Looking for techniques to understand the impact of a discrete events (interventions) on a continuous response variable in time series

Question

I am trying to model the effect of one or more discrete interventions (e.g., taking a pill, attending therapy) on a continuous outcome (e.g., pain level of a patient over time). The features are discrete binary events in time series. Here's an example of how the data might look:

timestamp  took_pill  attended_phys_therapy  pain_level 
------------------------------------------------------------------
1                                            4.1
2           true                             4.0
3                                            4.2
4                                            3.1
5                     true                   2.8
6                                            2.6
7                                            2.3
8                                            2.4

In this simple example, I'm trying to capture the fact that the interventions (the subject took a pill) at time t=2 led to a change in pain at time t={4..6}.

Here are some options I am considering:

1. Apply a decay function (e.g., Gaussian, exponential) to the binary events to create a continuous feature (took_pill_decayed), and do time lag regression of pain_level ~ took_pill_decayed + attended_phys_therapy_decayed

2. Aggregate both indep and dep variables to longer time windows that would capture both the event and the outcome (say, 6-hour windows). Make a "sliding window" for each time step.

A few additional notes/assumptions:

1. The effects of the interventions are non-permanent. I've looked into ITS (interrupted time series analysis) and paired t-test analyses . However, these seem to be tailored towards semi-permanent interventions such as economic policy changes.

2. Ideally, I would also like to understand how long after an intervention the outcome was influenced, not just whether it influenced it.

Would love any suggestions!

Answer 1

You can cast this as a time-dependent covariate problem (as in a Cox proportional hazards time-to-event model or as used in econometrics for normally distributed Y) or as a Markov state transition model. For either of these, interpretation is only easy if the treatment is an "external covariate" (pre-destined pill-taking time or randomized crossover study). Effects of internal time-dependent covariates are very difficult to interpret. In a state transition model if you knew that the tendency to take a pill is a function of previous pain levels and not based on unavailable (to the model) information, you still get a reasonable result. An introduction to Markov multi-state models (state transition models) for general Y (binary, ordinal, continuous) is here.

Answer 2

What you are looking for is the de facto period as compared to the de jure period of the interevention. This can be accomplished via Intervention Detection procedures which covers both pulse effects (your case) and permanent effects (intercept changes) . One needs to simultaneously incorporate (adjust for) auto-regressive structure. http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html and elsewhere pursues the detection of the true time period of the event. If the true period is not equal to the known period then we have the answer to your "how long" question.

If you wish to post a complete data set I will try to help further. The data of course must be fixed interval readings.