Comparative analysis of missing value imputation methods to improve clustering and interpretation of microarray experiments

General principle

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 [49, 51–54]].

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.

Error rate for each replacement method

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[29]: This method is always associated to very high RMSE values, which range in an interval from 0.6 to 0.7 for a rate τ ranging from 0.5 to 3.0% (see Figure 3b) and decrease for values from 0.30 to 0.40. Such a curved profile is observed for the datasets OS and GH₂O₂ (see Figure 3a). For the other dataset, RMSE increases as expected (see Figure 3a), but is always associated to high RMSE values.

• kNN[27]: Its RMSE values for all six data files always range between 0.3 and 0.4. The increase of τ only affects slightly the kNN approximation, at most 0.05 for the datasets B and OS. This constancy of RMSE values implies that for high rates of missing data (more than 20% of missing data) the RMSE values remain acceptable.

• SkNN[28]: Despite the fact that SkNN is an improvement of kNN, their RMSE values are surprisingly always higher than the one of kNN (from 0.01 to 0.08). Only with the dataset B, SkNN performs slightly better than kNN (RMSE difference of 0.076).

• LLSI[57]: The average RMSE values of LLSI ranges mainly from 0.34 to 0.41 for most of the dataset. Its performance could be considered as median and its effectiveness is close to the LSI_gene method. Its RMSE values increase gradually with the increase of τ, i.e., 0.1 from 0.5 to 50% of missing data. It is the less efficient method based on least square regressions. However for the dataset L, this method is the most powerful after the LSIs methods (see below).

• LSI_gene[29]: The effectiveness of LSI_gene is slightly affected by the increase in the percentage of missing data. For each data file, the values of RMSE range between 0.3 and 0.4. These results are close to those observed for methods LLSI and kNN, i.e., methods giving of the medium results ranging between the best (LSI_array) and the less efficient methods (EM_gene).

• Row Mean[29] and Row Average[57]: Low RMSE values are observed for L (0.23) and B (0.28) datasets. Only for dataset GHeat, the RMSE value is high (0.54). Strikingly this method shows equivalent and or better results than more elaborated approaches.

• BPCA[30]: For the OC, OS and GH₂O₂ datasets, and for τ comprises in the range 0.5 to 10-15% of missing data, BPCA appears to have one of the lowest RMSE values [see Additional file 2], only bypass by two other approaches. This method is powerful for low rates of missing values. However it should be noted that the efficiency of BPCA is strongly reduced when the rate of missing data increases. This is particularly notable in the case of the GHeat dataset. The values of the RMSE increases from 0.2 to 1.1 (see Figure 3a). For a τ value higher than 30%, BPCA performs worst than most of the imputation methods. This observation is less striking for the other datasets. For B and OS datasets, RMSE values increase by a maximum of 0.1 for τ increasing from 0.5 to 50%. It is a good illustration of the dataset specificity related to the quality of the imputation methods.

The different datasets influence the quality of the imputation

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.

Extreme values

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.

Clustering in question

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 [49, 58]. 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.

Distribution of the observations

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.

Imputation

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.

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., [24, 55, 56], 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.

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 [21, 22, 59], hierarchical clustering [19, 60], Self Organizing Tree Algorithm [61–63], 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 [68, 69], 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 [58, 75, 76]. Comparison of different clusters even obtained with the same classification approach is still a difficult task [see Additional file 4[69, 77–79]]. 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.

Datasets

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).

Datasets refinement: missing values enumeration

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.

Missing values generation

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].

Replacement methods

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].

Hierarchical Clustering

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:

(1)

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

An index for clustering results comparison: Conserved Pairs Proportion (CPP)

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).

Clustering Agreement Ratio (CAR)

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:

(2)

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.