Predicts the shape and scale parameters of a Weibull distribution that characterizes the absolute partial correlation edge weights of a psychometric network, given its number of nodes and sample size. Parameter estimates are derived from a Seemingly Unrelated Regression (SUR) model fitted to empirical network data from Huth et al. (2025), where absolute partial correlations were found to follow a Weibull distribution more consistently than Beta, Gamma, or log-normal alternatives.
weibull_parameters(nodes, sample_size, snr = 1, bootstrap = FALSE)A named numeric vector of length 2:
shapeThe predicted Weibull shape parameter (\(k > 0\)).
scaleThe predicted Weibull scale parameter (\(\lambda > 0\)).
Numeric (length = 1). The number of nodes (variables) in the network. Must be between 8 and 54, reflecting the range of the empirical networks used to fit the underlying SUR model.
Numeric (length = 1). The sample size of the dataset from which the network is estimated.
Numeric (length = 1).
Signal-to-noise ratio of partial correlations (\(\bar{|w|} / \mathrm{SD}(|w|)\)).
Values less than 1 indicate wider range of partial correlations (\(w\)) whereas
values greater than 1 indicate narrower range.
Defaults to 1 where the mean of the partial correlations (\(\bar{|w|}\))
is equal to the standard deviation (\(\mathrm{SD}(|w|)\))
Boolean (length = 1). If TRUE, a randomly sampled residual from
the SUR model fit is added to each predicted parameter, introducing
empirically grounded variability suitable for use in simulation or
bootstrapping contexts. Defaults to FALSE.
Alexander P. Christensen <alexpaulchristensen@gmail.com>
The shape and scale parameters are predicted from derived network descriptors that differ between the two SUR equations:
rlpThe reciprocal log of the number of nodes, \(1 / \log(p)\), capturing the diminishing marginal effect of network size on edge weight distributions. Used in both the shape and scale equations.
scalingThe standard error of partial correlations defined as \(\sqrt{1 / (n - p)}\), where \(n\) is the sample size and \(p\) is the number of nodes. Larger values indicate greater sampling uncertainty in the partial correlation estimates. Used in the scale equation only.
The two SUR equations have an asymmetric structure reflecting different
theoretical roles for sampling precision. Shape — which governs the
concentration of the edge weight distribution — is determined solely by
signal characteristics of the network via snr and rlp.
Scale — which governs the typical magnitude of edge weights — additionally
depends on scaling, as the expected size of partial correlations is
directly affected by estimation precision. This asymmetry is both
empirically supported (dropping scaling from the shape equation
costs \(\Delta R^2 < 0.006\)) and theoretically coherent.
These predictors enter two SUR equations whose coefficients are stored in
the internal weibull_weights dataset. SUR was used to account for the
correlated residuals between the shape and scale equations across networks
(residual correlation = 0.261). Model fit was strong: the shape equation
achieved \(R^2 = 0.887\) (RMSE = 0.048) and the scale equation achieved
\(R^2 = 0.885\) (RMSE = 0.011). 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. Heteroskedasticity was detected in both
the shape equation (Breusch-Pagan p < 0.001) and the scale equation
(Breusch-Pagan p < 0.001); robust standard errors (HC3) were used for
inference. Multicollinearity among predictors in the scale equation was
negligible (VIF \(\leq\) 1.60); the shape equation contains only two
predictors with no multicollinearity concern.
nodes influences predicted shape and scale via rlp.
sample_size influences predicted scale via scaling, and
both parameters via their joint contribution to snr when it is
estimated from data rather than supplied directly.
Empirically, shape values ranged from approximately 0.72 to 1.63 (M = 1.07, SD = 0.14) and scale values from approximately 0.03 to 0.19 (M = 0.10, SD = 0.03) across the 194 networks used to fit the model. Shape values near 1 indicate approximately exponential edge weight distributions; values above 1 indicate a rising hazard (mode-bearing distribution).
When bootstrap = TRUE, residuals are drawn via shuffle() --
a random sampling without replacement -- from the empirical SUR residuals,
preserving the observed marginal residual distribution.
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.
# Predict parameters for a 10-node network with n = 500
weibull_parameters(nodes = 10, sample_size = 500)
# With bootstrapped residuals for use in simulation
weibull_parameters(nodes = 10, sample_size = 500, bootstrap = TRUE)
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