Contradictory Conclusions and TimeIntervals
In what exact circumstances will the use of a different timeinterval in data collection lead to totally different substantive conclusions in a crosslagged design? This is the question Rebecca Kuiper and myself try to tackle in our latest paper, recently accepted to the teacher’s corner section of Structural Equation Modeling.
Many authors writing about continuoustime modeling have been at pains to point out that the estimates of crosslagged and autoregressive effects resulting from discretetime VAR(1) models can differ greatly depending on the timeinterval between observations (e.g. Gollob & Reichardt, 1987; Voelkle & Oud, 2013). As such, if two researchers are studying the same phenomenon, but one uses , for example, an interval of 2 hours between measurements, and the other uses say 4 hours, they may come to totally different substantive conclusions. However, the literature is so far lacking an accessible account of when this problem rears it’s head? Rebecca and me set out to make this issue clearer; in fact we show that it is only in very specific circumstances that this is guaranteed not to occur.
Types of conclusions
In this paper we consider three common types of conclusions researchers are interested in making in designs with many repeated measurement waves, and which are typically based on the estimates of discretetime VAR(1) models.^{1}
 Is process A more stable than process B? Typically based on the comparison of two autoregressive parameters
 Does process A have a bigger effect on process B, than process B has on process A? Typically based on comparing the absolute values of two crosslagged parameters
 Does process A have a positive or negative effect on process B? That is, the sign of the crosslagged parameter
Assuming that the datagenerating process is a continuoustime VAR(1) model, then these parameters will take on different values depending on the timeinterval. Let’s further focus on the case where researchers take equal timeintervals between measurements  if this isn’t the case, then the parameter estimates will be a mixture of different true parameters anyway.
When do contradictions (not) occur?
It turns out that there is only one very specific circumstance in which we are guaranteed not to make contradictory conclusions based on the choice of time interval; when there is a truly bivariate system (so also no unobserved third process!), which behaves in a stable, nonoscillating manner; the eigenvalues of the drift matrix are negative and real (and so the eigenvalues of the DTVAR(1) model are real and between 0 and 1)^{2}. Below we have taken an example of such a system and plotted how the laggedeffects matrix changes over different time intervals.
Contradictions: The general case
In general however, contradictions can occur in all other (multivariate) cases. For example, if we have three variables, but still a stable, nonoscillating (real eigenvalues only) system, all three conclusions can differ at different choices of time interval.
Furthermore, if we have a bivariate system which is oscillating (with complex eigenvalues for the drift matrix), we can also come to contradictory conclusions using different timeintervals.
Here, different conclusions are made based on the period of the process. In the figure above, the system has a period of 2: thus, no contradictory conclusions are made for different time intervals between 0 and 2, or between 2 and 4; however the conclusions made based on a timeinterval between 0 and 2 differ markedly from those made given a time interval between 2 and 4. More details on periodicequivalence is given in the full paper.
Final Remarks
In the full paper we provide a didactical walkthrough on the issue of timeinterval dependency of parameter estimates, more technical proofs regarding this issue in appendices for those interested, and a number of tools by which researchers can transform estimated effects matrices to make easier comparisons.
We have shown that the problem of timeinterval dependent effect estimates cannot be dismissed in general: in fact, it is only in very special circumstances that we can be sure that we are not making possibly misleading conclusions based on the (often somewhat arbitrary) choice of timeintervals. The extent to which, in psychology, we are ever observing a truly bivariate system is highly questionable; indeed in recent times much of the focus of psychology research has shifted towards highly multivariate systems, as evidenced by the (dynamic) network approach. In such contexts it seems that continuoustime modelling is even more desirable  the more dimensions the system has, the greater the risk that we would make entirely different conclusions about specific effects given a different interval of measurement.
References
Gollob, H. F., & Reichardt, C. S. (1987). Taking account of time lags in causal models. Child development, 58 (1), 8092. doi: 10.2307/1130293
Kuiper, R.M., & Ryan, O. (accepted) Drawing Conclusions from CrossLagged Relationships: Reconsidering the role of the timeinterval. Structural Equation Modeling.[PDF]
Voelkle, M. C., & Oud, J. H. L. (2013). Continuous time modelling with individually varying time intervals for oscillating and nonoscillating processes. British Journal of Mathematical and Statistical Psychology, 66 , 103126.
Footnotes

Throughout we frame conclusions based on the true parameters; that is, we do not take into account uncertainty or formal hypotheses tests (e.g., is the parameter significantly different from zero)? This is to show that misleading conclusions may be made even in ideal circumstances  that is, when we can estimate the discretetime model parameters with a very high precision. ↩

For a more detailed description of eigenvalues and how they relate to the behaviour of systems, I recommend reading our book chapter which discusses CT models and ESM data, preprint available here. ↩