Cluster-weighted modified Poisson regression for estimating risk ratios in longitudinal data with informative cluster sizes

Cluster-weighted modified Poisson regression for estimating risk ratios in longitudinal data with informative cluster sizes | medRxiv /* */ /* */ <!-- <!-- /*! * yepnope1.5.4 * (c) WTFPL, GPLv2 */ (function(a,b,c){function d(a){return"[object Function]"==o.call(a)}function e(a){return"string"==typeof a}function f(){}function g(a){return!a||"loaded"==a||"complete"==a||"uninitialized"==a}function h(){var a=p.shift();q=1,a?a.t?m(function(){("c"==a.t?B.injectCss:B.injectJs)(a.s,0,a.a,a.x,a.e,1)},0):(a(),h()):q=0}function i(a,c,d,e,f,i,j){function k(b){if(!o&&g(l.readyState)&&(u.r=o=1,!q&&h(),l.onload=l.onreadystatechange=null,b)){"img"!=a&&m(function(){t.removeChild(l)},50);for(var d in y[c])y[c].hasOwnProperty(d)&&y[c][d].onload()}}var j=j||B.errorTimeout,l=b.createElement(a),o=0,r=0,u={t:d,s:c,e:f,a:i,x:j};1===y[c]&&(r=1,y[c]=[]),"object"==a?l.data=c:(l.src=c,l.type=a),l.width=l.height="0",l.onerror=l.onload=l.onreadystatechange=function(){k.call(this,r)},p.splice(e,0,u),"img"!=a&&(r||2===y[c]?(t.insertBefore(l,s?null:n),m(k,j)):y[c].push(l))}function j(a,b,c,d,f){return q=0,b=b||"j",e(a)?i("c"==b?v:u,a,b,this.i++,c,d,f):(p.splice(this.i++,0,a),1==p.length&&h()),this}function k(){var a=B;return a.loader={load:j,i:0},a}var l=b.documentElement,m=a.setTimeout,n=b.getElementsByTagName("script")[0],o={}.toString,p=[],q=0,r="MozAppearance"in l.style,s=r&&!!b.createRange().compareNode,t=s?l:n.parentNode,l=a.opera&&"[object Opera]"==o.call(a.opera),l=!!b.attachEvent&&!l,u=r?"object":l?"script":"img",v=l?"script":u,w=Array.isArray||function(a){return"[object Array]"==o.call(a)},x=[],y={},z={timeout:function(a,b){return b.length&&(a.timeout=b[0]),a}},A,B;B=function(a){function b(a){var a=a.split("!"),b=x.length,c=a.pop(),d=a.length,c={url:c,origUrl:c,prefixes:a},e,f,g;for(f=0;f<d;f++)g=a[f].split("="),(e=z[g.shift()])&&(c=e(c,g));for(f=0;f<b;f++)c=x[f](c);return c}function g(a,e,f,g,h){var i=b(a),j=i.autoCallback;i.url.split(".").pop().split("?").shift(),i.bypass||(e&&(e=d(e)?e:e[a]||e[g]||e[a.split("/").pop().split("?")[0]]),i.instead?i.instead(a,e,f,g,h):(y[i.url]?i.noexec=!0:y[i.url]=1,f.load(i.url,i.forceCSS||!i.forceJS&&"css"==i.url.split(".").pop().split("?").shift()?"c":c,i.noexec,i.attrs,i.timeout),(d(e)||d(j))&&f.load(function(){k(),e&&e(i.origUrl,h,g),j&&j(i.origUrl,h,g),y[i.url]=2})))}function h(a,b){function c(a,c){if(a){if(e(a))c||(j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}),g(a,j,b,0,h);else if(Object(a)===a)for(n in m=function(){var b=0,c;for(c in a)a.hasOwnProperty(c)&&b++;return b}(),a)a.hasOwnProperty(n)&&(!c&&!--m&&(d(j)?j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}:j[n]=function(a){return function(){var b=[].slice.call(arguments);a&&a.apply(this,b),l()}}(k[n])),g(a[n],j,b,n,h))}else!c&&l()}var h=!!a.test,i=a.load||a.both,j=a.callback||f,k=j,l=a.complete||f,m,n;c(h?a.yep:a.nope,!!i),i&&c(i)}var i,j,l=this.yepnope.loader;if(e(a))g(a,0,l,0);else if(w(a))for(i=0;i (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0];var j=d.createElement(s);var dl=l!='dataLayer'?'&l='+l:'';j.src='//www.googletagmanager.com/gtm.js?id='+i+dl;j.type='text/javascript';j.async=true;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-P4HH5NV'); Skip to main content Home About Submit ALERTS / RSS Search for this keyword Advanced Search Cluster-weighted modified Poisson regression for estimating risk ratios in longitudinal data with informative cluster sizes View ORCID Profile Jemar R. Bather , View ORCID Profile Samuel Anyaso-Samuel , View ORCID Profile Yuyu Chen , View ORCID Profile Luther Elliott , View ORCID Profile Alex S. Bennett , View ORCID Profile Melody S. Goodman doi: https://doi.org/10.1101/2025.05.23.25328253 Jemar R. Bather 1 Center for Anti-racism, Social Justice & Public Health, NYU School of Global Public Health 2 Department of Biostatistics, NYU School of Global Public Health PhD Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Jemar R. Bather For correspondence: jemar.bather{at}nyu.edu Samuel Anyaso-Samuel 3 Division of Cancer Epidemiology & Genetics, National Cancer Institute PhD Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Samuel Anyaso-Samuel Yuyu Chen 2 Department of Biostatistics, NYU School of Global Public Health MS Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Yuyu Chen Luther Elliott 4 Department of Social and Behavioral Sciences, NYU School of Global Public Health 5 Center for Drug Use and HIV/HCV Research, NYU School of Global Public Health PhD Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Luther Elliott Alex S. Bennett 1 Center for Anti-racism, Social Justice & Public Health, NYU School of Global Public Health 4 Department of Social and Behavioral Sciences, NYU School of Global Public Health 5 Center for Drug Use and HIV/HCV Research, NYU School of Global Public Health PhD Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Alex S. Bennett Melody S. Goodman 1 Center for Anti-racism, Social Justice & Public Health, NYU School of Global Public Health 2 Department of Biostatistics, NYU School of Global Public Health PhD Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Melody S. Goodman Abstract Full Text Info/History Metrics Preview PDF Abstract Variation in binary outcomes over time by cluster size arises across various biomedical disciplines, including reproductive health, dental medicine, and psychiatric epidemiology. This study formally integrates modified Poisson regression with cluster-weighted generalized estimating equations (MP-CWGEE) for computing risk ratios in longitudinal studies with informative cluster sizes. Using a comprehensive Monte-Carlo simulation study, we empirically evaluated MP-CWGEE’s statistical properties against alternative modeling approaches: MP-GEE, log-binomial CWGEE (LB-CWGEE), and log-binomial GEE (LB-GEE). We conducted 1,000 simulations across varying sample sizes, risk ratios, and informativeness degrees. MP-CWGEE demonstrated superior performance in model convergence, empirical bias, average estimated standard error, coverage, and Type 1 error control. While LB-CWGEE showed comparable results, its convergence rates were slightly inferior. The benefits of cluster-weighted models (MP-CWGEE and LB-CWGEE) over unweighted models (MP-GEE and LB-GEE) were pronounced in scenarios with informative cluster sizes. We demonstrated MP-CWGEE’s practical application to a cohort study of people who used illicit opioids in New York City. We also provided implementation code for R, Stata, and SAS to facilitate wider adoption of the MP-CWGEE approach. 1 Introduction Quantifying the association between an exposure and a binary outcome is a common goal of biomedical researchers analyzing longitudinal data [ 1 ]. While the odds ratio is commonly used to characterize this association, it has two major limitations. First, it is frequently misinterpreted as the risk ratio (or relative risk) [ 2 ], providing an accurate approximation only when the outcome prevalence is rare (<10%) [ 1 ]. Second, the odds ratio is noncollapsible (sensitive to unmodelled covariates), complicating its interpretation and comparison across studies [ 3 , 4 ]. An alternative to the logit link function—which estimates the odds ratio in logistic regression— is the log link function, which directly estimates the risk ratio and provides a clinically meaningful association measure in longitudinal studies [ 1 , 5 ]. Risk ratios can be calculated using log-binomial regression via maximum likelihood (ML) estimation in a mixed-effects model [ 6 ]. This is a parametric approach with a cluster-specific interpretation (conditional on the random effect) [ 6 ]. Risk ratios can also be generated using log-binomial regression via generalized estimating equations (GEE), a semi-parametric marginal modeling technique with a population-averaged interpretation [ 7 ]. However, log-binomial regression frequently encounters convergence problems [ 5 , 8 , 9 ]. Zou and Donner [ 8 ] addressed this limitation by developing the modified Poisson (or robust Poisson) regression model using GEE (MP-GEE). This method employs the log link function to model binary outcomes as a function of explanatory variables while using a robust variance estimator to correct the standard error overestimation that would otherwise occur in conventional Poisson regression [ 8 ]. Simulation studies have demonstrated that the MP-GEE possesses desirable statistical properties [ 5 , 8 – 16 ]. MP-GEE exhibits greater robustness against model misspecification and outliers than the log-binomial ML model [ 8 , 9 , 11 – 13 ]. Additional simulation studies confirm that MP-GEE maintains adequate coverage, effectively controls type 1 error rates, and preserves asymptotic efficiency when estimating risk ratios [ 5 , 9 , 10 ]. However, despite these advantages, no published studies evaluate the performance of MP-GEE when cluster sizes are associated with the outcome (referred to as informative cluster size [ 17 , 18 ]). Outcome variation by cluster size arises across various biomedical disciplines, including sexual health [ 19 ], reproductive science [ 20 – 22 ], dental medicine [ 23 , 24 ], and psychiatric epidemiology [ 25 ]. For example, women with multiple pregnancies face higher risks of subsequent pregnancy complications than those with a single pregnancy [ 26 ]. Analytical approaches that re-strict the dataset to one pregnancy per person discard valuable information [ 17 , 27 ]. Conversely, including all available data without accounting for cluster informativeness overweights individuals contributing more pregnancies, which biases regression parameter estimates [ 17 , 27 ]. Two models have been developed to address this issue: within-cluster resampling and cluster-weighted generalized estimating equations (CWGEE). The within-cluster resampling approach randomly selects one observation per individual, fits a regression model, and then averages results across iterations to obtain parameter estimates [ 27 ]. While statistically valid, this method is computationally intensive. To overcome this limitation, Williamson and colleagues developed CWGEE [ 17 ], which inversely weights the GEE by cluster size. Williamson et al. [ 17 ] demonstrated that CWGEE is asymptotically equivalent to within-cluster resampling and requires less computational resources. Combining modified Poisson regression with cluster-weighted generalized estimating equations (MP-CWGEE) is a natural extension that can account for informative cluster sizes when computing risk ratios in longitudinal studies. However, a systematic evaluation of its statistical properties has yet to be reported despite its increasing application in recent studies [ 28 – 32 ]. Therefore, we evaluated MP-CWGEE’s performance in estimating risk ratios in longitudinal data with informative cluster sizes. The structure of this paper is as follows. Section 2 describes the mathematical notation, model specification, parameter interpretation, and robust variance estimator. Section 3 characterizes the simulation study comparing the MP-CWGEE model to alternative models for estimating risk ratios in longitudinal settings. Section 4 presents a real-world data application of the MP-CWGEE approach to a cohort study of people who use illicit opioids in New York City. Section 5 offers concluding remarks and discusses future research directions. Section 6 provides example codes for fitting the MP-CWGEE model in R, Stata, and SAS. 2 Modified Poisson regression using cluster-weighted generalized estimating equations 2.1 Model specification and parameter interpretation Let Y ij denote the binary outcome for the i th individual ( i = 1, …, N ) on the j th occasion ( j = 1, …, n i ). The total number of repeated measures for the i th individual is reflected by n i (cluster size), which can vary across study participants. Using the marginal modeling framework [ 7 ], we model the dichotomous response as a log-linear function of p explanatory variables: where is the primary exposure of interest, represent the covariates, and µ ij = E( Y ij | X ij ) is the marginal expectation of the binary outcome. The parameter exp ( β 1 ) represents the ratio of the risk in the exposed group to that of the unexposed group, after adjusting for covariates. 2.2 Cluster-weighted estimation 2.2.1 Parameter estimation The MP-CWGEE estimator of the regression coefficient vector, β = ( β 0 , β 1 , …, β p ) T , is obtained by solving the following score equation [ 17 ]: where α is the within-person correlation, R ( α ) is the working correlation matrix, is the working variance-covariance matrix, A i is the diagonal matrix with elements ϕv ( µ ij ), ϕ is the dispersion parameter, and v ( µ ij ) = µ ij is the variance function. The inverse weighting by the cluster size (1 /n i ) ensures that each study participant contributes equally to the estimation, regardless of their number of repeated measurements [ 17 , 19 ]. We assumed an independent working correlation structure, aligning with the seminal CWGEE paper [ 17 ], the seminal longitudinal modified Poisson regression paper [ 8 ], and the applied MP-CWGEE literature [ 31 , 32 ]. Assuming a non-independent correlation structure (e.g., exchangeable or first-order autoregressive) requires a generalized weight: [ 1 T R ( α ) 1 ] − 1 , where 1 represents an n i × 1 vector of 1s [ 17 , 19 ]. The asymptotic normality of β is derived in Section 6.2 . 2.2.2 Variance-covariance estimation The robust (sandwich) variance estimator for β takes the following form: where and 3 Simulation study 3.1 Design We conducted a simulation study to evaluate the performance of various regression approaches for estimating risk ratios in longitudinal data with informative cluster sizes. For the i th subject and the j th observation, a binary outcome was generated with probability: where is the total number of observations. We defined the vector of explanatory variables as , with denoting a binary exposure where the first N/ 2 individuals were exposed, and reflecting a continuous covariate. We set the true regression coefficients as β = ( β 0 , β 1 , β 2 ) T . We set β 0 = log(0.37) and β 2 = log(0.50). Cluster sizes n i were randomly drawn from {2, 4, 15}, with probabilities: ℙ ( n i = 2) = 0.5625, ℙ ( n i = 4) = 0.3750, and ℙ ( n i = 15) = 0.0625. This formulation ensures that the average number of repeated measures is approximately three. Each simulated dataset was analyzed using the following five models: MP-CWGEE, MP-GEE, log-binomial regression using cluster-weighted generalized estimating equations (LB-CWGEE), log-binomial regression using generalized estimating equations (LB-GEE), and log-binomial mixed effects model (LB-MM). Both GEE and CWGEE estimators used an independence working correlation structure. 3.2 Scenarios and performance metrics We explored a range of data-generating scenarios by varying the number of individuals, the risk ratio, and the informativeness of cluster sizes. The number of individuals, M , was 50, 100, and 200. The risk ratios were 1.2 (modest effect) and 1.5 (moderate effect). For the informativeness of cluster sizes, we defined where γ ∈ {0, 1, 1.5} controls the degree of informativeness [ 33 ]. Note that γ = 0 indicates non-informative cluster sizes. Using 1,000 Monte-Carlo simulations, we evaluated the following performance metrics: model convergence, empirical bias, average estimated standard error, coverage rates, and empirical Type 1 error rates. 3.3 Simulation study results 3.3.1 Model convergence Table 1 shows the convergence rates of all five models across all simulation settings. MP-CWGEE and MP-GEE consistently achieved perfect convergence (100%) across all simulation scenarios, regardless of the degree of informativeness, exposure effect, and sample size. LB-CWGEE also performed well, with convergence rates of at least 99%. LB-GEE exhibited reasonable convergence rates (all >88%), but these rates decreased as sample size decreased and informativeness increased. LB-MM demonstrated poor convergence across all settings (<36%), with the lowest rate (<1%) in a scenario with 200 study participants, a 1.5 exposure effect, and informativeness = 1.5. Given LB-MM’s poor convergence, we do not discuss its other performance metrics (e.g., bias) in the remainder of this report as these metrics would not be comparable to those of the models that showed good convergence. The non-convergence of the LB-MM was likely due to maximum likelihood estimates being on the parameter space boundary [ 11 , 34 ]. View this table: View inline View popup Download powerpoint Table 1. Model convergence rates (%) by sample size, exposure effect, and informativeness degree. 3.3.2 Empirical bias Fig. 1 illustrates the empirical bias of β 1 across varying informativeness degrees and sample sizes. When cluster sizes were non-informative ( γ = 0), all four models showed comparable empirical bias patterns, regardless of sample size or β 1 value. However, with informative cluster sizes ( γ > 0), the CWGEE approaches (MP-CWGEE and LB-CWGEE) produced substantially less biased estimates than their unweighted counterparts. Download figure Open in new tab Figure 1. Comparison of bias in parameter estimates for four risk ratio estimation models under varying conditions. The γ parameter reflects the informativeness degree ( γ = 0 is non-informative). Note that exp(0.18) = 1.2 and exp(0.41) = 1.5, denoting the true risk ratios. MP-CWGEE = modified Poisson regression using cluster-weighted generalized estimating equations; MP-GEE = modified Poisson regression using (unweighted) generalized estimating equations; LB-CWGEE = log-binomial regression using cluster-weighted generalized estimating equations; LB-GEE = log-binomial regression using (unweighted) generalized estimating equations. 3.3.3 Average estimated standard error Fig. 2 visualizes patterns in the average estimated standard error across 1,000 Monte-Carlo simulations. For all four models, regardless of informativeness degree and magnitude of β 1 , standard errors decreased as the sample size increased. At γ = 0 (non-informative cluster sizes), unweighted models (MP-GEE and LB-GEE) had smaller standard errors than cluster-weighted models (MP-CWGEE and LB-CWGEE). This trend eventually reversed in the presence of informative cluster sizes ( γ > 0), with cluster-weighted models showing better precision than unweighted models. Download figure Open in new tab Figure 2. Average estimated standard error comparison across four risk ratio estimation models under varying conditions. The γ parameter reflects the informativeness degree ( γ = 0 is non-informative). Note that exp(0.18) = 1.2 and exp(0.41) = 1.5, denoting the true risk ratios. MP-CWGEE = modified Poisson regression using cluster-weighted generalized estimating equations; MP-GEE = modified Poisson regression using (unweighted) generalized estimating equations; LB-CWGEE = log-binomial regression using cluster-weighted generalized estimating equations; LB-GEE = log-binomial regression using (unweighted) generalized estimating equations. 3.3.4 Coverage Table 2 summarizes the coverage rates for β 1 by sample size, exposure effects, and informativeness degree. Coverage rates decreased for the unweighted models as the informativeness parameter, γ , increased from 0.0 to 1.5. For example, at N = 50 and risk ratio = 1.2, the MP-GEE coverage decreased from 93.9% ( γ = 0.0) to 88.9% ( γ = 1.0) to 86.9% ( γ = 1.5). Increasing the sample size mitigated these coverage issues. For instance, at γ = 1.5 and risk ratio = 1.2, MP-GEE coverage improved from 86.9% (N = 50) to 91.0% (N = 100) to 93.1% (N = 200). By contrast, the cluster-weighted approaches maintained coverage rates close to the nominal 95% level across all simulated scenarios. View this table: View inline View popup Download powerpoint Table 2. Coverage rates (%) for β 1 across different simulation settings. 3.3.5 Empirical Type 1 error rate Fig. 3 summarizes the results from the analysis of Type 1 error rates. When γ = 0 (uninformative cluster sizes) and N = 50, all models yielded slightly inflated Type 1 error rates (range: 6.3-7.0%), but this improved as sample size increased to 100 (range: 3.6-4.7%) and 200 (range: 3.9-4.8%). As informativeness increased, the unweighted models exhibited inflated Type 1 error rates, particularly at smaller sample sizes. For example, at γ = 1.5 and N = 50, MP-GEE had a 12.5% Type 1 error rate, and LB-GEE had an 11.9% Type 1 error rate. In contrast, the cluster-weighted models controlled the Type 1 error rate as informativeness increased. Download figure Open in new tab Figure 3. Empirical Type 1 error rates for four risk ratio estimation models under varying conditions. The γ parameter reflects the informativeness degree ( γ = 0 is non-informative). MP-CWGEE = modified Poisson regression using cluster-weighted generalized estimating equations; MP-GEE = modified Poisson regression using (unweighted) generalized estimating equations; LB-CWGEE = log-binomial regression using cluster-weighted generalized estimating equations; LB-GEE = log-binomial regression using (unweighted) generalized estimating equations. 3.3.6 Additional simulation results The simulation results presented up to this point primarily focus on the performance of the estimators for β 1 , the exposure effect. However, prior studies [ 23 , 35 ] have shown that, when informative cluster sizes are not properly accounted for, the estimation of other regression parameters can also be biased. To provide a more comprehensive evaluation of the regression models, we report simulation results for the estimation of the intercept β 0 and the continuous covariate effect β 2 . Supplemental Figures 1 and 2 in the Supplementary Material present the relative bias of the estimators for β 0 and β 2 , respectively, across varying levels of informativeness and sample sizes. For both regression parameters, the results reveal consistent patterns across simulation settings. When cluster sizes are non-informative, all models yield approximately unbiased estimates, with bias decreasing as the number of samples increases. In contrast, under informative cluster size scenarios, the GEE models exhibit substantial bias, particularly for β 0 . The CWGEE models are markedly more robust to informativeness, with bias decreasing as sample size grows. Conversely, the bias in the GEE estimators tends to worsen as the number of samples increases. Supplemental Figures 3 and 4 of the Supplementary Material show the average estimated standard errors for β 0 and β 2 , respectively. Across all scenarios, GEE-based models report smaller standard errors than their CWGEE counterparts, irrespective of the degree of informativeness. This reflects the well-known trade-off between bias and variance in model estimation [ 36 ]. Coverage probabilities for β 0 and β 2 are reported in Supplemental Tables 1 and 2. For β 2 , all models achieve coverage rates close to the nominal 95% level, even under informative cluster sizes. However, for β 0 , the GEE models consistently fall short of the nominal coverage level when cluster sizes are informative. Notably, this undercoverage does not improve with increasing sample size. In contrast, the CWGEE models maintain coverage rates near 95%, further underscoring their robustness in the presence of informativeness. 4 Application to a study of people who use illicit opioids in New York City 4.1 Data source We analyzed repeated measures data from a prospective cohort study of over 400 adults (aged 18+) who used illicit opioids in New York City [ 37 , 38 ]. The dataset comprised 7,365 observations on 423 adults collected during monthly study engagements between 2019 and 2022. Complete details regarding the study design and recruitment procedures are described elsewhere [ 37 , 38 ]. The New York University Grossman School of Medicine Institutional Review Board approved this study protocol. 4.2 Measures The binary outcome of interest was whether the participant had any clinical indicators of major depressive disorder (1 = any, 0 = none). This was measured by the Cross-Cutting Depression Sub-scale from the Diagnostic and Statistical Manual of Mental Disorders Version 5 [ 39 ]. Race/ethnicity was the independent variable of interest and categorized as non-Hispanic White, non-Hispanic Black, Hispanic, and non-Hispanic Other. Covariates included age (measured continuously), educational attainment (did not complete high school, high school diploma/GED, some college or more), and having an opioid-using network size ≥ 5 (yes vs. no). 4.3 Statistical analysis Exploratory data analysis indicated that study participants had variable engagement with the research protocol, with considerable heterogeneity in study engagements (mean: 17 engagements, SD: 7, Table 3 ). Given the well-established clinical observation that individuals experiencing depression are less likely to participate in follow-up protocols [ 40 , 41 ], we hypothesized that the number of study engagements (cluster size) would be informative of the outcome. To test this hypothesis, we modeled having clinical indicators of major depressive disorder as a function of study engagement frequency (3-4 engagements, 5-10 engagements, 11-16 engagements, 17-24 engagements) using an MP-GEE model with an independent correlation structure. Cut-offs for the engagement frequency variable were derived based on exploratory analyses and subject matter expertise. View this table: View inline View popup Download powerpoint Table 3. Baseline characteristics of 423 adults who used illicit opioids in New York City. To quantify the adjusted association between race/ethnicity and having clinical indicators of major depressive disorder, we compared adjusted risk ratios (and 95% CIs) across different statistical models: MP-CWGEE, MP-GEE, LB-CWGEE, LB-GEE, and LB-MM. The MP-CWGEE and LB-CWGEE models were weighted by the inverse of study engagements (measured continuously). All models controlled for age, educational attainment, and having an opioid-using network size ≥5. Analyses used R version 4.4.3 (R Core Team, R Foundation for Statistical Computing), with statistical significance assessed as a 2-sided p <0.05. 4.4 Results Table 3 displays the baseline characteristics of the study population. Of the 423 participants (mean age: 48 years, SD: 11), 54% had clinical indicators of major depressive disorder, 41% identified as Hispanic, 24% did not graduate high school, and 76% had an opioid-using network size ≥5. Table 4 presents the associations between study engagement frequency and having clinical indicators of major depressive disorder. Compared to those with 3-4 study engagements, individuals with 11-16 study engagements were 30% (RR: 0.70, 95% CI: 0.51-0.95, p =0.024) less likely to have clinical indicators of major depressive disorder, indicating nonignorable cluster size. Individuals with 17-24 study engagements were 21% (RR: 0.79, 95% CI: 0.62-1.02, p =0.068) less likely to have clinical indicators of major depressive disorder than those with 3-4 study engagements, but this association was not statistically significant. View this table: View inline View popup Download powerpoint Table 4. Associations between study engagement frequency and having clinical indicators of major depressive disorder among 423 adults who used illicit opioids in New York City. Table 5 presents the comparison of statistical models that quantify the adjusted associations between race/ethnicity and having clinical indicators of major depressive disorder. Consistently across all models, individuals who identified as non-Hispanic Black were significantly less likely to have clinical indicators of major depressive disorder compared to their non-Hispanic White counterparts: MP-CWGEE (RR: 0.66, p =0.004), MP-GEE (RR: 0.67, p =0.012), LB-CWGEE (RR: 0.67, p =0.005), LB-GEE (RR: 0.67, p =0.013). The cluster-weighted models yielded narrower 95% CIs, reflecting greater precision in their estimates relative to unweighted models: MP-CWGEE (95% CI: 0.50-0.88) vs. MP-GEE (95% CI: 0.49-0.92); LB-CWGEE (95% CI: 0.50-0.88) vs. LB-GEE (95% CI: 0.49-0.92). The MP-CWGEE and LB-CWGEE models produced similar risk ratios and 95% CIs, which was expected since both theoretically estimate the same association parameter. We observed a similar finding when comparing the MP-GEE and LB-GEE models’ risk ratios and 95% CIs. The LB-MM model failed to produce standard errors even after using the other models’ estimates as initial starting values. View this table: View inline View popup Download powerpoint Table 5. Comparison of different statistical models quantifying the adjusted associations between race/ethnicity and having clinical indicators of major depressive disorder among 423 adults who used illicit opioids in New York City. 5 Discussion Accounting for informative cluster sizes when computing risk ratios in longitudinal data arises in various medical research areas, including reproductive epidemiology [ 28 , 30 , 31 ] and infection prevention [ 29 ]. Despite its potential utility, MP-CWGEE remains underused in published studies, with limited information regarding its operating characteristics. Using a comprehensive Monte-Carlo simulation study, we empirically evaluated MP-CWGEE’s statistical properties against alternative models, including MP-GEE, LB-CWGEE, and LB-GEE. Results indicated that MP-CWGEE consistently exhibited superior performance across key metrics: model convergence, empirical bias, average estimated standard error, coverage, and Type 1 error control. While LB-CWGEE performed similarly to MP-CWGEE, it demonstrated slightly poorer convergence rates. The advantages of cluster-weighted models (MP-CWGEE and LB-CWGEE) over unweighted models (MP-GEE and LB-GEE) became particularly evident when informative cluster sizes were present. Given that MP-CWGEE is nested within the GEE framework [ 7 ], it possesses several notable advantages over ML models. Compared to ML approaches, MP-CWGEE is less computationally intensive and requires fewer assumptions about the exposure-outcome relationship [ 7 ]. MP-CWGEE does not assume a random effect distribution, proceeds without requiring residuals to be normally distributed, and eliminates the need for correctly specifying the working correlation matrix [ 7 ]. From a health science perspective, MP-CWGEE provides additional benefits. It addresses the noncollapsibility of the odds ratio, resulting in a more clinically intuitive and meaningful association measure [ 1 , 3 , 42 ]. In addition, the MP-CWGEE model can be implemented in standard statistical software packages (e.g., R, SAS, Stata), making it accessible for applications across diverse epidemiologic fields such as reproductive medicine, psychiatric epidemiology, and hospital infection prevention. Sample code for fitting the MP-CWGEE model in R, SAS, and Stata is provided in Section 6.1 . Despite these strengths, MP-CWGEE has two major limitations. First, MP-CWGEE may compute predicted probabilities exceeding one [ 42 ], as the Poisson distribution’s support is non-negative. Consequently, MP-CWGEE may be more suitable for association and causal studies than prediction studies. Second, since MP-CWGEE is a semi-parametric approach, model selection cannot rely on parametric test statistics such as the likelihood ratio test, Akaike information criterion, or Bayesian information criterion [ 43 ]. In practice, subject matter expertise should guide the final model selection. The present study lays the groundwork for future research. One priority area is empirically estimating the impact of missing data patterns (e.g., missing at random, missing not at random) and evaluating corrective approaches such as inverse probability weighting and multiple imputation within the MP-CWGEE framework. Additionally, studies are needed to assess MP-CWGEE’s sensitivity to outliers and model misspecification [ 11 , 12 ]. It is unknown how robust the MP-CWGEE approach is to omitted-variable bias [ 44 ]. In other words, how does MP-CWGEE perform when an important interaction or quadratic term is not included in the model [ 11 ]? Small sample performance represents another crucial research gap. Future work should investigate finite sample bias-correction methods for MP-CWGEE when analyzing fewer than 30 clusters [ 45 – 47 ]. Lastly, the present study assumed that only the outcome depended on cluster size, but this relationship may be more complex. Comprehensive simulation studies should explore various potential relationships between outcomes, exposures, confounders, and cluster size. In settings where cluster sizes are non-informative, both CWGEE and GEE models yield approximately unbiased estimates; however, GEE tends to be more efficient due to its smaller standard errors. This highlights the importance of assessing informativeness prior to model selection, as unnecessary adjustments for informativeness can lead to a loss of efficiency. Several formal statistical procedures have been developed to test for informative cluster size across a variety of outcome types, including both continuous and non-continuous data [ 48 – 52 ]. Future work could explore the performance and applicability of these tests specifically within the context of modified Poisson regression, where such evaluations remain limited. Conclusions This study evaluated the performance of the MP-CWGEE model for estimating risk ratios in longitudinal data with informative cluster sizes. Monte-Carlo simulations demonstrated that MP-CWGEE outperformed alternative modeling approaches regarding model convergence, empirical bias, precision, coverage, and Type 1 error control. The applied example using data from adults who use illicit opioids in New York City confirmed MP-CWGEE’s utility in social epidemiologic research, providing more precise estimates than unweighted models in the presence of informative cluster sizes. Despite limitations such as potentially predicted probabilities exceeding one, MP-CWGEE offers a practical and accessible solution for analyzing longitudinal binary data with informative cluster sizes. Declarations Author contributions JRB : Conceptualization, Methodology, Software, Formal analysis, Writing (original draft), Visualization. SA: Conceptualization, Methodology, Software, Formal analysis, Writing (original draft), Visualization. YC: Investigation, Writing (review & editing). LE: Investigation, Writing (review & editing), Funding acquisition. ASB: Investigation, Writing (review & editing), Funding acquisition. MSG: Conceptualization, Investigation, Writing (review & editing). Data availability statement The data used for the data application is available upon reasonable request. Ethics approval and consent to participate The study protocol was approved by the New York University Langone Health Institutional Review Board. All study participants provided informed consent. Funding This research was funded by the National Institute on Drug Abuse grant R01DA046653 and R01DA052426. Competing interests None declared. Acknowledgments We thank the study participants for contributing to the cohort study used for the data application. 6 Appendix 6.1 Implementation code for R, Stata, and SAS This section provides example code for fitting the MP-CWGEE model with an independent correlation structure using the geepack package in R , the PROC GENMOD procedure in SAS , and the xtgee command in Stata . The inverse cluster weight (one divided by the number of repeated measures) is denoted by inverse_n_measures. R code Download figure Open in new tab SAS code PROC GENMOD DATA = dataset; CLASS person _id; MODEL outcome = exposure covariate / DIST = POISSON LINK = LOG; WEIGHT inverse _n_measures; REPEATED SUBJECT = person _id / TYPE = IND; RUN; Stata code Download figure Open in new tab 6.2 Asymptotic normality of MP-CWGEE estimator Theorem 1. Under suitable regularity conditions, the MP-CWGEE estimator is asymptotically normal: where β is the vector of true parameter values , B is the limit of the expected derivative matrix, and M is the limit of the variance-covariance matrix of the estimating functions . Proof . The following proof employs a strategy and uses notation similar to Williamson et al. [ 17 ]. We consider the asymptotic distribution of the MP-CWGEE estimator that solves the estimating equation: where U ij ( β ) represents the contribution to the score function from the j -th observation in the i -th cluster, and n i is the size of cluster i . Define as the average score contribution from cluster i . Under the true parameter vector β , we have . for all i . Let denote the centered version. Note that are independent with mean zero. To establish asymptotic normality, we verify the conditions for Lyapunov’s central limit theorem. 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