Estimation of the linear mixed integrated Ornstein–Uhlenbeck model

Figures & data

Table 1. Comparing parameterizations of the IOU process for balanced data (m= 1000, n=20 and k=3) simulated under the random intercept IOU model with weak derivative tracking $(\alpha =20.7,\omega =1)$.

		Parameterizations
		$\alpha \tau$	$\alpha \omega$	$\ln \alpha \tau$	$\ln \alpha \omega$	${\alpha }^{-2}\tau$	${\alpha }^{-2}\omega$
Converged	N	626	1000	823	993	987	582
Opt. Iter.	Median (IQR)	$8\left(3\right)$	$9\left(12\right)$	$12\left(29\right)$	$9\left(7\right)$	$25\left(54\right)$	$8\left(4\right)$
$\text{Var}\left[{\hat{u}}_{0i}\right]$	Bias $\times 1000$ (MCSE $\times 1000$)	$9.3\left(0.78\right)$	$0.45\left(0.68\right)$	$3.9\left(0.73\right)$	$0.18\left(0.69\right)$	$1.3\left(0.67\right)$	$10\left(0.8\right)$
	Empirical SD	0.083	0.096	0.091	0.097	0.093	0.081
	Median SE (IQR)	$0.083\left(0.07\right)$	$0.2\left(0.66\right)$	$0.1\left(1.3\right)$	$0.2\left(0.74\right)$	$0.18\left(0.94\right)$	$0.081\left(0.053\right)$
	CP $\mathrm{\%}$	$91\mathrm{\%}$	$94\mathrm{\%}$	$93\mathrm{\%}$	$94\mathrm{\%}$	$94\mathrm{\%}$	$90\mathrm{\%}$
$\hat{\alpha }$	Bias $\times 1000$ (MCSE $\times 1000$)	$-1400\left(590\right)$	$9600\left(590\right)$	$4200\left(570\right)$	$11000\left(640\right)$	$6400\left(490\right)$	$-3700\left(500\right)$
	Empirical SD	0.55	0.70	0.64	0.73	0.63	0.47
	Median SE (IQR)	$0.4\left(0.85\right)$	$1.9\left(14\right)$	$0.71\left(27\right)$	$2\left(16\right)$	$1.5\left(17\right)$	$0.38\left(0.62\right)$
	CP $\mathrm{\%}$	$68\mathrm{\%}$	$80\mathrm{\%}$	$76\mathrm{\%}$	$80\mathrm{\%}$	$80\mathrm{\%}$	$66\mathrm{\%}$
$\hat{\omega }$	Bias $\times 1000$ (MCSE $\times 1000$)	$11\left(0.9\right)$	$6.4\left(0.70\right)$	$8.3\left(0.78\right)$	$6.3\left(0.70\right)$	$6.3\left(0.70\right)$	$12\left(0.94\right)$
	Empirical SD	0.022	0.022	0.022	0.022	0.022	0.022
	Median SE (IQR)	$0.025\left(0.00098\right)$	$0.024\left(0.0059\right)$	$0.025\left(0.006\right)$	$0.024\left(0.0059\right)$	$0.024\left(0.0059\right)$	$0.025\left(0.00084\right)$
	CP $\mathrm{\%}$	$95\mathrm{\%}$	$95\mathrm{\%}$	$95\mathrm{\%}$	$95\mathrm{\%}$	$95\mathrm{\%}$	$94\mathrm{\%}$
$\hat{\sigma }$	Bias $\times 1000$ (MCSE $\times 1000$)	$17\left(0.97\right)$	$-1.1\left(0.95\right)$	$5.9\left(1\right)$	$-1.8\left(0.98\right)$	$0.98\left(0.9\right)$	$20\left(0.88\right)$
	Empirical SD	0.1	0.13	0.12	0.13	0.12	0.089
	Median SE (IQR)	$0.089\left(0.18\right)$	$0.41\left(1.7\right)$	$0.16\left(3.5\right)$	$0.42\left(1.9\right)$	$0.33\left(2.3\right)$	$0.086\left(0.14\right)$
	CP $\mathrm{\%}$	$69\mathrm{\%}$	$81\mathrm{\%}$	$76\mathrm{\%}$	$81\mathrm{\%}$	$80\mathrm{\%}$	$67\mathrm{\%}$

Optimization performed using NR algorithm. Over 1000 simulated datasets: number converged, median and IQR of the number of optimization iterations (Opt. Iter.) and for each variance parameter ν: bias of the estimates of ν with Monte Carlo standard error (MCSE), standard deviation of the estimates of its transformation $\breve{\nu }$, median and IQR of the estimated standard error (SE) of $\breve{\nu }$ and empirical coverage percentage (CP) of the $95\mathrm{\%}$ confidence interval for ν. Results reported to two significant figures.

Table 2. Comparing random slope IOU model for balanced data (m=1000, n=20 and k=3) simulated with weak derivative tracking $(\alpha =20.7,\omega =1)$.

		Random slope IOU models
		RE1	RE2	RE3
Converged	N	878	1000	1000
Opt. Iter.	Median (IQR)	$20\left(14\right)$	$12\left(13\right)$	$11\left(13\right)$
$\text{Var}\left[{\hat{u}}_{0i}\right]$	Bias $\times 1000$ (MCSE $\times 1000$)	$0.3\left(0.72\right)$	$0.97\left(1.1\right)$	$0.91\left(1.8\right)$
	Empirical SD of $\breve{\nu }$	0.095	0.036	0.028
	Median SE of $\breve{\nu }$ (IQR)	$0.23\left(0.64\right)$	$0.055\left(0.12\right)$	$0.035\left(0.05\right)$
	CP $\mathrm{\%}$	$95\mathrm{\%}$	$97\mathrm{\%}$	$97\mathrm{\%}$
$\text{Cov}[{\hat{u}}_{0i},{\hat{u}}_{1i}]$	Bias $\times 1000$ (MCSE $\times 1000$)	$-0.26\left(0.31\right)$	$0.28\left(0.65\right)$	$0.44\left(1.1\right)$
	Empirical SD	2.1	0.076	0.053
	Median SE (IQR)	$0.51\left(5.1\right)$	$0.093\left(0.17\right)$	$0.059\left(0.068\right)$
	CP $\mathrm{\%}$	$95\mathrm{\%}$	$96\mathrm{\%}$	$96\mathrm{\%}$
$\text{Var}\left[{\hat{u}}_{1i}\right]$	Bias $\times 1000$ (MCSE $\times 1000$)	$0.59\left(0.3\right)$	$-2.4\left(0.68\right)$	$-2.8\left(1\right)$
	Empirical SD	0.97	0.043	0.032
	Median SE (IQR)	$0.57\left(0.831\right)$	$0.043\left(0.0031\right)$	$0.032\left(0.0011\right)$
	CP $\mathrm{\%}$	$93\mathrm{\%}$	$95\mathrm{\%}$	$95\mathrm{\%}$
$\hat{\alpha }$	Bias $\times 1000$ (MCSE $\times 1000$)	$9100\left(610\right)$	$8600\left(580\right)$	$8800\left(580\right)$
	Empirical SD	0.68	0.70	0.70
	Median SE (IQR)	$2.5\left(14\right)$	$1.7\left(14\right)$	$1.7\left(14\right)$
	CP $\mathrm{\%}$	$81\mathrm{\%}$	$79\mathrm{\%}$	$79\mathrm{\%}$
$\hat{\omega }$	Bias $\times 1000$ (MCSE $\times 1000$)	$5.9\left(0.79\right)$	$9\left(0.81\right)$	$9.1\left(0.82\right)$
	Empirical SD	0.023	0.025	0.025
	Median SE (IQR)	$0.025\left(0.008\right)$	$0.029\left(0.0088\right)$	$0.029\left(0.0089\right)$
	CP $\mathrm{\%}$	$96\mathrm{\%}$	$96\mathrm{\%}$	$95\mathrm{\%}$
$\hat{\sigma }$	Bias $\times 1000$ (MCSE $\times 1000$)	$-1.3\left(0.98\right)$	$0.16\left(0.95\right)$	$0.01\left(0.96\right)$
	Empirical SD	0.12	0.13	0.13
	Median SE (IQR)	$0.48\left(1.7\right)$	$0.37\left(1.7\right)$	$0.36\left(1.7\right)$
	CP $\mathrm{\%}$	$82\mathrm{\%}$	$79\mathrm{\%}$	$79\mathrm{\%}$

Note: Optimization performed using NR algorithm and IOU parameterization $\left[\alpha \omega \right]$. Over 1000 simulated datasets: number converged, median and IQR of the number of optimization iterations (Opt. Iter.) and for each variance parameter ν: bias of the estimates of ν with Monte Carlo standard error (MCSE), standard deviation of the estimates of its transformation $\breve{\nu }$, median and IQR of the estimated standard error (SE) of $\breve{\nu }$ and empirical coverage percentage (CP) of the $95\mathrm{\%}$ confidence interval for ν. Results reported to two significant figures.

Figure 1. Line plots of height measurements from 10 students who attended Christ's Hospital School between 1936 and 1969.

Table 3. Fixed effects and variance parameter results of three models fitted to longitudinal height measurements (in metres) from Christ's Hospital School: the standard linear mixed model with a random intercept and slope (RS RE model), the random intercept IOU model (RI IOU model) and the random slope IOU model (RS IOU model). Results reported to two significant figures.

	RS RE model		RI IOU model		RS IOU model
	Estimate	Confidence Interval	Estimate	Confidence Interval	Estimate	Confidence Interval
Age (years)	0.055	$[0.054,0.055]$	0.046	$[0.045,0.046]$	0.046	$[0.045,0.047]$
Constant	0.83	$[0.82,0.84]$	0.94	$[0.93,0.95]$	0.93	$[0.92,0.95]$
$\text{Var}\left[{\hat{u}}_{0i}\right]$	0.019	$[0.018,0.021]$	0.0073	$[0.0063,0.0084]$	0.011	$[0.0093,0.013]$
$\text{Cov}[{\hat{u}}_{0i},{\hat{u}}_{1i}]$	−0.00087	$[-0.00098,-0.00077]$	−	−	−0.00054	$[-0.00066,-0.00041]$
$\text{Var}\left[{\hat{u}}_{1i}\right]$	0.000063	$[0.000057,0.000070]$	−	−	0.000026	$[0.000019,0.000037]$
$\hat{\alpha }$	−	−	1	$[0.95,1.1]$	1	$[0.98,1.1]$
$\hat{\omega }$	−	−	0.0015	$[0.0014,0.0016]$	0.0015	$[0.0014,0.0016]$
$\hat{\sigma }$	0.026	$[0.025,0.026]$	0.0086	$[0.0086,0.0087]$	0.0086	$[0.0086,0.0087]$