Create a standardized scale of temporal consistency The sum scores of step e have several disadvantages. First, the values are not easy to interpret substantively. Second, the range of the sum scores are sensitive to the choice of temporal distance unit here we used hours.
Third, the values are not directly comparable between two different temporal scales i. Positive values of TC obs indicate temporal consistency in the data: summarizing across all offenders, we see evidence that they repeatedly commit offenses at similar hours of day and week. A value of zero—the original midpoint of the scale in step e —indicates that, summarized across all offenders, the timings of crimes exhibit no discernible consistency i. Footnote 5 Negative values of TC obs indicate that offenses are committed at dissimilar hours of day and week.
Note that because of the standardization, the outcome of step f directly indicates the degree of temporal consistency in the data and is comparable regardless of time scale day versus week. While the result of the previous steps shows the degree of temporal offending consistency in the observed data, we cannot yet be certain that any observed consistency is due to intra -offender consistency in offending times: the temporal patterning of crime can also be the result of time-varying opportunities for crime i.
Therefore, a certain degree of consistency observed in our data could also be due to other sources of temporal variation, which would make all offenders more likely to commit crimes at certain times of the day or week.
Because the observed data are the result of a mixture of these sources of variation, we need to perform a test that is able to isolate intra-offender temporal behavior. In order to do so, we use a Monte Carlo permutation test. The basic idea of our Monte Carlo permutation test is to compare the observed temporal consistency with a simulated reference distribution of temporal consistency values from which the intra-offender temporal variation is removed. We simulate the distribution because there is no other way of knowing the remaining temporal variation, and we use a sample of permutations of the data because a full permutation is virtually impossible to calculate even with moderately sized data Johnson et al.
We randomly permute the original dataset many times to generate a distribution of temporal consistencies derived under the null hypothesis.
We subsequently compare the observed temporal consistency with the distribution of permutated temporal consistencies to assess the likelihood of observing the former. Specifically, we take the following steps to perform the Monte Carlo permutation test:.
It is important to keep the connection between the temporal information and crime types fixed for each offense, because this resembles the temporal variation caused by other sources of variation in our data. Run the permutated data through steps a through f as described in paragraph 3. Footnote 6. Repeat steps 1 and 2 times This leads to values of temporal consistency that is to be expected given the overall temporal distribution of crime events, one for each of the permutated datasets.
We will refer to the mean of the temporal consistency values of the permutated datasets as TC perm. Under the null hypothesis, it is very unlikely that many of these temporal consistency values are larger than the observed temporal consistency.
Note that an estimate of effect size of the intra -offender temporal consistency can be calculated by taking the difference between the observed temporal consistency and the temporal consistency values of the permutated datasets. The mean of these values is the best point estimate of the intra-offender effect, hereafter TC intra we discuss this in more detail in the Results section, see Fig.
Footnote 7 In addition, because the TC values are standardized, they are also comparable across temporal scales here, the daily versus weekly cycle. In the next section, the results are shown for all offenders. However, Table 1 shows that there are clear differences in the total number of offenses per offender.
Most offenders were suspected of committing only a few crimes, and only a few were much more prolific. We also tested whether the results were sensitive to our choice of using the end dates and times of the offenses see paragraph 3. In additional analyses not shown here , we observed substantively similar findings when using the starting dates and times instead of the ending records, as well as when using the midpoint between the starting and ending times.
This section shows the results for the hour of day and hour of week consistency analyses. We start our presentation of the results with Fig. Half of the crime pairs consist of crimes committed with a distance of 0—4 h. In line with our hypotheses, we see a higher proportion of crime pairs at shorter temporal distances.
Because the hour of week distances have a range of 0—84 see Fig. The hour of week proportions show that there is much intra-week variability. Note, however, that the four peaks across the 84 h of the cyclical week steadily diminish in size. These results indicate that offenders are most likely to commit multiple crimes on a similar hour of day, but also that offenders are slightly more likely to commit multiple crimes on the same hour of week otherwise the four peaks would have the same height.
For each hour distance, we also display the total range of proportions of crime pairs across the permutated datasets. For the daily cycle, it is clear that the permutated datasets also exhibit some temporal consistency, i. For the weekly cycle, the temporal consistency seems to show no discernible pattern other than the intra-week variation, but is difficult to be certain from this figure.
We next turn to the results of our actual hypotheses tests, which formally examine how unlikely the observed temporal consistency—i. We first display visually the outcome of the first hypothesis test regarding temporal consistency in individual offending patterns, i. Figure 4 left shows the hour of day consistency observed in the original data black dashed line , the distribution of temporal consistency values across the permutated datasets in grey , and a dotted line at zero for reference indicating random temporal behavior.
Footnote 8. Visualization of the statistical significance test outcome for Hypothesis 1. The figure shows the hour of day consistency left and hour of week consistency right , and presents key concepts used in the paper. On a scale of 0 random temporal behavior to 1 all offenders commit their crimes at exactly the same hour of day , the observed temporal consistency TC obs equals.
The Monte Carlo procedure allows us to separate this value into a unique intra-offender temporal consistency part and other sources of variation. The possible temporal consistency values that is to be expected given the overall temporal distribution of crime are displayed by the grey distribution, which has a mean of.
Put differently, of the total observed temporal consistency, about. While Fig. We do think it will be helpful to present the cumulative proportions of crime pairs across the hour of day and week distances, i. Figure 5 presents these observed cumulative proportions for the daily and weekly cycle in dashed red, while the cumulative proportions for permutated datasets are presented as solid lines for readability, we only draw of the outcomes, and we draw continuous lines rather than steps or bars.
The key outcomes after taking steps e through f and steps 1 through 4 are shown bottom-right. For each of our remaining hypotheses and sensitivity analyses, we will plot the cumulative proportions of crime pairs by hour distance and provide the key values of our tests embedded in the figure. Key values are displayed in the bottom-right corner. The p -values indicate whether TC obs is significantly larger than expected in its respective plot Color figure online.
With regard to our first hypothesis, we conclude that offenses committed by the same offender are more temporally consistent than is to be expected based on the overall temporal distribution of crime. We now turn to Hypothesis 2, which refers to an effect difference : we expect temporal consistency to be stronger for crime pairs of the same type of crime than for pairs of different crime types.
To test this hypothesis, the basic significance test needs to be adjusted slightly. We need to compare the observed temporal consistency and the temporal consistency values of the permutated datasets between 1 pairs of crimes of the same type and between 2 pairs of crimes of different types.
It is important that each offender in the permutated data still commits the same types of crimes with the same frequency as in the observed data, so that the number of same crime-type pairs and the number of different crime-type pairs remains the same.
Therefore, we need to carry out the permutations independently within each crime type. That is, we first permute the offender IDs for violent crimes, then for property crimes, and so on. After carrying out these adjusted permutations for the seven different types of crime, we calculated the standardized sum of the observed cumulative proportions and the standardized sum of the cumulative proportions in each of the permutated datasets separately for same crime-type pairs and for different crime-type pairs.
The cumulative proportions for the same type and different type crime pairs are presented in Fig. The p -values in the figure refer to the deviation of the standardized sum score of the dashed red line from the distribution of standardized sum scores of solid lines.
The second hypothesis refers to the difference between the left and right parts of the figure, or put informally; is TC intra for same crime-type pairs significantly higher than TC intra for different crime-type pairs? For both hour of day Fig. Note, however, that the differences are very small:. Cumulative proportions of crime pairs for hour of day top and hour of week bottom , for crime pairs of the same crime type left and crime pairs of a different crime type right.
Importantly, the effect size differences are also substantively meaningful for these offender groups: the differences between TC intra same type and TC intra different type now approach a 0.
In line with our second hypothesis, we conclude that temporal consistency in offending is stronger for offenses of the same type of crime than for offenses of a different type of crime, especially when looking at offenders that commit either 2, 3, 4 or 5 crimes.
Hypothesis 3 also implies an effect difference: we expect that intra-offender temporal consistency is stronger for crimes that are committed within a shorter time span. Two tests are performed: a comparison of 0—30 days one month versus 31— days one month to half a year , and a comparison of 31— versus — days half a year to three years.
For hour of day, the cumulative proportions of crime pairs for the three different recency categories are presented in Fig. The effect is also substantively meaningful, with mean expectation of. Again, these sensitivity tests highlight the importance of separating prolific offenders from the majority of offenders. Cumulative proportions of crime pairs for hour of day within 0—30 days left , 31— days middle and — days right. As displayed in Fig. The mean size of the difference is.
Note that the TC intra values for hour of week are much higher for these less prolific offenders: crime pairs within the month show striking weekly consistency, with an overabundance of crime pairs with relatively similar hours of week, in contrast to our main result of Hypothesis 1.
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