In environmental and occupational research, when concentration measurements are taken from the same individual or study site, they are considered to be repeated measurements. Furthermore, when these samples fall below the limit of detection (LOD) of laboratory instruments, they are referred to as left-censored repeated measures data. These unquantified measurements typically represent low-level concentrations between zero and the LOD. There is an increasing need for statistical models that can effectively analyze left-censored environmental exposure and biomonitoring data with repeated measures, particularly among industrial hygienists. This is due to the growing importance of estimating the impact of exposure on disease risk and understanding the variability in occupational exposure both within and between workers. Analytical results from laboratories, including data on environmental contaminants (e.g., from hand wipes or personal breathing zone air samples) and occupational exposures (e.g., from biological media such as urine or serum) are often subject to non-detectable concentrations and tend to exhibit right-skewed distributions. The statistical modeling of exposure data that contain both repeated samples and samples below the LOD can be complex.

The author would like to introduce the R package ‘marlod’ [1] to analyze left-censored repeated measures exposure data when:

The ‘marlod’ package offers comprehensive functionality for univariable and multivariable analyses of environmental exposure and biomonitoring data. The reference manual for function documentation is accessible through https://cran.r-project.org/web/packages/marlod/marlod.pdf, while the vignette or user guide can be obtained either via https://cran.r-project.org/web/packages/marlod/vignettes/marlod.html or by using the command line ‘??marlod’ in the R console. In existing literature, exposure data with correlated outcomes are often assumed to follow a log-normal distribution, necessitating a natural logarithmic transformation [2]. Recently, small sample exposure data have been analyzed using a combination of marginal modeling with different substitution methods (i.e., single or multiple imputation techniques) to obtain mean regression estimates. The preferred approach uses simple working correlation structures, requiring the estimation of fewer nuisance covariance parameters [3]. Note that Bayesian methods are also suggested as viable alternatives for addressing left-censored data.

GEE is a specialized instance of marginal mean regression analysis, providing consistent regression parameter estimates even when the working correlation structure is mis-specified [4]. This property also allows marginal modeling to effectively handle data with highly right-skewed outcome distributions [3]. In addition to the GEE, the package incorporates the GMM method, a marginal analysis technique widely used in econometrics that leverages all available estimating equations [5]. Moreover, the package includes the QIF method, which demonstrates improved estimator efficiency relative to GEE, particularly when the working correlation structure is mis-specified and in large-sample settings [6].

The package also includes marginal quantile regression model for analyzing left-censored repeated measures exposure data [7]. When using traditional parametric mean regression, if the transformed exposure data do not adhere to a known distribution, then results may be suboptimal, as the estimated mean and standard deviation can be sensitive to large values. Additionally, the geometric mean, defined as the exponentiated mean of the log-transformed data, may become unstable when the distribution of the logged data is asymmetric [8]. Unlike parametric mean regression, quantile regression does not require assumptions about the underlying distribution and is robust against outliers, making it advantageous for skewed data. Moreover, quantile regression provides a comprehensive view of the entire conditional distribution of the dependent variable, eliminating the need for data transformation, regardless of the skewness of the data. The vignette includes a simulated dataset from the literature [3] to illustrate multivariable analyses in marginal mean and quantile regression models.

Lastly, this package allows users to develop either mean or quantile predictive regression models that address issues related to left censoring and time-dependent covariates, enabling a quantitative assessment of whether past and current covariates can predict current and/or future exposure levels [9]. In addition, a simulated dataset with a time-dependency mechanism is provided to facilitate selecting and applying a specific time-dependent type. Repeated measures data from clustered studies typically exhibit positive and uniform correlations among sampling outcomes (i.e., repeated measures) within the same cluster. Repeated measures data can also be longitudinal, where outcomes are measured for each subject over time. In such studies, certain covariates may vary over time, and incorrectly treating time-dependent covariates as time-independent can lead to reduced efficiency in regression parameter estimation [10].