Robust Rank Aggregation

 

Goal: Integrate multiple ranked gene sets into a single cogent list.

 

Normalized rank: $ \frac{\text{rank} - 1}{n - 1} $

This transforms ordinal rankings into a standardized scale between 0 and 1, so that they are comparable.

 

Robust rank aggregation (RRA) performs a statistical test for each gene. Each gene is associated with a rank vector $ \textbf{r} = (r_1, \cdots, r_n) $. So, $r_j$ would be the normalized rank of a single gene (GAPDH, for example) from study $j$. We can order the normalized ranks, creating an order statistic: $ r_{(1)} < \cdots < r_{(n)} $.

 

The null hypothesis asumes that all studies produced irrelevant rankings. Under the null hypothesis, normalized ranks are generated from an iid uniform distribution:

$$ r_j \stackrel{iid}{\sim} U(0, 1) $$

$$ f_r(x) = I(0 \leq x \leq 1) $$

$$ F_r(x) = x \cdot I(0 \leq x \leq 1) $$

 

Since we are testing for each gene, we only need to find $ f(r_{(k)}) $. The PDF of a single order statistic $ r_{(k)} $ is:

$$ \begin{align*} f_{r_{(k)}}(x) &= \frac{n!}{(k-1)!(n-k)!} f_r(x)[ F_r(x) ]^{k-1} [ 1 - F_r(x) ]^{n-k} \\ &= \frac{n!}{(k-1)!(n-k)!} x^{k-1}(1-x)^{n-k} \end{align*} $$

$$ \implies r_{(k)} \sim Beta(k, n+1-k) $$

 

Now, using the CDF of Beta distribution, we can calculate the p-values of our observed rankings. For positive integers $\alpha$ and $\beta$ parameters, the CDF of a beta random variable can be expressed as a binomial distribution:

$$ \begin{align*} F_{\text{beta}}(x; \alpha, \beta) &= F_{\text{binomial}}(k=\beta - 1; n=\alpha + \beta - 1, p=1-x) \\ F_{r_{(k)}}(x) &= \sum_{j=k}^n {n \choose j} x^j (1-x)^{n-j} \end{align*} $$

 

After derivation, I have realized that you can simply use the fact that the CDF of $k^{\text{th}}$ order statistic is:

$$ F_{X_{(r)}}(x) = \sum_{j=r}^n {n \choose j} [F_X(x)]^j [1 - F_X(x)]^{n-j} $$

 

which would have been exactly the same as the derivation above. It is what it is... ¯\_(ツ)_/¯

 

For the current gene, we now have $n$ p-values, derived from the null distribution:

$$ F_{r_{(k)}}(r_{(k)}^{\text{observed}}) = P(r_{(k)} \leq r_{(k)}^{\text{observed}}) = \beta_k $$

 

The original paper defines $ \rho(\textbf{r}) = \min\limits_{k=1, \cdots, n} \beta_k $, aggregating multiple p-values by their minimum.

 

Mininmum of multiple p-values, as a random variable, follows a beta distribution with parameters $\alpha=1$ and $\beta=n$ (Song and Tseng, 2014). In RRA, this approach is not possible as it only works under the condition that multiple p-values are independent. Because the p-values are derived from an order statistic, they are not independent.