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##  The Advantages of Hierarchical Linear Modeling. ERIC/AE Digest.              

Hierarchical, or nested, data structures are common in many areas of            research. Until recently, however, an appropriate technique for                 analyzing these types of data has been lacking. Now that several                user-friendly software programs and more readable texts and                     treatments on the topic have become available, researchers will                 benefit from a greater understanding of hierarchical modeling and               its applications. This Digest introduces hierarchical data                      structure, describes how hierarchical models work, and presents                 three approaches to analyzing hierarchical data. WHAT IS A                    HIERARCHICAL DATA STRUCTURE?People exist within organizational              structures such as families, schools, businesses, churches, towns,              states, and countries. In education, students exist within a                    hierarchical social structure that can include family, peer group,              classroom, grade level, school, school district, state, and country.            Many other communities exhibit hierarchical data structures as                  well. 
Bryk and Raudenbush (1992) discuss two other types of data              hierarchies that are less obvious: repeated-measures data and                   meta-analytic data. Data repeatedly gathered on an individual is                hierarchical because all the observations are nested within                     individuals. While there are other adequate procedures for dealing              with this sort of data, the assumptions relating to them are                    rigorous, whereas procedures relating to hierarchical modeling                  require fewer assumptions. When researchers are engaged in the task             of meta-analysis, or analysis of a large number of existing studies,            subjects, results, procedures, and experimenters are nested within              each experiment. WHY IS A HIERARCHICAL DATA STRUCTURE AN                      ISSUE?Hierarchical, or nested, data present several problems for            analysis. First, people or creatures that exist within hierarchies              tend to be more similar to each other than people randomly sampled              from the entire population. For example, students in a particular               third-grade classroom are more similar to each other than to                    students randomly sampled from the school district as a whole or                from the national population of third-graders because they are not              randomly assigned to classrooms from the population, but rather,                based on geographic factors. Thus, students within a particular                 classroom tend to come from a community or community segment that is            more homogeneous in terms of morals and values, family background,              socioeconomic status, race or ethnicity, religion, and even                     educational preparation than the population as a whole. Further,                students within a particular classroom share the same teacher and               physical environment and have similar experiences, which may lead to            increased homogeneity over time. 
The problem of independence of               observations. 
Because individuals drawn from the same classroom or            school  tend to share certain characteristics (environmental,                   background, experiential, demographic, or otherwise), observations              based on these individuals are not fully independent. However, most             analytic techniques require independence of observations as a                   primary assumption for the analysis. Because this assumption is                 violated in the presence of hierarchical data, ordinary least                   squares regression (OLS) produces standard errors that are too                  small (unless these so-called design effects are incorporated into              the analysis). In turn, this leads to a higher probability of                   rejection of a null hypothesis than if: (a) an appropriate                      statistical analysis were performed, or (b) the data included truly             independent observations. 
The problem of how to deal with                     cross-level data. 
Going back to the example of our third-grade                classroom, it is often the case that a researcher is interested in              understanding how environmental variables (e.g., teaching style,                teacher behaviors, class size, class composition, district policies             or funding, or even state or national variables) affect individual              outcomes (e.g., achievement, attitudes, retention). But given that              outcomes are gathered at the individual level, and other  variables             exist at the classroom, school, district, state, or nation level,               the question arises as to what the unit of analysis should be, and              how to deal with the cross-level nature of the data. 
One strategy             would be to assign classroom or teacher (or school, district, or                other) characteristics to all students (i.e., to bring the                      higher-level variables down to the student level).The problem with              this approach, again, is non-independence of observations, because              all students within a particular classroom assume identical scores              on a variable. 
Another strategy would be to aggregate up to the               level of the classroom, school, or district, thus enabling us to                talk about the effect of teacher or classroom characteristics on                average classroom achievement. However, this approach has two                   limitations: (a) up to 80 to 90 percent of the individual                       variability on the outcome variable is lost, which can lead to                  dramatic under- or over-estimation of observed relationships between            variables (Bryk & Raudenbush, 1992), and (b) the outcome variable               changes significantly and substantively from individual achievement             to average classroom achievement. 
Aside from these problems, both             strategies prevent the researcher from disentangling individual and             group effects on the outcome of interest. As neither one of these               approaches is satisfactory, the third approach, that of hierarchical            linear modeling (HLM), becomes necessary. HOW DO HIERARCHICAL                 MODELS WORK?The basic concept behind hierarchical modeling is               similar to that of OLS regression. On the base level (usually the               individual level, referred to here as level 1), an outcome variable             is predicted as a function of a linear combination of one or more               level 1 variables, plus an intercept, as so: 
Y_ij= b_0j +                     b_1jX_1 + ... + b_kjX_k +r_ij 
where b_0j represents the intercept             of group j, b_1j represents the slope of variable X_1 of group j,               and r_ij represents the residual for individual i within group j. On            subsequent levels, the level 1 slope(s) and intercept become                    dependent variables being predicted from level 2 variables: 
b_0j =            g_00 + g_01 W_1 + ... + g_0kW_k + u_0j 
b_1j = g_10 + g_11                     W_1 + ... + g_1kW_k + u_1j 
and so forth, where g_00 and g_10 are              intercepts, and g_01 and g_11 represent slopes predicting b_0j and              b_1j respectively from variable W_1. Through this process, we                   accurately model the effects of level 1 variables on the outcome,               and the effects of level 2 variables on the outcome. In addition, as            we are predicting slopes as well as intercepts (means), we can model            cross-level interactions, whereby we can attempt to understand what             explains differences in the relationship between level 1 variables              and the outcome. AN EMPIRICAL COMPARISON OF THE THREE APPROACHES              TO

ANALYZING HIERARCHICAL DATATo illustrate the outcomes                 achieved by each of the three possible analytic strategies for                  dealing with hierarchical data, disaggregation (bringing level 2                data down to level 1), aggregation, and multilevel modeling, data               were drawn from the National Education Longitudinal Survey of 1988.             This data set contains data on a representative sample of                       approximately 28,000 U.S. eighth graders at a variety of levels,                including individual, family, teacher, and school. The analysis we              performed predicted composite achievement test scores (math and                 reading combined) from student socioeconomic status (family SES),               student locus of control (LOCUS), the percent of students in the                school who are members of racial or ethnic minority                             groups (%MINORITY), and the percent of students in a school who                 receive free lunch (%LUNCH). Achievement is our outcome, SES and                LOCUS are level 1 predictors, and %MINORITY and %LUNCH are level 2              indicators of school environment. In general, SES and LOCUS are                 expected to be positively related to achievement, and %MINORITY                 and %LUNCH are expected to be negatively related to achievement. In             these analyses, 995 of a possible 1,004 schools were                            represented (the remaining nine were removed due to insufficient                data). 
Disaggregated analysis. 
In order to perform the                      disaggregated analysis, the level 2 values were assigned to all                 individual students within a particular school (which is how the                NELS data set comes). A standard multiple regression was performed              via SPSS entering all predictor variables simultaneously. The                   resulting model was significant, with R=.56, R2=.32,                            F (4,22899)=2648.54, p < .0001. The individual regression weights               and significance tests are presented in the following table. 
{See             Table at end of Digest} 
Note:  B refers to an unstandardized                  regression coefficient, and is used for the HLM analysis to                     represent the unstandardized regression coefficients produced                   therein, even though these are commonly labeled as betas and                    gamma's. SE refers to standard error. Bs with different subscripts              were found to be significantly different from other Bs within the               row at p< .05. 
All four variables were significant predictors of              student achievement. As expected, SES and LOCUS were positively                 related to achievement, while %MINORITY and %LUNCH were negatively              related. 
Aggregated analysis. 
In order to perform the aggregated            analysis, all level 1 variables (achievement, LOCUS, SES) were                  aggregated up to the school level (level 2) by averaging. A standard            multiple regression was performed via SPSS entering all predictor               variables simultaneously. The resulting model was significant, with             R=.87, R2=.75, F (4,999)=746.41, p < .0001.  As seen in Table 1,                both average SES and average LOCUS were significantly positively                related to achievement, and %MINORITY was negatively related. In                this analysis, %LUNCH was not a significant predictor of average                achievement. 
Multilevel analysis. 
In order to perform the                   multilevel analysis, a true multilevel analysis was performed via               HLM, in which the respective level 1 and level 2 variables were                 specified appropriately. Note also that all level 1 predictors were             centered at the group mean, and all level 2 predictors were centered            at the grand mean. The resulting model demonstrated goodness of                 fit (Chi-square for change in model fit =4231.39, 5 df, p <.0001).              This analysis reveals significant positive relationships between                achievement and the level 1 predictors (SES and LOCUS), and strong              negative relationships between achievement and the level 2                      predictors (%MINORITY and %LUNCH). Further, the analysis revealed               significant interactions between SES and both level 2 predictors,               indicating that the slope for SES gets weaker as %LUNCH and                     as %MINORITY increases. Also, there was an interaction between LOCUS            and %MINORITY, indicating that as %MINORITY increases, the slope for            LOCUS weakens. There is no clearly equivalent analogue to R and R2              available in HLM. COMPARISON OF THE THREE ANALYTIC STRATEGIES AND             CONCLUSIONSFor the purposes of this discussion, we will assume              that the third analysis represents the best estimate of what                    the "true" relationships are between the predictors and the outcome.            Unstandardized regression coefficients (Bs in OLS, betas and gamma's            in HLM) were compared statistically via procedures outlined in Cohen            and Cohen (1983). 
In examining what is probably the most common               analytic strategy for dealing with data such as these, the                      disaggregated analysis provided the best estimates of the level 1               effects in an OLS analysis. However, it significantly overestimated             the effect of SES, and significantly and substantially                          underestimated the effects of the level 2 effects. The standard                 errors in this analysis are generally lower than they should be,                particularly for the level 2 variables. 
In comparison, the                    aggregated analysis overestimated the multiple correlation by more              than 100%, overestimated the regression slope for SES by 79% and for            LOCUS by 76%, and underestimated the slopes for %MINORITY by 32% and            for %LUNCH by 98%. 
These analyses reveal the need for multilevel              analysis of multilevel data. Neither OLS analysis accurately modeled            the true relationships between the outcome and the predictors.                  Additionally, HLM analyses provide other benefits, such as easy                 modeling of cross-level interactions, which allow for more                      interesting questions to be asked of the data. With nested and                  hierarchical data common in the social and other sciences, and with             recent developments making HLM software packages more user-friendly             and accessible, it is important f or researchers in all fields to               become acquainted with these procedures. REFERENCES AND                       RECOMMENDED READINGBryk, A.S., & Raudenbush, S. W. (1992).                  Hierarchical linear models: Applications and data analysis methods.             Newbury Park, CA: Sage Publications. 
Cohen, J., & Cohen,                      P. (1983). Applied multiple regression/correlation analysis for the             behavioral sciences. Hillsdale, NJ: Lawrence Erlbaum Associates,                Inc. 
Draper, D. (1995). Inference and hierarchical modeling in the            social sciences. Journal of Educational and Behavioral Statistics,              20 (2), 115-147. 
Hoffman, D. A., & Gavin, M. B. (1998). Centering             decisions in hierarchical linear models: Implications for research              in organizations. Journal of Management, 24 (5), 623-641. 
Nezlek,             J. B., & Zyzniewski, L. E. (1998). Using hierarchical linear                    modeling to analyze grouped data. Group Dynamics, 2,                            313-320. 
Pedhazur, E. J. (1997). Multiple regression in behavioral            research (pp.675-711). Harcourt Brace: Orlando, FL. 
Raudenbush, S.            W. (1995). Reexamining, reaffirming, and improving application of               hierarchical models. Journal of Educational and Behavioral                      Statistics, 20 (2),  210-220. 
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This Digest is based on a               paper originally appearing in Practical Assessment, Research &                  Evaluation, 7 (1). [Available online:                                           http://ericae.net/pare/getvn.asp?v=7&n=1 
This publication was                 prepared with funding from the Office of Educational Research and               Improvement, U.S. Department of Education, under contract                       ED99CO0032.  The opinions expressed do not necessarily reflect the              positions or policies of OERI or the U.S. Department of Education.              Permission is granted to copy and distribute this ERIC/AE                       Digest.       
TABLE=====================================Comparison of Three Analytic StrategiesDisaggregated | Aggregated | Hierarchical----------------------------------------------------------Variable: B SE t B SE t B SE t----------------------------------------------------------SES 4.97a .08 62.11 7.28b .26 27.91 4.07c .10 41.29LOCUS 2.96a .08 37.71 4.97b .49 10.22 2.82 .08 35.74%MINORITY -0.45a .03 -15.53 -0.40a .06 -8.76 -0.59 .07 -8.73%LUNCH -0.43a .03 -13.50 0.03b .05 0.59 -1.32c .07 -19.17----------------------------------------------------------**Title:** The Advantages of Hierarchical Linear Modeling. ERIC/AE Digest.              **Note:** Based on a paper appearing in "Practical Assessment, Research &                 Evaluation," v7 n1 (available online:                                           http://ericae.net/pare/getvn.asp?=7&n=1).                                    **Document Type:** Information Analyses---ERIC Information Analysis Products (IAPs) (071); Information Analyses---ERIC Digests (Selected) in Full Text (073); **Descriptors:**  Least Squares Statistics, Models,  Statistical Analysis                     **Identifiers:** ERIC Digests,  Hierarchical Linear Modeling, National Education                 Longitudinal Study 1988, Nested Data                                         
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