Introduction

Ecological Studies

• Based on grouped data, with the groups in a spatial context corresponding to geographical areas
   
• Ecological studies have a long history in many disciplines in addition to epidemiology and public health:
  -  political science, geography, sociology
   
• Due to aggregation, ecological studies are susceptible to unique challenges, in particular the potential for ecological bias
  -  the difference between estimated association based on ecological- and individual-level data
   
• Ecological data can be used for a variety of purposes:
  -  mapping: ecological bias is not a big problem (with-in areas variations may be obscured by agggregation)
  -  cluster detection: small-area anomalies may be washed away when data are aggregated

Ecological Bias

• The fundamental problem with ecological inference is that the process of aggregation reduces information
  -  this information loss usually prevents identification of parameters of interest in the underlying individual-level model
   
• If there is no within-area variability in exposures and confounders, then there will be no ecological bias
  -  therefore, ecological bias occurs due to within-area variability in exposures and confounders
   
• Distinct consequences of this variability:
  -  pure specification bias
  -  confounding

Pure Specification Bias

• Also called model specification bias
   
• This bias arises because a nonlinear risk model changes its form under aggregation
   
• This type of bias has nothing to do with confounding

Pure Specification Bias (cont'd)

• In an ecological setting, the individual-level data are unavailable, and rather, we observe the aggregate data that correspond to the average outcome and exposure

# individuals in area i (i=1,2,...,m)

Outcome for individual j in area i

Exposure for individual j in area i

• On aggregation, we have

Pure Specification Bias (cont'd)

• Aggregate to sum:

• Dividing the left- and right-hand sides by ni

• When there is no within-area variability in exposure:

There is no ecological bias

• The pure specification bias is reduced if areas are smaller, since the heterogeneity of exposures within areas is decreased

Confounding

• It is challenging to characterize the within-area joint distribution of exposures and confounders with only aggregated data
   
• Two scenarios when we can address the confounding issue with aggregated data
  -  the exposure and confounders are independent (no interaction between exposure and confounders)
  -  if we have the confounders that are constant within areas (e.g. county-level policy)

Mixed-Effects Models

• We usually assume the samples drawn from targeted population are independent and identically distributed (i.i.d.).

• This assumption does not hold when we have data with multilevel structure:
  - clustered and nested data (i.e. individuals within areas)
  - longitudinal data (i.e. repeated measurements within individuals)
  - non-nested structures (i.e. individuals within areas and belonging to some subgroups such as occupations)
   
• Samples within each group are dependent, while samples between groups stay independent
   
• Two sources of variations:
  - variations within groups
  - variations between groups

• A longitudinal study:
  - n = 3
  - t = 3
   
• Complete pooling
  - poor performance
   
• No pooling
  - infeasible for large n
   
• Partial pooling

• An alternative solution: include categorical individual indicators in the traditional linear regression model.
   
• Why do we still need mixed-effects models?

1. Account for both individual- and group-level variations when estimating group-level coefficients.
    
2. Easily model variations among individual-level coefficients, especially when making predictions for new groups.
    
3. Allow us to estimate coefficients for specific groups, even for groups with small n

Fixed and Random Effects

• Random Effects: varying coefficients
• Fixed Effects: varying coefficients that are not themselves modeled

Fixed and Random Effects

What's the difference between no-pooling models and mixed-effects models only with varying intercepts?

• In no-pooling models, the intercept is obtained by least squares estimates, which equals to the fitted intercepts in models that are run separately by group.
• In mixed-effects models, we assign a probability distribution to the random intercept: 

Linear Mixed-Effects Model

git pull

Load the Packages and Data

1,000 participants

5 repeated measurements

bmi

time

id

age

race: 1=white, 2=black, 3=others

gender: 1=male, 2=female

edu: 1=<HS, 2=HS, 3=>HS

sbp

am: 1=measured in morning

ex: #days exercised in the past year

Varying-intercept Model with No Predictors

Varying-intercept Model with an individual-level predictor

Varying-intercept Model with both individual-level and group-level predictors

Varying Slopes Models

With only an individual-level predictor

Varying Slopes Models

Add a group-level predictor

Non-nested Models

Generalized Linear Mixed-Effects Model

Mixed-Effects Logistic Model

Empty model

Mixed-Effects Logistic Model

Add bmi and race

Mixed-Effects Poisson Model

Parameter Estimation Algorithms

• ML: maximum likelihood
   
• REML: restricted maximum likelihood
  - default in lmer()
• PQL: pseudo- and penalized quasilikelihood
   
• Laplace approximations
  - default in glmer()
• GHQ: Gauss-Hermite quadrature
   
• McMC: Markov chain Monte Carlo

Bolker BM, Brooks ME, Clark CJ, Geange SW, Poulsen JR, Stevens MHH, et al. 2009. Generalized linear mixed models: A practical guide for ecology and evolution. Trends in ecology & evolution 24:127-135.

Mixed-Effects Model vs. GEE

Project Proposal and Presentation

Project Proposal

• Cover Page: Include title and list of team members.
  Project description: One-page, and please include the following:
  -  Specific Aims/Objectives: for those choosing option A, please cite the article you’d like to reproduce and briefly summarize the specific aims/objectives of the article. For those choosing option B, please state your aims/objectives.
  -  Approach/Research Design
  -  Timeline
  -  Literature cited (no page limit); please follow the Vancouver style.
• Proposals must use single column and single spacing; Arial or Times New Roman font; font size no smaller than 11 point; tables and figure labels can be in 10 point; 0.5 inch margins.

Project Presentation

• Up to ten (10) slides and no more than 7 minutes of presentation with 3 minutes Q&A
   
• Please submit the slides on Canvas by 3/16