What is LMM

• Also known as a linear mixed effects model, it is a statistical model that combines both fixed and random effects to analyze data that exhibit both within-group and between-group variability.
• "linear": the model is a linear regression model, where the response variable is assumed to be a linear combination of the predictor variables.
• "mixed": the model includes both
  • fixed effects (population characteristics)
    • the same as in a standard linear regression model= the average effect of the predictor variables on the response variable across all levels of the random effects
  • random effects (individual characteristics)
    • account for the variability between subjects or groups; allow for the effects of the predictor variables to vary across different groups or clusters of data
• Formula:

- where $Y$ is a vector of responses, $X$ and $Z$ are design matrices, $\beta$ and $b$ are vectors of fixed and random effects, respectively, and $ϵ$ is a vector of residual errors.
- These coefficients represent the average effect of the predictor variables on the response variable across all levels of the random effects.
- Compared to Anova & Post-hoc, ANOVA does not have the random effect part:
- $$Y = X\beta + \epsilon$$

Why LLM matters

Compared to Anova & Post-hoc, it is more generalized:

• It can handle non-independent data that exhibit within-group correlation, such as repeated measures or longitudinal data, or data that involve different levels of grouping, such as hierarchical or nested data.
• It can handle all types of data: continuous, discrete, categorical, ordinal
• It can handle unbalanced or missing data

Assumptions for using LLM

• subjects are random sample from the population of interest
• the values of the dependent variable have a multivariate normal distribution with covariance structure

• data from different individuals are independent, while repeated measurements of the same individual are not assumed to be independent
• missing data are assumed to be ignorable

What to determine when using LLM

• the best covariance structure
• the best model among:
  • random intercepts + fixed slopes
  • fixed intercepts + random slopes
  • random intercepts + random slopes