Missing value imputation improves clustering and interpretation of gene expression microarray data.
ABSTRACT Missing values frequently pose problems in gene expression microarray experiments as they can hinder downstream analysis of the datasets. While several missing value imputation approaches are available to the microarray users and new ones are constantly being developed, there is no general consensus on how to choose between the different methods since their performance seems to vary drastically depending on the dataset being used.
We show that this discrepancy can mostly be attributed to the way in which imputation methods have traditionally been developed and evaluated. By comparing a number of advanced imputation methods on recent microarray datasets, we show that even when there are marked differences in the measurementlevel imputation accuracies across the datasets, these differences become negligible when the methods are evaluated in terms of how well they can reproduce the original gene clusters or their biological interpretations. Regardless of the evaluation approach, however, imputation always gave better results than ignoring missing data points or replacing them with zeros or average values, emphasizing the continued importance of using more advanced imputation methods.
The results demonstrate that, while missing values are still severely complicating microarray data analysis, their impact on the discovery of biologically meaningful gene groups can  up to a certain degree  be reduced by using readily available and relatively fast imputation methods, such as the Bayesian Principal Components Algorithm (BPCA).

Conference Paper: Evaluation of missing values imputation methods in cDNA microarrays based on classification accuracy
[Show abstract] [Hide abstract]
ABSTRACT: Many attempts have been carried out to deal with missing values (MV) in microarrays data representing gene expressions. This is a problematic issue as many data analysis techniques are not robust to missing data. Most of the MV imputation methods currently being used have been evaluated only in terms of the similarity between the original and imputed data. While imputed expression values themselves are not interesting, rather whether or not the imputed expression values are reliable to use in subsequent analysis is the major concern. This paper focuses on studying the impact of different MV imputation methods on the classification accuracy. The experimental work was first subjected to implementing three popular imputation methods, namely Singular Value Decomposition (SVD), weighted Knearest neighbors (KNNimpute), and Zero replacement. The robustness of the three methods to the amount of missing data was then studied. The experiments were repeated for datasets with different missing rates (MR) over the range of 020% MR. In applying supervised two class classification we adopted a twofold approach, introducing all genes expressions to the classifiers as well as a subset of selected genes. The feature selection method used for gene selection is Fisher Discriminate Analysis (FDA), which improved noticeably the performance of the classifiers. The retained classifiers accuracies using imputed data after applying the three proposed imputation methods show slight variations over the specified range of MR. Thus, assessing that the three imputation methods in concern are robust.Biomedical Engineering (MECBME), 2011 1st Middle East Conference on; 01/2011  SourceAvailable from: Massimo Zollo
Dataset: 2169239461SP
Marianeve Carotenuto, Pasqualino De Antonellis, Lucia Liguori, Giovanna Benvenuto, Daniela Magliulo, Alessandro Alonzi, Cecilia Turino, Carmela Attanasio, Valentina Damiani, Anna Maria Bello, [......], Luigi Terracciano, Antonella Federico, Alfredo Fusco, Jamie Freeman, Trevor C Dale, Charles Decraene, Gennaro Chiappetta, Francovito Piantedosi, Cecilia Calabrese, Massimo Zollo  SourceAvailable from: Kohbalan Moorthy[Show abstract] [Hide abstract]
ABSTRACT: Many bioinformatics analytical tools, especially for cancer classification and prediction, require complete sets of data matrix. Having missing values in gene expression studies significantly influences the interpretation of final data. However, to most analysts' dismay, this has become a common problem and thus, relevant missing value imputation algorithms have to be developed and/or refined to address this matter. This paper intends to present a review of preferred and available missing value imputation methods for the analysis and imputation of missing values in gene expression data. Focus is placed on the abilities of algorithms in performing local or global data correlation to estimate the missing values. Approaches of the algorithms mentioned have been categorized into global approach, local approach, hybrid approach, and knowledge assisted approach. The methods presented are accompanied with suitable performance evaluation. The aim of this review is to highlight possible improvements on existing research techniques, rather than recommending new algorithms with the same functional aim.Current Bioinformatics. 01/2014; 9(1):1822.
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BMC Bioinformatics
Open Access
Research article
Missing value imputation improves clustering and interpretation of
gene expression microarray data
Johannes Tuikkala*1, Laura L Elo2,3, Olli S Nevalainen1 and
Tero Aittokallio2,3,4
Address: 1Department of Information Technology and TUCS, University of Turku, FI20014 Turku, Finland, 2Department of Mathematics,
University of Turku, FI20014 Turku, Finland, 3Turku Centre for Biotechnology, PO Box 123, FI20521 Turku, Finland and 4Systems Biology Unit,
Institut Pasteur, FR75724 Paris, France
Email: Johannes Tuikkala*  jotatu@utu.fi; Laura L Elo  laliel@utu.fi; Olli S Nevalainen  olli.nevalainen@utu.fi;
Tero Aittokallio  teanai@utu.fi
* Corresponding author
Abstract
Background: Missing values frequently pose problems in gene expression microarray
experiments as they can hinder downstream analysis of the datasets. While several missing value
imputation approaches are available to the microarray users and new ones are constantly being
developed, there is no general consensus on how to choose between the different methods since
their performance seems to vary drastically depending on the dataset being used.
Results: We show that this discrepancy can mostly be attributed to the way in which imputation
methods have traditionally been developed and evaluated. By comparing a number of advanced
imputation methods on recent microarray datasets, we show that even when there are marked
differences in the measurementlevel imputation accuracies across the datasets, these differences
become negligible when the methods are evaluated in terms of how well they can reproduce the
original gene clusters or their biological interpretations. Regardless of the evaluation approach,
however, imputation always gave better results than ignoring missing data points or replacing them
with zeros or average values, emphasizing the continued importance of using more advanced
imputation methods.
Conclusion: The results demonstrate that, while missing values are still severely complicating
microarray data analysis, their impact on the discovery of biologically meaningful gene groups can
– up to a certain degree – be reduced by using readily available and relatively fast imputation
methods, such as the Bayesian Principal Components Algorithm (BPCA).
Background
During the past decade, microarray technology has
become a major tool in functional genomics and biomed
ical research. It has been successfully used, for example, in
genomewide gene expression profiling [1], tumor classi
fication [2], and construction of gene regulatory networks
[3]. Gene expression data analysts currently have a wide
range of computational tools available to them. Cluster
analysis is typically one of the first exploratory tools used
on a new gene expression microarray dataset [4]. It allows
researchers to find natural groups of genes without any a
priori information, providing computational predictions
Published: 18 April 2008
BMC Bioinformatics 2008, 9:202doi:10.1186/147121059202
Received: 6 August 2007
Accepted: 18 April 2008
This article is available from: http://www.biomedcentral.com/14712105/9/202
© 2008 Tuikkala et al; licensee BioMed Central Ltd.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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and hypotheses about functional roles of unknown genes
for subsequent experimental testing. Clusters of genes are
often given a biological interpretation using the Gene
Ontology (GO) annotations which are significantly
enriched for the genes in a given cluster. The success of
such an analytical approach, however, heavily depends on
the quality of the microarray data being analyzed.
Although gene expression microarrays have developed
much during the past years, the technology is still rather
error prone, resulting in datasets with compromised accu
racy and coverage. In particular, the existence of missing
values due to various experimental factors still remains a
frequent problem especially in cDNA microarray experi
ments. If a complete dataset is required, as is the case for
most clustering tools, data analysts typically have three
options before carrying out analysis on the data: they can
either discard the genes (or arrays) that contain missing
data, replace missing data values with some constant (e.g.
zero), or estimate (i.e. impute) values of missing data
entries. Many imputation methods are available that uti
lize the information present in the nonmissing part of the
dataset. Such methods include, for example, the weighted
knearest neighbour approach [5] and the local least
squares imputation [6]. Alternatively, external informa
tion in the form of biological constraints [7], GO annota
tions [8] or additional microarray datasets [9], can be used
to improve the accuracy of these traditional methods, pro
vided relevant information is available for the given
experimental system or study organism.
While most of the imputation algorithms currently being
used have been evaluated only in terms of the similarity
between the original and imputed data points, we argue
that the success of preprocessing methods should ideally
be evaluated also in other terms, for example, based on
clustering results and their biological interpretation, that
are of more practical importance for the biologist. The
motivation is that, even though there are substantial dif
ferences in the imputation accuracy between the methods
at the measurement level, some of these differences may
be biologically insignificant or simply originate from
measurement inaccuracies, unless they can also be
observed at the next step of the data analysis. Moreover,
the imputation methods have conventionally been devel
oped and validated under the assumption that missing
values occur completely at random. This assumption does
not always hold in practise since the multiple experimen
tal measurements (arrays) may involve variable technical
and/or experimental conditions, such as differences in
hybridization, media or time. Accordingly, the distribu
tion of missing entries in many microarray experiments is
highly nonrandom, which may have resulted conclu
sions regarding the relative merits of the different imputa
tion methods being drawn too hastily.
In this paper, we systematically evaluate a number of
imputation strategies based on their ability to produce
complete datasets for the needs of partitional clustering.
We compare seven imputation algorithms for which
readytouse implementation is freely available and easily
accessible to the microarray community. Beyond the
imputation accuracy on the measurement level, we evalu
ate the methods in terms of their ability to reproduce the
gene partitions and the significant GO terms discovered
from the original datasets. The imputation methods are
compared on eight real microarray datasets, consisting of
microarray designs often encountered in practice, such as
time series and steady state experiments. The diversity of
these datasets allows us to investigate the effects on the
imputation results of various data properties, such as the
number of measurements per gene, missing value rate and
distribution, as well as their correlation structure. Some
recommendations for the microarray data analysts on the
use of imputation methods in different situations are
given in the discussion.
Table 1: Datasets.
Name
NMMC
MF
MVSD
MV
Type PC1
Brauer05
Ronen05
Spahira04A
Spahira04B
Hirao03
Yoshimoto02
Wyrick99
Spellman98E
19
26
23
14
8
24
7
14
6256
7070
4771
4771
6229
6102
6180
6075
3924
4916
2970
3340
5913
4379
6169
5766
3066
2695
2090
2898
259
2323
3600
1094
4.0
3.2
3.9
4.2
0.7
1.9
0.0
0.4
6.7%
3.8%
2.7%
3.0%
0.9%
3.2%
0.0%
0.4%
MT
MT
TS
TS
SS
MT
TS
TS
54.9%
51.1%
62.0%
54.1%
43.3%
64.7%
61.3%
39.9%
N is the number of measurements (columns in the observation matrix), M is the number of genes (rows), MC is the number of genes without
missing values in the complete dataset, MF is the number of genes after applying the filtering, MVSD is the standard deviation of the missing value
distribution over the measurements, MV is the percentage of missing values, Type indicates whether the dataset is a time series (TS), steady state
(SS), or mixed type (MT), i.e., multiple time series measured under different experimental conditions, and PC1 gives the proportion of total
variance explained by the first principal component (high value indicates high correlation structure between the genes [31]).
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Results
We tested imputation algorithms on eight different cDNA
datasets (see Table 1). Theses datasets were first preproc
essed and filtered (see Methods), and then transformed to
complete datasets by removing the genes (rows of the
dataset) which contained at least one missing value (see
Figure 1). Missing values were then generated for these
complete datasets using the missing value distribution
estimated from the original data. Multiple missing values
were allowed in a gene at different measurement points.
Different rates of missing values were used, namely 0.5%,
1%, 5%, 10%, 15% and 20%. Missing value generation
was repeated 30 times for each dataset and each missing
value percentage, thus yielding a total of 1440 different
datasets with missing values for the comparison (Figure
1).
Seven imputation algorithms, namely ZERO, RAVG, kNN,
BPCA, LLS, iLLS, and SVR algorithms (see Table 2), were
applied to each of the missing value datasets (see Meth
ods), thus yielding a total of 10080 different imputed
datasets. The conventional normalized root mean squared
error (NRMSE) was used to evaluate the similarity
between the original and imputed data points (see Eq. 6
A schematic illustration of the comparison procedure
Figure 1
A schematic illustration of the comparison procedure. The testing procedure was repeated for each of the eight data
sets (see Table 1) and for each of the missing value rates (q = 0.5%, 1%, 5%, 10%, 15%, and 20%). The different imputation
methods (see Table 2) were evaluated in terms of their capability to reproduce the original data values (NRMSE), clustering
solutions (ADBP for gene clusters) and their biological interpretations (ADBP for GO terms). Clustering analyses were per
formed also directly on datasets with missing values (Nimp). Missing values were generated 30 times and the kmeans cluster
ing was performed for multiple numbers of clusters (k = 2, 3,..., 10).
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in Methods). Then clustering was performed on the
imputed datasets using the kmeans algorithm, for k = 2,
3,..., 10, and the clustering results after imputation were
compared to those obtained originally on the complete
datasets using the Average Distance Between Partition
(ADBP) error (Eq. 3). The results were summarized over
all k values using the average normalized ADBP error (Eq.
4; the results for each k are detailed in Additional Files 3
and 4). As a reference, the missing value datasets were also
clustered prior to any imputation (Nimp) using the aver
age Euclidean distance (see Eq. 2). The same evaluation
approach was also used for the significantly enriched GO
terms discovered from the clustering results. In this case,
the GO terms were used in place of the genes in the ADBP
measure. The critical pvalue below which a GO term was
considered significant was adjusted so that the number of
GO terms per cluster was between 1 and 20 (threshold of
0.01 for Brauer05 and Ronen05 datasets and 0.05 for all
other datasets). To study the effect of different missing
value mechanisms, we also generated missing values in
the Spahira04B dataset completely at random and com
pared the resulting NRMSE and ADPB error rates to those
obtained when the true columnwise missing value pro
portions were preserved. Spahira04B was used as an
example dataset in this comparison because of its most
nonrandom missing value distribution (see Table 1).
Running times of the different imputation methods in an
example dataset (Ronen05) are presented in Table 2 for
the missing value rate of 10%. The Bayesian inference
based BPCA imputation was the fastest among the more
advanced imputation methods (SVR, LLS, iLLS, and
BPCA). However, the simple kNN was still about 10 times
faster than BPCA. The running time of the original LLS
method was about one third of that of its iterative version
iLLS. The support vectorbased SVR and the iterative iLLS
algorithms were clearly the slowest imputation methods.
The iLLS method produced, on rare occasion, estimates
for missing values which were up to ten times larger than
the original values. This seems to suggest an anomaly in
the system's implementation or operation.
Agreement with the original data values
Imputation accuracies in terms of the NRMSEs of the dif
ferent imputation methods are shown in Figure 2. With
each method and dataset, the imputation accuracy
decreased with the increasing missing value rate. By
default, ZERO imputation always has an NRMSE of one.
As expected, the worst imputation accuracies were
obtained with the simple ZERO and RAVG imputations.
RAVG was better than ZERO imputation in all datasets
except in Spellman98E, where the global correlation
between genes was relatively weak (see Table 1). The next
worst performing method on the measurement level was
kNN. This method seemed to benefit somewhat from a
strong global correlation structure, as is present in the
Spahira04A and Yoshimoto02 datasets. kNN performed
at its worst when there was weak correlation between the
genes, i.e. in the Spellman98E and Hirao03 datasets.
The imputation accuracy of the more advanced imputa
tion methods (LLS, iLLS, BPCA, and SVR) depended heav
ily on the properties of the dataset being imputed, and
therefore it was difficult to find a clear winner among
these methods. However, if we count the number of times
that a method was best or second best across all of the
datasets, then iLLS and BPCA were the two best methods
when assessed in terms of the NRMSE imputation accu
racy. A limitation of the iLLS method is that its implemen
tation produces, in some cases, inconsistent results which
led to a large variation among the replicate datasets (see
e.g. Hirao03 and Ronen05 in Figure 2). As the BPCA
imputation was relatively fast and provided robust esti
mates over a wide range of different situations, it could be
recommended in cases where performance is measured
solely by imputation accuracy.
Agreement with the original clustering results
The imputation algorithms were next compared by meas
uring how well the clustering of the original complete
dataset was preserved when clustering the imputed data
sets with different levels of missing values (see Figure 3).
Again, the missing values had a serious effect on the abil
Table 2: Imputation methods. The running times were calculated for the Ronen05 dataset with 10% of missing values. (Intel C2D
T7200@2 GHz with 2 GB RAM was used).
Imputation methodImplementationRunning time ReferenceURL
Support Vector Regression (SVR)
Iterated Local Least Squares (iLLS)
Local Least Squares (LLS)
Bayesian Principal Component Algorithm (BPCA)
k Nearest Neighbor (KNN)
Row Average (RAVG)
Zero imputation (ZERO)
C++
Matlab
Matlab
Matlab
C++
Matlab
Matlab
940 s
938 s
334 s
197 s
16 s
< 1 s
< 1 s
[17]
[16]
[6]
[18]
[5]
[6]
[5]
[32]
[33]
[34]
[35]
[36]
[34]
[37]
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Agreement with the original data values
Figure 2
Agreement with the original data values. Capability of the imputation methods to reproduce the original measurements
in the datasets. The imputation accuracy is represented as the normalized root mean squared error (NRMSE) and the error
bars are the standard error of mean (SEM) values over the 30 replicate missing datasets.
0.5
0.6
0.7
0.8
0.9
1
1.1
0.51510 15 20
NRMS Error
Percentage of missing values
Ronen05
A
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
0.51510 1520
NRMS Error
Percentage of missing values
Wyrick99
B
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0.515 101520
NRMS Error
Percentage of missing values
Spahira04A
C
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
0.515 1015 20
NRMS Error
Percentage of missing values
Spahira04B
D
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0.515 101520
NRMS Error
Percentage of missing values
Spellman98E
E
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
0.515101520
NRMS Error
Percentage of missing values
Hirao03
F
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0.5151015 20
NRMS Error
Percentage of missing values
Brauer05
G
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0.515 101520
NRMS Error
Percentage of missing values
Yoshimoto02
H
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
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Agreement with the original clusterings
Figure 3
Agreement with the original clusterings. Capability of the kmeans algorithm to reproduce the original gene clusters in
the datasets with or without imputation. The results are presented as the average distance between partitions (ADBP) of genes
and the error bars are the standard error of mean (SEM) values over the number of clusters (k = 2, 3,..., 10).
0
0.2
0.4
0.6
0.8
1
0.515101520
ADBP error for genes
Percentage of missing values
Ronen05
A
Nimp
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0
0.2
0.4
0.6
0.8
1
0.515101520
ADBP error for genes
Percentage of missing values
Wyrick99
B
Nimp
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0
0.2
0.4
0.6
0.8
1
0.515101520
ADBP error for genes
Percentage of missing values
Spahira04A
C
Nimp
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0
0.2
0.4
0.6
0.8
1
0.515 101520
ADBP error for genes
Percentage of missing values
Spahira04B
D
Nimp
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0
0.2
0.4
0.6
0.8
1
0.51510 1520
ADBP error for genes
Percentage of missing values
Spellman98E
E
Nimp
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0
0.2
0.4
0.6
0.8
1
0.5 151015 20
ADBP error for genes
Percentage of missing values
Hirao03
F
Nimp
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0
0.2
0.4
0.6
0.8
1
0.515 101520
ADBP error for genes
Percentage of missing values
Brauer05
G
Nimp
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0
0.2
0.4
0.6
0.8
1
0.515 1015 20
ADBP error for genes
Percentage of missing values
Yoshimoto02
H
Nimp
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
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ity of the kmeans algorithm to reveal the original parti
tioning of the different datasets as assessed with the ADBP
error. It turned out that even a small proportion (0.5%) of
missing values had a noticeable impact on the clustering
results. If the proportion of missing values was 5% or
more, then the disagreement between the original and the
reproduced clustering became relatively high in many of
the datasets. In each case, however, any type of imputa
tion was a significantly better than leaving the data with
out imputation (the Nimp trace). This result indicates that
valuable information is lost when using the average Eucli
dean distance alone.
The most striking observation was that the clear differ
ences observed in the NRMSE between the different impu
tation methods were barely visible in the clustering
results. While, as expected, the ZERO and RAVG imputa
tion methods were still consistently the two poorest meth
ods also when assessed in terms of the ADBP error, the
differences between the more advanced methods became
almost negligible. This was true also for the simpler kNN
method, except for datasets with relatively low correlation
structure (Hirao03 and Spellman98E). The surprisingly
consistent ability to identify the original partitions across
all of the datasets was also supported by the results of Fig
ure 4, in which the ADBP measure was used to compare
the significantly enriched GO terms revealed from the
original and imputed datasets. This result suggests that the
biological interpretation of the clustering results can – up
to a certain missing value rate – be preserved provided
that any of the advanced imputation methods is used.
Effects of random missing value generation and cluster
initialization
The effect of generating missing values completely at ran
dom is highlighted in Figure 5 using an example dataset
(Spahira04B). By comparing the imputation results in Fig
ure 5A (uniform distribution) to those in Figure 2D (true
distribution), one can observe that the simplistic random
missing value model tends to increase the NRMSE levels
of the imputation methods, especially at the low missing
value rates. At higher missing value rates, the imputation
accuracies of LLS and especially that of iLLS seemed to
benefit from the uniform missing value distribution.
Under the uniform model, these two methods appeared
to be comparable to BPCA, which was found as the best
method under the true missing value model estimated
directly from the particular dataset. Like before, the differ
ences became negligible when the imputation results were
evaluated with respect to the reproducibility of the origi
nal clusterings (Figure 5B) or their biological interpreta
tion (data not shown) in terms of the ADBP error rate.
These results show that the mechanism by which the miss
ing values are generated can have a marked effect on the
imputation results. In particular, if NRMSE is solely used
as an evaluation criterion, then one has to pay particular
attention to the missing value distribution of the dataset.
Further, we studied whether the negligible differences in
the ADBP error rates between the advanced imputation
methods could be due to instability of the kmeans clus
tering. To test the impact of cluster initialization, we
repeated clustering on the same example dataset
(Spahira04B) using 10 random cluster initializations (k =
5). The average ADBP values from the 10 different initial
izations shown in Figures 6A and 6B were relatively simi
lar to those of the KKZ initialization in Figures 3D and 4D,
respectively, suggesting that the observed differences
between the imputation methods were not solely due to
the cluster initialization. The supplementary figures origi
nating from the individual initializations (Additional
Files 1 and 2) demonstrate that while the different initial
izations may lead to different ADBP errors, the relative
order of the imputation methods remained effectively the
same. The same was also true when investigating the cor
responding ADBP results in the datasets separately for
each number of clusters (Additional Files 3 and 4). Taken
together, these results suggest that other determinants
related to the clustering analysis, including the initializa
tion and number of clusters, can have substantially more
impact on the ADBP errors than the selection of the impu
tation method.
Discussion
We have carried out a practical comparison of recent miss
ing value imputation methods by following the typical
microarray data analysis work flow: the given dataset is
first clustered and then the resulting gene groups are inter
preted in terms of their enriched GO annotations. Our key
finding was that clear and consistent differences can be
found between the imputation methods when the impu
tation accuracy is evaluated at the measurement level
using the NRMSE; however, similar differences were not
found when the success of imputation was assessed in
terms of ability to reproduce original clustering solutions
or biological interpretations using the ADBP error on the
gene groups or GO terms, respectively. The observed
dependence of the NRMSE on the properties of a dataset
can seriously bias the evaluation of imputation algo
rithms. In fact, it enables one to select a dataset that
favours one's own imputation method. Another potential
pitfall lies with the missing value distribution. If the
imputation algorithm uses the assumption of completely
random missing values, for example in parameter estima
tion, when in fact this is not case, it may lead to subopti
mal imputation results.
Regardless of the evaluation approach used, our results
strongly support earlier observations that imputation is
always preferable to ignoring the missing values or replac
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Agreement with the original GO terms
Figure 4
Agreement with the original GO terms. Capability of the kmeans algorithm to identify the original GO terms of the clus
ters in the datasets with or without imputation. The results are presented as the average distance between partitions (ADBP)
of terms and the error bars are the standard error of mean (SEM) values over the number of clusters (k = 2, 3,..., 10).
0
0.2
0.4
0.6
0.8
1
0.515 1015 20
ADBP error for GO terms
Percentage of missing values
Ronen05
A
Nimp
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0
0.2
0.4
0.6
0.8
1
0.515 10 1520
ADBP error for GO terms
Percentage of missing values
Wyrick99
B
Nimp
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0
0.2
0.4
0.6
0.8
1
0.515 101520
ADBP error for GO terms
Percentage of missing values
Spahira04A
C
Nimp
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0
0.2
0.4
0.6
0.8
1
0.515 101520
ADBP error for GO terms
Percentage of missing values
Spahira04B
D
Nimp
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0
0.2
0.4
0.6
0.8
1
0.5151015 20
ADBP error for GO terms
Percentage of missing values
Spellman98E
E
Nimp
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0
0.2
0.4
0.6
0.8
1
0.5 15 1015 20
ADBP error for GO terms
Percentage of missing values
Hirao03
F
Nimp
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0
0.2
0.4
0.6
0.8
1
0.51510 1520
ADBP error for GO terms
Percentage of missing values
Brauer05
G
Nimp
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0
0.2
0.4
0.6
0.8
1
0.515101520
ADBP error for GO terms
Percentage of missing values
Yoshimoto02
H
Nimp
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
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ing them with zeros or average values. In addition to the
NRMSE validation, Jörnsten et al. examined the effect of
imputation on the significance analysis of differental
expression; they observed that missing values affect the
detection of differentially expressed genes and that the
more sophisticated methods, such BPCA and LinCmb, are
notably better than the simple RAVG or kNN imputation
methods [10]. Scheel et al. also studied the influence of
missing value imputation on the detection of differen
tially expressed genes from microarray data; they showed
that the kNN imputation can lead to a greater loss of dif
ferentially expressed genes than if their LinImp method is
used, and that intensitydependent missing values have a
more severe effect on the downstream analysis than miss
ing values occurring completely at random [11]. Wang et
al. studied the impact of imputation on the related down
stream analysis, disease classification; they discovered
that while the ZERO imputation resulted in poor classifi
cation accuracy, the kNN, LLS and BPCA imputation
methods only varied slightly in terms of classification per
formance [12]. Shi et al. also studied the effect of missing
value imputation on classification accuracy; they discov
ered that the BPCA and iLLS imputation methods could
estimate the missing values to the classification accuracy
achieved on the original complete data [13].
To our knowledge, only one other study has investigated
the effect of missing values and their imputation on the
preservation of clustering solutions. Brevern et al. concen
trated on hierarchical clustering and the kNN imputation
method only and did not consider biological interpreta
tions of the clustering results; their main findings were
that even a small amount of missing values may dramati
cally decrease the stability of hierarchical clustering algo
rithms and that the kNN imputation learly improves this
stability [14]. Our results are in good agreement with
these findings. However, our specific aim was to investi
gate the effect of missing values on the partitional cluster
ing algorithms, such as kmeans, and to find out whether
more advanced imputation methods, such as LLS, SVR
and BPCA, can provide better clustering solutions than
the traditional kNN approach. Our results suggest that
BPCA provides fast, robust and accurate results, especially
when the missing value rate is lower than 5%. None of the
imputation methods could reasonably correct for the
influence of missing values above this 5% threshold. In
these cases, one should consider removing the genes with
many missing values or repeating the experiments if pos
sible.
A number of different clustering algorithms have been
proposed for the exploratory investigation of gene expres
Effect of completely random missing values
Figure 5
Effect of completely random missing values. Summary of the results when generating missing values completely at ran
dom. (A) Agreement with the original data values (for details see the legend of Fig. 2). (B) Agreement with the original cluster
ings (see the legend of Fig. 3).
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
0.515101520
NRMS Error
Percentage of missing values
Spahira04B
A
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0
0.2
0.4
0.6
0.8
1
0.515 10 1520
ADBP error for genes
Percentage of missing values
Spahira04B
B
Nimp
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
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sion microarray data [4]. We chose the kmeans algorithm
since it is very fast, unlike e.g. hierarchical clustering, is
widely used among biologists, and produces results which
are relatively straightforward to interpret. The fast running
time of kmeans was especially important because of the
large number of clusterings performed. Recent compara
tive studies have also demonstrated that partitional clus
tering methods often produce more meaningful clustering
solutions than hierarchical clustering methods [4]. To
avoid using random starting points in kmeans clustering,
we used the KKZ initialization since it has been found to
perform better than many other methods [15]. Due to the
extensive computational requirements, we could not
afford to do many random cluster initializations. In future
studies, it would be interesting to compare more sophisti
cated clustering methods, such as fuzzy cmeans, together
with alternative approaches for selecting good initializa
tions and appropriate numbers of clusters for each data
set.
Conclusion
Missing values have remained a frequent problem in
microarray experiments and their adverse effect on the
clustering is beyond the capacity of simple imputation
methods, like ignoring the missing data points or replac
ing them with zeros or average values. Biological interpre
tation of the gene clusters can, to a certain missing value
rate, be preserved by using readily available and relatively
fast imputation methods, such as the Bayesian Principal
Components Algorithm (BPCA).
Methods
Imputation methods
Relative expression data produced by cDNA microarrays
can be represented as an M × N matrix , in
which the entries of G are the log2 expression ratios for M
N
g
=
=
()
1
genes and N measurements (i.e. the rows
represent the genes and the columns stand for the differ
ent time points or conditions). In order to estimate a miss
ing value in G, one typically first selects k genes that are
somehow similar to the gene gi with the missing entry. A
number of statistical techniques are available to estimate
the missing value on the basis of such neighbouring
genes. In one of the earliest methods, known as the
weighted kNearest Neighbour (kNN) imputation, the
estimate is calculated as a weighted average over the cor
responding values of the neighbouring genes [5]. The
average Euclidean distance (Eq. 2) is typically applied
both in the selection and weighting of the neighbours,
G =
=
(),
,
gij i j
M N
1
giij j
Effect of random initialization of kmeans
Figure 6
Effect of random initialization of kmeans. Summary of the results when initializing the kmeans clustering from 10 ran
dom initializations (k = 5). (A) Agreement with the original clusters (see the legend of Fig. 3). (B) Agreement with the original
GO terms (see the legend of Fig. 4). The error bars represent the standard error of mean (SEM) values over the 10 random
cluster initializations.
0
0.2
0.4
0.6
0.8
1
0.515101520
ADBP error for genes
Percentage of missing values
Spahira04B
A
Nimp
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
0
0.2
0.4
0.6
0.8
1
0.515 101520
ADBP error for GO terms
Percentage of missing values
Spahira04B
B
Nimp
ZERO
RAVG
KNN
BPCA
LLS
iLLS
SVR
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although in principle any other dissimilarity measure
between the genes could be used instead. In the compari
sons between different missing value estimation methods,
we used as references two simple imputation strategies: In
Zero Imputation (ZERO), missing values are always
replaced by a zero value, whereas Row Average (RAVG)
replaces missing values by the average value of the non
missing values of the particular gene. These simple
approaches are also being used as preprocessing steps in
the more advanced imputation algorithms detailed
below.
The Local Least Squares (LLS) imputation is a regression
based estimation method that takes into account the local
correlation structure of the data [6]. The algorithm can
also automatically select the appropriate number of
neighbouring genes. Its modified version, the iterated
Local Least Squares (iLLS) imputation, uses iteratively
data from each imputation step as an input for the next
iteration step [16]. Both the LLS and iLLS methods use
RAVG imputation of missing entries prior to the primary
missing value estimation if the proportion of missing val
ues is deemed too high to construct a reliable complete
data matrix. In the Support Vector Regression (SVR)
imputation, the neighbouring genes are first selected
using a specific kernel function, the socalled radial basis
function, and an estimate of the missing value is calcu
lated using a relatively timeconsuming quadratic optimi
zation [17]. SVR uses the ZERO imputation of missing
entries prior to the primary imputation but it also codes
the entries so that the relative importance of ZERO
imputed entries is attenuated when the SVR imputation
has been completed for the other missing entries. Among
these more advanced imputation methods, the Bayesian
Principal Components Algorithm (BPCA) is appealing
because, although it involves Bayesian estimation
together with the iterative expectation maximization algo
rithm, its application is relatively fast and straightforward
due to the fact that it does not contain any adjustable
parameters if default settings are used [18]. In the BPCA
method, all of the missing entries are first imputed with
the RAVG imputation to obtain a complete data matrix.
All of the imputation methods included in the compari
sons are freely available on the internet (the links to the
websites are listed in Table 2). BPCA, LLS, iLLS, ZERO and
RAVG are implemented with Matlab, whereas SVR and
kNN are implemented with C++. The kNN imputation
method was run with default parameter (k = 10), and SVR
with default parameters (c = 8, g = 0.125, r = 0.01), since
their optimization is not trivial and could have biased the
results. The automatic parameter estimator was used with
LLS and iLLS. The other methods do not contain any free
parameters.
Clustering method
We used the kmeans algorithm, which is one of the most
popular partitional clustering algorithms applied to gene
expression data [4]. The basic idea of the algorithm is to
iteratively find representative cluster centroids for the k
clusters:
1. The initial k centroids are chosen either randomly or
using a given initialization rule.
2. Each gene is assigned to the cluster whose centroid is
closest to its expression profile.
3. Centroids are recalculated as arithmetic means of the
genes belonging to the clusters.
Steps 2 and 3 of the algorithm are iterated until the clus
ters become stable or until a given number of steps have
been performed. As the standard kmeans algorithm does
not include any rules for the selection of the initial centro
ids, we used the socalled KKZ (named after the Katsavou
nidis, Kuo, and Zhang) initialization algorithm [19]. The
KKZ method selects the first centroid C1 as the gene with
maximal Euclidean norm, i.e. C1 ≡ argmaxgi . Each of
the subsequent centroids Cj , for j = 2,..., k, is iteratively
selected as the gene whose distance to the closest centroid
is maximal among the remaining genes. The KKZ method
has turned out to be among the best initialization rules
developed for the partitional clustering algorithms [15].
In addition to the KKZ initialization, we also tested ran
dom starting points in the kmeans clustering (these
results are provided in Additional Files 1 and 2). The k
means algorithm and KKZ initialization were imple
mented with the Java programming language.
Dissimilarity measures
The Euclidean distance and Pearson correlation are per
haps the two most commonly used dissimilarity measures
in microarray data clustering and analysis. The Euclidean
distance:
is the simple L2 norm on vectors and it measures the abso
lute squared difference between two gene profiles gi and gj
. Since we use standardized gene expression profiles, the
Euclidean distance and Pearson correlation yield identical
clustering results.
When there are missing values present in the dataset G,
one has to modify the distance measure. A typical way is
to use the average Euclidean distance:
d i j
( , )
gg
iljl
l
N
()
=−
=∑
2
1
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where δ(gil , gjl ) equals one if values gil and gjl are present
both in genes i and j, and zero otherwise. A is the number
of cases where there is a missing value in either of the
∑
1
genes, i.e., .
The average Euclidean distance is also used in the selec
tion of the neighbouring genes in the kNN imputation.
The problem with this modified measure is its significant
loss of information when the number of missing values
increases. In the worst case when, for a given gene pair (i,
j), N equals A (i.e. there is a missing value in either or both
genes at every position), it is not be possible to calculate
the distance, even though half of the values of each gene
may still be present. In this case, we define .
Validation methods
Cluster validation methods that compare clustering
results with respect to a known set of class labels (the 'gold
standard') are called external measures [20]. To compare
the different imputation strategies, a measure was needed
for quantifying the agreement between the partitioning of
the imputed data and the partitioning of the original data.
We chose the Average Distance Between Partitions
(ADBP) measure because it matches optimally each clus
ter in the partitioning of the imputed data with its best
pair in the partitioning of original data, before measuring
the dissimilarity between the two partitionings, and hence
goes beyond calculating any overall statistic for the over
lap only. The optimal match was determined between the
clusters of the two partitions U and V using the Hungarian
method [21], which finds the k cluster pairs
k
∈∈
=1
such that the average distance
between the cluster pairs is minimized. The distance
between a cluster pair was calculated as the normalized
Hamming distance:
Here, n(uj ) and n(vj ) are the cardinalities of the clusters uj
and vj , respectively, and I{g ∉ uj } equals one, if the gene
g does not belong to the cluster uj , and zero otherwise
[22]. The final ADBP error was obtained as the normal
ized average
The ADBP error D(U, V) is zero when the two partitions
are identical and one when the partitions are totally differ
ent, thus making it easy to interpret the results and make
comparisons between datasets. Once clustering has been
performed, a biologist typically looks for the GO terms
that are significantly enriched among the genes of each
cluster. These terms can be used to characterize the func
tional roles of genes and thus to interpret the results of the
microarray experiment. Enriched GO terms for a given
cluster were found by calculating the socalled pvalue for
each GO term and then selecting the terms whose pvalues
were lower than a given threshold [23]. This enrichment
pvalue for each GO term t and for each cluster was calcu
lated using the hypergeometric distribution
Here, b is the number of genes in the cluster, K is the
number of genes in the cluster annotated with GO term t,
B is the number of genes in the background distribution
(i.e. the whole dataset), and T is the total number of genes
in the background distribution annotated with GO term t
[23]. The pvalue can be interpreted as the probability of
randomly finding the same or higher number of genes
annotated with the particular GO term from the back
ground gene list. We used the enrichment pvalues to find
between 1 and 20 most enriched GO terms for each clus
ter of partitions U and V. These clusters of GO terms were
compared using the ADBP measure as for the clusters of
genes, that is, we quantified the proportion of GO terms
present in the original data that were also discovered in
the imputed dataset.
In addition to using the above cluster validation methods,
we also evaluated the imputation accuracy of the methods
using the conventional NRMSE measure [9]. The NRMSE
is the root mean squared difference between the original
y and imputed values y' of the missing entries, divided by
the root mean squared original values in these entries:
where mean() stands for the arithmetic mean of the ele
ments in its argument array. The normalization by the
d i j
( , )
N A
−
gggg
iljliljl
l
N
(,) (
⋅
) ,
=−
=∑
1
2
1
δ
Agg
iljl
l
N
=
=−
((,))1
δ
d i j
( , ) = ∞
{(,)}
uU vV
jjj
D u v
(
n u
(
n v
gvgu
jj
jj
j
g u
∈
j
g v
∈
jj
,)
) ( )(
{
I
}{
I
}).
=
+
∉+∉
∑∑
1
D U V
( , )
k
D u v
(
jj
j
k
,).
=
=∑
1
1
p
T
i
B T
b i
B
b
i K
b T
=
−
−
=∑
min( , )
.
NRMSE =
− ′
(
y
mean
mean
(( ) )
)
,
y y
2
2
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(page number not for citation purposes)
root mean squared original values results in the ZERO
imputation always having an NRMSE value of one, thus
providing a convenient reference error level when com
paring different imputation methods and making the
NRMSE results comparable among different datasets.
Datasets
We used eight yeast cDNA microarray datasets for this
study (see Table 1). The Brauer05 dataset is from a
recently published study of the yeast response to glucose
limitation and it contains multiple time series measured
under different experimental conditions [24]. The
Ronen05 dataset is from a study of the yeast response to
glucose pulses when limiting galactose and contains two
time series [25]. Spahira04A and Spahira04B are two dif
ferent time series datasets from a study of the effect of oxi
dative stress on the yeast cell cycle [26]. The Hirao03
dataset comprises nontime series data from a study of the
identification of selective inhibitors of NAD+dependent
deactylaces [27]. The Yoshimoto02 dataset comprises
multiple time series data from a study of the yeast
genomewide analysis of gene expression regulated by the
calcineurin signalling pathway [28]. The Wyric99 dataset
is from a study of the nucleosomes and silencing factor
effect on the global gene expression [29]. The
Spellman98E dataset is the elutriation part of a study of
the cell cycleregulated genes identification on yeast [30],
which has been used in many comparisons of microarray
analysis tools.
In gene expression clustering, a typical preprocessing step
is to remove those genes which remain constant over all
of the conditions under analysis, as the microarray exper
iment cannot reliably provide any insight into possible
functional roles of such genes. Accordingly, each dataset
was first filtered so that genes with low variation in expres
sion values were left out. More specifically, we filtered out
those genes i for which maxj gij  minj gij < 1.5. Datasets
were then normalized on a genebygene basis so that the
average expression ratio of each gene became 0 and the
standard deviation 1. As the Euclidean distance and Pear
son correlation are directly proportional to each other
after such standardization, and thus yield similar cluster
ing results, we used only the Euclidean distance measure
in the results.
Authors' contributions
JT implemented the algorithms, carried out the experi
ments, and participated in drafting the manuscript. LLE
participated in the design of the study and in editing the
manuscript. OSN assisted in the design of the study and in
editing the manuscript. TA participated in the design of
the study and in drafting and editing the manuscript. All
authors read and approved the final manuscript.
Additional material
Acknowledgements
The work was supported by the Academy of Finland (grant 203632) and the
Graduate School in Computational Biology, Bioinformatics, and Biometry
(ComBi). Part of the computations presented in this work were made with
the help of the computing environment of the Finnish IT Center for Science
(CSC). The authors thank Dr. Milla Kibble for the language review of the
manuscript.
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Additional File 1
Effect of random initialization of kmeans on the ADBP error for genes.
Click here for file
[http://www.biomedcentral.com/content/supplementary/1471
21059202S1.pdf]
Additional File 2
Effect of random initialization of kmeans on the ADBP error for GO
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SVR algorithm [http://202.38.78.189/downloads/svrimpute.html]
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iLLS algorithm [http://www.cs.ualberta.ca/~ghlin/src/WebTools/
imputation.php]
LLS algorithm [http://wwwusers.cs.umn.edu/~hskim/tools.html]
BPCA algorithm [http://hawaii.naist.jp/~shigeo/tools/]
KNN algorithm [http://function.princeton.edu/knnimpute/]
ZERO imputation [http://users.utu.fi/jotatu/zero.m]
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