1. Cross-classified and Multiple membership models
Lecture 21
Cross-classified and Multiple membership models

2. Lecture Contents Cross classified models AI example
Multiple membership models
Danish chickens example
More complex structures
ALSPAC educational example
Thanks to Jon Rasbash for slides!

3. Cross-classification
For example, hospitals by neighbourhoods. Hospitals will draw patients from many different neighbourhoods and the inhabitants of a neighbourhood will go to many hospitals. No pure hierarchy can be found and patients are said to be contained within a cross-classification of hospitals by neighbourhoods :
nbhd 1
nbhd 2
Nbhd 3
hospital 1 xx x
hospital 2
hospital 3
hospital 4
xxx
Hospital H H H H4
Patient P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12
Nbhd N N N3

4. Other examples of cross-classifications
pupils within primary schools by secondary schools.
patients within GPs by hospitals.
interviewees within interviewers by surveys.
repeated measures within raters by individual. (e.g. patients by nurses)

5. Notation
With hierarchical models we use a subscript notation that has one subscript per level and nesting is implied reading from the left. For example, subscript pattern ijk denotes the i’th level 1 unit within the j’th level 2 unit within the k’th level 3 unit.
If models become cross-classified we use the term classification instead of level. With notation that has one subscript per classification, that captures the relationship between classifications, notation can become very cumbersome. We propose an alternative notation introduced in Browne et al. (2001) that only has a single subscript no matter how many classifications are in the model.

6. Single subscript notation
Hospital H H H H4
Patient P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12
Nbhd N N N3
We write the model as i
nbhd(i)
hosp(i) 1 2 3 4 5 6 7 8 9 10 11 12
Where classification 2 is neighbourhood and classification 3 is hospital. Classification 1 always corresponds to the classification at which the response measurements are made, in this case patients. For patients 1 and 11 equation (1) becomes:

7. Classification diagrams
In the single subscript notation we lose information about the relationship(crossed or nested) between classifications. A useful way of conveying this information is with the classification diagram. Which has one node per classification and nodes linked by arrows have a nested relationship and unlinked nodes have a crossed relationship.
Hospital
Neighbourhood
Hospital
Patient
Neighbourhood
Patient
Cross-classified structure where patients from a hospital come from many neighbourhoods and people from a neighbourhood attend several hospitals.
Nested structure where hospitals are contained within neighbourhoods

8. Example : Artificial insemination by donor
1901 women
279 donors
1328 donations
12100 ovulatory cycles
response is whether conception occurs in a given cycle
In terms of a unit diagram:
Or a classification diagram:
Donor
Woman
Cycle
Donation

9. Model for artificial insemination data
We can write the model as
Results:
Parameter
Description
Estimate(se)
intercept
-4.04(2.30)
azoospermia *
0.22(0.11)
semen quality
0.19(0.03)
womens age>35
-0.30(0.14)
sperm count
0.20(0.07)
sperm motility
0.02(0.06)
insemination to early
-0.72(0.19)
insemination to late
-0.27(0.10)
women variance
1.02(0.21)
donation variance
0.644(0.21)
donor variance
0.338(0.07)

10. Multiple membership models
When level 1 units are members of more than one higher level unit we describe a model for such data as a multiple membership model.
For example,
 Pupils change schools/classes and each school/class has an effect on pupil outcomes.
Patients are seen by more than one nurse during the course of their treatment.

11. Notation
Note that nurse(i) now indexes the set of nurses that treat patient i and w(2)i,j is a weighting factor relating patient i to nurse j. For example, with four patients and three nurses, we may have the following weights:
n1(j=1)
n2(j=2)
n3(j=3)
p1(i=1)
0.5
p2(i=2) 1
p3(i=3)
p4(i=4)
Here patient 1 was seen by nurse 1 and 3 but not nurse 2 and so on. If we substitute the values of w(2)i,j , i and j. from the table into (1) we get the series of equations :

12. Classification diagrams for multiple membership relationships
Double arrows indicate a multiple membership relationship between classifications.
We can mix multiple membership, crossed and hierarchical structures in a single model.
patient
nurse
patient
nurse
hospital
GP practice
Here patients are multiple members of nurses, nurses are nested within hospitals and GP practice is crossed with both nurse and hospital.

13. Example involving nesting, crossing and multiple membership – Danish chickens
Production hierarchy
10,127 child flocks
725 houses
304 farms
Breeding hierarchy
10,127 child flocks
200 parent flocks
As a unit diagram:
As a classification diagram:
Child flock
House
Farm
Parent flock

14. Model and results Results: Parameter Description Estimate(se)
intercept
-2.322(0.213)
1996
-1.239(0.162)
1997
-1.165(0.187)
hatchery 2
-1.733(0.255)
hatchery 3
-0.211(0.252)
hatchery 4
-1.062(0.388)
parent flock variance
0.895(0.179)
house variance
0.208(0.108)
farm variance
0.927(0.197)
Results:

15. ALSPAC data
All the children born in the Avon area in 1990 followed up longitudinally.
Many measurements made including educational attainment measures.
Children span 3 school year cohorts(say 1994,1995,1996).
Suppose we wish to model development of numeracy over the schooling period. We may have the following attainment measures on a child :
m1 m2 m3 m m5 m6 m7 m8
primary school secondary school

16. Structure for primary schools
P School Cohort
Area
M. Occasion
Pupil
P. Teacher
Measurement occasions within pupils.
At each occasion there may be a different teacher.
Pupils are nested within primary school cohorts.
All this structure is nested within primary school.
Pupils are nested within residential areas.

17. A mixture of nested and crossed relationships
M. occasions
Pupil
P. Teacher
P School Cohort
Primary school
Area
Nodes directly connected by a single arrow are nested, otherwise nodes are cross-classified. For example, measurement occasions are nested within pupils. However, cohort are cross-classified with primary teachers, that is teachers teach more than one cohort and a cohort is taught by more than one teacher. T1 T2 T3
Cohort 1 95 96 97
Cohort 2 98
Cohort 3 99 00

18. Multiple membership
It is reasonable to suppose the attainment of a child in a particualr year is influenced not only by the current teacher, but also by teachers in previous years. That is measurements occasions are “multiple members” of teachers.
m m m m4
t t t t4
M. occasions
Pupil
P. Teacher
P School Cohort
Primary school
Area
We represent this in the classification diagram by using a double arrow.

19. What happens if pupils move area?
M. occasions
Pupil
P. Teacher
P School Cohort
Primary school
Area
Classification diagram without pupils moving residential areas.
If pupils move area, then pupils are no longer nested within areas. Pupils and areas are cross-classified. Also it is reasonable to suppose that pupils measured attainments are effected by the areas they have previously lived in. So measurement occasions are multiple members of areas.
M. occasions
Pupil
P. Teacher
P School Cohort
Primary school
Area
Classification diagram where pupils move between residential areas.
BUT…

20. If pupils move area they will also move schools
M. occasions
Pupil
P. Teacher
P School Cohort
Primary school
Area
Classification diagram where pupils move between areas but not schools.
If pupils move schools they are no longer nested within primary school or primary school cohort. Also we can expect, for the mobile pupils, both their previous and current cohort and school to effect measured attainments.
M. occasions
Pupil
P. Teacher
P School Cohort
Primary school
Area
Classification diagram where pupils move between schools and areas.

21. If pupils move area they will also move schools cnt’d
And secondary schools…
M. occasions
Pupil
P. Teacher
P School Cohort
Primary school
Area
We could also extend the above model to take account of Secondary school, secondary school cohort and secondary school teachers.

22. Other predictor variables
Remember we are partitioning the variability in attainment over time between primary school, residential area, pupil, p. school cohort, teacher and occasion. We also have predictor variables for these classifications, eg pupil social class, teacher training, school budget and so on. We can introduce these predictor variables to see to what extent they explain the partitioned variability.

23. Information for the practicals
We have two MLwiN practicals taken from chapters of Browne (2003).
We firstly look at a cross-classified model for education data (primary schools and secondary schools.
We secondly look at a multiple membership model for a (simulated) earnings dataset.