Improved confidence intervals for nonlinear mixed-effects and nonparametric regression models

Abstract

Statistical inference for high dimensional parameters (HDPs) can be based on their intrinsic correlation; that is, parameters that are close spatially or temporally tend to have more similar values. This is why nonlinear mixed-effects models (NMMs) are commonly (and appropriately) used for models with HDPs. Conversely, in many practical applications of NMM, the random effects (REs) are actually correlated HDPs that should remain constant during repeated sampling for frequentist inference. In both scenarios, the inference should be conditional on REs, instead of marginal inference by integrating out REs. In this paper, we first summarize recent theory of conditional inference for NMM, and then propose a bias-corrected RE predictor and confidence interval (CI). We also extend this methodology to accommodate the case where some REs are not associated with data. Simulation studies indicate that this new approach leads to substantial improvement in the conditional coverage rate of RE CIs, including CIs for smooth functions in generalized additive models, as compared to the existing method based on marginal inference.

Figures (5)

• Figure 1: Simulated squared bias for estimators of the random walk $\Psi$'s, averaged over 500 simulated $\Psi$'s. For each set of $\Psi$'s, the squared bias is based on 1000 simulated data sets $Y_{t,i}|\Psi$ with $i=1,...,n$ and $t = 1,...,T$. Panel rows indicate choices for $n$ and columns indicate $T$. The colors correspond to the "posterior mode" estimator ($\hat{\Psi}$) and the bias-corrected estimator ($\hat{\Psi}_{\mathrm{BC}}$).
• Figure 2: Simulated coverage of 95% CIs for the random walk $\Psi$'s. Points indicate average CI coverages, and shaded regions indicate 5% and 95% quantiles, from the 500 simulated $\Psi$'s. Panels are described in Fig. \ref{['fig:Bias2_RW_EX']}.
• Figure 3: Simulated coverage of 95% CIs for an illustrative example of random walk $\Psi$'s. Red lines indicate CIs based on the "posterior mode" estimator ($\hat{\Psi}$) and conditional-$\Psi$ standard errors (SEs), green lines indicate CIs based on $\hat{\Psi}$ and random-$\Psi$ SEs, and blue lines indicate CIs based on the bias-corrected estimator ($\hat{\Psi}_{\mathrm{BC}}$) and its conditional-$\Psi$ SEs.
• Figure 4: Simulated coverage of 95% CIs for the random walk $\Psi$'s. $y$ observations for the last three time points are missing. Points indicate average CI coverages, and shaded regions indicate 5% and 95% quantiles, from the 500 simulated $\Psi$'s. Panels are described in Fig. \ref{['fig:Bias2_RW_EX']}.
• Figure 5: Simulated 95% CI coverage rates and squared biases for estimating the yearly mean global temperature anomalies obtained by fitting the global annual temperature anomalies rohde2020berkeley using a GAM. Upper panel: the black lines are the coverage rates for the true mean annual temperature anomalies of the 95% CIs constructed using $\hat{\Psi}(\hat{\theta})$ and marginal MSEs, and the red lines are those based on $\hat{\Psi}_{\mathrm{BC}}(\hat{\theta})$ and (\ref{['EQ:CI_g_October_23_2022']}); the blue dashed reference line is at 95%. Lower panel: the black and red lines are the simulated squared biases for estimating the true mean annual temperature anomalies based on $\hat{\Psi}(\hat{\theta})$ and $\hat{\Psi}_{\mathrm{BC}}(\hat{\theta})$ respectively, and the blue dashed reference line is at 0.