3.1.1 AES-PCA Method Sources

Adaptive, elastic-net, sparse PCA is a combination of the Adaptive Elastic-Net of Zou and Zhang (2009) and Sparse PCA of Zou et al. (2006). This method was applied to pathways association testing by Chen (2011). Accoding to Chen (2011), the “AES-PCA method removes noisy expression signals and also account[s] for correlation structure between the genes. It is computationally efficient, and the estimation of the PCs does not depend on clinical outcomes.” This package uses a legacy version of the LARS algorithm of Efron et al. (2003) to calculate the PCs.

3.1.2 Calculate Pathway-Specific Model p-Values

For the AES-PCA method, pathway p-values can be calculated with a permutation test. Therefore, when testing the relationship between the response and the PCs extracted by AES-PCA, the accuracy of the permuted p-values will depend on how many permutations you call for. We recommend 1000. Be warned, however, that this may be too few permutations to create accurate seperation in pathway significance p-values. You could increase the permutations to a larger value, should your computing resources allow for that. For even moderately-sized data sets (~2000 features) and 1000 pathways, this could take half an hour or more. If you choose to calculate the pathway p-values non-parametrically, about 20-30% of the computing costs will be extracting the AES-PCs from each pathway (though this proportion will increase if the LARS algorithm has convergence issues with the given pathway). The remaining 70-80% of the cost will be the permutation test (for 1000 permutations).

3.1.3 AES-PCA Pros and Cons

Pros:

• The AES-PCA method can handle some missingness in the response.
• The p-values can be calculated non-parametrically.

Cons:

• The AES-PCA algorithm requires optimization over two tuning parameters and can therefore be considerably slower than using the singular value decomposition or eigendecomposition to extract PCs.
• The p-values calculated may be too discrete for fewer than 10,000 permutations, which can affect the behavior of the adjustment procedures.

3.2.1 Survival Response

We will use two of our available cores with the parallel computing approach. We will adjust the p-values with the Hochberg (1988) and Sidak Step-Down FWER-adjustment procedures. We will now describe the computational cost for the non-parametric approach.

For the tiny 250 × 656 assay with 15 associated pathways, calculating pathway p-values with 1000 replicates completes in 28 seconds. If we increase the number of permutations from 1000 to 10,000, this calculation takes 222 seconds (7.9× longer). Even though we increased the permutations tenfold, the function completed execution less than 10 times longer (as we mentioned above, roughly a quarter of the computing time is extracting the PCs from each pathway, which does not depend on the number of permutations).

In the example that we show, we will calculate the pathway p-values parametrically, by specifying numReps = 0. Furthermore, the AES-PCA and Supervised PCA functions give some messages concerning the setup and progress of the computation.

colonSurv_aespcOut <- AESPCA_pVals(
  object = colon_OmicsSurv,
  numReps = 0,
  numPCs = 2,
  parallel = TRUE,
  numCores = 2,
  adjustpValues = TRUE,
  adjustment = c("Hoch", "SidakSD")
)
#> Part 1: Calculate Pathway AES-PCs
#> Initializing Computing Cluster: DONE
#> Extracting Pathway PCs in Parallel: DONE
#> 
#> Part 2: Calculate Pathway p-Values
#> Initializing Computing Cluster: DONE
#> Extracting Pathway p-Values in Parallel: DONE
#> 
#> Part 3: Adjusting p-Values and Sorting Pathway p-Value Data Frame
#> DONE

3.2.2 Regression Response

We can also make a mock regression data set by treating the event time as the necessary continuous response. For this example, we will adjust the p-values with the Holm (1979) FWER- and Benjamini and Hochberg (1995) FDR-adjustment procedures (as an aside, note that this type of multiple testing violates the independence assumption of the Simes inequality). For 1000 permutations, this calculation takes 17 seconds. For 10,000 permutations, this calculation takes 102 seconds (6.1× longer).

colonReg_aespcOut <- AESPCA_pVals(
  object = colon_OmicsReg,
  numReps = 0,
  numPCs = 2,
  parallel = TRUE,
  numCores = 2,
  adjustpValues = TRUE,
  adjustment = c("Holm", "BH")
)

3.2.3 Binary Classification Response

Finally, we can simulate a mock classification data set by treating the event indicator as the necessary binary response. For this example, we will adjust the p-values with the Sidak Single-Step FWER- and Benjamini and Yekutieli (2001) FDR-adjustment procedures. For 1000 permutations, this calculation takes 30 seconds. For 10,000 permutations, this calculation takes 226 seconds (7.6× longer).

colonCateg_aespcOut <- AESPCA_pVals(
  object = colon_OmicsCateg,
  numReps = 0,
  numPCs = 2,
  parallel = TRUE,
  numCores = 2,
  adjustpValues = TRUE,
  adjustment = c("SidakSS", "BY")
)

4.1.1 Supervised PCA Method Sources

While PCA is a commonly-applied unsupervised learning technique (i.e., response information is unnecessary), one limitation of this method is that ignoring response information may yield a first PC completely unrelated to outcome. In an effort to bolster this weakness, Bair et al. (2006) employed response information to rank predictors by the strength of their association. Then, they extracted PCs from feature design matrix subsets constructed from the predictors most strongly associated with the response. Chen et al. (2008) extend this technique to subsets of biological features within pre-defined biological pathways; they applied the Supervised PCA routine independently to each pathway in a pathway collection. Chen et al. (2010) built on this work, testing if pathways were significantly associated with a given biological or clinical response.

4.1.2 Calculate Pathway-Specific Model p-Values

As thoroughly discussed in Chen et al. (2008), the model fit and regression coefficient test statistics no longer come from their expected distributions. Necessarily, this is due to Supervised PCA’s strength in finding features already associated with outcome. Therefore, for the Supervised PCA method, pathway p-values are calculated from a mixture of extreme value distributions. We use a constrained numerical optimization routine to calculate the maximum likelihood estimates of the mean, precision, and mixing proportion components of a mixture of two Gumbel extreme value distributions (for minima and maxima of a random normal sample). The p-values from the pathways after permuting the response is used to estimate this null distribution, so result accuracy may be degraded for a very small set of pathways.

4.1.3 Supervised-PCA Pros and Cons

Pros:

• The Supervised PCs are extracted without numerical optimization, so calculating the PCs for each pathway is considerably faster than calculating AES-PCs.
• The p-values are calculated parametrically, so calculating the p-values is considerably faster than the non-parametric AES-PCA option, while holding better distributional properties than the parametric AES-PCA option.

Cons:

• In rare cases, numerical routines used to find the maximum likelihood estimates for the mixture distribution needed to calculate the p-values in Supervised PCA can fail to converge.
• The Supervised PCA method cannot have missing values in the response.

4.2.1 Survival Response

We will use two of our available cores with the parallel computing approach. We will adjust the p-values with the Hochberg (1988) and Sidak Step-Down FWER-adjustment procedures. For the tiny 250 × 656 assay with 15 associated pathways, this calculation is completed in 6 seconds. If we compare this to AES-PCA at 1000 permutations, Supervised PCA is 4.6× faster; for 10,000 permutations, it’s 36.1× faster.

colonSurv_superpcOut <- SuperPCA_pVals(
  object = colon_OmicsSurv,
  numPCs = 2,
  parallel = TRUE,
  numCores = 2,
  adjustpValues = TRUE,
  adjustment = c("Hoch", "SidakSD")
)

#> Initializing Computing Cluster: DONE
#> Calculating Pathway Test Statistics in Parallel: DONE
#> Calculating Pathway Critical Values in Parallel: DONE
#> Calculating Pathway p-Values: DONE
#> Adjusting p-Values and Sorting Pathway p-Value Data Frame: DONE

4.2.2 Regression Response

We can also make a mock regression data set by treating the event time as the necessary continuous response. For this example, we will adjust the p-values with the Holm (1979) FWER- and Benjamini and Hochberg (1995) FDR-adjustment procedures. This calculation took 5 seconds. If we compare this to AES-PCA at 1000 permutations, Supervised PCA is 3.4× faster; for 10,000 permutations, it’s 20.7× faster.

colonReg_superpcOut <- SuperPCA_pVals(
  object = colon_OmicsReg,
  numPCs = 2,
  parallel = TRUE,
  numCores = 2,
  adjustpValues = TRUE,
  adjustment = c("Holm", "BH")
)

4.2.3 Binary Classification Response

Finally, we can simulate a mock classification data set by treating the event indicator as the necessary binary response. For this example, we will adjust the p-values with the Sidak Single-Step FWER- and Benjamini and Yekutieli (2001) FDR-adjustment procedures. This calculation took 8 seconds. If we compare this to AES-PCA at 1000 permutations, Supervised PCA is 3.7× faster; for 10,000 permutations, it’s 27.6× faster.

colonCateg_superpcOut <- SuperPCA_pVals(
  object = colon_OmicsCateg,
  numPCs = 2,
  parallel = TRUE,
  numCores = 2,
  adjustpValues = TRUE,
  adjustment = c("SidakSS", "BY")
)