階層線性模式 (Hierarchical linear Model，簡稱 HLM)，HLM所發展的階層模型(Hierarchical
Linear and Nonlinear Modeling)軟體，包含線性和非線性部分，HLM可以讀取大部份統計
軟體的檔案如 SPSS, SAS, SYSTAT及STATA等等。
問題的，HLM提供的模型包括2-level models、3-level models、Hierarchical Generalized Linear
Models (HGLM)和Hierarchical Multivariate Linear Models (HMLM)等。
In social research and other fields, research data often have a hierarchical structure. That is, the individual subjects of study may be classified or arranged in groups which themselves have qualities that influence the study. In this case, the individuals can be seen as level-1 units of study, and the groups into which they are arranged are level-2 units. This may be extended further, with level-2 units organized into yet another set of units at a third level and with level-3 units organized into another set of units at a fourth level. Examples of this abound in areas such as education (students at level 1, teachers at level 2, schools at level 3, and school districts at level 4) and sociology (individuals at level 1, neighborhoods at level 2). It is clear that the analysis of such data requires specialized software. Hierarchical linear and nonlinear models (also called multilevel models) have been developed to allow for the study of relationships at any level in a single analysis, while not ignoring the variability associated with each level of the hierarchy.
The HLM program can fit models to outcome variables that generate a linear model with explanatory variables that account for variations at each level, utilizing variables specified at each level. HLM not only estimates model coefficients at each level, but it also predicts the random effects associated with each sampling unit at every level. While commonly used in education research due to the prevalence of hierarchical structures in data from this field, it is suitable for use with data from any research field that have a hierarchical structure. This includes longitudinal analysis, in which an individual's repeated measurements can be nested within the individuals being studied. In addition, although the examples above implies that members of this hierarchy at any of the levels are nested exclusively within a member at a higher level, HLM can also provide for a situation where membership is not necessarily "nested", but "crossed", as is the case when a student may have been a member of various classrooms during the duration of a study period.
The HLM program allows for continuous, count, ordinal, and nominal outcome variables and assumes a functional relationship between the expectation of the outcome and a linear combination of a set of explanatory variables. This relationship is defined by a suitable link function, for example, the identity link (continuous outcomes) or logit link (binary outcomes).
HLM 7 offers unprecedented flexibility in modeling multilevel and longitudinal data. With the same full array of graphical procedures and residual files along with the speed of computation, robustness of convergence, and user-friendly interface of HLM 6, HLM 7 highlights include three new procedures that handle binary, count, ordinal and multinomial (nominal) response variables as well as continuous response variables for normal-theory hierarchical linear models:
Four-level nested models:
Four-level nested models for cross-sectional data (for example, models for item response within students within classrooms within schools).
Four-level models for longitudinal data (for example items within time points within persons within neighborhoods).
Four-way cross-classified and nested mixtures:
Repeated measures on students who are moving across teachers within schools over time, or item responses nested within immigrants who are cross-classified by country of origin and country of destination.
Repeated measures on persons who are simultaneously living in a given neighborhood and attending a given school.
Hierarchical models with dependent random effects:
Spatially dependent neighborhood effects.
Social network interactions.
HLM 7 also offers new flexibility in estimating hierarchical generalized linear models through the use of Adaptive Gauss-Hermite Quadrature (AGH) and high-order Laplace approximations to maximum likelihood. The AGH approach has been shown to work very well when cluster sizes are small and variance components are large. the high-order Laplace approach requires somewhat larger cluster sizes but allows an arbitrarily large number of random effects (important when cluster sizes are large)
New HTML output that supplies elegant notation for statistical models including visually attractive tables is also now available, allowing the user to cut and paste output of interest into manuscripts.