STATISTICS AND THE SOCIAL SCIENCES

Hierarchical Linear Modeling with HLM/2/3

Robert A. Yaffee

To help researchers perform hierarchial linear modeling, NYU's Academic Computing Facility obtained a license for HLM/2/3 from Scientific Software in Chicago. HLM/2/3 consists of two statistical packages, HLM2 and HLM3, designed to perform two- and three-level modeling, respectively.

Consider educational situational analysis. An educational sociologist may want to investigate the effect of both individual (micro-level) and school (macro-level) variables on academic achievement. Surveying a sufficiently large selection of schools, the researcher may study achievement as a function of the personality characteristics of the students at the lower level and as a function of organizational characteristics of the schools at the higher level. For each school surveyed, a regression equation is computed for reading achievement as a function of psychological characteristics. Each equation possesses an intercept and slope for each independent psychological variable. For the whole set of schools, there is a complete set of regression equations. Using this complete set, micro-level psychological effects on educational achievement (reading, in this case) can be assessed.

At the higher, macro level of analysis, saved intercept and slope values form randomly distributed outcome variables of statistical models of higher-level effects. If there is one intercept and one slope coefficient, there are two equations at the higher level. One set explains the random intercept at the school level and the other set explains the random slopes at the school level. These equations may be regressions, analyses of variance, or analyses of covariance. If students are the units of analysis at the individual level, schools might be the unit of analysis at a higher level. Examples of school-level variables might be urban or rural, private or public, and proportion of minority students. If there is a yet higher level of analysis, the variables at this level might pertain to the whole school district, such as climate, population density, or region of the country.

Separating the individual effects from either contextual or compositional effects at a higher level is not difficult. A researcher may indicate contextual effects by using two effects. For one effect, he may subtract the school mean from the individual score on an independent variable. This effect provides for the individual deviation from the school mean. By including the group mean and that individual deviation score, the researcher may perform contextual or situational analyses. The compositional effect (ßc) is the extent to which the organizational-level relationship (ßo) differs from the individual-level relationship (ß I) : ßc = ßo - ß I .

Estimating the parameters of the models is performed by several algorithms. Randomly varying level-1 coefficients are estimated with empirical Bayes, while level-2 coefficients with complex error structures are estimated with generalized least squares. Maximum likelihood estimation via the EM algorithm is used to deal with the unbalanced nature of the variance components. Residuals of the level-1 coefficients are based on ordinary least squares, so that residual analysis is required.

As a computer program, HLM/2/3 has a few drawbacks that might daunt new users. Real limitations with file input, missing data, multilevel weighting, and graphical analysis of residuals could plague a student. The program restricts input to only two types of files: ASCII or SYSTAT input files. HLM/2/3 also has limitations in handling missing data. It assumes complete data and can tolerate missing data only on level-1. It has no provision for replacing missing data. Although HLM/2/3 provides for either listwise or pairwise deletion of cases in computations, pairwise deletion of data can lead to insurmountable statistical problems. Consequently, the manual for this version suggests using listwise deletion. It is necessary before entering the data to be sure that the level-2 file contains no missing data, for the HLM/2/3 program will read such missing data, as legitimate values.

Owing to the multilevel analyses, the user will have to know how to design the weighting for his study, which will depend on the nature of the sampling plan as well as the conceptual nature of the project. The program allows for two sets of weights that may be used together and may allow nesting of lower levels within upper levels. If complex sampling plans are used, the weights will have to be computationally tailored to the sampling plan and study design by the statistician in the data file (serious researchers should consider the use of Survey Data Analysis (SuDaAn) for this purpose). Another problem with HLM/2/3 is the necessity of residual analysis with ordinary least-squares residuals computed on level 1. HLM/2/3 has no provision for graphical analysis of the residuals, which are subject to homogeneity and normality requirements. For these types of tests, the researcher will have to interface the data files with other statistical programs in order to be sure that his models are valid.

Notwithstanding these limitations, HLM/2/3 is a program that allows sophisticated multilevel contextual analyses, including repeated measures to perform analyses of individual growth, as well as designs to perform meta-analyses. It may well eventually become part of the standard statistical repertoire of social scientists. If you want to use this package or need help with its application, contact me by phone at (212) 998-3402 or by e-mail to robert.yaffee@nyu.edu .

Posted 24 September 1996