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Data in medical sciences often have a hierarchical structure with lower level units (e.g. children) nested in higher level units (e.g. departments). Several specific but frequently studied settings, mainly in longitudinal and family research, involve a large number of units that tend to be quite small, with units containing only one element referred to as singletons. Regardless of sparseness, hierarchical data should be analyzed with appropriate methodology such as, for example linear-mixed models. Using a simulation study, based on the structure of a data example on Ceftriaxone consumption in hospitalized children, we assess the impact of an increasing proportion of singletons (0-95%), in data with a low, medium, or high intracluster correlation, on the stability of linear-mixed models parameter estimates, confidence interval coverage and F test performance. Some techniques that are frequently used in the presence of singletons include ignoring clustering, dropping the singletons from the analysis and grouping the singletons into an artificial unit. We show that both the fixed and random effects estimates and their standard errors are stable in the presence of an increasing proportion of singletons. We demonstrate that ignoring clustering and dropping singletons should be avoided as they come with biased standard error estimates. Grouping the singletons into an artificial unit might be considered, although the linear-mixed model performs better even when the proportion of singletons is high. We conclude that the linear-mixed model is stable in the presence of singletons when both lower and higher level sample sizes are fixed. In this setting, the use of remedial measures, such as ignoring clustering and grouping or removing singletons, should be dissuaded.

In a linear multilevel model, significance of all fixed effects can be determined using F tests under maximum likelihood (ML) or restricted maximum likelihood (REML). In this paper, we demonstrate that in the presence of primary unit sparseness, the performance of the F test under both REML and ML is rather poor. Using simulations based on the structure of a data example on ceftriaxone consumption in hospitalized children, we studied variability, type I error rate and power in scenarios with a varying number of secondary units within the primary units. In general, the variability in the estimates for the effect of the primary unit decreased as the number of secondary units increased. In the presence of singletons (i.e., only one secondary unit within a primary unit), REML consistently outperformed ML, although even under REML the performance of the F test was found inadequate. When modeling the primary unit as a random effect, the power was lower while the type I error rate was unstable. The options of dropping, regrouping, or splitting the singletons could solve either the problem of a high type I error rate or a low power, while worsening the other. The permutation test appeared to be a valid alternative as it outperformed the F test, especially under REML. We conclude that in the presence of singletons, one should be careful in using the F test to determine the significance of the fixed effects, and propose the permutation test (under REML) as an alternative.

In chemical risk assessment, it is important to determine the quantiles of the distribution of concentration data. The selection of an appropriate distribution and the estimation of particular quantiles of interest are largely hindered by the omnipresence of observations below the limit of detection, leading to left-censored data. The log-normal distribution is a common choice, but this distribution is not the only possibility and alternatives should be considered as well. Here, we focus on several distributions that are related to the log-normal distribution or that are seminonparametric extensions of the log-normal distribution. Whereas previous work focused on the estimation of the cumulative distribution function, our interest here goes to the estimation of quantiles, particularly in the left tail of the distribution where most of the left-censored data are located. Two different model averaged quantile estimators are defined and compared for different families of candidate models. The models and methods of selection and averaging are further investigated through simulations and illustrated on data of cadmium concentration in food products. The approach is extended to include covariates and to deal with uncertainty about the values of the limit of detection. These extensions are illustrated with (134) cesium measurements from Fukushima Prefecture, Japan. We can conclude that averaged models do achieve good performance characteristics in case no useful prior knowledge about the true distribution is available; that there is no structural difference in the performance of the direct and indirect method; and that, not surprisingly, only the true or closely approximating model can deal with extremely high percentages of censoring.

Count data are very common in health services research, and very commonly the basic Poisson regression model has to be extended in several ways to accommodate several sources of heterogeneity: (i) an excess number of zeros relative to a Poisson distribution, (ii) hierarchical structures, and correlated data, (iii) remaining "unexplained" sources of overdispersion. In this paper, we propose hierarchical zero-inflated and overdispersed models with independent, correlated, and shared random effects for both components of the mixture model. We show that all different extensions of the Poisson model can be based on the concept of mixture models, and that they can be combined to account for all different sources of heterogeneity. Expressions for the first two moments are derived and discussed. The models are applied to data on maternal deaths and related risk factors within health facilities in Mozambique. The final model shows that the maternal mortality rate mainly depends on the geographical location of the health facility, the percentage of women admitted with HIV and the percentage of referrals from the health facility.

In this paper we present and discuss a novel, simple and easy to implement parametric modeling approach to assess synergy. An extended three parameter log-logistic model is used to analyse the data and calculate confidence intervals of the interaction indices. In addition the model corrects for the bias due to plate-location effects. The analysis is performed with PROC NLMIXED and SAS-code is provided. The approach is illustrated using data coming from an oncology study in which the inhibition effect of a combination of two compounds is studied using 96-well plates and a fixed-ratio design.

All models for incomplete data either explicitly make assumptions about aspects of the distribution of the unobserved outcomes, given the observed ones, or at least implicitly imply such. One consequence is that there routinely exist a whole class of models, coinciding in their description of the observed portion of the data but differing with respect to their predictions of what is unobserved. Within such a class, there always is a single model corresponding to so-called random missingness, in the sense that the mechanism governing missingness depends on covariates and observed outcomes, but given these not further on unobserved outcomes. We employ these results in the context of so-called shared-parameter models where outcome and missingness models are connected by means of common latent variables or random effects, to devise a sensitivity analysis framework. Precisely, the impact of varying unverifiable assumptions about unobserved measurements on parameters of interest is studied. Apart from analytic considerations, the proposed methodology is applied to assess treatment effect in data from a clinical trial in toenail dermatophyte onychomycosis. While our focus is on longitudinal outcomes with incomplete outcome data, the ideas developed in this paper are of use whenever a shared-parameter model could be considered.