# Module 5: Introduction to Multilevel Modelling SPSS Practicals Chris Charlton 1 Centre for Multilevel Modelling

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1 Module 5: Introduction to Multilevel Modelling SPSS Practicals Chris Charlton 1 Centre for Multilevel Modelling Pre-requisites Modules 1-4 Contents P5.1 Comparing Groups using Multilevel Modelling... 4 P5.1.1 A multilevel model of attainment with school effects... 4 P5.1.2 Examining school effects (residuals)... 8 P5.2 Adding Student-level Explanatory Variables: Random Intercept Models P5.3 Allowing for Different Slopes across Schools: Random Slope Models P5.3.1 Testing for random slopes P5.3.2 Interpretation of random cohort effects across schools P5.3.3 Examining intercept and slope residuals for schools P5.3.4 Between-school variance as a function of cohort P5.3.5 Adding a random coefficient for gender (dichotomous x ) P5.3.6 Adding a random coefficient for social class (categorical x) P5.4 Adding Level 2 Explanatory Variables P5.4.1 Contextual effects P5.4.2 Cross-level interactions P5.5 Complex Level 1 Variation P5.5.1 Within-school variance as a function of cohort (continuous X) P5.5.2 Within-school variance as a function of gender (dichotomous X) P5.5.3 Within-school variance as a function of cohort and gender P5.6 Appendices P5.6.1 Appendix P5.6.2 Appendix This SPSS practical is adapted from the corresponding MLwiN practical: Steele, F. (2008) Module 5: Introduction to Multilevel Modelling. LEMMA VLE, Centre for Multilevel Modelling. Accessed at Centre for Multilevel Modelling

2 Some of the sections within this module have online quizzes for you to test your understanding. To find the quizzes: EXAMPLE From within the LEMMA learning environment Go down to the section for Module 5: Introduction to Multilevel Modelling Click "5.1 Comparing Groups Using Multilevel Modelling" to open Lesson 5.1 Click Q 1 to open the first question Introduction to the Scottish Youth Cohort Trends Dataset You will be analysing data from the Scottish School Leavers Survey (SSLS), a nationally representative survey of young people. We use data from seven cohorts of young people collected in the first sweep of the study, carried out at the end of the final year of compulsory schooling (aged 16-17) when most sample members had taken Standard grades 2. In the practical for Module 3 on multiple regression, we considered the predictors of attainment in Standard grades (subject-based examinations, typically taken in up to eight subjects). In this practical, we extend the (previously single-level) multiple regression analysis to allow for dependency of exam scores within schools and to examine the extent of between-school variation in attainment. We also consider the effects on attainment of several school-level predictors. The dependent variable is a total attainment score. Each subject is graded on a scale from 1 (highest) to 7 (lowest) and, after recoding so that a high numeric value denotes a high grade, the total is taken across subjects. The analysis dataset contains the student-level variables considered in Module 3 together with a school identifier and three school-level variables: Variable name CASEID SCHOOLID SCORE COHORT90 Description and codes Anonymised student identifier Anonymised school identifier Point score calculated from awards in Standard grades taken at age 16. Scores range from 0 to 75, with a higher score indicating a higher attainment The sample includes the following cohorts: 1984, 1986, 1988, 1990, 1996 and The 2 We are grateful to Linda Croxford (Centre for Educational Sociology, University of Edinburgh) for providing us with these data. The dataset was constructed as part of an ESRC-funded project on Education and Youth Transitions in England, Wales and Scotland Further analyses of the data can be found in Croxford, L. and Raffe, D. (2006) Education Markets and Social Class Inequality: A Comparison of Trends in England, Scotland and Wales. In R. Teese (Ed.) Inequality Revisited. Berlin: Springer. Centre for Multilevel Modelling

3 FEMALE SCLASS SCHTYPE SCHURBAN SCHDENOM COHORT90 variable is calculated by subtracting 1990 from each value. Thus values range from - 6 (corresponding to 1984) to 8 (1998), with 1990 coded as zero Sex of student (1=female, 0=male) Social class, defined as the higher class of mother or father (1=managerial and professional, 2=intermediate, 3=working, 4=unclassified). School type, distinguishing independent schools from state-funded schools (1=independent, 0=state-funded) Urban-rural classification of school (1=urban, 0=town or rural) School denomination (1=Roman Catholic, 0=non-denominational) There are students in 508 schools. Open the worksheet to From within the LEMMA Learning Environment Go to Module 5: Introduction to Multilevel Modelling, and scroll down to SPSS Datafiles Click 5.1.sav You will see the Data Editor Window, after switching to Variable View: Centre for Multilevel Modelling

4 P5.1 Comparing Groups using Multilevel Modelling P5.1.1 A multilevel model of attainment with school effects We will start with the simplest multilevel model which allows for school effects on attainment, but without explanatory variables. This null model may be written (5.1) where is the attainment of student i in school j, β 0 is the overall mean across schools, is the effect of school j on attainment, and is a student level residual. The school effects, which we will also refer to as school (or level 2) residuals, are assumed to follow a normal distribution with mean zero and variance. To run this model in SPSS we will use the MIXED command. Immediately after MIXED there is the response variable. The /FIXED option specifies the variables to include in the fixed part, in this case this is empty as the intercept is automatically included, and there are no other predictors in the fixed part. The /METHOD option allows selection of the estimation method, in this case maximum likelihood. /PRINT=SOLUTION requests that the parameter estimates are displayed after fitting. The /RANDOM option specifies which variables are included in the random part, as well as specifying the variable that defines the grouping (here, schools). Finally the /SAVE option specifies that we want to save the fixed-part prediction (FIXPRED), the prediction including random effects (PRED) and the standard errors of the prediction (SEPRED) back into the data set. Note that unlike software such as MLwiN the data does not have to be sorted a specific way in order to fit the model. MIXED score /FIXED= /METHOD=ML /PRINT=SOLUTION /RANDOM=INTERCEPT SUBJECT(schoolid) /SAVE=FIXPRED PRED SEPRED. Alternatively: Analyze>Mixed Models>Linear Add schoolid to Subjects Click Continue Add score to Dependent Variable Click Random Tick Include intercept Add schoolid to Combinations Click Continue Click Estimation Tick Maximum Likelihood (ML) Click Continue Centre for Multilevel Modelling

5 Click Statistics Tick Parameter estimates Click Continue Click Save Tick Predicted values under Fixed Predicted Values Tick Predicted values under Predicted Values & Residuals Tick Standard errors under Predicted Values & Residuals Click Continue Click OK We will see the following tables of results (the Model Dimension and Type III Tests of Fixed Effects tables are not of interest for this analysis, so we will omit them from subsequent results): Model Dimension a Number of Covariance Number of Subject Levels Structure Parameters Variables Fixed Effects Intercept 1 1 Random Effects Intercept 1 Variance 1 schoolid Components Residual 1 Total 2 3 Information Criteria a -2 Log Likelihood Akaike's Information Criterion (AIC) Hurvich and Tsai's Criterion (AICC) Bozdogan's Criterion (CAIC) Schwarz's Bayesian Criterion (BIC) The information criteria are displayed in smaller-is-better forms. Type III Tests of Fixed Effects a Source Numerator df Denominator df F Sig. Centre for Multilevel Modelling

6 Intercept Estimates of Fixed Effects a 95% Confidence Interval Parameter Estimate Std. Error df t Sig. Lower Bound Upper Bound Intercept Estimates of Covariance Parameters a Parameter Estimate Std. Error Residual Intercept [subject = Variance schoolid] The overall mean attainment (across schools) is estimated as The mean for school j is estimated as , where is the school residual which we will estimate in a moment. A school with > 0 has a mean that is higher than average, while < 0 for a below-average school. (We will obtain confidence intervals for residuals to determine whether differences from the overall mean can be considered real or due to chance.) Partitioning variance The between-school (level 2) variance in attainment is estimated as = 61.02, and the within-school between-student (level 1) variance (labelled Residual in the output) is estimated as = Thus the total variance is = The variance partition coefficient (VPC) is 61.02/ = 0.19, which indicates that 19% of the variance in attainment can be attributed to differences between schools. Note, however, that we have not accounted for intake ability (measured by exams taken on entry to secondary school) so the school effects are not valueadded. Previous studies have found that between-school variance in progress, i.e. after accounting for intake attainment, is typically close to 10%. Testing for school effects Centre for Multilevel Modelling

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