HLM 分層線性和非線性模型分析軟體 at卡貝軟體

原標題：HLM 分層線性和非線性模型分析軟體 at卡貝軟體

HLM 分層線性和非線性模型(Hierarchical Linear and Nonlinear Modeling)分析軟體

HLM 8版已經發布！

在社會研究和其它領域中,研究的數據通常是分層(hierarchical )結構的.也就是說,單獨研究的課題可能會被分類或重新劃分到具有不同特性的組中.在這種情況下,個體可以被看成是研究的第一層(level-1)單元,而那些區分開他們的組也就是第二層(level-2)單元.這可以被進一步的延伸,第二層(level-2)的單元也可以被劃分到第三層單元中.在這個方面很典型的示例,比如教育學(學生位於第一層,學校位於第二層,學校分布是第三層),又比如社會學(個體在第一層,相鄰的個體在第二層).很明顯在分析這樣的數據時,需要專業的軟體.分層線性和非線性模型(也稱為多層模型)的建立是被用來研究單個分析中的任意層次間的關係的,而不會在研究中忽略掉分層模型中各個層次間相關的變異性.

HLM程序包能夠根據結果變量來產生帶說明變量(explanatory variable,利用在每層指定的變量來說明每層的變異性)的線性模型.HLM不僅僅估計每一層的模型係數,也預測與每層的每個採樣單元相關的隨機因子(random effects).雖然HLM常用在教育學研究領域(該領域中的數據通常具有分層結構),但它也適合用在其它任何具有分層結構數據的領域.這包括縱向分析( longitudinal analysis),在這種情況下,在個體被研究時的重複測量可能是嵌套(nested)的.另外,雖然上面的示例暗示在這個分層結構的任意層次上的成員(除了處於最高層次的)是嵌套(nested)的,HLM同樣可以處理成員關係為"交叉(crossed)",而非必須是"嵌套(nested)"的情況,在這種情況下,一個學生在他的整個學習期間可以是多個不同教室里的成員.

HLM程序包可以處理連續,計數,序數和名義結果變量(outcome varible),及假定一個在結果期望值和一系列說明變量(explanatory variable)的線性組合之間的函數關係.這個關係通過合適的關聯函數來定義,例如identity關聯(連續值結果)或logit關聯(二元結果).

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).