Stefan Arnborg, KTH and SICS

http://www.nada.kth.se/~stefan

Brain Research and Data Mining

Visualization or Statistics?

• A good visualization strikes the investigator between the eyes with the truth J. Tukey

• The human perception system is biased towards wishful thinking - we normally see what we want to see.

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The human eye finds structureBayes factor is 1.2 in favor of no structure vs structureThe points are generated completely randomly

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Cumulative plots of x- and y-coordinate

Variables in test matrix

scanid Diagnosis (A or C) Demographics:Gendre Height Weight BMI Hand Age-MRI Birth-Mon Age-PmaBlood tests:B-MCV-01 S-ALAT-K1 S-ASAT-K1 S-CDT-02 S-CDT-B1 S-GLU-K2 S-GT-K1 S-K-K1 S-KOL-K1 S-LDL-B1 S-PROL-K1 S-TG-K1

MR Volumes

MR Volumes in test matrix(144 subjects)

BrsCSF BrsGrey BrsWhite Cer-CSF CerGrey CerWhite FroCSF FroGrey FroWhite OccCSF OccGrey OccWhite ParCSF ParGrey ParWhite SubCSF SubGrey SubWhite TemCSF TemGrey TemWhite VenCSF VenGrey VenWhite Rel-volGrey Total-intr Total-CSF Total-Grey Total-White Int-nocl Int-blood CSF/Grey Grey/Total White/Total CSF/Total

Vermis (manually traced, 109 subjects):CH TV AV PSV PIV

Thomas Bayes (1703-1762)

• If I suspect that a coin used for betting is unbalanced, how should I test it? - inverse probability.

• Prior: Before the experiment my probability for heads is uniformly distributed between 0 and 1.

• Posterior: After the experiment my probability is described by a ‘beta distribution’.

C. S. Peirce (1839 - 1914).Pragmaticism: How does our understanding of Nature develop? How should Sciencebe developed?

Semeiotics: Meaning is created by signs sent around in the nervous system. Scientific knowledge is created in a never-ending process of discontent with the current theories which force new models of thought.

C. S. Peirce (1839 - 1914).

A person is not absolutely anindividual. His thoughts are what he is ‘saying to himself’,that is, saying to that other selfthat is coming into life in the flow of time. When one reasons it is that critical self that one is trying to persuade.It is a necessity of Logic that every logical evolution of thought should be dialogic.Every thought is a sign.

The Sign of ThreeUmberto Eco, Thomas SebeokIndiana University Press, 1983.

Love, Chance and Logic:C.S. Peirce, 1923

Sherlock Holmes: common senseinference

Techniques used by Sherlock are modelled on Conan Doyle’s professor in medical school, who followed the methodological tradition of Hippocrates and Galen. Abductive reasoning, first spelled out by Peirce, is found in 217 instances in SherlockHolmes adventures - 30 of them in the first novel, ‘A study in Scarlet’.

Bayes’ factor

• Choice between two hypotheses, H1 and H2,given experimental/observational data D

P(H1|D) P(D|H1) P(H1)P(H2|D) P(D|H2) P(H2)

Posterior odds Bayes factor prior odds

Bayes factor 8 issignificant,32 is strong

Hierarchical models• Model parametrized:

H f( x | ) , f( | x) ~ f( x | ) f(),

• Modell hierarchical or composite: prior f() for H1: f( x | ) and f(),

P(D|H1) = ∫ f(D|) f() d∫ ∏f(d |) f() d

i

Hypothesis in test matrix

• H1: (no effect) a data column is generatedindependently of diagnosis (composite model)

• H2: the data for controls are generated by one composite model, for affected by another one.

Effect for TemGrey, not for CerGrey

Bayes factor 0.4 - weakly no effect Bayes factor 4- weak sign of effect + - affected o - controls

Difference women-men

BF 0.1: no effect for women BF 4 : weak sign of effect,men

Mass testing effects, confounders

• In a 1000-column table there are ~5000 accidental associations on the 1% level.

• Bayesian analysis, properly applied, avoids all problems of overfitting and mass testing.

• Observational studies are prone to misleadingconclusions from known & unknown confounder

• Causal graphical models is a tool to avoid this

Compensating mass testing

• Bonferroni 1937: For level and n tests, use level /n

• Benjamini 1996: Control False Discovery Rate

• Composite Bayes model (1763-2001): Theoretically optimal procedure, but requiresan explicit (composite) alternative to the nullhypothesis (with nuisance parameters).

Dependence on sampleBrsCSF 7.4TemCSF 6.6VenWhite 6.3Total-CSF 4.2SubWhite 3.5VenCSF 3.1FrCSF 2.9SubCSF 2.6BMI 1.9Rel-volGrey 1.6S-GLU-K2 1.6CerGrey 1.4

(144 subject sample)

PSV 9.5BrsCSF 7.7TemCS 7.3VenCSF 4.8TV 4.7Total-CSF 4.4SubCSF 3.5VenWhite 3.4FrCSF 3.4age 3.4AV 2.3SubWhite 2.2OccWhite 1.1S-GLU-K2 1.0(109 subject sample)

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109 Sample not matched wrt age!

Gendre differences

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Men

Women

Graphical models

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Xf(x,y,z)=f(x)f(y)f(z)

f(x,y,z)=f(x,z)f(x,y)/f(x)

f(x,y,z)

SubCSF

SubWhiteTemCSFVenWhite

Diagnosis FrCSFFrWhiteParCSF

Totalwhite

TotalCSF

BrsCSF

144 sample without Vermis variables

106 sample with Vermis variables

Diagnosis PSV

TV

PIV

TotalGrey

Graphical models, directed

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Xf(x,y,z)=f(x)f(y)f(z)

f(x,y,z)=f(x)f(y|x)f(x|y)

f(x,y,z)=f(x)f(y)f(x|y,z)

Experimental vs observational data

• Is there an association between treatment and recovery?

• Is there a causal link? - or a backdoor path(confounder)?

• Can we decide if a patient had recovered with a different treatment?

• Can we decide which treatment has best chance of recovery for patient?

Cause or effect?

• Association between drinking red wine and good health is known since long. Drinking 1 litrea day is equivalent, for life insurance purposes, to temperance (Skandia-If statistics, 1998)

• Does drinking red wine promote health?

• Does sound lifestyle promote drinking red wine?

• Or both? Causes are today only hypotheses!(Svenska Dagbladet Sept 3 2001)

Causal graphs-Bayesian networks

• Statistical DAG: f(x,y) = f(y|x) f(x)

• Causal graph: Arrow means causation: y <— F(x,)

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Controlling eelworms by fumigants(Cochran 1981)

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X: FumigantsY: YieldB: Birds

Z0: Eelworms in winterZ1: Eelworms at treatmentZ2: Eelworms after treatmentZ3: at end of season

Z2

P(y|x)= P(y|x z0) P(z0)

Must condition on Z0, or on Z1 and B.

Classification (Cheeseman, Stutz)

cases

Variables hiddenclass

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ab..bba

Model assumption:within each class,columns are generatedindependently of eachother. (Other options existfor numerical data)

Classification explains data!

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HW

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Autoclass1

Autoclass10

Autoclass10 vs CSF

Autoclass100

Mining causal chainsAre there pairs of variables where the association is differentfor controls than for affected? Can this indicate a regulation paththat is disturbed for affected?

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VenGrey

Strong difference in association

Rel-volGrey SubWhite 11.3VenGrey SubWhite 10.5Cer-CSF SubWhite 10.5Total-intr SubWhite 10.4FroCSF SubWhite 10.3Int-nocl SubWhite 10.2TemGrey SubWhite 9.9ParGrey Total-CSF 9.6