Answers to Exercises and questions for Discussion

Is the ‘fuzzification’ of ordered categories just another way of scaling a variable?

Ordered category variables that are a result of self-reported ratings (like Likert items from strongly agree to strongly disagree, or degrees of satisfaction) are direct measurements, so converting these mechanically into fuzzy values, as illustrated in Figure 7.1 in the text, can be seen as a simple rescaling in order to be able to use the variable in fuzzy set analysis software like fsQCA. However, this is a somewhat mechanical approach that may or may not ‘make sense’ in terms of set memberships and runs counter to the idea that set memberships should, ideally, be determined in a process of researcher calibration. It is also likely to produce too many values of exactly [0.5]. However, where ordered categories are used in summated rating scales, then the summations can be used by researchers to determine totals above which, for conceptual reasons, can be considered as ‘full membership’ of a set, the values below which constitute full non-membership and a crossover value of maximum ambiguity.

Why is it sometimes better to treat a nominal variable with two categories (a ‘dichotomy’) as two separate binary sets?

A dichotomy represents two contrasting groups, like privately owned and publicly owned organizations. If there really are only two possibilities, then being not privately owned means being publicly owned and can be treated as a single binary set. However, if there are other possibilities, like charities or political parties that are neither privately not publicly owned, then being not privately owned does not necessarily imply publicly owned, so we need two binary sets: privately owned/not privately owned and publicly owned/not publicly owned.

Try minimizing the following five expressions for sufficiency for the outcome Y:

A~B~C + ~A~BC + ~AB~C + ~ABC + A~BC      ----------     Y

The first two expressions ~A~B~C + ~A~BC reduce to ~A~B since Y happens irrespective of the presence of C. Similarly, ~AB~C + ~ABC reduces to ~AB. So the expression becomes

~A~B + ~AB + A~BC       ----------     Y

This may be further reduced since ~A~B + ~AB reduces to ~A. So the final minimized expression is

~A + A~BC         ----------     Y

This is logically equivalent to the original expression, but in more parsimonious form.

Why is it necessary to avoid, as far as possible, fuzzy membership values of [0.5]?

Any fuzzy set values of exactly [0.5] are excluded from the truth table since it cannot be determined whether they are more ‘in’ or ‘out’ of the set. Fuzzy set membership values of exactly [0.5] are best avoided wherever possible because, although they are included in fsQCA calculations of logical sufficiency, cases where any membership value in the configuration is exactly [0.5] are not included in the frequency column in the truth table. Box 7.2 in the text explains how to avoid these values.

How does counterfactual analysis help to focus the results produced by fsQCA?

All causal statements make assumptions about what would have occurred if the causal condition or set of conditions had not manifested itself. Thus, if we say that A causes B, we are making the assumption that, in the absence of A, B will not happen. A counterfactual is an assumption that is made about the impact of an unobserved event. In the context of fuzzy set analysis it is an assumption that is made about what would have been the outcome if there had been any empirical cases for a configuration that is a logical remainder – a row in a truth table without enough cases in it. Probing the historical and logical consistency of counterfactuals is the essence of counterfactual analysis.

The fsQCA analysis will make whatever counterfactual assumptions result in minimizations, as in Exercise 3 above. However, the software allows the researcher to restrict the use of counterfactuals by asking researchers to include their directional expectations about each causal condition. This will ask the researcher, for each causal condition, whether its presence or absence is expected to contribute to the outcome. Thus ~A~B~C + ~A~BC will not be reduced to ~A~B if the researcher has indicated that C is expected to contribute to the outcome. When diversity is limited, there will be many remainders and excluding them from the analysis by using the complex solution will result in little or no Boolean simplification. Likewise, a parsimonious solution, which makes all remainders available as simplifying assumptions (and which ignores any absent or present conditions entered in the intermediate solution), can be unrealistic, over-simple and may make assumptions that are untenable. Intermediate solutions strike a balance between parsimony and complexity by incorporating the researcher’s substantive and theoretical knowledge. In general, intermediate solutions are preferred since they are the most interpretable and incorporate only those assumptions that can plausibly be made.

In what ways is the output from fsQCA analyses closer to ‘reality’ than the output from variable-based analyses?

Reality is often complex and messy. There will often be more than one way in which an outcome may come about. Case characteristics may contribute to outcomes in different ways depending on what other characteristics they are combined with. Some properties may be essential for an outcome – they are necessary conditions; some combinations of properties may be sufficient or largely sufficient to produce an outcome. The output or ‘solutions’ from fsQCA allow for all these possibilities. There is not a single result as in many variable-based analyses, for example a multiple regression of R2 = 0.46. Even looking at beta coefficients for each independent variable entered into the equation assumes that each makes a fixed contribution to the outcome, all the other variables being held constant.

Download the alcohol marketing fuzzy set dataset, Fuzzysetalcoholmarketing.dat, from and save it onto your system. Download fsQCA. Just Google fsQCA and select fs/QCA software. Download fsQCA 2.0. Select File|Open|Data and browse to your saved.dat file and click on Open. Rerun the alcohol marketing data (a) using 0.8 as the consistency cut-off, then (b) the negation of the outcome intention to drink alcohol in the next year, then (c) trying both higher and lower frequency thresholds, and then (d) indicating in the intermediate solution all the conditions as absent or present. Repeat (a)–(d) but include drink status in the conditions.

(a) To obtain the completed truth table shown in Figure 7.22 in the text, select Analyze|Fuzzy Sets|Truth Table Algorithm. Put fsintention into the Outcome box and fsaware, fsinvolve, fslikeads, fslikeschool and cssibsdrink into Causal Conditions. Click on Run. To choose six as the minimum frequency threshold, click on 5 under number, then Edit|Delete current row to last row. Under raw consist, click on the top cell then Sort|Descending. To make 0.8 the consistency cut-off, enter 1 under fsintention for the first four configurations then 0 for all those below 0.8. (I think that 0.797599 can be considered to round off to 0.8.) Now click on Standard Analysis. For the intermediate solution, indicate that having siblings who drink, liking alcohol ads, involvement in alcohol advertising and being aware of such advertising all contribute to the outcome, but that the absence of liking school is likely to so contribute. Click on OK. The intermediate solution gives only one solution: that fsaware, fsinvolve and cssibsdrink jointly are sufficient for the outcome to an acceptable level of consistency. Compare this with Figure 7.23 in the text, which has used a consistency cut-off of 0.75 and results in two solutions, but the first has low consistency and the second has low unique coverage.

(b) To rerun the analysis, it is, unfortunately, necessary to go right back to Analyze|Fuzzy Sets|Truth Table Algorithm – there is no ‘back’ button. Put fsintention as Set Negated in the Select Variables screen. Select the same causal conditions as before. Make the frequency threshold 6 and revert to the original consistency threshold of 0.75. You will see from the truth table that levels of consistency are much higher for this negation of the outcome. It is easier to see what configurations might lead to a lack of intention to drink alcohol in the next year.

(c) If you lower the frequency threshold, for example to 1, then many more rows are included in the analysis. The result includes the two solutions for the higher threshold, but adds three more solutions. This result is, probably, less helpful. Raising the threshold, for example to 12, reduces the solutions to only one and with a lower overall consistency. The original frequency threshold was probably about right.

(d) In the Standard Analysis screen indicate all the conditions as absent or present (this is the default setting). The solutions under Intermediate Solution now turn out to be the same as under the Complex Solution; in other words, no counterfactual assumptions are made.

Not surprisingly, if you include drink status among the causal conditions, then this becomes the only condition that meets an acceptable level of consistency. For the most part, having already had an alcoholic drink is sufficient for the outcome intend to drink alcohol in the next year.

Go to the website Here you will find lots of interesting information about social surveys created by John Hall, previously Senior Research Fellow at the UK Social Science Research Council (1970–6) and Principal Lecturer in Sociology and Unit Director at the Survey Research Unit, Polytechnic of North London (1976–92). Download the ‘Quality of Life: 2nd Main Survey’. Select Subjective Social Indicators (Quality of Life)|SSRC Survey Unit Quality of Life Surveys in Britain 1971–1975|Abstracts, data and documentation. For the 2nd National Survey 1975, download the data from SPSS saved file and save onto your system. Download and print off the Abstract and the Questionnaire.

This survey covers many variables, but several of the questions like QA13 ask respondents to give a 0–10 rating. These can be readily converted into fuzzy set values so that 0=[0], 1=[0.1], 2=[0.2] up to 10=[1]. Copy all 13 items in QA13 (these are labelled var136-var148), copy all the items in QA18 and QA19 into a separate SPSS file and convert to fuzzy sets using SPSS Recode. Give the file a name like FuzzysetQOL. At this stage it is better to rename the variables so that you can recognize them in fsQCA, so var136 could become kitchen and so on (this can be done in the Variable View). This SPSS file now needs to be saved as a.dat file. Select File|Save As. Under Save as type select Tab delimited (*.dat). Check that the box Write variable names to spreadsheet is ticked. Click on Save. Now download fsQCA. Just Google fsQCA and select fs/QCA software. Download fsQCA 2.0. Select File|Open|Data and browse to your saved.dat file.

QA19 can be taken as the ‘outcome’ to be studied. This is an overall measure of satisfaction. The items in QA13 can be taken as potentially sufficient causal conditions. A truth table analysis will show which combinations of conditions may be sufficient for overall satisfaction. Follow the instructions in Boxes 7.3 and 7.4. See if you can interpret the results. You can try the same for QA18.

This is an exercise that gives you the opportunity to think about how you might transform variables into set memberships for fsQCA analysis. There are many different ways in which this can be done, so there are no ‘answers’ or correct solutions.