This paper illustrates the use of exact tests for identifying if a series of events display nonrandom
patterns with respect to the day of the week. The test developed is useful to a crime
analyst who often is examining a small number of related crime events and wants to establish
whether those events show any clustering with respect to the day of the week. A common
application is to establish whether a serial offender, one who commits multiple crimes over
an extended period of time (Rossmo, 2000), has a tendency to commit crimes more often
on certain days of the week. Such serial events frequently only have a few observations,
and the test detailed in this paper allows an objective statistical criteria to establish whether
there is a day of the week clustering whilst controlling Type I error rates.
If the day of the week patterns for a serial offender are non-random, it suggests a strategy,
such as targeted patrol or surveillance is feasible on those particular days that are more
likely for an offence to occur to either catch the suspect in future acts or at least deter the
offender. The utility of such tests becomes more obvious by the fact that an analyst will
need to make such decisions on a regular basis, and an analyst needs to triage the regular noise that occurs in identifying patterns. That is, an analyst both has to determine when a
pattern is random, to prevent taking action that is wasteful of resources, and make quick
decisions about when a pattern is non-random, because they typically monitor multiple
crime series near constantly.
Repeated criminal behaviour is not likely to be random. For one example, an offender’s
behaviour may be influenced by their routine activities. If on an offender’s way home from
work they observe a work-site for a vacant home, it provides a clue as to a place that could
be victimised. In that example, the offender would only observe the potential target on days
that they travel to-and-from work. Other cycles of victim behaviour may precipitate
offending on certain days of the week, such as a larger number of intoxicated individuals
leaving a bar on weekends or children leaving school during weekdays. Clustering in
spatial patterns of offending is well established (Hering & Bair, 2014; Rossmo, 2012),
and here the temporal analysis of repeat offending is of interest (Felson & Poulsen,
2003; Haberman & Ratcliffe, 2012; Johnson, Bowers, & Pease, 2012).
More common statistical tests rely on larger samples, but a crime analyst often only has a
few example cases to work with when trying to identify patterns. So in place of typical asymptotic
statistics, this paper shows how to generate exact test statistics given all potential
permutations of the N crimes in seven day-of-week bins.
The goal of such a test is to provide a simple measure of whether clustering occurs in the
data. Thus, it is a bit distinct from tests such as geographic offender profiling, which
provide predictions of where the offender is most likely to live (Block & Bernasco,
2009). The test is more akin to deciding whether clustering occurs at the onset, which is
much more difficult to ascertain in day of week patterns than it is in offence locations on
a map. Operationally, this is similar to some of the original goals of null hypothesis significance
testing to determine deviations from random that are worthy of further investigation
(Student, 1908). Using the exact distribution has the benefit that the analyst can specifically
control Type I error rates, as well as articulate simple look up tables. This is opposed to
relying on simulations (van Koppen, Elffers, & Ruiter, 2011), or relying on inappropriate
asymptotic approximations. Also it is a simple, ideographic test (Canter, Hammond,
Youngs, & Juszczak, 2013) that can easily be calculated by hand or in a spreadsheet.
This research finds that the likelihood ratio G-test is quite powerful (more so than the typical
chi-square test) and that an analyst only need to observe three crimes occurring on the
same day to conclude the patterns are non-random with a p-value of less than 0.05. This research
also shows how to calculate the G-test statistic and provide a spreadsheet to allow
an analyst to easily calculate the statistic for their own data or use a reference table in the Appendix
to see examples of non-random patterns. Three case studies of using the test statistic in
situations that are typical of a crime analyst are then discussed. The first is a series of thefts of
catalytic converters that are interval censored, for example, a crime occurred sometime between
Sunday and Monday (Ferson, Kreinovich, Hajagos, Oberkampf, & Ginzburg, 2007).
The second is a series of gang shootings over a year. The third is a set of arsons by year
and police zone. The first is an example of examining serial behaviour of one individual,
whilst the latter two try to identify clustering in a wider set of data.

At the onset, it seems safe to assume that repeated offending behaviour according to
weekly cycles is not likely to be reflective of individual personality characteristics but is more likely a function of the overall context in which that offender commits crime (Canter,
2004). Whilst days of the week are purely a human construct, it is one that is pervasive in
deciding the daily activities of individuals. Thus, the same type of rituals that defines most
all of society’s daily routines are likely to have significant influence on a variety of criminal
behaviour as well. As such, the theoretical reasons why certain crime behaviours may show
cyclical day of the week patterns are entirely based on how routine activities shape the
behaviour of both offenders and victims.
For an example, Rengert and Wasilchick (2000) describe the target search of some
serial burglars as being conditioned on identifying potential target homes whilst travelling
to-and-from work. Such a pattern was used to explain the spatial distribution of
offence locations but could also be used to explain temporal patterns as well. In a trivial
sense, offenders can only commit crime when they are free from other obligations. So an
offender may have the most opportune time to commit a burglary whilst travelling to or
from work, but at the least an offender cannot commit the burglary whilst they are
expected to physically be at work. Rengert and Wasilchick (2000) also use this to theorise
that women have more limited temporal opportunities to commit crime because of
increased familial burdens. Juveniles in school are likely an additional sub-group that
is more limited in their temporal opportunities to commit crime. Ratcliffe (2006) describes
this as temporal constraint theory.
Such consideration of routine activities not only applies to offenders but also applies to
victims as well. Offences such as interpersonal assaults are frequently increased on weekends,
when more individuals are out at social gatherings such as bars. Burglaries on the
other hand are frequently perpetuated when the house is empty, which occurs most often
during the daytime on weekdays (Cohen & Felson, 1979). It is likely the case that temporal
offending patterns are a priori a possibility for a wide variety of offences, simply because
many individuals have such constraints for all of their daily activities, including committing
crime.
Whilst examination of geographic patterns of individual crime series is well studied,
temporal patterns have received much less attention. Whilst geographic clustering is often
obvious when making simple pin maps, temporal patterns are likely much more difficult
for an analyst to easily discern in an ad-hoc manner. The temporal characteristic of crime
events most oft studied is that they are frequently nearby in space and time. Near-repeat
victimization patterns have been established for a variety of different crime types, such
as robbery, motor-vehicle theft, burglary, and shootings (Youstin, Nobles, Ward, & Cook,
2011). Also the time in between offences being shorter is a good indicator that cases are
linked to the same offender as well (Markson et al., 2010).
Assessment of seasonal cycles is another popular topic in criminology and has mainly
focused on whether inter-personal violence is increased in hotter weather (McDowall,
Loftin, & Pate, 2012). Assessment of time of day and day-of-week patterns is much less
prevalent though. This is despite the fact that time of the day is likely the best predictor
for the probability that a crime will occur (Felson & Poulsen, 2003).
There is not as much discrimination between temporal patterns of crimes between days
of the week as there are within the time of day, but it is generally accepted that certain
crimes are more-or-less likely to occur during weekdays versus weekends. One example
is that Andresen and Malleson (2015) identify that burglaries and assaults have distinct
weekly patterns in Vancouver, with assaults having a higher probability of occurring on
the weekends and burglaries having a higher probability occurring during the week (and
robberies showing no obvious weekly pattern).

Asimple test for evaluating whether crimes occur randomly across different days of the week
is the chi-square test. For the current application, it is expected that only a few crimes are observed,
and the conservative rule-of-thumb for the goodness of fit Pearson χ2 test to be valid is
to have at least an expected number of five observations per cell. This would mean to use the
asymptotic distribution for testing randomness in day-of-week patterns an analyst would need
to observe a total of 35 events. This sample size is not realistic for most serial crime events a
crime analyst would be interested in, especially for examining serial offending.
Although other sample size recommendations for the χ2 test exist, the small samples of
interest here are unlikely to meet any of them (Roscoe & Byars, 1971). An alternative to
using the asymptotic distribution though is to generate an exact distribution of the null
hypothesis based on articulating all potential permutations and assigning them a particular
probability based on the null hypothesis. Tate and Hyer (1973) find that the exact probabilities
diverge greatly from the asymptotic distribution for χ2 tests with a small number of
bins and observations, so simply using the asymptotic distribution may be misleading.
The number of potential permutations for N crimes in M bins is given by the binomial
formula:

This results in relatively few potential permutations that can be easily tabulated by the
computer for smaller values of M and N. For M= 7 (days of the week) and 35 crimes,
the total number of permutations is still under 5 million (4,496,388 to be exact).
The procedure to generate the null exact reference distribution is as follows:
• Generate all potential permutations of N crimes in M= 7 weekday bins.
• For each of the permutations, calculate the probability of observing that outcome under
the null hypothesis. Here, the null is that each crime has a multinomial distribution in
which the days of the week are the outcomes, and all days are equiprobable (e.g. each
day has a probability of 1/M= 1/7 of being selected). If the number of crimes observed
on each day of the week equal xi for x1 to x7, the multinomial probability mass function
is N!
• Generate the test statistic for all of the permutations.
• Sort the test statistics in ascending order and calculate the cumulative probability of the
test statistic under the null distribution.
From this information you can generate critical values for the test statistic under the null
using any statistical test. The most frequent test used is the Pearson χ2 test, but conducting
power analysis, this research finds that the likelihood ratio G-test (Cressie & Read, 1984) is
more powerful when there are eight or more crimes and is equal in power for fewer crimes.
Because this is likely a novel test, it is introduced here. The formula for the G statistic is:

where Oi is the observed count in the day of the week, and Ei is the expected value in that
day of the week. The sum is subsequently only taken over days of the week in which at least one value is observed. This test has the same asymptotic chi-square distribution as does
the Pearson χ2 test, which is six in this application (one minus seven days of the week bins).
For an example of generating the exact null distribution of the test statistic, with three
crimes, there are only three different possible G values out of the 84 different combinations
(because order does not matter for the G statistic). A set of crimes, {Monday, Monday,
Monday}, produce the same G statistic as {Wednesday, Wednesday, Wednesday}. With
three crimes Ei = 3/7, and all bins have the same expected value. So the day with two crimes
would generate a value of 2  ln(2/[3/7])≈3.08, the day with one crime would generate a
value of 1  ln(1/[3/7])≈0.85, and the other five days with zero crimes do not factor into
the equation. So the sum of all days equals 3.93, and so the G statistic is 2  3.93 = 7.86.
The three possible outcomes for the G statistic for three crimes in seven days are:
(A) All three events in the same day, for example, {Mon, Mon, Mon}. G= 11.68
(B) Two events in one day and one event in another day, for example, {Wed,Wed,Fri}.
G= 7.86
(C) All three events in different days, for example, {Mon,Tue,Thu}. G= 5.08
Table 1 shows the cumulative distribution function (CDF) forG-test and the three possible
values it takes. The null hypothesis that the probability of a crime on each weekday is equal can
be rejected at a test level of 0.05 if three crimes occur on the same day, because this outcome
falls past the 95th percentile of the null distribution.1 In practice, an analyst would want to
make sure that the window of possible days actually spans a whole week, for example, over
a month a specific offence was committed on three separate Mondays. Offenders can also have
patterns of near-repeat offences in space and time (Johnson, Summers,&Pease, 2009), and so
a spree of events all in one day is just as likely to indicate the following day more crimes will
occur as it is a proclivity to committing offences on that day of the week.
As stated prior, upon conducted power analysis it was found that theG-test was more powerful
than the Pearson χ2 test for this application.2 The power of a test is the probability that a
test rejects the null hypothesis when the alternative hypothesis is true (Kutner, Nachtsheim,&
Neter, 2004).One minus the power of the test is subsequently the probability of making a Type
II error. It is important to consider the power of tests before recommending its usage. If the test
is severely underpowered, not only is an analyst likely tomakemore Type II errors but when a
test does reject the null, an analyst is likely tomakeerroneous conclusions about the strength of
the relationship. This is because the test can only reject the null in extreme, outlying situations.
A test with little power has little utility in practice.
To calculate the power of a particular statistical test, an alternative under which the
data are generated needs to be specified. Here, a realistic set of alternative data generating
processes to consider are if the offender has zero probability of committing crimes on
certain days of the week. For the set of alternative processes used, the remaining days
with a positive probability will be equally split, for example, if there is a probability of
committing crime on three days, the probability on those three particular days equals 1/3. Distributing the probabilities unequally over the days results in more power to reject
the null, so these are conservative estimates.
Table 2 displays the power of the exact G-test using a critical level of α = 0.05. The rows
are the total number of crimes observed, and the columns are for under different alternative
hypotheses with the probability split across 2, 3, 4, and 5 days of the week.
Because of the discrete nature of the test statistics, the power oscillates at lower numbers.
With at least three crimes, the G-test has 100% power for an alternative process of crime
being committed only on one day during the week. For only two days of the week having
a positive probability, the G-test has low power when only observing three crimes but
reaches 100% for five crimes. When conducting randomised experiments, typically the analyst
strives for at least 80% power. This shows that the test is quite powerful in a set of
varied circumstances for a number of crimes as low as 18. Significant probabilities over
five or more days of the week are probably not of much interest to analysts, as that covers
too large a time period to focus attention on specific hot times.

The utility of such tests is illustrated using three example case studies from the same
jurisdiction. These examples were chosen to be typical of the types of data a crime analyst
usually have available at their disposal; they are not intended to make any generalizable
statement about broader criminal behaviour. They are short series idiosyncratic to this particular
jurisdiction, but these are the types of events that a crime analyst regularly encounters.

The first case study is an example of four thefts of catalytic converters. Catalytic converters
are devices that reduce the toxicity of exhaust on automobiles. They are a popular target for theft because they contain the metal platinum, and metal prices have been increasing in the
recent years (Ashby, Bowers, Borrion, & Fujiymam, 2014) making them continually more
expensive. In this case, the thefts were targeted towards larger vans or moving trucks
(which have larger and subsequently more valuable catalytic converters), and the offender
likely cut the catalytic converter very quickly with a power tool, such as a portable reciprocating
saw.
Table 3 displays a set of four crimes that occurred between September and October in
2013 in one city. Thefts of catalytic converters are otherwise rare occurrence in this particular
jurisdiction (besides a few other spurts of similar thefts earlier in the year). Thus, it
seems a fairly safe inference to link these events to the same offender simply based on
the modus operandi, despite the lack of other evidence. Being able to commit the crime
quickly, know that larger trucks are more valuable targets and have an established fence
to dispose of the goods all take a level of sophistication not typical of most thefts.
Dashes represent that the time of the theft within the day is entirely unknown; only the begin
and the end dates are known. Two other catalytic converter thefts that had dates that spanned
over two weeks in the time period are not included in the analysis—it seems unlikely that such
a wide occurrence will be able to provide any substantive information on day ofweek patterns.
Uncertainty in thefts is often associated with the routine activities of the victims. For
example, large uncertain times often occur over-night when the victim is sleeping, or
during the day when the victim is at work. If the series of crimes all occur overnight, an
analyst might shift the unit of analysis to overnight instead of days, that is, two days spanning
[12:00–12:00) instead of one day spanning [00:00–00:00) in military time. Felson and
Poulsen (2003) generally suggest that cut off for delimiting days starts at 05:00 instead of
midnight. But in this example, such a shift would not help, as two of the incidents have
completely unknown begin and end times.
How is an analyst supposed to handle testing for randomness when there is uncertainty
in the days? One approach is to generate all the potential permutations for day of week patterns,
and conduct the same test for each potential set of dates that the crimes occurred.
If d equals the number of the days of the week that a particular crime incident covers for
one crime, the total number of day-of-week permutations for a set of i to n crimes equals
Π di, the product of each di. For this particular data set, this product equals a total of twelve
permutations. Table 4 shows those twelve possible permutations for the days of the week
and their corresponding G-test statistics. The appendix contains tables for critical values of
N crimes in seven day-of-week bins for three through 35 crimes for the G-test. For four
crimes and a significance level of 0.05, the critical value for the G-test is 15.57. In this
example, all possible arrangements of the dates fail to reject the null hypothesis of each
day being chosen with the same probability. So despite there being partial overlap in the
first three crimes, there is no possible arrangement of this data, which rejects the null.

For four crimes and a significance level of 0.05 for the G-test, the test would need to
observe all four crimes on the same day to reject the null.
For other datasets with interval censoring, possible results are either that all of the
permutations would reject the null or that some permutations reject and others fail to reject.
In the first case, the interpretation is clear; the latter has an amount of uncertainty about
how to interpret the data. In the second case, either the analyst needs to reduce the uncertainty
in the times of the events (e.g. obtain surveillance footage limiting the potential dates
of the offences) or wait for more crimes to occur in the series to make inferences given the
uncertainty in the data.
When making recommendations based on particular patterns, it is important for an
analyst to identify the differences between random data and non-random data. Recommendations
that focus on the day-of -week should not be made when the patterns fail
to reject the null that the data are random with respect to the day of the week, as any
response conditional on day of week differences would be inefficient. For example,
surveillance on a particular day would not be recommended based on these observed
events, presuming the department would rather have a higher probability than random
on picking the right day of week.
In the case that only a few crimes are observed in the series but the patterns do not reject
the null the data are random, the analyst should still continue to monitor the temporal patterns
in the series if more crimes occur. It may be the case that when more events are reported
in the series, the sample may reject the null. But with the power of the test being
quite high for a variety of circumstances, when around seven cases are reached and the null
is not rejected, it is quite strong evidence that there is not clustering in at least three or
fewer days of the week. Shootings are often retaliatory in nature, and thus, the risk of a shooting is often increased
in a short time span following a shooting (Ratcliffe & Rengert, 2008). But the ability to victimise
a particular individual is conditioned on them being available, often in a public
space. Thus, it may be expected that there are potential day-of-week patterns guided by
the routine activities of offenders and/or victims. Or it may be the case that patterns are random with respect to the day-of-week and are based on serendipitous circumstances,
which bring potential gun offenders and victims in the same area (Jacobs, 2010).
Unlike many profiling cases, examining such a series of shootings involves a heterogeneous
number of individuals, albeit ones that may be connected with one another in a
myriad of ways.3 The same goals of the analyst apply though whether the series involves
one or many individuals—does the series show sufficient day-of-week patterns to justify a
particular response by the police agency? Or similarly, is there any evidence of a particular
day-of-week pattern that an analyst may further explore or exploit to identify when
shootings are most likely to occur in the future? Identifying cases in which there are a
weekly pattern may encourage the analyst or detective to dig down into the records to
identify other relationships, specifically focusing on the shootings that occur on the same
days of the week. Also having a quick test when data patterns are not random is useful,
as this prevents wasting time exploring random patterns in the data.
The nature of the gang conflict in this city is that there was one local gang that accounted
for the majority of violence and was often victims of shootings by one gang in a nearby
city. Field intelligence and monitoring of social media suggested that these two gangs
had continual conflict over multiple disputes throughout the time period. Whilst there have
been incidents of direct retaliation between the two groups within the same day or the day
after, each group had potential motivation to prompt a shooting throughout the year.
In this jurisdiction, there was a total of 31 gang related shootings in 2014. The first
criterion to define a gang shooting was that physical evidence was collected at the scene,
that is, the presence of shell casings, a victim, or a bullet hole in property. Cases with just
a report of shots fired were not considered. The second criterion was that the location of the
shooting was in the one hot spot area associated with the majority of the gang shootings or
that a gang member had some involvement (either as a suspect, a victim or was seen
loitering in the area before or after the shooting). Of these 31 shootings using incident
narratives and subsequent intelligence gathered, seven incidents of the local gang were
the intended targets, and six incidents of the local gang were the suspects. Table 5 shows
the total that each day of the week occurred for each type of event, either victim, suspect,
or unclassified, along with the calculated G statistic. Unlike in the prior example, there is
no uncertainty of the date and times of the events.
Each test fails to reject the null hypothesis that each day of the week is chosen with equal
probability. For the tests of the seven victimizations and the six suspects, the tests have
potentially small power if the shooters pick from three or more days of the week to commit
their crime. But the incidents with the unclassified shootings, as well as all gang related shootings provide strong evidence that there is not sufficient evidence to suggest one day
of the week is more likely than any other to have a shooting. The observed repeats on any
particular day of the week are consistent with random data for each of the series. The unknown
shootings and all gang related shootings have 18 and 31 incidents, respectively, so
do not suffer from being underpowered.
Given the sample sizes, this is very good evidence that particular interventions by the
police department to prevent shootings, such as hot spot patrols, should not be allocated
to specific days of the week, which is counter to the typical wisdom that more crimes occur
on weekends. The police department (or other agencies) should develop strategies to
reduce shootings that operate across all days of the week, and not, for example, just focus
on weekends. Or the police department should not specifically divert resources during the
weekend, when typical patrolmen may be busier with other calls. Diverting resources
during the weekdays is likely to be just as effective as the weekend. The final case study presents a series of arsons in 2014. Citywide, there were a total of 27
reported arsons, with the majority being burnings of vacant buildings. Arsons ranged
between zero and three per month, with the exception of July, which had five. Ex ante analysis
suggested that the total number of arsons citywide were consistently higher than
typical throughout the year, and the five in July were the highest monthly total recorded
since electronic records were regularly available starting in 2004. Patterns in 2015 have
subsequently regressed to more typical numbers of arsons, so analysis here is restricted
to cases in 2014.
Whilst again this involved analysis of a heterogeneous set of incidents, one that most
likely includes several offenders, the goal of the test has similar objectives for the analyst.
Upon observing that a series of events are geographically clustered, an analyst or detective
is likely to spend more time identifying whether those incidents are possibly linked to the
same offender(s). The same is true for other consistent patterns, including the day of the
week, but seeing a list of the days of the week are unlikely to provide as clear a visual
pattern as clustering on a map.
Table 6 displays the day-of-week patterns for each of the cities four police zones. Zone 4
was the area that had the most incidents and was under heavy scrutiny as to whether they
were caused by a serial arsonist. The G-test shows that for Zones 1 through 3, along with
the total citywide numbers, the test fails to reject the null hypothesis that each day of the week are equally likely to be chosen. But of the 15 incidents specifically in Zone 4, one
would reject the null hypotheses that arsons are equally likely to occur on each day of
the week. The clustering of seven events on Monday and four events on Tuesday is sufficient
to provide enough evidence to reject the null. This is a particular instance that knowing
the test is reasonably powered, even with only 15 crimes, has great utility. One might
simply look at the aggregate citywide statistics and not discern a pattern, thinking the beat
wide statistics are likely to be underpowered and not worthy of specific investigation. It is
also a case in which several crimes do occur on other days of the week, but there is sufficient
numbers in only two days to suggest a potential pattern.
For this particular example, identification of statistical significance in day-of-week
patterns for the arsons is the start of the analyst’s job, not the end. Because of the nature
and rarity of the event, traditional police responses like extra patrols, or surveillance are
unlikely to be an efficient response, so other tactics will need to be explored.
An analyst may drill down into the cases in Zone 4 and attempt to identify other particular
patterns in the series. For example, regular suspects in arsons in the city were homeless
squatters or adolescents vandalising properties. An analyst may see if there are other events
occurring nearby that might attract either or see if other crime incidents on those same days
of the week and in the same area identify related subjects. For example, an analyst may
look for instances of loitering in vacant buildings or field intelligence cards of suspicious
individuals for the same days of the week in which arsons are more prevalent. Canter
and Fritzon (1998) associate arsons that occur on weekdays more often with instrumental
motives and give adolescents breaking into buildings as an example.
Further, if an analyst can identify an external event that has a similar weekly cycle that is
associated with the arsons, it may lead investigators into possible suspects. Thus, the rejection
of one hypothesis that arsons occur with equal probability on each day of the week
lead to further hypotheses an analyst may be interested in testing (Chainey, 2012).

The job of an analyst is to identify patterns, and sifting through a series of related cases and
identifying characteristics to help predict future behaviour is a regular task. A regular
application of such a test would be to examine whether crime events by a serial offender
display clustering with respect to the day of the week but can be applied to any broader
setting in which an analyst may expect to see clustering in day-of-week patterns.
To aid this task, this paper shows a simple hypothesis test that allows an analyst to
quickly identify whether a series is likely to be random with respect to the days of the week
and derives the test statistics null distribution when only a few crimes are observed. This
can prevent analysts from thinking a pattern is significant when in reality it is not
(Guilfoyle, 2015), as well as identify cases that even with small samples can provide strong
evidence to inform the response by the police agency (Chainey, 2012).
This research finds that the likelihood ratio G-test is more powerful than the typical
Pearson chi-square test and shows that it is quite powerful under a wide variety of
circumstances. Sample sizes of 20 should be large enough to make the test very powerful
for the majority of circumstances analysts would be interested. For more specific alternatives
of having the majority of probability in only three days of the week, observing as
few as seven crimes is quite powerful. As such, this simple test for day-of-week clustering can help guide analysts in making objective decisions that have a controlled
Type I error rate as opposed to those based on gut feelings.
An example analysis of thefts of catalytic converters with interval censoring is given. In
all possible permutations given the begin and end dates in the particular example fails to
reject the null hypothesis of crimes occurring with equal probability on all days of the
week. So in this example, there is not enough evidence to suspect the offender had a proclivity
to commit crimes on certain day(s) of the week over others. For the intended uses of
small sample serial crime events, it is fairly simple to articulate all potential permutations.
But when evaluating the distributions of the times for larger samples, such as for aoristic
analysis (Ratcliffe, 2002), generating all potential permutations is prohibitive, and so it is
unclear how to effectively characterise the uncertainty in the times. Future research may
consider how to effectively characterise such uncertainty in circular interval censored data,
similar to how density estimates for interval censored data on the circular have been
proposed (Braun, Duchesne, & Stafford, 2005).
The second application is testing day-of-week-patterns for gang related shootings. In all
cases, where the one main gang was either a suspect (six cases) or a victim (seven cases), as
well as the overall total number of gang shootings (31), the test fails to reject the null that crimes
were equally likely to occur on every day of the week. This particular example runs counter to
the majority of crime, in that more interpersonal violent events occur on the weekends.
The last analysis presented is days-of-week of arson events, broken down by police
zones. This analysis showed that the police zone with the majority of arsons (15) showed
significant differences from randomness, whilst the other zones and the citywide aggregate
analysis did not.
A limitation of this particular test is that it does not provide predictions for the day-of-the
week when the next crime is most likely to occur. Whilst examining the counts in the days
of the week, an analyst can make simple predictions about which days are more likely, most
importantly the likelihood ratio G-test does not provide error around those predictions. A useful
future research endeavour may attempt to see the typical error in such predictions, both theoretically
and in actual applications, that result from using small samples of observed events.
Also, future applications may consider evaluating the power of such tests for testing
randomness in time of day of events. Identifying non-randomness in time of day patterns
has the same utility for police planners and analysts as do day of week patterns. Evaluating
time of day patterns will likely require more than seven bins to be useful, so it will decrease
the power of such tests. Interval censored recording times will also become more prevalent
at the smaller temporal units, making solutions tailored for interval censored data more
important. But given the differentiation of within day crime patterns (Felson & Poulsen,
2003), it may be the case that time of day patterns are under used in profiling cases, similar
to that of day-of-week patterns.
In any of the presented cases, an analyst could use ad-hoc decision making to determine
whether a series is random or not. The G-test shown here is a simple and objective criterion
with which analysts can use to identify if day of week patterns are non-random.