Abstract
Background
Missing values commonly occur in the microarray data, which usually contain more than 5% missing values with up to 90% of genes affected. Inaccurate missing value estimation results in reducing the power of downstream microarray data analyses. Many types of methods have been developed to estimate missing values. Among them, the regressionbased methods are very popular and have been shown to perform better than the other types of methods in many testing microarray datasets.
Results
To further improve the performances of the regressionbased methods, we propose shrinkage regressionbased methods. Our methods take the advantage of the correlation structure in the microarray data and select similar genes for the target gene by Pearson correlation coefficients. Besides, our methods incorporate the least squares principle, utilize a shrinkage estimation approach to adjust the coefficients of the regression model, and then use the new coefficients to estimate missing values. Simulation results show that the proposed methods provide more accurate missing value estimation in six testing microarray datasets than the existing regressionbased methods do.
Conclusions
Imputation of missing values is a very important aspect of microarray data analyses because most of the downstream analyses require a complete dataset. Therefore, exploring accurate and efficient methods for estimating missing values has become an essential issue. Since our proposed shrinkage regressionbased methods can provide accurate missing value estimation, they are competitive alternatives to the existing regressionbased methods.
Background
Nowadays microarray technique has become an important and useful tool in functional genomics research. This high throughput technique allows the characterization of the gene expression of the whole genome by measuring the relative transcript levels of thousands of genes in various experimental conditions or time points [1]. Microarray data analyses have been widely used to investigate various biological processes such as the cell cycle process [28] and the stress response [9,10].
Although the microarray technology has been developed for more than a decade, typical microarray data still contain more than 5% missing values with up to 90% of genes affected [11]. Missing values could be generated by various reasons, including technological failures, administrative error, insufficient resolution, image corruption, dust or scratches on the slide [12]. As many downstream analysis methods (such as gene clustering, disease classification and gene network reconstruction) require complete datasets, missing value estimation becomes an important preprocessing step in the microarray data analysis [1113].
The missing values in the microarray dataset are traditionally estimated by repeating the microarray experiments or simply replacing the missing values with zero or the row average (the average expression over the experimental conditions). Because these approaches are either timeconsuming or leading to serious estimation errors, more advanced missing value imputation methods are needed to solve the missing value problems. In 2001, Troyanskaya et al. published the first two missing value imputation algorithms based on the knearest neighbors (kNNimpute) and the singular value decomposition (SVDimpute) [12]. Since then, a lot of missing value imputation methods have been proposed such as Bayesian principal component analysis (BPCA) [14], Gaussian mixture clustering imputation (GMCimpute) [11], conditional ordered list imputation [15], randomforestbased imputation [16] and so on.
Among the existing missing value imputation methods, the regressionbased methods are very popular and contain many algorithms, including least squares imputation (LSimpute) [17], local least squares imputation (LLSimpute) [18], sequential local least squares imputation (SLLSimpute) [19], and iterated local least squares imputation (ILLSimpute) [13]. LSimpute estimates the missing values in the target gene by using a weighted average of the k estimates from the k most similar genes. Each estimate is attained by constructing a single regression model of the target gene by a similar gene. LLSimpute represents the target gene as a linear combination of k similar genes by a multiple regression model and uses the regression coefficients to estimate the missing values. SLLSimpute modifies the LLSimpute by estimating the missing values sequentially from the gene containing the fewest missing values and partially utilizing these estimated values. ILLSimpute modifies the LLSimpute by not choosing the similar genes with a fixed number k but defining the similar genes as the genes whose distances from the target gene are less than a distance threshold and then runs LLSimpute iteratively.
In this study, we focus on the regressionbased methods because these methods have been shown to have better performances than the other existing methods in many testing microarray datasets [20,21]. To further improve the performance of the regressionbased methods, we propose shrinkage regressionbased methods which use a shrinkage estimator to replace the least square estimator for the estimation of the regression coefficients in the regression model. The shrinkage estimator such as the JamesStein estimator has been shown to dominate the least square estimator in many statistical models [22,23]. By adopting our new regression coefficients in the regressionbased methods, we showed that an improvement on missing value estimation in six testing microarray datasets could be achieved.
Methods
In this study, we propose using the wellknown shrinkage estimation approach to improve three existing regressionbased methods (LLSimpute [18], SLLSimpute [19], and ILLSimpute [13]) for missing value estimation. We call our proposed methods the shrinkage regressionbased methods (see Figure 1). In the following subsections, we first introduce the shrinkage estimation approach and then describe the proposed shrinkage LLSimpute, shrinkage SLLSimpute, and shrinkage ILLSimpute.
Figure 1. The shrinkage regressionbased methods.
Shrinkage estimation approach
One of the shrinkage estimators, the JamesStein estimator, for the normal distribution is introduced here. Suppose that Y_{1}, Y_{2}, ..., Y_{k }are independent normal random variables and these k random variables all have a common known variance, but their means are unknown and different. Let Y_{i }~ N(θ_{i}, σ^{2}) and Y = (Y_{1}, ..., Y_{k}). Then we have Y ~ N(θ, σ^{2}I), where θ = (θ_{1}, ..., θ_{k}) and I is a k × k identity matrix. Let d(Y) = (d_{1}(Y), ..., d_{k}(Y)) be an estimator of θ. Under the squared error loss function
we are interested in finding estimators of θ such that the mean squared error E_{Y }[L (θ, d(Y))] is minimized. An intuitive estimator of θ is Y (i.e. ). However, Stein [22] showed that when k ≥ 3, there exists other estimators with smaller mean squared error than the intuitive estimator Y. For k ≥ 3, under the squared error loss, the intuitive estimator Y is dominated by the estimator
where [23]. The estimator in (2) is called the JamesStein estimator in the literature [23]. With the form in (2), the JamesStein estimator of θ_{i }is
It is worth noting that the estimator of θ_{i }in (3) depends on not only the random variable Y_{i}, but also the other variables Y_{1}, ..., Y_{i}_{1}, Y_{i + 1}, ..., Y_{k }because of the term . On the contrary, the intuitive estimator does not use the other variables Y_{1}, ..., Y_{i  1}, Y_{i + 1}, ..., Y_{k }but only uses Y_{i }to estimate θ_{i}. It has been shown that estimators using other variables' information provide more accurate estimation for θ than the intuitive estimator does [22]. In fact, except for the estimator in (3), the estimators of the form
all have uniformly smaller mean squared error than the intuitive estimator Y_{i}, for k ≥ 3 and 0 <c <2 (k  2). Among all the estimators of the form in (4), the estimator in (3) has the minimized mean squared error. The shrinkage estimation approach has also been shown to have good performance in interval estimation [24,25]. Based on the JamesStein estimator in (3), we developed shrinkage regressionbased imputation methods.
Notations
In a typical microarray data matrix, the rows are the genes under investigation and the columns are the experimental conditions or time points. The microarray data matrix is obtained by performing a series of experiments on the same set of genes. We use to represent a microarray data matrix with m genes and n experiments, and assume which is true for microarray data. In the matrix G, a row represents the expressions of the ith gene in n experiments:
where denotes the transpose of a column vector g_{i}. If there is a missing value in the lth position of the ith gene, we denote it as , i.e. G_{i},_{l }= g_{il }= .
Shrinkage local least squares imputation (Shrinkage LLSimpute)
In the LLSimpute method [18], a target gene with missing values is represented as a linear combination of k similar genes. Rather than using all genes in the dataset, only k genes with high similarity to the target gene are used. The procedure of selecting k similar genes is as follows. Suppose that the target gene is the first gene and has a missing value in the first position, i.e. = g_{11 }in the matrix . The Pearson correlation coefficient is used to find the k similar genes. These k similar genes are called the knearest neighbor genes, which have the k largest absolute values of the Pearson correlation coefficients. The Pearson correlation coefficient between the target gene and the jth gene is defined as
where and σ_{j }denote the average and the sample standard deviation of the vector (g_{j2}, ..., g_{jn}). When computing the correlation coefficients, g_{j1 }is not used because it corresponds to the position of the missing value in the target gene. Based on these selected knearest neighbor genes, a matrix and two vectors and can be formed as follows
where is the missing value in the target gene g_{1 }and are the knearest nieghbor genes of the target gene g_{1}. Each row of matrix A consists of the last n  1 elements of one knearest neighbor gene , 1 ≤ i ≤ k. The elements of the vector b comprise of the first elements of all these knearest neighbor genes and the elements of the vector w are the last n  1 elements of the target gene g_{1}. With the matrix A, and the vectors b and w, the least squares problem is formulated in LLSimpute as
Solving the above problem, the least square regression coefficients are acquired as
In the LLSimpute, the missing value is then estimated by
In this study, we want to improve the performance of LLSimpute by adjusting the regression coefficients in (8). Our shrinkage LLSimpute associates the LLSimpute method with the shrinkage estimator to impute the missing values. Our method replaces the regression coefficient estimators in (8) by the shrinkage estimator, and then use the new estimator to estimate the missing value in (9). However, we found that applying the existing shrinkage estimator in (3) did not always improve the performance of LLSimpute. Therefore, we tested different forms of the shrinkage coefficient estimators and conceived a feasible coefficient estimator to improve the LLSimpute method. We proposed using the shrinkage regression coefficients
to replace the conventional coefficients in (8), where σ^{2 }is the variance of the coefficients S is the norm of the coefficients (i.e. ), k is the row number of the matrix A and is the column number of the matrix A, which equals n  1 in this case. Finally, the missing value is estimated as
Shrinkage sequential local least squares imputation (Shrinkage SLLSimpute)
In the LLSimpute, it does not use the information of genes with missing values since the existence of missing values hinders the use of the other observed values of that gene. In the SLLSimpute method, it estimates the missing values sequentially from the gene containing the fewest missing values and partially utilizes these estimated values. The details of SLLSimpute [19] is described as follow. First, the microarray matrix is divided into two submatrices: a complete matrix consisting of genes without missing values and an incomplete matrix consisting of genes with missing values. In the incomplete matrix G_{2}, the genes are sorted by their missing rates. The first gene has the smallest missing rate and the last gene has the largest missing rate. The missing rate is calculated by
where c_{i }is the number of missing values in ith gene. The imputation is executed sequentially from the first gene of G_{2}. That is, the first gene of G_{2 }which has the smallest missing rate is selected as the target gene firstly. Then LLSimpute is applied to estimate the missing values in the target gene by finding the knearest neighbour genes from the complete matrix G_{1 }and then using the formula in (9) to estimate the missing values. After filling all the missing values in the target gene, it is moved to G_{1}. Then the second gene of G_{2 }is selected as the target gene and repeat the same process again. By moving the genes whose missing values have been imputed to the complete matrix, the previous target genes with imputed values can be utilized for the missing value estimation of the following target gene. However, too many missing values in a gene will result in big estimation error and reusing a gene with too many imputed values will reduce the imputation performance. Therefore, only the genes with missing rates less than a threshold r_{0 }are reused, where r_{0 }is set as the average missing rate of all genes containing missing values, i.e.,
By a similar argument as for the shrinkage LLSimpute, we apply the shrinkage estimator to SLLSimpute. The shrinkage SLLSimpute adjusts the coefficients of the regression model by the formula in (10) and use the formula in (11) to estimate the missing values.
Shrinkage iterated local least squares imputation (Shrinkage ILLSimpute)
LLSimpute and SLLSimpute methods select knearest neighbor genes for a target gene, where k is a fixed number. However, in the ILLSimpute method [13], it does not fix the number of similar genes selected. Alternatively, it defines the similar genes as the genes whose distances to the target genes are less than a distance threshold . The rationale of using a distance threshold rather than using a fixed number of similar genes is that some of the knearest neighbor genes are already far away from the target gene and are not very similar to the target gene.
The procedure of ILLSimpute is as follows. In the first iteration, missing values of each target gene are filled with the row average. Then a distance threshold is used to select the similar genes of each target gene. Finally, LLSimpute method is used to estimate the missing values of each target gene. In the later iteration, ILLSimpute method uses the imputed results from the previous iteration to reselect the similar genes of each target gene (using the same distance threshold) and applies LLSimpute method to reestimate the missing values.
By a similar argument as for the shrinkage LLSimpute, we apply the shrinkage estimator to ILLSimpute. The shrinkage ILLSimpute adjusts the coefficients of the regression model by the formula in (10) and use the formula in (11) to estimate the missing values.
Results and Discussion
We conducted several experiments to compare the performances of our shrinkage regressionbased methods and the original regressionbased methods under different scenarios. In the first subsection, we introduce the benchmark datasets. In the second subsection, we describe how we measure the performance of various imputation methods. In the following three subsections, we report the comparison results for different number of similar genes used, different missing rates, and different noise levels. Finally, we further compare the performances of our shrinkage regressionibased methods and three existing nonregressionbased methods.
Datasets
Considering the effects of dataset selection and types of microarray experiments on the performance of an imputation method, six representative datasets (three nontime series and three time series) were used in our simulations. They were Ogawa's data from the study of phosphophate accumulation and polyphosphophate metabolism (denoted as Ogawa, nontime series) [26], Bohen's follicular lymphomas data (denoted as BohenSH, nontime series) [27], the data from a lymphoma study (denoted as Lymphoma, nontime series) [28], the data from Brauer's experiments which studied the physiological response to glucose limitation in batch and steadystate cultures of yeasts (denoted as Brauer05, time series) [29], and Shapira's oxidative stress data (denoted as Shapira04A and Shapira04B, time series) [30]. We divided Shapira's data into two datasets because the authors used one kind of oxidative chemical in the experiment in Shapira04A, but they used another kind of oxidative chemical in the experiment in Shapira04B. The six microarray datasets were used as benchmark datasets in numerical experiments to compare the performances of our shrinkage regressionbased methods and the original regressionbased methods. Each dataset was processed by deleting the genes with missing values to generate a complete data matrix, and the details of these datasets were listed in Table 1.
Table 1. Benchmark datasets.
The performance measure
A common criterion used to compare the performances of different imputation methods is the normalized root mean squared error (NRMSE) [1113,1719]. From a microarray dataset, we can obtain an original data matrix M_{0 }with m genes and n experiments, and then we can construct a complete matrix by deleting the genes with missing values. After the complete data matrix M_{1 }is established, we randomly select a specific percentage of the elements of M_{1 }and regard these elements as missing values. Then we estimate the missing values using various imputation methods and compare their performances using NRMSE which is shown below:
where y_{guess }and y_{ans }are vectors whose elements are the estimated values by an imputation method and the known answers for all missing entries, respectively.
Performance comparison for different k values
A parameter k, the number of similar genes used, has to be determined before using two regressionbased methods (LLSimpute and SLLSimpute). Since the performance of both algorithms is known to be affected by the k value used and different microarray datasets may have different optimal k values [18,19], we tested several possible k values (50, 100, 150, 200, 250 and 300) on six benchmark datasets. Table 2 listed the optimal k values for LLSimpute and SLLSimpute on each of the six benchmark datasets. Another regressionbased method (ILLSimpute) does not have the parameter k and therefore was not considered in this numerical experiment.
Table 2. The optimal k value for each benchmark dataset.
For each of the six benchmark dataset, we also compared the performances of the proposed shrinkage regressionbased methods and the original regressionbased methods for several possible k values (50, 100, 150, 200, 250 and 300). In our numerical experiments, missing rate for each benchmark dataset was set to be 5%. Namely, for each dataset, we randomly removed 5% entries of the complete matrix to generate a matrix with missing values, and then estimated the missing values using the shrinkage and the original regressionbased methods. The same procedure was run for five independent rounds and the average NRMSE of these five simulations was used to compare the performances of different imputation methods.
As shown in Figure 2, the proposed shrinkage LLSimpute outperforms LLSimpute for all k values and all benchmark datasets. Similarly, the proposed shrinkage SLLSimpute outperforms SLLSimpute for all k values and all benchmark datasets (see Figure 3). The simulation results suggest that utilizing a shrinkage estimation approach to adjust the coefficients of the regression model can improve the performances of the original regressionbased methods.
Figure 2. Performance comparison between shrinkage LLS (shr_LLS) and LLS for different k values.
Figure 3. Performance comparison between shrinkage SLLS (shr_SLLS) and SLLS for different K values.
Performance comparison for different missing rates
In real applications, different microarray data may have different missing rates to be imputed. It is informative to know how an imputation method performs for different missing rates. Therefore, we compared the performances of the shrinkage regressionbased methods and the original regressionbased methods on the microarray data with different missing rates (1%, 5%, 10%, 15% and 20%). Namely, for each of the six benchmark dataset, we randomly removed x% (x = 1, 5, 10, 15 or 20) entries of the complete matrix to generate a matrix with missing values, and then estimated the missing values using the shrinkage and the original regressionbased methods. The same procedure was run for five independent rounds and the average NRMSE of these five simulations was used to compare the performances of different imputation methods. Note that the optimal k value used for each benchmark dataset was listed in Table 2.
Figure 4 shows that the proposed shrinkage LLSimpute outperforms LLSimpute for all missing rates and all benchmark datasets. Figure 5 shows that the proposed shrinkage SLLSimpute outperforms SLLSimpute for all missing rates and all benchmark datasets. Figure 6 shows that the proposed shrinkage ILLSimpute outperforms ILLSimpute for all missing rates and all benchmark datasets. The simulation results suggest that utilizing a shrinkage estimation approach to adjust the coefficients of the regression model can improve the performances of the original regressionbased methods.
Figure 4. Performance comparison between shrinkage LLS (shr_LLS) and LLS for different missing rates.
Figure 5. Performance comparison between shrinkage SLLS (shr_SLLS) and SLLS for different missing rates.
Figure 6. Performance comparison between shrinkage ILLS (shr_ILLS) and ILLS for different missing rates.
Performance comparison for different noise levels
In real applications, different microarray data may contain different levels of noises. It is informative to know how an imputation method performs for different levels of noises inherent in the microarray data. Therefore, we compared the performances of the shrinkage regressionbased methods and the original regressionbased methods on the microarray data with different noise levels. For each of the six benchmark dataset, we added Gaussian noises with different levels into the data. The magnitudes of the noises were set in terms of the standard deviations ranging from 0 to 0.25 with a step size 0.05. In our numerical experiments, missing rate for each benchmark dataset was set to be 5% and the optimal k value used for each benchmark dataset was listed in Table 2. Namely, for each dataset (after adding Gaussian noises into the data), we randomly removed 5% entries of the complete matrix to generate a matrix with missing values, and then estimated the missing values using the shrinkage and the original regressionbased methods. The same procedure was run for five independent rounds and the average NRMSE of these five simulations was used to compare the performance of different imputation methods.
Figure 7 shows that the proposed shrinkage LLSimpute outperforms LLSimpute for all noise levels and all benchmark datasets. Figure 8 shows that the proposed shrinkage SLLSimpute outperforms SLLSimpute for all noise levels and all benchmark datasets. Figure 9 shows that the proposed shrinkage ILLSimpute outperforms ILLSimpute for all noise levels and all benchmark datasets. The simulation results suggest that utilizing a shrinkage estimation approach to adjust the coefficients of the regression model can improve the performances of the original regressionbased methods.
Figure 7. Performance comparison between shrinkage LLS (shr_LLS) and LLS for different noise levels.
Figure 8. Performance comparison between shrinkage SLLS (shr_SLLS) and SLLS for different noise levels.
Figure 9. Performance comparison between shrinkage ILLS (shr_ILLS) and ILLS for different noise levels.
Performance comparison with three existing nonregressionbased methods
We have shown that our shrinkage regressionbased methods perform better than the existing regressionbased methods. Still, it would be interesting to know whether our shrinkage regressionbased methods provide more accurate missing value imputation than the existing nonregressionbased methods do. Therefore, we compared the performances of our shrinkage regressionbased methods and three existing nonregressionbased methods (kNNimpute [12], SVDimpute [12], and BPCA [14]) on the six benchmark microarray datasets. As shown in Figures 10, 11, 12, the proposed shrinkage regressionbased methods outperform these three existing nonregressionbased methods for almost all missing rates and all benchmark datasets. Taken together, our shrinkage regressionbased methods are competitive alternatives to the existing methods for microarray missing value imputation.
Figure 10. Performance comparison between shrinkage LLS (shr_LLS) and three nonregressionbased methods for different missing rates.
Figure 11. Performance comparison between shrinkage SLLS (shr_SLLS) and three nonregressionbased methods for different missing rates.
Figure 12. Performance comparison between shrinkage ILLS (shr_ILLS) and three nonregressionbased methods for different missing rates.
Conclusions
Imputation of missing values is a very important aspect of microarray data analyses because most of downstream analyses require a complete dataset. Therefore, exploring accurate and efficient methods for estimating missing values has become an essential issue. In this study, regressionbased methods associated with a shrinkage estimation approach are proposed to estimate missing values in the microarray data. Our methods take the advantage of the correlation structure existing in the microarray data and select similar genes for the target gene by Pearson correlation coefficients. Besides, our methods incorporate the least squares principle, utilize a shrinkage estimation approach to adjust the coefficients of the regression model, and apply the new coefficients of the regression model to estimate missing values. Simulation results show that the proposed shrinkage regressionbased methods provide more accurate missing value estimation for various types of datasets than the original regressionbased methods do. Since our proposed methods can be applied to modify any kind of regressionbased methods and can provide accurate missing value estimation, they are competitive alternatives to the existing regressionbased methods.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
WSW conceived the research topic and provided essential guidance. HW developed the alogirthm. CCC did all the simulations. HW, CCC, YCW, and WSW wrote the manuscript. All authors have read and approved the final manuscript.
Acknowledgements
This study was supported by the National Cheng Kung University and Taiwan National Science Council NSC 992628B006015MY3 and NSC 1012118M009006MY2.
Declarations
The full funding for the publication fee came from Taiwan National Science Council and College of Electrical Engineering and Computer Science, National Cheng Kung University.
This article has been published as part of BMC Systems Biology Volume 7 Supplement 6, 2013: Selected articles from the 24th International Conference on Genome Informatics (GIW2013). The full contents of the supplement are available online at http://www.biomedcentral.com/bmcsystbiol/supplements/7/S6.
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