Linear Mixed Model (LMM)

 What is linear mixed model design and how can we apply it?

A Linear Mixed Model (LMM) is a statistical modeling approach that is used to analyze data that have both fixed and random effects, and where the observations are correlated. LMMs are a generalization of linear regression models, and they allow for the modeling of complex relationships between the response variable and the predictor variables. LMMs are commonly used in various research areas, such as social sciences, medical research, and engineering. For example, in social sciences, LMMs may be used to analyze longitudinal data where the same individuals are measured multiple times over a period of time. In medical research, LMMs may be used to analyze data from clinical trials where patients are randomized to different treatments. The application of LMM involves the specification of a statistical model that includes fixed and random effects, as well as an error term. The fixed effects are the predictor variables of interest, while the random effects account for the variation in the data that cannot be explained by the fixed effects. The error term represents the variability in the data that cannot be accounted for by the model. LMMs can be applied using various software packages, such as R, SAS, and SPSS. The analysis involves fitting the model to the data and estimating the parameters of the model using maximum likelihood or restricted maximum likelihood estimation. Once the model has been fitted, hypothesis tests and confidence intervals can be calculated for the fixed effects, and the model can be used for prediction and simulation. LMMs are a powerful statistical modeling approach that can handle complex data structures and relationships between variables. However, they can be computationally intensive and may require a large sample size to estimate the parameters of the model accurately. Additionally, the interpretation of the results of an LMM can be challenging, as the fixed and random effects are often correlated.

Example

In a Linear Mixed Model (LMM), fixed effects are predictor variables that are of primary interest to the researcher, and their effects are assumed to be constant across all experimental units. Random effects, on the other hand, are variables whose effects are assumed to be random and vary across experimental units.

An example of fixed and random effects can be illustrated in a study that aims to investigate the effect of different doses of a drug on blood pressure. In this study, the fixed effect would be the dose of the drug, as this is the predictor variable of primary interest. The different doses of the drug would be assigned to the experimental units (e.g., patients), and their effects would be estimated by the model.

The random effect in this study could be the effect of the patient, as the response variable (blood pressure) may vary across different patients due to individual differences such as genetics, lifestyle, and other factors. The random effect of the patient would account for this variability and allow for a more accurate estimate of the fixed effect of the drug.

In summary, in this example, the fixed effect would be the dose of the drug, and the random effect would be the effect of the patient on the response variable (blood pressure). By including both fixed and random effects in the LMM, we can obtain more precise estimates of the treatment effects and account for the variability due to individual differences between the experimental units.