Interrupted Time Series Analysis with multiple Intervention timepoints

Question

I was wondering if it is possible to do a interrupted time series analysis with an intervention that is introduced and removed at specific intervals throughout the time-series.

All of the literature I am reading is saying that for a time-series analysis the assumption is that without the intervention the dependant variable will remain unchanged in the post-intervention compared to the pre-intervention.

However, my issue is that the intervention is introduced during a specific time period at set intervals only where it is expected to impact the dependant variable, then removed, and then introduced and removed again (and so on).

Is the interrupted time series analysis appropriate for this type of study design?

I've attached a picture for clarification. please let me know if I need to explain further

Answer 1

A starting point for the concept of ITSA is Shadish, Cooke, & Campbell (2002) and a starting palce for the mathematical procedures for ITSA is Glass, Wilson, & Gottman (1975).

Some researchers may recommend a dummy code moderator in a multiple regression, where 0 represents no intervention and 1 represents intervention:

$$\hat{y}=b_0+b_1(time)+b_2(intervention)$$

This is a simple solution and may work if you do not care about autocorrelation.

However, the interrupted time series (ITSA) allows you to include autoregressive and moving average components:

$$y_t=z_t=AR+I+MA+a_t$$

where $z_t$ is the observed value of teh DV at time point $t$, $AR$ is the order of autoregression of the series, $I$ is the order of differencing required to create a stationary series, $MA$ is the order of moving average of the series, and $a_t$ is the error.

Alternatively, and more precisely, an ARIMA (p, d, q) process may be modeled by:

$$y_t=Δz_t=ϕz_tΔz_{t−1}θzt$$ where $ϕ$ is the autocorrelation coefficient, $θ$ is the moving average coefficient, and $Δzt=zt−zt−d$ when $d>0$. When $d=0$, $Δzt=zt−1$ or simply $Δzt$ is ignored, depending on the order of p and q.

You can identify $ϕ$, $θ$, and $Δ$ using software, such as Rob Hyndman's auto.arima in R. The models are all different given the order of the coefficients and I do not know of any comprehensive source for all possible ITSA or a generalization thereof. Generally, though, there is a level at baseline, $L$, and a change from that level in the treatment phase, $\delta$, where the level of the treatment phase is $L+\delta$. This is similar to the dummy coding solution, but now you are incorporating the ARIMA model. You may need to derive the model yourself, as I did for an ARIMA(1,1,0) in a submitted manuscript where most of this information comes from (Raadt, in-press).

Glass, G. V., Willson, V. L., Gottman, J. M. (1975). Design and analysis of time-series experiments. Boulder, CO: Colorado Associated University Press.

Shadish, W. R., Cook, T. D., & Campbell, D. T. (2002). Experimental and quasi-experimental designs for generalized causal inference. Boston, MA, US: Houghton, Mifflin and Company.

Answer 2

I'd never heard of interrupted time series analysis before this question, and I can't really speak to how it's different from regular time series analysis that involves sophisticated-enough techniques to handle "outliers" and seasonality.

So I'd suggest looking more widely at time series literature in regards to seasonality effects -- which are sometimes confounding effects that researchers want to factor out, or are sometimes the focus of research -- and also "outliers".

Your problem sounds a lot like marketing time series problems, where marketers want to know if ad campaigns are having an impact on consumer behavior. In that case, ad campaigns are often aperiodic -- i.e. when HQ thinks sales need a boost -- but may be periodic and repeating like "Our Big Summer Sale!". And, of course, there are always seasonal effects like Christmas, summer vacations, etc, that affects marketing/sales.

You may run into issues where finance/marketing-related techniques are focused on monthly or quarterly time series and don't work very well on other timescales. State Space models are sophisticated but sometimes hard to use. ARIMA-based models might be useful. In the "outlier" literature you may see labels like AO (Additive Outlier, a one-time kind of thing), TC (Temporary Change, which sounds like your interventions), or LC (Level Change, a permanent shock).

What is your goal, actually? Do you want to evaluate the impact of interventions? Do you want to look at the time series, "factoring out" interventions? Right now it feels like your question is a bit theoretical and perhaps caught in a particular genre of time series analysis that may be too restrictive for what you want to do.