SuperMix是由DonaldHedeker教授和RobertGibbons教授以及SSI共同开发的处理混合模型又称阶层模型的软件。该软件可以对二层和三层数据进行线性处理。SuperMix适用于对连续(continuous)因变量、二元(binary)因变量、计数(count)因变量、顺序(ordinary)因变量和名义因变量进行模型分析。这一类分析被广泛应用于社会学、医药领域、经济学、工商业等各个领域。

SuperMix集合了四种混合效果程序的功能，MIXREG, MIXOR, MIXNO, and MIXPREG,由Donald Hedeker and Robert Gibbons开发成一种单独应用，为混合效果的回归模型提供估计。混合效果模式也被称为多层的，分层的，或者随机作用模型。这些模型可以用为纵向数据的分析，每个个体可能在不同数据场合测量。他们也能用于分类数据，比如为在诊所的病人。

● 简单直观的窗口使复杂的模型设计变得简便易行。

● 允许连续因变量(continuous outcome)混合模型中，因变量有自相关残差(auto-correlated residuals)。

● 在顺序因变量混合模型中可以包含scaling effects并进行non-proportional odds模型分析。

● 进行泊松分布混合模型分析。

● 名义因变量的 logistic regression模型分析。

● 分组时间序列的grouped-time survival混合模型分析。

● 二层和三层混合模型分析。

● 直观的图表。

SuperMix is a statistical software package, that deals with mixed-effects models, also known as multilevel, hierarchical, or random-effects models. It handles the following types of outcome variables: continuous, count, ordinal, binary and nominal. These models can be used for the analysis of cross-sectional hierarchical data and longitudinal data, where each individual may be measured on and at a different number of occasions.

Currently, if the outcome variable is count, ordinal, binary or nominal, SuperMix uses maximum likelihood estimates of the model parameters via numerical quadrature (integration). HLM uses approximate methods to obtain parameter estimates and also does not produce a deviance statistic for comparing nested models.

In the case of continuous outcomes, both HLM and SuperMix use ML and yield identical estimates. SuperMix, however, also allows users to impose constraints on the covariance matrices of the random effects. HLM offers both restricted and full ML. SMIX is full ML all the way.

The data set is from a study described in Reisby et. al., (1977) that focused on the longitudinal relationship between imipramine (IMI) and desipramine (DMI) plasma levels and clinical response in 66 depressed inpatients (37 endogenous and 29 non-endogenous). Following a placebo period of 1 week, patients received 225 mg/day doses of imipramine for four weeks. In this study, subjects were rated with the Hamilton depression rating scale (HDRS) twice during the baseline placebo week (at the start and end of this week) as well as at the end of each of the four treatment weeks of the study. Plasma level measurements of both IMI and its metabolite DMI were made at the end of each week. The sex and age of each patient were recorded and a diagnosis of endogenous or non-endogenous depression was made for each patient.

Although the total number of subjects in this study was 66, the number of subjects with all measures at each of the weeks fluctuated: 61 at week 0 (start of placebo week), 63 at week 1 (end of placebo week), 65 at week 2 (end of first drug treatment week), 65 at week 3 (end of second drug treatment week), 63 at week 4 (end of third drug treatment week), and 58 at week 5 (end of fourth drug treatment week). The sample size is 375. Data for the first 10 observations of all the variables used in this section are shown below in the form of a SuperMix spreadsheet file, named reisby.ss3.

The variables of interest are:

• Patient is the patient ID (66 patients in total).

• HDRS is the Hamilton depression rating scale.

• Week represents the week (0, 1, 2, 3, 4 or 5) at which a measurement was made.

• ENDOG is dummy variable for the type of depression a patient was diagnosed with (1 for endogenous depression and 0 for non-endogenous depression).

• WxENDOG represents the interaction between Week and ENDOG, and is the product of Week and ENDOG.

**Mathematical Model**

A general two-level model for a continuous response variable depending on a set of predictors can be expressed as

where denotes the value of for the level-1 unit nested within the thelevel-2 unit for and , the scalar product is the fixed part of the model, and and denote the random part of the model at levels 2 and 1 respectively. For the fixed part of the model, is a typical row of a design matrix while the vector contains the fixed, but unknown parameters to be estimated. In the case of the random part of the model at level 2, represents a typical row of a design matrix , and the vector of random level-2 effects to be estimated. It is assumed that are independently and identically distributed (i.i.d.) with mean vector 0 and covariance matrix. Similarly, the are assumed i.i.d., with mean vector . The elements of are typically a subset of those.

**The random intercept and slope model**

The random intercept and slope model for the response variable HDRS may be expressed as

where denotes the average expected depression rating scale value, denotes the coefficient of the predictor variable Week (slope) in the fixed part of the modelHDRS value over patients and between patients respectively.

**The random intercept and slope with a covariate and an interaction model**

The random intercept and slope model for the response variable HDRS with the variable ENDOG as a covariate and with an interaction effect between Week ENDOG may be expressed as

where Week and ENDOG in the fixed part of the model denotes the coefficient of the interaction and ENDOG in the fixed part of the model, and denote the variation in the average expected value over patients and over measurements (i.e., between patients) respectively.

**Preparing the data**

The random intercept and slope model above is fitted to the data in reisby.ss3. The first step is to create the ss3 file shown above from the Excel file reisby.xls. This is accomplished as follows.

Use the Import Data File option on the File menu to load the Open dialog box.

Browse for the file reisby.xls in the Examples, Continuous folder.

Select the file and click on the Open button to open the following SuperMix spreadsheet window for reisby.ss3.

After selecting the File, Save option, we are ready to fit the random intercept and slope model for HDRS to the data in reisby.ss3.

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