Understanding Random Effects and Fixed Effects in Statistical Analysis

Statistical analysis like multilevel modelling, panel data analysis, and linear mixed models are widely used in various fields like social science, economics, epidemiology, biostatistics, business analytics, psychology, and many more. In these fields, the terms fixed effect and random effect are frequently encountered, and as we delve deeper into the technical differences and equations, it can become increasingly confusing. Well, in this article, we will gain a clear understanding of these terms and their practical applications through simplified explanations and illustrative examples.

I am assuming that you, as a reader, have a basic understanding of statistical analysis, including key concepts such as dependent variables, independent variables, and regression.

Fixed effect

Fixed effects play a fundamental role in statistical analysis, providing a way to account for specific variables or factors that remain constant across observations. These effects allow us to capture the individual characteristics of entities under study and control for their impact on the outcome of interest.

To understand fixed effects, let’s consider a hypothetical research study. The study aims to examine the impact of different teaching methods (lecture-based vs project-based) on student performance across five schools (A, B, C, D, and E). Each school may have unique characteristics that could potentially affect student performance, such as school size, funding levels, and teacher-student ratio. In this study, a common aptitude exam is administered to all students from the five schools, and their test scores, ranging from 0 to 100, are collected.

To analyze the data, a regression model is employed. The dependent variable is the test score, while the independent variable is a binary variable indicating the teaching method (lecture-based coded as 0, project-based coded as 1).

In this model, the ‘teaching method’ is a fixed effect because the researchers assume the effect of the teaching method on the outcome variable is constant across all the observations (irrespective of school).

The equation for the model can be written as:

Yi = β0 + β1X + εi

In this equation, Yi represents the students’ scores, β0 is the intercept, X is the teaching method variable, β1 is its coefficient, and εi is the error term.

Suppose the coefficient of the independent variable is found to be 16 and statistically significant. This coefficient suggests that, on average, students taught using the project-based method obtained 16 units higher scores than those taught using the lecture-based method. The positive effect of the teaching method remains constant for all students across the schools, indicating that it does not vary among different schools.

Furthermore, we can incorporate school fixed effects into this model. We can create five dummy variables, one for each school, and include four of them in the model.

The updated equation would look like this:

Yi = β0 + β1X + β2A + β3B + β4C + β5D + β6D + εi

In this equation, A, B, C, and D represent the dummy variables for schools A, B, C, D, and E, respectively, and β2, β3, β4, β5, and β6 are their respective coefficients.

By including these fixed effects in our statistical model, we effectively control for all school-specific factors that remain constant, such as school resources, culture, or policies. The fixed effect variables capture the individual characteristics of each school that may affect student performance but are held constant across observations within each school.

Suppose we obtain the following equation:

Yi = 52.8 + 14.6X + 5.2A + 4.1B - 3.6C + 2.8D + εi

Now, after controlling for the effect of schools on average, students taught using the project-based method obtain a score that is 14.6 units higher than those taught using the lecture-based method. Students from School A, School B, and School D, on average, obtain scores that are 5.2 units, 4.1 units, and 2.8 units higher than School E, respectively. Students from School C, on the other hand, have scores that are, on average, 3.6 units lower than School E.

By introducing a school-level fixed effect in the model, we have effectively controlled for the characteristics specific to each school. These school-level fixed effects capture the unique attributes and factors that are inherent to individual schools but remain constant across observations within each school.

By incorporating these school fixed effects into our statistical model, we are able to isolate the impact of the teaching method (the independent variable) on student performance (the dependent variable) while holding constant the school-level factors that could confound the relationship. In other words, we control for the effects of school-specific characteristics that might affect test scores but remain unchanged within each school.

Here both ‘teaching method’ and ‘School’ are examples of fixed effects.

Random Effect

Unlike fixed effects, which capture specific characteristics that remain constant across observations, random effects are used to account for variability and differences between different entities or subjects within a larger group.

Continuing with the previous example, let us suppose the research says that the effect of teaching method is not constant across schools, rather it varies by school due to school-level characteristics. In such cases, a mixed effects model can be employed to meet the objectives. This model incorporates both fixed effects for the teaching method and random effects at the school level. The model structure can be represented as follows:

Yij = β0 + β1jXij + u0j + u1jXij + εij

Here: Yij represents the test score of the ith student in the jth school. β0 is the intercept, representing the overall average test score. β1j is the coefficient of the teaching method for the ith school, indicating the effect of the teaching method specific to each school. Xij is the teaching method variable for the ith student in the jth school. u0j is the random intercept, capturing the school-specific effect on the intercept (β0). u1j is the random slope, capturing the school-specific effect on the coefficient of the teaching method (β1). εij is the error term, representing the residual variation not accounted for by the fixed and random effects.

Let us say we run this model in statistical software and get the following result:

The results show that the estimated variance of the random effect for schools (var(teaching method)) is 3.5. This represents the variation in the effect of the teaching method across schools, capturing the school-specific differences in the impact of the teaching method on test scores. var(_constant) represents the variance of the intercept across the schools.

Talking in statistical language, each level of a random effect can be seen as a random variable originating from an underlying distribution. Estimating random effects allows for drawing inferences not only about the specific levels themselves (similar to fixed effects) but also about the population level and absent levels. This concept is known as exchangeability, where the levels in a random effect are not treated as separate and independent, but rather as representative samples from a larger collection of levels, some of which may not even be observed. It is important to note that despite the term “random,” random effects do not imply inherent randomness, but rather a way to consider each level as a draw from a random variable. Hence, the levels or groups within a random effect can be conceptualized as a sample from a larger population of levels, including some that may not be explicitly represented in the statistical model. Understanding random effects provides valuable insights into capturing variations and accounting for unobserved heterogeneity in statistical analysis.

Advantages and Disadvantages

When conducting statistical analysis, researchers have the option to utilize either random effects or fixed effects approaches. Each approach offers distinct advantages and disadvantages, depending on the research question, data structure, and underlying assumptions. Let’s discuss the benefits and limitations of using each approach.

Advantages of Random Effects:

★Accounting for Unobserved Heterogeneity: Random effects allow for the incorporation of unobserved factors that vary across entities within a dataset. This is particularly valuable in panel data or multilevel analysis, where individuals or groups are repeatedly observed. By capturing unobserved heterogeneity, random effects models account for variations in outcomes that cannot be explained by observed variables alone.

★Efficiency in Estimation: Random effects models can provide more efficient estimates when the random effects are correlated with the independent variables. This efficiency stems from exploiting the within-entity variation, which increases the precision of parameter estimates compared to fixed effects models.

★Generalizability: Random effects models are often considered more generalizable than fixed effects models. By allowing for entity-specific effects to vary across different groups, random effects models capture a wider range of variation in the population. This can enhance the external validity of the results, making them more applicable to broader contexts.

Disadvantages of Random Effects:

★Limited Interpretation of Entity-Specific Effects: While random effects models capture unobserved heterogeneity, they do not allow for the estimation and interpretation of entity-specific effects. The focus is on the average effects across entities, making it challenging to understand the variations and nuances within individual entities.

Advantages of Fixed Effects:

★Controlling for Time-Invariant Factors: Fixed effects models are well-suited for controlling time-invariant factors that may confound the relationship between independent and dependent variables. By including fixed effects, researchers effectively remove the influence of these unobserved factors, enabling a more accurate estimation of the effects of interest.

★Capturing Entity-Specific Effects: Fixed effects models provide a means to estimate and interpret entity-specific effects. By controlling for all time-invariant factors within each entity, researchers can analyze the within-entity variations, uncovering the unique impact of different variables on the outcome of interest.

Disadvantages of Fixed Effects:

★Limited Generalizability: Fixed effects models tend to focus on the within-entity variations, making the estimates less generalizable to populations outside the observed entities. The emphasis on controlling for time-invariant factors within each entity can limit the external validity of the findings.

In practice, the choice between random effects and fixed effects approaches depends on the specific research question, data structure, and assumptions underlying the analysis. Researchers should carefully consider the advantages and disadvantages of each approach and select the most appropriate method based on the specific context of their study.