Two dimensional polyacrylamide gel electrophoresis (2D PAGE) is used to identify differentially expressed proteins and may be applied to biomarker discovery. A limitation of this approach is the inability to detect a protein when its concentration falls below the limit of detection. Consequently, differential expression of proteins may be missed when the level of a protein in the cases or controls is below the limit of detection for 2D PAGE. Standard statistical techniques have difficulty dealing with undetected proteins. To address this issue, we propose a mixture model that takes into account both detected and non-detected proteins. Non-detected proteins are classified either as (a) proteins that are not expressed in at least one replicate, or (b) proteins that are expressed but are below the limit of detection. We obtain maximum likelihood estimates of the parameters of the mixture model, including the group-specific probability of expression and mean expression intensities. Differentially expressed proteins can be detected by using a Likelihood Ratio Test (LRT). Our simulation results, using data generated from biological experiments, show that the likelihood model has higher statistical power than standard statistical approaches to detect differentially expressed proteins. An R package, Slider (Statistical Likelihood model for Identifying Differential Expression in R), is freely available at http: //www. cebl. auckland. ac. nz/slider. php.
Two-dimensional polyacrylamide gel electrophoresis (2D PAGE) [1] separates thousands of proteins within a sample by their isoelectric points (pI) in the first dimension and their molecular weights in the second dimension. Gels are scanned and spot detection performed using commercial or in-house software packages. These programs convert gel images into vectors of matched spot volumes and most analyses are subsequently performed on these data [2]. 2D PAGE may be used to identify proteins that differentiate or characterize certain patient groups or sample sets. For instance, by comparing specimens from patients with a specified disease to a control group, statistical differences in the levels of proteins can be determined to identify proteins associated with a disease state that may serve as diagnostic or prognostic biomarkers [3]. Several statistical tests have been applied to detect differences in protein expression. These include the use of classical Student' s t-test, Analyses of Variance [2], principle component analysis and partial least squares analysis [4], [5]. A key disadvantage with these methods is their failure to adequately address the difficulties of dealing with non-expressed or undetected proteins in some or all subjects within a group [6], [7]. There are three broad reasons to explain why a given protein may not be detected in 2D PAGE experiments: (1) the lack of sensitivity of the experimental setup or software to detect the presence of an expressed protein, usually a consequence of some threshold of detectable concentration [8]; (2) the true absence or non-expression of a protein; and (3) software-induced error, when proteins are incorrectly designated as being absent [9]. Some researchers have developed methods that impute missing values from the existing data [6]. However, without knowing the true causes of these missing values, imputation may introduce additional errors to the dataset [5], [7]; in particular, by ignoring the possibility that a protein may not be expressed in a certain group of subjects, imputation may lead to an elevation in the numbers of false negatives. The problem of missing values may be addressed through the incorporation of missing observations into a statistical model of the data. Under the principle of likelihood, estimates of parameters (such as the mean expression intensity or the probability of expression) may then be obtained by computing the probability of obtaining the observed data, given different values of these parameters. The best estimates are those that maximize this probability, which is also called the maximum likelihood. Wood and co-workers [10] first proposed a statistical method to compute the likelihood for expressed proteins which simultaneously takes missing data into account along with expression profiles. Their method does not distinguish the processes that may account for why a protein is undetected. This means that the probability associated with non-detection is a composite of the probabilities of protein non-expression or expression below the level of detection. In our paper, a new likelihood model is proposed that extends the approach of Wood et al. and is specifically applicable to situations where subjects belong to either a Case group or a Control group, in keeping with a case-control experimental design. This extended model allows for non-detected proteins and classifies them into two categories: either (a) the protein truly is not expressed, or (b) the protein is expressed but the expression level is below the limit of detection. We show how our proposed new method performs under simulations and compare results with standard statistical approaches commonly applied to detect differences in protein expression between groups. We also present an example using a subset of spots from a Case-Control 2D PAGE experiment.
In this paper, we developed a likelihood-based approach by using two statistical distributions to describe the data (i. e. , a mixture model) to identifying proteins that are differentially expressed between two groups. True differential expression, under our definition, implies either a difference in the probabilities of expression between the two groups, or a difference in the mean expression levels, or both. Several standard statistical approaches only consider the difference in mean expression intensities. For any 2D PAGE experiments we should attempt to find the maximum number of truly differentially expressed spots and minimize both false positives and false negatives. The likelihood model classifies proteins that are undetected in some gels either as potentially expressed proteins that fall below the level of detection, or proteins that are not expressed. In so doing, the model tries to build a well-defined and biologically plausible picture of comparative protein expression. In contrast, standard statistical analyses (e. g. Student' s t-tests) are forced to ignore “missing” proteins, or require some ad hoc pre-processing of data such as the replacement of missing values by a global constant or some other more sophisticated imputation process [6]. However, attempting to impute missing values when a protein is truly not expressed effectively increases the error. Inappropriate analytical methods can lead to loss of important information and potentially incorrect conclusions. For example, if a protein is expressed only in Cases, or only in Controls, application of standard statistical approaches may result in failure to recognize that the protein is a potential biomarker for that disease. Our simulations highlight the contrast between the likelihood-based approach and the use of Student' s t-tests. The performance of these approaches is summarized in Table 6. This table illustrates that LRT performs well under all four analyses, while the performance of Student' s t-test varies between each analysis. In particular, when there are proteins that have not been identified in some gels, and are classed as “missing”, there are two kinds of t-test one may apply: one can choose to exclude “missing” values or one can replace these values with a global minimum. In two of our four sets of simulations, the Student' s t-test in which missing values were replaced by a global constant had higher power than the LRT. This is because the estimated variance is artificially deflated as a consequence of replacing many expression intensities with the same constant. In contrast, the LRT performs better than the Student' s t-test in Simulation 4, when the probabilities of protein expression are the same for the two groups, but the group-mean expression intensities differ. We expect that this situation, or one close to it, is more likely to mirror real experimental outcomes. Indeed, our application of the LRT on a small selection of proteins from a real biological experiment suggests that this is the case. We think that the likelihood model more realistically identifies the causes associated with “missing” data, and in so doing, provides a framework that is interpretable and intuitive. Mixture models are not new in the statistical literature and have been used in several other fields [16]. Similar models had been developed in other proteomic studies to handle missing values [17]; however, they have not been routinely applied to 2D PAGE experiments. The likelihood-based approach developed here can be applied to 2D PAGE experiments regardless of the physical or chemical system employed to generate the gel image and data. It also can be easily extended to allow multiple gels per patient and other, more complex, designs. What is required in these settings is to formulate appropriate distributional descriptions of the variances between gels within patients, and between patients within groups. In this regard, the process is no different from the parameterization under standard generalized linear mixture models. We are also developing extensions of this model for other proteomic data systems, including difference gel electrophoresis (DIGE) [18]. When we apply the same statistical test repeatedly, it is essential that multiple comparisons correction is applied after the analysis. Otherwise we are likely to discover large number of false positive differentially expressed proteins. In our analyses, we did not apply any correction for multiple tests, because our aim was to obtain estimates of the power and the false positive rates under different conditions. In practice, different multiple comparison procedures, such as the one proposed by Newton et. al. [19] can be implemented depending on the downstream analysis.
Many researchers use two dimensional polyacrylamide gel electrophoresis (2D PAGE) to identify proteins with different concentrations under different conditions. Several statistical methods have been used to identify these proteins, ranging from standard statistical tests to complex image analysis. Most of these methods fail to address the limitation of this technology, which is that when the concentration of a protein is too low, 2D PAGE is unable to detect this particular protein. Standard methodologies implemented in most software packages ignore these proteins completely. We propose an alternative approach based on the likelihood framework, which takes into account when the concentration of protein is above the detection level and below the threshold. Our results show that this model allows us to identify more proteins with different concentration levels under different conditions than the standard statistical approaches.
genetics and genomics/genetics of disease
biochemistry/bioinformatics
genetics and genomics/gene discovery
mathematics/statistics
A Statistical Model to Identify Differentially Expressed Proteins in 2D PAGE Gels