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Lecture 21 Cross-classified and Multiple membership models

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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!

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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 1nbhd 2Nbhd 3 hospital 1xxx hospital 2xx hospital 3xxx hospital 4xxxx Hospital H1 H2 H3 H4 Patient P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 Nbhd N1 N2 N3

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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)

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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 ith level 1 unit within the jth level 2 unit within the kth 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.

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Single subscript notation Hospital H1 H2 H3 H4 Patient P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 Nbhd N1 N2 N3 inbhd(i)hosp(i) We write the model as 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:

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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.

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

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Model for artificial insemination data We can write the model as ParameterDescriptionEstimate(se) intercept-4.04(2.30) azoospermia *0.22(0.11) semen quality0.19(0.03) womens age> (0.14) sperm count0.20(0.07) sperm motility0.02(0.06) insemination to early-0.72(0.19) insemination to late-0.27(0.10) women variance1.02(0.21) donation variance0.644(0.21) donor variance0.338(0.07) Results:

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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.

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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)0.50 p2(i=2)100 p3(i=3)00.5 p4(i=4)0.5 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 :

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Classification diagrams for multiple membership relationships Double arrows indicate a multiple membership relationship between classifications. patient nurse We 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.

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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 Child flock House Farm Parent flock As a unit diagram: As a classification diagram:

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Model and results ParameterDescriptionEstimate(se) intercept-2.322(0.213) (0.162) (0.187) hatchery (0.255) hatchery (0.252) hatchery (0.388) parent flock variance0.895(0.179) house variance0.208(0.108) farm variance0.927(0.197) Results:

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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

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

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M. occasions PupilP. 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. T1T2T3 Cohort Cohort Cohort A mixture of nested and crossed relationships

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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 PupilP. Teacher P School Cohort Primary school Area We represent this in the classification diagram by using a double arrow. Multiple membership

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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 M. occasions Pupil P. Teacher P 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?

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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 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

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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 cntd

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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

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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.

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