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Crisp-set Qualitative Comparative Analysis (csQCA) and Criminology

Published onNov 01, 2017
Crisp-set Qualitative Comparative Analysis (csQCA) and Criminology


This article presents Qualitative Comparative Analysis (QCA; Ragin, 1987, 2000) as a useful methodological approach for criminological research. The aim is to introduce QCA’s logic and assumptions and a step-by-step guide to its application of crisp-set QCA using an illustrative hypothetical example. While QCA is a relatively new method and not widely used within criminological research thus far, it offers a unique approach that is particularly well suited to the field. This article is intended to serve as an introduction to those researchers interested in QCA and to demonstrate how they may incorporate it into their research.


As with all fields, criminology is host to many analytical techniques and methods. The focus of this manuscript is on Qualitative Comparative Analysis (QCA; Ragin, 1987, 2000) and is aimed at those researchers within criminology who either have not been exposed to the technique and its benefits, or those who are curious about its advantages and how it might be incorporated into their own research. The motivation for this manuscript arose from the frustration that although there are numerous books and articles available that discuss QCA, the books are lengthy and, therefore, gaining a basic understanding of the method can be time consuming. Furthermore, available journal articles generally lack either the discussion of the logic and assumptions or the clear steps involved. The aim of this article, therefore, is to present the method and its application in a straightforward, step-by-step manner to aid researchers in the understanding and decision of whether to use QCA in future research projects. The steps provided demonstrate a very simple example of an analysis and are not intended to replace formal training or a manual (such as that of Schneider & Wagemann, 2012) but to assist in the decision of whether QCA is a viable option for criminological researchers.

QCA is a research approach and technique that has become popular within the social sciences, by and large through the work of Charles Ragin (1987, 2000). Since Ragin’s seminal book, The Comparative Method (1987) first introduced and outlined QCA, use of the method has seen a marked increase within many disciplines, in particular sociology, comparative politics, business, and economics (Rihoux, Álamos-Concha, Bol, Marx, & Rezsöhazy, 2013). Criminology, however, has yet to fully embrace it as a mainstream method. According to the Compasss website (—which is the main information hub for QCA that maintains an up-to-date database of publications having utilized the method—articles filed under the Legal Studies and Criminology heading account for only 13 of the 571 total entries (as of January, 2017). This manuscript will present the assumptions and logic that underpin QCA and then present a hypothetical study investigating the causal conditions that lead to a higher homicide rate to demonstrate the steps involved in a crisp-set QCA. It will then conclude with a discussion of the benefits of the method, including a real example of how QCA has been used in criminological research.

What is QCA?

QCA is a holistic and predominantly case-oriented, set-theoretic approach that assumes outcomes are produced by combinations of conditions working together (Ragin, 1987, 2000). It is a comparative method in that it examines whether cases that share the same outcome also share the same conditions and vice versa (Ragin, 2008). This is based on the “Direct Method of Agreement” and “Method of Difference” proposed by John Stuart Mill (1967 [1843]). These methods involve matching cases and removing irrelevant variables to determine common causal relationships (Rihoux & Ragin, 2009).

The goal of QCA is to identify the group of variables, or conditions, that are shared by the cases that display the same outcome in order to understand the way in which they combine to produce it (Mahoney & Goertz, 2006). The influence a condition has on an outcome depends on the other variables within the case and the way in which they co-exist and interrelate, which enables the identification of descriptively meaningful patterns of interaction (Ragin, Mayer, & Drass, 1984). QCA is not concerned with the influence single independent variables have on a dependent variable but rather how the conditions combine to produce the outcome. Therefore, QCA moves us towards a deterministic understanding of the world in terms of the consistent patterns that occur (or do not occur) when outcomes are present, or are indeed absent.

Traditionally there are two general methodological approaches to research, qualitative and quantitative, and QCA falls somewhere in the middle with the aim of integrating the advantages of both (Fritzsche, 2013; Rihoux & Ragin, 2009). On the one hand, cases are understood holistically and are defined according to their observed configuration of conditions that they are made up of (Fritzsche, 2013; Ragin, 2000). The integrity of the cases is maintained because they are treated holistically and observed in their causal entirety with the parts of each case being understood in relation to the other parts (Ragin, 1987). On the other hand, the cases are treated in a systematic and variable-oriented manner. Cases are evaluated in terms of their theoretically derived and observed constitutive conditions, which are then systematically and formally analyzed.

QCA should not only be considered a technique for analysis but also as   a methodological approach to research. This refers to “the processes before and after the analysis of the data, such as the (re-)collection of data, (re-) definition of the case selection criteria, or (re-)specification of concepts, often based on preliminary insights gained through QCA-based data analysis” (Schneider & Wagemann, 2012, p. 11). The iterative nature of QCA allows  the researcher to become closely acquainted with, and to obtain an intimate knowledge of the data to make transparent decisions. This formalizes and systematizes the qualitative process and is important for research in that it aids in the development of theory and concept formation and leads to a deep understanding of the empirical associations under investigation.

QCA allows the logic and depth that is associated with qualitative comparison, which traditionally uses smaller case numbers, to be applied to research that employs medium to large sample sizes (Ragin, 2000; Rihoux & Ragin, 2009). By incorporating these medium case numbers, QCA provides sufficient variation for cross-case analysis while still allowing for the in-depth, case-by-case understanding that is required to answer certain research questions. It formalizes what comparative research inherently sets out to achieve and, therefore, offers transparency and the ability for replication, which is often withheld in some qualitative research methods (Fritzsche, 2013; Rihoux & Ragin, 2009).

For the purpose of this manuscript, the original variant of QCA, crisp-set QCA, is used. In set theory, a “crisp-set” allows only for full case membership or full non-membership to the set. This requires that all conditions and outcome for each case be dichotomized to in and out, or true and false, depending on the pre-determined set values. Other variants of QCA have been developed for situations that involve conditions that are not ideally dichotomized. One variant employs fuzzy-sets (fuzzy-set QCA), which allows varying degrees of membership to a set rather than the dichotomized treatment in crisp-sets. Others allow for sets with multinomial categorical data (multi-value QCA) and the temporal ordering of conditions (temporal QCA). For more information on these variants, refer to Schneider and Wagemann (2012) who have explored these in detail.

Logic and assumptions

QCA is based on set theory and Boolean algebra in that it tests for setsubset relationships. Dependent on the observed environment, QCA understands the real world in terms of the membership of cases and conditions in and relationships between sets. With these foundations, there are four major logical assumptions that underlie its basis and differentiate it from other traditional statistical methods. These are necessary and sufficient conditions, conjunctural causality, equifinality and multifinality, and causal asymmetry. The following section will outline these concepts. To demonstrate their applicability to criminology, this manuscript will draw upon Polk’s (1993) scenarios of masculine violence, which Polk (1993) defined as homicides involving both male offenders and male victims. Typologies like the scenarios described by Polk (1993; although it should be noted that he used the term “scenario” to denote a more fluid concept rather than the strict dimensions that typologies, categories, or types assume) are useful to evaluate these assumptions because they are inherently, if not explicitly and intentionally, based on set-theoretic principles. They essentially assign concepts to themes in much the same way as cases are assigned to sets in set-theory.

Necessary and sufficient conditions

Set-theoretic methods share three features: first, they assign case membership to sets which represent concepts; second, relationships between those sets are analyzed; and third, those relationships are explained in terms of necessity and sufficiency, along with causes that can be interpreted from them (Schneider & Wagemann, 2012). Many social science research questions implicitly, and often unknowingly, address the concepts of necessity and sufficiency but are hidden in verbal terminology that is not common to set-theory (Mahoney, 2004; Schneider & Wagemann, 2012); criminology is no exception. Criminal and rapist typologies, such as those of Groth (1979), Douglas, Burgess, Burgess, and Ressler (1992), and Turvey (1999) are inherently based on the logic underpinning set-theory and the relations of necessity and sufficiency. As can be seen in Figure 1, necessity and sufficiency represent the relationships between the two sets, condition X and outcome Y.

Figure 1. Visual representation of sufficiency and necessity between sets X and Y

In set-theoretic terms, necessary conditions are supersets of the outcome while sufficient conditions are subsets. If condition X is necessary for outcome Y, every time Y occurs then X must be present, and is written as X ß Y. In contrast, for condition X to be sufficient for outcome Y, then every time X is present, Y will occur, or X à Y. Stated another way, if a condition is sufficient, it is a guarantee that the condition can, by itself produce the outcome. For example, air is necessary for humans to live—without it, a person would die. However, it is not sufficient because it cannot, by itself maintain human life as there are other factors required for survival, like water and food. Similarly, if an advertised job requires that the candidate holds a Masters degree, the qualification is necessary but will not guarantee they get the job, and is, therefore, not sufficient. On the other hand, jumping from a plane at 20,000 feet without a parachute is sufficient for death; however, there are many other ways one might die and it is, therefore, not necessary. Likewise, being a mother is sufficient but not necessary for being female in the same way that a room being square is sufficient but not necessary for it having four walls (it could be rectangular).

Polk’s (1993) scenarios of masculine violence are essentially the search for necessary and sufficient conditions that produce each scenario. As an example, masculine violence can be conceived as the outcome and male offender as the condition of interest to investigate the set relationships between them. The male offender is a necessary condition because for a scenario of masculine violence to occur, there must be a male offender involved, as per Polk’s (1993) definition. The male offender, however, is not sufficient for masculine violence because, as already stipulated, not every incident for which the offender is male involves male victims. There are incidents for which the male offender’s victim is female and the incident is not a scenario of masculine violence. Since the definition of masculine violence specifically dictates that it involves male offenders and male victims, the male offender is a necessary but not sufficient condition of masculine violence.

QCA formally and systematically addresses the degree to which these two set-theoretic relationships exist between the outcome and conditions rather than correlational relationships. By identifying the necessary and sufficient conditions for an outcome, QCA offers a deterministic understanding of causality and is ideal for answering questions that are interested in cause and effect. It is this feature that offers a particularly unique and compelling perspective to criminological research.

Conjunctural causality

Ragin (1987) noted that outcomes rarely have an isolated single cause and QCA extends beyond just looking for individual necessary and sufficient conditions and looks for combinations of those conditions. The likelihood of a fight ensuing in a bar, for instance, might depend on factors such as (but not limited to) the level of alcohol consumed by patrons and the dynamics  of the group of people present. The mix of those two factors may increase or decrease the likelihood of the fight; this is referred to as conjunctural causality. It rests on the fundamental premise that is characteristic of social phenomena, that in order to bring about an event, its causes must combine in both time and place (Miethe & Regoeczi, 2004; Ragin, 1987, 2000). In other words, the causal role a condition has is dependent on the other conditions that it is present or not present with. Consider the statement that “[s]ome compliments come off like insults; some insults come off like jokes” (Ragin, 1987, p. 23), which means that it is the complex interplay of sender, receiver, message, and the situation that determines whether a compliment or joke is funny or insulting. Likewise, the example from the previous section of air being necessary for human survival additionally suggested that eating food and drinking water are also necessary. These three factors are individually necessary but it might be claimed that taken together they are sufficient for human survival. In other words, human survival is only guaranteed so long as there is air, food, and water (note that this makes no statement regarding death—this will be explored in the causal asymmetry section below). This is an important foundation of QCA in that it looks at the way in which the conditions combine and work together to produce an outcome. Researchers interested in how several conditions combine to produce an outcome rather than how much a single variable influences a dependent variable might consider QCA as a viable and valuable research approach and method.

QCA uses Boolean logic and algebra to assess which combinations of conditions are sufficient and/or necessary for the outcome. A summary of the simplest set of conditions that explains an outcome is established by way    of logical minimization (Caves, Meuer, & Rupletta, 2015; Miethe & Regoeczi, 2004). To do so, QCA uses the Boolean operators logical “AND” (*) and logical “OR” (+). Consider a case consisting of conditions A, B, and C that displays outcome D, and a second case, also displaying outcome D that consists of only conditions A and B (C is not present). This means that the two pathways (A, B, and C, &, A and B) both lead to the presence of outcome D. Consequently, is does not matter whether condition C is present or not because outcome D will occur so long as conditions A and B are present (recall Mill’s (1967 [1843]) methods). This process is repeated until no further reductions are possible. In the above scenario, using Boolean operators, the complex solution is displayed as A*B*C + A*B —> D. This expression can be then logically minimized to A*B —> D, which is known at the parsimonious solution.

Conjunctural causation is at the heart of a lot of qualitative research in that many researchers seek the thematic structure of their outcome of interest. With regards to Polk’s (1993) scenarios, for instance, he suggested that the confrontational scenario of masculine violence is dominated by homicides that occur between males, in typically “open” areas, and with an audience of male peers. It is the combination of these conditions that characterizees the confrontational scenario, rather than the presence of any one of them alone. In Boolean terms, this may be expressed as:

male-to-male AND (*) open area AND (*) audience of male peers —> confrontational scenario of masculine violence.

Being based on set-theory, QCA is an especially useful tool for “concept formation, the creation of typologies, and causal analysis” (Schneider & Wagemann, 2012, p. 8) which all rely on the exploration of factors that combine to produce an outcome. In a field awash with profiles and typologies, the ability of QCA to formalize conjunctural causation makes the method particularly useful.

Equifinality and multifinality

Equifinality is essentially the idea that “all paths lead to Rome.” It is the notion that different conditions, or combinations of conditions, can lead to the same outcome. The two Boolean pathways from the section above demonstrate equifinality in that they both lead to outcome D (A*B*C OR (+) A*B —> D). In order to listen to music, for example, it does not matter whether it is performed live from a stage, emitted from speakers, or sent through earphones, only that it is heard. Similarly, consider two highly successful clothing companies. One might sell only via physical store-fronts, have exorbitant prices, and thrives on face-to-face consultancy, while the other is fully online, is cheap, and is successful because of their fast mail service. While they both have comparable profits, these are accomplished in two different ways demonstrating the alternative paths for both companies to achieve the similar outcome. With regards to Polk’s (1993) confrontational scenario of masculine violence, a more variable aspect is the relationship between the victim and offender. Polk and Ranson (1991) originally suggested that these confrontations occur typically between strangers; however, this was expanded by Polk (1993) to also include some friend/acquaintance homicides. Therefore, the masculine confrontational scenario can be produced by the combination of masculine violence with either stranger OR (+) friends/acquaintances relationships, not just strangers.

Multifinality, on the other hand, is the assumption that a condition, or a combination of conditions, can lead to different outcomes. We often observe this pattern in our social world which leads to interesting research questions, like why two people who grow up in the same environment go on to experience very different life paths. One might lead a life in accordance with our social norms while the other goes on to pursue a criminal career. Polk (1993) distinguished between three different homicide scenarios that are characterized by masculine violence—confrontational, conflict resolution, and during the course of another crime. The condition of the male offender and male victim taken together can lead to all three of these scenarios, demonstrating the notion of multifinality. By incorporating equifinality and multifinality, QCA is a flexible method that is reflective of the real world and its contradictions and complexities.

Causal asymmetry

In set-theoretic methods, there is not always a symmetrical relationship between the presence and absence of an outcome; they may have different explanations that must be explored separately (Ragin, 1987). In other words, the conditions that explain the occurrence of an outcome cannot be inverted and do not have the equal and opposite effect when the outcome is not present. This is in contrast to some traditional correlational approaches, which observe and assume symmetrical connections, meaning a variable’s influence is the same with an equally opposite effect, be it positive or negative, and strong or weak (Caves et al., 2015). For example, in set-theoretic terms, the fact that there are people that drive cars without a license does not undermine the claim that car drivers must legally have a valid drivers license. From a correlational position, however, every instance of a person driving without a valid license weakens the connection between valid license and driving a car. Furthermore, returning to the notion that humans need air, food, and water to survive makes no statement regarding how a person may die (or not survive, that is, the absence of the outcome). Their death may have nothing to do with the lack of air, food, or water, like jumping from a plane with no parachute. Therefore, from a set-theoretic position, no claim can be made or explanation be automatically derived about the absence of an outcome from its presence. This feature is explicitly set-theoretic and will be further demonstrated in the next section.

Performing Crisp-Set QCA

There are a number of computer programs available to use in conducting a QCA. These are obtained, free of charge, via the Compasss website ( and are available with either a graphical user interface (GUI; as is experienced with the point and click interface of SPSS) or command line interface (CLI; as with the syntax view of SPSS). GUI programs, such as fs/ QCA (Ragin & Davey, 2014) and Tosmana (Cronqvist, 2011), are typically more user-friendly, however, are limited in their capabilities. CLI programs, such as RStudio (RStudio Team, 2014), are far more flexible and powerful in their capabilities than the GUI programs, however, are much more difficult to use because they are run using a specific source code which must first be learned.

A crisp-set QCA involves four major steps: calibration of the data, analysis of necessary conditions, construction and analysis of the truth table, and analysis of sufficient conditions. To demonstrate the steps involved in crispset QCA, an illustrative example utilizing a hypothetical dataset will be used. Suppose a researcher is interested in what differentiates countries with very high and very low homicide rates, but correlational analyses are inconclusive. Following a review of the relevant literature and theories, they determine that the homicide rate can be explained by several possibilities (equifinality) and is dependent on the arrangement (conjunctural causation) of the level of the median population age, rate of homicide by gun, gross national income (GNI) level, and amount of infrastructure being developed for each country. By performing a QCA, the researcher can establish the combination of those conditions, or causal pathways, that are associated with both very high and very low homicide rates (asymmetry) to further develop and test the identified and relevant theories. By studying the cases that represent the extremes (very high and very low homicide rates) of the total population is to suggest that they are qualitatively different from one another, which is   an assumption that differs from approaches that assume causal symmetry. Once all the data has been collected, the researcher can turn to the first major step—calibration.

Calibration of the data for high homicide rate

The process of assigning set membership to empirical data is called calibration. The researcher’s first step, therefore, is to assign the raw variables to set membership scores for both the outcome and four conditions. The definitions for each condition and outcome set must be clearly determined followed by the outer limits and threshold cross-over points, which indicate whether a case will be fully in or fully out of the set. Ordinarily, the type of variable—binary, interval, or continuous—determines the method of calibration (Ragin, 2008). For the purpose of this demonstration, the original variant crisp-set QCA will be used, and, therefore, all variables will be dichotomized, and each country will be assigned as either fully in or fully out, or one and zero. It is important to note that the numbers assigned to each set (one or zero) are qualitative indicators or descriptors only and do not represent quantitative values. The level, or threshold, at which a country becomes fully in or out should be set from sources external from the observed data rather than the data itself, although there are several ways in which an anchor can be set (Schneider & Eggert, 2014; Schneider & Wagemann, 2012). At this point it should be noted that there are many ways a set can be determined or defined and that this is entirely dependent on the research question. The outcome in this hypothetical scenario is the homicide rate for each country, which will become the crisp-set labeled “RATE.” The 16 countries with the highest and lowest homicide rates in the world (World Health Organization, 2016; high n=9 and low n=7) comprise this set, and those countries with the very high homicide rates are deemed as fully in, and coded as 1, whilst those with the lowest are deemed as fully out, and coded as 0.

The first condition for this hypothetical study is the median population age, which is labeled “AGE”. There are a number of ways this data can be calibrated, but for this crisp-set scenario, the cutoff is set halfway between the top and bottom median ages for every country included in The World Factbook (Central Intelligence Agency, 2014). Therefore, those countries with a median age 33.1 years and above are considered fully in the set, and those below as fully out.

The second condition is the rate of homicide by gun for each included country, labeled “GUNS.”. The gun homicide rates are all per 100,000 and were gathered from the (2016) database. The cutoff for this condition was determined by the mean of the gun homicide rates. The countries for which the rate is 16.97 or above are assigned as fully in the set, and those that are below as fully out.

The gross national income (GNI) is the third condition of interest and is labeled “GNI”. The values for this condition represent the gross national income per capita at purchasing power parity in international dollars for 2013. A country is considered fully in the set if their value is 17,240 or above, which is the mean for all countries as compiled by The World Bank (2016).

The last condition is labeled “INFRA” and represents the ranking of each country’s level of infrastructure. The figures were compiled by Statistica (2016) and are represented on a Likert scale ranging from 1 (underdeveloped) to 7 (extensively developed by international standards). The median, or midpoint (4.95), according to Statistica (2016), is set as the cutoff, with those countries above classified as fully in, and those below as fully out.

Table 1 presents the raw and calibrated data for the outcome and four condition sets.

Table 1. Raw and Calibrated Data for Homicide Rate Outcome and Four Conditions

Note: n=16. AGE = median age; GUNS = rate of homicide by gun per 100,000; GNI = gross national income per capita at purchasing power parity; INFRA = ranking of infrastructure level; RATE = homicide rate.

Analyze necessary conditions for the high homicide rate

Necessary conditions are important to social research because they imply that for the outcome to occur, the condition must always be present. This has important policy and intervention implications because, if that necessary condition can be identified and manipulated, there is the opportunity to change the outcome (Ragin, 2000). Due to the asymmetrical feature of QCA, when investigating necessary conditions, the focus is only on those cases for which the outcome is present, not absent (Schneider & Wagemann, 2012). For this scenario, the researcher looks at all the cases for which the homicide rate is high (the outcome is present) and determines if there are any conditions that are consistently present (or absent depending on the calibration). For this particular scenario, there is one necessary condition. The seventh column of Table 1 indicates that the overall younger median age of a country is necessary for the high homicide rate (age <— RATE).

The consistency of this condition is 1.00 meaning that every country that observed a high homicide rate was associated with a younger median age.  In reality, perfect set-subsets are not common (Braumoeller & Goertz, 2000; Ragin, 2006) and, therefore, analyses often accept that a condition is necessary if it is present in most cases, or above a threshold of around 0.9 (Ragin, 2006). It is important to evaluate whether or not the condition makes sense as a necessary condition rather than to rely on the output of the analysis since high consistency is not a guarantee that the relationship is meaningful (Ragin, 2006). The coverage score essentially describes how empirically relevant and important the connections between the sets are (Ragin, 2008). It represents how well the particular configuration of conditions fits the outcome, in other words, how many cases that observe this configuration also display the outcome. Coverage, therefore, examines how close the subset is between the conditions and outcome and is analogous with R2 in correlational analysis. This is easier to understand visually; consider the circles from Figure 1—the closer the circles are in size to one another, the higher the coverage is, and the more the condition accounts for the occurrence of the outcome. In this case, the coverage was 0.75 (or 75%) because, of the total 12 countries that displayed the younger median age, nine were associated with the higher homicide rate (that is, 9/12=75%). The benchmark minimum value for the coverage for necessity is generally set at 0.5 (Schneider & Wagemann, 2012).

Truth table construction

The researcher’s next step is to construct the truth table from the observed data. The truth table looks like a data matrix, the difference lying     in the meaning of the rows. Each row in the truth table represents a logically possible configuration of the incorporated conditions and outcome, and, therefore, “denotes a qualitatively different combination of conditions” (Schneider & Wagemann, 2012, p. 92). The truth table essentially provides a way of formalizing what comparative research inherently sets out to achieve, in that its aim is to examine whether cases that share the same conditions share the same outcome and vice versa (Ragin, 2008). Table 2 presents the reduction of the calibrated data from Table 1 into a truth table. The number of possible rows is calculated with 2c, where c equals the number of incorporated conditions. Therefore, there are 16 possible rows (24=16), although there are only six for which there were observed cases in this hypothetical example. The other 10 possible rows are logically possible, but unobserved combinations of conditions and outcome, and are called logical remainders.

Table 2. Truth table for homicide rate outcome and four conditions

Note: n=16. n= number of cases; Consist = raw consistency.

In order to produce the truth table, the information from Table 1 is condensed and summarized, such that cases with the same configurations of conditions are combined forming the same row in the truth table. For instance, row one from Table 2 denotes cases (or countries) 15 and 16 from Table 1. They share the same conditions and outcome and, therefore, the same row in the truth table. Column “n” indicates how many of the cases exhibited the specific combination of conditions and outcome expressed in that row. Row four indicates that for three countries (cases 1, 2, and 3), the combination of a lower country-wide median age, higher rate of gun homicides, lower GNI, and lower level of infrastructure are associated with a higher homicide rate. The consistency for that row is 1.00, meaning that every instance of that configuration of conditions was associated with the high homicide rate. On the other hand, row one saw two countries (cases 15 and 16) display a younger median age, lower rate of gun homicides, higher GNI, and higher level of infrastructure, which were associated with a lower homicide rate. The consistency is 0.00 because, for this dataset, that configuration never leads to the higher homicide rate (when the outcome is present, but recall that this does not imply anything about the lower homicide rate). Row three presents a contradictory row; the same configuration of the conditions leads to both the higher and lower homicide rate and has a consistency of 0.67 (countries six, seven, and 14). This means that 67% of the countries displaying this configuration of conditions displayed the higher homicide rate (n=2 of 3 cases).

Contradictory rows are interesting in that they reflect the reality of social research and require further investigation and explanation. Post hoc investigation of the countries involved in that row may provide an explanation for the contradiction, such as an important condition having been omitted from the analysis that helps explain the difference in outcome, or the outcome or condition being incorrectly calibrated. These cases that are in conflict need to be resolved before testing for sufficiency by either recalibrating the data or dropping a case from the analysis (all of which is based on theory and by returning to the data, see Schneider & Wagemann [2012] for more details). For this study, the researcher went back to the data and realized that two cases had been incorrectly calibrated and changed the case membership for those cases. Note that this iterative process is based on and justified from prior knowledge and the decisions are completely transparent. Table 3 presents the new raw data matrix and Table 4 presents the new truth table with no contradictions.

Table 3. Revised raw and calibrated data for homicide rate outcome and four conditions

Note: n=16. AGE=median age; GUNS=rate of homicide by gun per 100,000; GNI = gross national income per capita at purchasing power parity; INFRA = ranking of infrastructure level; RATE=homicide rate.

Table 4. Revised truth table for homicide rate outcome and four conditions

Note: n=16. n = number of cases; Consist = raw consistency

Reanalyze necessary conditions for the high homicide rate

Having changed the original raw data, the tests for necessity must be rerun because the configurations of conditions have changed. Following the steps from the section above, only younger median age emerged again as a necessary condition with a consistency of 1.00 and coverage of 0.82 (9/11=0.82); therefore the coverage (the empirical relevance) has risen. Had there not been conflicting cases, the data would not have been revised and this  step would not have been required.

Truth table analysis

Having revised the truth table, tests for sufficiency are conducted. There are five rows relevant to the analysis of sufficiency because they display the outcome; rows one, two, three, four, and five. These rows from the truth table can be expressed in Boolean terms using uppercase letters for the presence of the conditions and outcome, lowercase for the absence, and the Boolean operators AND (*) and OR (+). Thus, rows one, two, three, four, and five can be expressed as:

age*GUNS*gni*infra + age*guns*gni*infra + age*GUNS*gni*INFRA + age*guns*GNI*infra + age*GUNS*GNI*infra —> RATE

Therefore, row one (the first pathway) represents countries with a very high homicide rate that have a lower median age, higher gun homicide rate and have a lower GNI and level of infrastructure. This solution from the analysis of the truth table represents the five pathways that are sufficient for the higher homicide rate prior to any minimizing procedures. In other words, so long as one of those combinations is present, the homicide rate is higher for this sample.

Reduce the solution

In order to express the five solutions from the truth table in a simpler and more economical manner, the researcher subjects the truth table to Boolean minimization, which uses the logic of Boolean algebra and Mill’s (1967 [1843]) methods of agreement and difference. The configurations from rows one and two from Table 4, for instance, both include younger median age, lower GNI, and lower levels of infrastructure, but differ in terms of the rate of gun homicides. Therefore, it does not matter whether the gun homicide rate is high or low with this particular configuration of conditions to produce  the higher homicide rate. Further minimization between rows one and three reveals that they differ in terms of the level of infrastructure but share the remaining conditions. Following full minimization procedures, there are two reduced sufficient pathways for the higher homicide rate, as expressed in Table 5. The minimized configurations are called the parsimonious solutions and are logically equivalent to the complex solution, only simpler.

Table 5. Minimized sufficient pathways for higher homicide rate

Note: Cov=Coverage.

In Boolean terms, Table 5 can be expressed as age*infra + age*GUNS*gni—> RATE. Either of the configurations from Table  5 is sufficient for reach ing a high homicide rate. In the first solution, a younger mean age and infrastructure are sufficient for the higher homicide rate, regardless of the rate of gun homicides and GNI. In the second, a younger median age, higher rate of gun homicides, and lower GNI is sufficient, and it does not matter what the level of infrastructure is. So long as the conditions in a country meet either of those two sufficient solutions, the country will experience a very high homicide rate.

Low homicide rate

As previously discussed, one of the features of QCA is causal asymmetry, in that the opposite effect and relations cannot be assumed for the absence of the outcome from its presence, in this case the low homicide rate. Because the researcher was interested in the differences between very high and very low homicide rates, it is therefore, important to also investigate the necessary and sufficient conditions that contribute to the low rate. By recoding the outcome so that the cases with the lower homicide rate are considered fully in (low homicide rate=1), the tests for necessity and sufficiency can be rerun.

Following the recoding, one necessary condition emerges; higher GNI is necessary for a low homicide rate (GNI <— rate; consistency=1.00; coverage=0.78). Recall that the  necessary condition  for  a  high homicide  rate was a younger median age (age <— RATE). There are also two conditions that are very close to being necessary for the lower homicide rate; high level of infrastructure (consistency=0.86, coverage=0.75) and low gun homicide rate (consistency=0.86, coverage=0.75). While it shouldn’t be claimed that these conditions are necessary, they might be of interest to the researcher and warrant further investigation. Subsequent analyses for sufficient conditions for the low homicide rate reveal two pathways: guns*GNI*INFRA+AGE*GUNS*GNI —> rate. Again, compare this solution to that produced for the high homicide rate (age*GUNS*gni+age*gni*infra —> RATE). Both tests for necessity and sufficiency are not directly opposite to the tests conducted for the higher homicide rate, demonstrating the importance of the asymmetrical feature of QCA. In a real study, these resultant pathways might indicate which conditions should be focused on to reduce the homicide rate for a country.

As a point of difference, had these conditions been investigated via correlational analyses, the researcher would have found statistically significant results between all conditions and the homicide rate (median age, r=-.688, p<.01, gun rate, r=.809, p<0.01, GNI, r=-.591, p<0.05, and infrastructure, r=-.529, p<0.05). However, the results of the QCA are far more nuanced and detailed than these correlations and indicate the causal pathways of condition combinations that account for the higher or lower homicide rate.

Being a qualitative method, it is important that the results of the analyses are not simply reported but that their meaning is interpreted. The pathways by themselves are less meaningful without an exploration and explanation of the context of the cases, which in turn can minimize the relevance and point of the research. The epistemological foundation of QCA is that it is fundamentally case oriented and the data should be treated in a qualitative way. Although numbers are attributed to sets throughout the QCA process, these are qualitative indicators rather than quantitative values. In this hypothetical example, for instance, exploring whether cases cluster for each path (such as type of, or region of country) enables a far deeper understanding of the data than looking at the conditions comprising the pathway alone. Furthermore, examining which conditions are causally relevant to the outcome allows for policy analysis and intervention programming to determine what influences a desired result, such as a low homicide rate. But these must be examined within the context of the data.

A real example

Now that the steps in QCA have been outlined, a real example of how QCA has been utilized in criminological research will be presented here. For her doctoral thesis, Parker (2017) employed crisp-set QCA to investigate the qualitative make-up of, and differences between, the seven homicide event motives. The conditions that were incorporated were: relationship between offender and victim; whether the offender and victim were the same gender; location of the homicide; cause of death; and whether the offender was over the age of 35 years. Her analysis for thrill-motivated homicides (defined as “homicides committed for pleasure, curiosity, boredom, or catharsis”; Parker, 2017, p. 103) incorporated 21 cases for which the outcome was always present (the aim was not to investigate the causal role of the conditions but to determine each of the motives’ structure). Thrill produced the necessary condition that the offender and victim were not related to one another (consistency=0.952). This suggests that, for a homicide to be motivated by thrill, the victim, in most cases, is not related to their attacker. This is in contrast to the homicides motivated by love (that is, “homicides committed in order to remove a person they love from a situation they perceive as being ‘worse than death’ including extended suicides and altruistic reasons”; Parker, 2017, pp. 110 111), which found the necessary condition that the offender and victim were related to one another. One conclusion that was proposed is that the bond that connects the victim and offender in the love homicides perhaps protects offender’s loved ones from homicides committed for thrill. Put another way, an offender who is looking to kill for pleasure or excitement, for instance, generally looks beyond the family unit for their victim. Analysis of the truth table revealed two dominant configurations of the incorporated conditions for thrill-motivated homicides (a cutoff of 3 cases per configuration was required for the pathway to be included). QCA’s strength lies in the ability for the researcher to have an intimate relationship with the data, and post hoc investigation of the cases associated with each configuration leads to one of the most interesting and enlightening elements of these homicides motivated by thrill. It was revealed that the thrill cases that involved multiple offenders also mostly involved victims who knew their attackers. On the other hand, in the cases where the victim was a stranger, the offender generally worked alone. The “main correlational effect” from the descriptive analyses for this sample was that strangers accounted for the highest proportion of the relationships observed for thrill, yet as is clear from the QCA results, this was dependent on the other conditions it was presented with, namely the number of offenders involved. The practical implications of such a finding, with further testing, is that it may indicate another avenue of questioning and focus for investigators rather than focusing on and working with the assumption that thrill kills are most often perpetrated by strangers (Holmes & Holmes, 2009; Howard, 2014).


The aim of this manuscript has been to demonstrate the procedure involved in a crisp-set QCA so to present it as a methodological option for criminological research. To date, its use within criminology has been limited, but it has many advantages that may benefit the discipline and its researchers. Within criminology, there are so many levels at which research can be focused (macro or individual, for example) and QCA has the ability to handle these differing levels and data types well. Being a bridge between qualitative and quantitative approaches, it allows for a holistic and broader approach  as is often desired with social research to consider the context in which the behaviors occur. On the other hand, it also allows for an intimacy with the data that is necessary for a deeper and more refined understanding of the concepts being investigated.

If a researcher’s motivation is to investigate conjunctural causality and equifinality, then QCA can provide a viable methodological approach and research design. QCA is particularly well suited to research that is interested in configurations of causal conditions, such as the assessment of how multiple influences achieve certain outcomes. For instance, it can be useful to determine which conditions (or combinations of) have contributed to a successful intervention. Whereas net effects might indicate that a condition may not assist, it could be that it is dependent on which other conditions it combines with. Similarly, the highly complex nature of our social world means that what influences an area’s crime rate (for example) most likely depends on the combination of conditions in that area rather than one condition alone. The set-theoretical principles upon which QCA is based are also useful for processes such as data analyses, theory development and testing, creating evidence-based typologies, developing causal hypotheses based on empirically observed cases, testing hypotheses, and being used as a robustness test. Importantly, it is systematic and transparent in that it cannot be conducted without following formalized steps that incorporate clear and theoretically driven decisions and definitions, which allows for replication and validity. This is a particularly beneficial advantage with regards to a qualitative approach and data.

There are, of course, limitations to this method which must be considered before QCA is utilized. For instance, in crisp-set QCA, the process of dichotomizing the sets can lead to the danger of losing some of the more complex information. If a case falls into both sets, for example, a judgement must be made as to which set is most suitable. Furthermore, some data is simply not suited to being treating as fully in or out of a set, in which case either another variant of QCA, or another method altogether should be used. Another potential limitation is that all data must be available or the condition or case must be dropped. This clearly may reduce the number of included cases or that a condition which might be important to the research is simply not investigated. QCA is also time-consuming because it is highly iterative.

As was previously illustrated in the broad steps above, the back and forth between analysis and calibration can be a lengthy process and if a condition is changed following reassessment of its value and definition, then the data for the new condition must be recollected for all cases. And lastly, the number of conditions incorporated should be kept to a minimum (ideally between four and six; Schneider & Wagemann, 2012) due to the exponentially increasing number of cases that is required for each condition that is included. Therefore, conditions must be carefully chosen based on theoretically derived decisions rather than testing for every conceivable configuration. This point may, in turn, limit the scope of the research.

The aim of this manuscript has been to introduce and outline QCA to a criminological audience where the method is not currently commonly used. The steps presented were provided only as a general guide to demonstrate the procedure rather than being stringently followed. When being incorporated into research, the process is far more iterative and specific. This manuscript by no means claims that QCA is the answer to every research question, but that the choice of method is dependent on the question being asked. QCA is based on set theoretical thinking and Boolean algebra, whose logic and assumptions differ from the traditional statistical approaches. It represents a different way to think about and approach data and their meaning. Research designs that involve hypotheses about necessity and sufficiency rather than correlations, focus on conjunctural causation, equifinality, and asymmetry, might, therefore, benefit from QCA as a viable methodological option.


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Belinda is currently a PhD student at the Queensland University of Technology. She has attained a Bachelor of Music (Performance) and Bachelor of Social Science (Psychology) with honours before turning to her higher education in criminology. Her area of interest is motive, particularly with regards to homicide, and her current research is exploring the situational characteristics associated with the motives in order to determine whether they differ qualitatively in terms of their victim, offender, and offence characteristics.


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