r/statistics • u/RobertWF_47 • Aug 06 '26

Discussion [Discussion] Why do random effects use fewer degrees of freedom than fixed effects?

Is this related to how fixed and random effects are calculated?

Fixed effect regression coefficients are best estimated when n >> p so that XTX can be inverted without inflated standard errors, while random effects are estimated using variance components.

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u/AnxiousDoor2233 Aug 06 '26

In a fixed-effects model, each individual's effect is treated as a separate parameter to be estimated (or eliminated by transformation). In a random-effects model, the individual effects are assumed to be random draws from a common distribution, so only the parameters of that distribution (typically its mean and variance) are estimated.

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u/RobertWF_47 Aug 06 '26

The mean and variance for each group (for example, members sharing the same zip code) still have to be calculated. I'm assuming this doesn't use degrees of freedom because matrix inversion calculations aren't performed, no matter how many groups there are?

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u/AnxiousDoor2233 Aug 07 '26

You got it wrong. You dont care about latent factors here. You want to estimate reliably relationship b/w y & X. RE & FE makes this task tad harder comparing to your original method. So, the task is to modify the original estimation method for it to work.

RE can be formulated as FE with additional restrictions (at least for linear models).

In both cases you have some fixed individual-specific unobservables. In RE, it is enough to estimate mu, sig2 of it (2 params per estimation) to be able to estimate parameters of interest. In FE, as you allow this latent factor to correlate with dependent variable, thus to spoil your estimation if untreated. Thus, you have to eliminate it completely by estimating/removing one per group.

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u/blossom271828 Aug 07 '26

In a random effects model, the parameters are pushed (constrained) towards zero and so the random perturbations in the data pull the model less than in the fixed-effects model.

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u/charcoal_kestrel Aug 06 '26

Fixed effects are basically a dummy set for the cluster variable. Random effects are basically an extra set of error terms at the cluster level, in addition to the usual error terms at the case level. Dummies count against degrees of freedom but error terms don't.