Intracluster correlation coefficients in the Greater Mekong Subregion for sample size calculations of cluster randomized malaria trials

Defining a new P. vivax infection from longitudinally collected data is more complicated than P. falciparum as P. vivax infections recur frequently. Recurrences of P. vivax infections after treatment of the blood stage infection can be due to recrudescence, relapse or reinfection. The ICCs are based on baseline prevalence of P. vivax/falciparum infections and the cumulative incidence of detected P. vivax/falciparum parasitaemias at each quarterly surveys based on uPCR results.

The cumulative incidence of P. vivax/falciparum parasitaemias over the 12-month period was calculated based on uPCR results collected at month 0, 3, 6, 9 and 12. A participant was considered to have a recurrent P. vivax infection if there were two or more positive uPCR results during the 12 months follow up period. As consecutive positive uPCR tests could be due to a re-infection following a new mosquito bite or a continuous infection which is likely to be due to persistence in P. falciparum infections and a relapse in P. vivax infections. To address this uncertainty, an “episode” was defined in two ways. In the first approach each positive uPCR test was considered as a separate episode (i.e. reinfections). In the second, consecutive positive uPCR results were considered to belong to the same continuous infection (persistent or relapsing asymptomatic parasitaemia). The ICCs for the second approach are presented in the Additional file 1.

The outcomes of interest for the ICC estimation are the prevalence and the incidence of P. falciparum and P. vivax infections. A logistic regression model is the most relevant model for prevalence while the Poison model is the most natural model when modeling incidence as the outcome of interest. The basic ICC formula presented above (\(\rho = \frac{{\sigma_{b}^{2} }}{{\sigma_{b}^{2} + \sigma_{w}^{2} }}\)) refers to a case where two-level hierarchical data are of interest. In practice, often several levels of clustering are available and of interest. The hierarchical structure of the data in the TME study included 4 levels: longitudinal data on infection status (level 1) collected repeatedly for each individual (level 2) who belonged to a village (level 3) which was located in a country (level 4). However, country specific ICCs were estimated because there was considerable heterogeneity in baseline P. falciparum/P. vivax prevalence between countries. In this case, the level of country is not considered. The model for estimating ICC for prevalence is reduced to 2 levels in each country because ICCs were estimate at baseline only and, therefore, each individual contributes only one observation at baseline. By contrast, for the estimation of ICC for incidence, multiple outcomes were aggregated, i.e. each individual had one observation for the outcome counts over time and the exposure time was aggregated for each individual. Two-level hierarchical models were fitted to estimate country specific ICCs for both prevalence and incidence with a village as unit of randomization.

Methodological approaches have been developed describing procedures used to compute ICC values applicable to models with multiple hierarchical levels that include logistic regression models as well as other generalized linear models such as the Poisson regression models [8, 10, 11]. The ICC values can be estimated from model equations as exact estimation methods or through use of simulations. A latent variable approach is another method for estimating ICC from logistic (logit link-scale) regression models. The link-scale is often considered to be of interest for prevalence, because the estimates of the individual outcomes are performed on the underlying latent scale [17, 18]. Nakagawa et al. provided comprehensive methods for calculating ICCs using both exact and latent variable methods for logistic and Poisson models [18]. However, Austin et al. have shown that there is no latent response formulation for Poisson models and such a model is, therefore, not included in this paper. The exact estimation method and the simulation method were utilized in the estimation of the ICC values for both incidence (Poisson model) and prevalence (logit model) of P. falciparum and P. vivax. In addition, the latent variable estimates of ICC for logit model are provided in the additional material for the estimation of ICCs for prevalence of P. falciparum and P. vivax. ICC values for the model are provided with and without the covariates sex and age because they were independently associated with the outcome [12]. The estimation of the country specific ICC is the main focus of this article. However, the overall ICC is also included for prevalence of P. falciparum and P. vivax using the latent variable method. For prevalence ICC uses the baseline prevalence as this is the time-point measure most often used in sample size calculations. The statistical methodology for estimating ICC from a random effects logistic model using the exact calculation, simulation-based and latent variable method is introduced. In addition the methodology for estimating ICCs from the random effect Poisson model using exact calculation and simulation-based methods is described [8].

For prevalence outcome, consider a logistic model for the outcome \(Y_{ij}\), where \(i\) denotes an individual and \(j\) denotes a cluster then:where \(p_{ij}\) is the probability of experiencing an outcome for individual \(i\) in cluster \(j\). And the logistic regression model is fitted as a generalized linear mixed model (GLMM) with logit link function as:where \(X_{ij}\) refers to covariates such as age and sex measured on the individual \(i\) in cluster \(j\) and \(\alpha_{j}\) is a cluster-specific random effect such that \(\alpha_{j}\) follows a normal distribution with a mean of 0 and variance equal to \(\sigma_{\alpha }^{2}\).

The exact ICC for prevalence using logistic model is calculated as follows [10] where \(\sigma_{\alpha }^{2}\) (the random effect variance for the level of interest) and \(\beta\) [the log (odds ratio)] are estimated from the model and \(X\) refers to covariates such as age and sex.

The calculation of ICC as a postestimation estimate from software is provided using the latent variable approach for logistic model. In the logistic model, the underlying logistic error distribution has a constant variance \(\frac{{\pi^{2} }}{3}\) which was used as residual variance when calculating ICC from a logistic model and then the latent ICC is given by \(ICC = \frac{{\sigma_{\alpha }^{2} }}{{\sigma_{\alpha }^{2} + \frac{{\pi^{2} }}{3}}}\), where \(\sigma_{\alpha }^{2}\) is the between cluster variance for the binary outcome. Goldstein et al. [11] suggested that the latent variable approach to estimate the ICC is only appropriate when the binary outcome can be an underlying continuous latent variable. However, this is the version of ICC that is readily obtained in most software including Stata. In a Poisson model, the error variance is not constant and depends on the covariates included in the model. Consider a Poisson model for the outcome \(Y_{ij}\), where \(i\) denotes an individual and \(j\) denotes a cluster then:

And the Poisson regression model is fitted as a generalized linear mixed model (GLMM) with log link function as:where \(X_{ij}\) refers to covariates such as age and sex measured on the individual \(i\) in cluster \(j\); \(\lambda_{ij}\) is an estimate of the expected number of outcome events for individual \(i\) in cluster \(j\); and \(\alpha_{j}\) is a cluster-specific random effect such that \(\alpha_{j}\) follows a normal distribution with a mean of 0 and variance equal to \(\sigma_{\alpha }^{2}\). The estimate of the ICC from exact calculation is calculated as follows [8]:where \(\sigma_{\alpha }^{2}\) (the random effect variance for the level of interest) and \(\beta\)(the log (incidence rate ratio)) are estimated from the model and \(X\) refers to covariates such as age and sex. The simulation procedures are detailed in Austin et al. [8].

The simulation-based algorithm for both prevalence and incidence proceeds as follows:

The confidence intervals for the ICCs were calculated by using bootstrapped samples to estimate the standard error. All analyses including simulations and bootstrapping were performed in Stata 15.

The ICC values for the prevalence of P. falciparum were less than 0.10 in all countries for a model without covariates as well as the model with age and sex as covariates (Table 1). Laos had the highest ICC values for the prevalence of P. falciparum infection with a value of 0.08 (95% CI 0.06 to 0.11) in either model. However, these ICCs are practically similar in all the four countries.

Similarly, as shown in Table 2, the ICC values for the prevalence of P. vivax infection at baseline were less than 0.10 for all countries for a model without covariates as well as the model with age and sex as covariates. Laos had the highest ICC for the prevalence of P. vivax infection of about 0.06 (95% CI 0.05 to 0.08). Again, these ICCs are very similar across the four countries.

Using an exact calculation approach from the Poisson model for incidence of P. falciparum, the country specific ICC values were less than 0.02 in Vietnam, Cambodia and Myanmar for a model without covariates as well as the model with age and sex as covariates (Table 3). Laos had the highest ICC for P. falciparum infection 0.71 (95% CI 0.52 to 0.89). Similarly, as shown in Table 4 below the ICC values for P. vivax infection were very low in Vietnam, Cambodia and Myanmar i.e. ICC of less than 0.10. Laos had the highest ICC for the incidence of P. vivax infections of around 0.81 (95% CI 0.59 to 1.00) for a model without covariates as well as the model with age and sex as covariates.

The actual ICC values from simulation and latent variable approaches are presented in the Additional file 1: Tables S1–S6. Simulations gave consistently higher ICC values than the corresponding exact calculation method for prevalence (Fig. 1). However, exact calculation gave lower ICC than latent variable approach for prevalence. In fact, the latent variable gave the highest ICCs compared to both the exact and the simulation methods for prevalence. The same trend was observed for estimation of ICCs for incidence with simulations giving consistently higher ICC values than the corresponding the exact calculation method (Additional file 2: Figures S1).

The overall estimated ICC from the latent variable approach for the prevalence are 0.26 (95% CI 0.13 to 0.45) and 0.21 (95% CI 0.10 to 0.38) for P. falciparum and P. vivax (in the 16 villages), respectively.

As shown in the Additional file 1: Tables S7–S10, the estimates of ICC are generally similar to corresponding scenarios under definitions of incidence.

In order to illustrate the number of clusters/villages (sample sizes) that would be needed to design new malaria pre-elimination trial, the overall ICC was estimated for the four countries using latent method for the comparison of prevalence of P. falciparum and P. vivax between the control and the intervention. The overall estimated ICCs from the latent variable approach are 0.26 (95% CI 0.13 to 0.45) and 0.21 (95% CI 0.10 to 0.38) for P. falciparum and P. vivax (in the 16 villages) respectively. Baseline prevalence observed in the TME trial for each country was used as the control prevalence. In line with the parent trial, the aim was to detect at least a 95% decline in prevalence of P. falciparum following administration of MDA plus a single low dose primaquine. The mean village size was set at 500 participants in line with the observed number of participants per village in the TME trial. Table 5 summarizes the expected number of villages per arm required in each country for stand-alone studies. For the design of a multicentre cluster randomized trial for the Greater Mekong Subregion the overall sample size is provided based on the average prevalence across the four countries. Separate cluster randomized trials, need to recruit 134 villages per arm for Cambodia due to the low prevalence.

The required number of clusters varies with varying villages sizes. As expected, the number of clusters increase with increasing ICC for a constant number of individuals per village. Figure 2 also shows that for ICCs between 0 and 0.4, with the given effect size, increasing the number of resident per village (village size) above 50 does not result in an increased benefit in terms of the number of clusters (villages) required to have sufficient statistical power.