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Analysis of Microarray Data

Rhys Price Jones

Anne Haake

Bioinformatics Computing II

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Some Basic Statistics

• dot product• mean• standard deviation• log base 2• etc.• util.ss

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Gene chips

• Spots representing thousands of genes• Two populations of cDNA

– different conditions to be compared

• One colored with Cy5 (red)• One colored with Cy3 (green)• Mixed, incubated with the chip• Figures from Campbell-Heyer Chapter 4

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Red/Green Intensity measurements

• (define redgreens '((2345 2467) (3589 2158) (4109 1469) (1500 3589) (1246 1258) (1937 2104) (2561 1562) (2962 3012) (3585 1209) (2796 1005) (2170 4245) (1896 2996) (1023 3354) (1698 2896)))

• Shows (red green) intensities for 14 (out of 6200!) genes

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Should we normalize?

• Average of reds is 2386.9• Average of greens is 2380.3• What does John Quackenbush say? (page

420)• Calculate standard deviations.• Return to this issue• For now, no normalization

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Ratios of red values to green

• (define redgreenratios (map (lambda (x) (round2 (/ (car x) (cadr x)))) redgreens))

• Produces (0.95 1.66 2.8 0.42 0.99 0.92 1.64 0.98 2.97 2.78 0.51 0.63 0.31 0.59)

• Which genes are expressed more in red than green?• Should these values be normalized?

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Yet another Color scheme

• (0.95 1.66 2.8 0.42 0.99 0.92 1.64 0.98 2.97 2.78 0.51 0.63 0.31 0.59)

• Highly expressed Neutral Less expressed• >2.0 >1.3 close to 1.0 >0.5 <0.5

– Seems arbitrary?– Log scale??

• Why oh why did they re-use red and green?• Clustering? Meaning?

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Larger experiment

• 12 Genes• Expression values at 0, 2, 4, 6, 8 and 10

hours

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Table 4.2 of Campbell/Heyer

• Name 0 hrs 2 hrs 4 hrs 6 hrs 8 hrs 10 hrsC 1 8 12 16 12 8 D 1 3 4 4 3 2 E 1 4 8 8 8 8 F 1 1 1 .25 .25 .1 G 1 2 3 4 3 2 H 1 .5 .33 .25 .33 .5 I 1 4 8 4 1 .5 J 1 2 1 2 1 2 K 1 1 1 1 3 3 L 1 2 3 4 3 2 M 1 .33 .25 .25 .33 .5 N 1 .125 .0833 .0625 .0833 .125

• Normalized how?

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Take logs

• C 0 3.0 3.58 4.0 3.58 3.0

D 0 1.58 2.0 2.0 1.58 1.0 E 0 2.0 3.0 3.0 3.0 3.0 F 0 0 0 -2.0 -2.0 -3.32 G 0 1.0 1.58 2.0 1.58 1.0 H 0 -1.0 -1.6 -2.0 -1.6 -1.0 I 0 2.0 3.0 2.0 0 -1.0 J 0 1.0 0 1.0 0 1.0 K 0 0 0 0 1.58 1.58 L 0 1.0 1.58 2.0 1.58 1.0 M 0 -1.6 -2.0 -2.0 -1.6 -1.0 N 0 -3.0 -3.59 -4.0 -3.59 -3.0

• Compare

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How Similar are two Rows?

• How similar are the expressions of two genes?• First we’ll normalize each row

(define normalize ; substract mean and divide by sd (lambda (l) (let ((m (mean l))

(s (standarddeviation l))) (map (lambda (x) (/ (- x m) s)) l))))

• What are the new mean and standard deviation?

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How Similar are two Rows?

• Calculate the Pearson Correlation between pairs of rows

(define pc ; pearson correlation (lambda (xs ys) (/ (dotproduct (normalize xs) (normalize ys))

(length xs))))

> (pc '( 1 2 3 4 3 2 ) ; row G '( 1 2 3 4 3 2 )) ; row L 1.0> (pc '( 1 2 3 4 3 2 ) ; row G '( 1 3 4 4 3 2 )) ; row D0.8971499589146109

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Some other pairs

• Name 0 hrs 2 hrs 4 hrs 6 hrs 8 hrs 10 hrsC 1 8 12 16 12 8 D 1 3 4 4 3 2 E 1 4 8 8 8 8 F 1 1 1 .25 .25 .1 G 1 2 3 4 3 2 H 1 .5 .33 .25 .33 .5 I 1 4 8 4 1 .5 J 1 2 1 2 1 2 K 1 1 1 1 3 3 L 1 2 3 4 3 2 M 1 .33 .25 .25 .33 .5 N 1 .125 .0833 .0625 .0833 .125

> (pc '( 1 3 4 4 3 2) ; row D '( 1 .33 .25 .25 .33 .5)) ; row M-0.9260278787295065> (pc '( 1 2 3 4 3 2) ; row G '( 1 .5 .33 .25 .33 .5)) ; row H-0.9090853650855358

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Correlation is sensitive to relative magnitudes

• pc(G,L) = 1 -- identically expressed genes• pc(G,D) = .897 -- similarly expressed genes• pc(D,M) = -.926 -- reciprocally expressed• pc(G,H) = -.909 -- also reciprocally expressed

• What happens if, instead of using the expression data we use the log transforms?– pc(G,L) = 1.0– pc(G,D) = 0.939– pc(D,M) = -1.0– pc(G,H) = -1.0

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Hierarchical Clustering

• Repeat– Replace the two closest objects by their combination

• Until only one object remains

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What are the objects?

(define objects (map (lambda (x) (cons

(symbol->string (car x))(cdr x)))

logtable42))

• Initially, the objects are the genes with the log transformed expression levels

• Typical object• ("E" 0 2.0 3.0 3.0 3.0 3.0)

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Combining objects

(define combine ; (lambda (xs ys) (cons (string-append (car xs) (car ys))

(map (lambda (x y) (/ (+ x y) 2.0)) (cdr xs) (cdr ys)))))

• combine names• average the entries• Typical combined pair:• ("EG" 0 1.5 2.29 2.5 2.29 2.0))

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Manual Hierarchical Clustering

• Let’s go to emacs

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K-means Clustering -- Lloyd’s Algorithm

Partition data into k clustersREPEAT

FOR each datapoint {Calculate its distance to the centroid of each clusterIF this is minimal for its own cluster

Leave the datapoint in its current cluster

ELSEPlace it in its closest cluster

}

UNTIL no datapoint is moved• Goal: minimize sum of distances from datapoints to

centroids

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Analysis of k-means clustering

• There are always exactly k clusters• No cluster is empty (why?)• The clusters are not hierarchical• The clusters do not overlap• Run time with n datapoints:

• Partitioning O(n)• FOR loop is O(nk)• REPEAT loop is ???• kanungo et al

Partition data into k clusters

REPEAT

FOR each datapoint {

Calculate its distance to the centroid of each cluster

IF this is minimal for its own cluster

Leave the datapoint in its current cluster

ELSE

Place it in its closest cluster

}

UNTIL no datapoint is moved

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Pro and Con

• Pro• With small k, may be faster than hierarchical• Clusters may be “tighter”

• Con• Sensitive to initial choice of k• Sensitive to initial partition• May converge to local, rather than global minimum• Not clear how good resulting clusters are

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Other Methods for Clustering

• Self Organizing Maps• SWARM technology• SOM/SWARM hybrid