Linking serial homicide - towards an ecologically valid application

Abstract

Purpose – Crime linkage analysis (CLA) can be applied in the police investigation-phase to sift through a database to find behaviorally similar cases to the one under investigation and in the trial-phase to try to prove that the perpetrator of two or more offences is the same, by showing similarity and distinctiveness in the offences. Lately, research has moved toward more naturalistic settings, analyzing data sets that are as similar to actual crime databases as possible. One such step has been to include one-off offences in the data sets, but this has not yet been done with homicide. The purpose of this paper is to investigate how linking accuracy of serial homicide is affected as a function of added hard-to-solve one-off offences.

Design/methodology/approach – A sample (N = 117–1160) of Italian serial homicides (n = 116) and hard-to-solve one-off homicides (n = 1–1044, simulated from 45 cases) was analyzed using a Bayesian approach to identify series membership, and a case by case comparison of similarity using Jaccard’s coefficient. Linking accuracy was evaluated using receiver operating characteristics and by examining the sensitivity and specificity of the model.

Findings – After an initial dip in linking accuracy (as measured by the AUC), the accuracy increased as more one-offs were added to the data. While adding one-offs made it easier to identify correct series
(increased sensitivity), there was an increase in false positives (decreased specificity) in the linkage decisions. When rank ordering cases according to similarity, linkage accuracy was affected negatively as a function of added non-serial cases.

Practical implications – While using a more natural data set, in terms of adding a significant portion of non-serial homicides into the mix, does introduce error into the linkage decision, the authors conclude
that taken overall, the findings still support the validity of CLA in practice.

Originality/value – This is the first crime linkage study on homicide to investigate how linking accuracy is
affected as a function of non-serial cases being introduced into the data.

Full text

Linking serial homicide –towards an
ecologically valid application
AQ:au Tom Pakkanen, Jukka Sirén, Angelo Zappalà, Patrick Jern, Dario Bosco,
Andrea Berti and Pekka Santtila
AQ: 1 Abstract
Purpose –Crime linkage analysis (CLA) can be applied in the police investigation-phase to sift through a
database to find behaviorally similar cases to the one under investigation and in the trial-phase to try to
prove that the perpetrator of two or more offences is the same, by showing similarity and distinctiveness
in the offences. Lately,research has moved toward more naturalistic settings, analyzing data sets that are
as similar to actual crime databases as possible. One such step has been to include one-off offences in
the data sets, but this has not yet been done with homicide. The purpose of this paper is to investigate
how linking accuracy of serial homicide isaffected as a function of added hard-to-solve one-off offences.
Design/methodology/approach –A sample (N = 117–1160) of Italian serial homicides (n = 116) and
hard-to-solve one-off homicides (n = 1–1044, simulated from 45 cases) was analyzed using a Bayesian
approach to identify series membership, and a case by case comparison of similarity using Jaccard’s
coefficient. Linking accuracy was evaluated using receiver operating characteristics and by examining
the sensitivity and specificity of the model.
Findings –After an initial dip in linking accuracy (as measured by the AUC), the accuracy increased as
more one-offs were added to the data. While adding one-offs made it easier to identify correct series
(increased sensitivity), there was an increase in false positives (decreased specificity) in the linkage
decisions. When rank ordering cases according to similarity, linkage accuracy was affected negatively
as a function of added non-serial cases.
Practical implications –While using a more natural data set, in terms of adding a significant portion of
non-serial homicides into the mix, does introduce error into the linkage decision, the authors conclude
that taken overall, thefindings still support the validity of CLA in practice.
Originality/value –This is the first crime linkage study on homicide to investigate how linking accuracy is
affected as a function of non-serial cases being introduced into the data.
Keywords Behavioral crime linking, Serial homicide, One-off homicide, Hard-to-solve homicide,
Ecological validity
Paper type Research paper
Introduction
Behavioral crime linking refers to the practice of analyzing the crime scene behavior of two
or more offences to determine if the offender could be the same (Woodhams and Bennell,
2014). The concept rests on two theoretical assumptions:
1. Offenders behave in a consistent way from one crime to another (Canter, 1995).
2. Offenders behave in a distinctive manner from other offenders committing similar
crimes (Alison et al., 2002).
A good number of studies has found support for the consistency and distinctiveness
hypotheses across a range of crimes from violent interpersonal crime (robbery, rape and
homicide) to property crime (car theft, burglary and arson) (Bennell et al., 2014). Moreover,
results from these studies suggest that crime linkage analysis (CLA) can be used to
Tom Pakkanen is based at
the Department of
Psychology, A
˚bo Akademi
University, Turku, Finland.
Jukka Sire
´n is based at the
Institute of Biotechnology,
Helsinki Institute of Life
Science, University of
Helsinki, Helsinki, Finland
and the Department of
Computer Science, Helsinki
Institute for Information
Technology HIIT, Aalto
University, Espoo, Finland.
Angelo Zappala
`is based at
(CRIMELAB), IUSTO,
Pontifical Salesian
University, Turin, Italy and
the Department of
Psychology, A
˚bo Akademi
University, Turku, Finland.
Patrick Jern is based at the
Department of Psychology,
A
˚bo Akademi University,
Turku, Finland. Dario Bosco
is based at Court of Naples,
Naples, Italy. Andrea Berti
is based at the Department
of Scientific Investigation,
Arma dei Carabinieri,
Roma, Italy. Pekka Santtila
is based at the Department
of Psychology, NYU
Shanghai, Shanghai,
China.
Received 31 January 2020
Revised 22 May 2020
Accepted 11 June 2020
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successfully identify series of crimes: Bennell and colleagues (2014) summarized the
findings of 19 crime linking studies, concluding that moderate levels of linking accuracy (as
measured by the AUC) were achieved.
The benefit of successfully linking crimes is that individual crimes can be investigated and
prosecuted as a whole, pooling valuable police resources (Grubin et al., 2001). Also, one
advantage of linking crimes using behavior is that technical evidence (such as DNA and
fingerprints) is not always available at the crime scene. Once automated computer systems
for CLA are in place, the analysis itself is fast and cheap by comparison to technical
evidence (Davies and Woodhams, 2019). CLA can be carried out in a number of ways, but
during a criminal investigation, a common scenario of CLA would be an investigator asking
an analyst: “I’m investigating this one offence, can you find anything behaviorally similar in
your database?” (Rainbow, 2014; for a recent and comprehensive review of the literature on
the practice of crime linking, see Davies and Woodhams, 2019).
Over the past decade, increased attention has been given to the methodology used in
linkage research, both in an attempt to compare the efficiency of different statistical
methods (Tonkin et al., 2017;Winter et al., 2013), and in trying to increase the ecological
validity of the research samples (Tonkin et al., 2011;Winter et al., 2013;Woodhams et al.,
2019;Woodhams and Labuschagne, 2012). The latter is a critical issue for the application
of CLA in the courtroom, as shown by a number of cases where expert opinions on crime
linkage have been rejected as evidence (Bosco et al., 2010;Pakkanen et al., 2014; see also
HMA v. Thomas Ross Young, 2013). In many cases, a key concern has been the
generalizability of crime linkage research on CLA to actual cases, as the data sets used in
research are often not representative of actual police databases. The risk here is that this
distort the probability of two cases being linked, which in reality may be lower, and this in
turn may result in a false conviction for a case where CLA has been relied upon.
Previous crime linkage research has likely overestimated behavioral similarity because
study samples have typically consisted of only solved and serial offences (Bennell and
Canter, 2002;Woodhams et al., 2007). Scholars have attempted to estimate bias arising
from studying solved offences by looking at crime linkage accuracy for unsolved (but linked
by DNA) rape cases. For example, Woodhams and Labuschagne (2012) found that in
cases of South African serial rape (N= 119), series that were linked by similarity in modus
operandi (Jaccard’s similarity coefficient of M= 0.51, SD = 0.11) displayed significantly
more similarity in crime scene behavior than cases linked by DNA only (M= 0.47, SD =
0.11). Still, they found that linked pairs of crimes displayed a greater behavioral similarity
than unlinked pairs and could be successfully differentiated even when the sample
included unsolved series. It should, however, be noted that the number of unsolved crimes
(n= 14) in the sample was low. Similarly, Woodhams et al. (2019) found, in a large (N=
3,364) international sample of rape cases, that the predictive accuracy of whether crime
pairs were linked did not decrease significantly when unsolved series were added to the
data (AUC = 0.87 to AUC = 0.85). Their sample had the same limitation, though, including a
relatively small number of unsolved cases (n= 92). When comparing the unsolved to the
solved series, predictive accuracy was significantly lower (AUC = 0.79 vs AUC = 0.86).
The aforementioned issue of solved vs unsolved offences is likely more pressing with cases
of rape than homicide, as the overall clearance rate of homicide is significantly higher than
that of rape (Aebi and Linde, 2012;Liem et al., 2019). The more critical issue in regard to
ecological validity of crime linking research on serial homicide is thus the typical exclusion
of apparent one-off offences (“apparent”, because absolute certainty can rarely be
achieved regarding whether an offender has offended only once). Traditionally research on
crime linkage has looked at data sets comprising exclusively of serial offences (Bennell and
Canter, 2002;Woodhams, 2008), while police databases (on which CLA is done in real life)
include both serial and one-off offences. Tonkin et al. (2011) were the first ones to consider
this in their study of Finnish burglaries. They analyzed 508 burglaries with a 3:1 ratio of serial
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to one-off offences and found no significant differences in prediction accuracy (of whether a
crime pair is linked) when adding one-offs to the sample. Lately, a few more studies (Slater
et al., 2015;Winter et al., 2013;Woodhams et al., 2019) have looked at data sets
comprising both serial and apparent one-off rapes.
Winter et al. (2013) analyzed 90 serial and 129 one-off rapes (at a 0.7:1 ratio) and found that
linking accuracy using a naı
¨ve Bayesian classifier was not diminished (series only AUC =
0.84, one-offs added AUC = 0.89). Specifically, the sensitivity (as measured by Youden’s
index) of the model (i.e. the ability to correctly identify linked crimes) rose from 0.78 with
only the serial rapes to 0.87 with added one-offs, while the specificity (i.e. the ability to
correctly reject unlinked crimes) went down from 0.83 to 0.66.
In their study of serial (n= 194) and one-off (n= 50) rapes (ratio 3.9:1), Slater et al. (2015)
found, contrary to Winter et al. (2013), that linking accuracy (as measured by the AUC) was
not significantly affected as one-off offences were introduced to the data (series only AUC =
0.87; one-offs added AUC = 0.86). However, looking at optimal decision thresholds (as
measured by Youden’s index), the results were similar to that of Winter et al. (2013): when
adding one-offs, sensitivity (ability to correctly identify links) remained the same (0.79) while
specificity (ability to correctly reject unlinked crimes) worsened (0.81 to 0.79). The authors
noted that as the proportion of one-off offences in their sample was low, future research
should investigate if an increased proportion of one-off offences causes decrements in
linking accuracy.
Woodhams et al. (2019) similarly looked at both serial and apparent one-off stranger rape
cases in a large international sample (2,173 serial and 1,191 one-off rapes, at a 1.8:1 ratio).
Much in line with Winter et al. (2013),Slater et al. (2015), Woodhams found the addition of
one-offs to have a negligible effect on linkage accuracy (series only AUC = 0.86; one-offs
added AUC = 0.85). While the study is a step in the right direction in terms of mirroring the
contents of an actual police database, the authors point out that the exact proportion of one-
off offences to serial offences is unknown. As the exact proportion is nearly impossible to
find since a database can both include seemingly one-off offences where the offender has
been caught for only one offence, and undetected series, where the link between an
offenders’ offences has not been successfully identified, the authors suggest research be
done on data sets where the proportion is varied. Only one study has tested how a varying
proportion of one-off offences to serial offences affects linking accuracy. Haginoya (2016)
developed samples from 840 serial and 630 one-off offences to do a series of comparisons
between linked and unlinked crime pairs. The results indicated that the AUC between
samples with varied proportions of one-off offences (0, 25, 50 and 75%) were comparable.
To date, no behavioral crime linking studies on homicide have included one-off homicides.
For stranger rapes and burglary, it appears feasible that one-offs are more scarce than
serial offences, whereas for homicides, one-off offences are expected to be more prevalent
than serial homicides. The FBI estimates that less than one per cent of the homicides
committed in the US annually are committed by serial killers (Morton and Hilts, 2008).
International estimates put the amount of serial homicides in similar proportions:
approximately one per cent in Australia (Mouzos and West, 2007) and in the UK (Wilson,
2007), and 1.6% in Sweden (Sturup, 2018). It is therefore of considerable importance to
conduct studies including one-off offences in crime linking research on homicide.
As most one-off homicides are committed by people close to the victim (Fox and Levin,
1998;Kraemer et al., 2004), they are commonly faster and easier to solve than serial
homicides. As crime investigators are unlikely to seek profiling advice in these “easier to
solve” homicides, and also prosecutors and defense lawyers are unlikely to ask for expert
advice on crime linking in these cases, it does not make sense to model crime linkage
advice after them. Indeed, the crime linkage database maintained by the Serious Crime
Analysis Section (SCAS) in the UK, for example, populates their database in accordance
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with the same logic: the database excludes the easier to solve homicides where the
offender and victim know each other, and there is no sexual element in the killing (HMIC and
HMICPS, 2012). Hence, it would be prudent to identify hard-to-solve one-off homicides for
the modeling.
Aim
We investigated how the prediction of series membership is affected when a data set
includes both serial homicides and hard-to-solve one-off homicides. As the exact proportion
of serial to hard-to-solve one-off homicides is unknown, statistical analyses were carried out
so that we varied the proportions of one-off homicides in the sample in an attempt to get a
clearer understanding of how the added one-offs influence predicted series membership.
We expected that adding hard-to-solve one-off homicides would decrease linkage
accuracy as a function of the proportion of included hard-to-solve one-off homicides. The
accuracy was expected to decrease because the added none-serial homicides were
expected to add noise to the data, making the series less distinguishable.
Method
The present study used a sample of Italian homicides. Much like elsewhere, there has been
a steady decline in the homicide rates in Italy. Homicides have gone down from 1.78/
100,000 in 1995 to 0.67/100,000 in 2016. Since 2014, there have been fewer than 500
homicides annually, with an all-time low of 400 in 2016 (UNOCD, 2018). In 2014, three
quarters (73.7%) of the homicides were solved (Italian National Institute of Statistics, 2017).
A previous study (Pakkanen, 2006) that sought to identify hard-to-solve homicides amongst
all (Finnish) homicides during a 10-year period, found that 7 in 100 cases could be
considered hard-to-solve. The prevalence of serial homicides in Italy was assumed to lie
somewhere between 1 and 2% of all homicides, as estimated elsewhere in the Western
world (Morton and Hilts, 2008;Mouzos and West, 2007;Sturup, 2018;Wilson, 2007). Thus,
based on estimates of clearance rates for homicide (Ministry of the Interior, 2018;Italian
National Institute of Statistics, 2017), homicides without an apparent motive, and homicides
committed by a stranger to the victim (Italian National Institute of Statistics, 2017), our
estimate is that the proportion of hard-to-solve to serial homicides in Italy is roughly 10 to 1.
As this is a rough approximation and the exact proportion is unknown, the study used a
design that varied this proportion in the analyses.
The serial homicides
The definition of a serial killer used was an offender who had killed more than one victim with
at least 24 h between the homicides, in accordance with a widely accepted definition
(Adjorlolo and Chan, 2014;Morton and Hilts, 2008;Yaksic, 2015). A total of 23 individual
killers were identified from Italian newspapers, internet searches and microfilms of journals
in libraries. The offenders had committed a total of 116 homicides, between 1970 and 2001
(most of them between 1980 and 2001). For two-thirds (n= 15) of the offenders, court files
were used, and for a third (n= 8), criminological literature and newspaper articles were
consulted. All killers had been convicted in an Italian court of law.
Eight of the offenders committed their offences together with another offender. One offender
committed one of his homicides with a co-offender and the rest by himself. The serial
homicides included 25 victims who were killed together with another victim. The homicide
series varied in terms of number of victims, and period during which the series was
committed. The median number of victims was 6 (M= 6.7, SD = 4.5); the most extensive
series included 17 victims (n= 1), while the shortest series encompassed two victims (n=
3). The median duration of a series was three years (M= 4.83, SD = 5.11). In the temporally
longest series, the time between the first and the last homicide was 16 years, while the
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offences in the shortest series took place within a single year. For an elaboration of the
sample and series characteristics, please see Santtila et al. (2008).
The hard-to-solve one-off homicides
A total of 45 hard-to-solve killings were sampled for the one-off homicides. The
operationalization of “hard-to-solve” was adapted from earlier studies on homicide
(Pakkanen, 2006;Pakkanen et al., 2015). Cases where the offender was caught at the
scene of the crime, or the police knew the offender at the onset of the police investigation
were excluded. Additionally, for inclusion, the time period between when the case was first
reported to the police and when the offender was either questioned for the first time as a
suspect or caught, had to be at least 72h. Cases that fit the aforementioned criteria were
identified mainly by enquiry from the Reparto Investigazioni Scientifiche di Roma (the
department of scientific investigation within the Arma dei Carabinieri). The enquiry was
supplemented by the identification of a few additional hard-to-solve cases from Italian news
media. The actual case files that were used to code the data were collected from each local
police district where the homicides had taken place.
The hard-to-solve one-off homicides had 56 individual offenders and 48 victims and were
committed between the years 2001 and 2014. There were five cases in which more than
one offender had killed one victim; in three cases three offenders worked together, and in
two cases, two offenders had committed the homicide together. Three cases had two
victims. In the first of these, there were three offenders; in the second, two offenders; and in
the third, a lone offender had killed two victims.
Sample
The present study used 116 cases of serial homicide and 45 hard-to-solve one-off
homicides. The non-serial homicides were used to simulate 1–1044 hard-to-solve one-off
homicides for a total N= 117–1160. This was done to account for the fact that the exact
proportion of hard-to-solve to serial cases is unknown, and also because we wanted to see
how linking accuracy was affected as a function of added hard-to-solve one-off cases.
Some cases had more than one offender or more than one victim. All offender-victim
pairings were coded separately (serial n= 155; one-off n= 62), but for the statistical
analyses, only a single offender-victim pairing (one pairing per case) were used. This was
done to reduce possible error of inflation of certain crime scene behaviors. In cases where a
single offender had killed two or more victims, one victim was randomly chosen for
inclusion.
Coding scheme
All cases (both serial and hard-to-solve one-off) were coded using the same coding
scheme. The files from which the data were coded were extensive, often containing
hundreds of pages. For the most part they included a summary of the findings of the
pathologist, witness statements and interrogations with the suspect (Salfati, 1998;
Pakkanen, 2006;Santtila et al., 2008). The coded variables included offence-related
information (such as where the body was found, what kind of injuries the victim had, what
weapon, if any and was used), socio-demographic information on the victim and offender
and situational variables (e.g. what time of day the killing occurred). Additionally, other
general information, such as the age of the offender and victim, was included. Most of the
variables were dichotomous and coded as either present (1) or absent (0). For most of the
serial homicides court transcripts were available, and for the rest of the cases major news
outlets were consulted to confirm a guilty verdict in the case. Also, the one-off offenders had
all received a guilty verdict for their homicide.
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For the analyses of the present study, only dichotomous offence-related variables and
victim related variables (N= 89) were used (see Appendix 1 for a listing of the variables and
their definitions). Variables with no variation (i.e. no present observations or all present
observations) were eliminated, as they would not have contributed to the statistical
analyses. The remaining variables had missing values ranging from 0 to 47.8% (Md =
5.6%). For the main analysis, the missing values were left as such, as the method is able to
coherently handle missing information and use it in the modeling. For the additional analysis
(the case wise comparison using Jaccard’s coefficient), the missing values were
substituted with a zero.
Inter-rater reliability
The inter-rater reliability of the coding scheme was estimated using Cohen’s
k
(Brennan
and Hays, 1992;Cohen, 1960), with a mean
k
of 0.72 (SD = 0.13; Minimum = 0.59,
Maximum = 0.89), which is generally considered to be good (Cicchetti, 1994). For the serial
homicides, the data coding was conducted by two research assistants under the
supervision of a senior researcher (the third author) with prior experience of the coding
scheme. To clarify the coding scheme (and improve inter-rater reliability), the first and third
authors discussed and refined the definitions of the variables at length before and during
the coding process. For the one-off homicides, the aforementioned senior researcher (the
third author) coded all of the data.
The simulation
The main reason for simulating hard-to-solve one-off homicides was pragmatic: it would
have been prohibitively laborious and time-consuming to gather and code a sample of
1000þhard-to-solve homicides, as a natural database (such as a national police register)
was not available. In addition, the simulation also made it easy to vary the proportion of
serial to one-off offences, which was necessary, as the exact ratio of serial to one-off
offences is unknown.
The simulation was carried out using all of the coded one-off homicides (N= 45). Each new
case was created one variable at a time; for each variable, one of the original 45 cases was
randomly picked and the value for that particular variable [in the original case] was chosen.
This way the expected frequency of the variables in the sample of simulated one-offs
matched the original one-off cases. We considered 100 distinct values for the number of
simulated one-offs, with the values chosen from an equally spaced grid on logarithmic scale
between 1 and 1044. The total number of samples Nvaried therefore between 117 and
1160, and the ratio of serial to one-off between 116:1 and 1:10. To account for the
randomness of the simulations, 100 replicate simulated samples were created for each
number of simulated one-offs. In total, we created 10,000 data sets with simulated one-offs
and original serials for the analyses.
To assess the validity of the simulation, we analyzed subsets of the original one-off cases. For
each number of simulated one-offs between 1 and 45, we created 100 random subsets of the
original one-offs. The created data sets with the subset of original one-offs and all serials (N
between 117 and 161) were then analyzed similarly as the data sets with simulated one-offs.
Comparison of the results from the analyses of data sets with simulated and original one-offs
were used as a measure of validity for the simulation. The simulation was deemed valid, if data
sets with simulated one-offs yielded similar results as data sets with original one-offs.
Analyses
For the main analysis, the Bayesian crime linking method developed by Salo et al. (2013)
was used. The method is based on modeling series-specific probabilities of presence for
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the variables, and using Bayes theorem to turn these into probabilistic predictions that a
given homicide is part of a series of homicides in the data. One of the main advantages of
the method developed by Salo and colleagues over traditional statistical methods for linking
crimes is that it models each series’characteristics separately (for an in depth description,
see Salo et al., 2013), instead of trying to look for global patterns over the whole data, which
in data this heterogeneous are very hard to find. A central disadvantage of this particular
method is that it assumes an ideal data set in which the linkage status of each case is
known. In reality this is not the case: in a typical police database, only a small portion of the
cases are identified as being linked to one another (i.e. the same perpetrator).
For the “learning” phase of the modeling, each homicide series was modeled separately,
using the dichotomous variables (N= 89). The one-off homicides were considered as a
class of their own (i.e. treated as their own series in the analysis). This assumes that the one-
off offences are more similar to each other (in terms of the crime scene behavior and victim
characteristics) than they are with the serial offences, an assumption that has some support
from Pakkanen et al. (2015) finding that there is a qualitative difference between serial
homicides and hard-to-solve one-off homicides. This difference is, further, quantifiable and
large enough to allow reliable differentiation: any given hard-to-solve one-off homicide and
serial homicide could be identified as such with a good accuracy (AUC = 0.88) (Pakkanen
et al., 2015). Next, the probability for each case to belong to every series was calculated
separately using a leave-one-out method of cross-validation. The highest probability of
series membership for each case was considered the best assessment of which series
each case belonged to. To get a better sense of the predictive accuracy of the model, and
for ease of comparison to earlier studies (Slater et al., 2015;Winter et al., 2013;Woodhams
et al., 2019), receiver operating characteristics (ROC) analyses were carried out. In these
analyses, we plotted the predicted probability against accuracy (i.e. whether the prediction
was correct). The resulting areas under the curve (AUC) were used to determine how well
the model fared with the addition of an increasing amount (from 1 to 1044) of hard-to-solve
one-off homicides, thus linearly modeling how linking accuracy is affected when one-off
homicides are added to the mix in increasing proportions. To find the optimal thresholds for
sensitivity and specificity of the ROC curves (i.e. to maximize the amount of correct links
and correct rejections), Youden’s index was calculated, in line with previous research
(Bennell and Jones, 2005;Slater et al., 2015;Tonkin et al., 2011;Winter et al., 2013).
Finally, a secondary analysis was carried out to further the understanding of the practical
implications of the main result (i.e. what happens to linking accuracy when one-offs are
added to the mix?), and also to counter the problem (outlined above) of assumed perfect
knowledge of series membership in the data. A method developed by Craig Bennell (2002)
(and subsequently used in several crime linking studies, Bennell and Canter, 2002;Burrell
et al., 2012;Tonkin et al., 2017;Woodhams and Labuschagne, 2012) was used to measure
the similarity, in terms of offender crime scene behavior, between all of the individual
homicides. This was done by calculating Jaccard’s coefficient of similarity for each pair of
crimes; J = a/(a þbþc), where a is the number of behaviors present in both crimes in the
pair, b the number of behaviors present in crime one but not in crime two, and c the number
of behaviors absent in crime one but present in crime two. A coefficient of 0 would thus
indicate no similarity whatsoever between the two cases, while 1 would indicate perfect
similarity, that is, that all the same behaviors are present in both offences. Using a leave-
one-out principle, each homicide was then compared to a ranked list (from most similar to
least similar) of all other homicides with and without the addition of the hard-to-solve one-off
homicides. That is, in a situation where a homicide investigator would be investigating a
particular case, and would ask the crime analyst for the most similar cases in the database,
the main research question could be formulated as “How far up in the ranking is the first
correctly linked offence with and without the one-offs added to the database?” The
simulation and all the analyses were carried out using Matlab[1].
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Results
One way to inspect the validity of the simulated one-off data is to compare the results of the
simulated cases to the original (n= 45) one-offs in the results of the linkage analysis. The
changes in the AUC as a function of both the added simulated, and real one-offs are very
similar: the curves (and their confidence intervals) mostly overlap ( F1Figure 1). In other words,
the AUCs produced by the simulated and the original hard-to-solve one-off offences did not
differ from each other significantly.
The AUC typically ranges from 0.50 (indicating that the model is no better than chance at
identifying which series an offence belongs to) up to 1.00 (indicating that the model predicts
series membership perfectly). The starting point for the AUC –prediction accuracy when
the data consisted of only serial homicides –was 0.88 (95% CI = 0.81–0.93). When adding
hard-to-solve one-off offences to the mix, there was an initial dip in the AUC (with 30 one-off
offences added AUC = 0.85; 95% CI = 0.76–0.91), after which the AUC steadily increased
all the way to AUC = 0.90 (95% CI = 0.81–0.94) with the maximum of 1044 added one-offs
(N= 1160) (Figure 1). According to commonly used criteria for interpreting the area under
the curve (Swets, 1988), all the AUCs represent a moderate to high level of accuracy.
One possible explanation for the initial dip in the AUC when one-offs are added, is that with
a low number of one-offs (approximately as many as the mean number of offences in a
series), the model fails to distinguish the one-offs from other series (or as a series of their
own), thus decreasing predictive accuracy of the model. When the number of added one-
off offences grow, it becomes increasingly easier for the model to distinguish them as a
separate class, and the predictive accuracy rises.
Next the sensitivity and specificity were examined as a function of added one-offs.
Youden’s index was calculated to find the optimal decision threshold, maximizing sensitivity
(identifying correct series) and specificity (rejecting false series). There is an initial decrease
in sensitivity (from 0.76 to 0.70), as one-offs are added to the data, most likely because the
model cannot yet identify the specific characteristics of the one-offs. This means that some
of the added one-offs are mistakenly linked to different series, and possible some divergent
serial cases are erroneously linked to the one-off category. This is also reflected by the
Figure 1 AQ: 2
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larger margins of error (Figure 2). When the number of one-offs increases, sensitivity
increases substantially (to 0.89), and similarly, the uncertainty becomes smaller. The
specificity decreases steadily with the added first 100 one-off cases (from 0.92 to 0.80),
after which it starts increasing some (to 0.84). In other words, as a function of the added
one-offs it becomes easier to identify series correctly (i.e. correct positives increase), but at
the same time, a larger number of series are identified erroneously (i.e. also false positives
increase) (
F2 Figure 2).
In the last stage of the analysis, the offences were compared case-by-case, using
Jaccard’s similarity coefficient. One by one the cases were compared to a ranking of all the
other cases, from most similar to least similar to the query case. A linked offence with the
rank 1 would be a perfect result (a “hit”): the most similar case in the data set (as measured
here by Jaccard’s similarity coefficient) is linked to the index offence (Yokota and
Watanabe, 2002). The hit rate in this study was 85.3% with no one-offs in the data, and
82.8% with all the 1044 one-offs added. In other words, for over 80% of the cases, the most
similar case could be found at the top of the ranking list, regardless whether the data
included one-offs.
As this does not take into account the varying length of the different series, an additional
comparison was made, where a corrected median rank of each series was used, and the
proportion of series where the median rank was among the 5 most similar cases, and 20
most similar cases were calculated. With the median rank correction, 56.0% of the cases
could be found among the 5 most similar, when no one-offs were in the data. This dropped
down to 50.7% with all the one-offs added. Similarly, 76.5% of the cases could be found
among the 20 most similar cases with no-offs included, and a respective 63.2% with all the
one-offs included (F3 Figure 3).
Discussion
Linking accuracy, as measured by the AUC, when hard-to-solve one-off homicides were
added to the data, decreased initially and started increasing steadily again, when the one-
offs were more than 30. If compared with the same proportion of serial cases to one-off
Figure 2
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cases in the data, the present findings are most like Woodhams et al. (2019) and Slater
et al. (2015) with a slight decrease in linking accuracy (Woodhams et al. 0.86 to 0.85; Slater
et al. 0.87 to 0.86; present study 0.88 to 0.86 and 0.88 to 0.85, respectively). With the same
proportion of serial cases to one-off cases as Winter et al. (2013), the present study had a
slight decrease in accuracy (0.88 to 0.87), while Winter and colleagues saw an increase in
accuracy (0.84 to 0.89). When we increased the number of one-offs significantly, to a
proportion more realistic in terms of what a national police database might look like (10 one-
offs: 1 serial), there was an increase in linking accuracy (0.88 to 0.90).
When examining sensitivity and specificity of the linkage predictions (setting cut-offs using
Youden’s index), the findings of the present study are in line with Winter et al. (2013) and
Slater et al. (2015). While identifying correct links becomes easier (sensitivity increases),
rejecting false links becomes harder (specificity decreases), introducing false positive
errors into the linkage decisions. In other words, as more hard-to-solve one-off homicides
are added to the data, the model makes more mistakes by identifying links that are not real.
This finding supports the hypothesis that the added non-serial cases increase error in the
linkage decisions. For the investigation-phase, sensitivity is likely to be preferred over
specificity, as the investigator would want to make sure not to miss any possible links. For
the trial-phase, on the other hand, this (using Youden’s index as a cut-off) might pose a
bigger problem, as false positives could arguably be considered worse in a court of law,
than false negatives.
Taking a closer look at the probabilities produced by the modeling in the present study; the
magnitude of the probabilities is considerably smaller for the one-offs than for the serial
cases. In other words, the model finds less similarity between the one-offs and any series,
than within the serial homicides. This finding is line with Pakkanen et al. (2015) finding, that
hard-to-solve one-offs are qualitatively different from serial homicides, and perhaps do not
interfere with linkage accuracy, as they are identified as belonging to a separate class by
the model. Research comparing serial homicides to one-off homicides (Fox and Levin,
1998;Kraemer et al., 2004;Pakkanen et al., 2015) have noted that serial killers tend to be
more maladjusted and pathological, that their motives (e.g. overrepresentation of sexual
Figure 3
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motives amongst serial killers), and crime scene behavior (e.g. serial killers display a higher
level of forensic awareness at the crime scene) differ from each other. The findings of the
present study would seem to suggest that the differences between these two types of killers
are quite distinguishable by crime linkage models.
For the case-by-case comparison and ranking, the results would suggest a bigger practical
(and negative) effect of adding one-offs to the data, than linking accuracy measured by the
AUC. The drop in the proportion of cases to be found among the top most similar cases is
noticeable. From a practical standpoint this would suggest that a crime investigator would
have to go through more homicide cases than previously thought, to maximize their
chances of finding linked cases in the crime database.
Conclusion
Against expectation, overall crime linking accuracy (as measured by the AUC) of homicides
increased (after an initial decrease), as hard-to-solve one-offs were added to the data.
Examining sensitivity and specificity more closely and looking at the case by case similarity
comparison we found that, in line with the hypothesis, adding hard-to-solve one-offs does
introduce error into the linkage decisions by increasing false positives. Taken as a whole,
the effects seem manageable in scope, and do not thus invalidate the viability of crime
linkage, even when the data used more closely mirrors real crime databases.
Replication is needed with other samples of serial and hard-to-solve one-off homicide,
though. Future research should strive to include one-offs, because of the increased
ecological validity, and because according to the present findings, their addition does
affect CLA. For the practice of crime linking, we would say that the present findings take us
one step closer to refining and fine-tuning automated algorithms to help sift through police
databases to be used in the investigation-phase, and to establish more accurate error rates
for our estimates of CLA in the trial-phase.
Limitations
A major methodological limitation of the current study is that the linkage status is known for
all the cases. This is obviously not the case with real crime databases, a fact, which may
lead to overestimation of linkage accuracy when predicting series membership in the
present study. To circumvent for this limitation, the method would need to be developed so
that either the “learning” phase of the modeling is done using only a part of the linked
series, or that series membership (or linkage status) is estimated in other ways.
The generalizability of the results of the present study, as such, to other countries and
databases are likely limited. The homicides themselves, both serial and hard-to-solve one-
offs, may very well have some cultural specificities (e.g. mafia related killings in Italy), that
may eventually affect the results of the modeling. This would also be the case regarding the
nature of the crime databases and the culture of crime investigation with the police in
different countries. Thus, future research should strive to replicate the modeling in the
countries and with databases where the CLA is to be applied.
Future research
To further increase the ecological validity of the data, and the research, the next natural step
would be to conduct the analyses on natural crime databases. This would give an even
more realistic estimate of the applicability and limitations of CLA in practice.
Another avenue that would be beneficial to explore further is the overlap of (serial) homicide
with other crimes, specifically rape, and also perhaps burglary and robbery. From a
behavioral standpoint, the distinction between rape and homicide (that the present study
has compared), may be serendipitous; for example, the offender brought a knife along and
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the victim tried to flee, which turned an intended rape into a homicide. Or a robber that tries
to extort the PIN-number for their victim’s credit card and ends up killing the person in a
crime where the initial intent was financial gain. While some comparative research like this
exists, models like the one used in the present study could be tweaked and modified to be
applied on, for example, naturalistic databases of other crime types to develop and extend
CLA even further.
Note
1 The code is freely available at https://github.com/jpsiren/CL1off
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Corresponding author
Tom Pakkanen can be contacted at: tom.pakkanen@abo.fi
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AppendixT1
Table A1 The coding scheme: variables used in the analyses (N= 89) and their definitions for being coded as present (1)
Variable Definition
Offence related variables
Point of fatal encounter Point of fatal encounter (where the killer and the victim initially met) and the murder scene are
different places
Abduction Victim forcibly removed from one place to another while still alive
Body moved Body was not found at the scene of the murder
Body in building Body was found inside a building
Body outside Body was found outside
Body in vehicle Body was found inside a vehicle
Body in water Body was found immersed in water
Body covered Body was covered by something but not inside an object
Body in bag Body was covered by putting it inside a bag or a suitcase
Body buried Body was buried in the ground
City Area where the body was found was inner city
Suburb Area where the body was found was a suburb
Countryside Area where the body was found was in the countryside
Uninhabited Area where the body was found was uninhabited
Rape Killing occurred in association with a rape
Robbery or burglary Killing occurred in association with a robbery or burglary
Other crime Killing occurred in association with another crime
Murder scene burned Murder scene was burned to destroy the body or evidence
Murder scene victim’s home Murder scene was the victim’s home
Disguise Offender wore a disguise of some kind
Blindfold Use of any physical interference with the victim’s ability to see
Bound Hands or legs of the victim bound during the attack
Binding to scene Object used in binding brought by the offender
Binding at scene Object used in binding found at the scene by the police
Binding from scene Object used in binding taken from the scene by the offender
Gagging Object used in prevention or noise (not manual gagging of the victim)
Gag to scene Object used in gagging brought by the offender
Gag from scene Object used in gagging taken from the scene by the offender
Single violence Only a single act of violence was directed at the victim by the offender
Clothed Victim found fully clothed
Naked Victim found fully naked
Partially unclothed Victim found partially unclothed
Genitals exposed Victim found with genitals exposed
Sex with victim Any evidence of achieved or attempted vaginal penetration, oral penetration, anal penetration, or
that the offender had ejaculated
Object penetration Victim penetrated with an object
Necrophilia Any evidence of postmortem sexual activity
Picquerism Any evidence of picquerism
Forensic awareness Steps taken by the offender to ensure no evidence can be obtained
Firearm Handgun, shotgun or rifle used in the killing
Touch shot Victim shot so that the firearm has touched the body when fired
Sharp weapon Sharp weapon (such as a knife or an axe) used in the killing
Multiple stab same Several stab wounds to the same body area
Multiple stab several Stab wounds to several body areas
Strangulation object Victim strangled with an object
Strangulation hands Victim strangled manually
Suffocation Victim suffocated by other methods than strangulation
Kick or hit Victim was kicked or hit (without a weapon)
Multiple hit Victim was hit (without a weapon) several times
Blunt weapon Blunt weapon used in the killing
Multiple blunt Victim hit with a blunt weapon several times
Excessive blunt Blunt weapon used excessively (more than needed to kill the victim)
(continued)
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Table A1
Variable Definition
Torture or humiliation Victim was tortured or publicly humiliated
Objects thrown Objects were thrown at the victim
Weapon to scene Weapon brought by the offender
Weapon at scene Weapon found by the police at the scene
Weapon from scene Weapon taken from the scene by the offender
Body parts removed Body parts removed from the victim
Removed parts found Removed body parts were found
Head area Injuries on the victim to the eyes, nose, mouth or head
Throat Injuries on the victim to the throat
Torso Injuries on the victim to the torso
Extremities Injuries on the victim to the hands, arms, legs or feet
Genitals Injuries on the victim to the genitals
Back Injuries on the victim to any of the areas on the backside of the body
Victim-related variables
Victim gender Victim was male
Alcohol Victim was under the influence of alcohol during the attack
Drugs Victim was under the influence of drugs or medicines during the attack
Student Victim was a student or pupil
Employee Victim was an employee
Unemployed Victim was unemployed
Prostitute Victim was a prostitute
Handicapped Victim had a mental of physical handicap
Health problems Victim had physical health problems
Foreigner Victim was a foreigner, refugee, or immigrant
Gay Victim was known to have engaged in same-sex sexual behavior
Relationship Victim was married, had a same-sex registered relationship, or was currently in a serious
relationship
Divorced Victim was divorced
Children Victim had children
Drug habit Victim had a drug habit
Psychiatric medication Victim had current or former psychiatric medication
Homeless Victim was without accommodation at the time of the killing
Institution Victim lives in an institution (hospital, youth home, prison)
Alone Victim lives alone
Cohabitation Victim lived with an intimate partner, (a) parent(s), children, other relatives, or with a flat mate
Owns apartment Victim owns his or her apartment
Rent Victim lives in a rented accommodation
Council flat Victim lives in a rented accommodation owned by the city council
Night shelter Victim used night shelters
Other’s flat Victim is staying at someone else’s flat
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