% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/weibull_weights.R
\docType{data}
\name{weibull_weights}
\alias{weibull_weights}
\title{SUR Model Coefficients and Residuals for Weibull Parameter Prediction}
\format{
A named list with two elements, \code{shape} and \code{scale},
each containing:
\describe{
  \item{\code{coefficients}}{A named numeric vector of regression
  coefficients from the SUR model. The equations are asymmetric: the shape
  equation includes an intercept and two predictors (\code{snr}, \code{rlp});
  the scale equation includes an intercept and three predictors
  (\code{snr}, \code{rlp}, \code{scaling}).
  Predictors are defined as follows:
  \describe{
    \item{\code{snr}}{Signal-to-noise ratio of the absolute partial
    correlations, computed as the mean divided by the standard deviation
    of the absolute edge weights. Used in both equations.}
    \item{\code{rlp}}{Reciprocal log of node count, \eqn{1 / \log(p)},
    where \eqn{p} is the number of nodes. Used in both equations.}
    \item{\code{scaling}}{Standard error of partial correlations,
    \eqn{\sqrt{1 / (n - p)}}, where \eqn{n} is the sample size and
    \eqn{p} is the number of nodes. Used in the scale equation only.}
  }}
  \item{\code{residuals}}{A numeric vector of residuals from the fitted SUR
  equation, used by \code{\link{weibull_parameters}} to introduce
  empirically grounded variability when \code{bootstrap = TRUE}.}
}
}
\usage{
data(weibull_weights)
}
\description{
A named list encoding the Seemingly Unrelated Regression (SUR) model fitted
to Weibull shape and scale parameters derived from 194
empirical networks. The object is consumed internally by \code{\link{weibull_parameters}}
to generate data-driven Weibull parameter estimates given a network's
number of nodes and sample size.
}
\details{
Absolute partial correlations from 222 deduplicated empirical networks
(Huth et al., 2025) were fitted to Beta, Gamma, log-normal, and Weibull
distributions via maximum likelihood. Weibull provided the best fit most
consistently: it outperformed each alternative by more than 2 log-likelihood
units far more often than the reverse (vs. Beta: 56--0; vs. Gamma: 15--2;
vs. log-normal: 155--13; vs. Exponential: 54--0).

The resulting Weibull shape and scale parameters were then jointly modelled
as a function of network descriptors using Seemingly Unrelated Regression
(\code{systemfit}), which accounts for correlated residuals between the
shape and scale equations across networks (residual correlation = 0.261).
Prior to fitting, networks with fewer than eight nodes (\eqn{p < 8}), more
than 300,000 observations, or fewer than one observation per edge
(\eqn{\text{ope} \leq 1}) were excluded, as Huth et al. (2025) demonstrated
that networks in this regime show the weakest statistical evidence for edge
presence or absence, yielding unstable parameter estimates (n = 28 excluded).
This left n = 194 networks for analysis. Shape and scale parameters were
modelled on their original scales.

The two equations have an asymmetric structure. Shape — which governs the
concentration of the edge weight distribution — is predicted from \code{snr}
and \code{rlp} only, reflecting that the shape of the distribution is a
property of the network's signal structure independent of sampling precision.
Scale — which governs typical edge weight magnitude — additionally includes
\code{scaling}, as the expected size of partial correlations is directly
affected by estimation precision. Dropping \code{scaling} from the shape
equation costs \eqn{\Delta R^2 < 0.006} and yields cleaner inference; all
shape predictors are significant under HC3-robust standard errors
(both \eqn{p < 0.001}).

Variance inflation factors for the scale equation (\eqn{\leq} 1.60)
confirmed the absence of problematic multicollinearity; the shape equation
contains only two predictors with no multicollinearity concern.
Breusch-Pagan tests indicated statistically significant heteroskedasticity
in both equations; however, all predictors remained significant under
HC3-robust standard errors, indicating no material effect on inference.
Shape residuals were normally distributed (Shapiro-Wilk W = 0.990,
p = 0.173). Scale residuals showed a modest departure from normality
(Shapiro-Wilk W = 0.973, p < 0.001), consistent with slight right skew
in the scale outcome and test sensitivity at n = 194 rather than a
substantive violation, as confirmed by visual inspection of the residual
histogram. Model fit was strong: shape \eqn{R^2 = 0.887} (RMSE = 0.048);
scale \eqn{R^2 = 0.885} (RMSE = 0.011).
}
\examples{
data("weibull_weights")

# Inspect SUR coefficients for each equation
weibull_weights$shape$coefficients
weibull_weights$scale$coefficients

# Predict Weibull parameters for a new network
weibull_parameters(nodes = 12, sample_size = 500)

}
\references{
Huth, K. B. S., Haslbeck, J. M. B., Keetelaar, S., Van Holst, R. J., &
Marsman, M. (2025). Statistical evidence in psychological networks.
\emph{Nature Human Behaviour}.
}
\keyword{datasets}
