# Combining the outputs of driver identification methods¶

## Rationale¶

Our goal is to provide a catalogue of driver elements which appropriately reflects the consensus from the DIMs we run.

To combine the results of individual statistical tests, p-value combination methods continue to be a standard approach in the field: e.g., Fisher 1, Brown 2, 3 and Stouffer Z-score 4 methods have been used for this purpose. These methods are useful for combining probabilities in meta-analysis, hence to provide a ranking based on combined significance under statistical grounds. However, the application of these methods may bear some caveats:

The ranking resulting from p-value combination may lead to inconsistencies when compared to the individual rankings, i.e., they may yield a consensus ranking that does not preserve recurrent precedence relationships found in the individual rankings.

Some methods, like Fisher’s or Brown’s method, tend to bear anti-conservative performance, thus leading to many likely false discoveries.

Balanced (non-weighted) p-value combination methods may lead to faulty results just because of the influence of one or more DIM performing poorly for a given dataset.

Weighted methods to combine p-values, like the weighted Stouffer Z-score, provide some extra room for proper balancing, in the sense of incorporating the relative credibility of each DIM. We reasoned that any good operational criteria to allocate weights should satisfy the following requirements: i) provide weighting on a cohort-specific basis, thereby allowing the relative credibility of a DIM to depend on the cohort; ii) reflect prior knowledge about known bona-fide driver genes; iii) reflect prior knowledge about the criteria that each DIM employed to yield its output.

Our approach works independently for each cohort: to create a consensus list of driver genes for each cohort, we first determine how credible each DIM is when applied to this specific cohort, based on how many bona-fide cancer genes reported in the COSMIC Cancer Gene Census database (CGC) are highly ranked according to the DIM. Once a credibility score has been established, we use a weighted method for combining the p-values that each DIM gives for each candidate gene: this combination takes the DIMs credibility into account. Based on the combined p-values, we conduct FDR correction to conclude a ranking of candidate driver genes alongside q-values.

## Weight Estimation by Voting¶

The relative credibility for each method is based on the ability of the method to give precedence to well-known genes already collected in the CGC catalogue of driver genes. As each method yields a ranking of driver genes, these lists can be combined using a voting system –Schulze’s voting method. The method allows us to consider each method as a voter with some voting rights (weighting) which casts ballots containing a list of candidates sorted by precedence. Schulze’s method takes information about precedence from each individual method and produces a new consensus ranking 5.

Instead of conducting balanced voting, we tune the voting rights of the methods so that we the enrichment of CGC genes at the top positions of the consensus list is maximized. We limit the share each method can attain in the credibility simplex –up to a uniform threshold. The resulting voting rights are deemed the relative credibility for each method.

## Ranking Score¶

Upon selection of a catalogue of bona-fide known driver elements (CGC catalogue of driver genes) we can provide a score for each ranking \(R\) of genes as follows:

\(E(R)\ = \sum_{i=1}^N \frac{p_{i}}{\log(i + 1)}\)

where \(p_{i}\) is the proportion of elements with rank higher than \(i\) which belong to CGC and N is a suitable threshold to consider only the N top ranked elements. Using \(E\) we can define a function \(f\) that maps each weighting vector \(w\) (in the 5-simplex) to a value \(E(R_{w})\) where \(R_{w}\) denotes the consensus ranking obtained by applying Schulze’s voting with voting rights given by the weighting vector \(w\).

## Optimization with constraints¶

Finally we are bound to find a good candidate for \(\widehat{w} = \textrm{argmax}(f)\). For each method to have chances to contribute in the consensus score, we impose the mild constraint of limiting the share of each method up to 0.3.

Optimization is then carried out in two steps: we first find a good candidate \(\widehat{w_{0}}\) by exhaustive search in a rectangular grid satisfying the constraints defined above (with grid step=0.05); in the second step we take \(\widehat{w_{0}}\) as the seed for a stochastic hill-climbing procedure (we resort to Python’s scipy.optimize “basinhopping”, method=SLSQP and stepsize=0.05).

## Estimation of combined p-values using weighted Stouffer Z-score¶

Using the relative weight estimate that yields a maximum value of the objective function f we combined the p-values resorting to the weighted Stouffer Z-score method. Thereafter we performed Benjamini-Hochberg FDR correction with the resulting combined p-values, yielding one q-value for each genomic element. If the element belongs to CGC, we computed its q-value using only the collection of p-values computed for CGC genes. Otherwise, we computed the q-value using all the computed p-values.

## Tiers of driver genes from sorted list of combined rankings and p-values¶

To finalize the analysis we considered only genes with at least two mutated samples in the cohort under analysis. These genes were classified into four groups according to the level of evidence in that cohort that the gene harbours positive selection.

The first group, named as TIER1, contained genes showing high confidence and agreement in their positive selection signals. Given the ranked list of genes obtained by the Schulze voting, TIER1 comprises all the ranked genes whose ranking is higher than the first gene with combined q-value lower than a specific threshold (by default threshold=0.05). The second group, name as TIER2, was devised to contain known cancer driver genes, showing mild signals of positive selection, that were not included in TIER1. More in detail, we defined TIER2 genes as those CGC genes, not included in TIER2, whose CGC q-value was lower than a given threshold (default CGC q-value=0.25). CGC q-value is computed by performing multiple test correction of combined p-values restricted to CGC genes. The third group, are genes not included in TIER1 or TIER2 with scattered signals of positive selection, frequently coming from one single method. Particularly, given the ranked list of genes by the Schulze voting, TIER3 was composed of all the ranked genes with q-value lower than a given threshold (by default threshold=0.05) whose ranking is higher than TIER1 last gene position and lower than the rejection ranking position. The rejection ranking position is defined as the ranking position for which all elements have a q-value lower than the input threshold (by default threshold=0.05). Finally, other genes not included in the aforementioned classes are considered non-driver genes.

## Combination benchmark¶

We have assessed the performance of the combination compared to i) each of the six individual methods and ii) other strategies to combine the output from cancer driver identification methods.

### Datasets for evaluation¶

To ensure a reliable evaluation we decided to perform an evaluation based on the 32 Whole-Exome cohorts of the TCGA PanCanAtlas project (downloaded from *https://gdc.cancer.gov/about-data/publications/pancanatlas*). These cohorts sequence coverage, sequence alignment and somatic mutation calling were performed using the same methodology guaranteeing that biases due to technological and methodological artifacts are minimal.

The Cancer Genes Census –version v87– was downloaded from the COSMIC data portal (*https://cancer.sanger.ac.uk/census*) and used as a positive set of known cancer driver genes.

We created a catalog of genes that are known not to be involved in cancerogenesis. This set includes very long genes (HMCN1, TTN, OBSCN, GPR98, RYR2 and RYR3), and a list of olfactory receptors from Human Olfactory Receptor Data Exploratorium (HORDE) (https://genome.weizmann.ac.il/horde/; download date 14/02/2018). In addition, for all TCGA cohorts, we added non-expressed genes, defined as genes where at least 80% of the samples showed a RSEM expressed in log2 scale smaller or equal to 0. Expression data for TCGA was downloaded from *https://gdc.cancer.gov/about-data/publications/pancanatlas*.

### Metrics for evaluation¶

We defined a metric, referred to as CGC-Score, that is intended to measure the quality of a ranking of genes as the enrichment of CGC elements in the top positions of the ranking; specifically given a ranking \(R\) mapping each element to a rank, we define the CGC-Score of \(R\) as

\(\text{CGC-Score}(R)\ = \sum_{i=1}^N\frac{p_{i}}{log(i + 1)} \; /\; \sum_{i=1}^N\frac{1}{log(i + 1)}\)

where \(p_{i}\) is the proportion of elements with rank \(\leq i\) that belong to CGC and \(N\) is a suitable threshold to consider just the top elements in the ranking (by default N=40).

We estimated the CGC-Score across TCGA cohorts for the individual methods ranking and the combined Schulze ranking.

Similarly, we defined a metric, referred to as Negative-Score, that aims to measure the proportion non-cancer genes among the top positions in the ranking. Particularly, given a ranking \(R\) mapping each element to a rank, we define the Negative-Score of \(R\) as:

\(\text{Negative-Score}(R)\ = \sum_{i=1}^N \frac{p_{i}}{log(i + 1)}\; /\; \sum_{i=1}^N \frac{1}{log(i + 1)}\)

where \(p_{i}\) is the proportion of elements with rank \(\leq i\) that belong to the negative set and \(N\) is a suitable threshold to consider just the top elements in the ranking (by default N = 40). We estimated the Negative-Score across TCGA cohorts for the individual methods ranking and the combined Schulze ranking.

### Comparison with individual methods¶

We compared the CGC-Score and Negative-Score of our combinatorial selection strategy with the individual output from the six driver discovery methods integrated in the pipeline.

As a result we observed a consistent increase in CGC-Score of the combinatorial strategy compared to individual methods across TCGA cohorts (see Figure below panel A-B). Similarly, we observed a consistent decrease in Negative-Score across TCGA cohorts (see Figure below panel C). In summary, the evaluation shows that the combinatorial strategy increases the True Positive Rate (measured using the CGC-Score) preserving lower False Positive Rate (measured using the Negative-Score) than the six individual methods included in the pipeline.

### Leave-one-out combination¶

We aimed to estimate the contribution of each method’s ranking to the final ranking after Schulze’s weighted combination. To tackle this question, we measured the effect of removing one method from the combination by, first, calculating the CGC-Score of the combination excluding the aforementioned method and, next, obtaining its ratio with the original combination (i.e., including all methods). This was iteratively calculated for all method across TCGA cohorts. Methods that positively contributed to the combined ranking quality show a ratio below one, while methods that negatively contributed to the combined ranking show a ratio greater than one.

We observed that across TCGA cohorts most of the methods contributed positively (i.e., ratio above one) to the weighted combination performance. Moreover, there is substantial variability across TCGA cohorts in the contribution of each method to the combination performance. In summary, all methods contributed positively to the combinatorial performance across TCGA supporting our methodological choice for the individual driver discovery methods (see Figure below panel E).

### Comparison with other combinatorial selection methods¶

We compared the CGC-Score and Negative-Score of our combinatorial selection strategy against other methods frequently used employed to produce ranking combinations, either based on ranking information –such as Borda Count 6 – or based on statistical information –such as Fisher 1 or Brown 2, 3 methods. Hereto, we briefly describe the rationale of the four methods we used to benchmark our ranking. For the sake of compact notation, let’s denote the rank and p-value of gene \(g\) produced by method \(m_{i}\) as \(r_{i,g}\) and \(p_{i,g}\), respectively.

*Borda Count:* For each ranked item \(g\) and method \(m_{i},\)
it assigns a score \(s_{i,g} = N - l_{i,g},\) where \(N\)
stands for the total number of items to rank and \(l_{i,g}\) is
the number of items ranked below \(g\) according to method
\(m_{i}\). For each item \(g\) an overall score
\(s_{g}= s_{1,g} + \ldots + s_{k,g}\) can then be
computed for each \(g,\) whence a ranking is obtained by descending
sort.

*Fisher:* It is based on the p-values \(p_{i,g}\). For each item
\(g\) the method produces a new combined p-value by computing the
statistic:

\(F_{g} = - 2\log\ p_{i, g} \sim \chi_{2k}^{2}\).

Under the null hypothesis, \(F_{g}\) are distributed as a chi-square with \(2k\) degrees of freedom, whence a p-value, which in turn yields a raking by ascending sort. Its applicability is limited by the assumption that the methods provide independent significance tests.

*Brown:* This method overcomes the independence requirement of Fisher’s
method by modeling the dependencies between the statistical tests
produced by each method, specifically by estimating the covariance
\(\Omega_{i,j} = \textrm{cov}( - 2\log p_{i,g}, - 2\log p_{j,g}).\)
Brown’s method 2 and its most recent adaptation 3 have been
proposed as less biased alternatives to Fisher.

We then computed the CGC-Score and Negative-Score based on the consensus ranking from the aforementioned combinatorial methods and compared them to our Schulze’s weighted combination ranking across all TCGA cohorts. Our combinatorial approach met a larger enrichment in known cancer genes for 29/32 (90%) TCGA cohorts (see Figure below panel D).

- 1(1,2)
Fisher R.A. (1948) figure to question 14 on combining independent tests of significance. Am. Statistician , 2, 30–31.

- 2(1,2,3)
Brown, M. B. 400: A Method for Combining Non-Independent, One-Sided Tests of Significance. Biometrics 31, 987 (1975). DOI: 10.2307/2529826

- 3(1,2,3)
William Poole, et al. Combining dependent P-values with an empirical adaptation of Brown’s method, Bioinformatics, Volume 32, Issue 17, 1 September 2016, Pages i430–i436, https://doi.org/10.1093/bioinformatics/btw438

- 4
Zaykin, D. V. Optimally weighted Z-test is a powerful method for combining probabilities in meta-analysis. Journal of Evolutionary Biology 24, 1836–1841 (2011). doi: 10.1111/j.1420-101.2011.02297.x

- 5
- 6
Emerson P. The original Borda count and partial voting. Soc Choice Welf (2013) 40:353–358. doi 10.1007/s00355-011-0603-9