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In the models discussed here, there is a hierarchy of variation that corresponds to groupings within the data. For example, students may be sampled from different classes, that in turn are sampled from different schools. Or, rather than being nested, groups may be crossed. Important notions are those of fixed and random effects, and variance components. Analysis of data from designs that have the balance needed to allow an analysis of variance breakdown are a special case. Further types of mixed models are generalized linear mixed models and repeated measures models. Repeated measures models are multilevel models where measurements consist of multiple profiles in time or space, resulting in time or spatial dependence. Relative to the length of time series that is required for a realistic analysis, each individual repeated measures profile can and often will have values for a few time points only.
This chapter shows how we can integrate inferences across models. We provide four examples of situations in which, by combining models, researchers can learn more than they could from any single model. Examples include situations in which researchers seek to integrate inferences from experimental and observational data, learn across settings, or integrate inferences from multiple studies.
This accessible and practical textbook gives students the perfect guide to the use of regression models in testing and evaluating hypotheses dealing with social relationships. A range of statistical methods suited to a wide variety of dependent variables is explained, which will allow students to read, understand, and interpret complex statistical analyses of social data. Each chapter contains example applications using relevant statistical methods in both Stata and R, giving students direct experience of applying their knowledge. A full suite of online resources - including statistical command files, datasets and results files, homework assignments, class discussion topics, PowerPoint slides, and exam questions - supports the student to work independently with the data, and the instructor to deliver the most effective possible course. This is the ideal textbook for advanced undergraduate and beginning graduate students taking courses in applied social statistics.
In this chapter we introduce developing and interpreting multilevel models. We first define multilevel models and explore how this approach is an improvement on disaggregation and aggregation of data across multiple levels. We then work through four different multilevel models. We provide examples of what kinds of questions can be answered by each model and how to interpret the statistical output. We then explore some additional issues in fitting multilevel models in Stata and consider additional applications of multilevel models.
Difficulties with emotion regulation are integral to borderline personality disorder (BPD) and its hypothesized developmental pathway. Here, we prospectively assess trajectories of emotion processing across childhood, how BPD symptoms impact these trajectories, and whether developmental changes are transdiagnostic or specific to BPD, as major depressive (MDD) and conduct disorders (CD) are also characterized by emotion regulation difficulties. This study included 187 children enriched for those with early symptoms of depression and disruptive behaviors from a longitudinal study. We created multilevel models of multiple components of emotional processing from mean ages 9.05 to 18.55 years, and assessed the effect of late adolescent BPD, MDD, and CD symptoms on these trajectories. Linear trajectories of coping with sadness and anger, and quadratic trajectories of dysregulated expressions of sadness and anger were transdiagnostic, but also exhibited independent relationships with BPD symptoms. Only inhibition of sadness was related to BPD symptoms. The quadratic trajectories of poor emotional awareness and emotional reluctance were also independently related to BPD. Findings support examining separable components of emotion processing across development as potential precursors to BPD, underscoring the importance of understanding these trajectories as not only a marker of potential risk but also potential targets for prevention and intervention.
The calibration of probability or confidence judgments concerns the association between the judgments and some estimate of the correct probabilities of events. Researchers rely on estimates using relative frequencies computed by aggregating data over observations. We show that this approach creates conceptual problems, and may result in the confounding of explanatory variables or unstable estimates. To circumvent these problems we propose using probability estimates obtained from statistical models—specifically mixed models for binary data—in the analysis of calibration. We illustrate this methodology by re-analyzing data from a published study and comparing the results from this approach to those based on relative frequencies. The model-based estimates avoid problems with confounding variables and provided more precise estimates, resulting in better inferences.
Wrongful convictions are an increasing salient feature of criminal justice discourse in the United States. Many states have adopted reforms to mitigate the likelihood of wrongful convictions, discover errors, and provide redress in the wake of exonerations, yet we know little about why some are seemingly more committed to reducing such errors than others. We argue that public opinion is consequential for policy reform, but its effects are contingent on the electoral vulnerability of state lawmakers. We also suggest that advocacy organizations play a critical role in policy adoption. Incorporating data from all 50 states from 1989 to 2018, we investigate the adoption of five types of wrongful conviction reforms: (1) changes to eyewitness identification practices, (2) mandatory recording of interrogations, (3) the preservation of biological evidence, (4) access to postconviction DNA testing, and (5) exoneree compensation. Our results highlight a more nuanced view of how public opinion shapes policy.
When working with grouped data, investigators may choose between “fixed effects” models (FE) with specialized (e.g., cluster-robust) standard errors, or “multilevel models” (MLMs) employing “random effects.” We review the claims given in published works regarding this choice, then clarify how these approaches work and compare by showing that: (i) random effects employed in MLMs are simply “regularized” fixed effects; (ii) unmodified MLMs are consequently susceptible to bias—but there is a longstanding remedy; and (iii) the “default” MLM standard errors rely on narrow assumptions that can lead to undercoverage in many settings. Our review of over 100 papers using MLM in political science, education, and sociology show that these “known” concerns have been widely ignored in practice. We describe how to debias MLM’s coefficient estimates, and provide an option to more flexibly estimate their standard errors. Most illuminating, once MLMs are adjusted in these two ways the point estimate and standard error for the target coefficient are exactly equal to those of the analogous FE model with cluster-robust standard errors. For investigators working with observational data and who are interested only in inference on the target coefficient, either approach is equally appropriate and preferable to uncorrected MLM.
Although modern lines for dealing with missing data are well established from the 1970s, today there is a challenge when researchers encounter this problem in multilevel models. First, there is a variety of existing software to handle missing data based on multiple imputation (MI), currently pointed out by experts as the most promising strategy. Second, the two principal paradigms of MI are joint modelling (JM) and fully conditional specification (FCS), one more complication because they are not equally useful depending on the combination of multilevel model and the estimated parameters affected by missing data. Technical literature do not contribute to ease the number of decisions that researcher has to do. Given these inconveniences, the present paper has three objectives. (1) To present a thorough revision of the most recently developed software and functions about multiple imputation in multilevel models. (2) We derive a set of suggestions, recommendations, and guides for helping researchers to handle missing data. We list a number of key questions to consider when analyzing multilevel models. (3) Finally, based on the previous relevant questions, we present two detailed examples using the recommended R packages to be easy for the researcher applying multiple imputation in multilevel models.
Interventions to reduce adolescents’ non-core food intake (i.e. foods high in fat and sugar) could target specific people or specific environments, but the relative importance of environmental contexts v. individual characteristics is unknown.
Design
Cross-sectional.
Setting
Data from 4d food diaries in the UK National Diet and Nutrition Survey (NDNS) 2008–2012 were analysed. NDNS food items were classified as ‘non-core’ based on fat and sugar cut-off points per 100g of food. Linear multilevel models investigated associations between ‘where’ (home, school, etc.) and ‘with whom’ (parents, friends, etc.) eating contexts and non-core food energy (kcal) per eating occasion (EO), adjusting for variables at the EO (e.g. time of day) and adolescent level (e.g. gender).
Participants
Adolescents (n 884) aged 11–18 years.
Results
Only 11 % of variation in non-core energy intake was attributed to differences between adolescents. In adjusted models, non-core food intake was 151 % higher (ratio; 95 % CI) in EO at ‘Eateries’ (2·51; 2·14, 2·95) and 88 % higher at ‘School’ (1·88; 1·65, 2·13) compared with ‘Home’. EO with ‘Friends’ (1·16; CI 1·03, 1·31) and ‘Family & friends’ (1·21; 1·07, 1·37) contained 16–21 % more non-core food compared with eating ‘Alone’. At the individual level, total energy intake and BMI, but not social class, gender or age, were weakly associated with more non-core energy intake.
Conclusions
Regardless of individual characteristics, adolescents’ non-core food consumption was higher outside the home, especially at eateries. Targeting specific eating contexts, not individuals, may contribute to more effective public health interventions.
The aim of this study was to establish the association of maternal, family, and contextual correlates of anthropometric typologies at the household level in Colombia using 2005 Demographic Health Survey (DHS/ENDS) data.
Methods.
Household-level information from mothers 18–49 years old and their children <5 years old was included. Stunting and overweight were assessed for each child. Mothers were classified according to their body mass index. Four anthropometric typologies at the household level were constructed: normal, underweight, overweight, and dual burden. Four three-level [households (n = 8598) nested within municipalities (n = 226), nested within states (n = 32)] hierarchical polytomous logistic models were developed. Household log-odds of belonging to one of the four anthropometric categories, holding ‘normal’ as the reference group, were obtained.
Results.
This study found that anthropometric typologies were associated with maternal and family characteristics of maternal age, parity, maternal education, and wealth index. Higher municipal living conditions index was associated with a lower likelihood of underweight typology and a higher likelihood of overweight typology. Higher population density was associated with a lower likelihood of overweight typology.
Conclusion.
Distal and proximal determinants of the various anthropometric typologies at the household level should be taken into account when framing policies and designing interventions to reduce malnutrition in Colombia.
Does direct democracy strengthen popular control of public policy in the United States? A major challenge in evaluating policy representation is the measurement of state-level public opinion and public policy. Although recent studies of policy responsiveness and congruence have provided improved measures of public opinion using multilevel regression and poststratification (MRP) techniques, these analyses are limited by their static nature and cross-sectional design. Issue attitudes, unlike more general political orientations, often vary considerably over time. Unless the dynamics of issue-specific public opinion are appropriately incorporated into the analyses, tests of policy responsiveness and congruence may be misleading. Thus, we assess the degree of policy representation in direct democracy states regarding same-sex relationship recognition policies using dynamic models of policy adoption and congruence that employ dynamic MRP estimates of attitudes toward same-sex marriage. We find that direct democracy institutions increase both policy responsiveness and congruence with issue-specific public opinion.
This article presents results from survey experiments investigating conditions under which Britons are willing to pay taxes on polluting activities. People are no more willing if revenues are hypothecated for spending on environmental protection, while making such taxes more relevant to people – by naming petrol and electricity as products to which they will apply – has a modestly negative effect. Public willingness increases sharply if people are told that new environmental taxes would be offset by cuts to other taxes, but political distrust appears to undermine much of this effect. Previous studies have argued that political trust shapes public opinion with respect to environmental and many other policies. But this article provides the first experimental evidence suggesting that the relationship is causal, at least for one specific facet: cynicism about public officials’ honesty and integrity. The results suggest a need to make confidence in the trustworthiness of public officials and their promises more central to conceptualizations of political trust.
This study examined the socioeconomic pathways linking partnership status to physical functioning, assessed using objective measures of late life physical functioning, including peak flow and grip strength. Using Wave 4 of the Survey of Health, Ageing and Retirement in Europe (SHARE), we ran multilevel models to examine the relationship between partnership status and physical function in late life, adjusting for social-network characteristics, socioeconomic factors, and health behaviours. We found a robust relationship between partnership status and physical function. Incorporating social-network characteristics, socioeconomic factors, and health behaviours showed independent robust relationships with physical function. Co-variates attenuated the impact of cohabitation, separation, and widowhood on physical function; robust effects were found for singlehood and divorce. Sex-segregated analyses suggest that associations between cohabitation, singlehood, divorce, and widowhood were larger for men than for women. Results suggest that social ties are important to improved physical function.
The link between inequality and negative social outcomes has been the subject of much debate recently, brought into focus by the publication of The Spirit Level. This article uses multilevel modelling to explore the relationship between inequality and five crime types at sub-national level across England. Controlling for other factors, inequality is positively associated with higher levels of all five crime types and findings are robust to alternative inequality specifications. Findings support the sociological – but not economic – theories and highlight the importance of policies to tackle broader social and economic inequalities.
Objective. Unravelling the determinants of parasite life-history traits in natural settings is complex. Here, we deciphered the relationships between biotic, abiotic factors and the variation in 4 life-history traits (body size, egg presence, egg number and egg size) in the fish ectoparasite Tracheliastes polycolpus. We then determined the factors affecting the strength of the trade-off between egg number and egg size. Methods. To do so, we used 4-level (parasite, microhabitat, host and environment) hierarchical models coupled to a field database. Results. Variation in life-history traits was mostly due to individual characteristics measured at the parasite level. At the microhabitat level (fins of fish hosts), parasite number was positively related to body size, egg presence and egg number. Higher parasite number on fins was positively associated with individual parasite fitness. At the host level, host body size was positively related to the individual fitness of the parasite; parasites were bigger and more fecund on bigger hosts. In contrast, factors measured at the environmental level had a weak influence on life-history traits. Finally, a site-dependent trade-off between egg number and egg size existed in this population. Conclusion. Our study illustrates the importance of considering parasite life-history traits in a hierarchical framework to decipher complex links between biotic, abiotic factors and parasite life-history traits.
This report attempts to give nontechnical readers some insight into how a multilevel modelling framework can be used in longitudinal studies to assess contextual influences on child development when study samples arise from naturally formed groupings. We hope to achieve this objective by: (1) discussing the types of variables and research designs used for collecting developmental data; (2) presenting the methods and data requirements associated with two statistical approaches to developmental data—growth curve modelling and discrete-time survival analysis; (3) describing the multilevel extensions of these approaches, which can be used when the study of development includes intact clusters or naturally formed groupings; (4) demonstrating the flexibility of these two approaches for addressing a variety of research questions; and (5) placing the multilevel framework developed in this report in the context of some important issues, alternative approaches, and recent developments. We hope that readers new to these methods are able to visualize the possibility of using them to advance their work.
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