Multiple imputation

Multiple imputation (MI) was originally designed to handle missingness in public-use large data sets [12]. The application of the MI process has been extended to various big data sets including microarrays [13]. The method replaces each missing value with multiple substitute values, say m, that represent the probability distribution of the missing value. A completed dataset is created by each set of draws. So the m imputations for each missing value create m complete data sets. They are stored in an auxiliary matrix, multiply-imputed data sets with one row for missing value and m columns. The first row of this matrix corresponds to the first set of imputed values of the missing values, and so on. As the complete-data analyses are applied to each multiply-imputed dataset (treating imputed values as fully observed and independent), m different sets of the parameter estimates and their variance-covariance matrices are generated. To combine the inferences from them, [12] suggests to take an average of all results, except the standard error (SE) term. The SE is constructed by the within variance of each dataset as well as the variance between imputed items on each dataset. These two variances are added together and the square root of them determines the SE. The author recommends to use no more than 5 imputations and sometimes as small number as 2 or 3 to generate useful statistical inferences. We use m = 5 for MI techniques in our analysis. It is important to note that the complete-data analyses in MI treat the imputed data as though they had been fully observed. This approach does not consider any dependence of the imputed data on the actual fully observed data.

MI using Markov chain Monte Carlo (MCMC)

Multiple imputed data sets can be generated by the MCMC method, which is applied to an arbitrary missing data pattern that assumes multivariate normality. MCMC has been used to explore posterior probability distributions to express unknown parameters in Bayesian inferences. Using this method, the entire joint posterior distribution of the unknown quantities is simulated and the parameter estimates based on the simulation are generated [14].

This process can be described in two steps. The first step is the imputation I-step which randomly draws values for missing values from the assumed distribution of missing values given observed values using the estimated mean vector and variance-covariance matrix, i.e. it draws values for $Y_{mis}^{(t+1)}$ from p(Y_mis|Y_obs, θ^t), where Y_mis and Y_obs are variables with missing values and observed values, respectively, and θ^t is a parameter estimate at the t^th iteration.

The posterior P-step randomly simulates the population mean vector and variance-covariance matrix from the complete sample estimates, i.e. it draws θ^(t+1) from $p(\theta |Y_{obs},Y_{mis}^{(t+1)})$. These new estimates are then used in the I-step. This creates a Markov chain $(Y_{mis}^{(1)},{\theta }^{(1)})$, $(Y_{mis}^{(2)},{\theta }^{(2)})$, …, which converges in distribution to p(Y_mis, θ|Y_obs). Enough iterations are carried out to have reliable results for a multiply imputed dataset and to converge to its stationary distribution from which we can simulate an approximately random draw of the missing values [15].

MI using Expectation-Maximization (EM) with bootstrapping algorithms

The EM algorithm is a very general iterative algorithm for maximum likelihood estimation of missing data [9]. One assumes a model for the data, maximizes the likelihood under the assumed model, obtains parameter estimates, and makes inferences based on the parameter estimates. The explicit form of parameter estimates does not usually exist for missing data. Here numerical methods like the Newton-Raphson algorithm are very complicated to use. Thus one can apply the EM algorithm which is an iterative method for maximizing the likelihood in missing data [10]. Compared to the Newton-Raphson algorithm, the EM algorithm is slower, but it increases the likelihood with each iteration and surely converges to a maximum for the distribution with one mode. The EM algorithm converges to a local maximum or a saddle point for the distribution with multiple modes.

The EM algorithm consists of two steps, the Expectation (E) and the Maximization (M) steps. The algorithm calculates the conditional expectation of missing values given non-missing values and current parameter estimates in the expectation step. In the maximization step the calculated expected values are used to maximize the likelihood of the complete data. These steps are iterated until the maximum likelihood of data converges. The EM algorithm may not have an explicit form. In this case, the maximization could be theoretically obtained using iterations in the maximization step.

The maximization step can be computationally expensive, which can make the EM algorithm unattractive. Fortunately, the EM with bootstrapping algorithm resolves this problem. It uses the conventional EM algorithm on multiple bootstrapped samples of the original missing data to draw values of the complete-data parameters. Then it draws imputed values from each set of bootstrapped parameters, replacing the missing values with these draws. The EM with bootstrapping algorithm can impute missing values in much less time than the EM algorithm itself [11].

K-Nearest Neighbors (KNN): modified and accounting for dependence KNN in general

The conventional KNN method replaces missing values using k-most similar non-missing subjects’ values [16, 17]. It can impute both discrete attributes (using the most frequent value among the k-nearest neighbors) and continuous attributes (using the mean among the k-nearest neighbors).

[8] implemented the KNN method that weights the contribution of each nearest neighbor by its similarity to the subject with the missing value. In our CRC study, the weights of the nearest neighbors in the imputation of missing value are measured by the Euclidean distance metrics of demographic and lifestyle variables such that the nearer neighbors to the subject contribute more to its imputation than the more distant ones. Based on the weighting method of [8], we briefly outline our weight calculations here. Let k be the chosen number of nearest neighbors, D_i1 ≤ … ≤ D_ik be the sorted distances of the k nearest neighbors from normal-missing subject i, and $D_i^{(max)}$ be the maximum distance (among all fully-observed subjects) from subject i. Then the weights a_i1, …, a_ik among the k nearest neighbors for subject i are obtained as follows: (1) These weights are used by the weighted KNN method to impute missing expression values of a particular gene as in Eq (2).

Our proposed imputation method accounts for the dependence induced by weighted KNN and can use the additional covariates such as demographic, general health, genetic, and lifestyle variables, as well as other biologically related information. The proposed imputation method takes advantage of the conventional KNN [16, 17] and further developed weighted KNN [8] imputation methods’ robustness to missing data, non-parametric approach, and speed in estimating missing values for microarray data, while considering the correlation structure of the data. In order to impute missing samples in the above mentioned motivating CRC case study, the proposed method has been modified to impute expressions for all miRNA of missing normal samples based on multivariate covariates (demographic and lifestyle variables) and to account for the dependence of the imputed data in subsequent differential expression tests. The demographic and lifestyle variables considered in this paper are five continuous (age, number of cigarettes/day, calories, BMI (Body mass index), and lutein and zeaxanthin concentration) and five binary (gender, recent aspirin/NSAID (Non-steroidal anti-inflammatory drug) use, recent smoker, menopause, and post menopause taking HRT (Hormone replacement therapy) within 2 years statuses) variables.

This modified KNN technique imputes all miRNA expression levels of missing normal samples by finding the k most similar subjects, not gene expression levels as in conventional KNN-based methods, based on the distance matrices of demographic and lifestyle covariates of patients and produces the variance-covariance matrices for each miRNA. For example, we can estimate the miRNA expression levels in missing normal tissues from a particular subject, based on the expression levels of scanned normal tissues from subjects who have similar demographic and lifestyle covariates.

Another advantage of this method is that it can integrate simultaneously multivariate covariates by aggregating and normalizing their distance matrices (Euclidean, Manhattan, Minkowski, and etc.) to find the nearest neighbor subjects. Specifically, two between-subject distance matrices are constructed based on the fully observed continuous and discrete covariates separately, using Euclidean and Manhattan distances, respectively. These two distance matrices are normalized by scaling between 0 and 1 [18] and aggregated by taking the weighted average of each distance matrix to achieve a single between-subject distance matrix.

Choice of optimal k

There have been many studies carried out to determine the optimal choice (parameter) of k for the KNN algorithm. [17] suggest to use the square root of the average number of complete cases after missing data removal, rounded to the nearest odd integer. The simulation studies of different k on Likert data [19] show the square root of the number of complete cases which is rounded to the nearest odd integer is a suitable choice for k. Moreover, [20] report on k = 10 for large data like from microarrays. [8] argue that the imputation method is fairly insensitive to the choice of k in the range 10–20. As k gets larger, the average distance to the neighbors increases which implies that the imputed value could be less accurate and the imputation time will increase.

However, the choice of a small k diminishes the KNN performance because the imputation process overemphasizes a few dominant genes (or subjects in our modification) in estimating the missing values. On the other hand, a large k may include genes (or subjects) that are significantly different from the missing values that may result in degrading the imputation performance.

Accounting for dependence of KNN-imputed data

Because the weighted KNN-imputed expression values are linear combinations of expression values of the fully observed subjects’ expression values, the imputed values are not necessarily independent of the fully observed values. The modified KNN-based imputation method has an advantage of considering this dependence induced by weighted KNN by providing variance-covariance matrices of each miRNA, which can be used when searching for differentially expressed miRNAs. We refer to this method as “KNN dependent”, while referring to the KNN imputation method that ignores the dependence as “KNN independent” in this paper. Its algorithm works almost the same as the algorithms of the conventional KNN-based methods, except it treats the rows as subjects or samples, and the columns as miRNAs.

To see how the proposed imputation method estimates the miRNA expression levels in missing normal samples and accounts for the dependence induced by the weighted KNN, suppose that in the CRC study of N subjects, we want to estimate expression levels of G miRNAs for normal samples of missing S subjects using demographic and lifestyle covariate data. For each normal-missing subject i, we find the k most similar subjects with non-missing normal samples (say subjects i₁, …, i_k), and impute the missing miRNA expression values by multiplying the miRNA expressions from normal samples of the k subjects with their corresponding weights, which are generated from the between-subject distance matrix. The imputation of the expression level of miRNA j in missing normal sample i will be produced as in Eq (2): (2)

Here, i = 1, …, S and j = 1, …, G. x_lj is the observed expression value of miRNA j in the observed normal sample of subject l, and a_lj is the weight of the subject in the imputation. The weights a_i1, …, a_ik are obtained as outlined in Eq (1) above. We can generalize Eq (2) to Eq (3): (3)

Here, $\widehat{\underset{\sim }{X}}$ is an S × G matrix of imputed normal tissue expression values, $\underset{\sim }{A}$ is a (N−S) × S matrix of weights a, and $\underset{\sim }{X}$ is a (N−S) × G matrix of observed normal tissue expression values. In column i of $\underset{\sim }{A}$, the only non-zero elements are in rows i₁, i₂, …, i_k, and are the coefficients a_i1, a_i2, …, a_ik in Eq (2).

The variance-covariance matrix of the normal tissue expression for miRNA j will be calculated as in Eq (4), assuming the order in the data is the fully observed N−S subjects followed by the S normal-missing subjects: (4)

Here, ${\sigma }_j^2$ is the variance of miRNA j and $\underset{\sim }{I}$ is the (N−S) × (N−S) identity matrix of non-missing subjects to represent the independence among non-missing subjects. The matrix part of the right-hand side of Eq (4) is denoted by ${\underset{\sim }{\Sigma }}_j$.

Testing for differential expression (DE) of miRNA while accounting for dependence

The paired t-test [21] may be used to check whether the miRNAs are differentially expressed in paired normal-tumor samples while accounting for the dependence induced by the imputation method. The paired t-test can be simplified to a one sample t-test of the difference of normal and tumor samples. The per-miRNA null hypothesis is that the difference of mean expression levels of miRNAs between normal and tumor samples is equal to zero. The test statistic for miRNA j can be found beginning with the following equation, as discussed in chapter 3 of [22]. (5)

Here, ${\underset{\sim }{D}}_j$ is a N × 1 vector of the difference of the j^th miRNA expressions for normal and tumor samples, μ_j is a single parameter representing the difference of mean expression levels of miRNA j between normal and tumor samples, and $\underset{\sim }{1}$ is N × 1 vector of 1’s. $Var(\underset{\sim }{\mathit{ϵ}})={\sigma }_j^2{\underset{\sim }{V}}_j$, where ${\underset{\sim }{V}}_j$ is the variance-covariance matrix of the tumor-normal difference in miRNA expression values for miRNA j, i.e., ${\underset{\sim }{V}}_j=\underset{\sim }{I}+{\underset{\sim }{\Sigma }}_j$, and needs to be a positive definite matrix.

The mean tumor-normal difference for miRNA j can be estimated by Eq (6): (6)

The ${\widehat{\mu }}_j$ in Eq (7) can be substituted from Eq (6): (7)

Then, the estimated variance of ${\widehat{\mu }}_j$ would be calculated as in Eq (8): (8)

Finally, the test statistic will be found using Eq (9) with a degree of freedom of N−1. (9)

This paired t-test can be used with the other imputation methods by replacing ${\underset{\sim }{\Sigma }}_j$ with the identity matrix, which represents the assumed independence of imputed miRNA values.

Measuring performance

The performance of the imputation methods on miRNA data is evaluated through root mean squared error (RMSE). The RMSE-based evaluation technique is the most commonly used method to compare similarity between true expression values and imputed expression values. Various variants of RMSE measures are used in the literature: the non-normalized RMSE measure [23] and the normalized RMSE measure by different normalizing constants: average value over all observations in complete data [8], standard deviation of the values in complete data over missing entries [24, 25], and root mean square of the values in complete data over missing entries [26]. However, all above mentioned various RMSE measures provide highly similar results [27].

In the motivating CRC case study, all miRNA expression levels of up to 50% missing normal samples, i.e. up to 50% missing rows (samples) of miRNA data must be imputed. Thus, the non-normalized RMSE that measures the difference between the imputed part of matrix and the original part of matrix, divided by the number of missing cells, can be used. It is calculated as Eq (10): (10)

Here, i = 1, …, S and j = 1, …, G. x_ij is the original value for missing sample i and miRNA j, while ${\widehat{x}}_{ij}$ is the imputed value for missing sample i and miRNA j.

Optimal number of nearest neighbor subjects (k)

Fig 1 shows the effect of the number of neighbor subjects, k, used in the KNN imputation method on the RMSE values for simulated data sets with different number of subjects and percent of normal-missing subjects. The RMSE decreases, i.e. the performance of KNN imputation increases, while the value of k increases. The falling of RMSE values slows down after k value of 10, and becomes approximately the same for the rest of k values. The imputation performance becomes approximately insensitive to the value of k within the range of 10–25 neighbor subjects. Thus, we used 10 nearest neighbor subjects to estimate the miRNA expression levels of normal samples for missing subjects.

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Fig 1. The RMSE values for different number of neighbor subjects (k).

https://doi.org/10.1371/journal.pone.0119876.g001

Simulation data sets

While we have complete normal and tumor sample data for more than 400 subjects in the CRC study, we compare imputation methods using simulated data to have clearly defined power and Type I errors. The imputation analyses were performed on normally distributed paired data matrices of G = 2000 miRNA features (columns) for each of the normal and tumor samples with sample sizes of N = 50, 100, 200, and 400 subjects (rows). We simulated expression levels of miRNAs for normal and tumor samples by controlling true differentially expressed miRNAs of tumor samples across all simulations. Particularly, all miRNA features of normal samples and only non-differentially expressed miRNA features of tumor samples were simulated based on μ = 2 and σ = 1.25, while the differentially expressed miRNA features of tumor samples, which consisted of 20% of all miRNA features of tumor samples, were simulated based on μ = 2.5 and σ = 1.25. This 20% differential expression rate as well as this mean tumor-normal difference of 2.5 and standard deviation of 1.25 were chosen based on characteristics of the motivating CRC study. We randomly applied missingness from 10 to 50 percent of the normal data rows. We performed 25 simulations for each sample size with different percent missingness.

To ensure that the simulated data sets reflected the characteristics of the CRC study, and that the demographic and lifestyle variables carried some useful information for imputation, the multivariate covariate data sets with demographic and lifestyle variables of subjects were simulated based on z randomly selected true differentially expressed miRNA expression levels using the characteristics of the CRC case study covariate data. For example, a continuous variable such as age of subjects was simulated as in Eq (11): (11)

Here, j = 1, …, z, $\widehat{C}$ is a simulated value of age, β₀ is the mean age of the patients in CRC case study, and β_j is uniformly distributed with a minimum and a maximum of up to 5% of the minimum and the maximum of the CRC case study patients’ age, respectively. In this paper, we used 2% of the minimum and the maximum of the continuous variables with z = 20, which was selected for computational simplicity, to simulate variables with similar characteristics of CRC case study covariates. x_j is the expression of truly differentially expressed miRNA j in tumor, and the error term ϵ is normally distributed with zero mean (μ = 0) and variance of 10% of variance of the patients’ age (${\sigma }^2=0.1*{\sigma }_{age}^2$).

The binary variables such as gender of subjects was simulated using a logistic regression model in Eqs (12) and (13): (12)

Here, p is the probability of gender = female, say.

Eq (12) can be rewritten as Eq (13): (13)

Here, $\widehat{P}$ is a simulated probability of gender = female, β₀ is the mode of the patients’ gender in the CRC case study, and β_j is uniformly distributed as U[−0.5, 0.5]. To ensure variability in simulated binary variables, we calculate ${\widehat{P}}^{\prime }$ as in Eq (14): (14)

In our simulated study, we had denoted as a male if the value of ${\widehat{P}}^{\prime }$ was between 0 and 0.5, and as a female if the ${\widehat{P}}^{\prime }$ was bigger than 0.5 but less than or equal to 1.

Demographic and lifestyle variables were thus simulated based on characteristics of five continuous (age, number of cigarettes/day, calories, BMI, and lutein and zeaxanthin concentration) and five binary (gender, recent aspirin/NSAID use, recent smoker, menopause, and post menopause taking HRT within 2 years statuses) variables from the CRC study.

We carried out the performance analyses as follows: First, we called arbitrarily the subjects with missing normal samples. Then, we imputed expression levels of the missing normal samples using the imputation methods mentioned in the Methods section. We evaluated the performance of these imputation methods against the initial generated data matrices by calculating the RMSE for such simulated data set. Moreover, we carried out the differential expression (DE) analyses on the imputed data sets to check whether the KNN dependent method has an equal statistical power in finding differentially expressed miRNA as other imputation techniques.

Performance of imputation techniques

The performance of the modified KNN method was assessed over MI techniques using MCMC and EM with bootstrapping algorithms for data matrices with different number of subjects and different percents of normal missing subjects. In Fig 2, the modified KNN method shows consistently better performance than other imputation techniques (systematically lower RMSE values) for sample sizes of 50, 100, 200, and 400 subjects, with missing percentages of 10–50.

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Fig 2. The RMSE values for different imputation techniques.

https://doi.org/10.1371/journal.pone.0119876.g002

The KNN imputation method also shows a robustness to increasing the percent of missing normal samples and the number of subjects in miRNA data sets. It keeps relatively the same performance for all levels of missing percents and number of subjects.

Moreover, the KNN imputation method required much less computational expense than the MI techniques using MCMC and EM with bootstrapping algorithms. For example, to impute the expressions of 50% missing normal samples in 400 subjects on a machine with CPU speed of 1.86 GHz and 2 GB RAM, the KNN method took approximately 35 minutes, whereas MCMC and EM with bootstrapping algorithms took approximately 10 and 5 hours, respectively.

Differential expression (DE) testing

We applied the paired t-test to the data sets, which were imputed by various imputation methods, to see how well we could identify differentially expressed miRNAs. First, we obtained a test statistic and a p-value for each miRNA feature in each imputed data set by controlling the false discovery rate (FDR) at 0.05 within each simulation. Then, we calculated the true positive rate (TPR), the false positive rate (FPR), and the false discovery rate (FDR) based on the miRNAs which were controlled as truly differentially expressed in the simulations. The TPR and FPR were defined and calculated as in [28] and [29], and the FDR was defined as in [30].

Fig 3 shows the performance (including power and FDR control) of the paired t-tests on the data sets imputed by the KNN dependent, the KNN independent, the MI using MCMC and MI using EM algorithms, the case deletion technique, as well as on the full data set for the number of subjects of 50, 100, 200, and 400 with the missing percent of normal samples of 10%–50%. The scatter plots of TPR and FPR for the similar conditions are represented in S1 Fig (see Supplemental materials).

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Fig 3. TPR and FDR for sample sizes of 50, 100, 200, and 400 with missingness of 10%–50%.

https://doi.org/10.1371/journal.pone.0119876.g003

From Fig 3 we can see that the power (i.e., the TPR values) increases with larger sample sizes. For 400 subjects and 50% missing normal samples, which are the characteristics of the CRC case study, there are clear clusterings of TPR and FDR values, separately for full, for KNN dependent and case deletion, and for KNN independent, MCMC, and EM methods. Although the KNN dependent has slightly lower power than the other imputation methods (the TPR values are in the range of 0.93–0.98 for 400 subjects and 50% missing), it controls the FDR values below the threshold of 0.05, which is represented by red dotted lines in the figures. The KNN independent, the MCMC, and the EM with bootstrapping algorithms have the highest power (the TPR values are in the range of 0.985-1 for 400 subjects and 50% missing), but lack control of the FDR, i.e. the FDR values cross the threshold of 0.05 for all number of subjects and missing percentages. The case deletion method shows the lowest power, but maintains control of the FDR for all number of subjects and percentages of missing normal samples.