Mixed-Effects Model

The mixed effects model (MEM) is a statistical modeling technique used for analyzing data in which the observations are not entirely independent of each other and both fixed and random effects are combined. The purpose is to understand the relationship between one or more dependent variables while accounting for the correlations or dependencies among the observations.

It is commonly used in various fields, including biology, psychology, social sciences, etc. These models can also handle unbalanced data, where some groups or clusters may have more observations than others. This statistical model incorporates both fixed effects and random effects. They provide a flexible framework for analyzing data with dependencies and varying levels of grouping.

A mixed effect model, specifically a mixed error-component model, combines fixed effects and random effects within a statistical framework. It represents an extension of simple linear models. Hence, it finds applications across various scientific disciplines. These models serve as a potent tool for addressing linear regression challenges when datasets exhibit both global and group-level trends. In situations involving repeated measurements on the same statistical units, such as longitudinal studies, or when observations cluster within related statistical units, researchers find them particularly valuable.

Moreover, let us understand the two effects of this model:

• Fixed effect: Fixed effects are parameters that represent average relationships between variables across the entire population. Researchers assume that the grouping factor remains constant across all levels, fixing them in place.
• Random effect: Random effects are additional parameters that account for variability at the group level. Hence, researchers assume that a probability distribution draws them, capturing unobserved or random variations among groups, making them random.

This is precisely why researchers developed mixed-effects model analyses – to handle intricate data and enable the utilization of the entire data set. However, linear mixed-effect models are valuable for capturing the nuances of real-world data. Here, observations are not independent and share commonalities within groups.

Adjustments in the mixed effect model involve additional fixed effects, covariates, or interaction terms in the model to better capture the complexity of the data. In addition, blocking of the mixed effect model is particularly useful. 

Researchers use mixed-effects models only when they know there are sources of variability that may affect the outcome, and the goal is to account for or control these sources.

Reliable and accurate results in mixed-effects model analysis depend on meeting a set of assumptions. These assumptions can be broken down into several critical criteria.

• Linearity: This assumption requires that the relationships between the variables in the model are linear. In simpler terms, if one plots these variables, they should exhibit a pattern that a straight line can well approximate.
• No Outliers: Outliers are data points that significantly deviate from the overall pattern of the data. In the context of mixed effects modeling, it's essential to have data without such extreme values, as linear regression is sensitive to them.
• Similar Spread across Range: This assumption is known as homoscedasticity. Moreover, it pertains to the uniformity of the spread of data points across the entire range of the variables. In other words, the variability in the data should be relatively consistent.
• Normality of Residuals: Residuals are the differences between the observed and predicted values. For reliable results, these residuals should follow a normal distribution, which resembles a bell-shaped curve. Therefore, this ensures that the model's predictions are applicable across the entire dataset without systematic bias.
• No Multicollinearity: When the independent variables in the model have a strong correlation with one another, this is known as multicollinearity. Hence, this can make it difficult to discern the individual effects of these variables and may lead to unstable regression coefficients and less reliable statistical significance. However, it only affects how well the model fits the data.

Let us look at the mixed effect model examples to understand the concept better: 

Example #1

Let's consider a simplified example with numeric dependent and independent variables in the context of a mixed-effects model.

Dependent Variable:

Let's assume our dependent variable is "Blood Pressure" (measured in mmHg). We are interested in how it changes over time for a group of patients.

Independent Variables:

• Time (in days): This variable represents the time points at which blood pressure measurements were taken. It could range from 0 to, say, 30 days.
• Drug Dosage (in mg): This variable represents the dosage of a hypertension drug administered to patients. It might take values like 0 mg (placebo), 50 mg, 100 mg, or 200 mg.
• Patient Age (in years): The age of each patient in the study, ranging from, for example, 30 to 70 years.
• Baseline Blood Pressure (mmHg): The initial blood pressure level of each patient at the beginning of the study.

Now, using this model, we can investigate how changes in drug dosage, patient age, and baseline blood pressure influence the blood pressure measurements over time. The fixed effects in the model would capture the relationships between these independent variables and the dependent variable. In contrast, the random effects would account for variations between patients and the potential clustering of patients within different medical centers (if applicable).

Moreover, the mixed effects model would allow us to examine how, on average, blood pressure changes with time, dosage, and patient characteristics. Thus also considering the specific characteristics of patients and any center-specific effects that might influence these changes. Therefore, this type of analysis provides a more comprehensive understanding of the factors affecting blood pressure and can be used to make informed decisions about the drug's efficacy and patient-specific responses.

Example #2

Let's consider the stock prices of companies within a specific industry. The fixed effects include factors like overall market trends, interest rates, or economic indicators that affect all companies uniformly. Meanwhile, random effects could account for company-specific variations that are not explained by fixed effects—for instance, management quality, corporate governance, or other distinctive factors. By employing a mixed-effects model, researchers can simultaneously analyze the broad industry trends affecting all companies while acknowledging and quantifying the unique characteristics that differentiate individual firms.

Hence, this approach allows for a more nuanced understanding of stock price movements, providing insights into both sector-wide influences and company-specific dynamics. It's particularly relevant in finance, where markets are inherently hierarchical. Where individual stocks are nested within sectors or industries and where accounting for both systematic and idiosyncratic factors is crucial for robust analyses and investment decision-making.

Here are the advantages and disadvantages of mixed effects model:

Advantages

• These models are specifically designed to handle hierarchical or nested data structures, such as repeated measures within subjects, patients within hospitals, or students within schools.
• They can account for the correlation or non-independence among data points within the same group or cluster.
• Moreover, they offer flexibility in modeling both fixed and random effects. Hence allowing for the inclusion of group-specific variability while still estimating the fixed effects of interest. Therefore, it is particularly useful when one wants to make generalizations beyond specific groups.
• Mixed effects models can handle unbalanced data, where some groups or clusters may have different sample sizes without excluding observations, making them suitable for real-world datasets.

Disadvantages

• These models can be more complex to set up and interpret compared to simpler linear models. They require a good understanding of both fixed and random effects, which can be challenging for some researchers.
• Like linear models, these models have their own set of assumptions. Therefore, failing to meet these assumptions can affect the validity of the results.
• Mixed effects models can be computationally intensive, especially when dealing with large datasets or complex models. Hence, this may require more advanced statistical software and hardware.