Factorgrams: A tool for visualizing multi-way associations in

http://www.psi.toronto.edu

Factorgrams: A tool for visualizing multi-wayassociations in biological data

Vincent Cheung, Inmar Givoni, Delbert Dueck, Brendan J. Frey

May 15, 2006 PSI TR 2006–44

Abstract

Effective visualization of biological data is often critical for subsequent analy-sis. The popular clustergram/dendrogram visualization rearranges rows andcolumns of a data matrix so as to highlight clusters of similar responses, butassumes each row or column belongs to only one cluster and cannot associateeach row or column with multiple clusters. Such multi-way associations oc-cur frequently, e.g., when a gene plays multiple biological roles. We describethe ’factorgram’ visualization, which rearranges the data into an expandedview, associating each row (or column) with multiple clusters of rows (orcolumns) and elucidating potentially new biological relationships. Factor-grams for mouse gene expression and yeast synthetic-lethal gene-interactiondatasets detect a larger number of statistically-significant clusters than clus-tergrams, plus a larger number of clusters enriched for gene ontology annota-tions. Experimentally-verified associations previously identified by manualrearrangement of rows and columns not grouped together by clustergrams,are readily identified by the factorgram.

Factorgrams: A tool for visualizing multi-way associations in biological data

Vincent Cheung1,2, Inmar Givoni1,2, Delbert Dueck1,2, Brendan J. Frey2,3,4

1. These authors contributed equally 2. Probabilistic and Statistical Inference Group, Departments of Electrical and

Computer Engineering and Computer Science, University of Toronto, 10 King’s College Rd., Toronto, ON, Canada, M5S 3G4

3. Banting and Best Department of Medical Research, University of Toronto, 112 College St., Toronto, ON, Canada, M5G 1L6

4. To whom correspondence should be addressed: Brendan J. Frey, [email protected], 416-978-7001 (office), 416-978-4425 (fax).

Effective visualization of biological data is often critical for subsequent analysis.

The popular clustergram/dendrogram visualization rearranges rows and columns of a

data matrix so as to highlight clusters of similar responses [1], but assumes each row

or column belongs to only one cluster and cannot associate each row or column with

multiple clusters. Such multi-way associations occur frequently, e.g., when a gene

plays multiple biological roles. We describe the ‘factorgram’ visualization, which

rearranges the data into an expanded view, associating each row (or column) with

multiple clusters of rows (or columns) and elucidating potentially new biological

relationships. Factorgrams for mouse gene expression and yeast synthetic-lethal gene-

interaction datasets detect a larger number of statistically-significant clusters than

clustergrams, plus a larger number of clusters enriched for gene ontology

annotations. Experimentally-verified associations previously identified by manual

rearrangement of rows and columns not grouped together by clustergrams, are

readily identified by the factorgram.

Factorgrams – Cheung et al.

2

High-throughput biological assays produce huge quantities of data, which can often be

organized into a matrix where rows and columns correspond to separate control variables,

and each element in the matrix is a binary- or real-valued measured response. For example,

the rows may correspond to microarray probes for genes, exons or tiled DNA segments, the

columns to time points or tissue samples, and each element to a real-valued transcript

abundance estimated using a microarray. Or, the rows and columns may correspond to

yeast genes and each element to a binary value indicating whether or not there is a

synthetic lethal interaction in the yeast strain with both genes deleted. While customized

techniques based on machine learning and statistical inference can be used to analyze this

kind of data, even simple rearrangements of the data or straightforward transformations of

the data can reveal biologically-significant patterns that are directly visualized in the

rearranged dataset. Currently, the most common visualization tool for biological matrix

data is the clustergram, in which the rows and the columns of the data matrix are re-ordered

using a clustering method such as hierarchical agglomerative clustering (HAC) [11] or k-

means clustering [12]. Similarities between nearby rows or columns may be indicated by

dendrogram. The rearranged matrix of data is shown as an image where the color at each

coordinate corresponds to the appropriate binary- or real-valued response [1].

A major deficiency of clustergrams is that rows (or columns) are grouped together

based on similarity of response across their entire columns (or rows). In many biological

assays, it is frequently the case that different biological processes will impact overlapping,

but not disjoint, sets of variables. For example, consider a yeast synthetic gene interaction

matrix in which rows correspond to gene deletion mutants of different non-essential yeast

genes, columns also correspond to gene deletion mutants of non-essential yeast genes, and

each element in the matrix measures the viability of a yeast strain with both genes knocked

out. If genes A and B exhibit similar viability patterns across a subset of other genes, they

can be grouped together, indicating that they may be part of the same functional pathway

involving all genes in the subset. However, gene B may additionally play a different

biological role as part of another pathway that does not involve gene A, but involves gene

C. Genes B and C may exhibit similar viability patterns across a different subset of genes,

namely those that are involved in the second pathway. In this case, there are two clusters


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corresponding to the two pathways and they overlap in the sense that they both contain

gene B. This creates a problem for clustering methods, which assume that clusters do not

overlap. Standard clustering techniques and the clustergram visualization do not provide a

general way to link gene B to both genes A and C, what we refer to as making ‘multi-way

associations’.

The failure of the clustergram to indicate multi-way associations is evident from the

data of Tong et al. [13], shown in Fig. 1A by ordering the rows and columns according to

the dendrograms produced by HAC (see Methods). Each row or column can only be

associated with nearby rows or columns, so overlapping clusters of genes (i.e., multi-way

associations) are not properly revealed. In Fig. 1B we highlight two examples of data

clusters that are significantly enriched for gene function annotations, but that were not

properly identified by hierarchical clustering and were thus broken apart in the clustergram.

In some cases, a row or column can be visually associated with two clusters if by

coincidence it is on the boundary between the two clusters or if a computational method is

used to move it to a position that is closer to another cluster. However, clustergrams cannot

generally reveal all such double-associations, because only a small number of cases can be

placed near cluster boundaries.

The ‘factorgram’ is a visualization that expands the input data matrix so as to enable

explicit visualization of data in overlapping clusters. We refer to each such overlapping

cluster as a ‘factor’, because techniques that explain each row or column of data as a

composition of multiple components are called ‘factorization’ methods in the machine

learning and statistics communities, and each component is called a ‘factor’ (c.f. [2]). The

output from a variety of factorization techniques [2-10] can be further processed to produce

the factorgram. The purpose of this paper is not to advocate a particular factorization

technique, but to illustrate how the factorgram visualization can be useful to biologists for

further analysis and detection of a larger number of statistically and biologically significant

associations than cannot be detected using clustergrams. Software for producing

factorgrams is available at http://www.psi.toronto.edu/factorgram.


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To give the reader a sense for how factorgrams work, in Fig. 2 we show a cartoon

illustrating one way to construct a factorgram. The clustergram of a synthetic binary data

matrix is shown in the upper-left corner, and includes blocks of data that have been broken

apart because specific rows and columns (shown with arrows) belong to multiple groups.

To produce the factorgram, the rows and columns are rearranged so that the dominant

factor appears as a contiguous block of data. This factor is then removed from the matrix

and placed in the factorgram. The rows and columns of the resulting matrix are again

rearranged so that the next dominant factor appears as a contiguous block of data. This

factor is removed and placed in the factorgram. This procedure is repeated until no more

significant factors remain in the data matrix. The factorgram is a figure showing the raw

data associated with all extracted factors and may additionally include appropriate row and

column labels. The factors may be shown along the diagonal of a matrix to reflect

similarities to the clustergram, or the factors can be arranged more compactly. Since the

same row (or column) can appear in multiple factors, the factorgram provides a way to

visualize clusters with overlapping variables.

Techniques for computing factorgrams should properly address how to efficiently

search over possible solutions and assess the statistical significance of the detected factors.

In contrast to clustergrams, where each row or column is associated with one cluster, in

factorgrams, each row or column can be associated with multiple factors through different

subsets of responses. This capability comes at the cost of an exponentially larger space of

possible solutions, since if there are k factors and each row or column can belong to n

factors at once, there are kn possible ways of assigning the row or column to the factors.

Approaches for efficiently searching over these assignments include computing linear

approximations and using more powerful probabilistic inference techniques. A binary

assignment variable with value 0 or 1 can be used to indicate whether or not a row or

column belongs to a factor. If these assignment variables are relaxed to be real-valued, data

elements are described by linear combinations of continuous variables. Continuous

solutions to the factorization problem can be computed using an eigen-vector method such

as principal components analysis [3] or an iterative technique such as factor analysis [2],

independent component analysis [4] and non-negative matrix factorization [5]. Once the


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real-valued ‘assignment’ variables have been computed, they can be thresholded to identify

components in the factorgram. One problem with these approaches is that a good binary

solution often cannot be obtained by thresholding the best continuous solution. An

alternative approach is to retain the binary representation of the assignment variables but

constrain the solution space. Bi-clustering [6] imposes the constraint that any two clusters

containing the same column (or row) must be defined by exactly the same set of columns

(or rows); this constraint makes extraction of factors easier, but is only appropriate when

different factors are defined by the same subsets of rows or columns.

Recently, methods have been proposed that retain the binary representation of the

assignment variables and avoid simplifying assumptions about the factors by using

techniques developed in the probabilistic and statistical inference communities to search for

the most probable setting of the assignment variables. The plaid method [7] works by

extracting one factor at a time using the linear approximation described above, but then

thresholds the assignment variables before proceeding to extract the next factor.

Probabilistic sparse matrix factorization [8,9] and matrix tile analysis [10] take a direct

approach and retain the binary representation, but find solutions by accounting for

uncertainties in the assignments and iteratively revisiting and refining factors until

convergence. The labelled latent Dirichlet allocation process analysis method [15] takes as

input the data matrix along with gene function annotations and performs Bayesian

inference in a hierarchical probabilistic graphical model to extract the factors.

The statistical significance of the factors in a factorgram can be assessed in a variety of

ways, but here we describe a general procedure that can be applied to any technique that

computes a factorgram. Denote the data matrix by X and the particular method used to

compute the factorgram by μ. We assume the method takes as input the data matrix and a

parameter θ that indirectly has an impact on the false detection rate. For example, θ could

be a real-valued threshold on a cost function that penalizes small factors (which are less

likely to be significant) but rewards factors containing highly similar data elements. For

method μ applied to data X using parameter θ, denote the number of extracted factors by

N(X,μ,θ). The null hypothesis is that a computed factor arose from random data with the


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same distribution as the source that produced X. Assuming the data matrix is large enough,

this distribution can be approximated by generating permuted data matrices (i.e., randomly

rearranging elements). Denote a data matrix obtained in this way by XR – since the data

elements were randomly rearranged, XR is a random variable. The expected number of

falsely detected factors can be estimated by E[N(XR,μ,θ)], where E[ ] denotes an

expectation w.r.t. the set of random matrices, XR. In this procedure, N(XR,μ,θ) is random

because XR is random, but also possibly because the method μ may produce different

solutions each time it is applied due to varying initial conditions.

We demonstrate the factorgram visualization using two publicly available datasets, the

yeast synthetic genetic array (SGA) dataset from Tong et al. [13] and the mouse gene

expression dataset from Zhang et al. [16]. The former is binary valued while the latter is

real-valued, and each dataset was factorized using a different method. However, the final

results are both readily visualized by the factorgram.

The SGA data is a binary-valued matrix of 135 by 1023 elements, where both rows

(query genes) and columns (array genes) correspond to yeast genes. Each element

represents the synthetic lethality of a double mutant whose corresponding row and column

genes have been knocked out. We applied HAC to both the rows and the columns of the

data matrix (see Methods) and in Fig. 1A we show the clustergram. The factorgram

generated from this data matrix (see Methods) contained 17 factors, 5.4 of which are

expected to be false detections based on the method described above, using ten random

permutation tests. The factorgram for the SGA data is shown in Fig. 3A, where each box

corresponds to one of the factors. While in the data matrix, only 3.35% of the gene pairs

exhibit synthetic lethality, in the identified set of factors 88.87% of the gene pairs exhibit

synthetic lethality. The identified factors account for 49.57% of the total number of

synthetic lethal interactions; 22.75% of the observed synthetic lethal interactions

correspond to array genes have three or less synthetic lethal interactions with query genes

and thus provide only weak evidence. In Fig. 3B, we show the clustergram obtained using

HAC (see Methods). We tried several clustering techniques and chose the one that

produced the largest number of biologically significant clusters (see below). In this figure,


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each element assigned to a factor is coloured so as to indicate the factor in Fig. 3A to which

it belongs. The clustergram breaks apart almost every identified factor, thus failing to

associate all genes within each factor.

An example of a well known biological pathway that is not readily identified in the

clustergram in Fig. 3A is the chitin synthase III pathway (SKT5, CHS3, CHS5, CHS7),

which is grouped together in the factorgram (dark green factor in Fig. 3). In some cases, the

clustergram correctly groups query genes, but fails to group array genes. For example,

query genes in the prefoldin complex (GIM3, GIM4, GIM5, PAC10, YKE2) were correctly

grouped in the clustergram, but corresponding array genes were not properly identified (red

factor in Fig. 3). The clustergram produced by Tong et al. used a different metric and

successfully grouped genes in the synthase III pathway, but failed to group together array

associated with the prefoldin complex. Instead, this complex was identified by laborious

manual rearrangement of the partial clusters. The large red factor shown in Fig. 3

successfully brings together similarities of array gene profiles and query gene profiles, thus

capturing both the row-based similarities and the column based similarities of the gene

profiles. Furthermore, the dark purple factor shows how the query genes of the prefoldin

complex are also similar based on their symmetric interaction as array genes, a relationship

that again was neither captured in our clustergram nor reported by Tong et al.

We next asked how many of the extracted factors are of biological significance in terms

of functional enrichment. We analysed whether the genes in each cluster were significantly

enriched for gene ontology (GO) annotations of biological process, molecular function, and

cellular component. We found 9 of the clusters to be enriched (p < 0.01) for at least one

annotation, with a total of 18 significant enrichments across all three categories (see

Methods). In comparison, when we used HAC to find the same number of clusters, we

were able to find only 4 clusters to be enriched for at least one annotation, with a total of

only 7 significant enrichments across all three categories. We tried a variety of different

clustering techniques, but these resulted in lower numbers of enriched clusters (data not

shown).


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The second dataset we analyzed and applied the factorgram visualization to is the

mouse mRNA expression dataset from Zhang et al. [16]. This dataset contains profiles for

over 22,709 known and predicted genes across 55 mouse tissues, organs, and cell types.

The factorgram generated from this dataset (see Methods) contained 37 factors and based

on the random permutation test we found that 0.5 factors are expected to be false

detections. The clustergram and factorgram are shown in Fig. 4. The factorgram correctly

identified many associations that are identified in the clustergram. For example, the cluster

containing nervous system tissues (inside the dark green bounding box in Fig. 4A) is

contained within the factor shown in Fig. 4D. Fig. 4C shows an example where the

discovered factor has been severely broken apart in the clustergram. This factor includes

genes that are expressed in both mature and embryonic nervous system tissues, and

includes genes not present in the previously described factor. Both of these factors have

statistically significant (p < 0.05) enrichment in gene annotations (see Methods).

Many factors reveal profile similarities across tissues which are not obviously similar,

and may seem surprising at first. However, this is one of the advantages of factorization

methods and of the factorgram, as they have the potential to enable the researcher to

scrutinize more interesting relationships. To answer the question of whether the additional

relationships detected in the factorgram are of biological significance, we analyzed the

factors for enrichment in GO biological process (GO-BP) annotations (see Methods). We

studied how many biologically significant groups were revealed in the factorgram

compared to other techniques for a fixed number of detected groups. In this case, parameter

θ controls the number of factors discovered in the data. We varied θ from 20 to 60 and

compared the genes in each factor with GO-BP annotations. Table 1 reports the number of

statistically significant factors (p < 0.05) for each θ-value; we also report corresponding

results for HAC. In general, the factogram reveals a larger number of significantly enriched

clusters.


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Table 1: Comparison of number of clusters or factors enriched for GO-

BP annotations for different numbers of extracted clusters/factors.

Number of clusters/factors, θ

Number of enriched factors in factorgram

Number of enriched clusters in clustergram

20 16 13 30 21 15 40 26 17 50 28 19 60 32 20

Just as Eisen et al. [1] demonstrated that the clustergram provides a visual tool enabling

biologists to gain leads to interesting associations, our goal in writing this paper was to

demonstrate the advantages of the factorgram visualization in revealing multi-way

associations and enabling biologists to detect multiple associations. Unlike clustergrams,

where the number of salient associations is limited by the two-dimensional arrangement of

the data, factorgrams extract many two-dimensional arrangements so as to identify multiple

associations. We are not advocating a specific computational technique for factorizing the

data matrix, but are instead introducing and advocating the factorgram as a way to visualize

the outputs from a variety of computational techniques [2-10], each of which is appropriate

in specific situations. Factorgrams can be used to visualize the output of these techniques,

regardless of whether each element in the generated factors belongs to only one factor or to

multiple factors. In our experiments on synthetic lethal yeast gene interaction data and

mouse gene expression data, we found that factorgrams reveal a larger number of

statistically significant and biologically significant clusters compared to the number

revealed in clustergrams by HAC.

As a general tool for analyzing matrices of data, the factorgram can potentially have

broad application in detecting associations in biological data. The factorgram has been

recently applied to chemical-genetic interaction data in yeast to identify associations

between chemical compounds and genes [17]. It has also been used to identify relationships

between hundreds of genes involved in transcription [18]. New large-scale microarray

datasets for the study of mammalian alternative splicing have also recently been analyzed


10

using factorgrams [19]. Software for automatically producing factorgrams is available at

http://www.psi.toronto.edu/factorgram. Due to its simplicity, ease of computation, and

potential for revealing novel biological insights, we believe the factorgram will prove to be

a useful tool for visualizing multi-way associations in high-throughput biological data.

Methods

1. Yeast gene deletion data analysis.

The clustergram was produced using the MATLAB implementation of hierarchical

agglomerative clustering (HAC). We used Hamming distance, Euclidean distance,

correlation and city-block pseudo-distance functions with average and single linkage and

report results for Hamming distance with average linkage, which obtained the largest

number of clusters enriched for GO annotations.

The factorgram was produced using ‘matrix tile analysis’ based on an iterated

conditional modes technique [10]. Matrix tile analysis takes as input a matrix, where each

element is the log-ratio of probabilities that the element belongs in a factor and does not

belong in a factor. If a synthetic lethal interaction was observed between two pairs of genes

in the dataset of Tong et al. [13], we set the log-ratio to be log((1-ε)/ε) and otherwise we set

it to be log(ε/(1-ε)), where ε is the noise probability set to 0.0335, based on the average

number of observed synthetic lethal interactions. The parameter θ, which indirectly

determines the false detection rate by weighting the benefit of explaining the data to the

cost of introducing additional factors, was set to 0.15.

2. Mouse gene expression data analysis.

The clustergram was produced using the MATLAB implementation of HAC with

Pearson correlation and average linkage, which was selected for visualizing results by

Zhang et al. [16].

For the factorgram, we tried a different factorization technique from above called

‘probabilistic sparse matrix factorization’ (PSMF) [8,9]. Each expression measurement of a


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gene in each tissue in the input matrix is represented by the arcsinh (approximately the

logarithm) of the ratio of normalized intensity of the gene in the given tissue to the gene’s

median normalized intensity across all 55 tissues. Observing that the majority of genes

expressed in any tissue were expressed in less than half of the tissues, Zhang et al. set ratios

less than one equal to one, reasoning that these ratios represent noise rather than down-

regulation thus the data is entirely non-negative.

PSMF discovers a large predetermined number of factors (set to 50 here) and then

prunes them until every factor has a standard deviation less than an input threshold, θ,

which was set to 9.5. To determine the expected number of false detections, we reran the

analysis 135 times on randomly permuted data and computed the expected number of false

detections, which was 0.5. Based on the permutation tests, we also estimate the distribution

of factor element intensities for non significant factors (as those expected to be found on

permuted data) and discard of any factor elements found in the unpermuted data which

have an intensity below that of the 97.5th percentile of permutated factor elements. For

visualization purposes, Fig. 4B shows only those genes with individual reconstruction error

of less than θ.

3. Gene ontology (GO) enrichment analysis

GO annotation labels for the yeast genes were retrieved from Saccharomyces Genome

Database at ftp://genome-ftp.stanford.edu/pub/yeast/data_download/literature_curation/.

The functional category labels for the genes with known biological function in the Zhang et

al. database were derived from Gene Ontology Biological Process (GO-BP) category labels

assigned to genes by the European Bioinformatics Institute and Mouse Genome

Informatics. Bonferroni-corrected p-values for each factor/cluster were computed using the

hyper-geometric distribution, testing the probability of observing by chance the overlap of

a subset of genes in each factor or cluster with a subset of genes sharing a specific GO

annotation. We assign p-values for each factor/cluster by comparing it with all GO

annotations and choosing the most significant p-values.


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Acknowledgements

We thank C Boone, M Escobar, J Greenblatt, N Krogan, QD Morris and S Roweis for

helpful conversations. This work was supported by NSERC and CIAR.

Correspondence and requests for materials should be addressed to Brendan Frey at

[email protected].

Figure Legends

Figure 1. (A) A clustergram of the 135 x 1023 yeast gene interaction data from Tong et al.

[13], where rows correspond to ‘query’ genes, columns correspond to ‘array’ genes and

each data element is white if a double knockout of the two genes is lethal. For visual

clarity, only columns with a non-negligible number of interactions are shown. (B)

Synthetic lethal interactions in two groups of data that we detected and are significantly

enriched for gene function annotations are coloured; the remaining interactions are shown

in grey. The clustergram cannot associate genes in multiple ways, so these groups are

broken apart. For example, the array genes (columns) corresponding to the data shown in

red all have synthetic lethal interactions with query genes (rows) GIM3, GIM4, GIM5,

PAC10 and YKE2 (the prefoldin complex), but this association is broken apart because

some of the array genes also have synthetic lethal interactions with query genes BIM1,

CTF4, KAR3 and CIN8, while others do not.

Figure 2. A cartoon illustration of one way a factorgram can be constructed. The

clustergram of a binary data matrix is shown in the upper-left corner. The small arrows

indicate rows and columns having multi-way associations not evident in the clustergram.

The factorgram is created by recursively reordering rows and columns to identify a block

of data and then extracting the block of data. Each block is called a ‘factor’ to differentiate

it from a ‘cluster’, because more than one factor can contain the same row or column, e.g.,

the red factor and the green factor both include rows 15-19. The factorgram is a

visualization of the raw data in the form of factors that may be placed in an expanded


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matrix (where multiple rows or columns can have the same label) or shown as a collection

of sub-matrices with labelled rows and columns.

Figure 3. (A) A factorgram of the yeast genetic interaction data from Fig. 1, where each

factor has been colour-coded and non-synthetic lethal interactions are shown in black. (B)

The continuation of Fig. 1B, where each observed synthetic lethal interaction has been

coloured according to the colour code from A. Almost all factors are broken into multiple

parts by the clustergram.

Figure 4. (A) A clustergram of the 22,709 gene x 55 tissue mouse microarray dataset from

Zhang et al. [16]. (B) Factorgram visualization with the factors placed in a diagonal

pattern. Two of the factors are enlarged in (C) and (D), and the general areas in the data set

from which they came are indicated. Each factor is composed of a subset of rows and

columns of the data, and only in certain cases do these subsets correspond to rows and

columns that are adjacent in the clustergram. When this is not the case, the clustergram has

broken apart the factor, as in (C).

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Figure 1B, Cheung et al


ECM15ERV29

KTR3LRE1

GS C2MNL1

MNS1YLR057W

CTS1HST1

RA S2YUR1

RAD24DIE2ALG6

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MUS81HST3MID2LIA1

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BB C1YB R094W


AS E1KA R9CIN1

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Factorgram

(i) (ii) (iii) (iv) (v)

(vi)

(vii)(viii)(x) (ix)

Regroup toIdentify Block




ExtractFactor

ExtractFactor

ExtractFactor

ExtractFactor

Clustergram

Start

Figure 2, Cheung et al

Figure 3A, Cheung et al

CC

H1

MID

2 P

FA

4 Y

GL0

46W

R

LM1

DF

G16

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2 U

BC

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PA

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8 BI

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BFA1CIN4

RBL2KIP2

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1 T

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9 E

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167

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IL

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SA

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SH

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CC

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MN

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Figure 4, Cheung et al

Bra

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Documents

Factorgrams: A tool for visualizing multi-way associations in