DOCSLIB.ORG

The Multilevel Model Framework

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Multilevel Model Framework The SAGE Handbook of Multilevel Modeling Edited by Marc A. Scott, Jeffrey S. Simonoff and Brian D. Marx “Handbook_Sample.tex” — 2013/7/23 — 9:39 — page 52 1 The Multilevel Model Framework J e f f G i l l Washington University, USA A n d r e w J. W o m a c k University of Florida, USA 1.1 OVERVIEW then nested in clinics or hospitals, which are then nested in healthcare management sys- Multilevel models account for different lev- tems, which are nested in states, and so on. els of aggregation that may be present in In the classic example, students are nested data. Sometimes researchers are confronted in classrooms, which are nested in schools, with data that are collected at different lev- which are nested in districts, which are then els such that attributes about individual cases nested in states, which again are nested in the are provided as well as the attributes of nation. In another familiar context, it is often groupings of these individual cases. In addi- the case that survey respondents are nested in tion, these groupings can also have higher areas such as rural versus urban, then these groupings with associated data characteris- areas are nested by nation, and the nations in tics. This hierarchical structure is common regions. Famous studies such as the American in data across the sciences, ranging from National Election Studies, Latinobarometer, the social, behavioral, health, and economic Eurobarameter, and Afrobarometer are obvi- sciences to the biological, engineering, and ous cases. Often in population biology a physical sciences, yet is commonly ignored hierarchy is built using ancestral informa- by researchers performing statistical analyses. tion, and phenotypic variation is used to Unfortunately, neglecting hierarchies in data estimate the heritability of certain traits, in can have damaging consequences to subse- what is commonly referred to as the “animal quent statistical inferences. model.” In image processing, spatial relation- The frequency of nested data structures in ships emerge between the intensity and hue of the data-analytic sciences is startling. In the pixels.There are many hierarchies that emerge United States and elsewhere, individual vot- in language processing, such as topic of dis- ers are nested in precincts which are, in turn, cussion, document type, region of origin, or nested in districts, which are nested in states, intended audience. In longitudinal studies, which are nested in the nation. In health- more complex hierarchies emerge. Units or care, patients are nested in wards, which are groups of units are repeatedly observed over “Handbook_Sample.tex” — 2013/7/25 — 11:25 — page 3 4 THE MULTILEVEL MODEL FRAMEWORK a period of time. In addition to group hier- like a linear model or generalized linear archies, observations are also grouped by the model. The departure comes from the treat- unit being measured. These models are exten- ment of some of the coefficients assigned to sively used in the medical/health sciences to the explanatory variables. What can be done model the effect of a stimulus or treatment to modify a model when a point estimate is regime conditional on measures of interest, inadequate to describe the variation due to such as socioeconomic status, disease preva- a measured variable? An obvious modifica- lence in the environment, drug use, or other tion is to treat this coefficient as having a dis- demographic information. Furthermore, the tribution as opposed to being a fixed point. frequency of data at different levels of aggre- A regression equation can be introduced to gation is increasing as more data are generated model the coefficient itself, using information from geocoding, biometric monitoring, Inter- at the group level to describe the heterogeneity net traffic, social networks, an amplification in the coefficient. This is the heart of the mul- of government and corporate reporting, and tilevel model. Any right-hand side effect can high-resolution imaging. get its own regression expression with its own Multilevel models are a powerful and assumptions about functional form, linearity, flexible extension to conventional regression independence, variance, distribution of errors, frameworks.They extend the linear model and and so on. Such models are often referred to the generalized linear model by incorporating as “mixed,” meaning some of the coefficients levels directly into the model statement, thus are modeled while others are unmodeled. accounting for aggregation present in the data. What this strategy produces is a method of As a result, all of the familiar model forms for accounting for structured data through utiliz- linear, dichotomous, count, restricted range, ing regression equations at different hierar- ordered categorical, and unordered categor- chical levels in the data. The key linkage is ical outcomes are supplemented by adding that these higher-level models are describing a structural component. This structure clas- distributions at the level just beneath them for sifies cases into known groups, which may the coefficient that they model as if it were have their own set of explanatory variables itself an outcome variable. This means that at the group level. So a hierarchy is estab- multilevel models are highly symbiotic with lished such that some explanatory variables Bayesian specifications because the focus in are assigned to explain differences at the indi- both cases is on making supportable distribu- vidual level and some explanatory variables tional assumptions. are assigned to explain differences at the Allowing multiple levels in the same group level. This is powerful because it takes model actually provides an immense amount into account correlations between subjects of flexibility. First, the researcher is not within the same group as distinct from cor- restricted to a particular number of levels. The relations between groups. Thus, with nested coefficients at the second grouping level can data structures the multilevel approach imme- also be assigned a regression equation, thus diately provides a set of critical advantages adding another level to the hierarchy, although over conventional, flat modeling where these it has been shown that there is diminishing structures emerge as unaccounted-for hetero- return as the number of levels goes up, and geneity and correlation. it is rarely efficient to go past three levels What does a multilevel model look like? At from the individual level (Goel and DeGroot the core, there is a regression equation that 1981, Goel 1983). This is because the effects relates an outcome variable on the left-hand of the parameterizations at these super-high side to a set of explanatory variables on the levels gets washed out as it comes down the right-hand side. This is the basic individual- hierarchy. Second, as stated, any coefficient level specification, and looks immediately at these levels can be chosen to be modeled “Handbook_Sample.tex” — 2013/7/25 — 11:25 — page 4 1.2 BACKGROUND 5 or unmodeled and in this way the mixture of Lee and Bryk (1989). These applications con- these decisions at any level gives a combina- tinue today as education policy remains an torially large set of choices. Third, the form of important empirical challenge. Work in this the link function can differ for any level of the literature was accelerated by the development model. In this way the researcher may mix lin- of the standalone software packages HLM, ear, logit/probit, count, constrained, and other ML2, VARCL, as well as incorporation into forms throughout the total specification. the SAS procedure MIXED, and others. Addi- tional work by Goldstein (notably 1985) took the two-level model and extended it to sit- 1.2 BACKGROUND uations with further nested groupings, non- nested groupings, time series cross-sectional It is often the case that fundamental ideas in data, and more. At roughly the same time, statistics hide for a while in some applied a series of influential papers and applica- area before scholars realize that these are tions grew out of Laird and Ware (1982), generalizable and broadly applicable princi- where a random effects model for Gaussian ples. For instance, the well-known EM algo- longitudinal data is established. This Laird– rithm of Dempster, Laird, and Rubin (1977) Ware model was extended to binary out- was pre-dated in less fully articulated forms comes by Stiratelli, Laird, and Ware (1984) by Newcomb (1886), McKendrick (1926), and GEE estimation was established by Zeger Healy andWestmacott(1956), Hartley (1958), and Liang (1986). An important extension to Baum and Petrie (1966), Baum and Eagon non-linear mixed effects models is presented (1967), and Zangwill (1969), who gives the in Lindstrom and Bates (1988). In addition, critical conditions for monotonic conver- Breslow and Clayton (1993) developed quasi- gence. In another famous example, the core likelihood methods to analyze generalized lin- Markov chain Monte Carlo (MCMC) algo- ear mixed models (GLMMs). rithm (Metropolis et al. 1953) slept quietly Beginning around the 1990s, hierarchical in the Journal of Chemical Physics before modeling took on a much more Bayesian com- emerging in the 1990s to revolutionize the plexion now that stochastic simulation tools entire discipline of statistics. It turns out that (e.g. MCMC) had arrived to solve the result- hierarchical modeling follows this same sto- ing estimation challenges. Since the Bayesian ryline, roughly originating with the statistical paradigm and the hierarchical reliance on dis- analysis of agricultural data around the 1950s tributional relationships between levels have (Eisenhart 1947, Henderson 1950, Scheffé a natural affinity, many papers were produced 1956, Henderson et al. 1959). A big step and continue to be produced in the inter- forward came in the 1980s when education section of the two. Computational advances researchers realized that their data fit this during this period centered around customiz- structure perfectly (students nested in classes, ing MCMC solutions for particular problems classes nested in schools, schools nested (Carlin et al.
Load more

Recommended publications
  • A Primer for Analyzing Nested Data: Multilevel Modeling in SPSS Using
    REL 2015–046 The National Center for Education Evaluation and Regional Assistance (NCEE) conducts unbiased large-scale evaluations of education programs and practices supported by federal funds; provides research-based technical assistance to educators and policymakers; and supports the synthesis and the widespread dissemination of the results of research and evaluation throughout the United States. December 2014 This report was prepared for the Institute of Education Sciences (IES) under contract ED-IES-12-C-0009 by Regional Educational Laboratory Northeast & Islands administered by Education Development Center, Inc. (EDC). The content of the publication does not necessarily reflect the views or policies of IES or the U.S. Department of Education nor does mention of trade names, commercial products, or organizations imply endorsement by the U.S. Government. This REL report is in the public domain. While permission to reprint this publication is not necessary, it should be cited as: O’Dwyer, L. M., and Parker, C. E. (2014). A primer for analyzing nested data: multilevel mod­ eling in SPSS using an example from a REL study (REL 2015–046). Washington, DC: U.S. Department of Education, Institute of Education Sciences, National Center for Educa­ tion Evaluation and Regional Assistance, Regional Educational Laboratory Northeast & Islands. Retrieved from http://ies.ed.gov/ncee/edlabs. This report is available on the Regional Educational Laboratory website at http://ies.ed.gov/ ncee/edlabs. Summary Researchers often study how students’ academic outcomes are associated with the charac­ teristics of their classrooms, schools, and districts. They also study subgroups of students such as English language learner students and students in special education.
    [Show full text]
  • A Bayesian Multilevel Model for Time Series Applied to Learning in Experimental Auctions
    Linköpings universitet | Institutionen för datavetenskap Kandidatuppsats, 15 hp | Statistik Vårterminen 2016 | LIU-IDA/STAT-G--16/003—SE A Bayesian Multilevel Model for Time Series Applied to Learning in Experimental Auctions Torrin Danner Handledare: Bertil Wegmann Examinator: Linda Wänström Linköpings universitet SE-581 83 Linköping, Sweden 013-28 10 00, www.liu.se Abstract Establishing what variables affect learning rates in experimental auctions can be valuable in determining how competitive bidders in auctions learn. This study aims to be a foray into this field. The differences, both absolute and actual, between participant bids and optimal bids are evaluated in terms of the effects from a variety of variables such as age, sex, etc. An optimal bid in the context of an auction is the best bid a participant can place to win the auction without paying more than the value of the item, thus maximizing their revenues. This study focuses on how two opponent types, humans and computers, affect the rate at which participants learn to optimize their winnings. A Bayesian multilevel model for time series is used to model the learning rate of actual bids from participants in an experimental auction study. The variables examined at the first level were auction type, signal, round, interaction effects between auction type and signal and interaction effects between auction type and round. At a 90% credibility interval, the true value of the mean for the intercept and all slopes falls within an interval that also includes 0. Therefore, none of the variables are deemed to be likely to influence the model. The variables on the second level were age, IQ, sex and answers from a short quiz about how participants felt when they won or lost auctions.
    [Show full text]
  • Introduction to Multilevel Modeling
    1 INTRODUCTION TO MULTILEVEL MODELING OVERVIEW Multilevel modeling goes back over half a century to when social scientists becamedistribute attuned to the fact that what was true at the group level was not necessarily true at the individual level. The classic example was percentage literate and percentage African American. Using data aggregated to the state level for the 50 American states in the 1930s as units ofor observation, Robinson (1950) found that there was a strong correlation of race with illiteracy. When using individual- level data, however, the correlation was observed to be far lower. This difference arose from the fact that the Southern states, which had many African Americans, also had many illiterates of all races. Robinson described potential misinterpretations based on aggregated data, such as state-level data, as a type of “ecological fallacy” (see also Clancy, Berger, & Magliozzi, 2003). In the classic article by Davis, Spaeth, and Huson (1961),post, the same problem was put in terms of within-group versus between-group effects, corresponding to individual-level and group-level effects. A central function of multilevel modeling is to separate within-group individual effects from between-group aggregate effects. Multilevel modeling (MLM) is appropriate whenever there is clustering of the outcome variable by a categorical variable such that error is no longer independent as required by ordinary least squares (OLS) regression but rather error is correlated. Clustering of level 1 outcome scores within levels formed by a levelcopy, 2 clustering variable (e.g., employee ratings clustering by work unit) means that OLS estimates will be too high for some units and too low for others.
    [Show full text]
  • An Introduction to Bayesian Multilevel (Hierarchical) Modelling Using
    An Intro duction to Bayesian Multilevel Hierarchical Mo delling using MLwiN by William Browne and Harvey Goldstein Centre for Multilevel Mo delling Institute of Education London May Summary What Is Multilevel Mo delling Variance comp onents and Random slop es re gression mo dels Comparisons between frequentist and Bayesian approaches Mo del comparison in multilevel mo delling Binary resp onses multilevel logistic regres sion mo dels Comparisons between various MCMC meth ods Areas of current research Junior Scho ol Project JSP Dataset The JSP dataset Mortimore et al was a longitudinal study of around primary scho ol children from scho ols in the Inner Lon don Education Authority Sample of interest here consists of pupils from scho ols Main outcome of intake is a score out of in year in mathematics MATH Main predictor is a score in year again out of on a mathematics test MATH Other predictors are child gender and fathers o ccupation status manualnon manual Why Multilevel mo delling Could t a simple linear regression of MATH and MATH on the full dataset Could t separate regressions of MATH and MATH for each scho ol indep endently The rst approach ignores scho ol eects dif ferences whilst the second ignores similari ties between the scho ols Solution A multilevel or hierarchical or ran dom eects mo del treats both the students and scho ols as random variables and can be seen as a compromise between the two single level approaches The predicted regression lines for each scho ol pro duced by a multilevel mo del are a weighted average of the individual scho ol lines and the full dataset regression line A mo del with both random scho ol intercepts and slop es is called a random slop es regression mo del Random Slopes Regression (RSR) Model MLwiN (Rasbash et al.
    [Show full text]
  • 3 Diagnostic Checks for Multilevel Models 141 Eroscedasticity, I.E., Non-Constant Variances of the Random Effects
    3 Diagnostic Checks for Multilevel Models Tom A. B. Snijders1,2 and Johannes Berkhof3 1 University of Oxford 2 University of Groningen 3 VU University Medical Center, Amsterdam 3.1 Specification of the Two-Level Model This chapter focuses on diagnostics for the two-level Hierarchical Linear Model (HLM). This model, as defined in chapter 1, is given by y X β Z δ ǫ j = j + j j + j , j = 1, . , m, (3.1a) with ǫ ∅ Σ (θ) ∅ j , j (3.1b) δ ∼ N ∅ ∅ Ω(ξ) j and (ǫ , δ ) (ǫ , δ ) (3.1c) j j ⊥ ℓ ℓ for all j = ℓ. The lengths of the vectors yj, β, and δj, respectively, are nj, r, 6 and s. Like in all regression-type models, the explanatory variables X and Z are regarded as fixed variables, which can also be expressed by saying that the distributions of the random variables ǫ and δ are conditional on X and Z. The random variables ǫ and δ are also called the vectors of residuals at levels 1 and 2, respectively. The variables δ are also called random slopes. Level-two units are also called clusters. The standard and most frequently used specification of the covariance matrices is that level-one residuals are i.i.d., i.e., 2 Σj(θ)= σ Inj , (3.1d) where Inj is the nj-dimensional identity matrix; and that either all elements of the level-two covariance matrix Ω are free parameters (so one could identify Ω with ξ), or some of them are constrained to 0 and the others are free parameters.
    [Show full text]
  • Multilevel Analysis
    MULTILEVEL ANALYSIS Tom A. B. Snijders http://www.stats.ox.ac.uk/~snijders/mlbook.htm Department of Statistics University of Oxford 2012 Foreword This is a set of slides following Snijders & Bosker (2012). The page headings give the chapter numbers and the page numbers in the book. Literature: Tom Snijders & Roel Bosker, Multilevel Analysis: An Introduction to Basic and Applied Multilevel Analysis, 2nd edition. Sage, 2012. Chapters 1-2, 4-6, 8, 10, 13, 14, 17. There is an associated website http://www.stats.ox.ac.uk/~snijders/mlbook.htm containing data sets and scripts for various software packages. These slides are not self-contained, for understanding them it is necessary also to study the corresponding parts of the book! 2 2. Multilevel data and multilevel analysis 7 2. Multilevel data and multilevel analysis Multilevel Analysis using the hierarchical linear model : random coefficient regression analysis for data with several nested levels. Each level is (potentially) a source of unexplained variability. 3 2. Multilevel data and multilevel analysis 9 Some examples of units at the macro and micro level: macro-level micro-level schools teachers classes pupils neighborhoods families districts voters firms departments departments employees families children litters animals doctors patients interviewers respondents judges suspects subjects measurements respondents = egos alters 4 2. Multilevel data and multilevel analysis 11{12 Multilevel analysis is a suitable approach to take into account the social contexts as well as the individual respondents or subjects. The hierarchical linear model is a type of regression analysis for multilevel data where the dependent variable is at the lowest level.
    [Show full text]
  • Multilevel Linear Models: the Basics
    CHAPTER 12 Multilevel linear models: the basics Multilevel modeling can be thought of in two equivalent ways: We can think of a generalization of linear regression, where intercepts, and possi- • bly slopes, are allowed to vary by group. For example, starting with a regression model with one predictor, yi = α + βxi + #i,wecangeneralizetothevarying- intercept model, yi = αj[i] + βxi + #i,andthevarying-intercept,varying-slope model, yi = αj[i] + βj[i]xi + #i (see Figure 11.1 on page 238). Equivalently, we can think of multilevel modeling as a regression that includes a • categorical input variable representing group membership. From this perspective, the group index is a factor with J levels, corresponding to J predictors in the regression model (or 2J if they are interacted with a predictor x in a varying- intercept, varying-slope model; or 3J if they are interacted with two predictors X(1),X(2);andsoforth). In either case, J 1linearpredictorsareaddedtothemodel(or,toputitanother way, the constant− term in the regression is replaced by J separate intercept terms). The crucial multilevel modeling step is that these J coefficients are then themselves given a model (most simply, a common distribution for the J parameters αj or, more generally, a regression model for the αj’s given group-level predictors). The group-level model is estimated simultaneously with the data-level regression of y. This chapter introduces multilevel linear regression step by step. We begin in Section 12.2 by characterizing multilevel modeling as a compromise between two extremes: complete pooling,inwhichthegroupindicatorsarenotincludedinthe model, and no pooling, in which separate models are fit within each group.
    [Show full text]
  • Introduction to Multilevel Models for Longitudinal and Repeated Measures Data
    Introduction to Multilevel Models for Longitudinal and Repeated Measures Data • Today’s Class: Features of longitudinal data Features of longitudinal models What can MLM do for you? What to expect in this course (and the next course) CLDP 944: Lecture 1 1 What is CL(D)P 944 about? • “Longitudinal” data Same individual units of analysis measured at different occasions (which can range from milliseconds to decades) • “Repeated measures” data (if time permits) Same individual units of analysis measured via different items, using different stimuli, or under different conditions • Both of these fall under a more general category of “multivariate” data of varying complexity The link between them is the use of random effects to describe covariance of outcomes from the same unit CLDP 944: Lecture 1 2 Data Requirements for Our Models • A useful outcome variable: Has an interval scale* . A one-unit difference means the same thing across all scale points . In subscales, each contributing item has an equivalent scale . *Other kinds of outcomes will be analyzed using generalized multilevel models instead, but estimation will be more challenging Has scores with the same meaning over observations . Includes meaning of construct . Includes how items relate to the scale . Implies measurement invariance • FANCY MODELS CANNOT SAVE BADLY MEASURED VARIABLES OR CONFOUNDED RESEARCH DESIGNS. CLDP 944: Lecture 1 3 Requirements for Longitudinal Data • Multiple OUTCOMES from same sampling unit (person) 2 is the minimum, but just 2 can lead to problems: . Only 1 kind of change is observable (1 difference) . Can’t distinguish “real” individual differences in change from error . Repeated measures ANOVA is just fine for 2 observations – Necessary assumption of “sphericity” is satisfied with only 2 observations even if compound symmetry doesn’t hold More data is better (with diminishing returns) .
    [Show full text]
  • Multilevel Models for Repeated Binary Outcomes: Attitudes and Vote Over the Electoral Cycle
    Multilevel models for repeated binary outcomes: attitudes and vote over the electoral cycle by Min Yang,* Anthony Heath** and Harvey Goldstein* * Institute of Education, University of London; ** Nuffield College, Oxford ABSTRACT Models for fitting longitudinal binary responses are explored using a panel study of voting intentions. A standard multilevel repeated measures logistic model is shown to be inadequate due to the presence of a substantial proportion of respondents who maintain a constant response over time. A multivariate binary response model is shown to be a better fit to the data. SOME KEYWORDS Longitudinal binary data, multivariate multilevel model, multilevel, political attitudes, voting. ACKNOWLEDGEMENTS This work was carried out as part of the Multilevel Models Project funded by the Economic and Social Research Council (UK) under the programme for the Analysis of Large and Complex Datasets. We are very grateful to referees for comments on an earlier draft. 1. INTRODUCTION The electoral cycle has become an established feature of voting behaviour, both in Britain and in other European countries. After an initial ‘honeymoon’ between a new government and the electorate, disillusion often sets in and government popularity - whether measured by opinion polls, by-elections or midterm elections such as the European and local elections - tends to decline. In most cases, there is then some recovery in the government’s standing in the run-up to the next general election (Miller, Tagg and Britton, 1986; Miller and Mackie, 1973; Reif, 1984; Stray and Silver, 1983). During the 1987-92 British parliament, for example, the Conservative government lost seven by-elections but subsequently won all of them back at the 1992 general election.
    [Show full text]
  • User-Friendly Bayesian Regression Modeling: a Tutorial with Rstanarm and Shinystan
    ¦ 2018 Vol. 14 no. 2 User-friendly BaYESIAN REGRESSION modeling: A TUTORIAL WITH rstanarm AND shinystan Chelsea Muth a, B, Zita OrAVECZ A & Jonah Gabry B A Pennsylvania State University B Columbia University AbstrACT ACTING Editor This TUTORIAL PROVIDES A PRAGMATIC INTRODUCTION TO specifying, ESTIMATING AND interpret- De- ING single-level AND HIERARCHICAL LINEAR REGRESSION MODELS IN THE BaYESIAN FRamework. WE START BY NIS Cousineau (Uni- versité D’Ottawa) SUMMARIZING WHY ONE SHOULD CONSIDER THE BaYESIAN APPROACH TO THE MOST COMMON FORMS OF regres- Reviewers sion. Next WE INTRODUCE THE R PACKAGE rstanarm FOR BaYESIAN APPLIED REGRESSION modeling. An OVERVIEW OF rstanarm FUNDAMENTALS ACCOMPANIES step-by-step GUIDANCE FOR fiTTING A single-level One ANONYNOUS re- viewer. REGRESSION MODEL WITH THE stan_glm function, AND fiTTING HIERARCHICAL REGRESSION MODELS WITH THE stan_lmer function, ILLUSTRATED WITH DATA FROM AN EXPERIENCE SAMPLING STUDY ON CHANGES IN af- FECTIVE states. ExplorATION OF THE RESULTS IS FACILITATED BY THE INTUITIVE AND user-friendly shinystan package. Data AND SCRIPTS ARE AVAILABLE ON THE Open Science FrAMEWORK PAGE OF THE project. FOR READERS UNFAMILIAR WITH R, THIS TUTORIAL IS self-contained TO ENABLE ALL RESEARCHERS WHO APPLY regres- SION TECHNIQUES TO TRY THESE METHODS WITH THEIR OWN data. Regression MODELING WITH THE FUNCTIONS IN THE rstanarm PACKAGE WILL BE A STRAIGHTFORWARD TRANSITION FOR RESEARCHERS FAMILIAR WITH THEIR FREQUENTIST counterparts, lm (or glm) AND lmer. KEYWORDS TOOLS BaYESIAN modeling, regression, HIERARCHICAL LINEAR model. Stan, R, rstanarm. B [email protected] CM: n/a; ZO: 0000-0002-9070-3329; JG: n/a 10.20982/tqmp.14.2.p099 Introduction research.
    [Show full text]
  • Fundamentals of Hierarchical Linear and Multilevel Modeling 1 G
    Fundamentals of Hierarchical Linear and Multilevel Modeling 1 G. David Garson INTRODUCTION Hierarchical linear models and multilevel models are variant terms for what are broadly called linear mixed models (LMM). These models handle data where observations are not independent, correctly modeling correlated error. Uncorrelated error is an important but often violated assumption of statistical procedures in the general linear model family, which includes analysis of variance, correlation, regression, and factor analysis. Violations occur when error terms are not independent but instead cluster by one or more grouping variables. For instance, predicted student test scores and errors in predicting them may cluster by classroom, school, and municipality. When clustering occurs due to a grouping factor (this is the rule, not the exception), then the standard errors computed for prediction parameters will be wrong (ex., wrong b coefficients in regression). Moreover, as is shown in the application in Chapter 6 in this volume, effects of predictor variables may be misinterpreted, not only in magnitude but even in direction. Linear mixed modeling, including hierarchical linear modeling, can lead to substantially different conclusions compared to conventional regression analysis. Raudenbush and Bryk (2002), citing their 1988 research on the increase over time of math scores among students in Grades 1 through 3, wrote that with hierarchical linear modeling, The results were startling—83% of the variance in growth rates was between schools. In contrast, only about 14% of the variance in initial status was between schools, which is consistent with results typically encountered in cross-sectional studies of school effects. This analysis identified substantial differences among schools that conventional models would not have detected because such analyses do not allow for the partitioning of learning-rate variance into within- and between-school components.
    [Show full text]
  • Actuarial Applications of Hierarchical Modeling
    Actuarial Applications of Hierarchical Modeling CAS RPM Seminar Jim Guszcza Chicago Deloitte Consulting LLP March, 2010 Antitrust Notice The Casualty Actuarial Society is committed to adhering strictly to the letter and spirit of the antitrust laws. Seminars conducted under the auspices of the CAS are designed solely to provide a forum for the expression of various points of view on topics described in the programs or agendas for such meetings . Under no circumstances shall CAS seminars be used as a means for competing companies or firms to reach any understanding – expressed or implied – that restricts competition or in any way impairs the ability of members to exercise independent business judgment regarding matters affecting competition. It is the responsibility of all seminar participants to be aware of antitrust regulations, to prevent any written or verbal discussions that appear to violate these laws, and to adhere in every respect to the CAS antitrust compliance policy. Copyright © 2009 Deloitte Development LLC. All rights reserved. 1 Topics Hierarchical Modeling Theory Sample Hierarchical Model Hierarchical Models and Credibilityyy Theory Case Study: Poisson Regression Copyright © 2009 Deloitte Development LLC. All rights reserved. 2 Hierarchical Model Theory Hierarchical Model Theory What is Hierarchical Modeling? • Hierarchical modeling is used when one’s data is grouped in some important way. • Claim experience by state or territory • Workers Comp claim experience by class code • Income by profession • Claim severity by injury type • Churn rate by agency • Multiple years of loss experience by policyholder. •… • Often grouped data is modeled either by: • Pooling the data and introducing dummy variables to reflect the groups • Building separate models by group • Hierarchical modeling offers a “third way”.
    [Show full text]