Channel Width 2021b

Data Preparation

The data were provided by Hillcrest Geographics and New Graph Environment in the form an csv file and prepared for analysis using R version 4.1.2 (R Core Team 2020). The FWA data were excluded as the channel widths were truncated below approximately 10 m.

Key assumptions of the data preparation included:

• Data points with a channel width of 0 m, a channel width \(\geq\) 9999 m, a watershed area < 0.1 ha, a gradient < 0.01% or > 50% are unreliable and were excluded.

Statistical Analysis

Model parameters were estimated using Bayesian methods. The estimates were produced using JAGS (Plummer 2003) and STAN (Carpenter et al. 2017). For additional information on Bayesian estimation the reader is referred to McElreath (2016).

Unless stated otherwise, the Bayesian analyses used weakly informative normal and half-normal prior distributions (Gelman, Simpson, and Betancourt 2017). The posterior distributions were estimated from 1500 Markov Chain Monte Carlo (MCMC) samples thinned from the second halves of 3 chains (Kery and Schaub 2011, 38–40). Model convergence was confirmed by ensuring that the potential scale reduction factor \(\hat{R} \leq 1.05\) (Kery and Schaub 2011, 40) and the effective sample size (Brooks et al. 2011) \(\textrm{ESS} \geq 150\) for each of the monitored parameters (Kery and Schaub 2011, 61).

The parameters are summarised in terms of the point estimate, lower and upper 95% credible limits (CLs) and 95% prediction limits (PLs) and the surprisal s-value (Greenland 2019). The estimate is the median (50th percentile) of the MCMC samples while the 95% CLs are the 2.5th and 97.5th percentiles. The 95% PLs are the 2.5th and 97.5th percentiles of individual channel widths based on the residual variation. The s-value can be considered a test of directionality. More specifically it indicates how surprising (in bits) it would be to discover that the true value of the parameter is in the opposite direction to the estimate. An s-value of 4.3 bits, which is equivalent to a p-value (Kery and Schaub 2011; Greenland and Poole 2013) of 0.05, indicates that the surprise would be equivalent to throwing 4.3 heads in a row. The condition that non-essential explanatory variables have s-values \(\geq\) 4.3 bits provides a useful model selection heuristic (Kery and Schaub 2011).

Model adequacy was assessed via posterior predictive checks (Kery and Schaub 2011). More specifically, the number of zeros and the first four central moments (mean, variance, skewness and kurtosis) for the deviance residuals were compared to the expected values by simulating new residuals. In this context the s-value indicates how surprising each metric is given the estimated posterior probability distribution for the residual variation.

Where computationally practical, the sensitivity of the parameters to the choice of prior distributions was evaluated by increasing the standard deviations of all normal, half-normal and log-normal priors by an order of magnitude and then using \(\hat{R}\) to test whether the samples where drawn from the same posterior distribution (Thorley and Andrusak 2017).

The results are displayed graphically by plotting the modeled relationships between particular variables and the response(s) with the remaining variables held constant. In general, continuous and discrete fixed variables are held constant at their mean and first level values, respectively, while random variables are held constant at their typical values (expected values of the underlying hyperdistributions) (Kery and Schaub 2011, 77–82). When informative the influence of particular variables is expressed in terms of the effect size (i.e., percent or n-fold change in the response variable) with 95% credible intervals (CIs, Bradford, Korman, and Higgins 2005).

The analyses were implemented using R version 4.1.2 (R Core Team 2020) and the mbr family of packages.

Model Descriptions

Model Templates

.data {
  int nObs;

  real width[nObs];
  real area[nObs];
  real precipitation[nObs];
parameters {
  real b0;
  real bDischarge;

  real<lower=0> sWidth;
model {
  vector[nObs] eWidth;

  b0 ~ normal(0, 1);
  bDischarge ~ normal(0.375,  0.125);

  sWidth ~ normal(0, 1);

  for (i in 1:nObs) {
    eWidth[i] = exp(b0 + bDischarge * (log(area[i]) + log(precipitation[i])));
    width[i] ~ lognormal(log(eWidth[i]), sWidth);
  }

Tables

Figures