In simple terms, a multilevel model of fixed effects is a statistical approach used when you have data structured in groups (like students within schools, patients within hospitals, or observations within individuals over time) where you treat the characteristics of each specific group as constant parameters to be estimated directly.
Understanding Fixed Effects in a Multilevel Context
When data has a hierarchical or clustered structure, a "multilevel" or "hierarchical" model is often appropriate. Within this framework, you can model group-level variation in different ways:
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Fixed Effects: This approach treats the effect of each group as a unique, fixed parameter. It essentially adds a separate intercept (or slope) for every single group in your dataset.
- Implementation: This is often done by including a dummy variable for each group (except one reference group) in the regression model.
- Focus: Fixed effects models primarily analyze the variation within each group. They look at how changes in individual-level variables affect the outcome for a specific individual or unit within their group, controlling for the overall differences between groups.
The Key Limitation: Confounding Group-Level Predictors
A critical characteristic of the traditional fixed effects approach (especially when implemented with dummy variables in panel data) is its inability to estimate the effect of variables that are constant for each group.
According to the reference: "In a fixed effects model, the effects of group-level predictors are confounded with the effects of the group dummies, ie it is not possible to separate out effects due to observed and unobserved group characteristics."
This means:
- If you have a variable like "school type" (e.g., public vs. private) that doesn't change for any student within a particular school, you cannot include this variable in a fixed effects model alongside the school dummy variables.
- The effect of "school type" (an observed group characteristic) is indistinguishable from the fixed effect (the dummy variable) representing that specific school's overall difference (which captures both observed and unobserved group characteristics).
- Consequently, fixed effects models are excellent for controlling for unobserved time-invariant or group-invariant confounding, but they cannot tell you why groups differ based on time-invariant characteristics.
Fixed Effects vs. Multilevel (Random Effects) Models
The reference highlights the distinction between fixed effects and random effects in multilevel modeling regarding estimating group-level variables:
| Feature | Fixed Effects Model (in a Multilevel Context) | Multilevel (Random Effects) Model |
|---|---|---|
| Group Effects Treated As | Fixed, unique parameters for each group (often estimated via dummies) | Random variables drawn from a distribution (e.g., normal distribution) |
| Estimates Group-Level Predictors? | No, effects are confounded with group dummies. Cannot estimate effects of time-invariant group characteristics. | Yes, can estimate the effects of both observed individual-level and group-level variables. |
| Focus | Within-group variation, controlling for between-group differences. | Both within-group and between-group variation; models the structure of the group-level variance. |
| Primary Benefit | Controls for all unobserved time-invariant/group-invariant confounders. | More efficient estimates (if assumptions hold), can generalize findings to a population of groups. |
| Reference Point | "the effects of group-level predictors are confounded with the effects of the group dummies" | "In a multilevel (random effects) model, the effects of both types of variable can be estimated." |
While a fixed effects model effectively removes the influence of any stable group difference (observed or unobserved) from the estimation of lower-level effects, it does so at the cost of being unable to analyze the impact of those stable group differences themselves. A random effects multilevel model, on the other hand, makes assumptions about the distribution of group effects but allows for the estimation of group-level predictors.
In summary, when referring to a "multilevel model of fixed effects," one is typically describing an approach to hierarchical data analysis where group-specific intercepts (or slopes) are treated as fixed, distinct parameters, primarily focusing on within-group variation and inherently unable to disentangle the effects of observed time-invariant group characteristics from the unique fixed effect assigned to each group.
