Skip to content

Statistics

SOPRANO produces two sets of statistical constraints on the dN/dS values computed by the pipeline.

Analytic estimates

For each pipeline run, there is an analytic estimate of confidence intervals (68 and 98 percentiles) and p-value. This uses Katz analytic method of estimating the confidence intervals, which can then be used to estimate a pvalue. (For example, see this reference.)

Numerical estimates

SOPRANO computes dN/dS based on the inclusion/exclusion of genomic regions defined by an input bed protein file.

If we instead randomize these regions before executing the pipeline, the downstream results can be interpreted as a null hypothesis simulation.

By randomizing and running the SOPRANO pipeline we are effectively drawing a "sample" from the unknown distribution of null hypothesis dN/dS values. By building an ensemble of many samples, it is therefore possible to build a kernel density estimate (KDE) from the empirical distribution.

Once the kernel has been estimated, it is possible to assess the statistical significance of the non-randomized dN/dS result. A p-value can be computed by direct integration from the asymptotic tail of the kernel up until the observed non-randomized dN/dS value(s).

The Gaussian kernel

The kernel is derived by applying a Gaussian kernel to the distribution of dN/dS from null hypothesis simulations. The kernel itself is computed with scikit-learn, for which we apply a grid search to find an optimal bandwidth for the window function.

By default, the grid bandwidth search is performed in 250 logarithmic spaces between -3 and 10. This is fairly sensible given the order-of-magnitudes one might expect for dN/dS in ON and OFF target regions.

These values can be adjusted at runtime via setting the environment variables:

  • SOPRANO_CV_LOG_MIN - Base10 log value for minimum bandwidth.
  • SOPRANO_CV_LOG_MAX - Base10 log value for maximum bandwidth.
  • SOPRANO_CV_LOG_STEPS - Number of linear spaces in log space.

Integration parameters

Once the kernel has been determined, one needs to determine the lower and upper bounds for the integration from which, p-values are computed.

Based on the spread of dN/dS values from the null distribution, a step distance is defined as some percentage of that spread. To determine the integration bounds, the kernel is estimated iteratively by "stepping" away from distribution maximum/minimum, until the absolute and relative difference of the kernel evaluated at that step is below a critical threshold. These values define the lower and upper bounds for the integration. The integration itself is performed with the SciPy quad integrator.

By default, SOPRANO walks away from the null distribution in 1% steps; until the absolute and relative difference between steps is < 1e-8. This will continue for no more than 1000 steps.

These values can be adjusted at runtime via setting the environment variables:

  • SOPRANO_KDE_ABS_TOL - The absolute tolerance for convergence.
  • SOPRANO_KDE_REL_TOL - The relative tolerance for convergence.
  • SOPRANO_KDE_STEP_PCT - Percentage of distributions spread to use as a step.
  • SOPRANO_KDE_MAX_ITER - Maximum number of iterations before termination.