Exploring configurational causation in large
datasets with QCA: possibilities and problems
Barry Cooper & Judith Glaesser
School of Education, Durham University
3rd ESRC Research Methods Festival
St Catherine’s College
Oxford, 30 June – 3 July 2008
A note re these slides.
• Some of these slides will be used in our presentation itself but some
have been written to provide, as a context for the tables, etc., a preand post-festival web-based sketch of the method we have
employed (Ragin’s Qualitative Comparative Analysis, or QCA) for
any readers new to it.
• After a brief description of the background to Ragin’s development of
the set theoretic approach, and a list of what we see as its strengths,
we will illustrate its use with large n data, drawing on our experience
of using QCA (Cooper, 2005, 2006; Cooper & Glaesser, 2007, 2008,
in press; Glaesser, forthcoming).
• To keep things less complex than they would otherwise become, we
will not draw attention, during this part of our presentation, to the
more problematic issues that we wish to mention.
• Instead, we deal with this aspect of our presentation after the
illustration of the use of QCA in a large n context.
Concerns about the dominant regression approach in quantitative analysis
have a long history. Here, for example, are various remarks taken from
Peter Abell’s 1971 book, Model Building in Sociology:
• It is often (perhaps more often than not) the case that the
covariation between sociological variables is not linear (p.174).
• It was argued ... that interaction is a characteristic feature of
sociological covariation (p.183).
• Multicollinearity is pervasive in sociology; it is more often than not
the case that explanatory variables are intercorrelated (p.189).
• But from what was said earlier it might be expected that …
(cardinal) variables will be of relatively rare occurrence in sociology.
One is much more likely to encounter the situation where nominal
and ordinal variables are related (p.197).
• We have noted earlier that the typical causal situation in social
science is one of over-determination – many different clusters of
variables are sufficient for a given effect (p.236).
Abell’s book also includes considerable discussion of the logic of
necessary and sufficient conditions alongside his discussion of
linear modelling.
Several authors, from various perspectives, have raised important
concerns about regression and its uses. For example (see attached
bibliography for details):
•
•
•
•
•
•
•
•
•
•
•
•
Boudon (1974a,b)
Byrne (1998, 2002)
Freedman (1987, 1997)
Hedström (2005)
Lieberson (1985)
Morgan and Winship (2007)
Ormerod (1998)
Pawson & Tilley (1997)
Pearl (2000)
Ron (2002)
Sörensen (1998)
Taagepera (2005).
Andrew Abbott (2001) has summarised some of the key
assumptions of the linear model normally used in regression:
•
•
•
•
The social world is made up of fixed entities with varying
attributes (demographic assumption).
– Some attributes determine (cause) others (attribute
causality assumption).
What happens to one case doesn't constrain what
happens to others, temporally or spatially (casewise
independence assumption).
Attributes have one and only one causal meaning within a
given study (univocal meaning assumption).
Attributes determine each other principally as
independent scales rather than as constellations of
attributes; main effects are more important than
interactions (which are complex types) (main effects
assumption).
Charles Ragin’s work
Ragin (1987) shared many of the concerns of these various writers,
but, in particular perhaps, focussed on Abbott’s third and fourth
points, the relative neglect of causal heterogeneity and complex
interaction in regression models when used in practice[1]. Using set
theory rather than regression’s linear algebra as the basis for
developing a configurational approach to causal modelling, he began
to explore ways in which (i) complex interaction between causal
factors and (ii) causal heterogeneity (i.e. the existence of several
distinct types of cases in a ‘population’[2] and therefore of possible
multiple pathways to an outcome) could be described in Boolean or
configurational terms (Ragin, 1987, 2000, 2006a). In doing so, he
also aimed to shift researchers’ practices away from a focus on the
net average effects of variables (i.e. on which variables win the race
to explain most variance) and towards an approach that recognised
that events in the world are often caused by conjunctions of factors
(Ragin, 2006b). It is his Qualitative Comparative Analysis (QCA) on
which we focus in this paper.
[1] On Abbott’s second point, see Hedström (2005).
[2] The returns to cognitive capacity, for example, might differ
systematically between social classes.
Before introducing QCA in more detail, we might set out what
we regard as the strengths of Ragin’s approach:
1.
2.
3.
4.
5.
6.
A focus on cases and their constituent features rather than,
as in regression, on abstracted variables (and therefore net
– and often average – effects).
Analysis of multiple and conjunctural causation in terms of
necessary and/or sufficient conditions rather than in terms of
the linear additive model.
The recognition, up front, of the possibility of causal
heterogeneity.
The offer of a rigorous approach, drawing on set theory and
logic, to the analysis of these features of social reality.
Through a focus on INUS[1] conditions, the allowing, up
front, of complex interactions between causes.
The recognition of the problems resulting from limited
diversity in social datasets.
[1] An INUS condition is “an insufficient but non-redundant part of an
unnecessary but sufficient condition” (Mackie, 1974).
Boolean functional form: an example
• Ragin’s QCA and its associated software use Boolean algebra to
address conjunctural causation. Boolean equations have a different
functional form to the regression equations with which social scientists
are familiar. Here is an example taken from a paper contrasting the
approaches (Mahoney & Goertz, 2006):
• Y = (A*B*c) + (A*C*D*E)
• In these equations the symbol * indicates Logical AND (set
intersection),+ indicates Logical OR (set union), upper case letters
indicate the presence of factors, lower case 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 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. A is necessary but
not sufficient. The factor C behaves differently in the two configurations.
This non-probabilistic - or veristic - example, of course, assumes no
empirical exceptions to these relations.
QCA: Sufficiency and quasi-sufficiency
Sufficiency, understood causally or logically, involves a subset relation.
If, for example, a single condition is always sufficient for an outcome to
occur, the set of cases with the condition will be a subset of the set of
cases with the outcome. This is shown in Figure 1 (next slide) based on
a hypothetical relation between being of service class origin and
achieving a degree. Given the condition, we obtain the outcome. In
applications to real large n data, perfect sufficiency is unlikely to be
found, and a situation like Figure 2 (next slide) will often be found, where
most but not all of the set of cases with the condition also are members
of the outcome set.
Using conventional crisp sets, the proportion of the members of the
condition set who are also members of the outcome set can be used as
a measure of the degree of consistency of the empirical relation with a
relation of perfect sufficiency (here: the number in the yellow subset
divided by the number in the yellow and green subsets taken together).
Figure 2 illustrates a relation that might be described as only ‘nearly
always sufficient’. Alternatively, using a probabilistic view of causation,
being of service class origin here could be said to be a sufficient
condition, all else being equal, for raising the probability of achieving the
outcome to a level equal to this “consistency” proportion.
Figure 1: Perfect Sufficiency
Figure 2: Quasi-Sufficiency
QCA: Necessity & Coverage
In Figure 3 (next slide), another hypothetical relation between being of
service class origin and achieving a degree is shown. This is another
example of less than perfect sufficiency. Here the members of the yellow
fringe of the service class origin set are not also members of the outcome
set. However, most members of this condition set are. This example is also,
in fact, a special case in that being of service class origin is a necessary
condition for achieving a degree (and in the case of necessity the outcome
set is, as can be seen, a subset of the condition set, reversing the direction of
the subsethood relation that characterises sufficiency).
Venn diagrams can also illustrate Ragin’s concept of explanatory coverage
(Ragin, 2006a). The proportion of the outcome set that is overlapped by the
condition set can be used as a measure of the degree to which the outcome
is covered (‘explained’) by the condition. In Figure 1 (previous slide), the
coverage of the outcome of having a degree by the condition of being of
service class origin can be seen to be low, with only around 40% of the (blue)
outcome set covered by the (yellow) condition set. In Figure 3 (next slide), on
the other hand, it can be seen that the whole of the outcome set (again in
blue) is covered by the (yellow) condition set, and coverage is 100% (the
arithmetic mark of a necessary condition in this simple case).
Figure 3: Quasi-Sufficiency (but with perfect necessity)
QCA: Multiple conditions and the partitioning of coverage: I
In more complex set theoretic models with more than one condition,
coverage can be partitioned in a manner analogous to the partitioning
of variance explained in regression-based approaches (Ragin,
2006a). The partitioning of coverage into raw and unique components
can be illustrated, again using imaginary data, by reference to a more
complex Venn diagram (Figure 4, next slide). Here we have added
the condition of being of high ability. In this fictional case we now
have two crisp sets representing the conditions, ‘SERVICE CLASS
ORIGIN’ and ‘HIGH ABILITY’, and the outcome is the achievement of
a degree. The Boolean solution can be written as
DEGREE = SERVICE CLASS ORIGIN + HIGH ABILITY.
Either being of service class origin or of high ability is sufficient for the
outcome (since both condition sets, considered separately, are
subsets of the outcome set). Greater coverage of the outcome is
achieved by having both of these factors in the analysis rather than
either alone.
Figure 4: Perfect Sufficiency (two conditions)
QCA: Multiple conditions and the partitioning of coverage: II
We can also see here how coverage can be partitioned straightforwardly in the case of
crisp sets. In the case of the relations illustrated in Figure 4 (previous slide) it is easy to
see that the total coverage can be broken into three components:
– That due to being of service class origin while not being of high ability (the yellow
subset as a proportion of the blue outcome set)
– That due to being of high ability while not being of service class origin (the orange
subset as a proportion of the blue outcome set)
– That due to being of service class origin and being of high ability (the red subset as
a proportion of the blue outcome set).
If we take service class origin as an example, Ragin (2006a) would describe the first of
these three (the yellow subset as a proportion of the outcome set) as the unique
coverage due to being from this social class background. On the other hand, the
coverage due to being of this class origin, whether or not this is conjoined with other
causal conditions in the model (the yellow and red subsets taken together as a
proportion of the outcome set), he would describe as the raw coverage due to
membership in this set (being of service class origin).
Parallel arguments apply to being of high ability.
From this point on we employ real large n data in illustrating QCA in use.
We can use data from the National Child Development Study (NCDS),
comprising children born in one week in March 1958, to illustrate a
multifactor conjunctural explanation[1]. Of course, we will not expect to find
perfect sufficiency in the empirical world and our example will show how the
method embodied in the software addresses this problem. We explore the
relations between highest qualifications achieved by age 33 and a number
of factors which might be seen as either causal or as summarising possible
causes of achievement.
To begin with we will take, as our outcome measure, having a highest level
of qualification of at least ‘A’ level or its equivalent (HQUAL_ADVANCED).
We wish to capture something more, when referring to social class origin,
than one point in time, and so, for illustrative purposes, we will take
father’s[2] social class at two points. We also include a measure of
mother’s education and sex of the respondent. We will not include any
measure of ability in this first example, in order to keep things simpler.
[1] We will begin by using a subset of the data containing 3826 cases chosen to include no missing values on four
measures of father’s class at different times and on mother’s education as well as other key variables.
[2] We use father’s class because there are many more cases of missing/not-applicable data for mother’s class.
However, we include a maternal influence via mother’s education.
An illustrative Boolean analysis.
We will address the Boolean equation:
HQUAL_ADVANCED =
function(MALE, PMT_FATHER_AT_BIRTH[1], PMT_FATHER_AT_AGE_11,
MOTHER_POST_16_EDUCATED)
where:
HQUAL_ADVANCED
MOTHER_POST_16_EDUCATED
MALE
PMT_FATHER_AT_BIRTH
PMT_FATHER_AT_AGE_11
refers to having qualifications of at least
‘A’ level standard by age 33.
refers to the mother having stayed on in
education after age 16.
refers to being male rather than female.
refers to the mother’s husband being in a
professional, managerial or technical
position[2] at the time of the respondent’s birth.
refers to the respondent’s father being in
a professional, managerial or technical
position when the respondent was aged 11.
We should stress that we are not claiming that we have anything like a properly
specified model of educational achievement here. Our purpose here is to illustrate
QCA in use with large n data.
[1] This is actually a measure of the mother’s husband in 1958, but to avoid unnecessary complexity
(and given that this is usually the respondent’s father) we have used this description.
[2] The PMT grouping used here comprises Classes I and II of the contemporary Registrar General’s
scheme.
Table 1: Proportions achieving HQUAL_ADVANCED by class origin, sex and
mother’s education (NCDS data; n=3826) : a crosstabulation
MOTHER’S EDUCATION AFTER 16
No
PMT FATHER AT BIRTH PMT FATHER AT AGE 11
FEMALE
MALE
Yes
Mean
Count
Mean
Count
No
No
.25
1193
.56
184
No
Yes
.44
158
.67
61
Yes
No
.37
51
.77
35
Yes
Yes
.60
120
.78
168
No
No
.40
1138
.57
164
No
Yes
.58
132
.78
64
Yes
No
.66
53
.82
34
Yes
Yes
.71
111
.79
160
QCA: Moving from the crosstab via a truth table to a Boolean solution
The first step required is to reconfigure this as a truth table (next slide) where a
“1” is entered to indicate the presence of a condition and a “0” to indicate its
absence. In this table, where the rows are ordered by the measure of
consistency with sufficiency, the first row (1101), for example, represents the
causal configuration:
MALE*PMT_FATHER_AT_BIRTH*pmt_father_at_age_11*MOTHER_POST_16_EDUCATED
with the upper case letters indicating membership in a set and lower case
letters non-membership. The proportion of the 34 cases in this configuration
who achieve the outcome, i.e. 0.824, appears in the consistency column.
The second step is to determine a threshold for quasi-sufficiency and, in the
light of this decision, to enter a “1” into the empty outcome
(HQUAL_ADVANCED) column against each row (or causal configuration) for
which the consistency proportion in the final column passes the threshold set.
This decision determines which configurations are allowed into the final
solution.
Table 2: Truth table for achieving HQUAL_ADVANCED (NCDS data, n=3826)
MALE
PMT_FATHER_AT
_BIRTH
PMT_FATHER_AT
_AGE_11
MOTHER_POST_
16_EDUCATED
1
1
0
1
1
1
1
1
0
0
number
HQUAL_
ADVANCED
Consistency
34
1
0.824
1
160
1
0.794
1
1
64
1
0.781
1
1
1
168
1
0.780
0
1
0
1
35
1
0.771
1
1
1
0
111
1
0.712
0
0
1
1
61
1
0.672
1
1
0
0
53
1
0.660
0
1
1
0
120
0
0.600
1
0
1
0
132
0
0.576
1
0
0
1
164
0
0.573
0
0
0
1
184
0
0.560
0
0
1
0
158
0
0.437
1
0
0
0
1138
0
0.397
0
1
0
0
51
0
0.373
0
0
0
0
1193
0
0.247
Three types of cases?
The decision re a threshold also effectively determines which cases, seen as
captured by configurations of conditions, will be grouped together in the final
solution. In this illustration we will assume that there are three levels of
outcome that we wish to understand in configurational terms:
–
–
–
Those configurations – or sets of cases – in which more than 60% of the
cases achieve the outcome. Passing this consistency level might be
argued to be consistent with this level of outcome approaching being
more or less the norm for these configurations. These configurations are
also those we might want to allow forward into a solution for quasisufficiency.
Those configurations (sets of cases) in which fewer than 40% of the
cases achieve the outcome. This level might be seen as making not
achieving this level of outcome more or less the norm for these
configurations.
The remaining configurations (sets of cases) in which 40% - 60% of the
cases achieve the outcome. In these configurations neither achieving nor
not achieving the outcome is the norm.
Clearly, these decisions require judgements to be made. The reader will see
that it is easy to explore other analyses based on other boundaries.
The first group of cases.
Let us turn to the first group. These configurations have been picked out
by entering 1s and 0s in Table 2 in the HQUAL_ADVANCED column.
Table 3a (next slide) shows the solution that results when fs/QCA is asked
to minimise the configurations picked out by these 1s. These eight rows
(‘causal configurations’) are subjected to an algebraic process of Boolean
minimisation[1] (Quine, 1952; Ragin, 1987) in order to create the final
simplest solution:
MALE*PMT_FATHER_AT_BIRTH +
PMT_FATHER_AT_BIRTH*MOTHER_POST_16_EDUCATED+
PMT_FATHER_AT_AGE_11* MOTHER_POST_16_EDUCATED
The two final expressions pick out cases whose mothers had stayed on
after 16 and had a father figure in the PMT class at one point of two in
their childhood. Both males and females are included in these
expressions. The first expression picks out just males who were born into
a family setting with a father in the PMT class at birth.
[1] This proceeds as follows. Taking the first two rows as an example, we have 1101 and 1111.
Clearly, at the level of quasi-sufficiency we have chosen the presence or absence of the third element
makes no difference. We can therefore replace it with a dash to indicate this, giving 11-1. A similar
argument can be applied to the fourth and fifth rows (0111 and 0101) to give 01-1. Taking 11-1 and
01-1 together, and continuing the process we arrive at -1-1. This is PMT_FATHER_AT_BIRTH*
MOTHER_POST_16_EDUCATED, one of the terms in our final solution.
Table 3a: Solution for those for whom achieving at least ‘A’ level
qualifications is more or less the norm (> 0.60 do so in each row allowed
forward)
--- TRUTH TABLE SOLUTION --raw
unique
coverage coverage consistency
-------- ------- ---------MALE*PMT_FATHER_AT_BIRTH+
0.158
0.067
0.751
PMT_FATHER_AT_BIRTH*MOTHER_POST_16_EDUCATED+
0.184
0.016
0.788
PMT_FATHER_AT_AGE_11*MOTHER_POST_16_EDUCATED
0.206
0.054
0.770
solution coverage:
0.305
solution consistency: 0.755
Table 3b: Solution for those for whom not achieving at least ‘A’
level qualifications is more or less the norm (< 0.40 do so in each
row allowed forward)
--- TRUTH TABLE SOLUTION --raw
coverage
-------
unique
coverage
---------
consistency
-----------
pmt_father_at_birth*pmt_father_at_age_11
*mother_post_16_educated+
0.440
0.266
0.320
male*pmt_father_at_age_11
*mother_post_16_educated
0.185
0.011
0.252
solution coverage:
0.451
solution consistency: 0.322
Table 3c: Solution for those for whom neither achieving nor not
achieving at least ‘A’ level qualifications is the norm
( 0.60 &  0.40 do so in each row allowed forward)
--- TRUTH TABLE SOLUTION --raw
unique
coverage coverage consistency
-------- ------- ----------pmt_father_at_birth*pmt_father_at_age_11
*MOTHER_POST_16_EDUCATED+
0.116
0.116
0.566
MALE*pmt_father_at_birth*PMT_FATHER_AT_AGE_11
*mother_post_16_educated+
0.045
0.045
0.576
male*PMT_FATHER_AT_BIRTH*PMT_FATHER_AT_AGE_11
*mother_post_16_educated
0.042
0.042
0.600
solution coverage:
0.203
solution consistency: 0.575
QCA: an example of a quasi-necessary condition: I
It might be thought, at least for some hypothesised meritocracy,
that were academic ability to be appropriately defined and
measured then some minimum level of this factor ought to be a
necessary condition for anyone to achieve a degree. Table 4a
illustrates this, where one cell should be empty if the chosen level
of ability (X) is a strictly necessary condition for a degree to be
achieved. Here, we might be seen as assuming causal
homogeneity for the factor of ability.
Table 4a: Strict necessity of some level of ability (X) for achieving a degree
Achieves degree
Ability
No
Yes
<X
Cases possible
Empty
X
Cases possible
Cases possible
QCA: an example of a quasi-necessary condition: II
An examination by eye of the NCDS distribution of the proportions achieving a
degree at each point of the ability scale allows us to estimate what such a level of
ability might be empirically, for all respondents taken together. It is, in fact, around
the mean ability score and if we create a factor setting ability as either over or
under the mean score for our subset of 3826, we obtain Table 4b, showing that
the proportion of those obtaining a degree whose ability score is below the mean
is only 10.4%. Especially given that this proportion may include cases where the
measurement was low through either error or chance factors, we might be willing
to say that a score above the mean approaches being a necessary condition for
achieving a degree in this sample and is therefore a quasi-necessary condition.
Table 4b: Achieving a degree by ability below and above the mean row (column %)[1]
Achieves degree
No
Ability
[1]
Yes
< Mean
1748 (52.7%)
53 (10.4%)
> Mean
1568 (47.3%)
457 (89.6%)
As it happens this test only has discrete scores, from 0 to 80. The mean lies between two of these scores.
QCA: an example of a quasi-necessary condition: III
However, we can not be satisfied with this conclusion which, as
we said, effectively assumes causal homogeneity, with ability
operating in the same way across all types of cases and, of
course, leaves us wondering about the features of the cases
amongst the 10.4%.
We obviously want to know whether there are sets of cases –
perhaps, for example, differentiated by social class - for whom
being either above or below the mean, when conjoined with
other factors, is either necessary and/or sufficient or not for
achieving a degree (or quasi-necessary or quasi-sufficient),
especially as apparent returns to ability vary by class, as Figure
5 (next slide), produced using a slightly different class origin
categorisation, clearly shows.
Figure 5: Proportions gaining a degree by ability at age 11 and social class
Proportion gaining degree (subset of NCDS, n=3826)
0.700
0.600
Proportion
0.500
service
0.400
intermediate
0.300
working
0.200
0.100
0.000
<=17 1824
2531
3238
3945
4652
5359
Ability Score (age 11)
6066
>66
QCA: an example of a quasi-necessary condition: IV
• To explore these questions, we might undertake an analysis that
includes a measure of ability being over the mean, given what we
found in Table 4b. Let us undertake an analysis of:
HQUAL_DEGREE =
function (ABILITY_ABOVE_MEAN, MALE, PMT_FATHER_AT_BIRTH,
PMT_FATHER_AT_AGE_11, MOTHER_POST_16_EDUCATED).
•
The relevant truth table is shown in Table 5 (next slide), with the rows
ordered by consistency. We can see that the first five rows have a
consistency level of 0.40 or above, which we might label as implying that for
these cases, gaining a degree is, all else being equal, a definite possibility,
something that is a pretty common occurrence in their milieus. Each of
these configurations is characterised by having ability above the mean, but
conjoined with several supportive paternal and maternal ascriptive factors,
and, in most cases, with male sex. The minimised solution for these rows is
shown in Table 6 (two slides on) where ABILITY_ABOVE_MEAN appears,
as a necessary condition should, in each expression.
• We will return to the somewhat paradoxical threshold-dependent
sense which the term “necessary” has in this claim after a
subsequent example.
Table 5
ABILITY_
ABOVE_MEAN
PMT_FATHER_AT
_BIRTH
MALE
PMT_FATHER_
AT_AGE_11
MOTHER_POST_16
_EDUCATED
HQUAL_
DEGREE
number
consistency
1
1
1
1
0
68
0.485
1
1
1
1
1
126
0.484
1
1
1
0
1
25
0.440
1
0
1
1
1
144
0.424
1
1
0
1
1
48
0.417
1
0
0
1
1
47
0.340
1
0
1
0
1
24
0.292
1
0
1
1
0
93
0.290
1
0
0
0
1
127
0.283
1
1
0
1
0
75
0.253
1
1
0
0
1
101
0.238
0
1
1
1
1
34
0.206
1
0
0
1
0
99
0.192
1
1
1
0
0
28
0.179
1
1
0
0
0
451
0.175
0
1
0
0
1
63
0.143
0
0
1
1
1
24
0.125
0
1
0
1
1
16
0.125
1
0
1
0
0
35
0.114
0
1
1
0
0
25
0.080
0
0
0
1
1
14
0.071
1
0
0
0
0
534
0.066
0
0
1
0
0
16
0.063
0
1
0
1
0
57
0.053
0
0
0
1
0
59
0.034
0
1
1
1
0
43
0.023
0
1
0
0
0
687
0.019
0
0
0
0
1
57
0.018
0
0
0
0
0
659
0.012
0
0
1
1
0
27
0.000
0
0
1
0
1
11
0.000
0
1
1
0
1
9
0.000
Table 6: Minimised solution for Table 5, for first five rows
--- TRUTH TABLE SOLUTION --frequency cutoff:
9.000
consistency cutoff: 0.417
raw
coverage
unique
coverage consistency
--------
---------- -----------
ABILITY_ABOVE_MEAN*MALE*PMT_FATHER_AT_BIRTH
*PMT_FATHER_AT_AGE_11+
0.184
0.065
0.485
ABILITY_ABOVE_MEAN*MALE* PMT_FATHER_AT_BIRTH
*MOTHER_POST_16_EDUCATED +
0.141
0.022
0.477
ABILITY_ABOVE_MEAN*MALE*PMT_FATHER_AT_AGE_11
*MOTHER_POST_16_EDUCATED +
0.159
0.039
0.466
ABILITY_ABOVE_MEAN*PMT_FATHER_AT_BIRTH
*PMT_FATHER_AT_AGE_11*MOTHER_POST_16_EDUCATED
0.239
0.120
0.452
solution coverage:
0.365
solution consistency: 0.453
QCA: an example of a quasi-necessary condition: V
A further inspection of Table 5 shows, as we might expect, that
having this level of ability characterises the top half of the ordered
table (14 out of the 16 rows). However, there are exceptions. The
first, in the twelfth row, is the configuration, with only 34 cases
ability_above_mean*MALE*PMT_FATHER_AT_BIRTH
*PMT_FATHER_AT_AGE_11*MOTHER_POST_16_EDUCATED
This conjunction of lower ability with supportive ascriptive factors is
associated with some 20.6% achieving a degree, some way above
the mean of 13.3%.
QCA: an example of a quasi-necessary condition: VI
We might be especially interested in exploring what it is about those with lower
than mean ability that might explain their achieving proportionally more degrees
than expected. It is likely, as we can see from this example, to be the presence
of supporting ascriptive factors. However, the numbers become very small in
some of the relevant rows in Table 5. For this reason, we will explore this
question using a different boundary within the ability scale. Sixty-one percent of
those achieving degrees in the 3826 have ability in the top 20% of the overall
distribution in the NCDS (see Table 7). We can use the remaining 39% to
explore what factors, conjoined with being outside the top 20% are associated
with raising the proportion gaining a degree. We will define, for current purposes,
ability in the top 20% as “high ability”.
Table 7: Degrees by High Ability (i.e. ability in top 20%) (column %)
Not of High
Ability
Of High Ability
No Degree
Has Degree
2693 (81.2%)
199 (39.0%)
623 (18.8%)
311 (61.0%)
QCA: an example of a quasi-necessary condition: VII
Therefore let us undertake a Boolean analysis parallel to the earlier one but that
excludes the top 20% of the ability range. Table 8 (next slide) is the relevant truth
table, ordered by consistency. A glance at this shows that, for these cases,
mother’s education is a key factor in raising the likelihood of a degree.
If we set a 0.20 threshold to explore this (having noted the jump from 0.16 to 0.20 in
the consistency column), we obtain the solution in Table 9 (two slides on). Within
the confines of this analysis, i.e. for those not of high ability as defined,
MOTHER_POST_16_EDUCATED is necessary to raise the proportion obtaining a
degree to 20%, as is also a father’s class position in the PMT classes for at least
one of the two points included.
However, the low coverage figure for the solution should be noted (0.296). Amongst
those not of high ability as defined, more degrees (140) are gained by individuals
outside of the configurations included in this solution than by those within them
(59). It must therefore be stressed that the sense of necessary here is necessary to
raise the proportion for a configuration to 0.2 or better and not the sense that it is
not possible for an individual to gain a degree without a suitably educated mother.
Many do precisely the latter.
Table 8: Degree by sex, class and mother’s education (only for those whose ability is outside the top 20%)
MALE
PMT_FATHER_
AT_BIRTH
PMT_FATHER_
AT_AGE_11
MOTHER_POST_
16_EDUCATED
number
HQUAL_
DEGREE
consistency
1
1
1
1
88
0.261
1
0
1
1
39
0.256
0
0
1
1
29
0.241
0
1
1
1
74
0.203
0
1
0
1
20
0.200
1
0
0
1
109
0.156
1
1
0
0
42
0.143
1
1
1
0
72
0.139
0
0
0
1
125
0.120
1
1
0
1
17
0.118
0
1
1
0
73
0.110
1
0
1
0
93
0.097
0
0
1
0
120
0.083
0
1
0
0
41
0.073
1
0
0
0
963
0.047
0
0
0
0
987
0.015
Table 9: Degree by sex, class and mother’s education (only for
those whose ability is outside the top 20%)
--- TRUTH TABLE SOLUTION ---
frequency cutoff: 17.000
consistency cutoff: 0.200
raw
unique
coverage coverage consistency
-----------------------------
PMT_FATHER_AT_AGE_11
*MOTHER_POST_16_EDUCATED+
0.276
0.201
0.239
male*PMT_FATHER_AT_BIRTH
*MOTHER_POST_16_EDUCATED
0.095
0.020
0.202
solution coverage:
0.296
solution consistency: 0.236
QCA: Limited Diversity in Datasets and Counterfactual Reasoning
In the examples we have used above, and with the number of conditions
employed in those models, we did not experience the problem of very
small numbers in some rows of the truth table that can arise with more
conditions as a consequence of (i) the exponential increase in the number
of rows as more conditions are included and (ii) the relations – or
correlations - between conditions in the empirical world (Ragin & Sonnett,
2005).
Small numbers of cases in some configurations constitute a problem
because it is difficult to make a valid statement about a group of cases
who, empirically, only appear in small numbers. In regression analyses,
since the weight of the various combinations of scores on variables is
taken into account in calculating average net effects, this problem is
effectively dealt with mechanically, partly via the use of significance tests.
Ragin has suggested a range of ways of using counterfactual reasoning
to address the problems caused by limited diversity. For our use of these
approaches with the NCDS data, which we will not have time to discuss,
see Cooper & Glaesser (2008).
QCA: Some Problems in its Use With Large Datasets
We will introduce here some of the problems and issues
that arise for us in using QCA with large n data.
We will begin with problems that are not peculiar to QCA
since they parallel the correlation / causation problem in
conventional quantitative analyses.
We will then discuss some problems that are more QCAspecific, though, to some extent, it must be remembered,
these may be a consequence of its relatively recent
development. Unlike regression, QCA has not been under
development for more than a century!
Although we may, and certainly should, have inserted
some ‘cautious’ words (‘potentially’, ‘possible’, etc.)
before the word causal at various places in this talk,
we have not yet addressed the question of whether
QCA, as an analytic tool, is able to avoid analogous
problems to those associated with moving from
correlations to causal claims in the regression
approach. Clearly, we might enter into a Boolean
model a ‘condition’ that we then found to be logically
necessary, for example, for some outcome, but which
we would not want to regard as truly causal.
Two types of such conditions are worth distinguishing.
QCA: non-causal conditions I
Alcohol might be a necessary (and causal) condition for
drunkenness, but, in a society in which it was always mixed
with tonic water, we would want to be able to reject a claim
(which QCA could obviously deliver, if used mechanically)
that tonic water was a necessary causal condition for
drunkenness.
We would do this, presumably, by reference to existing
theoretical knowledge, preferably of the mechanisms and
processes involved in the production of drunkenness and/or
by comparisons with other sets of findings where tonic water
was not mixed with alcohol, etc[1].
[1]
Cartwright (2007) provides a formal treatment of this correlation/causation problem in the context of QCA.
QCA: non-causal conditions II
To avoid problems of infinite regress, we would want to be
able to distinguish some types of causal necessary
conditions from others. It may well be necessary for oxygen
to be present in order for degrees to be achieved, but we
wouldn’t normally expect to address this in an analysis of
educational achievement.
Mackie’s (1974) concept of the “causal field” provides a
way of addressing this potential problem. This field acts as
a background context which absorbs the causal factors we
would not expect to see referred to as part of an
explanation of some particular outcome under examination.
QCA: non-causal conditions III
Having noted these problems, we would
nevertheless want to argue that, in our earlier
analyses, there are plausible mechanisms implied by
such summarising conditions as social class. These
conditions (class, ability, etc.) or, at least, the more
specific factors they summarise, are plausible
causal factors.
Furthermore, when addressing some evaluative
questions (e.g. is Britain a meritocracy?), the
question itself, once its constituent terms are
defined, usually points to the relevant factors to
include in a configurational analysis (Cooper, 2005,
2006).
QCA: Underdetermination of theory by data, etc.
We might find in some population that being in the set
male*WORKING_CLASS is perfectly sufficient for NOT achieving a
given level of educational qualification.
However, whether this is due to working class females lacking
some capacity or disposition required to cope with the appropriate
curriculum or whether, on the other hand, some form of educational
apartheid ensures that no working class female is allowed to enter
the institution offering the curriculum, clearly can not be read off
from the Boolean expression.
Of course, other Boolean models perhaps could be used to provide
part of the answer (exploring what happens to other females, to
working class males; including dispositional factors) but, ideally, we
need knowledge of the processes and mechanisms that generate
the observed outcomes. Nothing in Ragin’s work, we should note,
suggests that he thinks otherwise.
QCA: problems to do with randomness
We might find that the configuration HIGH_ABILITY *
SERVICE_CLASS has a consistency with sufficiency of, say, 0.90,
for achieving some outcome, thereby reaching a level that Ragin
would regard as indicating quasi-sufficiency. However, is this gap
between 1.00 and 0.90 to be explained by our having the
equivalent of an underspecified model in a regression analysis (e.g.
perhaps some missing ascriptive factors or a lack of factors
concerning ‘choice’) or by the existence of stochastic elements in
the social world (and/or measurement or sampling error)?
In the former case, there exists some causal heterogeneity yet to
be picked out by the conditions entered in the model. It might be
that HIGH_ABILITY * SERVICE_CLASS * MALE has perfect
consistency with sufficiency, for example. This would leave us,
however, with HIGH_ABILITY * SERVICE_CLASS * male having a
lower consistency than 0.90 and return us to the same question
again, but this time just for females.
QCA and counterfactualist perspectives of causation
A counterfactualist perspective on causation (e.g. Morgan &
Winship, 2007) could be used to raise questions about some
QCA-derived claims re causality in the same way it raises
questions about some regression-based forms of analysis that
basically use a branch of mathematics to describe relations in
datasets[1].
On the other hand, a move from a net effects perspective (one
assuming independently manipulable independent variables) to
one emphasising conjunctural causation might be expected to
make it less likely that unjustified counterfactual claims are
made by policy makers on the basis of research findings,
especially about the effects of intervening to change a single
factor without taking account of its context.
[1]
For a relevant and interesting exchange of views, see Ragin & Rihoux, 2004a,b; Lieberson,
2004; Seawright, 2004; Mahoney, 2004.
More QCA-specific issues: inference from samples to populations I
The first point concerns work that uses samples from some
population. This is usually the situation we find ourselves in when
working with large datasets. Although attempts have been made
(e.g. in earlier version of the fs/QCA software) to incorporate
significance testing (see also Ragin, 2000, and Smithson and
Verkuilen, 2006), this is an area requiring more work. Especially
when numbers become small in some rows of a truth table, and
especially when survey data are being used, a critic will always be
able to ask whether sampling (or measurement) error has been
taken into account. Although we have considerable sympathy with
the view that judgement should play a role in these situations –
especially as significance tests are frequently employed when the
conditions for their use are not met – we also recognise that more
work on incorporating significance testing into QCA would be useful,
simply because chance always offers a potential threat to any
analytic claim we might make.
But, note that Ragin (1987, 2000) has a different perspective on
‘populations’ to the one implied here.
More QCA-specific issues: inference from samples to populations II
A related problem we have ignored during the talk so far is that of
missing data. Can we assume that the Boolean solutions we have
presented, often based on smallish subsets of the whole NCDS
(because of the missing data problem) would hold for the NCDS as
a whole? This would seem unlikely unless the missing data have
been generated by random rather than systematic processes.
Of course, it is possible to undertake some simple checks to see
whether any bias is likely to have been introduced. It is also possible
to use sophisticated techniques (multiple imputation, etc.) to replace
missing data, but such approaches require considerable faith in the
very linear models that Ragin and others have argued are often
unhelpful in the social world. This is a difficult problem to which we
intend to give further thought.
More QCA-specific issues: case knowledge (or its lack) in
large n contexts
We lack, in the traditional sense, the detailed case knowledge
that Ragin argues is required to undertake QCA.
The NCDS, in one sense, does contain a mass of data on
each individual respondent but, for example,
(i) it is collected via techniques that are likely to generate
considerable error and,
(ii) it is not possible for us to return to the respondent to correct
likely errors or to seek new data from earlier periods as
analyses develop.
More QCA-specific issues: quasi as opposed to perfect
necessity and sufficiency
Repeating what we said earlier there is the question of
whether and when it makes sense to ever stop at quasilevels of consistency, i.e. to ignore the deviant cases in a
row (or to allow a ceteris paribus clause).
More generally, the use of weak implication (quasisufficiency and quasi-necessity as opposed to sufficiency
and necessity) deserves more discussion (but see Abell,
1971, and also Goertz, 2005; Waldner, 2005; Sekhon, 2005
for a recent exchange).
We’ve raised a lot of problems here, though we ourselves
believe QCA to be a very important addition to the armoury
of the social scientist interested in exploring potentially
causal relations.
The fuzzy set variety of QCA allows the conjunctural
perspective to be brought to bear more finely than the crisp
set version we have discussed here, but, inevitably, given
the nature of fuzzy sets and logic, brings along some
additional problems (many addressed in Ragin’s own
account in Fuzzy Set Social Science).
We are looking forward to further developments of these
methods and, in particular, to Ragin’s forthcoming new book
Redesigning Social Inquiry: Fuzzy Sets and Beyond.
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