Multilevel structures and classifications

This document aims to give a pictionary of basic structures and classifications that underlie multilevel models. We give pictures of common structures, as unit diagrams as classification diagrams, and as words. Note that term classification and level can be used somewhat interchangeably but level implies a nested relationship between sets of units where as classification does not.

You need to decide what are the levels in your data set. Have a look through the sets of examples (1-6) in section A, as you progress through the examples the structures get more complex. Most people will probably have structures similar to one of examples structures given example sets (1 or 2). Then read section B - what is a level? From this you should be able to decide if you have an appropriate data set for multilevel modelling.

Section A: Example Data Structures

Example set 1: Two level nested

For example

Unit diagram (nodes in diagram are population units)

unit diagram

The same structure as Classification diagram (nodes in diagram are classifications, arrow indicates nested relationship)

pupil-school

Other examples of a two level nested structures:

Classification diagram      Unit Diagram

classification-unit

Note that with multilevel repeated measures models individuals can have different numbers of measurement occasions and they can be measured at different times/ages.

subject-pupil    multivar-respons

Example Set 2 Three level Nested

For example

Unit diagram

unit diagram

Classification diagram

pupil-school-area

Other examples of a 3 level nested structure

pupil-cohort-school

Here the repeated measures are made on the schools. In this case we have scores for groups of pupils who entered school in 1990 and a further group who entered in 1991. The model can be extended to handle an arbitrary number of cohorts. Again the data need not be balanced : every school does not have to have children from both cohorts, within school cohort groups can be different sizes and total number of pupils per school can differ from school to school.

Example set 3 - Four level Nested

Classification diagram      Unit Diagram

pupil-class-school-LEA

Other examples of four level nested

msmnt occ

Cohorts are now repeated measures on schools and tell us about stability of school effects over time. Measurement occasions are repeated measures on pupils and can tell us about pupils' learning trajectories. To clarify this consider extending the above diagram to have 4 cohorts pupils starting secondary school in (1990, 1991, 1992, 1993) and children within cohorts are measured every year between the ages of 12 and 16.

Cohort arrow   time    arrow
1990 1991 1992 1993 1994 1995 1996 1997
1990 12 13 14 15 16      
1991   12 13 14 15 16    
1992     12 13 14 15 16  
1993       12 13 14 15 16

Here the black numbers (12..16) are the ages of kids from a cohort at a particular time.

Example set 4 : Non-Nested Models

Unit diagram where pupils lie within a cross-classification of school by area.

school-pupils-area

In this structure schools are not nested within areas. For example

Classification diagram

arrow from pupil to school and from pupil to area

Pupil is nested within school and pupils are nested within areas, however school and area are cross-classified so the nodes are not connected.

Other examples of two-way cross-classifications:

patient clinician 1 clinician 2 clinician 3 clinician 4
patient 1 m1,m2     m3
patient 2 m2 m1    
patient 3 m3   m1 m2
patient 4   m1,m2 m3  

So patient 1 is seen on measurement occasions 1 and 2(m1 and m2) by clinician 1 and by clinician 4 on measurement occasion 3(m3) and so on.

Example set 5 Multiple Membership Models

imiage showing 3 areas at bottom, 12 pupils in centre and four areas at top with lines showing links between area and pupils, also pupils and schools

Lets take the structure of pupils within a cross-classification of school by area. But now suppose pupil 1 moves in the course of the study from residential area 1 to 2 a nd from school 1 to 2.

Now in addition to schools being crossed with residential areas pupils are multiple members of both areas and schools.

Classification diagram (double arrow specifies multiple membership relationship)

double arrows from pupil to school and from pupil to area

Other examples of multiple membership relationships

patient at bottom, two arrows up to nurse

Example set 6

Complex population structures containing mixtures of hierarchies, crossings and multiple membership relationships.

Taking the example from section 11 of the workshop notes

In Denmark Child flocks (10,127) of chickens are reared in houses (725) within farms (304). The child flocks are slaughtered and then another flock of hatched chicks enter the house.

The hatched chicks come from parent flocks (200) who are bred to produce good quality stock. A child flock is made up of chicks from multiple parent flocks.

complex 4-level diagram showing 5 parent flocks at bottom, 12 child flocks, 4 houses and two farms at the top

As a classification diagram:

child flock at bottom, one arrow to house, 2 arrows to parent flock, and one arrow from house to farm

Section B: What is a level?

The examples in section A have individuals (eg students, doctors, animals), institutions (eg schools, hospitals), area and time (measurement occasion or cohort) as levels. These are some of the more common levels in a multilevel structure.

The key thing that defines a variable as being a level is that its units can be regarded as a random sample from a wider population of units. For example, if we have a multilevel data structure of students within schools, the students are a random sample from a wider population of students likewise the schools are a random sample from a wider population of schools.

Variables, like gender and social class are not levels. This is because they have a small fixed number of categories. For example, gender has only two categories male and female, there is no wider population of gender categories that male and female are a random sample from.

Generally multilevel models are useful for exploring how relationships vary across higher-level units, for example schools.

With nested relationships the higher the level the fewer the number of units at that level.

We number levels from 1, being the lowest, upwards.

In practice to do multilevel analysis you need to have at least 20 higher-level units. Note that in example 2.4 in section A of pupils: cohorts: schools, if we have 2 cohort groups per school and 100 schools we do not have 2 units at the cohort group level (level 2) we have 2*100 = 200 cohort groups.

To determine if your data sets is suitable for multilevel analysis firstly identify the levels in the multilevel structure and then check you have at least 20 units at higher levels.

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