Necessary Condition Analysis (NCA)

Dichotomous Necessary Condition

Figure 1 (top) shows two ways to graphically represent the dichotomous necessary condition. On the left, possible observations for a necessary condition are plotted in the center of cells of a contingency matrix (I will use the convention that the X axis is “horizontal” and the Y axis is “vertical” and that values increase “upward” and “to the right” ⁴ ). The contingency matrix is a common way to present dichotomous necessary conditions (e.g., Braumoeller & Goertz, 2000; Dul et al., 2010). On the right, the possible observations are plotted on a grid of a Cartesian coordinate system. This representation is used here to facilitate the generalization of necessary condition characteristics from dichotomous logic toward discrete and continuous logic (described later in the article). The dashed lines are the ceiling lines that separate the area with observations from the area without observations.

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Figure 1.

The dichotomous necessary condition with possible observations shown in the center of the cells of a contingency matrix (top left) and in the grid of a Cartesian coordinate system (top right).

The figure (top) shows that outcome Y = 1 is only possible if X = 1; however, if X = 1, two outcomes are possible, Y = 0 and Y = 1. Therefore X = 1 is necessary but not sufficient for Y = 1. However, if X = 0, then Y = 0, the absence of X is sufficient for the absence of Y. The necessity of the presence of X for the presence of Y and the sufficiency of the absence of X for the absence of Y are logically equivalent. Hume’s (1777) statement mentioned in the introduction section of this, namely, “if the first had not been, the second never had existed,” is the sufficiency of absence version of the necessary condition formulation.

The sufficiency of absence formulation of the necessary condition allows the value of X = 0 to be considered as a constraint or bottleneck for achieving the high value of the outcome (Y = 1). The empty cell for X = 0, Y = 1 indicates that a high value of Y = 1 cannot be achieved for the low value of X = 0. If the high Y value is to be achieved, the bottleneck must be removed by realizing a high value of X. On the other hand, the necessity of presence formulation of the necessary condition refers to the general organizational challenge of putting in place the right level of scarce resources or (managerial) effort to be able to reach a desired level of organizational performance (the outcome). Then X can be considered as an organizational ingredient (scarce event/characteristic/resource/effort) that is needed and therefore allows the desired organizational outcome Y (good performance) to occur.

The two versions of the same logic imply two possible ways of testing or inducing necessary conditions with data sets of observations (Dul et al., 2010). In the first way, different to traditional correlation or regression analyses, only successful cases are purposively sampled (for an everyday example, see Appendix A). Conversely, a correlation or regression analysis requires variance in the dependent variable, hence, then both successful and unsuccessful cases must be randomly sampled, and the analysis shows the extent to which determinants on average contribute to success. In NCA, the presence or absence of the condition is determined in cases with the outcome present (successful cases: Y = 1). The presence of the condition in successful cases is an indication of necessity of X = 1 for Y = 1. ⁵ If the condition is not present in the successful cases, then X is not necessary for Y. This approach was used by the current editor of Organizational Research Methods (ORM) before he took office as editor in July 2013. He performed a Google Scholar search to identify highly cited ORM papers and their common characteristics and found, for example, that such papers give practical recommendations to the organizational researcher (LeBreton, personal communication June 26, 2014). Additionally, in his first editorial, LeBreton (2014) states that “over the past 10 years, I have spent considerable time pondering what makes an impactful methods article. I have concluded that such articles share the following characteristics” (p. 114). The editor (implicitly) performed a necessary condition analysis: selecting cases purposively on the basis of the presence of the dependent variable (successful cases, i.e., highly cited papers) and identifying common determinants that were present in these cases. This resulted in necessary conditions for a highly cited paper in ORM, and several of these are listed in the editorial to stimulate authors, reviewers, and editors to produce impact papers in ORM by ensuring that the necessary conditions are in place (although this does not guarantee impact success). Necessary condition analysis is an effective and efficient way to find relevant variables that must be managed for success.

In the second approach to test necessary conditions, cases are selected in which the condition is absent (X = 0). The absence of the outcome in cases without the condition is an indication of necessity of X for Y. If success is present in cases without the condition, X is not necessary for Y. The editor could have searched for papers without practical recommendations to organizational researchers and—if the necessary condition holds—would have found that these papers are not highly cited. Normally, this second approach to test necessary conditions is much less efficient than the first one.

Two other approaches for selecting cases will not work. It is a common misunderstanding that necessary conditions can be detected by selecting and evaluating failure cases. There is no point in evaluating the characteristics of papers that are not highly cited. Figure 1 (top) shows that falsification of a necessary condition is not possible with failure cases (Y = 0) because the condition can be both absent (X = 0) and present (X = 1). In a similar way, an analysis of cases where the condition is present is not insightful either. A necessary condition is characterized by having no observations in the upper left corner. Therefore, the analysis of necessary but not sufficient conditions is about “the search for empty space” in the upper left corner. This applies not only to the dichotomous case but also to the discrete and continuous cases. ⁶

Illustrative Example of the Dichotomous Necessary Condition

Figure 1 (bottom) shows an example of Quantitative GRE scores of 342 students applying for admission to the Berkeley Sociology Graduate Program in 2009 (Goertz & Mahoney, 2012; Vaisey, 2009). A traditional data analysis to test the association between GRE score and admission would show a high correlation between the two variables (tetrachoric correlation coefficient = 0.6). The data justify the conclusion that success is more likely when the GRE score is at least 620 than when it is below 620. However, the necessary but not sufficient interpretation of the data is that a score of at least 620 is a virtual ⁷ (only one exception) necessary condition for admission. The exception is a student who was admitted based on a faculty member’s explicit testimony to the student’s quantitative abilities, which was regarded as superior information to the quantitative GRE score (Vaisey, personal communication, July 2, 2014). The upper left corner of Figure 1 is almost empty. Virtually all students (99%) who scored below 620 failed to be admitted. But scoring at least 620 did not guarantee success: Only 14% of these students succeeded in being admitted. Having a score below 620 is a bottleneck for being admitted. There may be other bottlenecks as well (e.g., a TOEFL score below 100). Advising students based on a traditional analysis would be to score high on GRE and TOEFL to increase your chances for success, whereas advising them based on a necessary but not sufficient analysis would add and make sure that you reach all minimum requirements to avoid guaranteed failure because you cannot compensate an inadequate score for one requirement with a high score for another one.

Discrete Necessary Condition

In the organizational sciences, variables are often scored by a limited number of discrete values, which usually have a meaningful order (ordinal scale), for example, a value can be higher/more/better than another value. Additionally, the differences between ordinal values can be meaningful, for example, the difference between high and medium may be the same as the difference between medium and low (interval scale). Many data in organizational sciences are obtained from measurements with respondents or informants using standardized questionnaires with Likert-type scales. In such an approach, a variable is usually represented by a group of items with several response categories. If interval properties are assumed, item responses can be combined (e.g., by adding or averaging) to obtain a score for the variable of interest. If items are scored on a discrete scale, the resulting score will be discrete as well. For example, if a variable is constructed from five items, each having seven response categories from 1 to 7, the number of discrete values of the variable is 31. ⁸

Figure 2 shows a trichotomous example of possible observations for a discrete necessary condition, where both X and Y have three possible values: low (0), medium (1), and high (2). The empty area in the upper left corner indicates the presence of a necessary condition. In this article, it is assumed that the ceiling line in a contingency table that separates the “empty” cells without observations from the “full” cells with observations as well as the ceiling line in a Cartesian coordinate system through the points on the border between the empty and the full zone are piecewise linear functions that do not decrease. ⁹ The question then is: “Which minimum level of the determinant X is necessary for which level of outcome Y”? Figure 2 (top) shows that for an outcome (performance) of Y = 2, it is necessary that the determinant (resource/effort) has value X = 2. However, X = 2 does not guarantee Y = 2. Therefore, X = 2 is necessary but not sufficient for Y = 2. For an outcome of Y = 1, it is necessary that the determinant has value of at least X = 1. However, X ≥ 1 does not guarantee Y = 1. Therefore, X ≥ 1 is necessary but not sufficient for Y = 1.

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Figure 2.

The discrete necessary condition with three levels, low (0), medium (1), high (2), in a contingency matrix (top left) and in the grid of a Cartesian coordinate system (top right).

Figure 2 (top) shows that the value X = 0 is a constraint for reaching both Y = 1 and Y = 2. The value X = 1 is a constraint for reaching only Y = 2. If the middle medium/medium cell (top left) or point (top right) were absent, the constraint of X = 1 would be stronger because X = 1 would then also constrain Y = 1. The size of the empty zone is an indication of the strength of the constraint of X on Y (more details follow in subsection Effect Size of the Necessary Condition). Also, for testing or finding discrete necessary conditions, cases can be selected purposively on the bases of the presence of the dependent variable. If one is interested in necessary conditions for Y = 2, then cases where this outcome is present (successful cases: Y = 2) are selected and the level of the condition X is observed. If in all successful cases no X is smaller than X = 2, X = 2 can be considered as necessary for Y = 2. Similarly, if one is interested in necessary conditions for Y = 1, then cases with this outcome are selected. If in all these cases no X is smaller than X = 1 (i.e., X ≥ 1), then it can be considered that X ≥ 1 is necessary for Y = 1. The aforementioned analyses of discrete necessary conditions can be applied to both ordinal variables (with Figure 2, top left) and interval variables (with Figure 2, top left or right; the Cartesian coordinate system assumes known order and distances between the category values) and can be extended with more discrete categories. A discrete necessary condition with a large number of categories nears a continuous necessary condition.

Illustrative Example of the Discrete Necessary Condition

In a multiple case study, Martin and Eisenhardt (2010) analyzed success of collaboration between business units in multi-business firms. From their data, a discrete (trichotomous) necessary condition on the relationship between “business unit [BU] representation” in the collaboration team and “collaboration performance” can be derived (see Appendix B), which is shown in Figure 2 (bottom). It turns out that a “high” level of BU representation is necessary but not sufficient for a “high” level of collaboration performance. All five highly successful cases (collaborations) have this high level of BU representation. There are also five other cases with “high” level BU representation, but these cases apparently did not achieve the highest level of performance. If only a “medium” level of collaboration performance were desired, then at least a “medium” level of BU representation is necessary, but again this is not sufficient because two collaborations had higher than medium levels of BU representation and showed “low” performance. This analysis results in two necessary (but not sufficient) propositions: “A high level of BU representation is necessary but not sufficient for a high-performing cross-business unit collaboration” and “At least a medium level of BU representation is necessary but not sufficient for a medium level of cross-business unit collaboration performance.” These propositions are different and more specific than the corresponding traditional proposition proposed by Martin and Eisenhardt (2010), which are implicitly formulated in terms of sufficiency: “BU representation is more likely to yield high collaboration performance.” Other examples of necessary condition hypotheses derived from Martin and Eisenhardt’s (2010) data set are presented in Appendix B.

In comparison to the dichotomous necessary condition analysis, the discrete necessary condition adds more variable levels to the analysis and can therefore provide more specific propositions. The trichotomous necessary condition in Figure 2 shows which levels of the determinant (medium or high) are necessary for reaching the desired level of collaboration performance (medium or high). Sometimes a medium level of the determinant may be enough for allowing the highest level of performance. Then, a higher level of the determinant may be “inefficient” (a further discussion on necessity inefficiency is presented in subsection Necessity Inefficiency).

Continuous Necessary Condition

In the case of a continuous necessary condition, the determinant X and the outcome Y are continuous variables (ratio variables). This means that the condition and the outcome can have any value within the limits of the lowest (0%) and highest (100%) values, allowing for even further detail. ¹⁰ Figure 3 contains graphs that correspond to the continuous necessary condition. The empty zone without observations is separated from the zone with observations by a straight ceiling line. In general, a ceiling is a boundary that splits a space in an upper part and a lower part, with only feasible points in the lower part of the space. In the two-dimensional Euclidian space (XY-plane), the expression X is necessary but not sufficient for Y is graphically represented by the ceiling line Y = f(X), with inequality Y ≤ f(X) representing the feasible points in the lower part of the space (on and below the ceiling line). For the linear two-dimensional case, the ceiling line is a straight line Y = aX + b (a = slope; b = intercept) with Y ≤ aX + b representing the lower part with the feasible XY-points. In this article, it is assumed that the ceiling line is linear and does not decrease. ¹¹

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Figure 3.

The continuous necessary condition with a large number of X and Y levels between low (0%) and high (100%), plotted in a Cartesian coordinate system. Note: The dashed line is the ceiling line. Top left: Idealized (“triangular”) scatterplot; the ceiling line coincides with the diagonal. Top right: Common (“pentagonal”) scatterplot; the ceiling line intersects X = X_Low and Y = Y_High axes at other points than (X_Low, Y_Low) and (X_High, Y_High). The upper left area between the lines X = X_cmax and Y = Y_cmin is the necessary condition zone where each X constrains Y and each Y is constrained by X. Bottom left: Example of a continuous necessary but not sufficient condition: Employee ambition is necessary but not sufficient for sales ability. The selected ceiling line technique (ceiling regression with free disposal hull) allows some data points above the ceiling line (see subsection Ceiling Techniques). The solid line through the middle of the data is the ordinary least squares regression line. Bottom right: Bottleneck table for the example illustrating what minimum level of the necessary conditions (X = Ambition) is required for different desired levels of the outcome (Y = Sales Ability). NN = not necessary. Source: Data obtained from Jeffrey Foster, Vice President of Science, Hogan Assessments, July 9-11, 2014.

The continuous necessary condition logic implies that a certain desired level of the outcome (Y = Y_c) can only be achieved if X ≥ X_c. This means that X_c is necessary for Y_c, but X_c is not sufficient for Y_c because there may be observations X = X_c with Y < Y_c, hence there are observations below the ceiling line for a given X_c. Within the scope of possible XY-values, the feasible necessary but not sufficient space in Figure 3 (top left) is the triangular space below the ceiling line: The ceiling line coincides with the diagonal of the scope. Although it is frequently stated that the necessary but not sufficient condition can be graphically represented by a triangular shape of the feasible area (Goertz & Starr, 2003; Ragin, 2000), this shape is just a special case of the more common pentagonal shape (Figure 3, top right). In the present article, we focus on the general linear ceiling line with a typical pentagonal feasible space. In traditional approaches, heteroskedasticity in such scatter plots (increasing variance with increasing X) is considered a concern for performing standard regression analysis, whereas it is inherent to the necessary but not sufficient logic. Traditional approaches draw average lines through the middle of the data (average trends), whereas NCA draws borderlines between the zone with and the zone without observations in the left upper corner. Therefore, traditional approaches to analyze pentagonal scatterplots may disconfirm X causes Y because the regression line may be virtually horizontal, but the NCA approach may confirm X causes Y because there is an empty zone in the upper left corner.

The presence and size of an empty zone is an indication of the presence of a necessary condition. The empty zone in the upper left corner reflects the constraint that X puts on Y. For example, the value X = X_c is a constraint (bottleneck) for reaching level of Y > Y_c. A higher level of Y is only possible by increasing X. Also, for testing or inducing continuous necessary conditions, cases can be selected purposively on the bases of the presence of the dependent variable. If one is interested in necessary conditions for Y > Y*, then cases where this outcome is present (successful cases: Y > Y*) are selected, the cases are plotted in a Cartesian coordinate system, and a ceiling line is drawn to represent the necessary condition X_c for Y_c (see the subsection Ceiling Techniques).

Illustrative Example of the Continuous Necessary Condition

The Hogan Personality Inventory (HPI) is a widely used tool to assess employee personality for predicting organizational performance (Hogan & Hogan, 2007). Figure 3 (bottom left) shows an example of the relationship between the HPI ambition score (one’s level of drive and achievement orientation) and supervisor rating of sales ability in a data set of 108 sales representatives from a large food manufacturer in the United States. Employees completed the HPI and supervisors rated employee’s performance in term of sales ability using a seven-item reflective scale (e.g., “seeks out and develops new clients”).

A traditional data analysis of the relationship between ambition and sales ability would consist of an ordinary least squares (OLS) regression line through the middle of the data (solid line in Figure 3, bottom left). This line runs slightly upwards, indicating a small positive average effect of ambition on sales ability (R ² = 0.07). The data justify the conclusion that on average more ambition results in more sales ability, but this is possibly not practically relevant because of the small effect size. However, the necessary but not sufficient interpretation of the data is that the empty zone in the upper left corner indicates that there is a necessary condition. The ceiling line indicates that there is a ceiling on sales ability depending on the sales representative’s level of ambition. The CR-FDH technique (fully explained in the Ceiling Techniques subsection) that is used in this example to draw ceiling lines allows some data points above the ceiling line. For a desired sales ability Level 4, it is necessary to have an ambition level of at least 30. About 20% of the sales representatives in the sample do not reach this level of ambition, which suggests that people with ambition levels below 30 will certainly fail to reach a sales ability level of 4 of higher. The other, about 80%, sales representatives who do have an ambition level of at least 30 could reach such a high level of sales ability, but this is not guaranteed; in fact, only about 10% of the sales representatives in the sample obtained this level of sales ability. In other words, an ambition level of at least 30 is necessary but not sufficient for a sales ability level of at least 4. An ambition level below 30 is a bottleneck for this high level of sales ability. Figure 3 (bottom right) shows the bottleneck technique, which is a table that indicates which level of the necessary condition (Ambition) is necessary for a given desired level of outcome (Sales Ability in %), according to the ceiling line.

Multivariate Necessary Condition

Until now, the analysis was bivariate with one X that is necessary for one Y. In this subsection, a more complex situation is considered: more than one necessary condition (X₁, X₂, X₃, etc.) and one Y. Multicausality is common in the organizational sciences. For example, Finney and Corbett (2007) reviewed empirical research on Enterprise Resource Planning (ERP) implementation success and identified 26 organizational determinants of success. Evanschitzky, Eisend, Calantone, and Jiang (2012) performed a meta-analysis of empirical research and identified 20 determinants of product innovation success. These examples can be extended with many other examples from virtually any subfield in the organizational sciences: A large set of determinants contribute to the outcome, and the underlying studies utilize traditional approaches such as correlation and regression to identify these determinants. Then, the size of the correlation or regression coefficient indicates the importance of a determinant. In this traditional logic, several determinants (X₁, X₂, X₃ … ) contribute to the outcome and can compensate for each other. In necessary but not sufficient logic, each single necessary determinant X_i always needs to have its minimum level X_ic to allow Y_c, independent of the value of the other determinants. If X_i drops below X_ic, the other determinants cannot compensate it. After the drop, the desired outcome Y_c can only be achieved again if X_i increases toward X_ic. This absence of a compensation mechanism once again shows the practical importance of identifying necessary conditions: putting minimum required levels of it in place and keeping these levels in place: “Satisfying necessary conditions … constitutes the foundation of achieving the goal” (Dettmer, 1998, p. 6).

Illustrative Example of the Multivariate Necessary Condition

As an example, the upper part of Figure 4 shows the scatterplots of four personality scores measured with the Hogan Personality Inventory (HPI) in the same sample as presented earlier. The plot for the first HPI score Ambition is the same as Figure 3 (bottom left). The other three scatterplots are for HPI scores Sociability, Interpersonal Sensitivity, and Learning Approach. The regression lines (solid lines through the middle) show that all four personality traits have a relatively small effect on Sales Ability. Ambition and Sociability have a small positive effect. The Interpersonal Sensitivity regression line is nearly horizontal. The Learning Approach regression line is slightly negative (!), indicating that on average Learning Approach has a small negative effect on Sales Ability. A traditional multiple regression analysis indicates that the four personality traits together hardly predict performance (adjusted R ² = 0.10). Yet, all four plots, including the plot for Learning Approach, show an empty space in the upper left corner, indicating that each personality trait may be necessary but not sufficient for Sales Ability. The necessary condition analysis predicts failure in these zones: The level of Y cannot be reached with the corresponding level of X. This applies to each determinant separately.

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Figure 4.

Example (see note 25) of the multivariate necessary condition with four necessary conditions. Note: The dashed lines are ceiling lines. The selected ceiling line technique (ceiling regression with free disposal hull) allows some data points above the ceiling line (see subsection Ceiling Techniques). The solid line is the ordinary least squares regression line. Upper part: Ceiling lines for four necessary conditions (Ambition, Sociability, Interpersonal Sensitivity, Learning Approach). Lower part: Bottleneck table: required minimum levels of the necessary condition for different desired levels of the outcome (Sales Ability). NN = not necessary.

The empty zones have different shapes because the ceiling lines (dashed lines drawn by using the CR-FDH ceiling technique, see Ceiling Techniques subsection) have different intercepts and slopes (see the ceiling equations in Figure 4, top). It implies that the necessary determinants put different constraints on the outcome. ¹² This is illustrated with the multivariate bottleneck technique. The bottleneck table in the lower part of Figure 4 shows that for a desired level of outcome Y_c = 80 (80% of the range of observed values; 0% = lowest, 100% = highest), the necessary levels of X₁, X₂, X₃, and X₄ are different: X_1c = 42, X_2c = 48, X_3c = 27, and X_4c = 35. A few important points are worth noting. First, we should recognize that each X variable must obtain these minimum ceiling values (X_i ≥ X_ic) before it is possible to obtain Y_c = 80. Thus, if even one of the X_i values falls below its ceiling value, Y_c = 80 will not be reached. This contrasts with traditional applications of the general linear model, where the X variables are allowed to operate in a compensatory manner (e.g., a low score on X₁ may be compensated for with a higher score on X₂). Second, it is also important to recognize that the ceiling values will change if we change the level of the desired outcome. For Y_c = 100, X_1c = 67, X_2c = 98, X_3c = 51, and X_3c = 68. Finally, it is also possible that for a given (lower) desired value of the outcome, one or more determinants are no longer necessary. For example, for Y_c = 60 or lower, determinant X₂ (Sociability) is no longer a bottleneck. For Y_c = 50 or lower, also X₃ (Interpersonal Sensitivity) and X₄ (Learning Approach) are no longer a bottleneck, and none of the determinants are necessary for Y_c ≤ 40.

It is typical for multivariate necessary conditions that no determinant is necessary at relatively low levels of the desired outcome, whereas several determinants become necessary at higher levels of desired outcome. Which specific determinant is the bottleneck depends on the specific levels of the determinants and the desired outcome. For example, if each of the four determinants has a score of 20, such an Ambition score and Sociability score allow a Sales Ability of at least 60, whereas such an Interpersonal Sensitivity score and Learning Approach score allow a Sales Ability of at least 70. Hence, a desired Sales Ability of 60 is possible with scores of 20 of the four HPI determinants, but a desired Sales Ability of 70 is not possible with such scores: The Ambition and Sociability scores are bottlenecks for such an outcome. The outcome of 70 is only possible if the Ambition score increases from 20 to at least 29 and the Sociability score from 20 to at least 23. The bottleneck table also shows that for a desired Sales Ability score below 40, no HPI score is a bottleneck, which illustrates “outcome inefficiency” and indicates that the outcome could be higher, even with low HPI scores (for a discussion on outcome inefficiency see subsection Necessity Inefficiency). The table also shows that for the highest outcome level (Sales Ability = 100), condition inefficiency is present: All HPI scores are smaller than100, with only the Sociability score close to 100. This indicates that for reaching the highest level of outcome, only about half to two-thirds of the maximum HPI scores for Ambition, Interpersonal Sensitivity, and Learning Approach are necessary, whereas for Sociability, the maximum HPI score is necessary (for a discussion on condition inefficiency see subsection Necessity Inefficiency).

Multivariate necessary condition analysis with the bottleneck technique identifies which determinants, from a set of necessary determinants, successively become the weakest links (bottlenecks, constraints) if the desired outcome increases. In other words, for a given level of the desired outcome, multivariate necessary condition analysis identifies the necessary (but not sufficient) minimum values of the determinants to make the desired outcome possible.

Ceiling Techniques

The starting point for the necessary condition analysis is a scatter plot of data using a Cartesian coordinate system, which plots X (the determinant and potential necessary condition) against Y (the outcome) for each case. If visual inspection suggests the presence of an empty zone in the upper left corner (with the convention that the X axis is “horizontal” and the Y axis is “vertical” and that values increase “upward” and “to the right”), a necessary condition of X for Y may exist. Then a ceiling line between the empty zone without observations and the full zone with observations can be drawn. A ceiling line must separate the empty zone from the full zone as accurately as possible. Accuracy means that the empty zone is as large as possible without observations in it. However, drawing the best ceiling line usually requires making a trade-off decision between the size of the empty zone and the number of observations in the empty zone (“exceptions,” “outliers,” “counterexamples”). Because the “empty” zone may not always be empty, it is named “ceiling zone” (C). The position of the data points around the ceiling line might suggest that the best ceiling line is not linear or increasing. Additionally, the best ceiling line may be a smooth line or a piecewise (linear) function. A ceiling function can be expressed in general terms as Y_c = f (X_c). For simplicity, the focus in this article is on continuous or piecewise linear, nondecreasing ceiling lines, which appears acceptable for first estimations of ceiling lines for most data sets. A nondecreasing (constant or increasing) ceiling line is typical and allows for a formulation that X ≥ X_c is necessary for Y_c.

Goertz et al. (2013) made the first attempts at drawing ceiling lines. They visually inspected the ceiling zone of a specific scatterplot and divided it manually into several rectangular zones. Rather than defining the ceiling line, the combination of rectangles resulted in a piecewise linear ceiling function with all observations on or below the ceiling. The advantage of this technique is that it implicitly only uses the location of upper left observations (where the border exists) to draw the ceiling line and that it can be made accurate with all observations on or below the ceiling line. One disadvantage of this technique is that it is informal and based on subjective selection of rectangles. Additionally, Goertz et al. introduced quantile regression to systematically draw continuous linear ceiling lines. Quantile regression (Koenker, 2005) uses all observations to draw lines. The 50th quantile regression line splits the observation in about half above and half below the line. This line is similar to a regular OLS regression line. However, for quantiles above, for example, 90, quantile regression results in a ceiling line that allows some points above the line. The advantage of this technique is that it is an objective, formal methodology to draw ceiling lines. However, a disadvantage is that it uses all observations below the ceiling line. For drawing a line between zones with and without observations, it may not be appropriate to use observations far away from this line.

For NCA, the advantages of the two approaches are combined by proposing two alternative classes of techniques, ceiling envelopment (CE) and ceiling regression (CR). Both are formal techniques and use only observations close to the ceiling zone. CE is a piecewise linear line and CR a continuous linear line (straight line).

Comparison of ceiling techniques

Table 2 compares the four different ceiling techniques. It also includes some other common techniques in operations research and econometrics for the estimation of efficiency frontiers that could be used for drawing ceiling lines as well. Corrected ordinary least squares (COLS) and stochastic frontier analysis (SFA) (e.g., Aigner & Chu, 1968; Bogetoft & Otto, 2011) have some disadvantages compared to the techniques selected previously. COLS shifts the OLS regression line toward the most upper point and therefore uses all observations. SFA focuses on the observations around the ceiling line but includes a stochastic term based on a probability assumption of the observations below the ceiling and thus also uses all observations. A last technique called low-high (LH) was created for the purpose of NCA as a rough first estimation of the ceiling line. It is the diagonal of the necessary condition zone defined by two specific observations in the data set: the observation with the lowest X (with highest Y) and the observation with the highest Y (with lowest X). However, this technique is sensitive to outliers and measurement error.

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Table 2.

Comparison of Techniques for Drawing Ceilings Lines.

Ceiling Line Technique	Observations That Are Used	Predefined Ceiling Function	Allows Points Above Ceiling	Drawing Procedure
CE-VRS	Upper left	Noa	No	Optimization
CE-FDH	Upper left	No	No	Optimization
CR-VRS	Upper left	Yes	Yes	Optimization + statistical
CR-FDH	Upper left	Yes	Yes	Optimization + statistical
COLS	All	Yes	No	Statistical
QR	All	Yes	Yes	Statistical
SFA	All	Yes	Yes	Statistical
LH	Lowest left and highest right	Yes	Yes	Mathematical

Note: LH = low-high; COLS = corrected ordinary least squares; QR = quantile regression; CE-VRS = ceiling envelopment with varying return to scale; CE-FDH = ceiling envelopment with free disposal hull; CR-VRS = ceiling regression with varying return to scale; CR-FDH = ceiling regression with free disposal hull; SFA = stochastic frontier analysis.

^aConvexity required.

The table shows that only five techniques use upper left points (LH, CE-VRS, CE-FDH, CR-VRS, and CR-FDH) to draw the border line between zones with and zones without observations. Within this group, LH uses only two observations, whereas CE-VRS, CE-FDH, CR-VRS, and CR-FDH use observations close to the border line. When comparing CE-VRS and CE-FDH, the latter has fewer limitations because it does not require convexity of the ceiling function and is therefore preferred. Because CR-VRS and CR-FDH are based on CE-VRS and CE-FDH, respectively, CR-FDH also has fewer limitations. Therefore, CE-FDH is the default as a nonparametric technique (no assumption about ceiling line function), and CR-FDH is the default for a parametric technique (linear ceiling line). When comparing CE-FDH and CR-FDH, the former is preferred when a straight ceiling line does not properly represent the data along the border between the empty and full zone and when smoothing considerably reduces the size of the ceiling zone. This holds particularly for dichotomous variables and for discrete variables with a small number of variable levels (e.g., max 5; see Figure 2). This may also happen if the number of observations is relatively low, namely, in small data sets. If a straight ceiling line is an acceptable approximation of the data along the border, the CR-FDH technique is preferred. Compared to CE-FDH, CR-FDH usually has some observations in the ceiling zone due to the smoothing approach, and the ceiling zone may be somewhat smaller. By definition, CE-FDH does not have observations above the ceiling line; it is therefore more sensitive to outliers and measurement errors than CR-FDH.

Example

In Figure 5, the eight techniques are applied to a data set of 28 countries with data about a country’s level of individualism according to Hofstede (1980) and a country’s innovation performance according to the Global Innovation Index (Gans & Stern, 2003).

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Figure 5.

Application of different ceiling techniques to a data set on a country’s individualism (X axis) and its innovation performance (Y axis).

The scatterplot shows that there is an empty space in the upper left corner indicating the presence of a necessary condition: Individualism (X) is necessary for innovation performance (Y). The ceiling lines differ considerably across techniques. The techniques that use all 28 observations (i.e., COLS, quantile regression [QR], and SFA) seem to underestimate (COLS and QR) or overestimate (SFA) the size of the ceiling zone. From the remaining techniques that do not allow observations in the ceiling zone (CE-VRS and CE-FDH), CE-FDH defines a considerably larger ceiling zone than CE-VRS because it is not restricted by a convexity requirement. From the remaining techniques that allow observations in the ceiling zone and have a predefined (linear) ceiling line (LH, CR-VRS, CR-FDH), LH is less accurate than CR-VRS and CR-FDH but defines a larger ceiling zone. In this example, CR-VRS and CR-FDH result in virtually the same ceiling line. For reference, the OLS line is shown in Figure 5 as well (the lowest straight line). It is clear that a line through the middle of the data that describes an average trend is not a proper candidate for a ceiling line (accuracy in this example is only 64%).

Analyzing a number of empirical samples and using simulated data sets indicate that CE-FDH and CR-FDH generally produce stable results with relatively large ceiling zones and with no (CE-FDH) or few (CR-FDH) observations in the ceiling zone. CE-FDH is similar to Goertz et al.’s (2013) rectangular approach and produces a piecewise linear function with no observations in the ceiling zone; CR-FDH is a technique developed for the present purpose of NCA. It produces a smooth linear ceiling function with normally a few observations in the ceiling zone. The ceiling techniques presented here are only linear nondecreasing ceiling lines. In future work, also nonlinearities will be explored, for example, triangular or trapezoidal piecewise linear functions, or nonlinear functions such as polynomials or power functions. The quality of ceiling lines needs to be systematically compared by evaluating fit, effect size, accuracy, and required assumptions.

Effect Size of the Necessary Condition

Perhaps in nearly all scatterplots, an empty space can be found in the upper left corner, which may indicate the existence of a necessary condition if this makes sense theoretically. The question then is: Is the necessary condition large enough to be taken seriously? Therefore, there is a need to formulate an effect size measure. An effect size is a “quantitative reflection of the magnitude of some phenomenon that is used for the purpose of addressing a question of interest” (Kelley & Preacher, 2012, p. 140). Applied to necessary conditions, the effect size should represent how much a given value of the necessary condition X_c constrains Y. The effect size measure for necessity but not sufficiency proposed here builds on the suggestion by Goertz et al. (2013) that “the importance of the ceiling … is the relative size of the no observation zone created by … the ceiling” (p. 4). The effect size of a necessary condition can be expressed in terms of the size of the constraint that the ceiling poses on the outcome. The effect (constraint) is stronger if the ceiling zone is larger. Hence, the effect size of a necessary condition can be represented by the size of the ceiling zone compared to the size of the entire area that can have observations. This potential area with observations is called the scope (S). The larger the ceiling zone compared to the scope, the lower the ceiling, the larger the ceiling effect, the larger the constraint, and therefore the larger the effect size of the necessary condition. The effect size can be expressed as follows: d = C/S, where d is the effect size, ¹³ C is the size of the ceiling zone, and S is the scope. Hence, d is the proportion of the scope above the ceiling. The scope can be determined in two different ways: theoretically, based on theoretically expected minimum and maximum values of X and Y, and empirically, based on observed minimum and maximum values of X and Y. Thus: S = (X_max – X_min) × (Y_max – Y_min).

Similar to other effect size measures such as the correlation coefficient r, or R ² in regression, this necessary condition effect size ranges from 0 to 1 (0 ≤ d ≤ 1). An effect size can be valued as important or not, depending on the context. A given effect size can be small in one context and large in another. General qualifications for the size of an effect as “small,” “medium,” or “large” are therefore disputable. If, nevertheless, a researcher wishes to have a general benchmark for necessary condition effect size, I would offer 0 < d < 0.1 as a “small effect,” 0.1 ≤ d < 0.3 as a “medium effect,” 0.3 ≤ d < 0.5 as a “large effect,” and d ≥ 0.5 as a “very large effect.” ¹⁴ Two recent papers with applications of necessary condition analysis have effect sizes between 0.1 and 0.3. One of these is in political science (Goertz et al., 2013), the other in business research (Van der Valk, Sumo, Dul, & Schroeder, 2015). In both papers, these effect sizes were considered as theoretically and practically meaningful. If a dichotomous classification is desired (e.g., for testing whether or not a necessary condition hypothesis should be rejected or for making an in kind qualification of the absence or presence of a necessary condition), I suggest to use effect size 0.1 as the threshold. Hence, an in kind necessary condition hypothesis in the continuous case (X is necessary for Y) is rejected if the effect size d is less than 0.1. A distribution of more than 150 continuous necessary conditions effect sizes from different studies indicates that roughly 50% of the effect sizes are small (d < 0.1), about 40% are medium (0.1 ≤ d < 0.3), and about 10% are large (0.3 ≤ d < 0.5). Very large effect sizes (d ≥ 0.5) can only be expected when the ceiling line is not a straight line (see the example in Figure 6).

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Figure 6.

Scatterplot of the relation between the number of white storks nesting pairs in a region (Storks) during a given year and the number of human births during that year (Births).

The effect size of the dichotomous necessary condition, as shown in Figure 1, can be either 0 or 1. Without observations in the upper left hand cell, X is necessary for Y, and the effect size is 1. If there were observations in the upper left hand cell, X would not be necessary for Y, and the effect size would be 0. However, if there are only a few observations in the upper left cell in comparison to the total number of observations, two interpretations are possible. In the deterministic interpretation, X is still not necessary for Y. In the more flexible stochastic interpretation, which I adopt in this article, after evaluation of the outliers in the “empty” cell, X could be considered as necessary. Outliers can be “substitutes,” as in the example of the GRE score in Figure 1 (bottom), or can simply represent error. With a few outliers and depending on the context, one may decide to denote the condition as “virtually” necessary. No general rule exists about the acceptable number of cases in the “empty” upper left cell. Dul et al. (2010) suggest (arbitrarily) that a maximum number of 1 out of 20 observations (5%) is allowed in the “empty” cell. Hence, this decision rule is not based on effect size but on accuracy: Below 95% accuracy, the in kind necessary condition hypothesis in the dichotomous case is rejected. ¹⁵

In the dichotomous case, no other effect size values than 0 or 1 are possible. Yet, the left side of Figure 1 suggests that the ceiling zone equals 1 (one empty cell) and that the scope equals 4 (four cells with possible observations), hence that the effect size is one-fourth. This is incorrect. The actual effect size of the dichotomous case can be observed more easily from the right side of Figure 1. This figure shows that the scope equals 1 × 1 = 1. Also, the area of the empty space equals 1 × 1 = 1, and therefore the effect size is 1. Similarly, in the discrete case of Figure 2 without medium-medium observations, both the empty space and the scope would be (2 – 0) × (2 – 0) = 4, thus the effect size equals 1. With medium-medium observations, the scope would be (2 – 0) × (2 – 0) = 4, and the ceiling zone is (1 – 0) × (2 – 0) + (2 – 1) × (2 – 1) = 3, hence, the effect size is three-fourths (and not three-ninths as suggested by the left side of Figure 2). ¹⁶

In the example shown in Figure 5, the minimum and maximum observed values of X (18, 91) and Y (1.2, 214.4) result in an empirical scope of 15,563.6. Consequently, the effect size calculated with CE-FDH is 0.42, and the effect size calculated with CR-FDH is 0.31.

Necessity Inefficiency

The effect size is a general measure that indicates to which extend the “constrainer” X constrains Y, and the “constrainee” Y is constrained by X. However, normally, not for all values of X, X constrains Y, and not for all values of Y, Y is constrained by X. When the feasible space is triangular, as in Figure 3 (top left), X always constrains Y, and Y is always constrained by X. But when the feasible space is pentagonal, as in Figure 3 (top right), the ceiling line intersects the Y = Y_High axis and the X = X_Low axis at other points than (X_Low, Y_Low) and (X_High, Y_High). As a result, for X > X_cmax, X does not constrain Y, and for Y < Y_cmin, Y is not constrained by X. The upper left area between the lines X = X_cmax and Y = Y_cmin in Figure 3 (top right) is the “necessary condition zone” where X always constrains Y and Y is always constrained by X. Within this zone, it makes sense to increase the level of the necessary condition in order to allow for higher levels of the outcome. Outside this zone, increasing the necessary condition level does not have such an effect. The area outside the necessary condition zone can be considered as “inefficient” regarding necessity. Necessity inefficiency has two components. Condition inefficiency (denoted here as i _x) specifies that a level of resource/effort X > X_cmax is not needed even for the highest level of performance Y. ¹⁷ Condition inefficiency can be calculated as i _x = (X_High – X_cmax)/(X_High – X_Low) × 100% and can have values between 0 and 100%. Outcome inefficiency (i _y) indicates that for a level of outcome Y < Y_cmin, any level of X allows for a higher value of the outcome (e.g., higher performance ambition). Outcome inefficiency can be calculated as i _y = (Y_cmin – Y_Low)/(Y_High – Y_Low) × 100%, with values between 0 and 100%. The effect size is related to inefficiency as follows: d = ½ (1 – i _x/100%) × (1 – i _y/100%). The larger the inefficiencies, the smaller the effect size. When both inefficiency components are absent, the idealized situation of Figure 3 (top left) is obtained, and the effect size is 0.5, which is the theoretical maximum for a straight ceiling line. When one of the inefficiencies equals 100%, there is no ceiling, and thus no necessary condition and zero effect size. When the X and Y axes are standardized (e.g., X_Low and Y_Low are 0 or 0% and X_High and Y_High are 1 or 100%), the angle of the ceiling line is 45 degrees if both inefficiencies are equal. When condition inefficiency is larger than outcome inefficiency, the ceiling line is steep (>45 degrees). A slope that is less steep (<45 degrees) indicates that outcome inefficiency is larger than condition inefficiency. In Figure 3 (bottom left), the condition inefficiency is 33%. This means that for an Ambition level (X) above 67% of 100 = 67, Ambition is not necessary for even the highest level of Sales Ability. In Figure 3 (bottom left), outcome efficiency is 46%. This means that for a desired Sales Ability level (Y) that is below 46% of 5.5 (the maximum observed level) = 2.5, Ambition is not necessary for Sales Ability. Because outcome inefficiency (46%) is larger than condition inefficiency (33%), the slope of the ceiling line is less than 45 degrees. ¹⁸

Comparison of NCA with QCA

Multicausal phenomena can also be analyzed with other approaches that use the logic of necessity and sufficiency. For example, qualitative comparative analysis (Ragin, 2000, 2008; for an introduction, see Schneider & Rohlfing, 2013) is gaining growing attention in organizational sciences (Rihoux, Alamos, Bol, Marx, & Rezsohazy, 2013). It aims to find combinations of determinants that are sufficient for the outcome while separate determinants may not be necessary or sufficient for the outcome (Fiss, 2007). NCA and QCA are similar in the sense that they both approach causality in terms of necessity and sufficiency and not in terms of correlation or regression. However, two major differences are that NCA focuses on (levels of) single determinants (and their combinations), whereas QCA focuses on combinations of determinants (configurations), and that NCA focuses on necessary determinants that are not automatically sufficient, whereas QCA focuses on sufficient configurations that are not automatically necessary (equifinality with several possible causal paths). ¹⁹ Although QCA centers on sufficient configurations, it is also used for necessary condition analysis (e.g., Ragin, 2003). But the QCA approach for “continuous” necessary condition analysis (fuzzy set QCA) is fundamentally different than the NCA approach presented here (Vis & Dul, 2015). Instead of a ceiling line, QCA uses the bisectional diagonal through the theoretical scope as the reference line for evaluating the presence of a necessary condition. Thus, QCA presumes that necessary conditions are only present in triangular scatter plots. ²⁰ If all observations are on and below the diagonal, the QCA reference line is the same as the NCA ceiling line. However, when a substantial number of data points show up above the reference line (e.g., because of inefficiencies, see examples in Figure 3 [top right and bottom] and in Figure 4), NCA “moves the ceiling line upward,” possibly with rotation, while the QCA reference line remains the same (the diagonal). Then NCA concludes that X is necessary for Y at lower levels of X, whereas QCA concludes that X is “less consistent” with necessity (necessity “consistency”; Bol & Luppi, 2013; Ragin, 2006) without specifying values of X where X is necessary for Y or where X is not necessary for Y. As a consequence, QCA normally finds considerably less necessary conditions in data sets than NCA Dul (2015a). Another difference is that QCA only analyzes in kind necessary conditions (condition X is necessary for outcome Y), whereas NCA can also analyze in degree necessary conditions (level of X is necessary for level of Y). NCA and QCA are also different with respect to evaluating the “importance” of a necessary condition. In NCA, importance is defined as the size of the empty zone in comparison to the scope, whereas in QCA (and other analytic approaches; e.g., Goertz, 2006a ²¹ ), importance of a necessary condition is defined as the extent to which the necessary condition is also sufficient (necessity “coverage”). In this view, a necessary condition is more important if it is more sufficient. ²² However, a fundamental notion in the current article is that single organizational ingredients are not sufficient for multicausal organizational phenomena, which explains and justifies the necessary but not sufficient expression. For a further discussion on the differences between NCA and QCA, see Vis and Dul (2015) and Dul (2015a).

Applying NCA instead of QCA for analyzing necessity can result is different conclusions. For example, Young and Poon (2013) applied QCA to identify the necessity of five IT project success factors: top management support, user involvement, project methodologies, high-level planning, and high-quality project staff. Using a sufficiency view of importance of necessity (Goertz, 2006a), they found importance levels of 0.97, 0.64, 0.70, 0.69, and 0.61, respectively. Applying the NCA approach to Young and Poon’s data set using CE-FDH gives necessary condition effect sizes of 0.30, 0.25, 0.19, 0.33, and 0, respectively. Comparing the rank orders of these two approaches shows that top management support is the most important in the sufficiency view, followed by project methodologies and high-level planning. However, in the necessary but not sufficient view, high-level planning has the largest effect size, followed by top management support and user involvement. The bottleneck analysis indicates that only project methodologies is necessary for a medium success level of 0.5 (the observed success levels ranged from 0.1 to 0.9), and the other four determinants are not. Yet, if the desired level of success increases, more conditions become necessary, and four out of five conditions are necessary for the maximum observed outcome level (0.9), with required levels of the five determinants of at least 88%, 67%, 25%, 100%, and 0% of the maximum range of observed levels of these determinants, respectively (the last determinant is not necessary). These percentages show necessity inefficiencies for all four determinants that are necessary, except for high-level planning that needs a level of 100% for maximum outcome. This NCA analysis adds to Young and Poon’s insights that top management support “is significantly more important for project success than factors emphasized in traditional practice” (p. 953). The NCA analysis shows that top management support is needed if maximum performance is desired, but not at maximum level. Top management support is not needed if only medium performance is desired (then project methodologies is needed). More importantly, for maximum performance, adequate levels of the other determinants, which Young and Poon call the “factors emphasized in traditional practice,” are also needed. Hence, high outcomes will not occur without properly balancing the minimum levels of the four necessary determinants. Therefore, NCA analysis does not support Young and Poon’s conclusion that “current practice emphasizing project methodologies may be misdirecting effort” (p. 953) because considerable levels of four necessary determinants including project methodologies are needed for high performance. This example of an additional NCA analysis is presented for purposes of illustration, and it is typical for the necessary but not sufficient approach. It may provide new insights about how organizations can efficiently put and keep organizational ingredients in place to achieve a desired outcome and by which level.

Limitations of NCA

Just like any other data analysis approach, NCA has several limitations. One fundamental limitation that it shares with other data analysis techniques is that NCA cannot solve the problem of “observational data cannot guarantee causality.” Observational studies are widespread in organizational research. All examples in Table 1 and all the examples used for illustration in this article are from observational studies. Observing a data pattern that is consistent with the causal hypothesis is not evidence of a causal connection. Hence, it is important that identified necessary conditions are theoretically justified, namely, that it is understood how X constrains Y and Y is constrained by X. Requirements for causal inference in empirical studies for building or testing necessary cause-effect relations are the same as for any other type of cause-effect relation. For example, a necessary cause is more plausible if the cause precedes the outcome and is related to the outcome and if an observed outcome cannot be explained by another cause (Shadish et al., 2002). If such requirements are not met, a necessary condition outcome that is empirically observed may, for example, be spurious, namely, caused by another variable (Spector & Brannick, 2010). This can be illustrated with a revision of the classic example of the relation between the number of storks in a region and the number of human newborns in that region (Box, Hunter, & Hunter, 1978). ²³ Figure 6 shows a scatterplot of the relationship between the annual number of white stork nesting pairs and the number of human births in the Oldenburg region in Germany over one decade. The scatterplot shows that there is an empty area in the upper left corner. The ceiling line is drawn with the CE-FDH rather than with the CR-FDH ceiling technique because the observations along the border between the empty zone and the full zone cannot be well represented by a straight line.

The data suggest the existence of a necessary condition with a very large effect size (0.73): A high number of human births in the region is only possible if the number of stork nesting pairs in the region is large. It is clear (unless one believes in fables or in the Theory of the Stork; Höfer & Przyrembel, 2004) that a theoretical explanation is missing in this example. A necessary relationship that is observed between X and Y can be spurious if a variable Z is sufficient for X and necessary for Y (Mahoney, 2007). In the storks example, one could state that when more people (Z) settle in a region and build houses with chimneys and other high human-made constructions, the number of stork nesting pairs (X) increases because storks favor high nesting places: Z produces X, hence is a sufficient cause of X. Additionally, Z is a necessary cause of Y because a high increase of the number of births is only possible when there are more people. In other words, the variable Z is the reason of the presumed “necessary” relation between storks and births. ²⁴ This example shows again that without theoretical support, a necessary condition cannot be presumed from observational data. This is not different from any other data analysis approach using observational data.

If the cause can be manipulated, a research design based on randomized experiments may be better equipped to fulfill requirements for causality than an observational study. A potential necessary cause of a multicausal phenomenon (e.g., storks for birth) can be tested in a randomized experiment where instances (cases) with the cause (regions with storks) are selected randomly from a defined population (all regions with storks) and then randomly assigned to the experimental group (regions that get a treatment) or control group (regions that do not get a treatment). In the experimental group, the presumed necessary causal condition is manipulated by taking it away (e.g., storks are chased away), whereas in the control group, the presumed causal condition is unchanged. Then we observe whether the effect disappears in the experimental group (reduction of birth rate) but not in the control group (birth rate constant). Such a design provides strong evidence for the (non)existence of a necessary cause. This type of experiment contrasts the common “sufficiency” experiment in which first cases without the cause (regions without storks) are randomly selected from a population (all regions without storks) and randomly assigned to the experimental or control group. Then the cause is added to the experimental group (storks are introduced), and we observe if the effect appears in the experimental group (increase of birth rate) and not in the control group (birth rate constant). ²⁵ The necessity experiment can be rephrased as a sufficiency experiment because a necessary condition that is formulated as “the presence of the cause is necessary for the presence the effect” can also be formulated as “the absence of the cause is sufficient for the absence of the effect.” Then in the sufficiency approach for testing a necessary cause, the “absence of the cause” (no storks) is added to the experimental group and we observe if the “absence of the effect” (reduction of births) appears. If the necessary cause cannot be manipulated (cannot be taken away), nonexperimental research designs such as natural experiments (where the necessary cause disappears naturally, and we observe if the effect disappears) or observational studies (where we observe if the necessary cause is present in cases with the effect or if the effect is absent in cases without the necessary cause) can be applied. Such approaches rely more on solid theoretical reasoning in order to infer necessity causality.

A limitation of NCA is that it may be more susceptible for sampling and measurement error than traditional data analysis approaches. The reason is that the proposed ceiling techniques (CE-FDH and CR-FDH) only use a small proportion of the observations in the sample for drawing the ceiling line (the observations near the ceiling line). The ceiling line (and therefore other quantities derived from it, e.g., effect size, accuracy) may be sensitive to selection bias and measurement error in the cases around the ceiling line. This particularly holds if the number of cases around the ceiling line is relatively small. Therefore, the accuracy of the ceiling line depends on having representative cases around the ceiling line with accurate measurement of the variables. Measurement precision and validity need to be ensured, in particular for the cases that are expected to be around the ceiling line (e.g., relatively successful cases). The ceiling line may also be susceptible to exceptions, outliers, or counterexamples in the upper left corner of the scatterplot. Therefore, observations near the ceiling line need to be evaluated before conclusions can be drawn. Exceptions (high outcome without the necessary condition) may be due to measurement error, unexplained stochasticity in the data, the case not being part of the theoretical domain where the necessary theory is supposed to hold, or the existence of a substitute condition that reflects the underlying concept (see example of Figure 1 [bottom] where GRE score and opinion of faculty member both indicate a student’s quantitative ability). The evaluation of cases near the ceiling line is not only desirable for reasons of internal validity; it also gives insight in “peers,” “best practices,” or “most efficient” cases. Cases near the ceiling line have the highest outcome in comparison with other cases with the same level of the necessary condition. These cases are therefore most efficient in the sense that they have the highest possible outcome for the given level of the necessary condition (outcome efficiency) and the lowest possible level of the necessary condition for the given outcome (condition efficiency).

NCA is a data analysis tool for calculating ceiling lines, effect sizes, and other relevant quantities. If the researcher is only interested in the specific data set, without a need for generalization to a wider population, this is perfectly acceptable. Also, if the data set is a census (all members of the population are in the data set), the calculated quantities are true representations of the population. However, if for reasons of statistical inference the data set is a probability sample from a population, the NCA quantities are only point estimates. Currently, NCA does not take sampling error into account (e.g., the NCA estimates have no confidence intervals). Future research could focus on adding confidence intervals to point estimates of NCA parameters. Building on developments within the field of production frontiers estimation, such confidence intervals could be based on resampling approaches such as jackknifing and bootstrapping (e.g., Simar & Wilson, 1998) or on Bayesian approaches (e.g., Pendharkar & Pai, 2013). Another sampling related limitation is introduced if purposive sampling is applied. Purposive sampling is an efficient way to select samples for calculating NCA quantities (see the example of selecting successful cases in Appendix A). The intention is that the sample is representative only for cases that are around the ceiling line (relatively successful cases). Then, relatively unsuccessful cases are not well represented in the sample, and therefore general sample descriptives (e.g., mean values of variables) or average sample trends (regression) do not represent the whole population. It should also be noted that a necessary condition may be trivial. When in a data set obtained by purposive or random sampling a necessary condition is identified, it is possible that this condition is trivial. Trivialness is the extent to which observations, on average, are clustered toward the maximum value of the necessary condition; hence, low values of the condition are rare. For example, if the required GRE score for admission would be for example 200, virtually all students would reach that score. Then such score would still be necessary for admission, but it would be trivial because a score below 200 hardly exists. Similarly, references are necessary for an impactful paper in ORM, but this condition is trivial because nearly all papers have references. A check on trivialness requires the identification of the existence of cases without (or with low levels of) the necessary condition (Dul et al., 2010).

Recommendations for Performing a Data Analysis With NCA Using a Stepwise Approach

Tables 3 and 4 show six steps for performing a necessary condition analysis. Table 3 refers to a situation where both X and Y are continuous variables or discrete variables with a large number (e.g., >5) of variable levels. Then the analysis is illustrated with a scatterplot as in Figures 3, 4, 5, and 6. Table 4 refers to a situation where both X and Y are dichotomous variables or discrete variables with a small number (e.g., <5) of variable levels. Now the analysis is illustrated with a contingency table as in Figures 1 and 2. Steps 1 and 2 and Steps 5 and 6 are relatively easy in both the scatterplot approach and the contingency table approach. Step 3 (draw ceiling line) and Step 4 (quantify NCA parameters) are relatively easy in the contingency table approach but more complex in the scatterplot approach. The NCA software tool has been developed (see the following) to assist in performing these steps. All steps in the contingency table approach can be performed by “manual/visual” analysis of the contingency table. This is particularly easy when the number of levels of the discrete variables is relatively low. The analysis for situations with dichotomous variables or discrete variables with a small number (e.g., <5) can also be done with the NCA software. Then the alternative data visualization with a Cartesian coordinate system (as in the right sides of Figures 1 and 2) and the scatterplot approach with the CE-FDH ceiling line technique must be used. This alternative analysis with NCA software (Table 3) gives the same results as the manual/visual contingency table approach (Table 4).

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Table 3.

Stepwise Approach for Data Analysis With Necessary Condition Analysis (NCA) for Continuous and Discrete Variables With a Large Number of Levels (Scatterplot Approach).

Step 1: Make Scatterplot	Step 2: Identify Empty Space	Step 3: Draw Ceiling Line (Use NCA Software)	Step 4: Quantify NCA Parameters (Use NCA Software)	Step 5: Evaluate Effect Size and Accuracy	Step 6: Formulate the Necessary Condition
a) Make a scatterplot by plotting the observations in an X-Y Cartesian coordinate system with X axis “horizontal” and Y axis “vertical” and values increase “to the right” and “upward.”	a) Visually inspect if the upper left corner is empty. Consider to allow some exceptions in the “empty space.” b) If there is “empty space,” continue with Step 3; if not, report that a necessary condition is not observed.	a) Draw a line in the upper left corner to separate the “empty space” and the full space using ceiling regression with free disposal hull (CR-FDH) if a straight line can fit the data points around the ceiling line or ceiling envelopment with free disposal hull (CE-FDH) if a piecewise linear function can fit the data points or use another ceiling technique (see Table 2).	a) Ceiling zone: Calculate the size of the “empty space.” b) Scope: Multiply the highest minus the lowest X values, with the highest minus the lowest Y values. c) Effect size: Divide ceiling zone by scope. d) Accuracy: the number of observations that are not in the “empty space,” divided by the total number of observations, multiplied by 100%. e) Condition inefficiency: (see Figure 3 top right) (XHigh – Xcmax)/(XHigh – XLow) × 100%. f) Outcome inefficiency: (see Figure 3 top right) (Ycmin – YLow)/(YHigh – YLow) × 100%.	a) Evaluate if the effect size (d) is theoretically or practically meaningful in the current context. b) Consider using the general benchmark 0 < d < 0.1 “small effect,” 0.1 ≤ d < 0.3 “medium effect,” 0.3 ≤ d < 0.5 “large effect,” and d ≥ 0.5 “very large effect.” c) Compare the accuracy with the benchmark of 95%. d) If effect size and accuracy are considered large enough, continue with Step 6. If not, report results.	a) Formulate the necessary condition in general terms (in kind) as “X is necessary for Y” or “X is necessary but not sufficient for Y.”a b) Additionally formulate the detailed necessary condition (in degree) by formulating the ceiling line (e.g., Yc = aXc + b) indicating which minimum level of Xc is necessary for a given level of Yc.

^aBecause normally there are also observations below the ceiling line, the necessary condition is usually not sufficient.

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Table 4.

Stepwise Approach for Data Analysis With Necessary Condition Analysis (NCA) for Dichotomous and Discrete Variables With a Small Number of Variable Levels (Contingency Table Approach).

Step 1: Make Contingency Table	Step 2: Identify Empty Space	Step 3: Draw Ceiling Line	Step 4: Quantify NCA Parameters	Step 5: Evaluate Effect Size and Accuracy	Step 6: Formulate the Necessary Condition
a) Make a contingency table with X axis “horizontal” and Y axis “vertical” and values increase “to the right” and “upward.” b) Write the number of observations in the cells of this table. c) Delete outer rows and outer columns if these are completely empty.	a) Visually inspect if the upper left corner is empty. Consider to allow some exceptions in the “empty space.” b) If there is “empty space,” continue with Step 3; if not, report that a necessary condition is not observed.	a) Draw a line in the upper left corner to separate the “empty space” and the full space.	a) Ceiling zone: Count the number of “empty cells”a in the “empty space.” b) Scope: Count the total number of cells minus the number of cells of a column, minus the number of cells of a row, plus 1. c) Effect size: Divide ceiling zone by scope. d) Accuracy: The number of observations that are not in the “empty space,” divided by the total number of observations, multiplied by 100%. e) Condition inefficiency: Divide the number of non-empty cellsa in the upper row minus one by the total number of cells in that row minus one and multiply the result by 100%. f) Outcome inefficiency: Divide the number of non-empty cells in the most left column minus one by the total number of cells in that column minus one and multiply the result by100%.	a) Evaluate if the effect size (d) is theoretically or practically meaningful in the current context. b) Consider using the general benchmark 0 < d < 0.1 “small effect,” 0.1 ≤ d < 0.3 “medium effect,” 0.3 ≤ d < 0.5 “large effect,” and d ≥ 0.5 “very large effect.” c) Compare the accuracy with the benchmark of 95%. d) If effect size and accuracy are considered large enough, continue with Step 6. If not, report the results.	a) Formulate the necessary condition in general terms (in kind) as “X is necessary for Y.”b b) Additionally formulate the detailed necessary condition (in degree): Select the row with level of the desired outcome for which the necessary condition can be formulated. c) Identify the critical cell in that row: the “non-empty” cell with the lowest level of X. d) Formulate the necessary condition: (at least) the level of X of the critical cell is necessary for the selected level of Y.

^aNote that an “empty cell” with an exception is an “empty cell” and not a “non-empty cell.”

^bTo evaluate if the necessary condition is also “not sufficient,” it can be observed if there are non-empty cells with lower levels of Y than the level of Y of the critical cell (the critical cell of a row is the “non-empty” cell with the lowest level of X) and that have the same or a higher level of X than the level of X of the critical cell. If yes, the relation is necessary but not sufficient. If not, the relation is just necessary.

Recommendations for Reporting NCA Results

In addition to the common standards for reporting research results, specific elements need to be added for describing the details of a necessary condition analysis. When a necessary but not sufficient theoretical statement is made, it is recommended that researchers should consistently formulate the theory to be tested or built as a necessary (but not sufficient) causal theory (e.g., by formulating propositions/hypotheses in terms of X is necessary [but not sufficient] for Y or by formulating a research question to which extend X is necessary but not sufficient for Y). Contrary to common practice (see Table 1), such theoretical statements should not be formulated (implicitly) as a sufficiency theory (e.g., by formulating traditional general propositions/hypotheses such as X affects Y or X has a positive effect on Y). In the Methods section of the report, it can be stated that a necessary condition analysis was applied to evaluate (test) or formulate (build) a necessary but not sufficient theoretical statement, and it could be specified how the six steps (Table 3 or 4) were performed. In the scatterplot approach, the specific ceiling technique (e.g., ceiling regression or ceiling envelopment) needs to be specified. Also, specification of the criteria for evaluation of the practical or theoretical significance of the effect size of the necessary condition is required. If specific criteria for the study’s specific context are not feasible, the general benchmark for necessary condition effect size may be selected (i.e., 0 < d < 0.1 “small effect”, 0.1 ≤ d < 0.3 “medium effect,” 0.3 ≤ d < 0.5 “large effect,” and d ≥ 0.5 “very large effect”), and for a dichotomous classification, an effect size threshold of 0.1 could be used. In the Results section of the report, first the contingency table or the scatter plot (including ceiling line) could be provided. Next an NCA results table with basic information about the NCA parameters (Step 4 in Tables 3 and 4) could be formulated for each necessary condition. As a minimum, this table should include: ceiling zone, scope, effect size, and accuracy. For discrete and continuous necessary conditions, also condition inefficiency and outcome inefficiency could be included. Table 5 shows the NCA results tables of the three illustrative examples in this article. In these examples, the general benchmark for effect size is used to appraise the results (see the asterisks). If the effect size is considered large enough, the necessary condition can be formulated in kind: X is necessary for Y. Subsequently, the ceiling line or bottleneck table can be used to formulate the necessary condition in degree: Level of X is necessary for level of Y.

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Table 5.

Necessary Condition Analysis (NCA) Result Tables for the Three Illustrative Examples.

	Dichotomous Examplea (Figure 1)	Discrete Exampleb (Figure 2)	Continuous Examplec (Figure 3)
	In kind formulation: A GRE score is necessary but not sufficient for admission. In degree formulation: A GRE score of at least 620 is necessary but not sufficient for admission success.	In kind formulation: Business unit (BU) representation is necessary but not sufficient for cross-business unit collaboration performance. In degree formulation: (1) A high level of BU representation is necessary but not sufficient for a high level of cross–business unit collaboration performance. (2) At least a medium level of BU representation is necessary but not sufficient for a medium level of cross–business unit collaboration performance.	In kind formulation: Ambition is necessary but not sufficient for Sales Ability. In degree formulation: At least a level Xc of Ambition is necessary but not sufficient for a level of Yc of Sales Ability, according to Yc = 0.045Xc + 2.53.
Ceiling zone	1	3	98.2
Scope	1	4	549.2
Ceiling line	Piecewise linear	Piecewise linear	Yc = 0.045Xc + 2.53
Accuracy (%)	99.7	100	96
Effect size	1***	0.75***	0.18*
Condition inefficiency (%)	NA	0	33
Outcome inefficiency (%)	NA	0	46

^a N = 342.

^b N = 12.

^c N = 108.

*d ≥ 0.1. ***d ≥ 0.5.

Additionally, the bottleneck table can be provided, in particular in the multivariate situation, showing which levels of the condition is a bottleneck for certain desired levels of the outcome. Figure 3 (bottom right) and Figure 4 (lower part) give examples of such tables. For convenience to the readers, the researcher can interpret the bottleneck table and the ceiling line as follows. First the researcher can classify the outcome (first column of the bottleneck table) into several classes, for example three classes (low, medium, and high outcome) on the basis of theoretical or practical considerations, measurement scale values (e.g., for a 7-point Likert scale low, <3; medium, 3-5; high, >5), the statistical distribution of the data (e.g., low, <25th percentile; medium, 25th-75th percentile; high, >75th percentile), or the structure of the data. Then for each class it can be indicated how many and which levels of determinants are required to allow that outcome. For example in Figure 4, based on the structure of the data, three classes are distinguished. In the low range outcome values (<50%), no determinant is necessary. In the middle outcome values (50%-60%), some but not all determinants are necessary and their levels need to be up to 20%. In the high range with outcome >70%, all four determinants are necessary for a high outcome with required condition levels of 15% to 73%. For the highest outcome (100%), all determinants must have a value of least 50% (and one nearly 100%). When the required condition percentage is below 100%, there is condition inefficiency.

A Software Tool for Necessary Condition Analysis

An NCA software tool is made available to identify necessary conditions in data sets. The tool was developed to facilitate the process of drawing ceiling lines, calculating NCA parameters, and creating bottleneck tables. The software, called NCA, is a package that runs with open source programming language R (Culpepper & Aguinis, 2011). The software can be obtained freely from http://cran.r-project.org/web/packages/NCA/index.html (Dul 2015b). A quick start guide is available at http://ssrn.com/abstract=2624981 or http://repub.eur.nl/pub/78323/ (Dul 2015c), which allows novice users without knowledge of R or NCA to perform an analysis within 15 minutes.

The NCA package does three main things:

It draws NCA plots, which are scatter plots with ceiling lines. The user can select up to eight different ceiling line techniques presented in Table 5. The defaults are CE-FDH (piecewise linear function) and CR-FDH (straight line).

The output includes scatterplots with ceiling line(s), a table with the NCA parameters (as well as other relevant information), and the bottleneck table. For running the NCA package, the user can change several parameters to fit the software to the specific situation.