Once a model of this type is specified, inferences can be drawn from
available data for the population means at any level (school, class,
district, etc.). These estimators, which can be regarded from a
Bayesian perspective as posterior means or from a frequentist
perspective as "Best Linear Unbiased Predictors" (BLUPs: see
Robinson, *Statistical Science*, 1991, pp. 15-32), often have
better properties than simple sample-based estimators using only data
from the unit in question. This makes them useful in the problem of
"small-area estimation," i.e. making estimates for units or domains
for which there is a very limited amount of information (see Ghosh and Rao,
*Statistical Science*, 1994, pp. 55-76).

Hierarchical models are often applicable to modeling of data from complex surveys, because usually a clustered or multistage sample design is used when the population has a hierarchical structure in the sense described above. Hierarchical modeling describes populations with complex structures in a way that differs fundamentally from the survey analysis software listed on our main page. Typically, classical survey data analysis estimates "finite population parameters" that describe a population of units at a given level (such as the mean test score of all students, or the regression of mean test scores on per-pupil expenditure across schools). The hierarchical structure of the population is considered because it affects the sample design and therefore the properties of the estimator used; for example, the variance of the mean student score is different when students are drawn as a cluster sample (where schools are clusters) than it would be if students are drawn by simple random sampling from a national population.

Hierarchical modeling, on the other hand, takes variation at different levels of the hierarchy as an object of inference, as described in the first paragraph of this page. A hierarchical model can involve estimands (separate regression coefficients at the school and student level, for example) that have no meaning without recognition of the hierarchical structure of the population. Obviously, the choice between these approaches must depend on the descriptive and scientific objectives of the study.

The following are **tutorial introductions** to hierarchical modeling for
practitioners:

- Bryk and Raudenbush,
*Hierarchical Linear Models*, Sage Publications, 1992. - Goldstein,
*Multilevel Statistical Models,*Edward Arnold, 1995. (Available for free download, http://www.arnoldpublishers.com/support/goldstein.htm.) - Longford,
*Random Coefficient Models*, Clarendon, 1993. - Snijders and Bosker,
*Multilevel Analysis: An Introduction to Basic and Advanced Multilevel Modeling*, Sage Publishers, 1999. - Kreft and de Leeuw,
*Introducing Multilevel Modeling*, Sage Publishers, 1998.

*Journal of Educational and Behavioral Statistics*, Vol. 20, No. 2 (1995). (Special issue.)*Sociological Methods and Research*, Vol. 22, No. 3 (1994). (Special issue.)- Kreft, de Leeuw, and van der Leeden, "A review of five multilevel
analysis programs: BMDP-5V, GENMOD, HLM, ML3, VARCL",
*American Statistician*, pages 324-335, 1994.

- Skinner, Holt and Smith,
*Analysis of Complex Surveys*, Wiley, 1989.