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Technical Note
Conjunction revisited
Karl J. Friston, William D. Penny, and Daniel E. GlaserT
The Wellcome Department of Imaging Neuroscience and Institute of Cognitive Neuroscience, University College London,
17 Queen Square, London, WC1N 3AR, UK
Received 23 August 2004; revised 23 December 2004; accepted 12 January 2005
Available online 3 March 2005
The aim of this note is to revisit the analysis of conjunctions in
imaging data. We review some conceptual issues that have emerged
from recent discussion (Nichols, T., Brett, M., Andersson, J., Wager,
T., Poline, J.-B., 2004. Valid Conjunction Inference with the Minimum
Statistic.) and reformulate the conjunction of null hypotheses as a
conjunction of k or more effects. Analyses based on minimum statistics
have typically used the null hypothesis that k = 0. This enables
inferences about one or more effects (k N 0). However, this does not
provide control over false-positive rates (FPR) for inferences about a
conjunction of k = n effects, over n tests. This is the key point made by
Nichols et al., who suggest a procedure based on supremum P values
that provides an upper bound on FPR for k = n. Although valid, this
is a very conservative procedure, particularly in the context of
multiple comparisons. We suggest that an inference on a conjunction
of k = n effects is generally unnecessary and distinguish between
congruent contrasts that test for the same treatment and incongruent
contrasts of the sort used in cognitive conjunctions. For congruent
contrasts, the usual inference, k N 0, is sufficient. With incongruent
contrasts it is sufficient to infer a conjunction of k N u effects, where u
is the number of contrasts that share some uninteresting effect. The
issues highlighted by Nichols et al., have important implications for
the design and analysis of cognitive conjunction studies and have
motivated a change to the SPM software, that affords a test for the
more general hypothesis k N u. This more general conjunction test is
described.
D 2005 Elsevier Inc. All rights reserved.
Introduction
The central distinction, which has been highlighted by recent
discussions, is whether conjunction refers to the activation (i.e.,
consistently large activation) or the underlying effects (i.e.,
consistently significant activation). Activation is an attribute of
the data, usually defined through a statistic. An effect is an
attribute of the real world, which we cannot observe. Declaring
a voxel to be dactivatedT allows one to infer the effect is
present with some sensitivity and specificity. The distinction is
formulated in Nichols et al. (2004) in terms of a global null
hypothesis and a conjunction null hypothesis about effects.
SPM tests the global null using the minimum T statistic. A test
for the conjunction null, proposed by Nichols et al. (2004), has
the same form but uses a much higher threshold. Put simply,
the difference rests on whether the conjunction refers to the
observed statistics or to the effects one is trying to infer. Both
approaches are valid but have different uses. Conjunction has
been clearly defined as bthe joint refutation of multiple null
hypothesesQ (Friston et al., 1999). In other words, a conjunction
of activations allows one to infer a conjunction of one or more
effects. However, Nichols et al. have pointed to examples in the
literature where the inference is misinterpreted as a conjunction
of all effects. This was their motivation for highlighting the
issue and proposing a new test.
In what follows, we make three points. First, conjunctions
based on the global null remain valid and exact. Second,
although valid, the alternative proposed in Nichols et al.
(2004) is conservative. In fact, it is often sufficiently
conservative to render it powerless in neuroimaging. This is
especially pronounced when considering the multiple compar-
isons problem. The third point is that rejection of the
conjunction null, although sufficient, is usually unnecessary.
We discuss this separately for congruent and incongruent
contrasts, testing the same and different treatments, respec-
tively. When the contrasts test the same effect, one can
assume a binomial prior on the number of effects. This was
the starting point for the meta-analysis presented in Friston et
al. (1999) and connects the current analysis with previous
work. The considerations for cognitive conjunctions are more
complicated and are usefully informed by Nichols et al.
(2004). The key revision here is that tests for a number k N
u of effects may be called for, depending on the conjunction
design and assumptions about the regional deployment of
treatment effects. We conclude with a section on how
conjunctions are specified in the next release of the SPM
software that allows one to test for k N u effects. This more
general specification subsumes tests of the global null k = 0
and tests of the conjunction null k N n ? 1.
1053-8119/$ - see front matter D 2005 Elsevier Inc. All rights reserved.
doi:10.1016/j.neuroimage.2005.01.013
T Corresponding author. Fax: +44 20 7916 8517.
E-mail address: daniel.glaser@ucl.ac.uk (D.E. Glaser).
Available online on ScienceDirect (www.sciencedirect.com).
www.elsevier.com/locate/ynimg
NeuroImage 25 (2005) 661–667
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Definitions
It is important to be clear on basic terminology used in this
discussion. Consider a test which is supposed to give a certain
false-positive rate a. If it gives exactly as many false-positives as
expected it is called an exact test. If it gives more false-positives
than expected, it is invalid. If it gives fewer false-positives than
expected, it is still valid but conservative.
Activations and effects
In classical imaging statistics, one declares a voxel to be
activated if its statistic exceeds some threshold. This statistic
reflects the likelihood of the effect being truly present as opposed
to being absent. Activation is therefore an attribute of the data. By
declaring a voxel activated, we infer the effect is present in a
probabilistic sense. The effect will be absent in some proportion of
activated voxels. The relationship between activation and effect is
best characterised in terms of conditional probabilities, namely the
specificity 1 ? a and sensitivity b and their complements false-
positive and negative rates. See the upper panel of Fig. 1. The
threshold is chosen to ensure the false-positive rate FPR = a is
small.
A conjunction of activations can be related to conjunctions of
effects in exactly the same way. See lower panel of Fig. 1. Here, we
have a variable k representing the number of effects that are truly
present in a conjunction of n contrasts. The lower row of the
probability table in Fig. 1 represents p(n|k) the probability of a
conjunction conditional on there being k effects. These can be used
to specify the FPR for any null hypothesis.
Global and conjunction null hypotheses
Conventionally, conjunction analyses are based on the global
null hypothesis that there are no activations k = 0. According to
Fig. 1, for a single test FPR = an. Therefore, a significant
conjunction allows one to say with a specificity of 1 ? anthat there
is a conjunction of one or more effects (i.e., k N 0). However, this
does not mean that all the effects are present. To make an inference
that there is a conjunction of all effects (i.e., k = n), one has to
include all the alternative outcomes under the conjunction null k b
n. This is the basis of an alternative procedure, advocated in
Nichols et al. (2004). The difference between the two procedures is
based on which outcomes the null hypotheses encompass (see Fig.
1). Below, we deal briefly with implementation and issues
surrounding both these null hypotheses.
Testing the global null
If the objective of conjunction analyses is to reject the global
null, why not use an F test spanning the contrasts in the
conjunction? The answer is that conjunctions allow one to focus
on a specific departure from the global null; namely, outcomes
that are consistent. This increases sensitivity markedly, when the
effects are consistent, as depicted schematically in the upper
panel of Fig. 2. Consider two contrasts. The bivariate distribu-
tion of two T values, under the global null, is shown in the
upper panels (for eight degrees of freedom). The false-positive
rate corresponds to the integral of this density over some region.
If the two T values fall in this rejection region, we reject the
global null. The deployment of this region defines the departure
from the null hypothesis that is considered interesting. For
example, an F test considers any departure interesting, as
reflected by the circular region surrounding the null distribution
on the upper left of Fig. 2. This is a large region and therefore
the threshold (radius of the circle) has to be high to maintain a
low FPR. The rejection region corresponding to the conjunction
is the upper [hyper-]quadrant, in which all the T values are
greater than some minimum T value. This is a more restricted
Fig. 1. Conditional probabilities for activation and their conjunction given the number of true effects. These probabilities are for a single voxel with known
specificity and sensitivity. For simplicity, we have assumed that sensitivity is the same for all effects, and that they are independent.
K.J. Friston et al. / NeuroImage 25 (2005) 661–667
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region and the thresholds can be more relaxed to maintain the
same specificity.
Heuristically, conjunctions generalise one-sided t tests to
multiple dimensions. For a single contrast, a two-tailed t test is
the same as an F test, and tests for both positive and negative
effects. A one-tailed t test, with the same specificity, uses a lower
threshold and has a higher sensitivity to positive [or negative]
effects. A conjunction of t tests is the multidimensional equivalent
of the one-sided t test and allows one to prescribe an increase in
sensitivity, to consistent effects, at the expense of missing
inconsistent responses (i.e., large positive and negative). In short,
testing the global null is like performing a one-tailed F test.
Like F tests, conjunctions are simply inference devices that
allow one to test a multidimensional hypothesis, specified by a
contrast weight matrix. The only difference is that the conjunction
is only sensitive to consistent effects. Conjunction analyses, using
the minimum T statistic, are useful when one knows, a priori, the
direction of the effect. For example, the search for bilateral effects
in voxel-based-morphometry is an established use of conjunctions
(Belton et al., 2003; Salmond et al., 2000). Because of their plastic
potential, children seldom develop severe neuropsychological
deficits unless homologous regions in both hemispheres are
damaged. A conjunction of effects (loss of grey matter density)
in homologous voxels has therefore been used to detect anatomical
correlates in developmental neuropsychology. Here, the contrasts
testing for changes in both hemispheres are orthogonal and test for
the same signed effect. In these analyses, finding a significant
region allows one to infer regional grey matter loss in one or both
hemispheres.
Conjunctions are not as sensitive as a single contrast testing
for the average effect over all contrasts (by the Neyman–Pearson
Lemma). However, rejecting a single hypothesis about the
average is not equivalent to rejecting a multidimensional
hypothesis (i.e., multiple null hypotheses) with a conjunction
analysis. This is because inferring the average effect is greater
than zero is not equivalent to a conjunction. In the example
above, declaring a significant reduction in grey matter density in
one or both hemispheres is not the same as saying the loss,
averaged over both hemispheres is significant. The latter could
occur with a profound decrease in one hemisphere and an increase
in the other. A conjunction analysis would not find this
inconsistent effect.
The nature of inference, afforded by conjunctions, is central
to the issues addressed here, particularly with the conventional
use of conjunctions to test the global null. These inferences mean
the evidence for consistent effects is significant, not the evidence
for significant effects is consistent. This distinction may be
semantic but speaks directly to people’s misconception about
conjunctions that Nichols et al. (2004) have highlighted. In the
example above, a significant conjunction does not mean that the
right hemisphere has lost grey matter and the left hemisphere has
lost grey matter. To infer this would require tests of each
hypothesis separately (c.f. reporting post hoc contrasts after
finding a significant effect with an F test). This separate
hypothesis testing is essentially what Nichols et al. are
proposing. In short, a significant conjunction is not a conjunction
of significances.
Minimum statistic tests
The fact that conjunctions can be formulated in terms of a
minimum test statistic is useful because the null distribution of the
minimum T statistic can be computed analytically. Furthermore,
there are analytic expressions for the maximum of this minimum T
statistic over a spatial search volume based on random field theory
(Worsley and Friston, 2000). These expressions can be used to
provide a FPR that is adjusted for the search volume in the usual
way.
The lower panels of Fig. 2 highlight the relative power of
conjunction analyses, using the minimum T statistic. The SPMs
Fig. 2. Schematic comparing conjunction analyses with those based on conventional F tests. The upper row shows bivariate T distributions under the global
null hypothesis and the rejection regions associated with each of the two tests. If the two T values fall in these regions, one can reject the null hypothesis.
Lower left: the contrasts used to specify the tests. Lower row: the ensuing SPMs based on a verbal fluency PET data set. The SPMs have been thresholded
at P = 0.001 (uncorrected).
K.J. Friston et al. / NeuroImage 25 (2005) 661–667
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were based on the verbal fluency data used to illustrate SPM
procedures over the years. They were acquired from 5 subjects
each responding to a heard letter by repeating the letter or
producing a word that started with that letter. Each of the two
conditions were repeated six times. The SPM{F} on the left
shows the F statistic testing for a condition-specific effect in
any (one or more) of the six replications. The SPM{min(T)}
shows the minimum T values over the six replications. Both
SPMs are testing the global null hypothesis and all are
thresholded to give the same false-positive rate (P = 0.001
uncorrected). The key thing to note is that the SPM{min(T)}
discloses more significant voxels than the conventional
SPM{F}. In fact, the SPM{F} found no activations at all
(all the voxels correspond to deactivations). The reason that the
conjunction SPM{min(T)} is more powerful is that the treat-
ments, in each of the six contrasts, were congruent. This was
assured by experimental design in which the same treatment
was delivered six times.
Congruent vs. incongruent effects
At this point, it is worth introducing a distinction between
contrasts that test the same thing and those that test different
treatments. We will refer to these as congruent and incongruent
contrasts. The verbal fluency example above used congruent
contrasts in the sense that the treatment was the same over each
replication. Another common example of congruent contrasts
would be contrasts testing for the same effect over subjects. In
the context of congruent contrasts, it is sufficient to test the
global null because its rejection allows us to infer that this
treatment effect was detected on one or more occasions. It does
not matter if the effect was not detected in some contrasts; a
significant effect has been demonstrated. The analysis of
congruent contrasts was the subject of Friston et al. (1999) and
will be reprised below.
Incongruent contrasts (e.g., a contrast for object naming and a
contrast for word naming) are more problematic. In this instance, it
may be relevant that the effect was absent in one of the contrasts.
Incongruent contrasts were the focus of cognitive conjunctions
(Price and Friston, 1997), the original motivation for conjunction
analyses. The idea here was to demonstrate region-specific
correlates of a cognitive component that was common to a set of
incongruent contrasts. Nichols et al. (2004) note that rejection of
the conjunction null, rather than the global null, is indicated in this
context.
Testing the conjunction null
If the objective is to infer a conjunction of effects, then it should
be sufficient to test each contrast separately and establish they are
all significant. This is precisely what Nichols et al. (2004)
conclude, although their derivation is a little more involved: in
rejecting the conjunction null one has to control the false-positive
rate over all outcomes that constitute that null. It may be useful to
consider a distribution over the number of effects.; c.f. a Bayesian
perspective, where we consider k as a random variable with prior
distribution p(k). From Fig. 1, this is:
FPRconj¼
X
k ¼ 0
n?1
p njk
ðÞp k ð Þ
¼
X
k ¼ 0
n?1
an?ibip k ð Þð1Þ
The problem here is that we do not know p(k). However, we can
establish an upper bound by noting
FPRconj b sup
i
an?ibi¼ a
ð2Þ
This means that if we set the specificity of each test to some
suitably small value, we can be assured that the FPR is controlled
for inferences about a conjunction of effects. This is exactly the
same as showing each contrast is significant in its own right.
Practically, the P value, or false-positive rate for the global null
is the probability of obtaining the minimum T value by chance
p(min(Ti)|k = 0). The corresponding FPR for the conjunction null
is the supremum P value over contrasts max(p(Ti)|k = 0)). From
now on, we will refer to the two approaches as the minimum
statistic and supremum P value approaches.
Fig. 3 tries to illustrate the heuristic behind the supremum P
value procedure. Again, consider two incongruent contrasts.
Because we want to infer both effects are present, the null includes
the situation where only one is present. The ensuing distribution of
T statistics is shown on the right. Clearly, to control FPR, in this
worst case scenario, the threshold adopted must be more
conservative than for the global null (left panel).
An over-conservative test
Although valid the alternative procedure is very conservative.
This is because Eq. (2) provides only an upper bound on the FPR.
Fig. 3. Schematic illustrating departure from the global null. Note that the density of the T statistics (shaded region) encroaches on the rejection region (bounded
by the dotted lines) when one of the effects is present (right panel). This is a violation of the global null but not of the conjunction null.
K.J. Friston et al. / NeuroImage 25 (2005) 661–667
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The actual FPR may be much smaller, depending on k. This can be
seen easily with the following example: assume we have two
incongruent contrasts (one for object-naming and one for word-
naming). We construct an SPM{min(T)} and identify the most
significant voxel as being in the left occipito-temporal region. The
minimum T value at this voxel was 2.4. The supremum P value
adjusted for the search volume was max(p(Ti|k = 0)) = p (T z
2.4|k = 0) = 0.99 after correction for multiple comparisons. We
have therefore failed to show a conjunction of effects. We now give
our data to a colleague who has never heard of the supremum
procedure. He examines the contrast for object-naming at a
corrected level and finds a significant activation with a T value
of 4.8 with an adjusted p(T z 4.8|k = 0) = 0.03 in the same region.
To see if word-naming activates this region, he examines the
second contrast, searching over a sphere of 8 mm radius, centred
on the maximum of 4.8. He then identifies our original voxel that
now has a P value of p(T z 2.4|k = 0) = 0.02 adjusted for the small
search volume. He concludes, properly, that both object- and word-
naming cause effects in this region. Where did we go wrong?
The problem is that the supremum P value assumes a worst
case scenario to provide the upper bound FPR. In the context of
neuroimaging, this is conservative because the FPR is based on the
null hypothesis that every voxel in the entire search volume
expresses an effect in all but one contrast. Not only is this
supremum very conservative, but it includes a null hypothesis that
cannot occur. It cannot occur because region-specific effects
cannot, by definition, exist everywhere. In other words, if all the
brain activated, there would be no region-specific response. This is
why whole-brain activation is treated as a confounding effect in
global normalisation procedures.
What do we want to test?
Congruent contrasts
The difficulty with the supremum P value approach is that we
do not know the probability distribution of the number of effects
p(k), therefore, we assume the worst case
p k ð Þ ¼
¼ 1
¼ 0
k ¼ n ? 1
k p n ? 1
?
ð3Þ
The problem is that this distribution could not be realised by any
plausible generative model. For example, suppose the treatment
tested by some congruent contrasts produced an effect with a
frequency or probability c. The ensuing distribution of effects
would have a binomial distribution
p k ð Þ ¼
n
k
?
1 ? c
ðÞkcn?k
?
ð4Þ
There is no value of c that gives a distribution conforming to the
null in Eq. (3). So what is the null distribution for k? In the case of
congruent contrasts, we want to know whether our treatment
produces an effect with non-zero probability. Therefore, the null
distribution obtains when c = 0. In this case
p k ð Þ ¼
¼ 1
¼ 0
k ¼ 0
k p 0
?
ð5Þ
This is simply the global null as used conventionally. In short, it is
entirely sufficient to use the minimum statistic test to reject k = 0 to
infer c N 0. In some cases, it may be interesting to supplement the
inference quantitatively, with a confidence interval on c. This was
described in some detail in Friston et al. (1999). However, in
practice, these confidence intervals have not been much used. Note
that, for congruent contrasts, we do not need to reject the
conjunction null that k b n.
Incongruent contrasts and cognitive conjunctions
In the case of incongruent contrasts, the situation is more
complicated. Here, the treatments differ and c will be treatment
specific. However, there are constraints which allow us to define
the null hypothesis. For example, the aim of conjunction analysis
is to identify responses to a common treatment (e.g., a cognitive
component of interest or CCI). This common component speaks
to the fact that the treatments are compound, with unique
components and common components. The logic of cognitive
conjunctions is that functionally specialised brain regions respond
selectively to one component. The aim is to find the brain region
that responds to the common component. Therefore, any voxel
will respond to the common component, to a unique component
or to no components. The null outcomes here are responses to a
unique component k = 1 or no component k = 0. It is therefore
sufficient to infer k N 1. Because we do not know the outcome
probabilities p(k), we can resort to the supremum approach of
Nichols et al. (2004). This entails assuming the worst case
scenario that every region responds to a unique component with
probability one, that is, k = 1. For a single voxel this ensures
FPR b an?1. In the context of statistical parametric mapping, this
corresponds to assuming a null distribution for the minimum
statistic based on n?1 contrasts. Heuristically, we still use the
minimum statistic over all n contrasts but assume one of them
was highly significant everywhere, a priori.
This approach can be generalised to conjunction designs in
which the cognitive components are unique to a subset of u
contrasts. The previous paragraph assumed u = 1. The example
presented in Nichols et al. (2004) used four working memory tasks
and a visual task. In this instance, unique components are found in
four of the five contrasts. This calls for an inference that k N u = 4.
This is, of course, a test of the conjunction null that Nichols et al.
were promoting. However, this design is very inefficient for
finding the responses to common components. A good conjunction
design should ensure that u is small.
In short, cognitive conjunctions allow one to test for effects
due to common treatment components. In carefully designed
conjunction studies, where each treatment comprises a common
component and unique components in u or fewer contrasts, it is
sufficient to infer that k N u by using the global null distribution
of the minimum statistic for n–u contrasts. Again, note that one
does not have to reject the conjunction null (unless u = n ? 1). It
is worth noting that defining the components of a treatment,
especially in cognitive science, is not always straightforward.
Debates about conjunction often rest on the task analysis and the
deeper issues usually pertain to interpretation, as opposed to
statistics.
Practical implications for analysis
In this section, we describe the changes to SPM that allow one
to test the null hypothesis that k N u. We then review the analysis of
K.J. Friston et al. / NeuroImage 25 (2005) 661–667
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congruent and incongruent contrasts. We conclude with comments
on how to qualify conjunction analysis, when reporting results.
Changes to SPM
The key contribution of Brett et al. (2004) and Nichols et al.
(2004) is to enforce a re-evaluation of inferences required in the
context of cognitive conjunctions. This has clarified the con-
straints on conjunction designs, namely that the components that
are not common should be unique to a small number of contrasts,
ideally u = 1. Furthermore, valid inference with the minimum
statistic requires one to infer that k N u. This can be effected
simply using the null distribution of the minimum statistic under
the global null for n–u contrasts. A special case of incongruent
contrasts, used in cognitive conjunctions, is when all the
treatments are the same and the contrasts are congruent. In this
instance, u = 0.
To enable tests of conjunctions of k N u effects, SPM will now
ask you to specify u after selecting the n contrasts. The prompt is
dnumber of effects under nullT. For a test of the global null, enter
d0T. For a test of the conjunction null, enter n?1. Notice that the
global and conjunction nulls are now both extreme cases of the
more general null hypothesis that k N u. We will therefore refer to
this as the conjunction null from now on. The ensuing SPM is
indexed by the number of contrasts assumed for the null
distribution of the minimum statistic. For example, SPM{T32
refers to an SPM whose P values are based on the minimum T
statistic with 32 degrees of freedom over 4 contrasts. This would be
obtained with 5 contrasts and u = 1. The actual changes to the code
are trivial and involve the addition of one line that effectively
reduces the number of contrasts SPM thinks are in the conjunction
4}
n p n?u. The nice thing about the revision to SPM2 is that the
both the conventional and alternative procedures can be imple-
mented within the same framework. A conventional test of k N 0 is
specified with u = 0. The analysis proposed by Nichols et al.
(2004) corresponds to making u = n ? 1.
When u = 0 inference is valid and exact. When u N 0, one is
implicitly invoking a supremum P value test, which provides a
conservative upper bound on the false-positive rate. As one might
expect, sensitivity falls quickly as the number of effects u grows.
This is illustrated in Fig. 4 using the verbal fluency data of Fig. 2.
Here we have shown the SPMs for u = 0, 1, .... It can be seen that
all voxels disappear by u = 4. If we treated this design as
congruent, then the test k N 0 is sufficient and allows us to declare
the verbal fluency treatment significant in voxels that show a
conjunction. However, we could pretend that the six contrasts
were incongruent with an unspecified contrast-specific component
to each treatment. In this instance, we would need to show k N 1.
The resulting voxels are seen in the second SPM (Fig. 4) and
show that this analysis is still more sensitive than a conventional
F test (c.f. Fig. 2).
Congruent contrasts
The procedure and inference for congruent contrasts remain
unchanged and is based on inferring that k N 0. This is an implicit
inference that c N 0, where c is the probability that the treatment
causes an effect. Operationally, one selects n contrasts and
specifies u = 0.
In the context of hierarchical observations (e.g., multi-subject
studies), one should not bsubstitute conjunction analyses for
random-effect analyses. Where the latter are indicated there is no
Fig. 4. Conjunction SPMs based on the PET data of Fig. 2. Here the threshold (P = 0.001, uncorrected) is based on null distributions for 6, 5, 3, etc contrasts,
using the minimum T statistic over all six. These represent tests of the conjunction of k N 0, k N 1 and k N 2 effects, respectively, specified by u = 0, u = 1 and
u = 2 (see main text).
K.J. Friston et al. / NeuroImage 25 (2005) 661–667
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alternativeQ (Friston et al., 1999). The argument in this paper, and
in Friston et al. (1999), concerning congruent contrasts, reduces to
the following assertion; if all subjects show an effect with
probability c and one or more subjects show an effect (i.e., k N
0), then c must be bigger than zero. However, positing the
existence of random effects, by invoking c, falls short of actually
estimating the between-subject variation in activation and making a
population-based inference.
Incongruent contrasts
Here, there are two ways to proceed. First, using small volume
adjusted P values centred on the maximum of the first contrast as
outlined in Testing the conjunction null. Second, one can use a
minimum statistic procedure with u N 0, where u is the number of
contrasts containing unique treatment components. The former
approach is exact but requires you to specify an order in which the
contrasts enter the conjunction. The second approach is based on a
supremum P value and is therefore conservative. It does however
retain some sensitivity for small u and is commutative.
Qualifying and reporting
In reporting subsequent conjunction analyses, it might be good
practice to describe the inference with something like the
following:
We performed a conjunction analysis using SPMs of the minimum
T-statistic over n orthogonal contrasts. Inference was based on P-
values adjusted for the search volume using random field theory.
The null distribution for the minimum statistic was based on n ? u
statistics. This enabled us to infer a conjunction of k N u effects at
significant voxels.
For those people who have used the global null for inferences
about cognitive conjunctions, and simply want to qualify their
inference. An appropriate passage might be:
It should be noted that our significant conjunction does not mean
all the contrasts were individually significant (i.e., a conjunction of
significance). It simply means that the contrasts were consistently
high and jointly significant. This is equivalent to inferring one or
more effects were present.
Conclusion
We hope that this note provides a clear framework for the range
of uses of conjunction analyses, and serves as a guide to their
interpretation.
Acknowledgments
We would like to acknowledge the patience and constructive
input of the SPM co-authors, especially Keith Worsley. This work
was funded by the Wellcome Trust and the ICN MRC Co-op
Imaging grant.
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