1. Model Based Monitoring in Continuous Time Han Oud
Radboud University Nijmegen, The Netherlands

2. Outline Longitudinal analysis methods in social science
- Autoregressive Cross-Lagged Panel (AR) model (Lazarsfeld, 1948)
- Latent Trajectory (LT) model (Rao, 1965; Bryk & Raudenbusch, 1992; Willett & Sayer, 1994)
- Autoregressive Latent Trajectory (ALT) model (Curran & Bollen, 2001; Bollen & Curran, 2004, 2006)
- All of them show serious problems which can be solved by continuous time
modeling (Delsing & Oud, 2008; Oud & Delsing, 2010; Oud, 2011; Voelkle,
Oud, Davidov, Schmidt, 2012; Voelkle & Oud, 2013)
- All of them stop before becoming practically useful in (a) monitoring and
(b) predicting development as well as (c) measuring intervention effects for
individual sample units
How both deficiencies can be met will be illustrated by means of a well-known ALT-example from the literature:
- Antisocial Behavior and Depressive Symptoms (Curran & Bollen, 2001)

3. - The analyses have been done by different less userfriendly programs but all will be done in the future by an extended version of the
CT- SEM program
(Voelkle & Oud, 2011)
- CT-SEM because continuous-time analysis by means of SEM
- Very flexible because based on R and SEM-package OpenMx
- Much more flexible than a previous on Mx based version in allowing (a) an
arbitrary number of time points, (b) an arbitrary number of variables, and
(c) offering different estimation methods for the highly nonlinear
continuous time model
- At the moment an extension of the program is constructed that does the
model-based monitoring, prediction and intervention effect measurement
in continuous time
- The optimal procedures used in monitoring, prediction and intervention
effect measurement are the Kalman filter and smoother

4. 8888 y x x x y Problems with AR model Δt = 0.75
Antisocial Behavior x
Δt = 0.75
Δt = 1.25
Depressive Symptoms
Problems with AR model x x y
Problem 1: In the case of unequal observation intervals, the AR effects (autoregressions and cross-lagged coefficients) of successive intervals cannot be related. Especially no equality constraints or comparisons are possible between the observation intervals, even if the actual underlying continuous time effects (auto-effects and cross-effects) are equal.
Because studies with different intervals cannot be compared, cumulativity of science is in jeopardy
Citing of previous studies with different intervals does not make sense

5. Cross-lagged effect or impulse-response functions
x1→ x2
x2→ x1

6. Problem 2: Oscillating movements often go unnoticed in discrete time,
as they are masked by the discrete-time screen

7. Problem 3, …

8. Illustrative ALT /CALT Example
Research Question:
Reciprocal associations between antisocial behavior and depressive symptoms: AB → DS, DS → AB or both ?
Results Curran & Bollen (2001)
Autoregressive Cross-Lagged Panel (AR) model
antisocial behavior → depressive symptoms
Latent Trajectory (LT) model
depressive symptoms (intercept) → antisocial behavior (slope)
Autoregressive latent trajectory (ALT) model

10. Continuous Time Results (Delsing & Oud, 2008)
Full CALT model
χ2(12) = 33.7, RMSEA=0.10
However, neither any of the slope variances nor any of the slope covariances with other variables was significant.
χ2 –difference test for eliminating slope variables gave:
χ2dif (11) = 22.8 n.s.
CALT model without slope variables
χ2(23) = 56.5, RMSEA=0.09

11. Oscillators and stabilizer Cross-lagged effect functions
| 1.1
Cross-lagged effect functions

12. Application of the model: predictions and interpolations
Kalman smoother estimates for this child (confidence intervals)
Subject-specific means for a child with 2, 4, 5 5
Means
←Interpolation Prediction→
| |

13. N=1 intervention analysis in continuous time
(hypothetical) actual development after 1992

14. Thank you for your attention!

15. Second-order continuous-time modelLT
slope a: -time-unspecific
change variable
-unstable
Second-order continuous-time model
(Oud, 2010)
derivative : -time-specific
change variable
-unstable or stable

16. auto- and cross-effectsAR
auto- and cross-effects LT
linear component
time-unspecific relations between time-unspecific curve factors (i.e. noncausal in this respect)
cov(αx,βx) = auto-effect
cov(αx,βy) = cross-effect t0 t1 t2 t3 1 2 3 βx βy αx αy
time-specific influences (Bollen & Curran, 2006, p. 208), nonanticipative (causality property) y x
yt = ayt-1+bxt-1
yt - yt-1= (a-1)yt-1+bxt-1
3. …

17. auto- and cross-effects
Problems with LT model AR
auto- and cross-effects + - σ2 μ
unstable by definition LT
linear component
1. stable or unstable:
can be assessed empirically
(by means of the eigenvalues
of autoregressive or drift matrix)
not only linear but polynomial components in general (Bollen & Curran, 2006, p )