Covariate-adjusted Matrix

Visualization via

Correlation Decomposition吳漢銘

淡江大學 數學系 資料科學與數理統計組

hmwu@mail.tku.edu.tw

http://www.hmwu.idv.tw

Outlines

Data/Information Visualization Two Demo Data Sets Generalized Association Plots (GAP) Related Works with Matrix Visualization Covariate-adjusted Matrix Visualization

For a discrete covariate: Within And Between Analysis (WABA) For a continuous covariate: Partial Correlations

Examples GAP Software Concluding Remarks

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Data/Information Visualization

Exploiting the human visual system to extract information from data.

Provides an overview of complex data sets. Identifies structure, patterns, trends,

anomalies, and relationships in data. Assists in identifying the areas of interest.

Matrix Visualization: reorderable matrix, the heatmap, color histogram, data image.

Visualization =

Data

information

Graphing for Data

+ Fitting

+ Graphing for Model

Raw Data Matrix Raw Data Map

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The Iris Data (Anderson 1935; Fisher 1936)

The iris data published by Fisher (1936) have been widely used for examples in discriminant analysis and cluster analysis.

Images source: http://www.stat.auckland.ac.nz/~ihaka/120/Lectures/lecture27.pdf

4 variables

50x3=150 subjects

Raw Data Matrix

1 covariate

setosa

versicolor

virginica

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Psychosis Disorder Data (Chen 2002)

All the symptoms are recorded on a six point scale

(0-5).

Scale for Assessment of Negative Symptoms

(SANS): 20 items, 5 subgroups.

Expression (NA1-7)Scale for Assessment of Positive Symptoms

(SAPS): 30 items, 4 subgroups.

Hallucinations (AH1-6)

69 schizophrenic

26 bipolar disorders

Speech (NB1-4)

Hygiene (NC1-3)

Activity (ND1-4)

Inattentiveness (NE1-2)

Behavior (BE1-4)

Delusions (DL1-12)

Thought disorder (TH1-8)

50 Variables

95 Subjects

正性症狀 : 行為的過量

負性症狀 : 行為的不足

精神分裂症

躁鬱症

精神疾病

幻覺

妄想行為

思考失序

表達語言

社交

做事的意志

衛生

Raw Data Matrix胡海國 國立臺灣大學 精神科教授

國立臺灣大學醫學院附設醫院 精神部主任

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(4) Sufficient

充分

Generalized Association Plots (GAP) (Chen, 2002)

Four Steps of Generalized Association Plots (GAP)

(1)Presentation

呈現

(2) Seriation

排序

(3) Partition

分割

Raw Data Matrix

Proximity Matrices for Rows and Columns

Clustering Summarization

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Presentation of Raw Data Matrix

0. Data Transformation

1. Selection of Proximity Measures

2. Color Spectrum

3. Display Conditions

The 1st Step of GAP

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Presentation of Raw Data Matrix: iris data

(2) Color Spectrum

(1) Selection of Proximity Measures

(3) Range Matrix Condition

Pearson Correlation Matrix for Variables

Eculidean Distance Matrix for Subjects

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Presentation of Raw Data Matrix: Psychosis Disorder Data

Pearson Correlation CoefficientCorrelation Matrix for Variables

Correlation Matrix for Subjects

Raw Data Matrix

(2) Color Spectrum

(1) Selection of Proximity Measures

(3) Range Matrix Condition

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Seriation of Proximity Matrices and Raw Data Matrix

Relativity of a Statistical Graph Global Criterion

GAP Rank-Two Elliptical Seriation

Local Criterion Tree Seriation Flipping of Tree Intermediate Nodes

The 2nd Step of GAP

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Relativity of a Statistical Graph

Placing similar objects at closer positions.Placing different objects at distant positions.

Seriation Methods

(1) Rank Two Ellipse Ordering (Chen, 2002)

(2) Hierarchical Clustering Tree (Average-Linkage)

Seriation Methods

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GAP Rank-Two Elliptical Seriation

The p objects fall on an ellipse and have unique relative position on the ellipse (Chen 2002).

Seriation Algorithms with Converging Correlation Matrices

First two Eigenvectors

Correlation Matrix (without ordering)

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Hierarchical Clustering Tree with a Dendrogram

Different Seriations Generated from Identical

Tree Structure

Tree seriation for proximity matrices

Tree seriation for raw data matrices

3 flips1 flipmany flips5 flips

ideal model

Tree seriation

Internal Tree Flips External Tree Flips Ziv Bar-Joseph, David K. Gifford, and Tommi S. Jaakkola, (2001), Fast Optimal Leaf Ordering for Hierarchical Clustering. Bioinformatics 17(Suppl. 1):S22–S29.

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GAP Rank-two elliptical seriation Michael Eisen (1998) tree seriation

Global vs. Local Seriation

Data: 517 genes by 13 arrays

Tien, Y. J., Lee, Y. S, Wu, H. M. and Chen, C. H.* (2008), Methods for Simultaneously Identifying Coherent Local Clusters with Smooth Global Patterns in Gene Expression Profiles. BMC Bioinformatics 9:155, 1-16.

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Related Works of Matrix Visualization

Concept:1. Bertin (1967): reorderable matrix.2. Carmichael and Sneath (1969): taxometric maps.

Clustering of data arrays:1. Hartigan (1972): direct clustering of a data matrix. 2. Tibshirani (1999): block clustering. 3. Lenstra (1974): traveling-salesman problem.4. Slagle et al. (1975): shortest spanning path.

Colour Representation:1. Wegman (1990): colour histogram.2. Minnotte and West (1998): data image.3. Marchette and Solka (2003): outlier detection.

1

12

1 2 3

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Exploring proximity matrices only:1. Ling (1973): shaded correlation matrix.2. Murdoch and Chow (1996): elliptical glyphs.3. Friendly (2002): corrgrams.

Integration of raw data matrix with two proximity matrices1. Chen (1996, 1999, and 2002): generalized association plots (GAP).

Reordering of variables and samples1. Chen (2002): concept of relativity of a statistical graph.2. Friendly and Kwan (2003): effect ordering of data displays.3. Hurley (2004): placing interesting displays in prominent positions.

Matrix Visualization (MV): reorderable matrix, the heatmap, color histogram, data image.

Related Works of MV (conti.)

1

2

3

1

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Covariate-adjusted

First two PCAs for Iris Data Psychosis Disorder Data

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A Model18/42

Correlation Decomposition19/42

Covariate-adjusted MV for Discrete Case

Correlation (Distance) for rows

based on(1) raw data matrix(2) fitted data matrix(3) residual data matrix

Correlations for

columns

Discrete Covariate

Y

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Within And Between AnalysisDansereau, F., Alutto, J. A., & Yammarino, F. J. (1984).

Total correlation

Between-group correlation Within-group correlation

Between-eta correlation Within-eta correlation

Between component Within component

WABA equation

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Three Steps to WABA

WABA I: Assessment of Variation: eta Each variable is assessed to determine whether the variable varies  

between group (suggesting within-group homogeneity). within groups (suggesting within-group heterogeneity). both between and within groups (suggesting individual differences

rather than within-group homogeneity or heterogeneity).

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Three Steps to WABA (conti.)

WABA II: Assessment of Covariation: RB, RW Relationship among variables are assessed to determine whether the

correlation between variables is primarily a function of between-group covariance within-group covariance within- and between-group covariance (suggesting individual differences).

Drawing Inferences: Combination of WABA I and WABA II: R, B, W

The results of the first two steps are assessed for consistency and combined to draw the best overall conclusion from the data.

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Covariate-adjusted MV for Continuous Case

Correlation (Distance) for rows

based on(1) raw data matrix(2) fitted data matrix(3) residual data matrix

Correlations for

columns

Continuous Covariate

Y

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Partial Correlations

Conditional correlation is equivalent to partial correlation under some assumptions (Kurowicka and Cooke, 2000).

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Assessing the Goodness of Fit of the Model Component

+=

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Significance Analysis of the Residual Component Dunn and Clark’s z test for the equality of two dependent

correlations in the case of N exceeds 20 (Steiger, 1980). Test whether the correlations between variables Xj and Xk are

different significantly before and after a covariate adjustment.

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z-score Significant Map

This z-score significant map is helpful identifying variable pairs with the most significant differences in correlation before and after a covariate adjustment.

R Radj

zDunn and

Clark’s z test

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Simulation Study29/42

Psychosis Disorder Data: R

Rank-two ellipse orderingFive symptom groups

identified by Chen (2002).

thought disorder ( 思考失序 )

Negative ( 負性症狀 )

auditory hallucination ( 聽幻覺 )

loss of ego boundary ( 分際喪失 )

Mania ( 狂躁 )

NOTE: the mania symptoms are negatively related to the negative symptoms and the auditory hallucination symptoms.

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Psychosis Disorder Data: R=B+W

By comparing B and R, the negative correlations between the mania symptoms

V5 (DL4, TH6-8) with the negative symptoms V2 (NC1-ND4) and the auditory

hallucination symptoms V3 are mostly due to the patients‘ subtypes.

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Psychosis Disorder Data: B

Average-linkage + GrandPa Flip

mania symptoms (DL4, TH6-8)

negative symptoms (NC1-ND4)

auditory hallucination symptoms

Delusions ( 妄想 )

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Psychosis Disorder Data: RB

Average-linkage + GrandPa Flip

All correlations are either positive one or negative one since there are only two subtypes for patients.

Two clusters (DL2-TH6) and (NA7-NA6) are formed and are negatively correlated.

For 50 between-eta correlations, symptom TH6 with the darkest between-eta has the most significant difference between schizophrenic and bipolar disorders.

話停不下來的 (Pressure of Speech)

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Psychosis Disorder Data: W

Rank-two ellipse ordering

Residual Patterns

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Psychosis Disorder Data: RW

Rank-two ellipse ordering

Four new symptom groups: (ND2-NE1), (TH5-TH7), (Th3-Th4) and (DL4-DL6).

Four symptoms NE1, DL2, BE1, and BE2 were grouped into the original negative symptoms group.

The symptoms in the TH (thought disorder) were grouped into two highly correlated subgroups (TH3-TH4, Th5-TH7).

All hallucination symptoms (AH1-6) and most of the delusion symptoms (except DL2, DL3) were clustered together.

negative symptoms

thought disorder

hallucination

delusion

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Psychosis Disorder Data: Z

Positive z scores: between and within the group of

the negative symptoms (V2, except NA6, DL3, and BE4) and the group of the auditory hallucination symptoms (V3, except DL6)

within the group of the mania symptoms (V5, except DL5 and BE3)

Negative z scores: between the group of mania

symptoms V5 and the group of the negative symptoms (V2, except DL3, and BE4) and the group of the auditory hallucination symptoms (V3, except DL6).

negative

auditory hallucination

mania

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Psychosis Disorder Data: Z

Changed significantly: the symptom TH4 ( 不合邏

輯 ) of V1 to the group of the negative symptoms (V2, except NA6, DL3, and BE4), the group of the auditory hallucination symptoms (V3, except DL6).

Without significant relationship with any other symptom for different patients' subtypes:

Eleven symptoms (TH5, NE2, DL2, BE1, BE2, DL3, BE4, AH6, AH5, DL5,and BE3)

Note: positive symptoms of behavior (BE1-BE4) are all included.

thought disorder

loss of ego boundary

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Psychosis Disorder Data: Z

Single-linkage+GrandPa Flip

Most significant difference A right slash: A reversed slash:

(AH1, DL4), (AH1, TH7), (DL1, DL4), (TH6, NC2), (TH7, NA1), (TH7, NA2), (TH7, NA3) (TH7, NA4), (TH7, NA5), (TH7, NB1), and (TH7, ND1).

Bipolar disorders patients tend to have higher distractible speech score (TH7).

Schizophrenic patients are more likely having higher negative symptoms scores.

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Psychosis Disorder Data: Z

Single-linkage+GrandPa Flip

Most significant difference A right slash: A reversed slash:

(AH1, NA5), (DL1, NA4), (DL7, NA1) and (DL7, NA5).

Bipolar disorders patients have lower scores on these symptoms than schizophrenic patients.

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GAP Software verison 0.2.7

Generalized Association Plots Input Data Type: continuous or binary. Various seriation algorithms and clustering analysis. Various display conditions.

Modules: Covaraite Adjusted. Proximity Modelling. Nonlinear Association Analysis. Missing Value Imputation.

http://gap.stat.sinica.edu.tw/Software/GAP

Statistical Plots2D Scatterplot, 3D Scatterplot (Rotatable)

Download

Wu, H. M., Tien, Y. J. and Chen, C. H.* (2010). GAP: A Graphical Environment for Matrix Visualization and Cluster Analysis, Computational Statistics and Data Analysis, 54, 767-778.

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Concluding Remarks

Suggestions A preliminary step in modern exploratory data analysis. A continuing and active topic of research and application. New generation of exploratory data analysis (EDA) tool.

Matrix Visualization Color order-based representation of data

matrices. Provide several levels of information.

Covariate-adjusted Matrix Visualization Decomposition of correlations. Working on fitted and residual data matrix. Interactive Software: GAP. Extension to multi-level data.

GAP

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Acknowledgment

cchen@stat.sinica.edu.tw http://gap.stat.sinica.edu.tw

hmwu@mail.tku.edu.tw http://www.hmwu.idv.tw

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