Mixed Effect Modeling. Mixed Effects Models. Synonyms. Definition. Description

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1 ixed Effects odels 4089 ixed Effect odeling Hierarchical Linear odeling ixed Effects odels atthew P. Buman 1 and Eric B. Hekler 2 1 Exercise and Wellness Program, School of Nutrition and Health Promotion Arizona State University, Phoenix, AZ, USA 2 Nutrition Program, School of Nutrition and Health Promotion, Arizona State University, Phoenix, AZ, USA Synonyms Hierarchical linear modeling; Individual growth modeling; ultilevel modeling; Random coefficients regression Definition ixed effects models are statistical models used to account for nonindependence among units. These models are useful to account for clustering of related units or in repeated measures (i.e., longitudinal) designs. Description ixed effects models had their start in the field of education. In education, students are grouped into meaningful clusters of related units; e.g., students are grouped into classes, and classes are grouped into schools. Students, classes, and schools all have characteristics of interest. Historically, techniques used to analyze these types of data were the following: 1. The first technique was to disaggregate all of the higher-level contextual data (e.g., classes, schools) down to the level of the individual. Standard ordinary least-squares (OLS) regression methods were then used to make inferences about individuals. The problem with this approach is that it is almost always the case that students within classes (and classes within schools) share certain characteristics. Thus, classic statistical techniques, which carry a basic assumption of independence of observations, are not tenable. 2. The second technique was to aggregate the individual-level characteristics up to higher levels and to conduct the analysis at the aggregated level. The main problem with this approach is that all the within-level (i.e., within-school/within-classroom) variation is discarded. While the amount of variation at this level varies by outcome of interest and sample characteristics, when hierarchical data is being considered, it is almost always non-ignorable (Hox, 2002). In both of the above techniques, researchers fail to account for the inherent hierarchies in the data. By ignoring these structures, researchers run the risk of both interpretational and statistical errors such as the ecological fallacy (i.e., making individual-level inferences based on aggregated data) or atomistic fallacy (i.e., making group-level inferences based on individual-level data). ixed effects modeling is a hierarchical extension of standard OLS regression methods which allows researchers to examine predictors at every level of existing hierarchies and thus overcome the limitations of these historical approaches. These methods can be used across a diverse range of research problems. It has been used to analyze clustered cross-sectional data; longitudinal, growth, and intervention data; and binary and other non-normal data. Each of these applications share three common assumptions: (a) a hierarchical structure to the dataset, (b) a single outcome or response variable measured at the lowest level of the hierarchy, and (c) explanatory variables at all levels of the hierarchy (Hox, 2002). Conceptually, mixed effects models can be applied to datasets with any number of

2 4090 ixed Effects odels levels within a hierarchy; however, in practice most analyses handle data with two or three levels. Components of ixed Effects odels Level-1 effects refer to effects that occur at the lowest level of the hierarchy. In most cases this refers to effects on an individual. These effects are sometimes called within-group effects. For example, if a research wanted to study an individual s self-rated quality of life, predictors such as self-efficacy, an individual s observed level of physical function, and gender would all be considered level-1 effects. These effects are what are most commonly estimated using tradition OLS regression methods and can be represented with the following equation: Y ij ¼ B 0j þ B 1j X ij þ e ij (1) In Eq. 1, B 0j represents the average (across all individuals, i) intercept, B 1j represents the average regression coefficient (or slope) for the predictor variable (X), and e ij represents the average residual error in the model. Subscripts i and j in this equation recognize that each of these terms varies by the individual (level-1 effects, i) and by higher level in the hierarchy (j), respectively. Level-2 effects refer to effects that occur at this higher level of the hierarchy. In two-level analyses these effects are sometimes called between-group effects and the same value applies to all individuals within the group. For example, predictors such as neighborhood-level income and worker s industry type (e.g., blue collar vs. white collar) would be considered level-2 effects. In simple cases, these effects are represented using the following two equations: B 0j ¼ g 00 þ g 01 Z j þ u 0j (2) B 1j ¼ g 10 þ g 11 Z j þ u 1j (3) Equation 2 predicts the average intercept for the higher level of the hierarchy by a given level-2 predictor (Z). Equation 3 describes how the relationship between the outcome variable (Y) and the level-1 predictor variable (X) varies depending on the level-2 predictor (Z). In both Eqs. 2 and 3, residual error terms (u 0j and u 1j ) are estimated for the higher level of the hierarchy. Equations 1 3 can be combined into a single complex regression equation to represent the complete multilevel model: Y ij ¼ g 00 þ g 10 X ij þ g 01 Z j þ g 11 X ij Z j þ u 1j X ij þ u 0j þ e ij (4) Equation 4 can also be partitioned into fixed (g 00 þ g 10 X ij þ g 01 Z j þ g 11 X ij Z j ) and random (u 1j X ij þ u 0j þ e ij ) components of the equation. Fixed effects refer to the known, measured predictors which are specified in the model. Random effects, also sometimes referred to as stochastic effects or variance components, contain the random error terms of the model that represent the unexplained portion of Y by the level-1 (as represented by e ij ) and level-2 (u 1j X ij þ u 0j ) predictors. As stated, the necessity of mixed effects models is to account for the inherent hierarchical structure of observations (i.e., nonindependence of data). The extent to which observations are dependent is expressed by the intraclass correlation coefficient (ICC), which is interpreted like any other correlation coefficient. There are many formulas offered to calculate the ICC, but conceptually the ICC represents the ratio of level- 2 variance to the total variance in the model (level-1 + level-2 variance) (Tabachnick & Fidell, 2007). In a null model, where no predictors are included to explain effects, total variance in Y is decomposed into level-1 variance (denoted as s 2 e) and level-2 variance (denoted as s 2 u0) components. The ICC is thus expressed in Eq. 5: ICC ¼ s 2 u0= s 2 e þ s 2 u0 (5) The ICC can be interpreted (in the null model) as the proportion of the total variance to be explained at the level-2, higher level of the hierarchy. The inverse of the ICC (1-ICC) can conversely be interpreted as the proportion of variance to be explained at the level-1, lower

3 ixed Effects odels 4091 level of the hierarchy. ICC values can be compared in subsequent models to provide estimates of variance explained relative to the null model (Singer & Willett, 2003). Applications to Clustered Cross-Sectional Data ixed effects models are commonly used to handle clustered cross-sectional data. Clustered data refers to data on units nested within naturally occurring hierarchies (e.g., residents nested within neighborhoods (Sallis et al., 2009), students nested within classrooms (Bryk & Raudenbush, 1992), doctors nested within hospital clinics (Velikova et al., 2004)). The basic assumption within clustered data models is that observations within the same cluster may be more similar than observations in separate units, on average. For example, in a study looking at the effects of functional status on quality of life, a researcher may sample across divergent regions of a country. An individual s quality of life is likely not to be independent of what region of the country in which they reside. ixed effects models account for variation in quality of life at both the individual level (level 1) and at the country region level (level 2), and predictors can be specified at each of these levels to predict this variation. Applications to Repeated easures (i.e., Longitudinal) Data (Also Known as Individual Growth odels or Growth Curve odeling) ixed effects models can be used to model change over time in a variable of interest. Repeated measures data is nested within persons, such as multiple occasions of the same measurement for each person over time (Cillessen & Borch, 2006; Kristjansson, Kircher, & Webb, 2007). The basic assumption within repeated measures models is that measurement over time within the same person is related. Each individual has their own unique trajectory of change, which when aggregated across individuals comprises group-level trajectories. For example, in a treatment study looking at how an exercise intervention may improve quality of life in a sample of cancer survivors, individuals might be assumed to show linear improvement over time. In reality, where individuals started (intercept) and how much they improved (slope) over the course of the intervention period varies. ixed effects models account for this variation both within and between individuals. The level-1 submodel refers to how each individual changes over time and the level-2 submodel describes how these changes differ across people. odels may incorporate time-constant (e.g., group assignment, gender) or time-varying (e.g., selfefficacy) covariates as predictors. Practical Considerations Sample size and power to detect meaningful effects are of utmost importance when applying a mixed effects modeling framework. General rules of thumb have been recommended. Eliason (1993) has suggested at least 60 observations if only five or fewer parameters are being estimated. aas and Hox (2005) have suggested at least 30 units at each level of analysis. These standard conventions are helpful but have not undergone extensive empirical testing or have been tested for special cases only. For example, the latter recommendation generally refers to the power needed to detect cross-level (level-1 level-2) interaction effects, which are not of particular interest in all cases. Unlike traditional OLS regression methods, sample size considerations must take into account the smallest unit of analysis and higher levels within the hierarchy. One shortcoming commonly seen in clustered data analyses is level- 2 sparseness, which refers to situations where there are a large number of level-2 units with few individuals per unit. A recent onte Carlo simulation study by Bell, Ferron, and Kromrey (2008) suggests that even under the most extreme condition tested (a small number of level-2 units: 50; a small number of level-1 units per cluster: average ¼ 10; high proportion of singletons [a single level-1 unit in a cluster]: 70 %), sparseness resulted in only small reductions in the accuracy of confidence intervals for level-2 predictors and did not impact the accuracy of the fixed or random estimates for level-1 predictors.

4 4092 ixed Effects odels issing data is an important consideration in both clustered cross-sectional and repeated measures applications of the mixed effects model. In general, missing data handling procedures are viewed as superior in mixed effects models relative to traditional OLS regression methods. Traditional regression relies on listwise deletion, which means that complete cases are dropped from the analysis when data is not present. While this may be acceptable when small amounts of data are missing, this can lead to serious biases and inefficiencies in most cases (Duncan, Duncan, & Strycker, 2006). ixed effects models rely on imputation-based procedures to replace missing data, resulting in a complete dataset, which provides greater power through increased sample size and reduced bias due to selective missingness, relative to listwise deletion procedures. While estimation procedures vary, mixed effects modeling uses direct or full maximum likelihood procedures to impute missing data. Briefly, as Schafer (2001, p. 358) explains, these methods treat missing data as an explicit source of random variability over which to be averaged. These methods have gained increased popularity in recent years given their accessibility in common statistical packages. Two limitations to imputation-based procedures should be mentioned. First, these methods assume data to be missing at random, which is both an untestable assumption and one that may seem implausible under some conditions (Schafer, 2001). In some cases it may be more conceptually plausible to rely on manual imputation procedures such as baseline or last observation carried forward procedures. Second, imputation procedures are only performed on the outcome variable and do not apply to predictor variables. In cases of missing predictor variables, cases are still listwise deleted. ixed effects modeling has increased in popularity in recent years given its growing accessibility through common statistical software packages. ixed effects models have been present for many years in software dedicated to these procedures such as hierarchical linear modeling (Bryk & Raudenbush, 1992) and LwiN (Rasbash, Charlton, Browne, Healy, & Cameron, 2005). A general latent variable framework of mixed models is available through plus (uthen & uthen, ). Finally, mixed effects model procedures are available through widely available software packages such as SAS PROC ixed (SAS Institute Inc., 2008) and SPSS ixed Procedure (SPSS Inc., 2005). Cross-References Ecological Fallacy Education Hierarchical Linear odeling Intraclass Correlation Coefficient (ICC) issing Data Ordinary Least-Squares (OLS) odel Power to Detect eaningful Effects Sample Size References Bell, B., Ferron, J.., & Kromrey, J. D. (2008). Cluster size in multilevel models: The impact of sparse data structures on point and interval estimates in two-level models. Paper presented at the Proceedings of the Joint Statistical eetings. Bryk, A. S., & Raudenbush, S. W. (1992). Hierarchical linear models for social and behavioral research: Applications and data analysis methods. Newbury Park, CA: Sage. Cillessen, A. H. N., & Borch, C. (2006). Developmental trajectories of adolescent popularity: A growth curve modelling analysis. Journal of Adolescence, 29, Duncan, T. E., Duncan, S. C., & Strycker, L. A. (2006). An introduction to latent variable growth curve modeling: Concepts, issues, and applications. ahwah, NJ: Lawrence Erlbaum Associates. Eliason, S. R. (1993). aximum likelihood estimation: Logic and practice. Newbury Park, CA: Sage. Hox, J. (2002). ultilevel analysis. ahwah, NJ: Lawrence Erlbaum Associates. Kristjansson, S. D., Kircher, J. C., & Webb, A. K. (2007). ultilevel models for repeated measures research designs in psychophysiology: An introduction to growth curve modeling. Psychophysiology, 44, aas, C. J.., & Hox, J. (2005). Sufficient sample sizes for multilevel modeling. ethodology: European Journal of Research ethods for the Behavioral and Social Sciences, 1(3),

5 obility 4093 uthen, L. K., & uthen, B. O. ( ). plus User s Guide. Fifth Edition. Los Angeles, CA: uthen & uthen. Rasbash, J., Charlton, C., Browne, W. J., Healy,., & Cameron, B. (2005). Lwin version Centre for ultilevel odeling, University of Bristol. Sallis, J., Saelens, B., Frank, L. D., Conway, T. L., Slymen, D. J., Cain, K., et al. (2009). Neighborhood built environment and income: Examining multiple health outcomes. Social Science & edicine, 34(1), SAS Institute Inc. (2008). SAS/STAT 9.2 user s guide. Cary, NC: Author. Schafer, J. L. (2001). ultiple imputation with PAN. In L. A. Collins & A. G. Sayer (Eds.), New methods for the analysis of change. Washington, DC: American Psychological Association. Singer, J. D., & Willett, J. B. (2003). Applied longitudinal data analysis: odeling change and event occurrence. New York: Oxford University Press. SPSS Inc. (2005). Linear mixed-effects modeling in SPSS: An introduction to the IXED procedure. Chicago: Author. Tabachnick, B. G., & Fidell, L. S. (2007). Using multivariate statistics (5th ed.). Boston: Allyn & Bacon. Velikova, G., Booth, L., Smith, A. B., Brown, P.., Lynch, P., Brown, J.., et al. (2004). easuring quality of life in routine oncology practice improves communication and patient well-being: A randomized controlled trial. Journal of Clinical Oncology, 22(4), ixed ethod eta-methodology ixed ethodology eta-methodology ixture Growth odel Latent Class odel ixture odel Latent Class odel ixture Regression Analysis Latent Class odel obility atteo Colleoni Sociology and Social Research, University of ilan Bicocca, ilano, Italy Definition obility is the objective and subjective propensity to be mobile in space with any mode, in order to reach places where social activities are to be carried out in everyday life. Description Before the invention of motorized transport means (train, bus, automobile, etc.), the study of the movement of people was not a major object of scientific inquiry. With the advent of motorized vehicles and the related problems in terms of infrastructure building, managing of flows, resource consumption, and urban sustainability (Newman & Kenworthy, 1999), the transport sciences were born and for several decades they were the main (and almost exclusive) approach to the study of movement. Implicit in this approach is a view of individuals as the objects (rather than subjects ) of transport and a focus on mass and motorized transport means. This has led to a neglect of slow modes of autonomous movement such as walking and cycling (Litman, 2003) and to an exclusive focus on the origins and destinations of travel, while overlooking the activities that cause people to move around. The shortcomings of this approach have led scholars to use the term mobility in order to consider a wider range of factors which are usually gathered under the label mobility demand : the sociodemographic characteristics