# Is interrupted time series analysis appropriate with these data?

The aim of my study is to investigate the effect of a change in legislation on violent behavior. The same intervention was introduced at different times in different countries.

For my analysis, I am looking at 6 different countries that have adopted a change in legislation at very different points in time, so I have conducted one ITS analysis for every country. Derived from theory, I have proposed a step-change model for all countries, as the intervention is identical. The dependent variable was measured yearly.

Unfortunately, the quality of the data varies strongly between countries and the observation time frames are of different length. For country 1 for example, I have a nearly equal amount of pre- and post-intervention data points and the data show a visible increasing slope with little variance, so my results make sense and are easy to interpret.

For country 2 however, the variance in the data is much greater and I feel like the predicted trend does not really represent the data.

For country 3, although in theory the step change model (red) makes the most sense, a step & slope change model (green) represents the data much better. Please find some exemplary graphs attached.

From the above mentioned problems I derive the following questions: 1. Considering country 2 and looking at the data, would it even be appropriate to conduct a ITS analysis? How can I tell if the criteria are actually met and is it okay to interpret the data accordingly? 2. Looking at country 3, is there any justification to propose a different model for one country than for the other 5? Obviously, the results obtained from using the step change model seem wrong to me.

I also have 2 more general question: 3. Does it make sense to correct for seasonality in my case? As I am looking at yearly data I cannot think of obvious seasonal influences on my DV like I would with monthly data. 4. When I add a slope-change component to my model, do I interpret the coefficients in the same way as in a step change only model? Because when looking at country 3, although the “step” seems to have become smaller, the effect according to the coefficients has increased.

Thank you very much for taking your time to read this; I would really appreciate your help!

Data: country1

time,absolute,rate,pop,intervent
1,258,7.41,34828170,0
2,238,6.75,35246374,0
3,217,6.09,35657429,0
4,226,6.27,36063459,0
5,266,7.29,36467218,0
6,261,7.08,36870787,0
7,252,6.76,37275652,0
8,241,6.4,37681749,0
9,301,7.9,38087868,0
10,403,10.47,38491972,0
11,407,10.46,38892931,1
12,467,11.89,39289878,1
13,493,12.42,39684295,1
14,519,12.95,40080160,1
15,500,12.35,40482788,1
16,583,14.26,40895752,1
17,604,14.62,41320500,1
18,630,15.09,41755196,1
19,568,13.46,42196030,1
20,555,13.02,42637511,1
21,593,13.77,43075416,1
22,515,11.84,43508460,1
23,593,13.5,43937140,1
24,701,15.8,44361150,1


data country2

time,absolute,rate,pop,intervent
1,23,6.99,3291053,0
2,21,6.35,3308012,0
3,32,9.64,3319736,0
4,37,11.13,3325473,0
5,46,13.83,3326040,0
6,50,15.04,3323668,0
7,56,16.86,3321476,0
8,64,19.27,3321803,0
9,82,24.66,3325401,0
10,60,18.01,3331749,0
11,63,18.86,3340221,0
12,64,19.11,3349676,0
13,49,14.59,3359275,0
14,67,19.89,3368934,0
15,54,15.98,3378975,0
16,59,17.41,3389443,1
17,68,20.00,3400436,1
18,45,13.19,3412009,1
19,57,16.65,3424129,1
20,67,19.5,3436641,1
21,77,22.32,3449285,1


data country3

time,absolute,rate,pop,consent
1,147,17.46,8421056,0
2,136,16.07,8464787,0
3,152,17.85,8514206,0
4,128,14.94,8567384,0
5,137,15.88,8625137,0
6,125,14.39,8686738,0
7,124,14.18,8746776,0
8,108,12.28,8798234,0
9,104,11.77,8836420,0
10,99,11.17,8859191,1
11,112,12.63,8868853,1
12,129,14.54,8870848,1
13,108,12.17,8873100,1
14,97,10.92,8881640,1
15,108,12.14,8897793,1
16,98,10.99,8920710,1
17,114,12.74,8951436,1
18,123,13.68,8990654,1
19,128,14.16,9038623,1
20,137,15.06,9096165,1
21,133,14.51,9162939,1
22,152,16.46,9236428,1
23,128,13.74,9313087,1
24,118,12.57,9390168,1
25,146,15.42,9466710,1
26,143,14.99,9542812,1
27,152,15.8,9618016,1
28,166,17.13,9692131,1
29,169,17.31,9764950,1
30,195,19.83,9836007,1
31,192,19.38,9904896,1
32,181,18.15,9971638,1


The seminal work of Mcleary and others was innovative but untreated pulses often effect conclusions and should always be identified and included where necessary . Forming a model with a de jure intervention point is often less effective than actually finding the de facto time point as was suggested here http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html due to unknown delays or anticipatory effects. Post your data in a csv file and I will try and help further.

EDITED AFTER RECEIPT OF YOUR DATA:

It appears that you wish to model ABS a count series as a function of population. Converting these two series to a rate and then modelling the rate is poor practice. What is preferable is to use population coded in millions a predictor series and to assess the need for a dummy series of 0's and 1's representing a possible latent determinstic structure.

I took your 3 problems (country1 , country2 and country3 ) and assessed the importance of Population and any evidence of a latent omitted deterministic variable.

Here are the 3 Actual, Fit and Forecasts

and and

The equation for country1 ( no population input series was needed )is here ... random walk with drift

The equaaion for country2 is here using the population variable

The equation for country3 is here simply a random walk with no drift.

In summary I didn't and couldn't answer your questions because you are using a derived series (rate) which can have unfortunate consequences much like fitting a regression line through the origin.

It is always better and cleaner to use the observed data. Early researches not knowing how exactly to deal with causa series ...sometimes used rate ... with consequences.

In summary I don't see any empirical suggestion that Level Shift indicators are warranted in any of the 3 cases. I do n't see a role for assuming the nature and form of a series reflecting "interruption" give that memory has been adequately treated. If one assumes no memory then that might cause one to suggest a remedy for the omitted memory .

Hope this helps.

• thank you for your answer! I don't know how to post a csv file here, but I could send it to you? – lisannceline Sep 11 '19 at 14:12
• since your data is short to add text to list the values – IrishStat Sep 11 '19 at 14:34
• thank you, I added the data – lisannceline Sep 11 '19 at 14:54
• what is the dependent series (y) ? – IrishStat Sep 11 '19 at 15:00
• the absolute number of crimes per year (absolute), but since I performed a poission regression, I used population (pop) as an offset variable to transform the absolute numbers in to rates. For plotting I used the calculated rates (rate) – lisannceline Sep 11 '19 at 15:06

The fact that the nature of the intervention is the same in all countries does not imply that its effect (if any) must be of the same type (e.g. a vertical shift) in all cases. I think it would be all right to test using in each case the model which seems best for the country at hand. In particular, for country 3 the red fit seems to me undesirable.

Anyhow, if for some reason you absolutely want to use a single model for all three countries, a model with (possibly) a vertical shift AND a change in slope at the time of the intervention might do. Then you might test that both effects are zero (=the intervention does not appear to have any effect whatsoever) versus the alternative that a change in level or slope (or both) has occurred. The extra parameters fitted will likely carry a penalty in terms of reduced power.

With some colleagues I recently completed the following work which you may want to look at. It is a sort of dynamic Poisson regression which may apply to your problem and accounts for the time series nature of the data:

Antibiotic susceptibility trend before and after long-term use of selective digestive decontamination: a 16 year ecological study, May 2019 Journal of Antimicrobial Chemotherapy 74(8) DOI: 10.1093/jac/dkz186

• Thank you for your reply, I'll have a further look at your paper! By reduced power, do you mean I should interpret the coefficients with more caution? Because I still can't really make sense of the increased "effect size" suggested by the step chance parameter when adding one for slope change. – lisannceline Sep 11 '19 at 14:16