The Multilevel Approach to Repeated Measures for Complete and Incomplete Data

Repeated measurements often are analyzed by multivariate analysis of variance (MANOVA). An alternative approach is provided by multilevel analysis, also called the hierarchical linear model (HLM), which makes use of random coefficient models. This paper is a tutorial which indicates that the HLM can be specified in many different ways, corresponding to different sets of assumptions about the covariance matrix of the repeated measurements. The possible assumptions range from the very restrictive compound symmetry model to the unrestricted multivariate model. Thus, the HLM can be used to steer a useful middle road between the two traditional methods for analyzing repeated measurements. Another important advantage of the multilevel approach to analyzing repeated measures is the fact that it can be easily used also if the data are incomplete. Thus it provides a way to achieve a fully multivariate analysis of repeated measures with incomplete data. This revised version was published online in June 2006 with corrections to the Cover Date.     

Analysis of longitudinal data using the hierarchical linear model

The hierarchical linear model in a linear model with nested random coefficients, fruitfully used for multilevel research. A tutorial is presented on the use of this model for the analysis of longitudinal data, i.e., repeated data on the same subjects. An important advantage of this approach is that differences across subjects in the numbers and spacings of measurement occasions do not present a problem, and that changing covariates can easily be handled. The tutorial approaches the longitudinal data as measurements on populations of (subject-specific) functions.     

Nested Unobserved Heterogeneity not Orthogonal to Regressors

Many social phenomena are of a nested nature and recordedin hierarchical data, e.g., repeated observations of siblings (individual level) within families(context level). In these phenomena, unobserved heterogeneity can occur at both levels and may becorrelated with the regressors. This article addresses nested unobserved heterogeneity notorthogonal to the regressors, which is rarely discussed in the methodology literature. The articleextends the econometric one-factor fixed-effects approach to handle nested fixed effects.F tests for model comparisons are used to test whether the total heterogeneity exists and whetherthe total heterogeneity consists solely of contextual heterogeneity. It then introduces methods todecompose the two levels of heterogeneity and provides formal tests for each level andtheir relative importance, which are developed from classical ANOVA. To provide a stronger testfor time-varying context-specific heterogeneity, the article develops an estimator using thedifference-in-differences method. An empirical example of a study on child behavior problems isused to illustrate the methods introduced in the article.     

Longitudinal Meta-analysis

The goal of meta-analysis is to integrate the research results of a number of studies on a specific topic. Characteristic for meta-analysis is that in general only the summary statistics of the studies are used and not the original data. When the published research results to be integrated are longitudinal, multilevel analysis can be used for the meta-analysis. We will demonstrate this with an example of longitudinal data on the mental development of infants. We distinguish four levels in the data. The highest level (4) is the publication, in which the results of one or more studies are published. The third level consists of the separate studies. At this level we have knowledge about the degree of prematurity of the group of infants in the specific study. The second level are the repeated measures. We have data about the test age, the mental development, the corresponding standard deviations, and the sample sizes. The lowest level is needed for the specification of the meta-analysis model. Both the way in which the multilevel model has to be specified (the Mln-program is used) as the results will be presented and interpreted.     

Regional Inequalities in Consumption Patterns: A Multilevel Approach to the Case of Italy

The main aim of this paper is to evaluate the disparities in the Italian regions on the demand side. In more detail, an attempt will be made to find if the consumption behaviour of Italian households is different in the regions. With this in mind, Istat's 2000 Italian Family Budget data set was analysed. The data in question, which were collected through a two‐stage sample over Italy's 20 regions, contains information regarding the expenses of approximately 23,000 households. In this analysis, both households and regions are considered as units: households are nested in the regions so that the basic data structure is hierarchical. In order to take this hierarchical structure into account, a multilevel model was used, making it possible for parameters to vary randomly from region to region. The model in question also made it possible to consider heterogeneity across different groups (regions), such as stochastic variation. First, regional inequalities were tested using a simple model in which households constituted the first level of analysis and were grouped according to their region (the second level). As a second step, and in order to investigate the interaction between geographical context and income distribution, another model was used. This was cross‐classified by income and regions. The most relevant results showed that there is wide fragmentation of consumption behaviour and, at the same time, various differentiated types of behaviour in the regions under analysis. These territorial differentials become clear from income class and items of consumption.     

Sample Size Considerations for Multilevel Surveys

Social and economic data commonly have a nested structure (for example, households nested within neighborhoods). Recently techniques and computer programs have become available for dealing with such data, permitting the formulation of explicit multilevel models with hypotheses about effects occurring at each level and across levels. If data users are planning to analyze survey data using multilevel models rather than concentrating on means, totals, and proportions, this needs to be accounted for in the survey design. The implications for determining sample sizes (for example, the number of neighborhoods in the sample and the number of households sampled within each neighborhood) are explored.     

物流管理专业中职-高职-应用本科衔接教育多层次培养物流人才模式探讨

文中首先对我国物流人才需求进行分析,并对物流人才进行了层次分类,然后结合职业教育中物流人才结构和职业教育的层次结构对应找出了我国物流人才培养中存在的瓶颈问题,提出采用中职-高职-应用本科＂直通车＂形式,将物流管理专业中作业、管理和规划三大系列人才与职业教育的层次结构相对应进行衔接教育,多层次培养人才模式。     

Using Single-Equation Models of Function-of-Functions Type in Sociological Research

This paper argues that single-equation explanatorymodels of many types of social phenomena should not be built in accordance with establishedsociological ways of thinking. In sociological research, the focus is often on the causal mechanisms behindphenomena, and it is often interesting to use models that show the hierarchical causal structure,that is, how influences are nested in the causal process. I propose such a model with a form that reflectsa two-step structure. According to this model, the dependent factor is a product of independentfactors that are linear functions of variables. The model, which should be used with the factor product inunexpanded form, can be assumed to have wide application. However, the models used insociological research and discussed in textbooks are generally very different. They do not have afunction-of-functions form, but take a form in which variables are directly entered. Furthermore,even if they take interaction into consideration, they are linear in an extended sense because they construeit as one or more terms that are products of single variables. In comparison with the proposed typeof model, these models are technically simpler. However, this paper argues that the proposed typeof model is superior in many contexts because it better reflects the causal process.     

A Tutorial on Multilevel Survival Analysis: Methods,Models and Applications

Data that have a multilevel structure occur frequently across a range of disciplines, including epidemiology, health services research, public health, education and sociology. We describe three families of regression models for the analysis of multilevel survival data. First, Cox proportional hazards models with mixed effects incorporate cluster‐specific random effects that modify the baseline hazard function. Second, piecewise exponential survival models partition the duration of follow‐up into mutually exclusive intervals and fit a model that assumes that the hazard function is constant within each interval. This is equivalent to a Poisson regression model that incorporates the duration of exposure within each interval. By incorporating cluster‐specific random effects, generalised linear mixed models can be used to analyse these data. Third, after partitioning the duration of follow‐up into mutually exclusive intervals, one can use discrete time survival models that use a complementary log–log generalised linear model to model the occurrence of the outcome of interest within each interval. Random effects can be incorporated to account for within‐cluster homogeneity in outcomes. We illustrate the application of these methods using data consisting of patients hospitalised with a heart attack. We illustrate the application of these methods using three statistical programming languages (R, SAS and Stata).     

An Analysis of WellBeing in Secondary School with Multilevel Growth Curve models and Multilevel Multivariate Models

This paper reports on an exploration of student wellbeing in secondary school. A wellbeing questionnaire was administered four times to the same students. Multilevel models were applied in which measurements are grouped within students within schools. Differences between students are large, but there are only minor differences between schools regarding the wellbeing. Two methods of analysis of longitudinal data are compared: a multilevel multivariate approach and a multilevel growth curve analysis. It is shown that the estimation of individual growth curves is an elegant and parsimonious way of modelling. The multivariate approach on the other hand is a more modest model. The assumptions, advantages and disadvantages of both perspectives are listed.     

Evaluating contemporary social exclusion in Europe: a hierarchical latent class approach

In recent years, the concept of social exclusion has received a renewed attention in scientific research, as well as in politics. In this contribution we propose a hierarchical Latent Class (LC) model for the analysis of differences and similarities about experiences and perceptions of social exclusion among European regions. Social exclusion is a situation that affects individuals, and derives from a multidimensional deprivation in several domains of life. In particular, we identify and define an economic, a social and an institutional dimension. The LCs, which structure the individuals with respect to a set of observed indicators, represent different typologies of social exclusion at individual level according to the three identified dimensions. The regional differences in the latent variable distribution are modeled following a nonparametric approach for the random effects. This multilevel extension leads to the identification of a typology of regions, allowing different social exclusion structures to stand out for different European areas. The hierarchical latent class approach proves to be profitable in investigating the relevance of different risk factors of social exclusion and their relationships, and in verifying whether, and to what extent, the same risks and disadvantages determine the same perception of marginalization and exclusion in different political, economic, social and cultural contexts. The analysis is carried out using the 56.1-2001 Eurobarometer Survey, which focused on poverty and social exclusion situations, from both a subjective and an objective point of view.     

Conducting multilevel studies in purchasing and supply management research

Purchasing and supply management (PSM) research commonly covers multiple levels of theory and analysis. The theorizing and simultaneous testing of hypotheses across multiple levels is referred to as multilevel analysis (MLA) and is commonly performed using hierarchical linear modeling (HLM). Researchers in the PSM domain have paid little attention thus far to the topic of multilevel studies. Although MLA holds the potential to yield novel insights into PSM issues, it also generates new challenges for authors and reviewers alike. We contribute to this methodological dialogue by examining reasons for conducting multilevel PSM research and offering practical guidance for increasing its methodological rigor.     

Measuring efficiency in education: The influence of imprecision and variability in data on DEA estimates

Many studies devoted to efficiency performance evaluation in the education sector are based on measures of central tendency at school level as, for example, the average values of students belonging to the same school. Although this is a common and accepted way of summarizing data from the original observations (students), it is not less true that this approach neglects the existing dispersion of data, which may become a serious problem if variability across schools is high. Additionally, imprecision may arise when experts on each evaluated subject select the battery of questions, with different levels of difficulty, which will be the base for the final questionnaires completed by students. This paper uses data from US students and schools participating in PISA (Programme for International Student Assessment) 2015 to illustrate that schools' efficiency measures based on aggregate data and imprecision may reflect an inaccurate picture of their performance if they are compared to measures estimated accounting for broader information provided by all students of the same school. In order to operationalize our approach, we resort to Fuzzy Data Envelopment Analysis. This methodology allows us to deal with the notion of fuzziness in some variables such as the socio-economic status of students or test scores. Our results indicate that the estimated measures of performance obtained with the fuzzy DEA approach are highly correlated with those calculated with traditional DEA models. However, we find some relevant divergences in the identification of efficient units when we account for data dispersion and vagueness.     

改进AHP主观性的DEA/AHP新模型

本文提出了一种DEA(数据包络分析)与AHP(层次分析法)结合的新模型。该模型有别于目前国内很多将AHP作为约束锥的DEA混合模型。通过对模型的阐述和算例的验证,得到了DEA/AHP方法相对于DEA的优势。并指出了以AHP作为约束锥的DEA混合模型的不足,以及本文提出的DEA/AHP方法对于AHP主观性的改进,并实现了DEA方法中决策单元的完全排序。     

Using crowdsourced data to analyze transport crime

ABSTRACT

Anecdotal evidence suggests that harsh social conditions in the road haulage industry are having an impact on transport crime. This paper analyses transport crime, and demonstrates how to use a combination of official statistics and crowdsourced data in the process. A hierarchical regression analysis was applied to investigate the relations among different factors in order to predict transport crime threats. A secondary data set on transport crime from the Swedish Police was combined with primary crowdsourced data from volunteer observations of trucks in Sweden from both high-wage and low-wage countries. The findings imply that transportation is more vulnerable to antagonistic threats in geographical areas where the low-wage hauliers operate more frequently. For policymakers and practitioners, these findings provide useful guidance for the planning of security measures. To the authors’ knowledge, this paper is the first exploratory study of its kind that uses a combination of official statistics and crowdsourced data.     

The Problem of Time Dependent Explanatory Variables at the Context-Level in Discrete Time Multilevel Event History Analysis: A Comparison of Models Considering Mobility Between Local Labour Markets as an Example

The paper deals with the question of how to include time dependent explanatory variables at the context-level in multilevel event history models. In general, context-level explanatory variables in multilevel models are assumed to be time constant. Only time constant context-level explanatory variables perform the task of reducing context-level error variance. Thus, it will be suggested that the analysis should be extended to a three-level model. In this model, time periods of persons constitute level 1 units, time periods of contexts constitute level 2 units and the contexts themselves constitute level 3 units – in which in turn level 2 units are clustered. Considering mobility between local labour markets as an example, four different ways of modelling time varying context-level variables are compared. The result is that the proposed three-level model leads to the most conservative results.     

基于土木结构递阶分散控制的稳定性研究

采用递阶分散控制的方法对土木结构进行振动控制,同时结构满足稳定性要求。控制作用分为局部和全局控制两级,局部控制器对子结构的振动控制,用李雅普诺夫方法来获得子结构稳定,而全局控制器来减小子结构之间相互作用而保证结构全局稳定性。对悬臂梁结构进行了控制稳定性仿真分析,将振动仿真结果与集中控制的结果进行对比分析。结果表明,基于递阶分散的控制方法保证大型土木结构的稳定性。     

Do LGBT‐supportive corporate policies enhance firm performance?

Prior research provides evidence that lesbian, gay, bisexual, and transgender (LGBT)‐supportive corporate policies are related to important human resource functions, such as enhanced recruitment and retention. In addition, prior research indicates that investors view the adoption of such policies positively. We examine the firm‐performance mechanisms underlying favorable stock‐market reactions based on an integration of perspectives from corporate social responsibility and the business case for diversity. Specifically, we estimate a hierarchical linear model (HLM) to account for the nested nature of our data (firms nested within states) and find that (1) the presence of LGBT‐supportive policies is associated with higher firm value, productivity, and profitability; (2) the firm‐value and profitability benefits associated with LGBT‐supportive policies are larger for companies engaged in research and development (R&D) activities; and (3) the firm‐value and profitability benefits of LGBT‐supportive policies persist in the presence of state antidiscrimination laws. In supplemental analyses, we find that firms implementing (discontinuing) LGBT‐supportive policies experience increases (decreases) in firm value, productivity, and profitability. We are among the first to link LGBT‐supportive policies specifically to financial performance outcomes as well as to develop and test a multilevel model of these relationships. Our results have important implications for theory and research on LGBT issues in organizations, human resource managers, and policymakers.     

Sustainable development of small‐ and medium‐sized enterprises in the European Union: A taxonomy of circular economy practices

This paper focuses on the implementation of circular economy (CE) practices in small‐ and medium‐sized firms in all 28 European Union (EU) countries. The analyses take into account the hierarchical nature of the collected data as firms are nested within EU countries, that is, the heterogeneity between different types of firms and countries according to practices and attitudes towards CE. The multilevel latent class model identifies groups of firms and groups of EU countries that are homogeneous in terms of CE, that is, how the homogeneous groups of small‐ and medium‐sized enterprises (SMEs) are distributed across the groups of EU countries. These results, together with the fact that firms with similar CE attitudes and practices have different demographic and business profiles across groups of countries, shed further light on the topic of green behavior in the EU with implications for businesses' environmental policies. Moreover, indications emerge that European policies favoring the implementation of CE practices should be targeted at least for subgroups of European countries, considering the different composition by typology of SMEs operating in their territories and that, at the same time, policies should be defined within each group of countries to account for the specific features of each of the four classes of SMEs.     

Extending Facet Design to Such Complex Psychological Structures as Piaget's Theory

This study is aimed at exploring whether Guttman's facet design could be extended into a hierarchical facet model, designed to map hypothesized relationships within such a complicated structure of a hierarchical theory as that of Piaget. The basis of the hierarchical facet model was a thorough conceptual analysis by means of facet design of 12 Piagetian experiments resulting in a differentiated classification model in the form of a mapping sentence. This mapping sentence elucidated the complex structures of the developmental sequence of Piaget's theory into a unified structure of content facets and their elements by hypothesizing two content facets, Conceptual Instrumentalities and Cognitive Strategies, underlying the content domain, i.e., the internal structure of logical operational thinking as it is represented by the logical operational structures at three developmental levels, and by deducing a set of elements of these facets. In order to outline the interrelationships between the operational structures a hierarchical facet model was elaborated representing both the step-by-step acquisition of the structures within and between the three developmental levels and which structures are necessary prerequisites for arriving at the next step in the developmental sequence. The hierarchical facet model was presented in graphical form giving an overview of the fundamental structures and their positions in the developmental sequence.