## Multilevel Approaches

Hui Hu Ph.D.

Department of Epidemiology

College of Public Health and Health Professions & College of Medicine

October 25, 2018

PHC6016 Social Epidemiology

# Introduction

• 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

Two extreme cases:

• when the group-level variation is very little
- reduce to traditional regression models without group indicators (complete pooling)
• when the group-level variation is very large
- reduce to traditional regression models with group indicators (no-pooling)

### Little risk to apply a mixed-effects model

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:

Intraclass Correlation (ICC)

shows the variation between groups

ICC ranges from 0 to 1:

• ICC -> 0: the groups give no information (complete-pooling)
• ICC -> 1: all individuals of a group are identical (no-pooling)

Intraclass Correlation (ICC)

ICC ranges from 0 to 1:

• ICC -> 0: "hard constraint" to
• ICC -> 1: "no constraint" to
• Mixed-effects model: "soft constraint" to

This constraint has different effects on different groups:

• For group with small n, a strong pooling is usually seen, where the value of     is close to the mean      (towards complete-pooling)
• For group with large n, the pooling will be weak, where the value of    is far away from the mean      (towards no-pooling)

# Linear Mixed-Effects Model

### 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

allows intercept to vary by individual

estimated intercept, averaging over the individuals

estimated variations

### Varying Slopes Models

With only an individual-level predictor

# Generalized Linear Mixed-Effects Model

Empty model

### Parameter Estimation Algorithms

• ML: maximum likelihood

• REML: restricted maximum likelihood
- default in lmer()
• PQL: pseudo- and penalized quasilikelihood

• Laplace approximations
- default in glmer()

• 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

Mixed-Effects Model Marginal Model with GEE
Distributional assumptions Yes No
Population average estimates Yes Yes
Group-specific estimates Yes No
Estimate variance components Yes No
Perform good with small n Yes No

By Hui Hu

# Multilevel Approaches - PHC6016

Slides for the Social Epidemiology guest lecture, Fall 2018

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