SuperMix是由DonaldHedeker教授和RobertGibbons教授以及SSI共同開發的處理混合模型又稱階層模型的軟件。該軟件可以對二層和三層數據進行線性處理。SuperMix適用于對連續(continuous)因變量、二元(binary)因變量、計數(count)因變量、順序(ordinary)因變量和名義因變量進行模型分析。這一類分析被廣泛應用于社會學、醫藥領域、經濟學、工商業等各個領域。

SuperMix集合了四種混合效果程序的功能，MIXREG, MIXOR, MIXNO, and MIXPREG,由Donald Hedeker and Robert Gibbons開發成一種單獨應用，為混合效果的回歸模型提供估計?；旌闲ЧＪ揭脖环Q為多層的，分層的，或者隨機作用模型。這些模型可以用為縱向數據的分析，每個個體可能在不同數據場合測量。他們也能用于分類數據，比如為在診所的病人。

● 簡單直觀的窗口使復雜的模型設計變得簡便易行。

● 允許連續因變量(continuous outcome)混合模型中，因變量有自相關殘差(auto-correlated residuals)。

● 在順序因變量混合模型中可以包含scaling effects并進行non-proportional odds模型分析。

● 進行泊松分布混合模型分析。

● 名義因變量的 logistic regression模型分析。

● 分組時間序列的grouped-time survival混合模型分析。

● 二層和三層混合模型分析。

● 直觀的圖表。

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 lon-gitudinal 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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