauMaeeiiaaIaeeyzauMaeeOCaiNaeeOCaiNaee4Ba8MaeeOCaiNaeeiiaaIaee4Ba8MaeeOzayMaeeiiaaIaeeyzauMaee4CamNaeeiDaqNaeeyAaKMaeeyBa0MaeeyyaeMaeeiDaqNaeeyzauMaeeiiaaIaeeOzayMaeeOCaiNaee4Ba8MaeeyBa0MaeeiiaaIaeeyuaeLaeeysaKKaeeOrayeaaaGcbaGaeyypa0tcfa4aaSaaaeaacqqG0baDcqqGYbGCcqqGHbqycqqGJbWycqqGLbqzcqqGGaaicqqGVbWBcqqGMbGzcqqGGaaicqqGJbWycqqGVbWBcqqG2bGDcqqGHbqycqqGYbGCcqqGPbqAcqqGHbqycqqGUbGBcqqGJbWycqqGLbqzcqqGGaaicqqGTbqBcqqGHbqycqqG0baDcqqGYbGCcqqGPbqAcqqG4baEcqqGGaaicqqGVbWBcqqGMbGzcqqGGaaicqqGWbaCcqqGHbqycqqGYbGCcqqGHbqycqqGTbqBcqqGLbqzcqqG0baDcqqGLbqzcqqGYbGCcqqGGaaicqqGLbqzcqqGZbWCcqqG0baDcqqGPbqAcqqGTbqBcqqGHbqycqqG0baDcqqGLbqzcqqGZbWCcqqGGaaicqqGMbGzcqqGYbGCcqqGVbWBcqqGTbqBcqqGGaaicqqGhbWrcqqGfbqrcqqGfbqraeaacqqG0baDcqqGYbGCcqqGHbqycqqGJbWycqqGLbqzcqqGGaaicqqGVbWBcqqGMbGzcqqGGaaicqqGJbWycqqGVbWBcqqG2bGDcqqGHbqycqqGYbGCcqqGPbqAcqqGHbqycqqGUbGBcqqGJbWycqqGLbqzcqqGGaaicqqGTbqBcqqGHbqycqqG0baDcqqGYbGCcqqGPbqAcqqG4baEcqqGGaaicqqGVbWBcqqGMbGzcqqGGaaicqqGWbaCcqqGHbqycqqGYbGCcqqGHbqycqqGTbqBcqqGLbqzcqqG0baDcqqGLbqzcqqGYbGCcqqGGaaicqqGLbqzcqqGZbWCcqqG0baDcqqGPbqAcqqGTbqBcqqGHbqycqqG0baDcqqGLbqzcqqGZbWCcqqGGaaicqqGMbGzcqqGYbGCcqqGVbWBcqqGTbqBcqqGGaaicqqGrbqucqqGjbqscqqGgbGraaaakeaacqGH9aqpjuaGdaWcaaqaaiabbohaZjabbwha1jabb2gaTjabbccaGiabb+gaVjabbAgaMjabbccaGiabbohaZjabbghaXjabbwha1jabbggaHjabbkhaYjabbwgaLjabbohaZjabbccaGiabb+gaVjabbAgaMjabbccaGiabbofatjabbweafjabbohaZjabbccaGiabbAgaMjabbkhaYjabb+gaVjabb2gaTjabbccaGiabbEeahjabbweafjabbweafjabbccaGiabbwgaLjabbohaZjabbsha0jabbMgaPjabb2gaTjabbggaHjabbsha0jabbwgaLjabbohaZbqaaiabbohaZjabbwha1jabb2gaTjabbccaGiabb+gaVjabbAgaMjabbccaGiabbohaZjabbghaXjabbwha1jabbggaHjabbkhaYjabbwgaLjabbohaZjabbccaGiabb+gaVjabbAgaMjabbccaGiabbofatjabbweafjabbohaZjabbccaGiabbAgaMjabbkhaYjabb+gaVjabb2gaTjabbccaGiabbgfarjabbMeajjabbAeagjabbccaGiabbwgaLjabbohaZjabbsha0jabbMgaPjabb2gaTjabbggaHjabbsha0jabbwgaLjabbohaZbaaaaaaaa@A56A@

provided the estimates are unbiased estimates of the parameters of interest 22. This definition of RE generalizes to situations in which there are multiple parameters to be estimated. Thus from Table 8 with AR(1) correlation structure, one obtains

RE = 1.1117.

Using exchangeable working correlation, RE is 1.3082 (see Table 9). This implies that QIF parameter estimates are more efficient than GEE estimates assuming AR(1) or exchangeable correlation structures. This is consistent with the simulation results obtained by Qu et al 3.