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This paper assesses the options available to researchers analysing multilevel (including longitudinal) data, with the aim of supporting good methodological decision-making. Given the confusion in the literature about the key properties of fixed and random effects (FE and RE) models, we present these models’ capabilities and limitations. We also discuss the within-between RE model, sometimes misleadingly labelled a ‘hybrid’ model, showing that it is the most general of the three, with all the strengths of the other two. As such, and because it allows for important extensions—notably random slopes—we argue it should be used (as a starting point at least) in all multilevel analyses. We develop the argument through simulations, evaluating how these models cope with some likely mis-specifications. These simulations reveal that (1) failing to include random slopes can generate anti-conservative standard errors, and (2) assuming random intercepts are Normally distributed, when they are not, introduces only modest biases. These results strengthen the case for the use of, and need for, these models.

This article challenges Fixed Effects (FE) modeling as the ‘default’ for time-series-cross-sectional and panel data. Understanding different within and between effects is crucial when choosing modeling strategies. The downside of Random Effects (RE) modeling—correlated lower-level covariates and higher-level residuals—is omitted-variable bias, solvable with Mundlak's (1978a) formulation. Consequently, RE can provide everything that FE promises and more, as confirmed by Monte-Carlo simulations, which additionally show problems with Plümper and Troeger's FE Vector Decomposition method when data are unbalanced. As well as incorporating time-invariant variables, RE models are readily extendable, with random coefficients, cross-level interactions and complex variance functions. We argue not simply for technical solutions to endogeneity, but for the substantive importance of context/heterogeneity, modeled using RE. The implications extend beyond political science to all multilevel datasets. However, omitted variables could still bias estimated higher-level variable effects; as with any model, care is required in interpretation.

Whilst some argue that a solution to the age-period-cohort (APC) 'identification problem' is impossible, numerous methodological solutions have been proposed, including Yang and Land's Hierarchical-APC (HAPC) model: a multilevel model considering periods and cohorts as cross-classified contexts in which individuals exist. To assess the assumptions made by the HAPC model, and the situations in which it does and does not work. Simulation scenarios assess the effect of cohort trends in the Data Generating Process (DGP) (compared to only random variation), and grouping cohorts in both DGP and fitted model. The model only works if either they can assume that there are no linear (or non-linear) trends in periods or cohorts, they control any cohort trend in the model's fixed part and assume there is no period trend, or they group cohorts in such a way that they exactly match the groupings in the (unknown) DGP. Otherwise, the model can arbitrarily reapportion APC effects, radically impacting interpretation.

This paper develops a methodology for extending multilevel modelling to incorporate spatial interaction effects. The motivation is that classic multilevel models are not specifically spatial. Lower level units may be nested into higher level ones based on a geographical hierarchy (or a membership structure--for example, census zones into regions) but the actual locations of the units and the distances between them are not directly considered: what matters is the groupings but not how close together any two units are within those groupings. As a consequence, spatial interaction effects are neither modelled nor measured, confounding group effects (understood as some sort of contextual effect that acts 'top down' upon members of a group) with proximity effects (some sort of joint dependency that emerges between neighbours). To deal with this, we incorporate spatial simultaneous autoregressive processes into both the outcome variable and the higher level residuals. To assess the performance of the proposed method and the classic multilevel model, a series of Monte Carlo simulations are conducted. The results show that the proposed method performs well in retrieving the true model parameters whereas the classic multilevel model provides biased and inefficient parameter estimation in the presence of spatial interactions. An important implication of the study is to be cautious of an apparent neighbourhood effect in terms of both its magnitude and statistical significance if spatial interaction effects at a lower level are suspected. Applying the new approach to a two-level land price data set for Beijing, China, we find significant spatial interactions at both the land parcel and district levels.

Analyses of health over time must consider the potential impacts of ageing as well as any effects relating to cohort differences. The British Household Panel Survey (BHPS) and Understanding Society longitudinal studies are employed to assess trends in mental ill-health over a 26-year period. This analysis uses cross-classified multilevel models in an exploratory, non-parametric approach to evaluate age and cohort effects net of each other. Mental ill-health evidences an initial worsening trend as people age which then reverses and exhibits improvement in late-middle-age, before declining again in the latter stages of life. There were less defined cohort trends. The modelling technique also reveals the relative importance of the temporal contexts in relation to inter- and intra-individual effects on mental ill-health, demonstrating that the ageing and cohort dimensions explain little variation compared to these more dominant within and between influences. Ultimately, we suggest that researchers would benefit from wider use of this exploratory modelling strategy when evaluating underlying health trends and more research is now needed to explore potential explanations of these baseline trajectories.

Background Trust is important for health at both the individual and societal level. Previous research using Western concepts of trust has shown that a high level of trust in society can positively affect individuals’ health; however, it has been found that the concepts and culture of trust in China are different from those in Western countries and research on the relationship between trust and health in China is scarce. Method The analyses use data from the national scale China General Social Survey (CGSS) on adults aged above 18 in 2005 and 2010. Two concepts of trust (“out-group” and “in-group” trust) are used to examine the relationship between trust and self-rated health in China. Multilevel logistical models are applied, examining the trust at the individual and societal level on individuals’ self-rated health. Results In terms of interpersonal trust, both “out-group” and “in-group” trust are positively associated with good health in 2005 and 2010. At the societal level, the relationships between the two concepts of trust and health are different. In 2005, higher “out-group” social trust (derived from trust in strangers) is positively associated with better health; however, higher “in-group” social trust (derived from trust in most people) is negatively associated with good health in 2010. The cross-level interactions show that lower educated individuals (no education or only primary level), rural residents and those on lower incomes are the most affected groups in societies with higher “out-group” social trust; whereas people with lower levels of educational attainment, a lower income, and those who think that most people can be trusted are the most affected groups in societies with higher “in-group” social trust. Conclusion High levels of interpersonal trust are of benefit to health. Higher “out-group” social trust is positively associated with better health; while higher “in-group” social trust is negatively associated with good health. Individuals with different levels of educational attainment are affected by trust differently.

Many ecological- and individual-level analyses of voting behaviour use multiple regressions with a considerable number of independent variables but few discussions of their results pay any attention to the potential impact of inter-relationships among those independent variables—do they confound the regression parameters and hence their interpretation? Three empirical examples are deployed to address that question, with results which suggest considerable problems. Inter-relationships between variables, even if not approaching high collinearity, can have a substantial impact on regression model results and how they are interpreted in the light of prior expectations. Confounded relationships could be the norm and interpretations open to doubt, unless considerable care is applied in the analyses and an extended principal components method for doing that is introduced and exemplified.

It is claimed the hierarchical-age–period–cohort (HAPC) model solves the age–period–cohort (APC) identification problem. However, this is debateable; simulations show situations where the model produces incorrect results, countered by proponents of the model arguing those simulations are not relevant to real-life scenarios. This paper moves beyond questioning whether the HAPC model works, to why it produces the results it does. We argue HAPC estimates are the result not of the distinctive substantive APC processes occurring in the dataset, but are primarily an artefact of the data structure—that is, the way the data has been collected. Were the data collected differently, the results produced would be different. This is illustrated both with simulations and real data, the latter by taking a variety of samples from the National Health Interview Survey (NHIS) data used by Reither et al. (Soc Sci Med 69(10):1439–1448, 2009) in their HAPC study of obesity. When a sample based on a small range of cohorts is taken, such that the period range is much greater than the cohort range, the results produced are very different to those produced when cohort groups span a much wider range than periods, as is structurally the case with repeated cross-sectional data. The paper also addresses the latest defence of the HAPC model by its proponents (Reither et al. in Soc Sci Med 145:125–128, 2015a). The results lend further support to the view that the HAPC model is not able to accurately discern APC effects, and should be used with caution when there appear to be period or cohort near-linear trends.

Kelley et al. argue that group-mean-centering covariates in multilevel models is dangerous, since—they claim—it generates results that are biased and misleading. We argue instead that what is dangerous is Kelley et al.’s unjustified assault on a simple statistical procedure that is enormously helpful, if not vital, in analyses of multilevel data. Kelley et al.’s arguments appear to be based on a faulty algebraic operation, and on a simplistic argument that parameter estimates from models with mean-centered covariates must be wrong merely because they are different than those from models with uncentered covariates. They also fail to explain why researchers should dispense with mean-centering when it is central to the estimation of fixed effects models—a common alternative approach to the analysis of clustered data, albeit one increasingly incorporated within a random effects framework. Group-mean-centering is, in short, no more dangerous than any other statistical procedure, and should remain a normal part of multilevel data analyses where it can be judiciously employed to good effect.

Much has been written in recent years about the claimed polarization of the U.S. electorate, with substantial differences as to whether there has been greater spatial polarization, at several geographical scales, over recent decades. To assess the veracity of those alternative views, a bespoke data set showing percentage support for the Democratic Party's presidential candidates at the county, state, and divisional scales has been analyzed using a robust, statistically based measure of polarization and segregation. The ecological results provide clear and compelling evidence of a trend toward greater polarization across the nine census divisions, across the forty-nine states within those divisions, and across the 3,077 counties within the states-with strong evidence that the differences over time at the last of those scales are highly statistically significant. Within those general trends, polarization has been greater in some states than others and also within some states more than others-identifying additional geographies calling for further research.