Individual effects of a panel model

fixef() returns the estimated individual effects alpha_i of a fixed-effects (model = "within") fit, and ranef() the empirical Bayes predictions of the random effects of a model = "random" fit.

Usage

fixef(object, ...)

ranef(object, ...)

# S3 method for class 'panglm'
fixef(object, ...)

# S3 method for class 'panglm_hurdle'
fixef(object, ...)

# S3 method for class 'panglm'
ranef(object, type = c("mean", "mode"), ...)

Arguments

object: a fitted panglm() model (or a panhurdle() fit for fixef())
...: unused
type: for ranef(), "mean" (posterior mean, default) or "mode" (posterior mode)

Value

fixef(): a named numeric vector of individual effects (names are the individual identifiers), or for two-way fits a list with components individual and time. ranef(): a matrix with one row per individual (row names are the identifiers) and one column per random effect, with attribute "type".

Fixed effects

Gaussian within fits: alpha_i = ybar_i - xbar_i' beta, the intercepts of the least-squares dummy-variable regression. They are not normalized: fixest::fixef() gives the same values for a model without an intercept.
Poisson within fits: the conditional (Poisson fixed-effects) estimator does not estimate alpha_i, but the individual intercept that maximizes the unconditional likelihood at the conditional beta has the closed form alpha_i = log(sum_t y_it / sum_t exp(x_it' beta)), the values reported by fixest::fepois(). Individuals with no positive outcome (alpha_i = -Inf) are omitted, as in fixest.
Negative binomial and gamma within fits: the dummy-variable intercepts of the joint maximum likelihood fit. The negative binomial fit drops the individuals whose outcome is always zero.
Two-way fits (Gaussian, Poisson, negative binomial): a list with the individual and time effects, normalized so that the time effects average zero.
The conditional logit (family = "binomial", one-way or two-way) and the fixed-effects ordered logit condition the individual effects out of the likelihood and do not estimate them: fixef() stops with an error. The individual intercepts are not consistently estimable with a fixed number of periods anyway (incidental parameters).
panhurdle() fits: the dummy-variable intercepts of the zero-truncated count part (individuals with at least one positive count); the binary part is a conditional logit.

Random effects

ranef() returns the posterior mean (type = "mean", default) or the posterior mode (type = "mode") of each individual's random effects given the individual's outcomes, at the estimated parameters (empirical Bayes).

Gaussian random intercept: the BLUP sigma_mu^2 T_i / (sigma_mu^2 T_i + sigma_e^2) * mean_t(y_it - x_it' beta) with the variance components of the fit (Swamy-Arora, or maximum likelihood for weighted fits). Mean and mode coincide.
Gaussian random slopes: the BLUPs stored by the fit.
Binomial, gamma, beta, Tobit and ordered random intercepts, which are normal on the link scale: the posterior mean is computed by adaptive Gauss-Hermite quadrature with the R nodes of the fit, centred at the posterior mode and scaled by the curvature there, the rule used for the likelihood. With R = 1 (Laplace) the mean is the mode.
Poisson and negative binomial random intercepts (Hausman, Hall and Griliches), whose mixing distribution is not normal: the effect is reported on the log scale of the multiplicative effect on the mean. For the Poisson-gamma model nu_i ~ Gamma(d, d) and the posterior is Gamma(d + sum_t y_it, d + sum_t exp(x_it' beta)), so the posterior mean of log(nu_i) is digamma(d + Y_i) - log(d + L_i) and its mode log((d + Y_i) / (d + L_i)). For the beta-mixed negative binomial model the effect is log(delta_i), with delta_i / (1 + delta_i) distributed as Beta(b, a), giving digamma(b + Y_i) - digamma(a + L_i) and log((b + Y_i) / (a + L_i)).
Random slopes of the Poisson, negative binomial, gamma and binomial families: posterior modes are those stored by the fit; posterior means use the adaptive product rule of the fit (R^q nodes).

Empirical Bayes predictions are shrunk towards zero, the more so the fewer observations an individual has: their spread is smaller than the estimated standard deviation of the random effect (for the Gaussian model, var(b_i) = sigma_mu^2 * shrinkage_i), so their histogram is narrower than the assumed normal density (see plot.panglm(), which = "effects").

Generics

The generics fixef() and ranef() are also defined by nlme, whose generics lme4 and fixest re-export. panglm exports its own generics so that it does not depend on nlme, and when nlme is loaded it also registers its methods for nlme's generics: whichever package is attached last, fixef(fit) and ranef(fit) dispatch to the panglm methods.

Examples

data(copd)
fe <- panglm(fev1 ~ crp, data = copd, index = c("id", "visit"),
             model = "within")
head(fixef(fe))
#>        1        2        3        4        5        6 
#> 3.072628 3.011334 2.915205 3.492994 3.210295 2.814040 

re <- panglm(fev1 ~ crp + treatment, data = copd,
             index = c("id", "visit"), model = "random")
head(ranef(re))
#>   (Intercept)
#> 1  -0.1462000
#> 2  -0.2056603
#> 3  -0.2981489
#> 4   0.2606700
#> 5   0.1480610
#> 6  -0.3970215

hosp <- panglm(hospitalized ~ crp + treatment, data = copd,
               index = c("id", "visit"), model = "random",
               family = "binomial")
head(ranef(hosp))
#>          [,1]
#> 1  0.02573863
#> 2  0.02580553
#> 3  0.03808140
#> 4  0.01191119
#> 5  0.56973142
#> 6 -0.43496147