2012 advanced statistical methods for the analysis of large data sets

Studies in Theoretical and Applied Statistics Selected Papers of the Statistical Societies For further volumes: http://www.springer.com/series/10104 Series Editors Spanish Society of Statistics and Operations Research (SEIO) Ignacio Garcia Jurado Soci´et´e Franc¸aise de Statistique (SFdS) Avner Bar-Hen Societ`a Italiana di Statistica (SIS) Maurizio Vichi Sociedade Portuguesa de Estat´ıstica (SPE) Carlos Braumann Agostino Di Ciaccio Mauro Coli Jose Miguel Angulo IbaQnez Editors Advanced Statistical Methods for the Analysis of Large Data-Sets 123 Editors Agostino Di Ciaccio University of Roma “La Sapienza” Dept of Statistics P.le Aldo Moro 00185 Roma Italy agostino.diciaccio@uniroma1.it Mauro Coli Dept of Economics University “G d’Annunzio”, Chieti-Pescara V.le Pindaro 42 Pescara Italy coli@unich.it Jose Miguel Angulo IbaQnez Departamento de Estad´ıstica e Investigaci´on Operativa, Universidad de Granada Campus de Fuentenueva s/n 18071 Granada Spain jmangulo@ugr.es This volume has been published thanks to the contribution of ISTAT - Istituto Nazionale di Statistica ISBN 978-3-642-21036-5 e-ISBN 978-3-642-21037-2 DOI 10.1007/978-3-642-21037-2 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2012932299 c Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable to prosecution under the German Copyright Law The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Editorial Dear reader, on behalf of the four Scientific Statistical Societies: SEIO, Sociedad de Estad´ıstica e Investigaci´on Operativa (Spanish Statistical Society and Operation Research); SFC, Soci´et´e Franc¸aise de Statistique (French Statistical Society); SIS, Societ`a Italiana di Statistica (Italian Statistical Society); SPE, Sociedade Portuguesa de Estat´ıstica (Portuguese Statistical Society), we inform you that this is a new book series of Springer entitled Studies in Theoretical and Applied Statistics, with two lines of books published in the series “Advanced Studies”; “Selected Papers of the Statistical Societies.” The first line of books offers constant up-to-date information on the most recent developments and methods in the fields of Theoretical Statistics, Applied Statistics, and Demography Books in this series are solicited in constant cooperation among Statistical Societies and need to show a high-level authorship formed by a team preferably from different groups to integrate different research points of view The second line of books proposes a fully peer-reviewed selection of papers on specific relevant topics organized by editors, also in occasion of conferences, to show their research directions and developments in important topics, quickly and informally, but with a high quality The explicit aim is to summarize and communicate current knowledge in an accessible way This line of books will not include proceedings of conferences and wishes to become a premier communication medium in the scientific statistical community by obtaining the impact factor, as it is the case of other book series such as, for example, “lecture notes in mathematics.” The volumes of Selected Papers of the Statistical Societies will cover a broad scope of theoretical, methodological as well as application-oriented articles, surveys, and discussions A major purpose is to show the intimate interplay between various, seemingly unrelated domains and to foster the cooperation among scientists in different fields by offering well-based and innovative solutions to urgent problems of practice On behalf of the founding statistical societies, I wish to thank Springer, Heidelberg and in particular Dr Martina Bihn for the help and constant cooperation in the organization of this new and innovative book series Maurizio Vichi v • Preface Many research studies in the social and economic fields regard the collection and analysis of large amounts of data These data sets vary in their nature and complexity, they may be one-off or repeated, and they may be hierarchical, spatial, or temporal Examples include textual data, transaction-based data, medical data, and financial time series Today most companies use IT to support all business automatic function; so thousands of billions of digital interactions and transactions are created and carried out by various networks daily Some of these data are stored in databases; most ends up in log files discarded on a regular basis, losing valuable information that is potentially important, but often hard to analyze The difficulties could be due to the data size, for example thousands of variables and millions of units, but also to the assumptions about the generation process of the data, the randomness of sampling plan, the data quality, and so on Such studies are subject to the problem of missing data when enrolled subjects not have data recorded for all variables of interest More specific problems may relate, for example, to the merging of administrative data or the analysis of a large number of textual documents Standard statistical techniques are usually not well suited to manage this type of data, and many authors have proposed extensions of classical techniques or completely new methods The huge size of these data sets and their complexity require new strategies of analysis sometimes subsumed under the terms “data mining” or “predictive analytics.” The inference uses frequentist, likelihood, or Bayesian paradigms and may utilize shrinkage and other forms of regularization The statistical models are multivariate and are mainly evaluated by their capability to predict future outcomes This volume contains a peer review selection of papers, whose preliminary version was presented at the meeting of the Italian Statistical Society (SIS), held 23–25 September 2009 in Pescara, Italy The theme of the meeting was “Statistical Methods for the analysis of large datasets,” a topic that is gaining an increasing interest from the scientific community The meeting was the occasion that brought together a large number of scientists and experts, especially from Italy and European countries, with 156 papers and a vii viii Preface large number of participants It was a highly appreciated opportunity of discussion and mutual knowledge exchange This volume is structured in 11 chapters according to the following macro topics: • • • • • • • • • • • Clustering large data sets Statistics in medicine Integrating administrative data Outliers and missing data Time series analysis Environmental statistics Probability and density estimation Application in economics WEB and text mining Advances on surveys Multivariate analysis In each chapter, we included only three to four papers, selected after a careful review process carried out after the conference, thanks to the valuable work of a good number of referees Selecting only a few representative papers from the interesting program proved to be a particularly daunting task We wish to thank the referees who carefully reviewed the papers Finally, we would like to thank Dr M Bihn and A Blanck from Springer-Verlag for the excellent cooperation in publishing this volume It is worthy to note the wide range of different topics included in the selected papers, which underlines the large impact of the theme “statistical methods for the analysis of large data sets” on the scientific community This book wishes to give new ideas, methods, and original applications to deal with the complexity and high dimensionality of data Sapienza Universit`a di Roma, Italy Universit`a G d’Annunzio, Pescara, Italy Universidad de Granada, Spain Agostino Di Ciaccio Mauro Coli Jos´e Miguel Angulo Ibanez Q Contents Part I Clustering Large Data-Sets Clustering Large Data Set: An Applied Comparative Study Laura Bocci and Isabella Mingo Clustering in Feature Space for Interesting Pattern Identification of Categorical Data Marina Marino, Francesco Palumbo and Cristina Tortora Clustering Geostatistical Functional Data Elvira Romano and Rosanna Verde Joint Clustering and Alignment of Functional Data: An Application to Vascular Geometries Laura M Sangalli, Piercesare Secchi, Simone Vantini, and Valeria Vitelli Part II 13 23 33 Statistics in Medicine Bayesian Methods for Time Course Microarray Analysis: From Genes’ Detection to Clustering Claudia Angelini, Daniela De Canditiis, and Marianna Pensky Longitudinal Analysis of Gene Expression Profiles Using Functional Mixed-Effects Models Maurice Berk, Cheryl Hemingway, Michael Levin, and Giovanni Montana A Permutation Solution to Compare Two Hepatocellular Carcinoma Markers Agata Zirilli and Angela Alibrandi 47 57 69 ix 470 E Raffinetti and P Giudici i X yj j D1 i X y.j / j D1 is intuitively true for all i , then we also have that n X i X yj i D1 j D1 n X i X y.j / I i D1 j D1 now, because of the aforementioned relationship (11) we have n X n.n C 1/MY iyi n.n C 1/MY n X i D1 which gives Pn Pn i D1 iy.i / i D1 Remark CY;X1 ;X2 ;:::;Xk the Lorenz curve iy.i / ; i D1 iyi t u D C1 if and only if concordance function overlaps with Proof Concordance function overlaps with the Lorenz curve if and only if Pi Pi u t j D1 y.j / D j D1 yj ) r.yi / D r.yi / for every i D 1; 2; : : : ; n Remark CY;X1 ;X2 ;:::;Xk the dual Lorenz curve D if and only if concordance function overlaps with Proof This remark can be proved, similarly to Remark 1, from the second system of page by first noticing that: n X n C i /y.i / D i D1 so n X i D1 iy.i / D n.n C 1/MY n X y.nC1 i /i i D1 n X y.nC1 i /i i D1 and therefore by applying this equivalence in the denominator of (8) we get an equivalent formulation of the concordance index based on LY : P niD1 iyi n.n C 1/MY P CY;X1 ;:::;Xk D n.n C 1/MY niD1 iy.nC1 i / P n iy n.n C 1/MY Pn i D1 i D : i D1 iy.nC1 i / n.n C 1/MY P Finally, since from the second system of equations of page we have ij D1 yj Ä Pi t u j D1 y.nC1 i / , 8i , then the result follows similarly to Remark proof Multivariate Ranks-Based Concordance Indexes 471 An alternative concordance measure, which provides a measure of distance between concordance function and the Y Lorenz curve, is the Plotnick indicator (see e.g Plotnick 1981) expressed by Pn Pn i D1 iy.i / i D1 iyi P IY;X1 ;X2 ;:::;Xk D Pn : (12) i D1 iy.i / n C 1/ niD1 y.i / Furthermore, one can verify that: IY;X1 ;X2 ;:::;Xk D , r.yOi / D r.yi / ) IY;X1 ;X2 ;:::;Xk D , r.yOi / D n C r.yi / ) n X n X iy.i / D i D1 n X i D1 iyi D iyi ; (13) i D1 n X n C i /y.i / : (14) i D1 2.2 Some Practical Results Suppose to have data concerning 18 business companies three characters: Sales revenues (Y ) (expressed in thousands of Euros), Selling price X1 / (expressed in Euros) and Advertising investments X2 / (expressed in thousand of Euros) These data are shown in Table Table Data describing Sales revenues, Selling price and Advertising investments expressed in Euros ID Business company Sales revenues Selling price Advertising investments 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 350 202 404 263 451 304 275 385 244 302 274 346 253 395 430 216 374 308 84 73 64 68 76 67 62 72 63 54 83 65 56 58 69 60 79 74 45 19 53 31 58 23 25 36 29 39 35 49 22 61 48 34 51 50 472 E Raffinetti and P Giudici Table Results Ordered yi 202 216 244 253 263 274 275 302 304 308 346 350 374 385 395 404 430 451 r.yi / 10 11 12 13 14 15 16 17 18 b yi 231.07 291.41 270.46 234.10 282.73 310.41 251.56 310.47 245.57 373.26 363.04 356.71 380.97 308.07 413.45 380.68 360.99 411.05 Ordered b yi 231.07 234.10 245.57 251.56 270.46 282.73 291.41 308.07 310.41 310.47 356.71 360.99 363.04 373.26 380.68 380.97 411.05 413.45 r.b yi / 10 11 12 13 14 15 16 17 18 yi ordered by r.b yi / 202 253 304 275 244 263 216 385 274 302 350 430 346 308 404 374 451 395 The model used to describe relations among the involved variables is based on linear regression The application of ordinary least square method leads to the following estimated regression coefficients ˇ0 Š 98:48, ˇ1 Š 0:63, ˇ2 Š 4:57 so the regression line is yOi D 98:48 C 0:63x1i C 4:57x2i Once getting the estimated Y values, we assign their ranks and finally order Y values according to yOi ranks All the results are summarized in Table 2: through all these information we can compute concordance index in a multivariate context using (8) recalling that yi represent the Y variable values ordered with respect to yOi ranks Concordance index assumes value 0.801 proving that there is a strong concordance relation among the response variable Y and the explanatory variables X1 ; X2 : this conclusion is well clear in Fig where concordance curve (denoted with the continuous black line), is very close to Y variable Lorenz curve (denoted by the dash dot line) A further verification of this result is provided by the Plotnick indicator (12), whose numerical value is very close to 0, meaning that the distance between concordance function and Lorenz curve is minimum Conclusion Through this analysis it has been proved that dependence study can be led in terms of concordance and discordance topics: the choice of a linear regression model is limited when one considers only quantitative variable In the described Multivariate Ranks-Based Concordance Indexes 473 context we referred to quantitative variables because we started from the source of the concordance problem involving the income amount before and after taxation intended as a quantitative character A future extension can regard the application of the concordance index analysis in cases when one of the considered variable is binary and the adopted model is a logistic regression Another important development is establishing if there exists a relation between the determination coefficient, intended as a dependence measure in a linear regression model, and the concordance index: our further research, focused on this topic, is in progress Acknowledgements The authors acknowledge financial support from the European grant EU-IP MUSING (contract number 027097) The paper is the result of the close collaboration among the authors, however, it has been written by Emanuela Raffinetti with the supervision of Prof Paolo Giudici References Leti, G.: Statistica descrittiva Il Mulino (1983) Muliere, P.: Alcune osservazioni sull’equit`a orizzontale di una tassazione Scritti in onore di Francesco Brambilla Ed by Bocconi Comunicazione 2, (Milano, 1986) Musgrave, R.A.: The Theory of Public Finance New York, Mc Graw Hill (1959) Petrone, S., Muliere, P.: Generalized Lorenz curve and monotone dependence orderings Metron Vol L, No 3–4 (1992) Plotnick, R.: A Measure of Horizontal Inequity The review of Economics and Statistics, 2, 283–288 (1981) This page intentionally left blank Methods for Reconciling the Micro and the Macro in Family Demography Research: A Systematisation Anna Matysiak and Daniele Vignoli Abstract In the second half of the twentieth century, the scientific study of population changed its paradigm from the macro to the micro, so that attention focused mainly on individuals as the agents of demographic action However, for accurate handling of all the complexities of human behaviours, the interactions between individuals and the context they belong to cannot be ignored Therefore, in order to explain (or, at least, to understand) contemporary fertility and family dynamics, the gap between the micro and the macro should be bridged In this contribution, we highlight two possible directions for bridging the gap: (1) integrating life-course analyses with the study of contextual characteristics, which is made possible by the emergence of the theory and tools of multi-level modelling; and (2) bringing the micro-level findings back to macro outcomes via meta-analytic techniques and agent-based computational models The Need to Bridge the Gap Between the Micro and the Macro Perspectives in Family Demography Research After mid-twentieth century the scientific study of population changed its paradigm from the macro to the micro so that the main focus of attention has been devoted to individuals as the agents of demographic action Event-history analysis was born from the need to develop a comprehensive theoretical framework for studying events A Matysiak ( ) Institute of Statistics and Demography, Warsaw School of Economics, ul Madali´nskiego 6/8, 02-513 Warsaw, Poland e-mail: amatys@sgh.waw.pl D Vignoli Department of Statistics “G Parenti”, University of Florence, Viale Morgagni 59, 50134, Italia e-mail: vignoli@ds.unifi.it A Di Ciaccio et al (eds.), Advanced Statistical Methods for the Analysis of Large Data-Sets, Studies in Theoretical and Applied Statistics, DOI 10.1007/978-3-642-21037-2 43, © Springer-Verlag Berlin Heidelberg 2012 475 476 A Matysiak and D Vignoli that occur within the life-course (Courgeau and Lelievre 1997) This new approach led to a much wider set of research into human behaviours than classical macrodemographic analysis It also allowed to shift the research from the mere description of phenomena to its interpretation, avoiding the risk of ecological fallacy (Salvini and Santini 1999) Apart from numerous benefits this shift from the macro to the micro brought also some disadvantages First, for many years the importance of the social and economic context in which individuals live was disregarded and its potential effect on fertility and family behaviours was ignored Second, the improvement in the access to the individual-level data and development of the techniques of event-history analysis led to an explosion in the number of micro-level studies These micro-level studies are generally fragmented, however, and often provide contradictory results Third, more progress is needed as regards the inference about the macro-level outcomes from the micro-level studies Drawing conclusions from the micro-level studies on macro-level phenomena risks atomistic fallacy as micro-level studies focus often on a specific situation, constituting only a piece in the overall puzzle of understanding contemporary fertility and family dynamics Additionally, inference can be complicated by possible interactions of micro-level processes Recently, a renewed interest in linking macro- and micro-level research has been recorded in many disciplines of social science (e.g Voss 2007) Scientists now emphasize that bridging the gap between micro- and macro-approaches in family demography research is a prerequisite for a deeper understanding of contemporary fertility and family dynamics This new trend is reflected in two international demographic research projects conducted within the EU Framework Programmes: Mic-Mac (Willekens et al 2005) and Repro (Philipov et al 2009) Sharing this view, in this contribution we outline the directions for research and the analytical methods which will facilitate successful reconciliation of the micro and the macro in family demography research In what follows we propose to bridge the macro-to-micro gap by: (1) integrating life-course analyses with contextual characteristics, feasible owing to the emergence of the theory and tools of multi-level modelling; and (2) bringing the micro-level findings back to macrooutcomes via meta-analytic techniques and agent-based computational models Before we proceed with our analytical suggestions, we briefly present the concept of methodological individualism which initially drove the shift from the macro to the micro level in family demography research Methodological Individualism The major inference of methodological individualism is that understanding individual behaviour is crucial for explaining the social phenomena observed at the macro level Various versions of this doctrine have developed across disciplines They range from the more extreme, which suggest that social outcomes are created exclusively by individual behaviours, to the less absolute, which additionally assign Methods for Reconciling the Micro and the Macro in Family Demography Research 477 an important role to social institutions and social structure (Udehn 2002) Such a moderate version of methodological individualism was proposed by Coleman (1990) and adopted in demography (De Bruijn 1999: 19–22) According to Coleman, the relation between an individual and society runs both from the macro to the micro level and from the micro to the macro level There are three mechanisms corresponding to this process are: (1) the situational mechanism in which context influences individual background; (2) the action formation mechanism within which individual background affects individual behaviour; and (3) the transformational mechanism which transforms individual actions into a social outcome (see also Hedstr¨om and Swedberg 1999; Billari 2006) Individual life choices are at the centre of this theoretical model Individuals not live in a vacuum, however, but are embedded in a social environment – i.e., in a macro-context This context is a multi-level and multidimensional “structure of institutions that embody information about opportunities and restrictions, consequences and expectations, rights and duties, incentives and sanctions, models, guidelines, and definitions of the world” (De Bruijn 1999: 21) Such information is continuously being transmitted to individuals who acquire, process, interpret, and evaluate it In this way, the context influences people’s life choices, reflected in occurrence or non-occurrence of demographic events, which are subsequently transformed into a social outcome that is observed at the macro level An improvement in the availability of longitudinal data as well as the development of event-history analysis tools allowed social researchers to achieve a deeper insight into the action-formation mechanism, or at least into the manner in which the individual background influences people’s behaviours Much less attention has so far been paid to exploring the situational and transformational mechanisms Below we elaborate on ways these macro-to-micro and micro-to-macro gaps can be closed in empirical research by using the most suitable analytical methods available Alongside the presentation of these methods, we document a series of examples from literature For consistency in the general reasoning of this paper, all illustrations refer to the field of family demography Bridging the Macro-to-Micro Gap: Multi-Level Event-History Analyses Life-course theory and event-history techniques, which aim to explore people’s life choices, have become standard practice in family and fertility research However, these approaches ignore the fact that individuals are by their very nature nested in households, census tracts, regions, countries, etc., and that these situational contexts affect people’s decisions In light of the conceptual framework proposed by Coleman (1990), this significantly limits our ability to understand human behaviours (Pinnelli 1995; De Rose 1995; Blossfeld 1996; Santini 2000; Rosina and Zaccarin 2000) 478 A Matysiak and D Vignoli Furthermore, such approaches also cause technical problems, as applying single-level models to hierarchically structured data leads to a bias in the model estimates The reason for this is that single-level models assume the independence of observations which are in fact dependent, as they are nested within one unit For instance, households residing within the same neighbourhood are likely to have similar characteristics The most influential approach that has been created to account for the hierarchical structure of the data is multi-level modelling Multi-level models see individuals as behavioural agents, embedded in social units (tracts, regions, countries, etc.) They allow the analyst to detect the effect of the context on individual behaviour as well as to identify the macro-characteristics which are mainly responsible for the contextual effect (Borra and Racioppi 1995; Micheli and Rivellini 2000; Zaccarin and Rivellini 2002) The natural implication of these methods is that they blur the artificial boundaries between micro and macro analyses (Voss 2007) Multi-level event-history analysis in particular represents a challenging and so far not much explored opportunity for bridging the gap between analysis of events unfolding over the life-course (the micro approach) and the contextual (macro) approach in family demography research However, while the methods (and corresponding software packages) are relatively well-established, data availability is a critical point In order to conduct a multi-level event-history analysis, longitudinal individual data should be linked with the time-series of contextual indicators This requires data on the migration histories of the individuals, together with all their other lifecourse careers, as well as time-series data for contextual indicators Consequently, this method has so far mainly been employed on cross-sectional data Only recently have some researchers started to investigate the influence of macrolevel factors on family-related behaviours from a longitudinal perspective Still fewer have allowed for a hierarchical structure by taking into account the unobserved community-level factors or even by introducing some contextual indicators into models in order to explicitly study their impact on family-related behaviours As an example we refer to the study by Adser`a (2005), who used a multi-level event-history model in order to explore the impact of regional unemployment on childbearing, employing data from the European Community Household Panel (ECHP 1994–2001) The study was conducted on a pooled dataset for thirteen European countries and included information on the country-level gender unemployment gap and the long-term unemployment rate, which was introduced into the model on a higher level than the individual one Adser`a’s results clearly indicate that a higher gender gap in unemployment and a higher long-term unemployment rate slow down the transition to motherhood and higher order births To summarise, the existing macro-to-micro studies generally make use of data from a national, a regional, or even a municipal level The available literature not only indicates the differences between countries or regions in the timing of fertility or in fertility intentions, but also demonstrates that a proper accounting for context may change the influence of individual-level factors (Philipov et al 2009) Consequently, future research should give better recognition to multi-level eventhistory approaches Methods for Reconciling the Micro and the Macro in Family Demography Research 479 Bridging the Micro-to-Macro Gap: Meta-Analyses and Agent-Based Computational Models Despite the problems with data availability, the contextual influence on action formation is already quite well understood By contrast, the transformational mechanism (the transfer from the micro to the macro level) is as yet largely unexplored At the same time, the rapid development of micro-level studies increases the need to summarize the existing individual-level empirical evidence and to relate them to the macro-level outcomes In this section, we elaborate on two possible ways of bridging the micro-macro gap from the bottom up, namely meta-analysis and agentbased computational models 4.1 Meta-Analytic Techniques Meta-analysis, also referred to as a quantitative literature review, can facilitate drawing general conclusions from micro-level findings This methodology, relatively new in the social sciences, was developed in order to synthesise, combine and interpret a large body of empirical evidence on a given topic It offers a clear and systematic way of comparing results of different studies, standardised for the country analysed, the method applied, the control variables employed, the sample selected, etc In order to conduct a meta-analysis, papers researching a topic of interest are collected in a systematic manner Estimated coefficients are selected across studies and recalculated in a standardised way into comparable indicators (i.e effect sizes) The effect sizes constitute the units of statistical analysis, and can be combined into single summary indicators or analysed using regression techniques The quintessence of this approach is quantifying the effect of interest on the basis of the available micro-level empirical studies Meta-analysis has only recently been adopted in family demography research The very few such studies in this field include meta-analyses of: the aggregate relationship between a population’s age structure and its fertility as hypothesised by Easterlin (Waldorf and Byun 2005), the impact of modernisation and strength of marriage norms on divorce risks in Europe (Wagner and Weiss 2006), and the microlevel relationship between fertility and women’s employment in industrialised economies (Matysiak and Vignoli 2008) In order to give a better insight into the meta-analysis method, we elaborate shortly on the meta-study by Matysiak and Vignoli (2008) It aimed to synthesise micro-level findings on the relationship between fertility and women’s employment in industrialised economies Two effects were analysed: that of women’s work on fertility (90 studies) and that of having young children on women’s employment entry (55 studies) The authors found that the micro-level relationship between the two variables is still negative, but its magnitude varies across countries, differing in their welfare policies, the labour market structures and the social acceptance of women’s work This variation in 480 A Matysiak and D Vignoli the magnitude of the micro-level relationship explains the existence of the positive cross-country correlation between fertility and women’s labour supply, which has been observed in OECD countries since the mid-1980s (Engelhardt et al 2004) Meta-analysis certainly is a useful tool for summarising and synthesising the abundant micro-level research Its unquestionable strength is that effect estimates produced within its framework have higher external validity than those obtained in individual studies owing to the generality of results across various research papers (Shadish et al 2002) Nevertheless, a weakness of this method lies in the assumption that the micro-to-macro transformation can be achieved through a simple summation of individual-level actions into a macro-level outcome According to Coleman (1990), the complex interactions between and within social groups, as well as the heterogeneity of individuals, preclude such a simple aggregation Since demographic choices are made by interacting and heterogeneous individuals, this assumption, implicit in meta-analysis, may not be valid 4.2 Agent-Based Computational Models Agent-based computational models come as a solution to this problem They seem to be the most powerful tool which is available for transforming the micro results to the macro-level outcomes and which allows to account for heterogeneity among individuals and for the complexity of individual-level interactions (Billari and Ongaro 2000; Billari 2006) It includes micro-simulation, which models macro processes on the basis of empirical models (i.e event-history models, or even multilevel event-history models), as well as formal models of demographic behaviours, which operationalise decision-making processes at the micro level and simulate their outcomes in terms of macro-level indicators The additional advantage of agent-based computational models is that they allow study of the impact of policy interventions on demographic behaviours, taking into account policy side effects as well as the interactions of policy with other elements of the social system (Van Imhoff and Post 1998) Below we give one example of micro-simulation that was run with the goal of, among others, assessing the macro-level consequences of an increase in women’s employment on fertility (Aassve et al 2006) The first study was conducted in two steps First, using the British Household Panel Study, the authors estimated a multi-process hazard model of five interdependent processes: childbirth, union formation, union dissolution, employment entry, and employment exit They found the employment parameter in the fertility equation to be strongly negative The micro-simulation conducted in the second step showed, however, that increasing the hazard of employment entry by 10% and decreasing the hazard of employment exit by another 10% led to a decline in the proportion of women having their second child before the age of 40 by only 0.2% points This was much less than one could have expected from the analysis of the parameter estimates in the fertility equation The underlying reason was that employment affected fertility also in an indirect way: it had a positive impact on the time spent Methods for Reconciling the Micro and the Macro in Family Demography Research 481 in a union, which in turn facilitated childbearing In short, the negative direct and the positive indirect effect of employment on fertility cancelled each other out, resulting in very small general effects of employment on fertility This study clearly demonstrated that interpreting parameters from a hazard model alone is not enough to conclude on the subsequent macro-level developments The interactions between the processes should also be taken into account Towards an Empirical Implementation of the Theoretical Model: Implications for Data Collection and an Avenue for Future Research The concepts and relationships presented in this paper are summarised in Fig 1, which illustrates the theoretical model of methodological individualism in the context of family demography research (see also Muszy´nska 2007: 169; Philipov et al 2009: 17) The scheme of the theory is supplemented with information on analytical methods that could support formation of a comprehensive explanation of the mechanisms and factors driving change in family-related outcomes, as observed at the macro-level In short, multi-level event-history models are suggested for operationalising the situational and action formation mechanisms, while metaanalyses and agent-based computational models are viewed to be the most suitable for quantifying the transformational mechanism We believe that in the future it will be possible to implement this full theoretical model in a single study in the field of family demography The major challenge to be faced at that stage will be collection of suitable data Today, in fact, the gap between the analytical tools available and the proper data seems to be the most important barrier preventing population scientists from following the research framework suggested Conducting a multi-level event-history analysis requires data on the migration histories of individuals together with all other life-histories, Fig Theoretical model for the explanation of family and fertility dynamics complemented with the most suitable methods for its implementation 482 A Matysiak and D Vignoli as well as time-series contextual data Similarly, performing a micro-simulation requires information on several individual life-histories that are often closely connected To date, such data are not available It should be noted, however, that substantial advancement in this direction has been made within the Generations and Gender Programme (GGP) (Vikat et al 2007; Kveder 2009) Its international harmonised database will include individual life-histories of respondents residing in over twenty developed countries It will additionally be supplemented by the Contextual Database, which contains high quality data at the national or regional level (Spielauer 2006) Furthermore, other contextual indicators can be found in the Family Database developed by the OECD or in the EDACWOWE Portal developed within the RECWOWE (Reconciling work and welfare in Europe) project A serious drawback of the GGP is its very limited scope of information on migration histories of the respondents, which impedes the possibilities of linking the longitudinal individual data with the time-series of contextual indicators In future data collection programmes, care should be taken to eliminate this shortcoming References Adser`a, A.: Vanishing 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