HLM 分层线性和非线性模型(Hierarchical Linear and Nonlinear Modeling)分析软件
HLM 8版已经发布！
在社会研究和其它领域中,研究的数据通常是分层(hierarchical )结构的.也就是说,单独研究的课题可能会被分类或重新划分到具有不同特性的组中.在这种情况下,个体可以被看成是研究的第一层(level-1)单元,而那些区分开他们的组也就是第二层(level-2)单元.这可以被进一步的延伸,第二层(level-2)的单元也可以被划分到第三层单元中.在这个方面很典型的示例,比如教育学(学生位于第一层,学校位于第二层,学校分布是第三层),又比如社会学(个体在第一层,相邻的个体在第二层).很明显在分析这样的数据时,需要专业的软件.分层线性和非线性模型(也称为多层模型)的建立是被用来研究单个分析中的任意层次间的关系的,而不会在研究中忽略掉分层模型中各个层次间相关的变异性.
HLM程序包能够根据结果变量来产生带说明变量(explanatory variable,利用在每层指定的变量来说明每层的变异性)的线性模型.HLM不仅仅估计每一层的模型系数,也预测与每层的每个采样单元相关的随机因子(random effects).虽然HLM常用在教育学研究领域(该领域中的数据通常具有分层结构),但它也适合用在其它任何具有分层结构数据的领域.这包括纵向分析( longitudinal analysis),在这种情况下,在个体被研究时的重复测量可能是嵌套(nested)的.另外,虽然上面的示例暗示在这个分层结构的任意层次上的成员(除了处于最高层次的)是嵌套(nested)的,HLM同样可以处理成员关系为"交叉(crossed)",而非必须是"嵌套(nested)"的情况,在这种情况下,一个学生在他的整个学习期间可以是多个不同教室里的成员.
HLM程序包可以处理连续,计数,序数和名义结果变量(outcome varible),及假定一个在结果期望值和一系列说明变量(explanatory variable)的线性组合之间的函数关系.这个关系通过合适的关联函数来定义,例如identity关联(连续值结果)或logit关联(二元结果).
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.
HLM fits 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.
HLM 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).
Due to increased interest in multivariate outcome models, such as repeated measurement data, contributions by Jennrich & Schluchter (1986), and Goldstein (1995) led to the inclusion of multivariate models in most of the available hierarchical linear modeling programs. These models allow the researcher to study cases where the variance at the lowest level of the hierarchy can assume a variety of forms/structures. The approach also provides the researcher with the opportunity to fit latent variable models (Raudenbush & Bryk, 2002), with the first level of the hierarchy representing associations between fallible, observed data and latent, "true" data. An application that has received attention in this regard recently is the analysis of item response models, where an individuals "ability" or "latent trait" is based on the probability of a given response as a function of characteristics of items presented to an individual.
In HLM 7, unprecedented flexibility in the modeling of multilevel and longitudinal data was introduced with the inclusion of 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. HLM 7 introduced four-level nested models for cross-sectional and longitudinal models and four-way cross-classified and nested mixture models. Hierarchical models with dependent random effects (spatial design) were added. Another new feature was 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).
In HLM8, the ability to estimate an HLM from incomplete data was added. This is a completely automated approach that generates and analyses multiply imputed data sets from incomplete data. The model is fully multivariate and enables the analyst to strengthen imputation through auxiliary variables. This means that the user specifies the HLM; the program automatically searches the data to discover which variables have missing values and then estimates a multivariate hierarchical linear model (”imputation model”) in which all variables having missed values are regressed on all variables having complete data. The program then uses the resulting parameter estimates to generate M imputed data sets, each of which is then analysed in turn. Results are combined using the “Rubin rules”.
Another new feature of HLM 8 is that flexible combinations of Fixed Intercepts and Random Coefficients (FIRC) are now included in HLM2, HLM3, HLM4, HCM2, and HCM3.A concern that can arise in multilevel causal studies is that random effects may be correlated with treatment assignment. For example, suppose that treatments are assigned non-randomly to students who are nested within schools. Estimating a two-level model with random school intercepts will generate bias if the random intercepts are correlated with treatment effects. The conventional strategy is to specify a fixed effects model for schools. However, this approach assumes homogeneous treatment effects, possibly leading to biased estimates of the average treatment effect, incorrect standard errors, and inappropriate interpretation. HLM 8 allows the analyst to combine fixed intercepts with random coefficients in models that address these problems and to facilitate a richer summary including an estimate of the variation of treatment effects and empirical Bayes estimates of unit-specific treatment effects. This approach was proposed in Bloom, Raudenbush, Weiss and Porter (2017).