The complexity of school and neighbourhood effects and movements of pupils on school differences in models of educational achievement

Summary.  Traditional studies of school differences in educational achievement use multilevel modelling techniques to take into account the nesting of pupils within schools. However, educational data are known to have more complex non-hierarchical structures. The potential importance of such structures is apparent when considering the effect of pupil mobility during secondary schooling on educational achievement. Movements of pupils between schools suggest that we should model pupils as belonging to the series of schools that are attended and not just their final school. Since these school moves are strongly linked to residential moves, it is important to explore additionally whether achievement is also affected by the history of neighbourhoods that are lived in. Using the national pupil database, this paper combines multiple membership and cross-classified multilevel models to explore simultaneously the relationships between secondary school, primary school, neighbourhood and educational achievement. The results show a negative relationship between pupil mobility and achievement, the strength of which depends greatly on the nature and timing of these moves. Accounting for pupil mobility also reveals that schools and neighbourhoods are more important than shown by previous analysis. A strong primary school effect appears to last long after a child has left that phase of schooling. The additional effect of neighbourhoods, in contrast, is small. Crucially, the rank order of school effects across all types of pupil is sensitive to whether we account for the complexity of the multilevel data structure.     

A multilevel exploration of the role of interviewers in survey non-response

This paper illustrates the use of multilevel statistical modelling of cross-classified data to explore interviewers' influence on survey non-response. The results suggest that the variability in whole household refusal and non-contact rates is due more to the influence of interviewers than to the influence of areas. The results from separate logistic regression models are compared with the results from multinomial models using a polytomous dependent variable (refusals, non-contacts and responses). Using the cross-classified multilevel approach allows us to estimate correlations between refusals and non-contacts, suggesting that interviewers who are good at reducing whole household refusals are also good at reducing whole household non-contacts.     

The effect of area of residence over the life course on subsequent mortality

Summary.  Life course epidemiology concentrates on the contribution that social or physical exposures have across the life course on adult health. It is known that the area of residence can affect health, but little is known about the effect of the area of residence across the life course. We examine the contribution that area of residence in 1960, 1970, 1980 and 1990 made on subsequent mortality for 49736 male inhabitants of Oslo in 1990. We compare the performance of multiple-membership and cross-classified multilevel models on these data with a correlated cross-classified model that was developed for this.     

Fitting logistic multilevel models with crossed random effects via Bayesian Integrated Nested Laplace Approximations: a simulation study

Fitting cross-classified multilevel models with binary response is challenging. In this setting a promising method is Bayesian inference through Integrated Nested Laplace Approximations (INLA), which performs well in several latent variable models. We devise a systematic simulation study to assess the performance of INLA with cross-classified binary data under different scenarios defined by the magnitude of the variances of the random effects, the number of observations, the number of clusters, and the degree of cross-classification. In the simulations INLA is systematically compared with the popular method of Maximum Likelihood via Laplace Approximation. By an application to the classical salamander mating data, we compare INLA with the best performing methods. Given the computational speed and the generally good performance, INLA turns out to be a valuable method for fitting logistic cross-classified models.     

Generalized linear mixed models for ordered responses in complex multilevel structures: effects beneath the school or college in education

Summary.  The complexities of educational processes and structure and the need for disentangling effects beneath the level of the school or college are discussed. Ordinal response multilevel crossed random-effects models for educational grades are introduced. Weighted random effects for teacher contributions are then added. Estimation methodology is reviewed. Specially written macros for quasi-likelihood with second-order terms are described. The application discusses General Certificate of Education at advanced level grades cross-classified by student and teaching group within a number of institutions. The methods handle teacher effects where several teachers contribute to provision and where each teacher deals with several groups. Some methodological lessons are drawn for sparse data and the use of extra-multinomial variation. Developments of the analysis yield conclusions about the sources of variation in educational progress, and particularly the effect of teachers.     

Analysis of repeated categorical responses from fully and partially cross-classified data

Many follow-up studies involve categorical data measured on the same individual at different times. Frequently, some of the individuals are missing one or more of the measurements. This results in a contingency table with both fully and partially cross-classified data. Two models can be used to analyze data of this type: (i) The multiple-sample model, in which all the study subjects with the same configuration of missing observations are considered a separate sample. (ii) The single-sample model, which assumes that the missing observations are the result of a mechanism causing subjects to lose the informtion from one or some of the measurements. In this work we compare the two approaches and show that under certain conditions, the two models yield the same maximum likelihood estimates of the cell probabilities in the underlying contingency table.     

A comparison of univariate and multivariate multilevel models for repeated measures of use of antenatal care in Uttar Pradesh

Summary.  We compare two different multilevel modelling approaches to the analysis of repeated measures data to assess the effect of mother level characteristics on women's use of prenatal care services in Uttar Pradesh, India. We apply univariate multilevel models to our data and find that the model assumptions are severely violated and the parameter estimates are not stable, particularly for the mother level random effect. To overcome this we apply a multivariate multilevel model. The correlation structure shows that, once the decision has been made regarding use of antenatal care by the mother for her first observed birth in the data, she does not tend to change this decision for higher order births.     

中国教育收益率的长期变动趋势分析

本研究采用中国健康与营养调查1989-2009年的追踪数据,使用多层线性交互分类模型估计了全国教育收益率的总体变化趋势,并通过引入宏观经济发展指标来考察教育收益率变化的影响因素.结果表明,21世纪以来全国教育收益率并没有延续以往快速稳定增长的势头,中国市场转型过程中的制度变革和经济结构变化对教育收益率的变动都有显著影响.由于我国城乡分割的二元经济结构,本研究还比较了城镇和农村地区教育收益率变动趋势的差异,农村地区教育收益率先高后低反映了我国先农村后城市的改革开放进程.     

Synthesizing categorical datasets to enhance inference

A common data analysis setting consists of a collection of datasets of varying sizes that are all relevant to a particular scientific question, but which include different subsets of the relevant variables, presumably with some overlap. Here, we demonstrate that synthesizing cross-classified categorical datasets drawn from an incompletely cross-classified common population, where many of the sets are incomplete (i.e., one or more of the classification variables is unobserved), but at least one is completely observed is expected to reduce uncertainty about the cell probabilities in the associated multi-way contingency table as well as for derived quantities such as relative risks and odds ratios. The use of the word “expected” here is the key point. When synthesizing complete datasets from a common population, we are assured to reduce uncertainty. However, when we work with a log-linear model to explain the complete table, because this model cannot be fitted to any of the incomplete datasets, improvement is not assured. We provide technical clarification of this point as well as a series of simulation examples, motivated by an adverse birth outcomes investigation, to illustrate what can be expected under such synthesis.     

Determinants of Contraceptive Use in Egypt: A Multilevel Approach

The increasing use of family planning methods seems to be the intermediate determinant which mostly influences the fertility decline in developing countries, and in particular in those countries which are in an advanced phase of demographic transition such as Egypt. Moreover large countries, like Egypt, are characterized by very different geographical realities and even by strong regional heterogeneities. The aim of this study is the analysis of the determinants of contraceptive use in Egypt, with particular reference to the differentials due to the socio-economic context and to the area of residence. To estimate each individual and regional factors’ effect on contraceptive use, a logistic two-level random intercept model is fitted to EDHS 2000 data; the use of a multilevel analysis is suggested by the two-level data structure: the first level units are the women, the second level units are their regions of residence.     

Testing for homogeneity in cross-lagged panel studies

Panel studies are statistical studies in which two or more variables are observed for two or more subjects at two or more points In time. Cross- lagged panel studies are those studies in which the variables are continuous and divide naturally into two effects or impacts of each set of variables on the other. If a regression approach is taken5 a regression structure Is formulated for the cross-lagged models This structure may assume that the regression parameters are homogeneous across waves and across subpopulations. Under such assumptions the methods of multivariate regression analysis can be adapted to make inferences about the parameters. These inferences are limited to the degree that homogeneity of the parameters Is 'supported b}T the data. We consider the problem of testing the hypotheses of homogeneity and consider the problem of making statistical inferences about the cross-effects should there be evidence against one of the homogeneity assumptions. We demonstrate the methods developed by applying then to two panel data sets.     

Estimating individual-level interaction effects in multilevel models: a Monte Carlo simulation study with application*

Moderated multiple regression provides a useful framework for understanding moderator variables. These variables can also be examined within multilevel datasets, although the literature is not clear on the best way to assess data for significant moderating effects, particularly within a multilevel modeling framework. This study explores potential ways to test moderation at the individual level (level one) within a 2-level multilevel modeling framework, with varying effect sizes, cluster sizes, and numbers of clusters. The study examines five potential methods for testing interaction effects: the Wald test, F-test, likelihood ratio test, Bayesian information criterion (BIC), and Akaike information criterion (AIC). For each method, the simulation study examines Type I error rates and power. Following the simulation study, an applied study uses real data to assess interaction effects using the same five methods. Results indicate that the Wald test, F-test, and likelihood ratio test all perform similarly in terms of Type I error rates and power. Type I error rates for the AIC are more liberal, and for the BIC typically more conservative. A four-step procedure for applied researchers interested in examining interaction effects in multi-level models is provided.     

An L1-type estimator of multivariate location and shape

The main goal of the paper is to specify a suitable multivariate multilevel model for polytomous responses with a non-ignorable missing data mechanism in order to determine the factors which influence the way of acquisition of the skills of the graduates and to evaluate the degree programmes on the basis of the adequacy of the skills they give to their graduates. The application is based on data gathered by a telephone survey conducted, about two years after the degree, on the graduates of year 2000 of the University of Florence. A multilevel multinomial logit model for the response of interest is fitted simultaneously with a multilevel logit model for the selection mechanism by means of maximum likelihood with adaptive Gaussian quadrature. In the application the multilevel structure has a crucial role, while selection bias results negligible. The analysis of the empirical Bayes residuals allows to detect some extreme degree programmes to be further inspected.     

Small area estimates for cross-classifications

Summary.  We develop a class of log-linear structural models that is suited to estimation of small area cross-classified counts based on survey data. This allows us to account for various associ- ation structures within the data and includes as a special case the restricted log-linear model underlying structure preserving estimation. The effect of survey design can be incorporated into estimation through the specification of an unbiased direct estimator and its associated covariance structure. We illustrate our approach by applying it to estimation of small area labour force characteristics in Norway.     

Age-period-cohort analysis: an illustration of the problems in assessing interaction in one observation per cell data

This paper discusses the specific problems of age-period-cohort (A-P-C) analysis within the general framework of interaction assessment for two-way cross-classified data with one observation per cell. The A-P-C multiple classification model containing the effects of age groups (rows), periods of observation (columns), and birth cohorts (diagonals of the two-way table) is characterized as one of a special class of models involving interaction terms assumed to have very specific forms. The so-called A-P-C identification problem, which results from the use of a particular interaction structure for detecting cohort effects, is shown to manifest itself in the form of an exact linear dependency among the columns of the design matrix. The precise relationship holding among these columns is derived, as is an explicit formula for the bias in the parameter estimates resulting from an incorrect specification of an assumed restriction on the parameters required to solve the normal equations. Current methods for modeling A-P-C data are critically reviewed, an illustrative numerical example is presented, and one potentially promising analysis strategy is discussed. However, gien the large number of possible sources for error in A-P-C analyses, it is strongly recommended that the results of such analyses be interpreted with a great deal of caution.     

Evaluation of missing data mechanisms in two and three dimensional incomplete tables

The analysis of incomplete contingency tables is a practical and an interesting problem. In this paper, we provide characterizations for the various missing mechanisms of a variable in terms of response and non-response odds for two and three dimensional incomplete tables. Log-linear parametrization and some distinctive properties of the missing data models for the above tables are discussed. All possible cases in which data on one, two or all variables may be missing are considered. We study the missingness of each variable in a model, which is more insightful for analyzing cross-classified data than the missingness of the outcome vector. For sensitivity analysis of the incomplete tables, we propose easily verifiable procedures to evaluate the missing at random (MAR), missing completely at random (MCAR) and not missing at random (NMAR) assumptions of the missing data models. These methods depend only on joint and marginal odds computed from fully and partially observed counts in the tables, respectively. Finally, some real-life datasets are analyzed to illustrate our results, which are confirmed based on simulation studies.     

Multilevel models for longitudinal data

Summary.  Repeated measures and repeated events data have a hierarchical structure which can be analysed by using multilevel models. A growth curve model is an example of a multilevel random-coefficients model, whereas a discrete time event history model for recurrent events can be fitted as a multilevel logistic regression model. The paper describes extensions to the basic growth curve model to handle auto-correlated residuals, multiple-indicator latent variables and correlated growth processes, and event history models for correlated event processes. The multilevel approach to the analysis of repeated measures data is contrasted with structural equation modelling. The methods are illustrated in analyses of children's growth, changes in social and political attitudes, and the interrelationship between partnership transitions and childbearing.     

Claim Dependence Induced by Common Effects in Hierarchical Credibility Models

In the usual credibility model, observations are made of a risk or group of risks selected from a population, and claims are assumed to be independent among different risks. However, there are some problems in practical applications and this assumption may be violated in some situations. Some credibility models allow for one source of claim dependence only, that is, across time for an individual insured risk or a group of homogeneous insured risks. Some other credibility models have been developed on a two-level common effects model that allows for two possible sources of dependence, namely, across time for the same individual risk and between risks. In this paper, we argue for the notion of modeling claim dependence on a three-level common effects model that allows for three possible sources of dependence, namely, across portfolios, across individuals and simultaneously across time within individuals. We also obtain the corresponding credibility premiums hierarchically using the projection method. Then we derive the general hierarchical structure or multi-level credibility premiums for the models with h-level of common effects.     

On elliptical multilevel models

Multilevel models have been widely applied to analyze data sets which present some hierarchical structure. In this paper we propose a generalization of the normal multilevel models, named elliptical multilevel models. This proposal suggests the use of distributions in the elliptical class, thus involving all symmetric continuous distributions, including the normal distribution as a particular case. Elliptical distributions may have lighter or heavier tails than the normal ones. In the case of normal error models with the presence of outlying observations, heavy-tailed error models may be applied to accommodate such observations. In particular, we discuss some aspects of the elliptical multilevel models, such as maximum likelihood estimation and residual analysis to assess features related to the fitting and the model assumptions. Finally, two motivating examples analyzed under normal multilevel models are reanalyzed under Student-t and power exponential multilevel models. Comparisons with the normal multilevel model are performed by using residual analysis.