Introduction

What is a mixed model?

Generalisation of linear models where observations are not independent. Mixed models model the covariance structure of data.

Three basic types:

• random effects: certain effects arise from a distribution and add
• additional source of variation
• random coefficients: a covariate varies randomly, used for repeated
• measures to model relationship with time, different from random effects
• because estimates are correlated
• covariance pattern: directly models covariance structure

Why use mixed models?

• more appropriate fixed effect estimates (covariance pattern)
• to make inference over wider population
• to deal with missing data
• fixed effect estimates shrunken so extreme estimates less likely with
• small samples
• can account for heterscedasticity between treatment groups

Useful definitions

Containment

Basically the same as nesting (?).

Balance

When a design is balanced estimate effects will equal raw means. Balance occurs for a fixed effect when:

• within each category of fixed effect, observations have equal proportions
• in categories of every other effect fitted at the same level
• if a fixed effect is contained within a random effect, there are an equal
• number of observations at each category of the random effect

It’s also important identify when fixed effect means will differ depending on whether fixed or mixed model is used. They will be the same provided:

• within each category of the effect, observations have equal proportions
• in categories of every other random effect fitted at the same level
• if the effect is contained within a random effect, there are an equal
• number of observations at each category of the random effect

Rules of thumb:

• if equal number of observations in every cell and no continuous
• covariates, then all fixed effects are balanced.

• missing observations give imbalance
• continuous effects give imbalance (balance across random effects may
• still occur)

Error strata

Error stratum defined by each random effect and by the residuals. The containment stratum of a fixed effect is defined as error strata that contains the effect – the residual stratum, unless contained within a random effect. Usually an effect has only one containment stratum, but can have more in more complicated situations.

Higher level strata are defined by random effects themselve contained within another random effect. If higher level strata are present and data are imbalanced across random effects, fixed effects will be estimated using information from the higher level strata as well.