L0ggm (version 0.1.2)

weibull_weights: SUR Model Coefficients and Residuals for Weibull Parameter Prediction

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 weibull_parameters to generate data-driven Weibull parameter estimates given a network's number of nodes and sample size.

Usage

data(weibull_weights)

Format

A named list with two elements, shape and scale, each containing:

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 (snr, rlp); the scale equation includes an intercept and three predictors (snr, rlp, scaling). Predictors are defined as follows:

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.

rlp

Reciprocal log of node count, \(1 / \log(p)\), where \(p\) is the number of nodes. Used in both equations.

scaling

Standard error of partial correlations, \(\sqrt{1 / (n - p)}\), where \(n\) is the sample size and \(p\) is the number of nodes. Used in the scale equation only.

residuals

A numeric vector of residuals from the fitted SUR equation, used by weibull_parameters to introduce empirically grounded variability when bootstrap = TRUE.

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 (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 (\(p < 8\)), more than 300,000 observations, or fewer than one observation per edge (\(\text{ope} \leq 1\)) 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 snr and 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 scaling, as the expected size of partial correlations is directly affected by estimation precision. Dropping scaling from the shape equation costs \(\Delta R^2 < 0.006\) and yields cleaner inference; all shape predictors are significant under HC3-robust standard errors (both \(p < 0.001\)).

Variance inflation factors for the scale equation (\(\leq\) 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 \(R^2 = 0.887\) (RMSE = 0.048); scale \(R^2 = 0.885\) (RMSE = 0.011).

References

Huth, K. B. S., Haslbeck, J. M. B., Keetelaar, S., Van Holst, R. J., & Marsman, M. (2025). Statistical evidence in psychological networks. Nature Human Behaviour.

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)