Figure 1 shows the general principle of the analysis. From the initial gene expression datasets, the series of observations with missing values are eliminated to create a Reference matrix. Then simulated missing values are generated for a fixed τ percentage and are included in the Reference matrix. In a second step, these simulated missing values are imputed using the different available methods. Difference between the replaced values and the original true values is finally evaluated using the root mean square error (RMSE) (see Methods). In this work, we chose 5 microarray datasets, very different one from the other, i.e., coming from yeasts and human cells, and with or without kinetics (see Table 2). The idea was to have the broadest possible vision types of expression data [see Additional file 1 for more details 4951525354].

Our goals were also (i) to evaluate methods that experimental scientists could use without intervention, (ii) to select only published methods, and (iii) to analyse influence of the gene clusters. Indeed, some studies have been done to compare numerous methods, e.g., 55, but does not go through the clustering; while less frequent researches goes through the clustering, but test only a limited number of imputation methods as 56. We so have searched all kinds of published imputation methods with available dedicated softwares or codes, whenever the Operating System, language or software. From this search, we selected 12 available replacement methods, which were compatible with high-throughput computation. Others methods had not been used due to the unavailability of the program despite the indication in the corresponding papers or to impossibility to modify the source code to used our microarrays data.

Figure 2 shows the dispersion of expected and true values, for three given imputation methods. On one hand, kNN and EM_gene approaches exhibit a high dispersion between expected and true values; the correlations R equal respectively 0.33 and 0.32 (see Figures 2a and 2b). On the other hand EM_array approach presents a highly better agreement with a R value of 0.97 (see Figure 2c). Figure 3 shows the evolution of RMSE values for τ ranging between 0.5 and 50% using the two datasets G_Heat and OS. These two examples are good illustrations of the different behaviours observed with the different replacement methods. Some have initial high RMSE values and remains quite consistent, while others have lower initial RMSE values but are very sensitive to an increased rate of MVs. Moreover, performances for the different methods appeared to be dependant of the used dataset.

•

EM_gene

•

kNN

•

SkNN

•

LLSI

•

LSI_gene

•

Row Mean

•

BPCA

•

LSI_array, LSI_combined, LSI_adaptative and EM_array

Table 4 shows the average RMSE values for each imputation methods. They are given as the average of all the simulations ranging from τ = 0.5 to 50% (50,000 independent simulations per imputation method). This table highlights the differences that were observed between the datasets. Nonetheless, it allowed us to rank the methods in term of efficiency. Roughly, we could identify three groups: The first one comprise four methods (EM_array, LSI_array, LSI_combined and LSI_adaptative) for which small RMSE values were always observed (EM_array always exhibited the best performances); (2) the second group comprised 4 methods, i.e., BPCA, Row Mean, LSI_gene and LLSI; (3) and finally the third group, which can be considered as the last group, comprised three methods, i.e., kNN, SkNN and EM_gene.

Notably, this order depends on the dataset, but still the changes are often limited. For instance, EM_gene performs better than kNN and SkNN for B dataset, but does not perform better than the others. Strong changes could be noted for OS that allows SkNN to be better than LLSI and LSI_gene. Nonetheless, it is mainly due to the poor quality of the estimation of these two methods with this dataset. For the L dataset, we observed that LLSI method performs well and remains better than other LSIs and EM_array methods. GHeat dataset that is associated to the highest average RMSE values has strong particularities as (i) kNN performs better than BPCA, Row Mean, LSI_gene and LLSI, and (ii) BPCA and Row Mean performs poorly compared to other methods, being only slightly better than EM_gene. Hence, it appears that GHeat is a more difficult dataset to impute.

The same methodology was followed to analyze the extreme values, i.e., 1% of the microarray measurements with the highest absolute values. They have major biological key role as they represent the highest variations in regards to the expression reference [see Additional file 3]. Figure 4 presents similar examples to these of Figure 3, but this time, only extreme values were used in the analysis. Thus, the percentage of missing values τ can be differently apprehend, i.e., τ = 10% corresponds to 10% of the extreme missing values, so 0.1% of the values of the dataset. At one exception, all the replacement methods decrease in effectiveness for the estimate of the extreme values. Performance of the methods also greatly depends on the used dataset and especially -in agreement with previous observation - in the case of the GHeat dataset. A description of the behaviour of each method is presented in Additional file 3. kNN 27 is the less powerful method in most of the case (see Figures 4a and 4b). Its average RMSE value is often 0.5 higher than the second poorest imputation method. Interestingly, in the case of the extreme values, SkNN improved greatly. EM_gene 29 remains one of the less powerful methods for the imputation of missing values. LLSI 57 method effectiveness remains similar compared to the other methods of its group. Row Mean 29 and Row Average 57 have RMSE values increased by 0.2 to 0.4 for the yeast dataset, which is correct in regards to other methods (see Figures 6). Their efficiencies are median compared to the other methods. BPCA 30 has a correct behaviour. But contrary to most of them, it is very sensitive to the datasets. LSI_gene 29 has the lowest RMSE values observed after EM_array, LSI_array, LSI_combined and LSI_adaptative. This result shows that LSIs, whatever the specificity of their implementations, are effective to impute the values missing.

EM_array method is again the most performing method (see previous section). Its RMSE values are almost identical to the ones previously computed. LSI_array, LSI_combined and LSI_adaptative are slightly less efficient than previously seen. Thus, the clustering we have proposed remains pertinent when only the extreme values are implicated. LSI_array, LSI_combined, LSI_adaptative and EM_array are always good, and the less efficient methods can be associated now to considerable RMSE values. Noticeably, kNN efficiency collapses and the influence of datasets on the imputation quality is sharpened.

A critical point in the analysis of DNA data is the clustering of genes according to their expression values. Missing values have an important influence on the stability of the gene clusters 4958. Imputations of missing values have been used both to do hierarchical clustering (with seven different algorithms) and k-means 20 (see Methods).

Figure 5a shows the Cluster Pair Proportions (CPP, 49 see Methods section) of OS using hierarchical clustering with complete linkage, average linkage, McQuitty and Ward algorithm. CPP values of average linkage ranges between 78 and 68%, those of McQuitty between 58 and 45%, those of Ward between 57 and 35% and finally those of complete linkage between 50 and 41%. We obtain for the 7 hierarchical clustering algorithms the same behaviours than previously observed 49: ranging from high CPP values for single linkage to low CPP values for Ward. This observation can be explained by the topology given by each algorithm, e.g., Ward gives well equilibrated clusters whereas single linkage creates few major clusters and numerous adjacent singletons.

For every hierarchical clustering methods the CPP values are different, but the general tendencies remain the same: (i) imputation of small rate τ of MVs has always a strong impact on the CPP values, and (ii) the CPP values slowly decreased with the increased of τ. Between 0.5 and 3% of MVs and the CPP values decrease by 1 to 3% per step of 0.5% of MVs. From τ equals 3.5 to 20% of MVs, the values of CPP decrease overall by 10%. For higher rate of MVs the decreasing of CPP is slower. This loss of stability is present in the case of the k-means method and for each type of hierarchical classification (except for the methods single linkage and centroid linkage, due to the building of the clusters).

Individual evaluation of the methods highlights the lack of efficiency of the EM_gene imputation method; it obtains always the lowest CPP values, i.e., 1.37 to 5.34% less than other approaches. At the opposite, EM_array, LSI_array, LSI_combined and LSI_adaptative are associated to the highest CPP values. In the case of the methods with a median efficiency, e.g., Row_Mean, their CPP values could be assigned as median compared to the values of the other methods. Figure 5b shows the particular example of OS dataset. CPP values of BPCA (average value equals 42.6%) are close to the most powerful methods (42.8% for the four methods). Moreover, in the classical range of τ less than 20%, it is the best. As seen in Table 4, BPCA is one of the best approaches for this dataset. Hence, common trends can be found between the quality of the imputation method and the gene cluster stability.

In addition, evaluation of imputation methods shows that the cluster quality depends on the dataset. For instance, with the dataset OS, imputation of missing values with kNN method gives an average CPP value (for the Ward algorithm) that equals 42.9%, while the average CPP values for all the other methods only equals 40.6% whereas its RMSE value is one of highest (see Table 4). The CPP differences are mainly bellow 5%. These results show that an improvement has been obtained since last study. Nonetheless, no new approaches had drastically improved the quality of the clustering. Interestingly, k-means approach had similar tendencies, underlining that this low improvement is not due to hierarchical clustering.

Another question is the comparison between hierarchical clustering algorithms and k-means. Nonetheless, comparison only between hierarchical clustering algorithms is already a difficult task. Comparison with k-means is so more difficult. Indeed, the use of the same number of clusters to compare the hierarchical clustering algorithms with k-means can leads to a wrong conclusion. Indeed, for an equivalent number of clusters, most of the CPP values of k-means are lower than CPP values obtained with hierarchical clustering algorithms. However, it is only due to the dispersion of observations within the clusters obtained by k-means approach. Thus, to have an unbiased comparison, the dispersion of genes within cluster between k-means and hierarchical clustering algorithms must be computed. It had been done, as previously described 49. Following this approach, Ward and complete linkages were defined as the best approaches to assess an unbiased comparison. They have both CPP values lower than k-means CPP values. The differences were often higher than 5% underlining the interest of k-means approach to cluster gene expression profiles.

When index CPP is calculated, only one group is taken into account. To go further, we used another index, named CPP_f that allows to take into account the five closer groups, and to check the pairs of genes remaining joint partners. The values of CPP_f are higher than those of the CPP, e.g., 20% for the Ward. Methods associated to high CPP values have also high CPP_f values, while methods with low CPP values have also a lower CPP_f values. These weak variations shows that often a part of the observations, not associated to the original cluster could be find in its vicinity. These results are entirely in agreement with our previous results 49. It shows here that the novel imputation methods have not permit to get closer related genes with better improvement.

The analysis of associations could also take into account the non-associations. For this purpose, Clustering Agreement Ratio (CAR, see Methods section) has been used which considers both associated and non-associated genes. CAR values are higher than the one of the CPP due to the calculation of the pairs of genes remaining dissociated. Indeed, it is more probable than the genes are dissociated than associated according to the number of treated genes and the number of generated groups. For the OS dataset, the highest values of the index CAR concerns Ward classification and are ranging between 88.2 and 91.2%. For the GHeat dataset, it ranges between 91.0 and 94.1%. Complete linkage, average linkage and McQuitty have lower CAR values (80%). For k-means classification, the values are higher 1 to 2% compared to Ward classification, 10% better than McQuitty and Complete linkage and 13% to average linkage. This results underlines that K-means allows so a better stability of gene clusters.

Since our previous analysis 49, numerous new MVs imputation methods have been proposed. Some appeared to be true improvements in regards to the computation of RMSE. In particular, EM_array is clearly the most efficient methods we tested. For τ < 35%, it is the best imputation method for 60% of the values, and for τ > 35%, in 80%. This feature was confirmed by the analysis of extreme values. LSI_array, LSI_combined and LSI_adaptative follow closely the efficiency of EM_array. We have unsuccessfully tried to combine these four different methods to improve the RMSE values. No combination performs better than EM_array.

We can underline four interesting points:

i. As expected, the imputation quality is greatly affected by the rate of missing data, but surprisingly it is also related to the kind of data. BPCA is a perfect illustration. For non-kinetic human dataset, MVs estimations were correct, whereas for the GHeat dataset the error rate appeared to be more important.

ii. The efficiency of Row_Mean (and Row_Average) is surprisingly good in regards to the simplicity of the methodology used (with the exception of GHeat dataset).

iii. Even if kNN is the most popular imputation method; it is one of the less efficient, compared to other methods tested in this study. It is particularly striking when analyzing the extreme values. SkNN is an improvement of kNN method, but we observed that RMSE values of SkNN were not better than ones of kNN. It could be due to the use of non-optimal number of neighbours (k), as for kNN. It must be noticed we used k_opt defined by 27, this choice has a direct impact on the imputation values.

Extreme values are the ones that are the most valuable for the experiments. The imputation of extreme value missing data shows that -except for EM_array- the effectiveness of all the methods is affected.

Our results are so in good accordance with the results of Brock and co-workers 55 who found that methods from Bo and co-workers 29, Kim and co-workers 57 and Oba and co-workers 30 are highly competitive. However, they consider "that no method is uniformly superior in all datasets" 29. Our results are simpler to summarize as we observe -thanks to our distance criteria- a grading between the effectiveness of the methods. LLSI of Kim and co-workers 57 has a correct behavior for all datasets while BPCA of Oba and co-workers 30 is strongly dependant of the dataset. At the opposite, the methods implemented by Bo and co-workers 29 remain the most efficient in all cases. Moreover, some implemented methods of Bo and co-workers 29 have not been tested by 55, but are the most efficient. All these results are reinforced by the analyses of extreme value imputations.

An important point must be not forgotten, we have, as the other authors, e.g., 245556, used the entire dataset, i.e., no specific selection of interesting profile gene had been done. It could have importance in terms of quality of the imputation values and consequence on the clustering.

A strong assumption of the microarray data analysis is that genes with similar expression profiles are likely to be co-regulated and thus involved in the same or similar biological processes. Different types of clustering and classification methods have been applied to microarray data, e. g., some classical as k-means clustering 20, self-organizing maps 212259, hierarchical clustering 1960, Self Organizing Tree Algorithm 616263, and some dedicated approaches as DSF_Clust 64, re-sampling based tight clustering 65, cluster affinity search technique 66, multivariate Gaussian mixtures 67, model-based clustering algorithms 6869, clustering of change patterns using Fourier coefficients 70, Nearest Neighbor Networks 71, Fuzzy clustering by local Approximation of membership 72 or Multi-Dimensional Scaling 73.

Given one particular dataset, different clustering algorithms are very likely to generate different clusters 74. This is true when large-scale gene expression data from microarrays are analyzed 587576. Comparison of different clusters even obtained with the same classification approach is still a difficult task [see Additional file 469777879]. Thus, to assess the relevance of missing value imputation methods, we observed the behaviours of different hierarchical clustering methods and k-means clustering using CPP, CPP_f 49 and newly introduce CAR index. Results follow exactly the observations done on RMSE values (see previous section). Only one method seems ambiguous: kNN. Indeed, its CPP and CPP_f are higher than expected. It is mainly due to the selection of the genes in the different datasets. We have decided at the beginning to not discard any genes, i. e., we have absolute no a priori. Thus very flat profiles have been conserved and empower kNN that prefers to predict values closer to zero than the other methods (see Figure 4 of 49). It generates clusters with lot of zero, these clusters are so stable. For the majority of the methods, the order of effectiveness of the methods for the maintenance of stability within the groups between various classifications is identical. Combination of CPP, CPP_f and CAR index underlines the interest of k-means clustering in regards to hierarchical clustering methods. For comparable clusters, k-means gives better values.

Wang and co-workers does not found a strong difference between the three imputation methods they used, i.e., kNN, BPCA and LLS, in the classification performance 24. The only comparable extensive study has been done by Tuikkala and co-workers 56, they have focussed interestingly on the GO term class and use k-means. They have tested six different methods with less simulation per missing value rates and less missing value rates. But, the important point is they have not tested the methods found the most efficient by our approach. We also slightly disagree with their conclusion about the quality of BPCA 56. It can be easily understand as only a very limited number of clusters has been tested (5 clusters); in our case, we have supervised the choice of cluster numbers (see Method section), leading to a higher number of clusters. This higher number is so more sensitive to the quality of clustering. It must be noticed we have used Euclidean distance and not Pearson correlation, it was mainly to (i) stay consistent with our previous research, and (ii) as we have not filtered the data, Pearson correlation could have aggregated very different profiles. As the time computation was very important, it was not possible to test the two possibilities.

We used 5 data sets for the analysis [see Additional file 1]; they were mainly coming from the SMD database 81. The first one, named Ogawa set, was initially composed of N = 6013 genes and n = 8 experimental conditions about the phosphate accumulation and the polyphosphate metabolism of the yeast Saccharomyces cerevisiae 51. The second one corresponds to various environmental stress responses in S. cerevisiae 52. This set, named Gasch set, contains N = 6153 genes and n = 178 experimental conditions. Due to the diversity of conditions in this set, we focused on two experimental subsets corresponding to heat shock and H₂O₂ osmotic shock respectively. Bohen and co-workers have analyzed the patterns of gene expression in human follicular lymphomas and the interest of treatment by rituximab 53. This dataset is composed of N = 16.523 genes and n = 16 experimental conditions. The last dataset has been obtained by Lucau-Danila, Lelandais and co-workers 54. To precisely describe the very early genomic response developed by yeast to accommodate a chemical stress, they performed a time course analyses of the yeast gene expression which follows the addition of the antimitotic drug benomyl. The dataset is a kinetic that comprised N = 5.621 genes for n = 6 kinetic time (30 seconds, 2, 4, 10, 20 and 40 minutes).

From the original datasets, we built complete datasets without MVs. All the genes containing at least one missing value were eliminated from the Ogawa set (noted OS). The resulting OS set contains N = 5783 genes and n = 8 experimental conditions. The second set without MVs was taken from Gasch et al. and called GS. The experimental conditions (column) containing more than 80 MVs were removed. The resulting GS matrix contains N = 5843 genes and n = 42 experimental conditions. Two subsets were generated from GS and have been noted GHeat and GH₂O₂. They correspond to specific stress conditions as described previously. GHeat and GH₂O₂ contain respectively N = 3643 genes with n = 8 experimental conditions and N = 5007 genes with n = 10 experimental conditions.

To test the influence of the matrix size, i.e., the number of genes, we built smaller sets corresponding to 1/7 of OS, GS, GHeat and GH₂O₂. Principles are described in 49. For the dataset of Bohen et al. (noted B), we have done the same protocol and used a subset representing 1/7 of B, i.e., N = 861 genes. For the dataset of Lucau-Danila et al.: 54, 11.4% of the genes have at least one missing values. The dataset with no missing values (noted L) was so composed of N = 4645 genes.

From the sets without MVs, we introduced a rate τ of genes containing MVs (τ = 1 to 50.0%), these MVs are randomly drawn. Each random simulation is generated at least 100 times per experiment to ensure a correct sampling. It must be notices that contrary to our previous work, each gene could contain more than one MV 49.

The different packages have been downloaded from the authors' websites (see Table 1). kNN has been computed using the well-known KNNimpute developed by Troyanskaya and co-workers 27. The determined k_opt value is associated with a minimal global error rate as defined by Troyanskaya and co-workers 27. BPCA was used without its graphical interface 30 as for the Bo et al. package (Java) 29. For LLSI and Row_Average, we have modified the original Matlab code to use our own microarray datasets 57. SkNN was performed with R software 28.

The hierarchical clustering (HC) algorithm allows the construction of a dendogram of nested clusters based on proximity information 19. The HC have been performed using the "hclust" package in R software 82. Seven hierarchical clustering algorithms have been tested: average linkage, complete linkage, median linkage, McQuitty, centroid linkage, single linkage and Ward minimum variance 83.

The distance matrix between all the vectors (i.e., genes) is calculated by using an external module written in C language. We used the normalized Euclidean distance d* to take account of the MVs:

v and w are two distinct vectors and m is the number of MVs between the two vectors. Thus, (v_i - w_i) is not computed if v_i and/or w_i is a missing value

To assess the influence of missing data rates and different replacement methods into clustering results (see Figure 1), we have analysed the co-associated genes of an original dataset (without MVs) compared to these genes location in a set with MVs. A similar approach has been used by Meunier et al. on proteomic data 84.

Hence, we realized in a first step the clustering with the data sets without MV by each aggregative clustering algorithm. The results obtained by these first analyses are denoted reference clustering (RC). In a second step, we generated MVs in data. The MVs are replaced by using the different replacement methods. Then we performed the hierarchical clustering for each new set. The results obtained by these second analyses are denoted generated clustering (GC).We compared the resulting clusters defined in RC and GC and assessed the divergence by using an index named Conserved Pair Proportions (CPP). The CPP is the maximal proportion of genes belonging to two clusters, one from the RC and the other one from the GC (cf. Figure 1 of 49 and Additional file 5 for more details).

The Clustering Agreement Ration (CAR) is the concordance index measuring the proportion of genes pairs, either belonging to a same cluster (resp. different clusters) in the reference clustering (RC) and found again in a same cluster (resp. different clusters) in the clustering (GC) obtained without or after replacing the MVs.

The index CAR is defined by the following equation:

where and specify the co-presence of two genes in a same cluster, i.e., they take the value 1 when the genes i and j belong to a same cluster in the clustering RC and GC respectively. The numbers of pairs in G genes is G.(G - 1)/2. The first term of the numerator corresponds to the co-presence of the pair (i, j) in a same cluster for RC and GC, and, the second term the co-absence of this pair in a same cluster.