Appropriate statistical model for this data

[Luis Mijares Castaneda]

Hello, I have data on stolen cars from a U.S. city for the year 2024. Below is a dataex sample. The variable id identifies individuals; stolen_cars is the number of cars stolen from each individual; and the date variable records when the theft occurred. I’m interested in testing whether more cars were stolen during April 18–28, 2024. I created an indicator variable, intervention, that denotes whether the theft occurred outside this window (i.e., prior to April 18, 2024 or after April 28, 2024). I initially considered an ARIMA model, but the data are panel rather than a single time series. I also considered an interrupted time-series approach but wasn’t sure which model would be most appropriate. Would a segmented negative binomial regression be suitable? Any suggestions would be appreciated.

* Example generated by -dataex-. For more info, type help dataex
clear
input byte(id stolen_cars) str9 date byte intervention
1 1 "1-Jan-24"  0
2 3 "1-Jan-24"  0
3 1 "1-Jan-24"  0
4 2 "2-Jan-24"  0
5 1 "18-Apr-24" 1
6 1 "18-Apr-24" 1
7 1 "30-Apr-24" 2
8 1 "7-May-24"  2
9 2 "7-May-24"  2
end

[Carlo Lazzaro]

Luis:
why not going:

egen new_time=group( date)
. poisson stolen_cars i.intervention i.new_time
note: 2.new_time omitted because of collinearity.
note: 5.new_time omitted because of collinearity.

Iteration 0:  Log likelihood = -11.021237  
Iteration 1:  Log likelihood = -11.021236  

Poisson regression                                      Number of obs =      9
                                                        LR chi2(4)    =   0.75
                                                        Prob > chi2   = 0.9447
Log likelihood = -11.021236                             Pseudo R2     = 0.0330

------------------------------------------------------------------------------
 stolen_cars | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
intervention |
          1  |  -.5108255     .83666    -0.61   0.541    -2.150649    1.128998
          2  |  -.1053605   .7302967    -0.14   0.885    -1.536716    1.325995
             |
    new_time |
          2  |          0  (omitted)
          3  |   .1823217     .83666     0.22   0.827    -1.457502    1.822145
          4  |   -.405465     1.1547    -0.35   0.725    -2.668636    1.857706
          5  |          0  (omitted)
             |
       _cons |   .5108256   .4472136     1.14   0.253    -.3656969    1.387348
------------------------------------------------------------------------------

.

[Felix Bittmann]

In addition to Carlo's helpful response, it would be interesting to know the distribution and descriptives of stolen_cars. This could help you decide whether poisson or nbreg are more appropriate.

[Luis Mijares Castaneda]

the summary of the stolen_cars variable, it seems there's underdispersion so negative binomial regression instead of poisson right?

   Percentiles      Smallest
 1%            1              1
 5%            1              1
10%            1              1       Obs               2,210
25%            1              1       Sum of wgt.       2,210

50%            1                      Mean           1.115385
                        Largest       Std. dev.      .4726934
75%            1              6
90%            1              7       Variance       .2234391
95%            2              7       Skewness        7.03407
99%            3              9       Kurtosis       77.41599

.

[Felix Bittmann]

When the conditional variance is greater than the conditional mean, nbreg seems better, but in your case you can go with poisson.
https://stats.oarc.ucla.edu/stata/da...al-regression/

You can also test:

tabstat stolen_cars, by(intervention) stats(mean var n)

What is also unclear to me now: do you mean more cars stolen per thief in a given time or more thefts, in total?

[Luis Mijares Castaneda]

I thought you could use negative binomial for under-dispersion for a moment, it seems you can't

[Luis Mijares Castaneda]

When the conditional variance is greater than the conditional mean, nbreg seems better, but in your case you can go with poisson.
https://stats.oarc.ucla.edu/stata/da...al-regression/

You can also test:

tabstat stolen_cars, by(intervention) stats(mean var n)

What is also unclear to me now: do you mean more cars stolen per thief in a given time or more thefts, in total?

I’m assessing whether the daily incidence of reported car thefts increased during a 10-day city festival (April 18–28, 2024). Because this window is much shorter than the rest of the year, I focus on the rate of thefts (e.g., thefts per day) rather than total counts, comparing the festival period with non-festival days.

[Felix Bittmann]

I am not sure if this works as you intend. As far as I understand, each row in your data contains information about how many cars a thief has stolen on a given day. I read that you want to know whether thefts, in general, go up in a given period. This can either be because thief is stealing more cars per day or that simply more thieves are active. To me it makes more sense to check whether more cars, in total, get stolen in the "treatment" period.

. clear all

. set seed 123

. set obs 90
Number of observations (_N) was 0, now 90.

. gen id = _n

. gen stolen = runiformint(1,4)

. recode id (1/20 = 1 "Pre") (21/70 = 2 "Treat") (71/90 = 3 "Post"), gen(intervention)
(89 differences between id and intervention)

. tab intervention     //20 thieves are active in the pre and post phase but 50 in the treatment phase

  RECODE of |
         id |      Freq.     Percent        Cum.
------------+-----------------------------------
        Pre |         20       22.22       22.22
      Treat |         50       55.56       77.78
       Post |         20       22.22      100.00
------------+-----------------------------------
      Total |         90      100.00

.
.
. poisson stolen i.intervention    //the results show that the average number of cars stolen per day per thief is identical in all three periods

Iteration 0:  Log likelihood = -144.80096  
Iteration 1:  Log likelihood = -144.80096  

Poisson regression                                      Number of obs =     90
                                                        LR chi2(2)    =   0.82
                                                        Prob > chi2   = 0.6633
Log likelihood = -144.80096                             Pseudo R2     = 0.0028

------------------------------------------------------------------------------
      stolen | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
intervention |
      Treat  |  -.1033424   .1620072    -0.64   0.524    -.4208706    .2141859
       Post  |  -.1786918   .1998023    -0.89   0.371     -.570297    .2129135
             |
       _cons |   1.011601     .13484     7.50   0.000     .7473194    1.275882
------------------------------------------------------------------------------

. margins intervention

Adjusted predictions                                        Number of obs = 90
Model VCE: OIM

Expression: Predicted number of events, predict()

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
intervention |
        Pre  |       2.75   .3708099     7.42   0.000     2.023226    3.476774
      Treat  |       2.48   .2227106    11.14   0.000     2.043495    2.916505
       Post  |        2.3   .3391165     6.78   0.000     1.635344    2.964656
------------------------------------------------------------------------------

.
. collapse (sum) stolen, by(intervention)

. list   // Here we see that much more cars got stolen in the treatment period that in the other ones, even if the length of each period is identical (by definition of this data example)

     +-------------------+
     | interv~n   stolen |
     |-------------------|
  1. |      Pre       55 |
  2. |    Treat      124 |
  3. |     Post       46 |
     +-------------------+

This example shows data for 3 periods. We just assume that each period is 10 days. While the average cars stolen per thief are identical (poisson shows this), the treatment period has overall more stolen cars. This is only shown then collapsing the data. Is this what you want to find out?

[Luis Mijares Castaneda]

Felix Bittmann Yes, but I'm wondering if there could be a way to deal with possible autocorrelation in the time series data? Could I use an ARIMA model of some sort?

[Carlo Lazzaro]

Luis:
you can replace the default standard errors with -vce(cluster id)- if you have at least 30 -id-.
This way, the -poisson- accomodates for potential overdispersion and autocorrelation of the epsilon.