Abstract. An important problem in multilevel modeling is what constitutes a sufficient sample size for accurate estimation. In multilevel analysis, the major restriction is often the higher-level sample size. In this paper, a simulation study is used to determine the influence of different sample sizes at the group level on the accuracy of the estimates (regression coefficients and variances) and their standard errors. In addition, the influence of other factors, such as the lowest-level sample size and different variance distributions between the levels (different intraclass correlations), is examined. The results show that only a small sample size at level two (meaning a sample of 50 or less) leads to biased estimates of the second-level standard errors. In all of the other simulated conditions the estimates of the regression coefficients, the variance components, and the standard errors are unbiased and accurate.

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Purpose -The opposite of absenteeism, presenteeism, is the phenomenon of employees staying at work when they should be off sick. Presenteeism is an important problem for organizations, because employees who turn up for work, when sick, cause a reduction in productivity levels. The central aim of the present study is to examine the longitudinal relationships between job demands, burnout (exhaustion and depersonalization), and presenteeism. We hypothesized that job demands and exhaustion (but not depersonalization) would lead to presenteeism, and that presenteeism would lead to both exhaustion and depersonalization over time. Design/methodology/approach -The hypotheses were tested in a sample of 258 staff nurses who filled out questionnaires at three measurement points with 1.5 years in-between the waves. Findings -Results were generally in line with predictions. Job demands caused more presenteeism, while depersonalization was an outcome of presenteeism over time. Exhaustion and presenteeism were found to be reciprocal, suggesting that when employees experience exhaustion, they mobilize compensation strategies, which ultimately increases their exhaustion. Research limitations/implications -These findings suggest that presenteeism can be seen as a risk-taking organizational behavior and shows substantial longitudinal relationships with job demands and burnout. Practical implications -The study suggests that presenteeism should be prevented at the workplace. Originality/value -The expected contribution of the manuscript is not only to put presenteeism on the research agenda but also to make both organizations and scientists attend to its detrimental effects on employees' wellbeing and (consequently) on the organization.

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A multilevel problem concerns a population with a hierarchical structure. A sample from such a population can be described as a multistage sample. First, a sample of higher level units is drawn (e.g. schools or organizations), and next a sample of the sub‐units from the available units (e.g. pupils in schools or employees in organizations). In such samples, the individual observations are in general not completely independent. Multilevel analysis software accounts for this dependence and in recent years these programs have been widely accepted. Two problems that occur in the practice of multilevel modeling will be discussed. The first problem is the choice of the sample sizes at the different levels. What are sufficient sample sizes for accurate estimation? The second problem is the normality assumption of the level‐2 error distribution. When one wants to conduct tests of significance, the errors need to be normally distributed. What happens when this is not the case? In this paper, simulation studies are used to answer both questions. With respect to the first question, the results show that a small sample size at level two (meaning a sample of 50 or less) leads to biased estimates of the second‐level standard errors. The answer to the second question is that only the standard errors for the random effects at the second level are highly inaccurate if the distributional assumptions concerning the level‐2 errors are not fulfilled. Robust standard errors turn out to be more reliable than the asymptotic standard errors based on maximum likelihood.

Analysis of variance (ANOVA) is an extremely important method in exploratory and confirmatory data analysis. Unfortunately, in complex problems (e.g., split-plot designs), it is not always easy to set up an appropriate ANOVA. We propose a hierarchical analysis that automatically gives the correct ANOVA comparisons even in complex scenarios. The inferences for all means and variances are performed under a model with a separate batch of effects for each row of the ANOVA table.We connect to classical ANOVA by working with finite-sample variance components: fixed and random effects models are characterized by inferences about existing levels of a factor and new levels, respectively. We also introduce a new graphical display showing inferences about the standard deviations of each batch of effects.We illustrate with two examples from our applied data analysis, first illustrating the usefulness of our hierarchical computations and displays, and second showing how the ideas of ANOVA are helpful in understanding a previously fit hierarchical model. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Statistics, 2005, Vol. 33, No. 1, 1-33. This reprint differs from the original in pagination and typographic detail. 1 2 A. GELMANin its more elaborate forms such as split-plot analysis. We believe, however, that the ideas of ANOVA are useful in many applications of statistics. For the purpose of this paper, we identify ANOVA with the structuring of parameters into batches-that is, with variance components models. There are more general mathematical formulations of the analysis of variance, but this is the aspect that we believe is most relevant in applied statistics, especially for regression modeling.We shall demonstrate how many of the difficulties in understanding and computing ANOVAs can be resolved using a hierarchical Bayesian framework. Conversely, we illustrate how thinking in terms of variance components can be useful in understanding and displaying hierarchical regressions. With hierarchical (multilevel) models becoming used more and more widely, we view ANOVA as more important than ever in statistical applications.Classical ANOVA for balanced data does three things at once: