Trend Model Variant: Interpretability Report

Testing for secular weight trends using linear δ parameter

Overview

This report analyzes a model variant that adds an explicit linear trend parameter δ to test whether weight changes over the study period show a secular trend independent of fitness effects. Comparing this model to the constrained AR(1) model helps determine whether strength/aerobic effects are robust or confounded with trends.

Key Question: Does adding a trend parameter change estimates of γ_s and γ_a? If yes, the fitness effects were partially confounded with the trend.

Model Specification

Trend Model Equation

$$weight[t] \sim \text{Student-t}(\nu, \mu[t], \sigma_w)$$

$$\mu[t] = weight\_intercept + \delta \cdot \frac{t - 0.5D}{D} + \gamma_s \cdot strength\_fitness[t] + \gamma_a \cdot aerobic\_fitness[t] + f\_{daily}[t] + \epsilon[t]$$

where:

  • δ: Linear trend parameter (per standardized time unit)
  • Time centering: (t - 0.5D) / D maps to [-0.5, 0.5] to reduce correlation with intercept
  • All other components identical to base model

Prior for Trend Parameter

delta ~ normal(0, 1.0)

Weakly informative: 1 std unit of δ corresponds to ~31.45 lbs change over full study period (924 days). If δ = ±0.01, that's 0.31 lbs per 924 days.

Posterior Parameters

Posterior means and 95% credible intervals. Focus on δ and comparison of γ_s, γ_a to base model.

Parameter Mean 2.5% CI 97.5% CI R-hat ESS (bulk)
delta 0.0340 nan nan — —
rho 0.2647 nan nan — —
sigma_epsilon 0.3768 nan nan — —
gamma_s 0.1537 nan nan — —
gamma_a -0.0873 nan nan — —
weight_intercept -1.9153 nan nan — —
nu 13.5329 nan nan — —
sigma_w 0.0947 nan nan — —
sigma_fourier 0.1796 nan nan — —
beta_s 0.3072 nan nan — —
beta_a 0.3326 nan nan — —
alpha_d_s 0.9948 nan nan — —
alpha_m_s 0.5019 nan nan — —
alpha_d_a 0.8097 nan nan — —
alpha_m_a 0.4978 nan nan — —

Trend Parameter Interpretation

Estimated Secular Trend (δ)

Point estimate: δ = 0.034039 (95% CI: [nan, nan])

Implied weight change over study period:

Interpretation:
  • If CI includes 0: No clear evidence of secular trend; fitness effects are likely genuine
  • If δ > 0 and significant: Upward weight trend exists; check if γ_s shrank compared to base model
  • If δ < 0 and significant: Downward weight trend exists; opposite interpretation

Comparison to Base Model

To assess whether adding the trend parameter changes fitness effect estimates, compare:

Parameter Base Model Trend Model Change Interpretation
γ_s ~0.143 See table above ↓ = confounding Strength effect less/more confounded with trend
γ_a ~-0.086 See table above Shift = trend impact Aerobic effect independent of trend
δ N/A See above — Estimated secular weight change

Model Comparison (LOO-CV): If elpd difference is <1 SE, both models are equivalent. If trend model has better predictive performance, the trend parameter provides real value for prediction.

Diagnostics

MCMC Sampling Configuration:

Convergence: Check R-hat values above. All should be <1.01 for good mixing.

Key Takeaways

Next Steps