Outlier Detection using Projection Quantile Regression for Mass Spectrometry Data with Low Replication

Background

Mass spectrometry (MS) data are often generated from various biological or chemical experiments. Such vast data is usually analyzed automatically in a computer process consisting of pre-processing, significance test, classification, and clustering. Elaborate pre-processing is essential for successful analysis with reliable results. One pre-processing step is required to detect outliers, which which are extreme due to technical reasons. The plausible outlying observations detected can be examined carefully, and then corrected or eliminated if necessary. However, as the manual examination of all observations for outlier detection is time-consuming, plausible outlying observations must be detected automatically.

Identification of statistical outliers is the subject of some controversy in statistics[1]. Several outlier detection algorithms have been proposed for univariate data, including Grubbs’ test[2] and Dixon’s Q test[3]. These tests were designed to analyze data under the normality assumption, so that they may produce unreliable outcomes in the case of few replicates. Furthermore, they are not applicable for duplicated samples. Another naive approach to detect outliers statistically constructs lower and upper fences of differences between two samples, Q₁ - 1.5 IQR and Q₃ + 1.5 IQR, where Q₁ is the lower 25% quantile, Q₃ is the upper 25% quantile, and IQR = Q₃ - Q₁. They are claimed to be outliers if they are smaller than the lower fence or larger than the upper fence. However, this may generate a spurious result because variability is heterogeneous in high-throughput data even generated from MS experiments.

Figure1 shows the log-scale scatter plot of the technically duplicated samples under the same biological condition from a MS experiment. The variability differs according to the intensity levels in the plot, so that the naive outlier detection method, ignoring the heterogeneity of variability, may often miss true outliers at high levels and select false outliers at low levels. If a number of technical replicates for each peptide under the same biological condition can be obtained in MS experiments, the examination of outliers can be conducted for each peptide. However, a small number of replicates is usually conducted for MS experiments due to the high cost of experiments and the limited supply of biological samples.

Cho et al.[4] proposed a more elaborate approach for detecting outliers with low false positive and negative rates in MS data to solve the problem when the number of technical replicates is two. The algorithm was developed by utilizing quantile regression for duplicate MS experiments. The R package (called OutlierD) that was also developed can only be used for duplicate experiments. Therefore, we here propose a new outlier detection algorithm for multiple high-throughput experiments, particularly those with few, but more than two replicates.

Classical Approaches

Suppose that there are n replicated samples and p peptides in MS data. Then let x _ij be the i th replicated sample from experiments under the same biological or experimental condition, where i = 1,…,n and j = 1,…,p. For convenience, let$y_{\mathit{\text{ij}}}=\underset{2}{\text{log}}\left(x_{\mathit{\text{ij}}}\right)$. Typically, n is small and p is very large in high-throughput data, i.e., p >> n. In addition, let y_(1)j ≤ y_(2)j≤⋯≤ y_(n)j be ordered samples for peptide j, where$y_{\left(1\right)j}=\underset{1\le i\le n}{\text{min}}{y}_{\mathit{\text{ij}}}$ and$y_{\left(n\right)j}=\underset{1\le i\le n}{\text{max}}{y}_{\mathit{\text{ij}}}$, the smallest and the largest observations, respectively.

The denominator is the difference between the largest and smallest observations and the numerator is the difference between the smallest two values or the largest two values. If the test statistic Q _j is smaller than the critical value given by Rorabacher[5], peptide j is flagged as an outlier. If n = 2, the statistic is always 1; thus, this test is applicable for n ≥ 3.

where${\bar{y}}_{·j}$ is the sample mean and s _j the standard deviation for peptide j. The denominator is the standard deviation and the numerator is the difference between the smallest (or largest) value and the sample mean. If T _nj or T_1j is smaller than the critical value, peptide j is flagged as an outlier. If n = 2, the statistic is always$1/\sqrt{2}$; thus, this test is also applicable for n ≥ 3.

Proposed Methods

In duplicated experiments (n = 2), two observed values, x_1j and x_2j for each j, should be theoretically identical, but are not identical in practice due to their variability. Even though they are not identical, they should not differ substantially. The tolerance of the difference between the two observed values from the same condition is not constant because their variability is heterogeneous. The variability of high-throughput data depends on intensity levels.

Cho et al.[4] proposed the construction of lower and upper fences using quantile regression in an MA plot with M and A values in vertical and horizontal axes, respectively, where M _j is the difference between replicated samples for j and A _j is the average, i.e.$M_j=y_{1j}-y_{2j}=\underset{2}{\text{log}}(x_{1j}/x_{2j})$ and$A_j=(y_{1j}+y_{2j})/2=(1/2)\underset{2}{\text{log}}\left(x_{1j}{x}_{2j}\right)$ to detect the outliers accounting for the heterogeneity of variability.

In multiple experiments (n ≥ 2), it is natural to investigate outliers based on all observed values in a high-dimensional space. An outlier will be a very large distance from the center of the distribution of a peptide. The cutoffs of distances for classification of outliers depend on the degree of variability from the center. The degree of variability is dependent on intensity levels and the center can be defined as the 45° line from the origin. More flexibly, the center can be obtained by principal component analysis (PCA), as seen in Figure2. The first principal component (PC) becomes the center of each intensity level, i.e., a new axis for intensity levels. The experiments are replicated under the same biological and technical condition; hence, most variation can be explained by the first PC. It implies that it is enough to use the first PC practically. An outlier will have a large distance from its projection. Following the notations for applying quantile regression, we can define the distance of peptide j to the projection as M _j and the length of the projection on the new axis as A _j . Then the first and third quantiles can be obtained by applying quantile regression on an MA plot with M and A in the vertical and horizontal axes, repectively; hence, the upper and lower fences can be constructed to classify the outliers.

where 0 < q < 1, and g(A _j ;θ₀θ₁) = θ₀ + θ₁A _j . Using Equation (3), the 0.25 and 0.75 quantile estimates, Q₁(A) and Q₃(A), are calculated depending on the levels A. Then, the lower and upper fences are constructed: Q₁(A) - kIQR(A) and Q₃(A) + kIQR(A), where IQR(A) = Q₃(A) - Q₁(A) and k is a tuning parameter. We set k to 1.5 as the default value in our algorithm and software program because the value is practically often used. A larger k value selects fewer peptides, while a smaller k selects more outliers. The value can be adjusted empirically according to the magnitude of the variation of the data.

where θ₁ is the asymptote, θ₂ is the log rate, and θ₃ is the value of A at which the response becomes zero. In addition, Self-starting, Frank, Asymptotic with Offset and Copula functions can be employed. For nonparametric quantile regression, we utilize smoothing spline with the total variation regularization for univariate data to our algorithm[10]. A smoothing parameter plays a role in adjusting the degree of smoothness. We set it to 1 as the default, but it can be changed by users. The algorithm using projection can be summarized as follows.

Proposed Algorithm

This projection approach utilizes all the replicates simultaneously, and a high-dimensional problem reduces to two-dimensional one that can easily be solved. Shifts from biased experiments can be ignored due to the use of PCA.

Simulated data

where B _j ∼Bernoulli(1/2) and Z _j ∼N(1/μ _j ,0.01). The relationships between the means and the variances are shown in Figure3. For 950 non-outliers (j = 1,…,950), we assumed that$Y_{\mathit{\text{ij}}}\sim N({\mu }_j,{\sigma }_j^2)$ for i = 1,…,n. For 50 outliers (j = 951,…,1000), we assumed that$Y_{\mathit{\text{ij}}}\sim N({\mu }_j^{\prime },{\sigma }_j^2)$ for one of the samples and$Y_{\mathit{\text{ij}}}\sim N({\mu }_j,{\sigma }_j^2)$ for the other samples, where μ _j ∼ U(5,35) and${\mu }_j^{\prime }$=μ _j + (2B _j −1)U(1,2) for constant variance and${\mu }_j^{\prime }$= μ _j + (2B _j −1)(120/μ _j )U(1,2) for other variances. Thus, an artificial data set for each n was generated with 950 non-outliers and 50 outliers. Then, the data were used to check the sensitivities (the probabilities of detecting outliers correctly), specificities (the probabilities of detecting non-outliers correctly), and accuracies (the probabilities of detecting outliers or non-outliers correctly) of the quantile and projection quantile approaches for n = 2 and the Dixon test, Grubbs’s test, and projection quantile approaches for n = 3,…,8. Constant, linear, nonlinear, and nonparametric quantile regressions were accounted for the quantile and projection quantile approaches. This procedure was repeated 1000 times independently.

Table1 presents the sensitivities, specificities, and accuracies of the quantile and projection quantile methods for the simulated data from duplicated experiments (n = 2) and Figure4 shows their confidence intervals. The classical methods were not applied because they work only for n > 2. Under the constant variance, all the methods performed well. Under the linear, nonlinear, and nonparametric variances, the quantile and projection quantile methods with constant quantile regression performed worse than those with the other quantile regression due to the heterogeneity of the variability, as shown in Cho et al.[4]. When comparing the quantile and projection quantile methods, the latter sometimes had somewhat lower sensitivities than the former. However, the quantile and projection quantile methods are mostly comparable.

		Simulated Under
n	Method	Constant	Linear	Nonlinear	Nonparametric
	Quantile
	Constant	(85.0, 99.5, 98.8)	(84.7, 93.1, 92.6)	(94.3, 87.6, 87.9)	(94.3, 87.7, 88.0)
	Linear	(85.0, 99.5, 98.8)	(83.7, 99.3, 98.5)	(87.7, 94.7, 94.4)	(87.3, 94.7, 94.3)
	Nonlinear	(85.0, 99.5, 98.8)	(83.3, 99.3, 98.5)	(87.7, 94.8, 94.5)	(86.9, 94.9, 94.5)
	Nonparametric	(79.0, 99.2, 98.2)	(81.6, 99.1, 98.2)	(84.8, 99.0, 98.3)	(84.8, 99.0, 98.3)
2	Projection Quantile
	Constant	(88.9, 99.1, 98.6)	(69.7, 97.0, 95.7)	(78.6, 94.1, 93.4)	(78.8, 94.1, 93.3)
	Linear	(88.8, 99.1, 98.5)	(86.5, 98.9, 98.3)	(88.5, 96.1, 95.7)	(88.2, 96.1, 95.7)
	Nonlinear	(88.8, 99.1, 98.5)	(86.5, 98.9, 98.3)	(88.3, 98.0, 97.6)	(87.9, 98.0, 97.4)
	Nonparametric	(83.2, 98.7, 97.9)	(84.4, 98.7, 98.0)	(86.6, 98.6, 98.0)	(86.0, 98.5, 97.9)

Real-life data

We here illustrate the projection quantile approach with real-life data obtained from three replicates of LC/MS/MS experiments with 922 peptides (n = 3 and p = 922). The details of the experiments can be found in Min et al.[11] and Cho et al.[4]. Here, the primary goal of the analysis is to detect outliers automatically in the pre-processing step prior to further analysis.

To use the projection approach, we first investigate how much the first PC explains the variation in the data. The first PC takes 96.9% of the variation and the second and third PCs take 1.73% and 1.34%, respectively. This supports that it is enough to use only the first PC. The projection approach with constant, linear, nonlinear, and nonparametric quantile regression selected 74, 69, 99, and 67, respectively. The 3-D scatter plot of the data, shown in Figure5, revealed the variability of the data to be heterogeneous. Constant quantile regression tended to select more peptides at low levels as outliers, whereas the others selected more peptides at the higher levels.

This implies that the projection approach assuming a constant variance can generate many false positives and/or false negatives and, therefore, that more flexible quantile regression is more appropriate than constant quantile regression.