Analysis of Longitudinal Multinomial Outcome Data

Purpose: To describe and demonstrate methods for analyzing longitudinal correlated eye data with a continuous outcome measure. Methods: We described fixed effects, mixed effects and generalized estimating equations (GEE) models, applied them to data from the Complications of Age-Related Macular Degeneration Prevention Trial (CAPT) and the Age-Related Eye Disease Study (AREDS). In CAPT (N = 1052), we assessed the effect of eye-specific laser treatment on change in visual acuity (VA). In the AREDS study, we evaluated effects of systemic supplement treatment among 1463 participants with AMD category 3. Results: In CAPT, the inter-eye correlations (0.33 to 0.53) and longitudinal correlations (0.31 to 0.88) varied. There was a small treatment effect on VA change (approximately one letter) at 24 months for all three models (p = .009 to 0.02). Model fit was better with the mixed effects model than the fixed effects model (p < .001). In AREDS, there was no significant treatment effect in all models (p > .55). Current smokers had a significantly greater VA decline than non-current smokers in the fixed effects model (p = .04) and the mixed effects model with random intercept (p = .0003), but marginally significant in the mixed effects model with random intercept and slope (p = .08), and GEE models (p = .054 to 0.07). The model fit was better with the fixed effects model than the mixed effects model (p < .0001). Conclusion: Longitudinal models using the eye as the unit of analysis can be implemented using available statistical software to account for both inter-eye and longitudinal correlations. Goodness-of-fit statistics may guide the selection of the most appropriate model.

In order to better understand aging, longitudinal studies are run in which participants are evaluated repeatedly and selected end-points (e.g., score on a cognitive screen, falls, occurrence/reoccurrence of a condition) are examined. The objective of the present paper is primarily to describe the methods available that take into account correlation between binary outcomes, and in particular to model the association of binary outcomes after controlling for covariates by using an implementation of generalized estimating equations (GEE) called ‘alternating logistic regression’ (ALR). In GEE, association within longitudinal outcomes is accounted for but not estimated. Alternating logistic regression, however, basically enables simultaneous estimation of pair-wise odds ratios of outcomes within a cluster, while accounting for the dependence of the outcome on covariates. A sub-sample (n = 2458) from a community-based sample of Duke Established Populations for Epidemiologic Studies of the Elderly is used. In the example used here, logistic regression using GEE and ALR is used to model binary outcomes at three time points (baseline, three and six years later) and to control for covariates in a representative community-based sample 65 years of age and older (n = 2458). The outcomes indicate any problem versus no problem on a five-item activities of daily living (ADL) scale in a community sample. The ALR model, however, provides insight into decline in ADL from baseline to each of the time-points whereas GEE does not. In both controlled and uncontrolled analyses, decline in ADL over three and six-year intervals (baseline to three years later, baseline to six years and three years post-baseline to six years post-baseline) is significant.

BackgroundAnalyses of multicenter studies often need to account for center clustering to ensure valid inference. For binary outcomes, it is particularly challenging to properly adjust for center when the number of centers or total sample size is small, or when there are few events per center. Our objective was to evaluate the performance of generalized estimating equation (GEE) log-binomial and Poisson models, generalized linear mixed models (GLMMs) assuming binomial and Poisson distributions, and a Bayesian binomial GLMM to account for center effect in these scenarios.MethodsWe conducted a simulation study with few centers (≤30) and 50 or fewer subjects per center, using both a randomized controlled trial and an observational study design to estimate relative risk. We compared the GEE and GLMM models with a log-binomial model without adjustment for clustering in terms of bias, root mean square error (RMSE), and coverage. For the Bayesian GLMM, we used informative neutral priors that are skeptical of large treatment effects that are almost never observed in studies of medical interventions.ResultsAll frequentist methods exhibited little bias, and the RMSE was very similar across the models. The binomial GLMM had poor convergence rates, ranging from 27% to 85%, but performed well otherwise. The results show that both GEE models need to use small sample corrections for robust SEs to achieve proper coverage of 95% CIs. The Bayesian GLMM had similar convergence rates but resulted in slightly more biased estimates for the smallest sample sizes. However, it had the smallest RMSE and good coverage across all scenarios. These results were very similar for both study designs.ConclusionsFor the analyses of multicenter studies with a binary outcome and few centers, we recommend adjustment for center with either a GEE log-binomial or Poisson model with appropriate small sample corrections or a Bayesian binomial GLMM with informative priors.

Introduction: Research suggests obesity may be transmitted through social networks. A possible pathway is social influence on food choice. We investigated peer influence on the healthfulness of worksite food choices using social network analysis in a large hospital employee population, hypothesizing that socially-connected employees’ food choices would be correlated. Methods: Data on all food purchases in 2016 were obtained from the hospital’s cash register database. The cafeteria system uses traffic light labels to mark foods as healthy (green), less healthy (yellow), or unhealthy (red). Employees’ food purchases were identified through the use of cafeteria debit cards; social ties among employees were inferred based on a validated algorithm using demographics and time/location of purchases. We used spatial autoregression (SAR) and generalized estimating equation (GEE) models to calculate associations between the proportions of employees’ and coworkers’ purchases that were labeled green (or red). SAR models assessed concurrent purchases of an employee and coworker, weighting the association between their purchases by the frequency and inferred strength of the social tie. GEE models assessed longitudinal relationships between purchases coworkers’ made in the presence of an employee in one 8-week period and the employee’s purchases in the next 8-week period. Food and beverages were analyzed separately. Models adjusted for employee and coworker confounders (age, sex, race/ethnicity, job type, education). Results: In all, 5,118 employees used cafeteria debit cards to make purchases. Up to 536,240 employee/coworker interactions were observed, depending on the model (SAR, GEE) and outcomes (green/red, food/beverages). SAR models showed that a 1 percentage point increase in the network-weighted average of coworkers’ green-labeled (healthy) food purchases was associated with a 0.39 percentage point increase in an employee’s concurrent green-labeled food purchases (p&lt;.001). Positive associations were also observed for red foods (0.20), and green (0.14) and red (0.31) beverages (all p&lt;.001). Longitudinal GEE models showed that employees, as a population, increased purchases of green-labeled items by 0.013 percentage points on average when coworkers with whom they visited cafeterias in the prior 8 weeks increased their purchases of green items by 1 percentage point (p&lt;.001). Similar associations were observed for red foods (0.013), and for green (0.006) and red beverages (0.020) (all p&lt;.003). Conclusions: Employees’ healthy and unhealthy food choices are correlated. Although one explanation is that people eat with others who have similar preferences, the longitudinal findings suggest that people create social norms for eating that influence peers. Worksite and other social networks may be novel targets for population-level interventions to promote healthy diet.

Acute kidney injury (AKI) is common in the intensive care unit, where it is associated with increased mortality. AKI is often defined using creatinine and urine output criteria. The creatinine-based definition is more reliable but less expedient, whereas the urine output based definition is rapid but less reliable. Our goal is to examine the urine output criterion and augment it with physiological features for better agreement with creatinine-based definitions of AKI. The objectives are threefold: (1) to characterize the baseline agreement of urine output and creatinine definitions of AKI; (2) to refine the urine output criteria to identify the thresholds that best agree with the creatinine-based definition; and (3) to build generalized estimating equation (GEE) and generalized linear mixed-effects (GLME) models with static and time-varying features to improve the accuracy of a near-real-time marker for AKI. We performed a retrospective observational study using data from two independent critical care databases, MIMIC-III and eICU, for critically ill patients who developed AKI in intensive care units. We found that the conventional urine output criterion (6 hr, 0.5 ml/kg/h) has specificity and sensitivity of 0.49 and 0.54 for MIMIC-III database; and specificity and sensitivity of 0.38 and 0.56 for eICU. Secondly, urine output thresholds of 12 hours and 0.6 ml/kg/h have specificity and sensitivity of 0.58 and 0.48 for MIMIC-III; and urine output thresholds of 10 hours and 0.6 ml/kg/h have specificity and sensitivity of 0.49 and 0.48 for eICU. Thirdly, the GEE model of four hours duration augmented with static and time-varying features can achieve a specificity and sensitivity of 0.66 and 0.61 for MIMIC-III; and specificity and sensitivity of 0.66 and 0.64 for eICU. The GLME model of four hours duration augmented with static and time-varying features can achieve a specificity and sensitivity of 0.71 and 0.55 for MIMIC-III; and specificity and sensitivity of 0.66 and 0.60 for eICU. The GEE model has greater performance than the GLME model, however, the GLME model is more reflective of the variables as fixed effects or random effects. The significant improvement in performance, relative to current definitions, when augmenting with patient features, suggest the need of incorporating these features when detecting disease onset and modeling at window-level rather than patient-level.

Introduction: Hypertension (HT) is a prevalent contributor to cardiovascular diseases. To evaluate the incidence of HT and its relationship with other cardio metabolic risk factors changes and lifestyle components, Healthy Heart Project follow up data for 8 years were analyzed. Methods: A total of 283 unique normotensive participants of Yazd Healthy Heart Project (1169 observations) attended blood pressure longitudinal study. Multi-level model and Generalized estimation equation (GEE) model with an unstructured correlation matrix were used to analyze the longitudinal data by Stata/MP 11.2. Results: The incidence rates of HT and prehypertension were 39.5 and 77.93 in each 1000 person-year followup, respectively. In addition, systolic blood pressure was significantly predicted by diastolic blood pressure (B=1.09, 95% CI=0.99-1.19, P<0.0001), uric acid (B=1.04, P=0.003), and waist circumference (B=0.131, P=0.033). Moreover, GEE model with logit link function showed that hypertension was significantly predicted by triglyceride (OR=1.003, P=0.044), cholesterol (OR=0.97, P=0.004), LDL-cholesterol (OR=1.02, P=0.003), uric acid (OR=1.19, P=0.023), body mass index (OR=1.09, P=0.028), and also significantly predicted by obesity (B=1.85, P=0.007), abdominal obesity (B=1.85, P=0.007), age (B=1.47, P=0.017), and diabetes mellitus (B=2.14, P=0.003). Conclusion: The study results showed that the incidence rate of HT was high in Yazd and the major predictors of systolic blood pressure were abdominal obesity and diastolic blood pressure. Besides, diastolic blood pressure was significantly determined by systolic blood pressure and general obesity. Moreover, both systolic and diastolic blood pressures were independently predicted by serum uric acid level.

BackgroundCluster randomised trials (CRTs) are commonly analysed using mixed-effects models or generalised estimating equations (GEEs). However, these analyses do not always perform well with the small number of clusters typical of most CRTs. They can lead to increased risk of a type I error (finding a statistically significant treatment effect when it does not exist) if appropriate corrections are not used.MethodsWe conducted a small simulation study to evaluate the impact of using small-sample corrections for mixed-effects models or GEEs in CRTs with a small number of clusters. We then reanalysed data from TRIGGER, a CRT with six clusters, to determine the effect of using an inappropriate analysis method in practice. Finally, we reviewed 100 CRTs previously identified by a search on PubMed in order to assess whether trials were using appropriate methods of analysis. Trials were classified as at risk of an increased type I error rate if they did not report using an analysis method which accounted for clustering, or if they had fewer than 40 clusters and performed an individual-level analysis without reporting the use of an appropriate small-sample correction.ResultsOur simulation study found that using mixed-effects models or GEEs without an appropriate correction led to inflated type I error rates, even for as many as 70 clusters. Conversely, using small-sample corrections provided correct type I error rates across all scenarios. Reanalysis of the TRIGGER trial found that inappropriate methods of analysis gave much smaller P values (P ≤ 0.01) than appropriate methods (P = 0.04–0.15). In our review, of the 99 trials that reported the number of clusters, 64 (65 %) were at risk of an increased type I error rate; 14 trials did not report using an analysis method which accounted for clustering, and 50 trials with fewer than 40 clusters performed an individual-level analysis without reporting the use of an appropriate correction.ConclusionsCRTs with a small or medium number of clusters are at risk of an inflated type I error rate unless appropriate analysis methods are used. Investigators should consider using small-sample corrections with mixed-effects models or GEEs to ensure valid results.

End-stage renal disease (ESRD) related to HIV is becoming a leading cause of renal replacement therapy requirement is some areas of the world. Our study aims to describe the incidence and renal outcomes of HIV-associated nephropathy (HIVAN), and immune-mediated kidney disease related to HIV (HIVICK) in Colombia. A retrospective cohort study was performed, including all HIVAN or HIVICK incident cases assessed by the infectious diseases division in a high complexity institution in Colombia, between 2004 and 2018. A longitudinal data model under the Generalized Estimating Equations (GEE) method was used to determine changes on the glomerular filtration rate (GFR) over time. Within a cohort composed by 1509 HIV-infected patients, we identified 22 with HIV-associated glomerular disease. Cumulative incidence was 1.45%. At diagnosis, GFR was above 30 mL/min in 90.8% of patients, and 77.2% displayed sub-nephrotic proteinuria. Factors associated with GFR at diagnosis were: level of CD4 (Coefficient 0.113, CI 95 %: 0.046, 0.179, p < 0.01), and the inverse of the CD4/CD8 ratio. The GEE model did not demonstrate significant changes in the GFR over a 3-year period. Findings were similar when comparing GFR at diagnosis with GFR at 12 (-3.9 mL/min/1.73m2, CI 95% -7.3, 0.4, p = 0.98), 24 (-2.47 mL/min/1.73m2, CI 95% -7.0, 2.1, p=0.85), and 36 months (0.39 mL/min/1.73m2, CI 95% -4.4, 5.2, p = 0.43) of follow-up. Patients with glomerular disease associated with HIV have stable GFR over a 3-year period, and low rates of progression towards dialysis requirement. Differences with previous reports could be related with early diagnosis and treatment with highly active antiretroviral therapy.

BackgroundMulticentre randomized controlled trials (RCTs) routinely use randomization and analysis stratified by centre to control for differences between centres and to improve precision. No consensus has been reached on how to best analyze correlated continuous outcomes in such settings. Our objective was to investigate the properties of commonly used statistical models at various levels of clustering in the context of multicentre RCTs.MethodsAssuming no treatment by centre interaction, we compared six methods (ignoring centre effects, including centres as fixed effects, including centres as random effects, generalized estimating equation (GEE), and fixed- and random-effects centre-level analysis) to analyze continuous outcomes in multicentre RCTs using simulations over a wide spectrum of intraclass correlation (ICC) values, and varying numbers of centres and centre size. The performance of models was evaluated in terms of bias, precision, mean squared error of the point estimator of treatment effect, empirical coverage of the 95% confidence interval, and statistical power of the procedure.ResultsWhile all methods yielded unbiased estimates of treatment effect, ignoring centres led to inflation of standard error and loss of statistical power when within centre correlation was present. Mixed-effects model was most efficient and attained nominal coverage of 95% and 90% power in almost all scenarios. Fixed-effects model was less precise when the number of centres was large and treatment allocation was subject to chance imbalance within centre. GEE approach underestimated standard error of the treatment effect when the number of centres was small. The two centre-level models led to more variable point estimates and relatively low interval coverage or statistical power depending on whether or not heterogeneity of treatment contrasts was considered in the analysis.ConclusionsAll six models produced unbiased estimates of treatment effect in the context of multicentre trials. Adjusting for centre as a random intercept led to the most efficient treatment effect estimation across all simulations under the normality assumption, when there was no treatment by centre interaction.

BackgroundOsteoporosis is characterized by low bone mass and compromised bone structure, heritable traits that contribute to fracture risk. There have been no genome-wide association and linkage studies for these traits using high-density genotyping platforms.MethodsWe used the Affymetrix 100K SNP GeneChip marker set in the Framingham Heart Study (FHS) to examine genetic associations with ten primary quantitative traits: bone mineral density (BMD), calcaneal ultrasound, and geometric indices of the hip. To test associations with multivariable-adjusted residual trait values, we used additive generalized estimating equation (GEE) and family-based association tests (FBAT) models within each sex as well as sexes combined. We evaluated 70,987 autosomal SNPs with genotypic call rates ≥80%, HWE p ≥ 0.001, and MAF ≥10% in up to 1141 phenotyped individuals (495 men and 646 women, mean age 62.5 yrs). Variance component linkage analysis was performed using 11,200 markers.ResultsHeritability estimates for all bone phenotypes were 30–66%. LOD scores ≥3.0 were found on chromosomes 15 (1.5 LOD confidence interval: 51,336,679–58,934,236 bp) and 22 (35,890,398–48,603,847 bp) for femoral shaft section modulus. The ten primary phenotypes had 12 associations with 100K SNPs in GEE models at p < 0.000001 and 2 associations in FBAT models at p < 0.000001. The 25 most significant p-values for GEE and FBAT were all less than 3.5 × 10-6 and 2.5 × 10-5, respectively. Of the 40 top SNPs with the greatest numbers of significantly associated BMD traits (including femoral neck, trochanter, and lumbar spine), one half to two-thirds were in or near genes that have not previously been studied for osteoporosis. Notably, pleiotropic associations between BMD and bone geometric traits were uncommon. Evidence for association (FBAT or GEE p < 0.05) was observed for several SNPs in candidate genes for osteoporosis, such as rs1801133 in MTHFR; rs1884052 and rs3778099 in ESR1; rs4988300 in LRP5; rs2189480 in VDR; rs2075555 in COLIA1; rs10519297 and rs2008691 in CYP19, as well as SNPs in PPARG (rs10510418 and rs2938392) and ANKH (rs2454873 and rs379016). All GEE, FBAT and linkage results are provided as an open-access results resource at .ConclusionThe FHS 100K SNP project offers an unbiased genome-wide strategy to identify new candidate loci and to replicate previously suggested candidate genes for osteoporosis.

Liang과 Zeger는 이산형 혹은 연속형 반복측정자료를 분석하기 위한 일반화 추정방정식 (GEE)을 제안하였다 GEE모형은 범주형 반복측정자료의 모형으로 확장될 수 있으며, 이 GEE추정량은 대표본인 경우 다변량 정규분포를 따른다. 그러나 GEE는 대표본근사이론에 기초한다. 본 논문에서는 소표본인 경우 반복 측정된 순서자료에 대한 GEE추정량의 성질을 연구한다. 우리는 두가지 방법을 사용하여 두그룹의 반복 측정된 순서자료를 생성하며 모의실험을 통하여 소표본인 경우 여러 개 범주를 갖는 순서반응 자료에 대하여 GEE추정량의 1종 오류율, 검정력, 상대효율, 두 그룹의 표본크기가 다를 경우 효과, 그리고 분산 추정량의 성질등을 연구한다. Liang and Zeger proposed generalized estimating equations(GEE) for analyzing repeated data which is discrete or continuous. GEE model can be extended to model for repeated categorical data and its estimator has asymptotic multivariate normal distribution in large sample sizes. But GEE is based on large sample asymptotic theory. In this paper, we study the properties of GEE estimators for repeated ordinal data in small sample sizes. We generate ordinal repeated measurements for two groups using two methods. Through Monte Carlo simulation studies we investigate the empirical type 1 error rates, powers, relative efficiencies of the GEE estimators, the effect of unequal sample size of two groups, and the performance of variance estimators for polytomous ordinal response variables, especially in small sample sizes.

During the COVID-19 pandemic, excess mortality has been reported worldwide, but its magnitude has varied depending on methodological differences that hinder between-study comparability. Our aim was to estimate variability attributable to different methods, focusing on specific causes of death with different pre-pandemic trends. Monthly mortality figures observed in 2020 in the Veneto Region (Italy) were compared with those forecasted using: (1) 2018-2019 monthly average number of deaths; (2) 2015-2019 monthly average age-standardized mortality rates; (3) Seasonal Autoregressive Integrated Moving Average (SARIMA) models; (4) Generalized Estimating Equations (GEE) models. We analyzed deaths due to all-causes, circulatory diseases, cancer, and neurologic/mental disorders. Excess all-cause mortality estimates in 2020 across the four approaches were: +17.2% (2018-2019 average number of deaths), +9.5% (five-year average age-standardized rates), +15.2% (SARIMA), and +15.7% (GEE). For circulatory diseases (strong pre-pandemic decreasing trend), estimates were +7.1%, -4.4%, +8.4%, and +7.2%, respectively. Cancer mortality showed no relevant variations (ranging from -1.6% to -0.1%), except for the simple comparison of age-standardized mortality rates (-5.5%). The neurologic/mental disorders (with a pre-pandemic growing trend) estimated excess corresponded to +4.0%/+5.1% based on the first two approaches, while no major change could be detected based on the SARIMA and GEE models (-1.3%/+0.3%). The magnitude of excess mortality varied largely based on the methods applied to forecast mortality figures. The comparison with average age-standardized mortality rates in the previous five years diverged from the other approaches due to the lack of control over pre-existing trends. Differences across other methods were more limited, with GEE models probably representing the most versatile option.

Although air pollution and weather changes have been identified as putative risk factors that precipitate acute attacks of bronchial asthma, so far there have been no studies that could conclusively establisha clear association. This study was carried out to investigate the effect of the concentrations of NO2, SO2, and particulate matter on exacerbation of bronchial asthma. A longitudinal study was conducted during February 2014-January 2015. Sixty-one known cases of asthma were recruited from the outpatient department of a chest hospital. Data regarding weather parameter (temperature, rainfall, and relative humidity) and environmental pollution (SO2and NO2) and respirable suspended particulate matter were procured from the Indian Meteorological Department and Delhi Pollution Control Committee, respectively. Association was examined using the generalized estimation equation (GEE). Separate models were developed for weather parameter and pollution parameters. This study could not find a significant association between any of the weather parameters and occurrence of asthmatic attacks. In the GEE model, where average values of SO2, NO2, and PM were used, the PM was found to be significantly associated with asthmatic attacks. PM was found to increase the risk of exacerbation of asthma three folds.

Maternal exposure to bisphenol A and anogenital distance throughout infancy: A longitudinal study from Shanghai, China

The present study aimed to assess infant and young child feeding (IYCF) practices and the tracking of dietary diversity score (DDS), intakes of Fe- and vitamin A-rich foods and meal frequency in a peri-urban area in Nepal. Furthermore, to explore whether sociodemographic factors were associated with tracking patterns of these IYCF practices. Longitudinal study. Monthly food intake was measured by 24 h recall. Four time slots were used (9-12, 13-16, 17-20 and 21-24 months). Tracking of IYCF practices was investigated using generalized estimating equations (GEE) models and Cohen's weighted kappa. Multinominal logistic regression was used to identify determinants for tracking of the IYCF practices. Bhaktapur municipality, Nepal. Children (n 229) aged 9-24 months, randomly selected. Prevalence of minimum meal frequency was higher than for minimum dietary diversity at all time slots. Tracking based on absolute measures (GEE models) was moderate for DDS (0·48) and meal frequency (0·53), and low for intakes of Fe- (0·23) and vitamin A-rich (0·35) foods. Tracking based on rank measured was moderate for DDS and meal frequency, and fair for Fe- and vitamin A-rich foods. Low socio-economic status significantly increased the odds (OR; 95 % CI) of tracking of low v. high DDS (3·31; 1·44, 7·60) and meal frequency (3·46; 1·54, 7·76). Low tracking for intakes of Fe- and vitamin A-rich foods implies that interventions to improve these IYCF practices must address underlying causes for irregular intake to have sustainable effects.