On Conjunction

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On Conjunction

• Gary Huang M.Phil
• CSLB, Univ. of Cambridge
• CBCS, Peking Univ.

March, 2007
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Preface

• This talk is dedicated to Prof. Paul Fletcher
  at University of Cambridge who was also my
  external dissertation examiner. My apologies
  should go to him due to my misinterpretation of
  conjunction in SPM on my examining date. He
  kindly let me pass the exam regardless of the
  mistakes I made. This talk, therefore, aims for a
  correct explanation of conjunction in
  neuroimaging to the public.

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What is conjunction?

• Layman term brain regions that respond to all
  of a set of different conditions, e.g. which
  brain regions are active in all three tasks?

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What is conjunction?

• Layman term brain regions that respond to all
  of a set of different conditions, e.g. which
  brain regions are active in all three tasks?
• Statistical term the joint refutation of
  multiple null hypotheses (Friston et al., 1999)

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Activation Effect

• Activation and underlying effect
• Activation attribute of data (SPM, statistic
  parametric map)
• Effect attribute of the real world
  (psychological/experimental effect)
• Activation allows one to infer the effect is
  present with some sensitivity and specificity.

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Common Misunderstandings

• 1) Conjunction is the literal common area
  activated by two tasks.
• e.g. task A activates ACC, task B also activates
  area ACC. Literally speaking, both A and B
  activate ACC regardless of the extent and
  location of the activation.

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Common Misunderstandings

• 1) Conjunction is the literal common area
  activated by two tasks.
• e.g. task A activates ACC, task B also activates
  area ACC. Literally speaking, both A and B
  activate ACC regardless of the extent and
  location of the activation.
• 2) Conjunction is the masking on one SPM with
  another.
• e.g. mask the activation map for task A with that
  for task B. This looks for the conjunction of
  activation maps, common activated area (e.g. ACC)
  in the map by both tasks, but cant conclude that
  those areas (e.g. ACC) are responsible for both
  EFFECTS, with no statistical inference being made
  with the sensitivity and specificity of those
  effects (aßerrors).

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Common Misunderstandings

• 1) Conjunction is the literal common area
  activated by two tasks.
• e.g. task A activates ACC, task B also activates
  area ACC. Literally speaking, both A and B
  activate ACC regardless of the extent and
  location of the activation.
• 2) Conjunction is the masking on one SPM with
  another.
• e.g. mask the activation map for task A with that
  for task B. This looks for the conjunction of
  activation maps, common activated area (e.g. ACC)
  in the map by both tasks, but cant conclude that
  those areas (e.g. ACC) are responsible for both
  EFFECTS, with no statistical inference being made
  with the sensitivity and specificity of those
  effects (aßerrors).

A significant conjunction is not a conjunction of
significances.
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First statistical attempt Conjunction in SPM96

• Conjunction algorithminteraction masking
• A is Task 1 Baseline B is Task 2 Baseline
• Main effect AB, including effect for A and B or
  either A or B.
• Interaction effect, B-A, B significantly larger
  than A
• Conjunction AB Sig.(B-A)

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First statistical attempt Conjunction in SPM96

• Conjunction algorithminteraction masking
• A is Task 1 Baseline B is Task 2 Baseline
• Main effect AB, including effect for A and B or
  either A or B.
• Interaction effect, B-A, B significantly larger
  than A
• Conjunction AB Sig.(B-A)
• Problem example A is very large, B is large but
  significantly smaller, interaction masking can
  reject this conjunction.

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Conjunction in Logic

• IF
• Significant effect for task A (MA, map for A)
• Significant effect for task B (MB, map for B)
• THEN
• Conjunction map is
• MA OR MB? (Global Null, GN)
• Null Hypo (not MA) AND (not MB)
• MA AND MB ? (Conjunction Null, CN)
• Null Hypo (not MA) OR (not MB)

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Problems of Null Hypothesis

• In statistics, H0 null hypothesis, H1 alternative
  hypothesis Valid for single null hypothesis
• But for multiple null hypotheses, rejecting H0
  has more to say There are many H1 that are valid
  with the refutation of H0.

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Problems of Null Hypothesis

• In statistics, H0 null hypothesis, H1 alternative
  hypothesis Valid for single null hypothesis
• But for multiple null hypotheses, rejecting H0
  has more to say There are many H1 that are valid
  with the refutation of H0.

H0 no common effect for A, B and C H1.1 common
effect for A and B, but not C (valid for
GN) H1.2 common effect for A, B and C (valid for
GN and CN)
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Conjunction in GN CN
FPR False-Positive Rate
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Minimum t Statistic (MS)

• Use the minimum t statistic across several
  comparisons to infer the conjunction effect.
• Example in SPM 99/2,
• Image a voxel where effect A give a t of 0.8 and
  effect B gives a t of 1.6.
• Neither effect alone is significant, but both
  effects are well above zero, suggesting a
  significant conjunction effect.
• Null hypothesis if there is no effect of A or B
  then both t will be drawn from a random (global
  null) t distribution.
• Minimum t (0.8) falls in top 5 for the
  distribution, so the conjunction is significant.

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Comparison of GN CN with MS
Significant
Not Sig.
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Reason underneath df, T distribution and t
statistic
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Intermediate Null (IN)

• Global Null (relatively easy but not always
  valid)
• At least one comparison shows an effect
• Conjunction Null (too conservative)
• All n comparisons show effects (no assumption on
  dependence of effects)
• Intermediate Null (between GN CN, controlled
  threshold)
• More than k (kltn) comparisons show effects

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FPR in GN, CN and IN
GN
CN
IN
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Conjunction in SPM5 (IN)
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Summary

• 1) A significant conjunction is not a conjunction
  of significances.
• 2) Global Null (in SPM 99/2) is easy to show a
  conjunction effect but not always valid.
• 3) Conjunction Null is always valid but too
  conservative (largest FPR).
• 4) Intermediate Null (in SPM 5) is a compromise
  between GN and CN.

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Question Time 4 U