1. THREE-LEVEL MODEL Two views The intractable statistical complexity that is occasioned by unduly ambitious three-level models (Bickel, 2007, 246) AND higher levels may have substantial effects, but without the guidance of well-developed theory or rich substantive literature, unproductive guesswork, data dredging and intractable statistical complications come to the fore (Bickel, 2007, 219) But technically, a three-level model is a straightforward development of 2-level model; substantively research problems are not confined to 2 levels!

2. THREE-LEVEL MODEL | Unit and classification diagrams, dataframes Some example of applied research Algebraic specification of 3 level random-intercepts model Various forms of the VPC Specifying models in MLwiN Residuals Applying the model - the repeated cross-sectional model; changing school performance Further levels: - as structures etc - in MLwin

3. Student Schoo l Class Student St1 St2 St3 St1 St2 St1 St2 St3 St1 St2 St3 St4 School Sc1 Sc2 Sc3 Class C1 C2 C1 C2 Student achievement affected by student characteristics, class characteristics and school characteristics Need more than 1 class per school; imbalance allowed Need lots of pupils in lots of classes in lots of schools! Three-level models Unit and classification diagrams

4. Data Frame for 3 level model Classifications or levels ResponseExplanatory variables NB categorical and continuous variables can be included at any level Student i Class j School k Current Exam score ijk Student previous Examination score ijk Student gender ijk Class teaching style jk School type k 1117556MFormalState 2117145MFormalState 3119172FFormalState 1216849FInformalState 2213736MInformalState 1126756MFormalPrivate 2128276FFormalPrivate 3128550FFormalPrivate 1135439MInformalState NB must be sorted correctly for MLwiN, recognises units by change in higher-level indices

5. Some examples (with references) West, B T et al (2007) Linear mixed models, Chapman and Hall, Boca Raton Dependent variable: students gain in Maths score, kindergarten to first grade Explanatory variables -1: Student (1190): Maths score in kindergarten, Sex, Minority, SES -2: Classroom (312) Teachers years of teaching experience, Teachers maths experience, teachers maths knowledge -3: School (107) % households in nhood of school in poverty NB lacks power to infer to specific classes/schools?

6. Some examples continued Bickel, R (2007) Multilevel analysis for applied research, Guildford Press, New York Dependent variable: Maths score for 8 th graders in Kentucky Explanatory variables -1: Student (50,000): Gender, Ethnicity, -2: Schools (347) School size, % of school students receiving free/reduced cost lunch -3: Districts (107) District school size

7. Some examples continued Ramano, E et al (2005) Multilevel correlates of childhood physical aggression and prosocial behaviour Journal of Abnormal Child Psychology, 33, 565-578 -individual, family and neighbourhood Wiggins, R et al (2002) Place and personal circumstances in a multilevel account of womens long-term illness Social Science & Medicine, 54, 827- 838 - Large scale study, 75k+ women in 9539 wards in 401 districts; used PCA to construct level-2 variables from census data

8. Algebraic specification of random intercepts model

9. Various forms of the VPC for random intercepts model

10. Correlation structure of 3 level model S 1112222333 C1121122112 P1231234123 1111 0000000 112 1 0000000 123 10000000 2110001 000 212000 1 000 223000 1 000 224000 1000 31100000001 3120000000 1 3230000000 1 Intra-class correlation (within same school & same class) Intra-school correlation (within same school, different class)

11. Example: pupils within classes within schools (Snijder & Bosker data)

12. Variance Partition Coefficients: pupils within classes within schools (Snijder & Bosker data)

13. Specifying models in MLwiN Three-level variance components for attainment

14. Specifying models in MLwiN Are there classes and/or schools where the gender gap is large, small or inverse? Student gender in fixed part and Variance functions at each level Level 2 variance Level 3 variance Level 1 variance

15. Specifying models in MLwiN Is the Gender gap differential according to teaching style? Cross-level interactions between Gender and Teaching style in the fixed part of the model IE main effects for gender & style, and first order interaction between Student Gender and Class Teaching Style Fixed part Cons: mean score for Male in Formally-taught class Female: differential for female in formal class Informal: differential for male in informal class Female*Informal: differential for female in informal class

16. Residuals Key notion is that highest level residual is a random, allowed-to-vary departure from general relationship Each lower level residual is allowed-to-vary random departure from the higher-level departure

17. Level 3 residuals: school departures from grand mean line

18. Level 2 residuals: class departures from the associated school line

19. Level-1 residuals: student departures from the associated class line

20. Applying the model: the repeated cross- sectional model; changing school performance Sc1 Sc2 Sc3.... 1985 1986 1987 1985 1987 1986 1987 St1…St9 St1… St25 St1 …St32 St1… St22 St1… St12 St1… St29 St1… St13 School Student Cohort Modelling Exam scores for groups of students who entered school in 1985 and a further group who entered in 1986. In a multilevel sense we do not have 2 cohort units but 2S cohort units where S is the number of schools. The model can be extended to handle an arbitrary number of cohorts with imbalance

21. Applying the model: the repeated cross- sectional model; changing school performance Modelling Exam scores aged 16 for Level 3 139 state schools from the Inner London Education Authority, Level 2 304 cohorts with a maximum of 3 cohorts in any one school, and Level 1 115,347 pupils with a maximum of 135 pupils in any one school cohort pupil level variables: Sex, Ethnicity, Verbal Reasoning aged 11 cohort-level variables: % of pupils in each school who were receiving Free-school meals in that year, % of pupils in the highest VRband in that year, the year that the cohort graduated school level variables: the sex of the school (Mixed Boys and Girls); the schools religious denomination (Non-denominational, CofE, Catholic)

22. Further levels - as structures, etc Some examples of 4-level nested structures: student within class within school within LEA people within households within postcode sectors within regions Finally, Repeated measures within students within cohorts within schools O1 O2 O1 O2 O1 O2 O1 O2 Cohorts are now repeated measures on schools and tell us about stability of school effects over time Measurement occasions are repeated measures on students and can tell us about students learning trajectories. 1990 1991 1990 1991 Sc1 Sc2... Cohort Msmnt occ student School St1 St2... St1 St2.. St1 St2.. St1 St2..

23. Further levels - in MLwiN Click on extra subscripts! Default is a maximum of 5 but can be increased