MIXED: Multilevel Modeling

As of version 11.0, SPSS can estimate hierarchical or multilevel models. Such models refer to data about individuals in contexts, such as pupils from several classes (and perhaps classes from several schools). Thus, individual data are correlated (as pupils from the same class and/or school are subject to the same influences), and in addition we may be interested in "higher level" effects, such as class or school characteristics, on individual variables (such as school achievement). Procedure MIXED can handle such data.

A null model (model with intercept only):

MIXED math_sco                                
  / RANDOM = intercept | SUBJECT(schoolid).

math_sco is the dependent variable, and schoolid is the variable that denotes the school to which each pupil belongs.

Example with a fixed-effect coviarate :

MIXED math_sco WITH homew
  / FIXED = homew
  / RANDOM = intercept | SUBJECT(schoolid).

Example with a random coefficient for the covariate (random slope model):

MIXED math_sco WITH homew
  / FIXED = homew
  / RANDOM = intercept homew | SUBJECT(schoolid).

Note that of course you can add many more variables, including context variables. Also, you may estimate models with repeated measurements from the same subjects. Such data exhibit a similar structure (several measurements from one subject are correlated due to traits that are exhibited in all single measurements). I cannot go into detail here; perhaps I will get back to this issue at some later time.

I will explain briefly the output from the last model. You will find that I use a German version of SPSS (small wonder given that I work at a German university).

The first table (after some introductory basic information) gives some "goodness of fit" measures. The value that is exhibited first is sometimes referred to as the "deviance".

Eingeschränkte -2 Log Likelihood 3657.197
Akaike-Informationskriterium (AIC) 3663.197
Hurvich und Tsai (IC) 3663.244
Bozdogan-Kriterium (CAIC) 3678.941
Bayes-Kriterium von Schwarz (BIC) 3675.941

The next table, in English "fixed parameter estimates", displays the estimates for the coefficients, their standard errors, the degrees of freedom, the t-values, the significance levels (rounded, there is no .000 level; ".000" actually means "less than .0005") and the lower and upper boundaries of the confidence interval.

(Sorry, I don't know how this dashed empty space arises. If someone finds out, please let me know.)

Schätzungen fester Parameter

Schätzung Standardfehler Freiheitsgrade T-Statistik Signifikanz Konfidenzintervall 95%
Parameter Untergrenze Obergrenze
Intercept 46.4644111 1.6089580 21.824 28.879 .000 43.1260796 49.8027427
HOMEW 1.9745172 .8314629 19.211 2.375 .028 .2355357 3.7134988

The third table presented here displays the random effects and their standard errors. The first line, "residual", contains the Level 1 variance, the next lines give the Level 2 variance of the intercept and the slope(s).

Schätzungen von Kovarianzparametern
Parameter Schätzung Std.-Fehler
Residuum 53.9433220 3.5539885
Intercept [Subjekt = SCHOOLID] ID: Diagonal 50.7356570 17.8784872
HOMEW [Subjekt = SCHOOLID] ID: Diagonal 13.7586332 5.0798734

Note: The example is taken from Kreft, I. & de Leeuw, Jan: Introducing Multilevel Modeling, London: Sage, 1998, p. 66-7. The estimates obtained from SPSS differ slightly (but not substantially) from those from the book, especially with respect to the variance components.

© W. Ludwig-Mayerhofer, IGSW | Last update: 21 Mar 2003