Bayesian integration of networks without gold standards

Abstract

Motivation: Biological experiments give insight into networks of processes inside a cell, but are subject to error and uncertainty. However, due to the overlap between the large number of experiments reported in public databases it is possible to assess the chances of individual observations being correct. In order to do so, existing methods rely on high-quality ‘gold standard’ reference networks, but such reference networks are not always available.

Results: We present a novel algorithm for computing the probability of network interactions that operates without gold standard reference data. We show that our algorithm outperforms existing gold standard-based methods. Finally, we apply the new algorithm to a large collection of genetic interaction and protein–protein interaction experiments.

Availability: The integrated dataset and a reference implementation of the algorithm as a plug-in for the Ondex data integration framework are available for download at http://bio-nexus.ncl.ac.uk/projects/nogold/

Contact: darren.wilkinson@ncl.ac.uk

Supplementary information: Supplementary data are available at Bioinformatics online.

1 INTRODUCTION

A significant proportion of knowledge about molecular biological processes is distributed over a large number of online databases (Stein, 2002). This knowledge has been obtained through experiments performed in laboratories all over the world. Overlaps often exist across the contents of these databases. The sub-discipline of integrative bioinformatics aims at collating this knowledge and making it accessible to both humans and computers.

A popular integration paradigm is the construction of functional networks (James et al., 2009; Lee et al., 2004; von Mering et al., 2003). Functional networks represent different types of relationships between biological entities in an abstract manner. Associations such as genetic interactions (GIs), protein–protein interactions (PPIs), gene regulation and co-expression are combined into simple abstract statements of functional relatedness, which are termed functional interactions.

An alternative paradigm is semantic data integration (Cerami et al., 2010; Cheung et al., 2005; Koehler et al., 2006; Smith et al., 2007). These approaches aim at representing the biological information (and as much of its meaning as possible) in a computationally accessible fashion. Rather than generalizing over all types of associations between entities to infer functional interactions, each type of association is considered separately.

An important question regarding such networks is how to assess the degree of confidence in each statement, that is, how likely the statement is to be correct. Several popular solutions to this problem exist for functional networks (Lee et al., 2004; Lycett, 2007). These methods assess the quality of each input dataset against one or more additional datasets of higher quality, usually manually-curated collections. Based on the confidence measures gained from this comparison it is then possible to calculate a confidence measure for each functional interaction. The high-quality datasets used in these comparisons are often referred to as ‘gold standards’.

The method described by Lee et al. evaluates each evidential dataset against such a gold standard and obtain a log likelihood score (LLS). Subsequently, for each interaction in question, a weighted sum is formed over the LLS scores of those datasets that report the interaction. The weights are chosen in a manner that represents the degree of dependency between the datasets (Lee et al., 2004).

Lycett describes a method that extends the original method of Lee et al. Not only one, but several different gold standards are used to generate LLS scores for the datasets. Furthermore, instead of creating a score for each interaction via the weighted sum described above, this method computes an existence probability from the original LLS scores and then averages over the different probabilities according to the different gold standards. The authors show that any bias inherent in the used gold standards can thus be overcome (Lycett, 2007).

These methods work very well for functional networks. However, inferring confidence assessments for semantic networks, rather than functional networks, is more challenging, because each single type of association must be scored separately. Reliable gold standards only exist for some of these types. The methods discussed above are thus only of limited use for assessing semantic networks. While solutions for specific types of data do exist, for example, PPIs (Bader et al., 2004; Braun et al., 2009; Troyanskaya et al., 2003; Venkatesan et al., 2009), these are again dependent on additional data. It would desirable to find a method that can infer confidence measures on biological networks in the absence of a gold standard, that is, based only on the existing experimental data.

To provide a more generic solution to this problem, we present a fully Bayesian method which calculates, for each statement in a semantically-integrated dataset, the probability that it is true. We have evaluated the method's effectiveness in comparison to related methods. The validity of any results of the method's application to real data is difficult to verify without knowing the absolute biological truth. Therefore, we have developed a tool that tests integration methods on simulated data with the same characteristics as real biological networks.

2 METHODS

2.1 Probabilistic integration

The complete set of interactions of a certain type within the cell (e.g. PPIs) can be modelled as a network G=(V, E), where entities, such as proteins, are nodes (vertices) V={v₁,…, v_N} and their associations are undirected edges . If one considers each pair of nodes to potentially have an edge, it is possible to model the process of the experimental prediction of such an edge as described below.

Let X={X₁,…, X_n} be a collection of n networks which have been experimentally derived from G. Considering a single potential edge e, each experiment X_i makes a statement about e's existence. Let D^e_i be a random variable that assumes realization 1 when the i−th experiment X_i predicts that the edge exists and 0 when it predicts that the edge does not exist. Let d^e_i be the measured realization from X_i, then (D^e_i=d^e_i) is the event that the measured realization in experiment i is d^e_i. Furthermore, let D^e_(n) be the vector of all n experimental measurement events (D^e_i=d^e_i) for the edge. Finally, let L^e be the event that the edge really does exist in G (e∈E). We are interested in ℙ(L^e|D^e_(n)), that is, the probability that the edge really exists given all our experimental measurements.

An important concept necessary to determine this probability is the Bayes factor (Kass and Raftery, 1995). For each of the n experiments a Bayes factor Λ_i can be determined, which is defined as

(1)

If experiment i predicts that the edge exists, then Λ_i is the probability of a true positive in i divided by the probability of a false positive in i. Otherwise, if i predicts that the edge does not exist, then Λ_i is the probability of a false negative in i divided by the probability of a true negative in i.

Then, under the assumption that all measurements are independent from each other, Bayes theorem can be used to show that

(2)

where 𝕆(·) is used to denote the odds of an event, and is defined for an arbitrary event F by

(3)

A full proof of Equation (2) is provided in the Supplementary Methods. This recursive equation can be expressed iteratively as

(4)

hat is, the odds of the edge existing, given all the experimental measurements, is the product of the Bayes factors for these measurements with the prior odds of edge existence. The specification of prior odds 𝕆(L^e) is described in the Supplementary Methods. Odds can obviously be converted into the corresponding probability using inversion.

As mentioned above, Equation (2) and thus also Equation (4) work under the assumption that experimental measurements are stochastically independent. This assumption is not valid for real data, and thus introduces a potential source of error into the methodology. Lee et al. address this problem by introducing dependency coefficients for their datasets (Lee et al., 2004), a solution which we argue is somewhat ad hoc. Instead we assume independence and focus on verifying that our method is robust to this assumption.

In order to calculate the Bayes factors above it is necessary to determine the rates of false positives and false negatives in each dataset. One approach is to compare each dataset to a gold standard and count the number of differences. However, due to the limited availability of gold standards as discussed above, this approach is not feasible here. Therefore, the only available option is to evaluate the datasets against a common consensus. A naive approach is to start with random values for the error rates, and to use these rates to create a candidate integrated network. Updated parameter values and the resulting integrated networks can then be computed iteratively. However, a series of networks produced by this method does not typically converge to any sensible result (data not shown).

To overcome this problem, a fully Bayesian approach was employed to generate samples from the full joint posterior distribution of π(θ, G|X), where θ={(α₁, β₁),…, (α_n, β_n)} is the vector of error rates associated with the members of the vector of experimental networks X, and where α_i is the false positive rate of X_i and β_i is the false negative rate of X_i.

To determine the joint posterior distribution π(θ, G|X), one may exploit the following equation:

(5)

Thus, the posterior distribution π(θ, G|X) is proportional to the joint distribution π(θ, G, X). The joint distribution may in turn be factored as:

(6)

In summary, we conclude that

(7)

As a consequence, three values need to be determined: the prior distribution π(G), the prior distribution π(θ) and the likelihood ℙ(X|θ, G). We define the prior distribution of π(G) as a random graph prior:

(8)

(9)

where q is the prior probability of an edge really existing.

To determine the prior distribution π(θ), we have to consider the nature of the error rates α_i and β_i as ‘success rates’ for misreading each potential edge. Modelling each observation event over a potential edge as a Bernoulli experiment with such a success rate, the number of false positives and false negatives in an experimental graph X_i would follow a binomial distribution. The Beta distribution is conjugate to this binomial likelihood, and is dependent on two parameters, a and b. We make the assumptions:

(10)

(11)

For the sake of simplicity, we will later assume that all prior parameters are equal to 1, giving _[0,1] priors for all rates. Since sampling from π(θ, G|X) directly would be very difficult, we instead employ a Gibbs sampling approach (Gelfand and Smith, 1990) and alternately sample from π(θ|G, X) and π(G|θ, X).

The algorithm proceeds in cycles. At the beginning of each new cycle, a potential true graph G needs to be sampled based on the error rate vector θ. The sampling is accomplished by using the Bayesian method discussed above to infer posterior existence probabilities for each edge. These probabilities can then be used to sample a potential G by lookup. That is, for each potential edge, a _[0,1] random number is sampled. If that random number is smaller than the posterior existence probability of that edge, G will contain the edge. Otherwise G will not contain the edge.

The second step in each cycle is the sampling of a new error rate vector θ based on G. As explained above, Beta distributions can be used to describe uncertainty about θ. One can compare each X_i to the currently assumed true graph G and use it to count the number of supposed true positives (tp), false positives (fp), true negatives (tn) and false negatives (fn). Then, the full conditionals for α and β are as follows:

(12)

(13)

To initiate the algorithm as a whole, we need to generate an initial error rate vector θ_init. It is sufficient to sample the initial values for each α_i and β_i from their prior distribution.

Pseudocode 1. —

The Gibbs sampling algorithm for sampling from π(θ, G|X).

2.2 Evaluation method

In order to evaluate the method described above and to compare it against other probabilistic integration methods, a simulation and testing environment was created. The testing tool creates a random graph according to a specified model. In the simulation, this graph assumes the role of a true biological graph. The tool then derives a set of graphs from the true graph with pre-determined error rates. In the simulation these graphs assume the role of experimental datasets. The simulated experimental datasets are subsequently passed to the integration method under investigation. The integrated graph resulting from the integration method is then compared with the original simulated true graph to evaluate the integration method's performance.

Such a testing workflow can be programmed to be executed a large number of times in order to measure a method's average behaviour. Furthermore, the testing tool allows for the automatic variation of different input parameters.

The simulated true graph is created as a scale-free graph using a preferential attachment algorithm, since many molecular-biological graphs have been shown to be approximately scale-free (Eisenberg and Levanon, 2003; Jeong et al., 2001). A description of the algorithm can be found in the Supplementary Methods. Not only does the choice of scale-free background graphs more closely match the topology of real biological graphs; it also poses an additional challenge for the new algorithm, since such graphs break with the assumption of a random graph prior distribution for G.

The next step consists of the simulation of experimental measurements on the true graph. This is the most crucial step of the artificial testing environment as it is responsible for replicating all the different faults and problems of real data. The simplest type of error occurring in experimental measurements is random noise. This type of error is easily simulated by randomly inserting edges that do not exist in the real data and removing edges that do exist in the real data until the desired error rates are reached.

The next problem is systematic error, also known as experimental bias. This phenomenon in particular leads to the violation of stochastic independence between datasets. We simulate this by sampling separate false negative probabilities for each interaction and false positive probabilities for each non-interacting node pair from Beta distributions. We then use these prepared probabilities to generate false positives and false negatives in the experiments, thus introducing the same bias/systematic error. More detail is provided in the Supplementary Methods.

After the simulated evidential graphs have been given to the integration method in question the resulting integrated network is evaluated against the original graph. There are a number of different quality measures that can be applied. One important aspect is to measure the accuracy of error rate estimates in the individual experiments. We can define the quadratic loss for the error estimates as follows:

(14)

where and are the estimates of the false positive and false negative rates for experiment i according to the integration method in question.

Also, to measure the accuracy of the final edge probabilities produced by the method, we can define further loss functions. Since we can expect a vast number of true negatives when working with sparse, scale-free graphs, it would be helpful to see the loss over interacting and non-interacting node pairs separately. These can be interpreted as analogue to the algorithm's false negative rate and false positive rate:

(15)

(16)

We have evaluated the new algorithm in comparison with two gold standard-based methods presented by Lee et al. (2004) and Lycett (2007). For our evaluation we have examined the following scenario: each integration method is given the task of processing a number (3, 5, 7, 9 and 11) of experimental scale-free networks with 500 nodes, an average false negative rate 0.15 and average false discovery rate of 0.15 (corresponds to false positive rate of 0.0006). For the two gold standard-based methods (Lee and Lycett), one bias-free input experiment with the same FN and FDR rates was assigned as the gold standard. The additional input parameters for the gold standard-based methods (such as the dependency factor) have been set to optimal values to ensure peak performance. We have run 5000 replicates of the above test and averaged over the results. As a basic benchmark, we have also executed the same workflow for a naive integration method. The naive method simply assigns the observed proportion of experiments that support a given edge as its existence probability.

2.3 Application to biological networks

In addition to the evaluation on artificial data we applied the new method to semantically integrated biological data. We used the data integration system Ondex (Koehler et al., 2006) to gather as much data as possible on the Saccharomyces cerevisiae PPI and GI network. We imported all S. cerevisiae data from the BioGRID (Breitkreutz et al., 2008), MINT (Chatr-aryamontri et al., 2007), IntAct (Hermjakob et al., 2004) and MIPS-MPACT (Guldener, 2006) via the PSI-MI 2.5 XML (Kerrien et al., 2007) format. The resulting Ondex dataset represented proteins and genetic features, as well as their interactions, the experiments in which the interactions have been observed and the publications in which the experiments were described. To identify and interlink equivalent entries we used a semantic merger method. This method carefully identifies and resolves redundancies between data from the different databases, while preserving separate any separate lines of evidence. Finally, we summarized small low-throughput experiments (<20 nodes) into larger groups according to their experimental type, since low sample sizes can be expected to lead to very imprecise error rate estimates. To determine the above cut-off, we analyzed the variance of the sampling distribution with respect to the sample size. The chosen cut-off excludes datasets for which the statistical power substantially drops. Further details on the semantic integration procedure can be found in the Supplementary Material.

Having established a semantic knowledge network of physical and GIs within S. cerevisiae we executed the new algorithm over all 51 006 contained physical interactions (PSI-MI ontology ID MI:0915) and all 15 006 contained synthetic GIs (PSI-MI ontology ID MI:0794), as an exemplary type of GI. For comparison purposes, we also executed the naive integration method described above on the physical interaction data.

3 RESULTS AND DISCUSSION

3.1 Evaluation on simulated data

Figure 1 shows the performances of the four tested methods (Naive integration, Lee et al., Lycett and the method proposed here) over different numbers of input datasets. It is clearly visible that the new fully Bayesian method overall outperforms both gold standard-based methods. For low numbers of input datasets, the fully Bayesian method's loss over existing interactions (L⁽⁺⁾) is substantially lower and is the only method to perform better than the naive approach for three input experiments. Given more than five experiments, all methods show comparably low loss of existing interactions. The variance in performance is at comparable levels in all methods.

Fig. 1.

Evaluation of the algorithm in comparison with the naive method as well as the GS-based methods by Lee et al. and Lycett given different numbers of experiments. Losses are averaged over 5000 replicates. Top: average loss over interacting node pairs (L⁽⁺⁾). Centre: average loss over non-interacting node pairs (L⁽⁻⁾). Bottom: average loss regarding error rate estimates (L_ER). Whiskers indicate one SD. The naive method does not estimate error rates and is thus excluded from this metric. The performance of the gold standard-based methods regarding error rate estimation cannot be expected to improve with the number of experiments, since their estimates are always based on the gold standard and not on the experiments.

Regarding non-existing interactions the fully Bayesian method constantly performs better than Lycett's method. Lee's method shows unexpected behaviour here as its performance on non-existing edges actually worsens with increasing numbers of datasets. Both the mean and the variance of its loss function increase steadily. This is most likely caused by the method not taking into account any available negative evidence. The Lee method's central weighted sum only comprises the positive LLS scores for datasets that support an edge, but no negative components for datasets that do not support the same edge. Thus, with a rising number of datasets it becomes increasingly likely for non-existing edges to be misclassified. The proposed method, on the other hand, improves strongly with a growing number of input datasets. As expected, the SDs regarding the L⁽⁻⁾ metric are ~100 times smaller than those for the L⁽⁺⁾ metric, since the number of existing edges is very small compared with the number of potential edges.

The average loss regarding the θ estimates (L_ER) shows the fully Bayesian method's superior precision, which improves in both mean and variance with the number of input experiments, while both gold standard-based methods stagnate at constant large amounts of loss. This observation is to be expected, since these methods do not use the input datasets in order to predict error rates, but instead only rely on the gold standard.

3.2 Interaction probabilities for S. cerevisiae

As discussed in Section 2.3, we have applied the proposed method to physical and GI datasets for the yeast S. cerevisiae. Figure 2 shows histograms of the resulting existence probabilities. Regarding physical interactions, 20 180 (39.03%) of the experimentally reported interactions have been assigned probabilities <0.1. A total of 22 090 (43.3%) have been assigned probabilities >0.9. The remaining interactions are widely distributed around a small peak near probability 0.3.

Fig. 2.

Histogram of existence probabilities for the interactions in the integrated dataset. Top: PPIs, bottom: synthetic GIs.

When looking at the probability distribution over synthetic genetic interactions it is apparent that with 14 386 (95.8%) a vast majority of interactions have been assigned values of <0.5. Of these, 8086 (53.9%) have a probability <0.1. Only 230 (1.5%) have a probability >0.9. This is most likely due to the much poorer coverage and data quality for synthetic GIs compared with physical interactions. An examination of the data shows that only 3.32% of the synthetic interactions are backed up by >2 experiments, whereas for physical interactions the same is true for 18.08%.

Even though it would be desirable to evaluate the correctness of these confidence assessments, without knowing the absolute truth regarding which interactions are real and which are not, a satisfying answer cannot be found. Apart from the discussed limitations of existing reference datasets often treated as gold standards, it would be problematic to evaluate the Lee and Lycett methods on the basis of the same gold standard data they received as input.

However, it is possible to illustrate the difference in results between the naive method and methods that take into account the reliability of each input dataset. Figure 3 shows a comparison between probabilities assigned to PPIs according to the naive method and according to the proposed method. It is clearly visible that only few agreements exist. A correlation of only 0.36 can be measured. This is to be expected. In contrast to the naive method, the proposed algorithm infers false positive and false negative error rates for each experiment and thus reaches more refined conclusions.

Fig. 3.

Probabilities assigned to PPIs by the naive integration method and the proposed method. As expected, a strong difference is clearly visible, as the proposed method takes into account the reliability of each experiment.

3.3 Summary and conclusion

We have presented a novel, fully Bayesian method for assessing the credibility of experimental network data in the absence of gold standards. We have evaluated the method's performance in comparison with two existing gold standard-based methods using a rigorous testing environment. The new method has shown excellent performance despite the testing environment being designed to favour the competing gold standard-based methods: both competing reference methods have been given optimal input parameters as well as unbiased gold standards, which would not normally be available. Additionally, the environment has been set up to simulate scale-free graphs, which emulate the topology of real biological graphs and thus do not meet the proposed algorithm's assumptions regarding the prior distribution of G. Finally, the simulation of experimental data has been designed to introduce bias and thus violate the algorithm's assumption of statistical independence of the datasets.

We have evaluated the performance of the new algorithm and the two reference methods using three different metrics. These metrics express the accuracy of error rate estimation as well as the accuracy of probability assignment to existing and non-existing edges. Other performance metrics could have been used as well (e.g. ROC–AUC or weighted sums of loss functions). However, we argue that such metrics would have provided less detail, since they represent a method's performance as a single number rather than yielding information on different categories of performance. Given biological networks or their simulated equivalents, it is crucial to highlight the difference between existing and non-existing edges, since their respective amounts are so vastly different. Thus errors on non-existing edges would always overshadow the errors on existing edges in summarizing metrics like ROC–AUC.

While we have shown that the new algorithm performs well on large input experiments, it remains yet to be shown how the method performs on smaller datasets originating from low-throughput experiments. A further limitation to the approach is that it does not take into account any potential pre-existing confidence assessments from the original experiments.

We conclude that the new fully Bayesian method is a valuable addition to the set of tools available for confidence assessment of experimental datasets. It is particularly useful for the application on semantically integrated knowledge networks that consist of heterogeneous data, since it allows for every sub-network to be addressed separately without need for reconfiguration or search for applicable gold standards. As shown in Section 2.3, we were able to easily apply the method to the PPI and synthetic GI sub-networks of the same dataset.

3.4 Outlook

Future work will include a further evaluation of the method on real biological data. It is possible to compare the algorithm's performance to other methods when using the calculated confidence values for protein function prediction.

Furthermore, we would intend to analyze the method's performance on small low-throughput type input datasets. Finally, it would be interesting to explore more of the huge simulation parameter space. A great number of combinations of different numbers of small and large input experiments with varying false positive and false negative rates remain to be surveyed.

Another useful feature would be the incorporation of potential pre-existing prior information. This endeavour is largely complicated by the lack of a common standard. For example, PPI data as available in PSI-MI format from the databases mentioned above contains various different confidence values in different metrics, such as probabilities, letter grades or bit-scores. If these difficulties can be successfully tackled, the resulting improvements will make the presented method even more reliable and useful.