4 Theoretical and Procedural introduction to Qualitative Comparative Analysis

4.1 Qualitative Comparative Analysis - a theoretical introduction

4.1.1 Set-theoretic approach

To better understand QCA, one of the most known set-theoretic method, an introduction to the whole perimeter of the set-theoretic approach is fundamental. This approach revolves around four main points:

1. They manage cases in sets with membership scores. Membership scores immediately invoke the notion of dichotomies: set membership score being 1 when the element is part of the set and 0 when the element is not part of the set. However, many concepts have a more complex nature and cannot be conceptualized as clear dichotomies, or crisp sets. Set-theories therefore went beyond this with the introduction of fuzzy sets where partial set membership is allowed. Fuzzy sets conceptualize fuzzy boundaries between sets and elements can fit into them while respecting their true gradient; (Schneider & Wagemann, 2012)

2. They see and treat relations between phenomena as set relations. This statement may be clearly explained with an empirical observation: all crows are black birds. Both “crow” and “black bird” represent sets for which different cases have different membership scores: - 1 if the bird is a crow, 0 if the bird is not a crown; - 1 if the bird is black, 0 if the bird is not black.

Although it can be said that all crows are black birds, we cannot affirm its contrary: all black birds are crows meaning that “crows” is a subset of the set of “black birds”. In return, this implies that “black birds” is a superset of “crows”. Rephrasings like these, may not seem of particular interest when applied to simple examples but are actually extremely important for set-theoretic methods since they lay the foundations for set relation interpretation in terms of sufficiency, necessity and of their more complex evolutions, the INUS condition; (Schneider & Wagemann, 2012)

3. They interpret set relations in terms of sufficiency, necessity and in terms of causes that derive from them (INUS conditions). To continue with our example, we can affirm that being a black bird is a necessary condition for being a crow since the latter is a subset of the former. At this point two analytical consequences are triggered: - there are other scenarios in which alternative factors can produce the same outcome, other cases in which a bird may be black without being a crow (principle of equifinality); - with more complex examples, one finds that the set relationships giving the wanted result is frequently to be found in combinations of various sets where single conditions do not display the effect alone but only combined with other conditions (conjunctural causation).

The theoretical combination of these two analytical consequences, equifinality and conjunctural causation, produce the concept of INUS condition, which states causal relevance of conditions in set relations. (Schneider & Wagemann, 2012) In other words, INUS condition means “causal conditions that are insufficient but necessary parts of causal recipes which are themselves unnecessary but sufficient for producing the outcome”. (Mahoney & Goertz, 2006; Ragin, 2008)

4. They conceptualize asymmetry of causal relations. The concept of asymmetry has to be defined more clearly than in non-set-theoretic approaches since it becomes fundamental for the creation of membership sets. (Goertz & Mahoney, 2012) In a set-theory analysis for instance, stating that “black” is not the opposite of “white” or that “rich” is not the opposite of “poor” is the first step to identify membership sets to capture two qualitatively different states of being. In a non-set-theoretic methodology, instead, only one indicator would be used, for instance, an inference of the degree of richness through, say, the monthly income. Asymmetry in terms of causal relations, pushes the interpretation towards the awareness that the “non-occurrence of the outcome cannot automatically be derived from the explanation for the occurrence of the outcome”(Schneider & Wagemann, 2012).

4.1.2 Causality in the Set-theoretic approach

As previously stated, conjunctural causation is one of the funding aspects of the concept of causality in set-theoretic methodologies, and therefore in a QCA. Doing so, they leave room for complexity where different conditions that combine into different causal paths -each being relevant in their way- may lead to the same outcome. (De Meur & Rihoux, 2002).

Causality in this context is freed from common assumption:

Additivity is not assumed. Each single factor is not thought of as having a separate and independent impact on the outcome, instead, it is the combination of those factors to be constituting the causation for the occurrence of the outcome.
Single causal combination is not assumed. Instead, researchers are encouraged to find all the possible causal combinations that can determine the same result.
Uniformity of causal effects is not assumed. A single condition, combined with others may produce an outcome, but not necessarily this still happens if the combinations of conditions change.
Causality is asymmetrical. Explaining the causal combinations of factors that determine a certain outcome does not necessarily explain also the absence of said outcome. (Berg-Schlosser et al., 2009)

4.1.3 Objective of a Qualitative Comparative Analysis

Qualitative Comparative Analysis (QCA) is one of the best known set-theoretic methodology. It’s logical foundations are laid on all the theoretical assumptions aforementioned on membership score, set relations and causality defined in the set-theoretic methodology and, according to Berg-Schlosser et al. (2009), additionally builds on Humme’s(1758) and Mill’s(1843) canons: - Method of Agreement: establishes that “if two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree is the cause (or effect) of the given phenomenon”; - Method of Difference: “establishes absence of a common cause or effect, even if all other circumstances are identical”.

Both the Method of Agreement and the Method of Difference work combined to establish the matching and contrasting of study cases in order to identify common causes while eliminating all other possibilities. In this sense, “if an instance in which the phenomenon under investigation occurs, and an instance in which it does not occur, have every circumstance in common save one, that one occurring only in the former; the circumstance in which alone the two instances differ, is the effect, or the cause, or an indispensable part of the cause, of the phenomenon.” (Mill, 1843) Although this might not always lead to new ground-breaking discoveries, this approach enables the elimination of false hypotheses, reducing complexity, and the channelling towards more precise ones while preserving the concept of plurality of causes. (Cohen & Nagel, 1934)

QCA stands on these premises in creating a systematic comparison of cases through formal tools and a specific conception of cases that guide configurational choices.(Berg-Schlosser et al., 2009) This approach is oriented towards fitting cases into sets by identifying their membership scores and further analyze them in terms of how sets interact between each other to then produce the outcome. To open and analyze the black box of variable association that happen in nature, QCA uses both a qualitative (case-oriented) and a quantitative (variable oriented) approach to achieve this goal. (Rihoux & Lobe, 2009) In particular, qualitative characteristics of the studied elements are turned into Boolean scores for set membership (crisp sets) while quantitative characteristics are turned into degrees of set membership (fuzzy sets). (Ragin, 2008)

In the QCA process, cases are dealt with in a holistic perspective and the researcher is encouraged to engage in a dialogue between cases and theory where cases triggers insights for theory and theory guides the operational configuration choices that make possible the study of cases. In terms of a data analysis technique, QCA can be considered to be based on standardized algorithms and the appropriate software (Schneider & Wagemann, 2012), in our case, that would be the QCA library in R Studio.

4.2 Qualitative Comparative Analysis - a summary of the procedure

The procedure of a QCA in operational terms is:

4.2.1 Step1. Calibration of variables.

Assigning set membership scores to empirical data, which is the very essence of calibration, is the starting point of every set-theoretic methodology.

Both qualitative and quantitative variables concurring to the outcome, are given a set membership score through their transformation into crisp sets (for qualitative variables) or fuzzy sets (for quantitative variables). For the first, the set membership score can be 0 or 1: 0 in case the quality defining the variable in question is absent, 1 in case it is present. For the second, the set membership score can vary between 0 and 1. In this case, it is important to define the so called cross-over point: the 0.5 membership score in which the element is neither in nor out of the set.

Even though, - crisp sets present only cases that can be a member of the set or not, - while fuzzy sets capture the degree of set membership of their cases,

both variants define qualitative difference between cases through the calibration process. In this sense, some cases will be part of the set for they present the quality defining the set or, in case of a fuzzy set, some cases will be more in than out of said set. (Schneider & Wagemann, 2012)

The best way to assign valid rules for set-membership values during the calibration process is based on the combination of theoretical knowledge and empirical evidence. (Ragin, 2000) Calibration should transcend the relativity of a specific data-set and be of its own absolute value, therefore it is good practice that set membership scores should not depend on arithmetic parameters (such as mean or median) to define cases “in the set” and the ones “out of the set”. (Schneider & Wagemann, 2012)

This first point, due to the absence of specific theoretical literature of ESG score forecasting and the lack of a precise data-set codebook, has impossible to be meticulously respected in this analysis.

4.2.2 Step2. Sufficient and necessary (or INUS) conditions.

Every set-theoretic method goal is to explore the potentially causal relations between combinations of conditions and the outcome under investigation. The main goal in these cases, according to Schneider and Wagemann, consists in “unraveling necessary and sufficient conditions and combinations for these two type of causes such as INUS conditions”. (2012)

4.2.2.1 Sufficient conditions for crisp sets

A statement such as “X is a sufficient condition to the outcome Y”: - implies that: “if X, then Y” = “X implies Y” = “X is a subset of Y” - does not imply that: “~X implies ~Y”

Statements like the previous, create expectations on the value Y only when X is present and say nothing about Y when X is not present (Schneider & Wagemann, 2012). This is due to the asymmetry of causal relations conceptualized in set-theoretic methodologies, and therefore in a QCA.

In particular, being X a sufficient condition to Y, we will expect three cases to happen: - we expect cases with both X and Y -> (X=1, Y=1); - we expect no cases with X and ~Y -> (X=1, Y=0); - we have no expectations for Y with ~X -> (X=0, Y=1).

If we have cases with X and ~Y, then our sufficiency claim of X on the outcome Y is falsified. With respect to the sufficiency of X for Y, this data matrix and two-by-two sufficiency table might come to help:

(Schneider & Wagemann, 2012)

Having this clear in mind, conditions applied to empirical cases may be reordered into a data matrix where: - each column is a condition or combination of conditions; - each row is an empirical case that show the presence (1) or absence (0) of each conditions or combination of conditions.

We will analyze this last matrix further, in the “Step3. Truth tables” chapter. In any case, it is important to bear in mind that for a condition or combination of conditions to be sufficient to the outcome, Y must occur every time said condition is present (1) while no said condition may be linked to ~Y.

4.2.2.2 Sufficient conditions for fuzzy sets

Crisp sets, as previously said, require the non-existence of cases where X = 1 and Y = 0 in order for the condition to be sufficient to the outcome. In a fuzzy set, the reasoning is exactly the same save for the fact that the dichotomous set membership score is replaced by the degree of set membership for cases to fit into conditions.

In this case, if X is sufficient for Y, set membership score in X must be equal to or smaller than its fuzzy-set membership in Y: X</=Y. (Ragin, 2000)

Here a XY plot visualization of what said:

(Schneider & Wagemann, 2012)

we expect cases with (X < Y): cases above the main diagonal contains cases with a membership in X smaller than in Y;
we expect cases with (X = Y): cases along the diagonal describes cases where membership in X and Y are identical;
we expect no cases with (X > Y): cases below the main diagonal contains cases with a membership in X greater than in Y.

4.2.2.3 Necessary conditions for crisp and fuzzy sets

The logic for necessary condition follows specularly the pattern for sufficiency condition, for this reason the discussion will be shorter. When investigating a statement of necessity, the focus shifts on the cases for which the outcome (Y) is present.

A statement such as “X is a necessary condition to the outcome Y”: - implies that: “if Y, then X” = “Y implies X” = “Y is a subset of X”

As previously explained, in sufficiency exploration, only cases that are member of condition X matter. For necessity exploration, instead, the issue is specular and cases that matter are members of outcome Y. (Schneider & Wagemann, 2012) In this sense, statements of necessity create expectations on the value X only when Y is present, in particular we expect that all cases with Y will also display a X. At the same time, cases that display ~Y are not covered by the claim of X being necessary to Y.

For crisp necessity claims, this would be the summary matrix combination: (Schneider & Wagemann, 2012)

While for fuzzy necessity claims, this would be the XY plot visualization: (Schneider & Wagemann, 2012)

4.2.2.4 INUS condition

INUS stands for “insufficient but necessary part of a condition which is itself unnecessary but sufficient for the result” (Mackie, 1974)

Put it other terms, a certain condition A has its effect on Y only when combined with another condition B. A, therefore, is insufficient on its own but it is necessary for the sufficient combination with B for the outcome. Still, the combined condition A*B may not be necessary for the outcome when it is not the only path to the occurrence of Y.

The INUS condition can be explored when all conditions of sufficiency and necessity are explored along with their relevant combinations.

4.2.3 Step3. Truth tables.

Creating truth tables is at the core of the QCA approach and exploration.

Truth tables are an indispensable tool for QCA for they enable to find and analyze configurations of conditions sufficient (and necessary) to produce the outcome. Here one might have to deal with contradictory configurations, or cases that display the same values on the condition variables but lead to different outcomes. (Berg-Schlosser et al., 2009)

A truth table is a data matrix with some particular features. These are: - each row present a qualitatively different condition or combination of conditions; - each condition or combination of conditions can take value of 1 (occurrence) or 0 (non occurrence); - the total number of rows is calculated by the expression 2^k where k is the number of conditions and 2 is two states in which said condition can be (1 or 0).

Having this clear in mind, the creation of a truth table follows three steps both for crisp and fuzzy sets: 1. identify all the logically possible combinations of conditions (k) that should correspond to the number 2^k; 2. assign the value for each empirical case from our data to the truth table row that corresponds to its condition; 3. compile the outcome column for each condition combination row with the empirical cases values for the outcome. (i.e. A~BC shows outcome Y, therefore Y will take value 1). (Schneider & Wagemann, 2012)

4.2.3.1 Truth table for fuzzy sets

While for crisp sets, truth tables value assignment is extremely straight-forward, for fuzzy sets there are a few things to bear in mind.

Firstly, for fuzzy sets, the truth table row to which the case best fits in is the one in which its partial set membership score is higher that 0.5. Moreover, while for crisp sets if A(1) then ~A(0), for fuzzy sets if A(0.9) then ~A(0.1).

The second aspect concerns the condition combinations value assignment: in this case, the fuzzy membership score will take the value of the minimum across conditions. (i.e. A(0.9), B(0.7), C(0.6) -> then ABC will have membership value of 0.6)

Finally, the third aspect concerns a mathematical property fundamental for set membership scores: “no matter how many fuzzy sets are combined, any given case has a membership of higher than 0.5 in one and only one of the 2k logically possible combinations”.(Schneider & Wagemann, 2012)

4.2.4 Step4. Minimization of the configurations.

Minimization of the configurations is actually an appendix of the truth table process, in particular the last part of the truth table analysis. It is run in order to eliminate redundancy information about sufficiency contained in the truth table. (Schneider & Wagemann, 2012)

Schneider and Wagemann in their “Set-Theoretic Methods for Social Sciences”, state that “if two truth table rows, which are both linked to the outcome, differ in only one condition – with that condition being present in one row and absent in the other – then this condition can be considered logically redundant and irrelevant for producing the outcome in the presence of the remaining conditions involved in these rows”.(2012) Therefore, the redundant condition can be omitted and the two rows merged into one (see table below).

minimized row	AB ->	Y=1
row1	ABC ->	Y=1
row2	AB~C ->	Y=1