The user is prompted to specify the formats of the level 1 and level 2 data files. The only requirement is that the level 2 identifier (appearing in the first columns of both the level 2 and level 2 data files) must be input as a character variable. The format statements for the level 1 and level 2 sample files shown above are (a2, 2x, f2.0, 2x, f2.0) and (a2, 2x, fl.O), respectively (where 'a2' indicates a 2-digit character variable, '2x' denotes 2 blank spaces between columns and 'f2.0' denotes a numeric variable occupying at most two columns with no digits following the decimal place). The user is also prompted for variable names and file locations.
Once the format of the data, variable names and the location of the ASCII data files are input into HLM/2L, an SSM file is constructed. The SSM file contains J matrices (one for each level 2 unit). Each matrix contains the number of level 1 units (tij), the means of the level 1 variables, the sums of cross products of level 1 variables as well as the level 2 observations. Once the SSM file is constructed, subsequent run starts with the SSM file and proceed quickly. HLM/2L can handle up to 300 level 2 units (J < 300), 25 level 2 (for example, physician) variables, and 25 level 1 (for example, patient) variables.
We describe the use of HLM/2L in interactive mode, as the logic of the modelling process is apparent to the user in this mode. (In batch mode, commands are stacked in a file and input. Batch mode can be more efficient for the more advanced user.) Once the program is initiated, the user is prompted to specify the name of the SSM file. The user is then prompted to choose the dependent or outcome variable, from the list of level 1 variables displayed on the screen. Next, the user is prompted to choose level 1 predictors or covariates from the list of level 1 variables displayed on the screen. Once the level 1 predictors are selected, the user is prompted to specify variables to be centred. HLM/2L offers three options: (i) centring about the grand mean (X..); (ii) centring about the group (level 2) means (X.j); or (iii) no centring. The user is then prompted to select level 2 predictors or covariates and offered two options for centring: (i) centring at the grand mean of the level 2 variable (W); or (ii) no centring. The user is also offered options related to specific tests of hypothesis. These include, for example a test for homogeneity of level 1 variances and multivariate tests of covariance components. There are a number of other options available in the HLM/2L execution (see Bryk et a/.1).
HLM/2L generates extensive output, including the specifications of the level 1 and level 2 models using the notation in Section 3 which is extremely useful for model checking. HLM/2L outputs the starting values for parameter estimates, estimates of fixed effects along with i-statistics and significance levels for hypothesis testing, estimates of the covariance components along with chi-square statistics and significance levels. The user can request that empirical Bayes estimates of the individual random effects be output into an ASCII file, a SAS file or a SYSTAT file for further analysis.
Proc Mixed is a SAS procedure designed to handle mixed effects linear models. Detailed documentation is available in SAS/STAT® Software: Changes and Enhancements through Release 6.119 and in Littell et al.2
Data can be input into SAS from ASCII files or from SAS data sets. Data for SAS Proc Mixed must reside in a single file containing one record for each level 1 unit. Both level 1 variables (for example, the patient-level outcome, Y, for example, satisfaction, and patient covariate age, X) and level 2 variables (for example, physician's years in medical practice, W) are contained in the same file. The following illustrates the proper format of the data file, using the sample files shown in Section 5.1.1. The level 2 identifier appears in the first column and is indicated in bold face. The level 2 variable, W, appears in the rightmost column. Data file (total number of records =
The SAS Proc Mixed syntax is similar to the SAS Proc GLM syntax. The user specifies classification variables in the 'class' statement and the outcome or dependent variable for the analysis along with the fixed effects in the 'model' statement. The user specifies random effects in a 'random' statement and may include statements to perform repeated measures or time series analysis. SAS allows the user to specify a stratification variable in the 'by' statement. Specific tests of hypothesis for fixed or random effects can be requested using 'contrast' and 'estimate' statements. The user can specify initial or starting estimates for model parameters in the 'parms' statement. The user can also request that any part of the output be saved in a SAS data set using the 'make' option. (We provide the SAS code used in our analyses in the tables of results and in the Appendix.)
SAS prints the output and also allows the user to save all or part of the output in a file for presentation. The output includes a description of the classification variables (for example,
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