What does it do?
The ‘gfilmm’ package allows to generate simulations from the generalized fiducial distribution of the parameters of a Gaussian linear mixed model with categorical random effects (numeric random effects are not supported) and interval data. It also provides some helper functions to get summary statistics and confidence intervals.
The algorithm implemented in ‘gfilmm’ is the one described in the paper Generalized fiducial inference for normal linear mixed models written by Jessi Cisewski and Jan Hannig. It is coded in C++ and the code is based on the original Matlab code written by Jessi Cisewski.
Fiducial inference has something similar to Bayesian inference: the uncertainty about the parameters are represented by a distribution, the fiducial distribution, with the help of which we conduct inference on the parameters in a way similar to the Bayesian way, based on the posterior distribution of the parameters. The main difference is that there is no prior distribution (so fiducial inference is similar to objective Bayesian inference). The fiducial inference yields results close to the ones of the frequentist inference.
First example: a non-mixed linear model
The data must be given as a dataframe. Here we simulate data from a simple linear regression model:
set.seed(666L)
n <- 30L
x <- 1L:n
y <- rnorm(n, mean = x, sd = 2)
y_rounded <- round(y, digits = 1L)
dat <- data.frame(
ylwr = y_rounded - 0.05,
yupr = y_rounded + 0.05,
x = x
)
Now we run the fiducial sampler:
library(gfilmm)
fidSims <- gfilmm(
y = ~ cbind(ylwr, yupr), # interval data
fixed = ~ x, # fixed effects
random = NULL, # random effects
data = dat, # data
N = 10000L # number of simulations
)
A summary of the fiducial simulations (the Pr(=0) column will be explained latter):
gfiSummary(fidSims)
#> mean median lwr upr Pr(=0)
#> (Intercept) 0.8696454 0.8742805 -0.9653809 2.720132 NA
#> x 0.9109923 0.9107001 0.8080415 1.015237 NA
#> sigma_error 2.4486778 2.4154403 1.8862475 3.211994 0
#> attr(,"confidence level")
#> [1] 0.95
The fiducial confidence intervals are close to the frequentist ones:
lmfit <- lm(y ~ x)
confint(lmfit)
#> 2.5 % 97.5 %
#> (Intercept) -0.9589356 2.70406
#> x 0.8076886 1.01402
The fiducial cumulative distribution function of the slope:
Fslope <- gfiCDF(~ x, fidSims)
plot(Fslope, main = "Slope", ylab = expression("Pr("<="x)"))
To get a fiducial density, I recommend the ‘kde1d’ package:
library(kde1d)
kfit <- kde1d(fidSims$VERTEX[["x"]], weights = fidSims$WEIGHT, mult = 4)
curve(dkde1d(x, kfit), from = 0.7, to = 1.1)
Fiducial predictive inference. The gfilmmPredictive function samples the generalized fiducial predictive distribution. All the functions seen above can be applied to the output.
fpd <- gfilmmPredictive(fidSims, newdata = data.frame(x = c(1, 30)))
gfiSummary(fpd)
#> mean median lwr upr
#> y1 1.801507 1.816828 -3.21099 6.922675
#> y2 28.155631 28.157729 22.85169 33.325925
#> attr(,"confidence level")
#> [1] 0.95
Compare with the frequentist approach:
predict(lmfit, newdata = data.frame(x = c(1, 30)), interval = "prediction")
#> fit lwr upr
#> 1 1.783417 -3.408471 6.975305
#> 2 28.198198 23.006310 33.390086
A mixed model
Now let us simulate some data from a one-way ANOVA model with a random factor:
mu <- 10000 # grand mean
sigmaBetween <- 2
sigmaWithin <- 3
n <- 8L # sample size per group
set.seed(666L)
groupmeans <- rnorm(2L, mu, sigmaBetween)
y1 <- rnorm(n, groupmeans[1L], sigmaWithin)
y2 <- rnorm(n, groupmeans[2L], sigmaWithin)
y <- c(y1, y2)
y_rounded <- round(c(y1, y2), digits = 1L)
dat <- data.frame(
ylwr = y_rounded - 0.05,
yupr = y_rounded + 0.05,
group = gl(2L, n)
)
We run the fiducial sampler:
fidSims <- gfilmm(~ cbind(ylwr, yupr), ~ 1, ~ group, data = dat, N = 10000L)
Observe that the between standard deviation sigma_group has a positive value in the Pr(=0) column:
gfiSummary(fidSims)
#> mean median lwr upr Pr(=0)
#> (Intercept) 10000.698525 10001.828832 9979.48611 10026.836248 NA
#> sigma_group 18.264835 3.399129 0.00000 83.980735 0.08605762
#> sigma_error 4.132655 4.008439 2.78631 6.267243 0.00000000
#> attr(,"confidence level")
#> [1] 0.95
What does it mean? The fiducial distributions of the variance components have a mass at zero, and this value is the probability that the between standard deviation equals zero. So you have to be careful if you are interested in the fiducial density of a standard deviation: if Pr(=0) is not null for the standard deviation you are interested in, the fiducial distribution of this standard deviation does not have a density. It has a mass at zero, and a density on the strictly positive real numbers.
Compare the fiducial confidence interval of the grand mean to its Kenward-Roger confidence interval:
library(lmerTest)
library(emmeans)
fit <- lmer(y ~ (1|group), data = dat)
emmeans(fit, ~ 1)
#> 1 emmean SE df lower.CL upper.CL
#> overall 10002 2.1074 1 9975.4 10029
#>
#> Degrees-of-freedom method: kenward-roger
#> Confidence level used: 0.95
With gfiConfInt we can get a fiducial confidence interval for any parameter of interest, for example the coefficient of total variation:
gfiConfInt(~ sqrt(sigma_group^2 + sigma_error^2)/`(Intercept)`, fidSims)
#> 2.5% 97.5%
#> 0.0003318767 0.0084440371