1 Department of Psychology and Human Development Vanderbilt University GCM, 2010
2 1 Multilevel Modeling - A Brief Introduction
3 Introduction In this lecture, we introduce the multilevel model for change. We begin with a quick review of multilevel models, and then specialize to the multilevel growth curve model.
4 In many situations in data analysis, the data are nested, which often implies a hierarchical structure. For example: soldiers are nested within platoons, which are nested within companies, etc. Students are nested within classes, which are nested within schools, which are nested within school districts, etc. Of special interest in our current application is how, in longitudinal data analysis, observations can be nested within individuals.
5 In his introductory text, Goldstein (1999) considers a sample of students from the Junior School Project (JSP) data. There were over 1000 students nested within 50 schools Each student took the Raven progressive matrices test (an ability measure) and also was assessed several times in English and Mathematics achievement. Suppose we consider, for the time being, just the relationship between the Raven test and the first test result in Mathematics.
6 Notational Choices There are many ways we can write a multilevel model. We ll look at several here. We ll start with a classic two-level approach used by Raudenbush and Bryk. Keep in mind that you have to process the ideas. The notational variations (use of different letters, etc.) that you will see in various books and articles are virtually endless.
7 The Level-1 Model At Level 1, we can analyze the relationship between the Raven test and Math Achievement within each school. In this case, if we fit a simple linear regression line to the data, the model can be written for the ith student in the j th school as Y ij = β 0j + β 1j X ij + r ij (1) The r ij are regression residuals and are assumed to be independently and identically normally distributed with a mean of 0 and a variance of σ 2.
8 The Level-2 Model Notice how each school has its own slope and intercept. The slope and intercept define the regression line. These slopes and intercepts can be viewed as arising through random normal variation around central value. Specifically, the Level-2 model is β 0j = γ 00 + u 0j (2) β 1j = γ 10 + u 1j (3) The β 0 s and β 1 s are allowed to correlate, so they have a covariance matrix T that is not necessarily diagonal.
9 The Combined Mixed Model By simply substituting the Level-2 model into the Level-1 expression, we get the combined (mixed) model which contains both fixed (the γs) and random (the βs) effects. We get Y ij = γ 00 + u 0j + (γ 10 + u 1j )X ij + r ij = γ 00 + γ 10 X ij + (u 0j + u 1j X ij + r ij ) (4) = γ 00 + γ 10 X ij + r ij (5) = γ 00 + γ 10 X ij + (u 0j + u 1j X ij ) + r ij (6) The above multilevel model is a special case of the linear mixed effects model. The γs are fixed effects, and the us are random effects, i.e., random variables having a distribution.
10 The Combined Model Matrix Notation As I already mentioned, we can express the above model in a wide variety of notations. Here is one matrix notation. First, let s simply convert the scalar notation directly into matrix notation using the same lettering. Let y j represent the vector of scores of the individuals in group j. Then y j = X j γ + X j u j + r j (7) Note that, in the above, the scores in X j can be either design codes or actual scores on observed variables, because of the longstanding tradition in regression modeling of treating observed scores on predictors as fixed regardless of whether they are actually fixed scores or random variables.
11 The Combined Model Matrix Notation Suppose we have just two schools and there are 3 students in school 1 and 4 students in school 2. The notation corresponds to the data as follows: [ ] y 1 y 2 Y 11 Y 21 Y 31 Y 12 Y 22 Y 32 Y 42 = = [ X 1 X 2 ] γ + 1 X 11 1 X 21 1 X 31 1 X 12 1 X 22 1 X 32 1 X 42 [ X 1 0 [ γ00 0 X 2 γ 10 ] + ] [ u 1 u 2 ] [ ] r + 1 r 2 1 X X X X X X X 42 u 01 u 11 u 02 u 12 + r 11 r 21 r 31 r 12 r 22 r 32 r 42
12 The General Linear Mixed-Effects Model The preceding can be seen (with some minor changes in lettering) to be a special case of the General Linear Mixed-Effects Model. Pinheiro and Bates (2000) in their well-known book, Mixed-Effects Models in S and S-Plus use the following notation for the GLMM for a single level of grouping, i.e., what Raudenbush and Bryk refer to as a two-level model. y i = X i β + Z i b i + ɛ i (8) b 1 N (0, Ψ), ɛ i N (0, σ 2 Λ i ) (9) where X i is the fixed effects regressor matrix for the ith unit, and Z i is the random effects regressor matrix, which usually contains a subset (perhaps all) of the columns of X i. The vector β contains fixed effects, while b i contains the random effects.
13 Fitting the Two-Level Model The two level model with random slopes and intercepts can be fit with two rather different programs. One program, R, is free, and very general. The other, HLM, is a well-known commercial program designed to fit hierarchical linear models. R requires that you compute the full mixed model, as shown in Equation 6. HLM allows you to input the Level-1 and Level-2 models separately, and declare whether the effects for each are fixed or random. To get a feel for how they work, we ll try both programs on the high school data.
14 To fit LME models in R, we need to use the lmer function in the lme4 package. We begin by loading in the data file, which is in long format. We ll convert it to wide format and remove missing data. > library(foreign) > jsp.long <- read.table("jsp.csv", header=true, + sep = ",") > jsp.wide <- reshape(data=jsp.long,direction="wide", + timevar="year",idvar="pupil") > jsp.wide <- na.omit(jsp.wide) > write.table(jsp.wide,"jsp.wide.txt",row.names=f, + col.names=t,sep=",")
15 Then, we enter the model, using the appropriate syntax. > full.model <- lmer(maths.0 ~ ravens (1 + ravens.0 school.0),data=jsp.wide) > full.model Linear mixed model fit by REML Formula: maths.0 ~ ravens.0 + (1 + ravens.0 school.0) Data: jsp.wide AIC BIC loglik deviance REMLdev Random effects: Groups Name Variance Std.Dev. Corr school.0 (Intercept) ravens Residual Number of obs: 887, groups: school.0, 48 Fixed effects: Estimate Std. Error t value (Intercept) ravens Correlation of Fixed Effects: (Intr) ravens
16 With HLM, rather than fitting the mixed model directly, we input the Level-1 and Level-2 models separately. This is a rather straightforward (if somewhat idiosyncratic) process.
17 HLM is a popular software program that makes construction of basic multilevel models relatively straightforward. In particular, it does not require combination of models from two or more levels into a single regression model. Consequently, many find it very convenient and (relatively) easy to use, which has contributed to its popularity. In this introduction, we will revisit the model that we examined, and set it up and analyze it in HLM. We assume that you have the HLM6 program (full or student version) installed on your computer.
18 and Input HLM has limited (and somewhat disguised) data input capabilities. In practice, you will probably input most of your data as either SPSS.sav files (if you have the Student Version), or comma-delimited ASCII files with a header containing column names (if you have the full version). Since R writes ascii files routinely using the write.table() function (and the sep = ',' option), and also has extensive data manipulation capabilities, you may find it convenient to use R to construct your HLM files.
19 and Input The link between the level-1 and level-2 models in the HLM parameterization is the subscript j, which refers in the current example to the school.0 variable. To set up the data for HLM, we need two files, one for the level-1 variables, one for the level-2 variables. Each file has to be sorted in ascending order of the ID variable. We need to include ravens.0, math.0, and school.0 in the level-1 file, and school.0 in the level-2 file. Since all the variables we need are in one file, we are ready to go.
20 Setting up the Multivariate Data Matrix (MDM) file is a key first step to using HLM2 to analyze a 2-level model problem. Make sure you have the file jsp.wide.txt available, or, if you are using the Student Version, the SPSS file jsp.wide.sav. We will show the procedure for the SPSS file. Begin by starting up HLM. Then click on the Make New MDM File -> Stat Package Input menu option, whether you are loading an SPSS file or a comma-delimited text file! (This is counterintuitive and very poor human factors design. One would expect this to be found under the ASCII file input node.)
21 Next, you will be asked to select a program. Select HLM2. Then click on the OK button.
22 A large dialog box will open. Go to the drop-down list for file type, and select SPSS/Windows.
23 If you are not already there, go to the directory where the data files are. Select jsp.wide.sav and click on the Open button.
24 Look in the dialog box for the grouping that is titled Level-1 Specification. I ve highlighted the group in red in the picture below. On the right side of that grouping is a button Choose Variables. Click on it.
25 A dialog box will open up that will allow you to select and classify level-1 variables. The variable that spans the two levels of your model is SCHOOL.0, and this variable is designated an ID variable. The variables MATHS.0 and RAVENS.0 are in the level-1 model, so they are checked off as being in the MDM. When you are ready to exit the dialog, it should look like this:
26 Next, go to the Level-2 Specification group and first, click on the Browse button. Again select the file jsp.wide.sav. Second, click on the Choose Variables button.
27 This will take you to another variable selection dialog. Again school is the ID variable, and we actually have no level-2 predictors in this model. However, HLM requires you to select at least one additional variable, so choose the same variables as for the level-1 file.
28 At this point, the HLM program again exhibits poor human factors design. The necessary next step is to save a mdmt template file. However in order to do that, you have to enter the name of the mdm file you want to save! You enter jsp.mdm in the appropriate edit field, then click on Save mdmt file.
29 Once you ve done saved the mdmt file, you can make the MDM file by clicking on the make MDM file at the bottom left.
30 At this point, you are strongly advised to Examine the basic statistics for the MDM file you have just created. You do this by clicking on the Check Stats button as shown below.
31 Examine the statistics, see if you have chosen the correct variables, and check whether the descriptive statistics make sense. Then click Done
32 Level-1 Specification The next step is to specify the model. Note that the Level-1 Button is highlighted.
33 Level-1 Specification our first step is to choose the Level-1 outcome variable. Click on MATHS.0 and a flying menu will open. Choose Outcome variable. You have now selected your outcome variable.
34 Next, you will see a window open up. This window will contain the current model specification in HLM notation. Note that a baseline Level-2 model has already been specified. Normally, you would next enter the Level-2 specification, but in this case, we are actually finished.
35 s The next step is to specify the characteristics of the outcome variable. Click on the Outcome button.
36 You ll see the dialog pictured below. Because we re assuming a normally distributed outcome, you don t have to do anything, although, if you wish, you could save residual files for analysis by another program. Just click Ok. Note: if you don t do this, your model will not be specified! Most modern software assumes a default (in this case a normal outcome variable) but HLM does not.
37 Adding a Level-1 Predictor In this case, we wish to predict MATHS.0 from RAVENS.0 at level 1. So we click on the variable name and add it, uncentered. Notice how the model is changed automatically to reflect the new predictor.
38 Adding a Level-2 Random Term The level-2 model includes random slopes and intercepts. Notice how the random effect u 1j is greyed out in the level-2 model. Click on it and it is immediately added to the level-2 model.
39 If you wish, HLM will automatically combine the two models into a single mixed model, which might be especially useful if you wish to use another program (like R) to analyze the model. Simply click on the Mixed button in the lower right corner of the main window. In this case we can see that this model is indeed the same as what we had before.
40 Analyzing the Model Click on the Run Analysis button. HLM will open a DOS window and the model will run. There will be a brief pause near the end of calculations. Don t interrupt! The window will eventually shut.
41 Viewing Output The output window will not open automatically. You need to select the File >View Output menu option.
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