- 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?

- Account for both individual- and group-level variations when estimating group-level coefficients.

- Easily model variations among individual-level coefficients, especially when making predictions for new groups.

- Allow us to estimate coefficients for specific groups, even for groups with small n

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

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)

**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)

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

*allows intercept to vary by individual*

*estimated intercept, averaging over the individuals*

*estimated variations*

With only an individual-level predictor

Add a group-level predictor

Empty model

Add bmi and race

- 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 | 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 |