### Student Notebook

# Time-Series Methods in Experimental Research

For many experimental psychologists, the go-to methodological designs are cross-sectional. Cross-sectional studies involve measuring the relationship between some variable(s) of interest at one point in time; some common examples include single-session lab studies and online surveys (e.g., via MTurk). These designs can be useful for isolating relationships between variables, establishing conditions of convergent and discriminant validity, and utilizing samples that are statistically representative of larger populations. Nevertheless, quantitative researchers have noted that attempts to measure and analyze interindividual variation are incomplete without an accompanying account of the underlying temporal dynamics that define these processes (e.g., Molenaar, 2008; Molenaar, Huizenga, & Nesselroade, 2002). This claim follows from the idea that cross-sectional designs, while potentially well-suited for large samples, are often underpowered, overgeneralized, and ill-approximated to the statistical assumptions implied by general linear methods. For these reasons, psychological scientists should consider supplementing their methodological toolkits with time-series techniques to explicitly investigate the time-dependent variation that can be observed within individual subjects.

The purpose of this article is to briefly discuss the importance of time-series methods in experimental research and to acquaint the reader with some statistical techniques that are easily accessible and can be employed when testing hypotheses with time-series data.

## Measuring Behavior as a Time Series

According to Daniel T. Kaplan and Leon Glass (1995), there are two critical features of a time series that differentiate it from cross-sectional data-collection procedures:

**Repeated measurements of a given behavior are taken across time at equally spaced intervals.**Taking multiple measurements is essential for understanding how any given behavior unfolds over time, and doing so at equal intervals affords a clear investigation of how the dynamics of that behavior manifest at distinct time scales.

**The temporal ordering of measurements is preserved.**Doing so is the only way to fully examine the dynamics governing a particular process. If we expect that a given stimulus will influence the development of a behavior in a particular way, utilizing summary statistics will completely ignore the temporal ordering of the data and likely occlude one’s view of important behavioral dynamics.

Linear computations such as mean and variance merely describe global properties of a data set and thus may fail to capture meaningful patterns that only can be identified by looking at the sequential dependency between time points. Consequently, time-series techniques provide a valuable approach in studying psychological processes, which are, by their very nature, fundamentally embedded in time. (For a more detailed treatment of this subject, see Deboeck, Montpetit, Bergeman, & Boker, 2009.)

## Analyzing Time-Series Data

Once you’ve collected a series of behavioral measurements on your variable(s) of interest, there are a variety of ways to explore and quantify the observed dynamics. Here are a few techniques that can be used to investigate patterns within time-series data:

**Autocorrelation/Cross-correlation.** An autocorrelation reflects the magnitude of time dependency between observations within a time series. An autocorrelation plot depicts correlations between measurements *X _{t}* and

*X*, such that each value represents the extent to which any given behavior is related to previous behaviors within the series. A cross-correlation involves relating two time series that are shifted in time at lag

_{t+n}*n*(i.e.,

*X*and

_{t}*Y*), and can reveal, for example, whether one process tends to “lead” the other’s behavior or whether they oscillate together.

_{t+n}**Recurrence quantification analysis (RQA).** RQA begins by simply plotting a time series against itself (i.e., *X _{t}* against

*X*) and then quantifies whether certain states of the behavior remain stable or recur in time, as well as what percentage of the series is constituted by deterministic patterns. Cross-RQA also can be used to analyze the degree of recurrence and deterministic patterning between two processes, and it has been applied to the study of interpersonal coordination and postural control (e.g., Shockley, Santana, & Fowler, 2003) as well as to the quantification of emotional synchrony in dyadic conflict discussions (Main, Paxton, & Dale, 2016).

_{t}**Phase space reconstruction (PSR).** When obtaining a behavioral time-series, one of your goals could be to determine what variables are involved in producing particular patterns of behavior and what the possible structure of the underlying dynamics may be. One way to accomplish this is to reconstruct the phase space, which is a multidimensional plot that represents all possible states within the process and can be used to approximate the number of variables involved in producing the observed behavioral changes. For example, we may interpret high trait self-esteem as representing a strong tendency for an individual to adopt and maintain positive self-evaluations. Collecting repeated measurements of state self-esteem and then performing a PSR could help describe the strength of that individual’s tendency to retain a positive image of herself as well as reveal the compensatory dynamics that follow from a negative self-evaluative state.

**Spectral analysis.** Mathematically, any time series can be transformed into a linear composition of sine and cosine waves with varying frequencies. One goal in analyzing

time-series data is often to find out what deterministic cycles (i.e., which of the component waves) account for the most variance within the series. Performing a spectral decomposition transforms a time series into a set of constituent sine and cosine waves that then are used to calculate the series’ power spectral density function (PSD). Plotting the series’ PSD reveals the squared correlations between each component frequency and the series as a whole, yielding a similarly intuitive interpretation to *R ^{2} *in multiple regression. In this vein, Gottschalk, Bauer, and Whybrow (1995) applied spectral analysis toward studying the changes in self-reported mood among bipolar patients and control subjects, finding that bipolar individuals tended to exhibit cyclical patterns of mood change that were significantly more chaotic and deterministic than the comparatively random fluctuations observed in control subjects.

**Differential equation modeling.** Essentially, differential equations allow one to study how different variables change with one another as well as how the state of one variable can be influenced by how it is changing (Deboeck & Bergeman, 2013). Derivative estimates of a single time series can be calculated by a number of different techniques from which differential equations then are constructed and tested based on the researcher’s predictions about how those variables are related. An intuitive example of this might be in considering a committed romantic relationship, in which changes in one person’s level of emotional satisfaction conceivably lead to changes in their partner’s level of satisfaction and vice versa. Each partner’s feelings might be coupled with the other’s in a complex manner, such that differential equations could be used to model their emotional relationship and show how changes in one person’s mood are inextricably linked with changes in the other’s mood.

## Applying These Techniques to Your Research

Though these methods may appear foreign and somewhat challenging at first, they quickly become more intuitive once seen in an applied context. The above list represents only some of the more common techniques used in time-series analysis, especially those that have been applied successfully within the psychological sciences.

## References and Further Reading

Deboeck, P. R., & Bergeman, C. S. (2013). The reservoir model: A differential equation model of psychological regulation. *Psychological Methods, 18*, 237–256.

Deboeck, P. R., Montpetit, M. A., Bergeman, C. S., & Boker, S. M. (2009). Using derivative estimates to describe intraindividual variability at multiple time scales. *Psychological Methods, 14*, 367–386.

Gottschalk, A., Bauer, M. S., & Whybrow, P. C. (1995). Evidence of chaotic mood variation in bipolar disorder. *Archives of General Psychiatry, 52*, 947–959.

Holden, J. G., Riley, M. A., Gao, J., & Torre, K. (2013). *Fractal analyses: Statistical and methodological innovations and best practices. *Retrieved September 13, 2016, from http://www.frontiersin.org/books/Fractal_Analyses_Statistical_And_Methodological_Innovations_And_Best_Practices/179

Kaplan, D., & Glass, L. (1995). *Understanding nonlinear dynamics*. New York, NY: Springer.

Main, A., Paxton, A., & Dale, R. (2016). An exploratory analysis of emotion dynamics between mothers and adolescents during conflict discussions. *Emotion*. Advance online publication. doi:10.1037/emo0000180

Molenaar, P. C. M. (2008). Consequences of the ergodic theorems for classical test theory, factor analysis, and the analysis of developmental processes. In S. M. Hofer & D. F. Alwin (Eds.), *Handbook of cognitive aging: Interdisciplinary perspectives* (pp. 90–104). Thousand Oaks, CA: SAGE Publications.

Molenaar, P. C. M., Huizenga, H. M., & Nesselroade, J. R. (2003). The relationship between the structure of interindividual and intraindividual variability: A theoretical and empirical vindication of Developmental Systems Theory. In U. M. Staudinger & U. Lindenberger (Eds.), *Understanding human development *(pp. 339–360). Dordrecht, the Netherlands: Kluwer.

Riley, M. A., & Van Orden, G. C. (2005). *Tutorials in contemporary nonlinear methods for the behavioral sciences*. Retrieved March 1, 2005, from http://www.nsf.gov/sbe/bcs/pac/nmbs/nmbs.jsp

Shockley, K., Santana, M. V., & Fowler, C. A. (2003). Mutual interpersonal postural constraints are involved in cooperative conversation. *Journal of Experimental Psychology: Human Perception and Performance, 29*, 326–332.