lecture 21 cross-classified and multiple membership models
TRANSCRIPT
Lecture 21
Cross-classified and Multiple membership models
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!
Cross-classificationFor 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 x x
hospital 3 xx x
hospital 4 x xxx
Hospital H1 H2 H3 H4
Patient P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12
Nbhd N1 N2 N3
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)
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.
Single subscript notationHospital H1 H2 H3 H4
Patient P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12
Nbhd N1 N2 N3
i nbhd(i) hosp(i)1 1 12 2 13 1 14 2 25 1 26 2 27 2 38 3 39 3 410 2 411 3 412 3 4
)1()3()(
)2()(0 iihospinbhdi euuy
We write the model as
1)3(
4)2(
3011
1)3(
1)2(
101
euuy
euuy
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:
Classification diagrams
Hospital
Patient
Neighbourhood
Hospital
Patient
Neighbourhood
Nested structure where hospitals are contained within neighbourhoods
Cross-classified structure where patients from a hospital come from many neighbourhoods and people from a neighbourhood attend several hospitals.
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.
Example : Artificial insemination by donor
Women w1 w2 w3 Cycles c1 c2 c3 c4… c1 c2 c3 c4… c1 c2 c3 c4… Donations d1 d2 d1 d2 d3 d1 d2 Donors m1 m2 m3
1901 women279 donors 1328 donations12100 ovulatory cyclesresponse is whether conception occurs in a given cycle
In terms of a unit diagram:
Donor
Woman
Cycle
Donation
Or a classification diagram:
Model for artificial insemination data
),0(~
),0(~
),0(~
)()logit(
),1(~
2)4(
)4()(
2)3(
)3()(
2)2(
)2()(
)4()(
)3()(
)2()(i
uidonor
uidonation
uiwoman
idonoridonationiwomani
ii
Nu
Nu
Nu
uuuX
Binomialy
We can write the model as
2)4(u
0
1
2
3
4
5
6
7
2)2(u
2)3(u
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)
Results:
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.
Notation
),0(~
)1(),0(~
)(
2
2)2(
)2(
)(
)2()2(,
ei
uj
inursejijjiii
Ne
Nu
euwXBy
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 0 0.5
p2(i=2) 1 0 0
p3(i=3) 0 0.5 0.5
p4(i=4) 0.5 0.5 0
i
i
i
i
euuXBy
euuXBy
euXBy
euuXBy
)2(2
)2(14
)2(3
)2(23
)2(12
)2(3
)2(11
5.05.0
5.05.0
1
5.05.0
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 :
Classification diagrams for multiple membership relationships
Double arrows indicate a multiple membership relationship between classifications.
patient
nurseWe can mix multiple membership, crossed and hierarchical structures in a single model.
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.
Example involving nesting, crossing and multiple membership – Danish chickens
Production hierarchy10,127 child flocks 725 houses 304 farms
Breeding hierarchy10,127 child flocks200 parent flocks
farm f1 f2… Houses h1 h2 h1 h2 Child flocks c1 c2 c3… c1 c2 c3…. c1 c2 c3…. c1 c2 c3…. Parent flock p1 p2 p3 p4 p5….
Child flock
House
Farm
Parent flock
As a unit diagram: As a classification diagram:
Model and results
),0(~
),0(~),0(~
)()logit(
),1(~
2)4(
)4()(
2)3(
)3()(
2)2(
)2(
)(.
)4()(
)3()(
)2()2(,i
uifarm
uihouseuj
iflockpjiifarmihousejjii
ii
Nu
NuNu
euuuwXB
Binomialy
0
1
2
3
4
5
2)2(u
2)3(u
2)4(u
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:
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 m4 m5 m6 m7 m8
primary school secondary school
ALSPAC data
•Measurement occasions within pupils.
M. Occasion
Pupil P. Teacher
•At each occasion there may be a different teacher.
P School Cohort
•Pupils are nested within primary school cohorts.
Primary school
Area
•All this structure is nested within primary school.• Pupils are nested within residential areas.
Structure for primary schools
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 96 97 98
Cohort 3 98 99 00
A mixture of nested and crossed relationships
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.
m1 m2 m3 m4
t1 t2 t3 t4
M. occasions
Pupil P. Teacher
P School Cohort
Primary school
AreaWe represent this in the classification diagram by using a double arrow.
Multiple membership
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. TeacherP School Cohort
Primary school
Area
M. occasions
Pupil
P. TeacherP School Cohort
Primary school
Area
Classification diagram without pupils moving residential areas.
Classification diagram where pupils move between residential areas.
BUT…
What happens if pupils move 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. TeacherP School Cohort
Primary school
Area
M. occasions
Pupil P. TeacherP School Cohort
Primary school
Area
Classification diagram where pupils move between schools and areas.
If pupils move area they will also move schools
And secondary schools…
M. occasions
Pupil P. TeacherP 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.
If pupils move area they will also move schools cnt’d
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.
Other predictor variables
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.