# 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 | |

/ PRINT = G SOLUTION | |

/ 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 | |

/ PRINT = G SOLUTION | |

/ RANDOM = intercept | SUBJECT(schoolid). |

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

MIXED math_sco WITH homew | |

/ FIXED = homew | |

/ PRINT = G SOLUTION | |

/ 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.)

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).

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