- Hierarchical data structures, e.g.
- individuals in larger units in ...
- students in classes in schools in boards in states ...
some classroom variables might be aggregated
student variables
- repeated measures: times in individuals in ...
- surveys: respondents in interviewers
- Modelling hierarchy, unsatisfactory methods:
1) Disaggregate higher-order variables to individual
level and analyze individuals. Problem: probable
lack of conditional independence of response,
due to: unmeasured variables.
2) Aggregate individual level variables to higher level
and analyze at higher level. Problem: lose within
group information, aggregated relationship might
be different from individual level relationship.
[e.g. ecological correlation, Simpson's Paradox]
- Using traditional linear models: assume linear, normal, homosc.
and independent. Keep 1st 2 but relax last 2.
Why?:
- not independent within groups because of unmeasured
variables similar within groups which "vanish"
in error term
- groups might vary in homogeneity -> heterosc.
- Formalization:
Each group has a different regression model.
Intercepts and slopes are random sample from popn of groups
- if intercepts only random: variance components model
- if slopes also: more complex. In mid 80s packages appeared
to handle this
- Limitations:
- linearity and normality.
- needed extensions: more levels, multivariate y, generalized
linear models, ...
Chapter 1
==============
Hierarchical data structures:
workers in firms
households in countries
times in persons: repeated measures, growth curves
meta analysis
educational data:
individual growth of students over time (Level 1)
effects of personal characteristics (Level 2)
effect of classroom organization, teacher, ... (Level 3)
...
Dilemmas in analysis:
Limitations of conventional techniques result in concern over:
- aggregation bias
many different meanings
- misestimated precision
target of inference
- unit of analysis problem
- impoverished conceptualization:
effects can be different at different levels
- measurement of change problems
Hierarchical models more homologous with basic phenomena
History of Statistical Theory for Hierarchical Models
sociology: multilevel linear models
biometrics: mixed-effects, random-effects
econometrics: random coefficient regression models
statistics: covariance components
"hierarchical linear models" used to encompass
Computation:
EM algorithm in 70s
Applied in early 80s
Research purposes:
1) improved estimation within individual units: borrowing strength
from other units,
- use of standardized test scores to select business school
applicants: what formula predicts with minorities?
can't get enough data from any one school.
2) cross-level effects: e.g. interactions between effects at different
levels
- Social research: Mason: urban-rural fertility differential
differs between countries in relationship with GNP
- Developmental: longitudinal vocabulary growth in children
3) partitioning variance between levels
- growth in math achievement from grade 1 to grade 3:
618 students in 86 schools: 83% of variance in growth
rate was "between" schools but only 14% of variance
in initial status was between schools.
Chapter 2
==================