Info

and the vector of ones and zeroes becomes the level-1 explanatory variable for the GLS estimation, in this case providing the coefficient that is the estimator of a2. Similarly, for a model where there is a single variance term at level 2, the level-2 component V2j written as a lower triangle vector is

Goldstein [13] shows that this procedure produces maximum likelihood estimates under Normality.

5. THE MLwiN SOFTWARE

MLwiN has been under development since the late 1980s, first as a command-driven DOS based program, MLn, and since 1998 in a fully-fledged windows version, currently in release 1.10. It is produced by the Multilevel Models Project based within the Institute of Education, University of London, and supported largely by project funds from the U.K. Economic and Social Research Council. The software has been developed alongside advances in methodology and with the preparation of manuals and other training materials.

Procedures for fitting multilevel models are now available in several major software packages such as STATA, SAS and S-plus. In addition there are some special purpose packages, which are tailored to particular kinds of data or models. MIXOR provides ML estimation for multi-category responses and HLM is used widely for educational data. See Zhou et al. [14] for a recent review and Sullivan et al. [3] for a description of the use of HLM and SAS. Many of the models discussed here can also be fitted readily in the general purpose MCMC software package WinBUGS [15].

MLwiN has some particular advanced features that are not available in other packages and it also has a user interface designed for fully interactive use. In later sections we will illustrate some of the special features and models available in MLwiN but first give a simple illustration of the user interface. We shall assume that the user wishes to fit the simple two-level model given by (1).

In this tutorial we cannot describe all the features of MLwiN, but it does have general facilities for data editing, graphing, tabulation and simple statistical summaries, all of which can be accessed through drop-down menus. In addition it has a macro language, which can be used, for example, to run simulations or to carry out special purpose modelling. One of the main features is the method MLwiN uses to set up a model, via an 'equation window' in which the user specifies a model in more or less exactly the format it is usually written. Thus to specify model (1) the user would first open the equation window which, prior to any model being specified, would be as shown in Figure 1.

This is the default null model with a response that is Normal with fixed predictor represented by Xfi and covariance matrix represented by fi. Clicking on the N symbol delivers a drop

Figure 1. Default equation screen with model unspecified.
Figure 2. Equation screen with model display.

down menu, which allows the user to change the default distribution to binomial, Poisson or negative binomial. Clicking on the response y allows the user to identify the response variable from a list and also the number and identification for the hierarchical levels. Clicking on the x0 term allows this to be selected from a list and also whether its coefficient is random at particular levels of the data hierarchy. Adding a further predictor term is also a simple matter of clicking an 'add term' button and selecting a variable. There are simple procedures for specifying general interaction terms.

Model (1), including a random coefficient for x1 in its general form as given by (3), will be displayed in the equation window as shown in Figure 2. Clicking on the 'Estimates' button will toggle the parameters between their symbolic representations and the actual estimates after a run. Likewise, the 'Name' button will toggle actual variable names on and off. The 'Subs' button allows the user to specify the form of subscripts, for example giving them names such as in the screen shown in Figure 3, where we also show the estimates and standard errors from an iterative fit.

In the following sections we will show some further screen shots of models and results.

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