Qualitative work and the testing and development of
theory: lessons from a study combining cross-case and
within-case analysis via Ragin’s QCA
Barry Cooper & Judith Glaesser
Durham University
Session Title: Challenging the Quantitative-Qualitative
Divide: Exploring Causal Analysis
ESRC Research Methods Festival
Oxford 2012
This work was supported by an
ESRC research fellowship [RES063-27-0240] awarded to JG.
Our talk will draw on chapter 6 of Cooper, Glaesser, Gomm &
Hammersley (2012) Challenging the Qualitative-Quantitative
Divide and the paper by Cooper & Glaesser (2012) in FQS. It
comprises:
• General comments on regularities and causation.
• Illustration of the ways in which the set theoretic description
of regularities contrasts with the linear correlational
approach.
• Problems in using QCA to gain causal knowledge
(analogous to the “correlation does not imply causation”
problem).
• Using qualitative data (interviews here) to undertake
process-tracing to understand the nature of regularities, i.e.
the processes that have generated them, and also to
improve predictive models (based on existing theory).
• The substantive context is the sociology of education.
"The basic idea of the covering-law account is very simple: An
explanation is a deductive (or statistical) argument that has a
description of the explanandum phenomenon as a conclusion
and one or more empirically validated general law statements
and a set of statements describing particular facts (the initial
conditions) as its premise. The core underlying idea is that
explanations make the explanandum expected [given the
explanans, etc.]. This means that explanation and prediction
are more or less the same thing; the only difference is that in
the case of explanation we already know the outcome"
(HEDSTRÖM & YLIKOSKI, 2010, ARS, pp.54-55).
But, for example: Barometer readings change prior to storms (a
regularity-based law). A storm can therefore be “explained” by
a barometer’s pressure reading falling. Not obviously
explanatory. Common cause is a drop in atmospheric pressure.
So: regularities themselves need explaining.
 In survey-based work on social class and educational
achievement, conventional approaches to exploring the
relationships between theoretically selected variables
employ measures of linear correlation, sometime
directly, and often as the building blocks of regression
models of varying degrees of sophistication. Basically,
here, the model is, more of X leads to more of Y.
 Our presentation will address an alternative approach,
Ragin’s Qualitative Comparative Analysis (QCA), based
in set theoretic mathematics rather than the linear
algebra that underlies the correlational approach.
 While the correlational approach focuses on the
relations between “independent” variables and some
dependent variable, the set theoretic approach focuses
on cases understood as configurations of features, and
explores what combinations of features are sufficient
and/or necessary for some outcome to be achieved.
Quotes from Pawson (2007), “Causality for Beginners”:
SUCCESSIONISTS locate and identify vital causal agents as
‘variables’ or ‘treatments’. Research seeks to observe the
association between such variables by means of surveys or
experimental trials. Explanation is a matter of distinguishing
between associations that are real or direct (as opposed to
spurious and indirect) and of providing an estimate of the size
and significance of the observed causal influence(s).
CONFIGURATIONISTS begin with a number of ‘cases’ of a
particular family of social phenomenon, which have some
similarities and some differences. They locate causal powers in
the ‘combination’ of attributes of these cases, with a particular
grouping of attributes leading to one outcome and a further
grouping linked to another. The goal of research is to unravel
the key configurational clusters of properties underpinning the
cases and which thus are able to explain variations in outcomes
across the family.
Quote from Pawson (2007), “Causality for Beginners”:
GENERATIVISTS, too, begin with measurable patterns and
uniformities. It is assumed that these are brought about by the
action of some underlying ‘mechanism’. Mechanisms are not
variables or attributes and thus not always directly
measurable. They are processes describing the human
actions that have led to the uniformity. Because they depend
on this choice making capacity of individuals and groups, the
emergence of social uniformities is always highly conditional.
Causal explanation is thus a matter of producing theories of
the mechanisms that explain both the presence and absence
of the ‘uniformity’.
Now, moving from these general perspectives to something
more concrete …
Much theory addressing the relationship between social class
and educational achievement has two features that make
regression methods based on linear correlation sometimes less
valuable than an alternative approach based in set theory that
focuses on (i) modelling the conditions necessary and/or
sufficient for some outcome and on (ii) causes conceptualised as
conjunctions of factors.
For example, Boudon’s model of the primary and secondary
effects of social class might be paraphrased as claiming that, in
some social classes, early achievement is necessary but not
sufficient for later achievement.
Bourdieu’s account of capitals often refers to conjunctions of
causes. Educational capital may need to be combined with
social capital in order to be sufficient for some occupational
destinations to be achieved.
The most common uses of regression do not set out to address these
logical and/or causal structures.
In a simple additive (and, for illustration, deterministic) equation:
OUTCOME = 2S + 2T
we can see that even if S is zero T can have an effect, and vice
versa.
Of course, regression equations can be written to capture the claims
of Boudon and Bourdieu. Re Bourdieu, consider:
DESTINATION = EDUCATIONAL CAPITAL X SOCIAL CAPITAL
Here both capitals must be present to produce a non-zero outcome.
Some of what we discuss can be modelled with such interaction
terms, dummy variables, variable transformations, etc. (but not all,
we think, see Vaisey, 2007, fn11, on fuzzy set QCA). However, we
claim that the set theoretic approach has a more obvious affinity with
these types of relation and aim to illustrate this.
We will now contrast correlational and set theoretic
accounts of a bivariate relation that varies by type of case.
Initially we will use invented data on the relation between
educational achievement and measured ability across three
social class groups.
Having shown what the set theoretic approach can capture,
we will then apply this approach to real data from the
National Child Development Study.
Figure 1: achievement by ability (invented data,194 cases in 3 social classes)
Note: the size
of the shape
indicates the
number of
cases at any
point.
Correlation:
r = 0.601
Class, ability, achievement: What might a scatterplot of all cases hide?
Figure 2: just the cases in social class 1 (invented data, n=43)
Note: Class 1 is
the highest
social class of
the three.
For class 1, we
obtain an upper
triangular plot.
r = 0.739
Note:
y=x
diagonal
line is for
later use.
Figure 3: Just class 2 cases (invented data, n=78)
r = 0.739
r = 0.739
Figure 4: Just class 3 cases (invented data, n=73)
While the scatter of points for class 2 has the elliptical form welldescribed by a linear correlation coefficient, the scatter of points for
classes 1 and 3 has a more triangular form not well-suited to a
description using this coefficient.
A look at the graph for
class 1 shows that the
(invented) cases with
higher ability achieve
highly but that so also
do some cases with
lower ability.
Ability, in a sense to be
described, is sufficient
but not necessary for
achievement for cases
in class 1.
Before we can systematically describe these
relationships in terms of sufficiency and necessity,
we need to introduce QCA.
Crisp and fuzzy set variants. See Ragin (e.g.
2008) for more detail.
Figure 6: Perfect sufficiency
of ability for achievement in
some imaginary world
In Figure 6, cases with “high ability”
are a subset of (are included in) the
set of “high achievers”. This is
equivalent to “high ability” being
(logically) a sufficient condition for
high achievement.
Figure 7: Quasi-sufficiency of
ability for achievement in
another imaginary world
In Figure 7, the proportion of
cases of the condition set that
is included in the outcome set
can be used as a measure of
consistency with perfect
sufficiency. This is high here.
Table 3: membership in the sets “high achiever” and “high ability”
Not high achiever
High achiever
High Ability
Cell 1
Cell 2
Not high ability
Cell 3
Cell 4
These subset relationships can also be discussed in the context of
crosstabulations of membership in sets. In the tradition of correlational
analysis, there is a concern with symmetry. For a high positive correlation
we would want cases mainly in both of cells 2 and 3 of this table.
Analysing sufficiency moves us away from a concern with symmetry. To
test whether high ability is sufficient for high achievement, we only need
to look at the first row, containing cells 1 and 2. The crucial thing is that
there be no (or very few) cases in cell 1, since these would contradict a
claim that being of high ability is sufficient (or quasi-sufficient) for high
achievement.
Similar Venn diagrams can be drawn, and arguments made, for necessity,
but, in this case, the outcome must be a subset of the condition.
Here the conjoined (intersected) set (in red) of the highly able and
the highly ambitious is a subset of the outcome set of high
achievers. Hence the two factors conjoined are logically sufficient for
high achievement.
Y = (A*B*c) + (A*C*D*E)
In these Boolean equations the symbol * indicates Logical AND (set
intersection), + indicates Logical OR (set union), upper case letters
indicate the presence of factors, lower case letters indicate their
absence.
In this fictional example of causal heterogeneity, the equation
indicates that there are two causal paths to the outcome Y. The first,
captured by the causal configuration A*B*c involves the conjoined
presence in the case of features A and B, combined with the
absence of C. The second, captured by A*C*D*E, requires the joint
presence of A, C, D and E. Either of these causal configurations is
sufficient for the outcome to occur, but neither is necessary,
considered alone. The factor A is necessary but not sufficient. The
factor C behaves differently in the two configurations.
Example taken from: Mahoney, J. & Goertz, G. (2006): A tale of two cultures:
contrasting quantitative and qualitative research. Political Analysis, 14, (3).
Another Boolean algebraic summary of the results of a cross-case
analysis:
A*B*C + D*E*f + B*D*F => Outcome
Note that, unlike regression equations, QCA retains skeletal cases –
as conjunctions of factors – in its algebraic formulations.
A*B*C is a type of case, as is D*E*f, as is B*D*F.
In variable-based analysis, employing correlation, as every student is
warned, correlation does not equal causation (for several reasons).
Similar issue arises with QCA. We will return to these later.
Producing these summary equations requires knowledge of “truth
tables” and “Boolean minimisation”. Let’s look at these …
QCA employs “truth tables”, where a 1 indicates the factor is
present, and 0 it is absent. Here is an invented truth table:
A
B
C
Number
1
1
1
1
0
0
0
0
1
1
0
0
1
1
0
0
1
0
1
0
1
0
1
0
120
110
100
120
100
100
800
720
Consistency (with sufficiency) Number with
for the outcome
the outcome
0.925
111
0.900
99
0.890
89
0.500
60
0.350
35
0.060
6
0.020
16
0.008
6
Table 1: Invented data for O (Outcome) = Function (A,B,C)
Three configurations (shaded) have consistencies high
enough to justify a claim that these are quasi-sufficient for the
outcome to be attained. They are ABC, ABc and AbC.
(Note: we sometimes drop the *s in A*B*C etc. for the sake of simplicity.)
The result can be minimised thus:
ABC + ABc + AbC => O becomes AB + AC => O,
(with an overall consistency with sufficiency of 0.906).
How? Here, e.g., the presence or absence of the
factor C makes no relevant difference, given AB, and
so we can collapse ABC + ABc to AB.
Similarly, ABC + AbC becomes AC.
“AB + AC => O” tells us a great deal. Given a new
case falling into the set “AB + AC”, we could claim
that there is a 0.906 chance of this case having the
outcome.
These examples all concerned crisp sets (a case is either fully in or
out of the set; equivalently, a factor is either present or absent).
Chapter 6 of our recent book (Cooper et al, 2012) employs fuzzy sets,
allowing partial membership in sets, ranging from zero (fully out) to
one (fully in), through such values as 0.25 (more out than in), 0.5 (as
much in as out), 0.75 (more in than out).
A good example is “adulthood” (Kosko, 1994). Most judges would
agree that an age of ten would rule out adulthood (giving a
membership score of zero) and one of 30 would rule it in (giving a
membership score of one) but there would be much more discussion
about the age range 15 to 21.
Here it would seem inappropriate to allocate a score of either zero or
one the only possibilities available in the crisp set context. In fuzzy
set based descriptions of cases a score of say 0.9 can be used for
the 20 year-old to indicate almost full, but not quite full, membership
of the set of `adults'. A nineteen year-old might be allocated a score of
0.8, and so on.
The invented data for class 1 again.
We noted earlier that the
scatter of points here
represented ability being
sufficient (but not necessary)
for achievement.
Note that, for each case, the
fuzzy ability score is lower
than or equal to the fuzzy
achievement score (i.e. x < =
y). This is in fact the simplest
test for (perfect) fuzzy
subsethood, i.e., for every
case, the condition score
must be less than or equal to
the outcome score. For real
datasets, a simple measure
(from 0 to 1) of closeness to
perfect sufficiency is the
proportion of cases that
passes this test.
For necessity the test is reversed (y<=x). The lower triangular plot for class
3 passes this test. Here ability is necessary (but not sufficient) for
achievement.
This is a case of perfect
necessity. No cases stray over
the y=x line.
Again, for real datasets, a
simple measure (from 0 to 1) of
closeness to perfect necessity
is the proportion of cases that
is on the correct side of this line
or on it.
The invented data for Class 3 again.
We can compare these set
theoretic measures of
sufficiency and necessity for
the three classes with the
correlation coefficients (which
were all equal, recall, at 0.739).
Table 4: set theoretic testing of the ability => achievement relationship
Simple inclusion
algorithm
Sufficiency Necessity Result: Ability is …
(0-1 scale) (0-1 scale)
All classes together
0.459
0.598
Neither sufficient nor
necessary
Class 1
1.000
0.023
Sufficient but not
necessary
Class 2
0.590
0.539
Neither sufficient nor
necessary
Class 3
0.000
1.000
Necessary but not
sufficient
While the correlation coefficients were identical by class, the
measures of nearness to perfect necessity and sufficiency clearly
are not. They have captured important differences between class 1
and class 3.
Now a brief account of an example using real data (see book
for the detail):
We explore the relation between
ACHIEVEMENT and measured ABILITY
as it varies by combinations of values of
FATHER’S CLASS, GRANDFATHERS’ CLASS (both), SEX.
We focus on comparing
the correlation between achievement and ability
with
the consistency with sufficiency of ability for achievement.
Our data come from the National Child Development Study (cases born in
one week in 1958). We use a sample of 5,272 cases from the NCDS with
no missing values on the variables we employ here.
Variables:
Measured ability (variable n920) is taken at age 11.
The measure of achievement, highest qualifications obtained, is taken at
age 33 and includes both academic and vocational qualifications (derived
variable HQUAL).
Because of the ways each grandfather’s class was recorded we will be
using the categories of the Registrar General’s scheme for these, but we
will be using an approximation to Goldthorpe’s class scheme for the
respondent’s father’s class (at the respondent’s age of 11). Given the
illustrative nature of the analysis here, this mixing of categories, though
undesirable, is of no consequence, we believe, for our arguments.
Ability and achievement are calibrated as fuzzy sets (see Cooper et al,
2012, chapter 6, for the details).
Consider two contrasting sets of cases (i.e. configurations of
factors):
• males with a class of origin towards the top of the social class
distribution who also had grandfathers towards the top of this
distribution
• females who had fathers and grandfathers towards the bottom
of the distribution
We can hypothesise that ability might have tended towards being
a sufficient condition for achievement for the former group (but
perhaps not a necessary one) and towards being a necessary
condition for the latter group (but perhaps not a sufficient one).
If this is so, we should expect an upper triangular plot of
achievement against ability in the first case and a lower triangular
plot in the second.
The cases here are the 60 males whose father was in Goldthorpe’s
class 1 (the upper service class) and whose grandfathers were
both in either Registrar General’s social class I or II.
Pearson correlation is
0.505.
Consistency with
sufficiency is 0.833.
Consistency with
necessity is 0.167.
This approaches the
form of an upper
triangular plot (i.e. it
tends, descriptively,
towards ability being
sufficient but not
necessary for
achievement).
The cases here are the 94 females whose own father was in
Goldthorpe’s class 7 (the semi and unskilled manual working class)
and whose grandfathers were both in either RG class IV or V.
Pearson correlation is
0.472.
Consistency with
sufficiency is 0.207.
Consistency with
necessity is 0.721.
This approaches the
form of a lower
triangular plot (i.e. it
tends, descriptively,
towards ability being
necessary but not
sufficient for
achievement).
The clear tendencies are that:
For the configuration
* = AND, meaning
together with
HIGH_GFs’_CLASS*HIGH_FATHER’S_CLASS*MALE
ability is sufficient, but not necessary, for later achievement.
For the configuration
Lower case => females
LOW_GFs’_CLASS*LOW_FATHER’S_CLASS*male
ability is necessary, but not sufficient, for later achievement.
Note: usual language is quasi-sufficient and quasi-necessary in these situations.
HQUAL BY ABILITY*CLASS*MALE*GFS (Ordered by consistency with sufficiency)
Sex
Grandfathers
both in RG I
or II
Goldthorpe
Class
Correlations
N
Measure of consistency
with sufficiency (of
ABILITY for HQUAL)
N
Male
Yes
1
0.505
60
0.833
60
Female
Yes
3
0.604
11
0.818
11
Male
Yes
7
0.556
9
0.778
9
Male
Yes
2
0.539
40
0.700
40
Female
Yes
2
0.531
56
0.679
56
Rows omitted for this presentation
Male
No
3
0.445
236
0.568
236
Male
No
6
0.517
758
0.485
757
Rows omitted for this presentation
Female
Yes
4
0.429
37
0.324
37
Female
No
6
0.483
820
0.319
818
Female
No
5
0.446
147
0.313
147
Female
No
7
0.503
463
0.257
460
Female
No
4
0.649
110
0.255
110
Note: “not in I/II” is rather heterogeneous.
Types still too heterogeneous?
We have now shown why we think QCA, and the
ideas of sufficiency and necessity, are useful for
describing social regularities.
Also have given examples of configurations of
conditions that are quasi-sufficient for some
outcome.
But if, say, G*h*I*K => some outcome, what can
we say from this re causal relations between the
conditions and the outcome?
Now we will address some of the ways in
which QCA results might be misinterpreted
from the perspective of a concern with
causality.
We will take a realist view, wanting to
understand the generative processes by
which these configurational regularities are
produced.
Let’s invent an example …
Let A stand for belonging to a family amongst the social elite in early
nineteenth century England. Let B stand for having parents who
value literacy. Let C stand for being male, with c then indicating
being female. Let the outcome O be achieving literacy.
“AB => literacy” then suggests that gender makes no difference
and, indeed, this may be true in the sense that, irrespective of
gender, the conjunction of A and B is quasi-sufficient for the
outcome. The actual causal routes to the outcome may,
nevertheless, of course, differ for ABC and ABc. Boys may receive
their education in schools, girls in the home. (The pedagogy may
differ by gender. The nature of the outcome “literacy” may therefore
also be heterogeneous by gender.)
Minimising configurations (ABC, ABc -> AB here) can deflect us
from important differences in causal paths to an outcome.
We explore this and several related issues at length in Cooper &
Glaesser (2012).
• The remainder of this talk continues to draw on our project
studying educational transitions and achievement in English
and German secondary schools.
• British longitudinal data (NCDS, BCS) and German panel
survey data (SOEP) have been analysed using a
configurational case-based approach (Ragin’s QCA) rather
than a variable-based approach (i.e. correlation-based
regression methods).
• We will focus on the necessary and/or sufficient conditions for
a particular outcome to be achieved.
• However, as noted earlier, like correlational methods, QCA
alone cannot ground causal claims.
• To address the causal processes generating the relationships
between conjunctions of conditions and educational outcomes,
some 80 interviews have been undertaken with young people
aged around 17.
• These interviewees were selected on the basis of a typology
created via our QCA analyses.
We now move to a QCA analysis, using German SOEP data, of the
conjunctions of conditions that are sufficient and/or necessary for
being in the Gymnasium at age 17. Our theoretically selected
potentially causal conditions are:
Gender:
At least one parent has the Abitur:
At least one parent in the service class:
Adolescent had Gymnasium recommendation:
(MALE / male)
(ABI_1P / abi_1p)
(SC_1P / sc_1p)
(GY_REC / gy_rec)
We find (for 790 cases born in the late 1980s) that having the
recommendation (GY_REC) is a quasi-necessary condition for the
outcome (with a consistency of 0.89).
Also, the following is one solution for quasi-sufficiency (with overall
consistency of 0.86):
ABI_1P*GY_REC
+ MALE*SC_1P*GY_REC => GY_17
Drawing on an earlier paper1 addressing theory development via combining
cross-case and within-case analysis, we can now list “typical” and “deviant”
cases (from Germany) in regard to both sufficiency and necessity:
Configuration
Outcome Type of case
1 ABI_1P*GY_REC
present
Typical with regard to
sufficiency
2 MALE*SC_1P*GY_REC
3 ABI_1P*GY_REC
present
absent
4 MALE*SC_1P*GY_REC
absent
5 gy_rec
present
Typical w.r.t. sufficiency
deviant with regard to
sufficiency
deviant with regard to
sufficiency
deviant with regard to
necessity
Available interviewees
(not taken from SOEP
dataset)
Alina, Anna, Lena,
Ludwig, Roman
Daniela?, Nicole?
Jonas, Ludwig
Samuel
None.
Sibel, Andreas, Julia,
Christian, Aynur, Tessa
(i.e. the quasi-necessary
condition GY_REC is
absent for these cases)
Table 6: Types of configurations
1. Glaesser, J. & Cooper, B. (2011) Selecting cases for in-depth study from a survey dataset:
an application of Ragin’s configurational methods, Methodological Innovations Online.
In Cooper & Glaesser (2012) we discuss three
German interviewees, two deviant re necessity and
one re sufficiency. Our purpose in focusing on deviant
cases is to develop causal understanding and/or
theory.
One discussed case is Andreas, deviant re necessity,
who is of the type
MALE*ABI_1P*SC_1P*gy_rec.
To make sense of such deviant cases, we aim to find
an X2 that might substitute for the quasi-necessary
condition X1 (here, the recommendation for
Gymnasium).
X2 might, of course, be a conjunction of factors.
Andreas attended a Realschule but is soon to enter the
Gymnasium, belatedly achieving this outcome.
The educational careers of his siblings seem key factors. They all
attended Gymnasium. Both his parents have the Abitur.
The lack of the recommendation could be calmly overcome given
relevant knowledge in his family of the system (see Cooper &
Glaesser, 2012, for evidence).
A candidate X2 here (as an alternative to the recommendation)
seems to comprise, then, a family where gaining the Abitur is a
normal element for both parents and their children conjoined with
the organisational possibility of moving from the Realschule to some
type of Gymnasium.
In fact, it turns out that having at least one parent with the Abitur can
play this role, as the following cross-case test shows:
the quasi-necessity of GY_REC + ABI_1P raises consistency with
necessity from 0.89 to 0.96.
We argued earlier some regularities can be fully represented in terms of
sufficiency and necessity. As we noted, however, regularities alone cannot
establish causation. A further step is needed. This might be theoretical
interpretation, if strong enough theory exists. Alternatively, within-case data
collection and analysis can help gain access to the processes that generate
quasi-regularities and the exceptions to them.
In Cooper & Glaesser (2012), we discuss deviant cases, with the goal of
understanding why it is, re sufficiency, that some cases with a normally
sufficient combination of conditions to gain our outcome did not and why, re
necessity, some cases lacking a normal necessary condition nevertheless
managed to achieve our outcome.
An important common factor for all three cases was the immediate and
extended familial context in terms of what it was “natural” or habitual to do,
educationally and occupationally.
We have focused in this paper on discussing “deviant” cases. We are also
analysing typical cases in order to asses the extent to which their
biographies can be captured by theories such as Bourdieu’s. We will report
on this ongoing work in future papers (see Glaesser & Cooper, 2012, BSA,
for a start).
References
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combining Cross-Case and Within-Case Analysis via Ragin's QCA. Forum: Qualitative Social Research 13(2): 4.
http://www.qualitative-research.net/index.php/fqs/article/view/1776
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