# Random effects model

A random effects model is a type of regression analysis technique used to analyze data with a hierarchical structure. It is used to explain the variance in a response variable by taking into account both fixed effects (which are constant across observations) and random effects (which vary across observations). In management, it can be used to assess the impact of certain interventions on employee or customer behavior, or to understand the varying effects of a certain product or service based on different demographic or geographic characteristics.

## Example of random effects model

• A random effects model can be used in human resources to assess the impact of an intervention on employee performance. For example, a company might use a random effects model to evaluate the effect of offering an additional bonus or incentive on employee satisfaction and turnover rates. The fixed effects might include the size of the bonus or the type of job, while the random effects could include the individual employee's motivation or experience level.
• Random effects models can also be used to evaluate the impact of a product or service on customer behavior. For example, a company might use a random effects model to assess the impact of a new loyalty program on the number of purchases customers make over a certain period of time. The fixed effects might include the type of loyalty program or the type of customer, while the random effects could include the customer's individual preferences or the timing of the loyalty program.
• In marketing, a random effects model can be used to evaluate the impact of different types of advertisements on consumer purchasing behavior. For example, a company might use a random effects model to compare the effectiveness of television and online advertisements. The fixed effects might include the type of advertisement or the type of product, while the random effects could include the consumer's individual preferences or the timing of the advertisement.

## Formula of random effects model

Random effects models can be expressed as a general linear mixed model. The general form of this model is given by:

\begin{align} Y_{ij} &= \beta_0 + \beta_1 x_{ij} + \beta_2 z_{ij} + \epsilon_{ij} \\ \epsilon_{ij} &= \alpha_i + \eta_{ij} \end{align}

This model is composed of two equations. The first equation is the fixed-effects equation and it expresses the mean response (Y) of an observation (i) as a function of two predictor variables (x and z). The second equation expresses the random effect (αi) associated with each observation (i) and is assumed to be normally distributed.

The parameters of the model (β0, β1, β2) are estimated using maximum likelihood estimation. This means that the parameters are estimated in such a way that the model best explains the observed data.

The random effects model is often used when there is evidence of unobserved heterogeneity in the data, i.e. when there are differences in the data that are not explained by the fixed effects. This model allows researchers to account for this heterogeneity and to better explain the variance in the response variable.

## When to use random effects model

Random effects models are useful when attempting to analyze data that are not easily modeled using a traditional linear regression approach. Applications include:

• Analyzing the effects of interventions on employee or customer behavior, such as measuring the impact of a new training program or customer service initiative.
• Modeling the effects of different demographic or geographic characteristics on a product or service, such as understanding the varying effects of a new food product across different regions.
• Assessing the relative importance of different factors in predicting an outcome, such as determining which marketing techniques are most effective in increasing sales.
• Identifying and understanding the relationships between different variables, such as understanding how a change in one variable affects another.

## Types of random effects model

A random effects model is a type of regression analysis technique used to analyze data with a hierarchical structure. It takes into account both fixed effects (which are constant across observations) and random effects (which vary across observations). There are several types of random effects models, including:

• Mixed effects models: This type of model allows for the inclusion of both fixed and random effects in the same model. It is useful for analyzing data from experiments or surveys where some variables are fixed (e.g. product type, gender) and others are random (e.g. location, age).
• Hierarchical linear models: This type of model is used to explore relationships between variables at different levels of a hierarchical structure. It is typically used to model data from nested designs, such as a school survey where the observations are nested within schools.
• Multilevel models: This type of model is used to analyze data with multiple levels of nesting. It is particularly useful for exploring relationships between an individual-level variable (e.g. age) and a group-level variable (e.g. school).
• Random coefficient models: This type of model is used to analyze data where the effects of some variables vary across observations. It is used to identify the sources of variability in a dataset, such as the effects of different demographic factors on customer behavior.

## Advantages of random effects model

Random effects models offer several advantages over other forms of regression analysis. They can provide more accurate estimates of the effects of certain interventions on customer or employee behavior, as well as better insight into the varying effects of a product or service based on different demographic or geographic characteristics. Additionally, they can help to identify sources of variation within the data and account for that variation in the model. Some of the key advantages of random effects models include:

• Increased accuracy: The inclusion of random effects in the model allows for more accurate estimates of the effects of certain interventions on customer or employee behavior.
• Improved insight: Random effects models can provide more insight into the varying effects of a product or service based on different demographic or geographic characteristics.
• Identification of sources of variation: Random effects models can help to identify sources of variation within the data and account for that variation in the model.
• Increased flexibility: Random effects models can be used to fit models with both fixed and random effects, allowing for increased flexibility when analyzing data.

## Limitations of random effects model

A random effects model has several limitations that should be taken into consideration when using it. These include:

• Model misspecification: Random effects models assume that the data are generated by a hierarchical structure, and if the data do not follow this structure, the model will not be able to accurately capture the data's true structure.
• Unobserved heterogeneity: Random effects models assume that the underlying population is homogeneous, which can lead to biased estimates if the underlying population is actually heterogeneous.
• Limited interpretability: It is difficult to interpret the results of a random effects model, as the effects of the random terms are generally not directly interpretable.
• Limited power: Random effects models are generally less powerful than fixed effects models, as they have fewer degrees of freedom.
• Computational complexity: Random effects models can be computationally intensive, and may require specialized software or hardware to properly analyze the data.

 Random effects model — recommended articles Analysis of covariance — Confirmatory factor analysis — Analysis of variance — Logistic regression model — Latent class analysis — Homogeneity of variance — Standardized regression coefficients — Maximum likelihood method — Three-Way ANOVA