Approximate Profile Maximum Likelihood

We propose an efficient algorithm for approximate computation of the profile maximum likelihood (PML), a variant of maximum likelihood maximizing the probability of observing a sufficient statistic rather than the empirical sample. The PML has appealing theoretical properties, but is difficult to compute exactly. Inspired by observations gleaned from exactly solvable cases, we look for an approximate PML solution, which, intuitively, clumps comparably frequent symbols into one symbol. This amounts to lower-bounding a certain matrix permanent by summing over a subgroup of the symmetric group rather than the whole group during the computation. We extensively experiment with the approximate solution, and find the empirical performance of our approach is competitive and sometimes significantly better than state-of-the-art performance for various estimation problems

Significant genes obtained by LASSO&Permuted based maxT algorithm for the three models (SNP, CPG, and Global) in the original dataset (EPICURO Study) and the replication dataset (TCGA).

<p>Significant genes obtained by LASSO&Permuted based maxT algorithm for the three models (SNP, CPG, and Global) in the original dataset (EPICURO Study) and the replication dataset (TCGA).</p

Deviance across the genome when applying LASSO and ENET to select SNPs, CpGs or both (Global model).

<p>The dots in the figure indicate the deviance of each gene located in the corresponding position in the genome. There are a total of 20,899 gene expression probes measured. Significant genes after applying the permutation-based MaxT method are tagged. The figures represent the deviance per gene expression probe using LASSO for the SNP model (A), the CpG model (B) and the Global model (C) and using ENET for the SNP model (D), the CpG model (E) and the Global model (F).</p

Scenario and workflow of the overall analysis implemented.

<p>The integrative framework proposed is based on three steps. Step 1 corresponds to the selection of SNPs and CpGs in 1MB window upstream and downstream from each probe in the gene expression array. Step 2 corresponds to the application of LASSO and ENET to each probe obtaining the deviance per probe. Step 3 corresponds to the permutation-based MaxT method application where gene expression levels within the individuals are permuted B = 100 times obtaining the deviance per probe.</p

Kaplan-Meier curve for recurrence and survival.

<p>Kaplan-Meier curve for recurrence and survival.</p

Most neoplastic cells express null or weak CK20. However, there are groups of cells expressing simultaneously CD 133 and CK 20 (encircled areas).

<p>Most neoplastic cells express null or weak CK20. However, there are groups of cells expressing simultaneously CD 133 and CK 20 (encircled areas).</p

Significant genes obtained by ENET&Permuted based maxT algorithm for the three models (SNP, CPG, and Global) in the original dataset (EPICURO Study) and the replication dataset (TCGA).

<p>Significant genes obtained by ENET&Permuted based maxT algorithm for the three models (SNP, CPG, and Global) in the original dataset (EPICURO Study) and the replication dataset (TCGA).</p

Statistically significant genes associated with SNPs and/or CpGs selected by LASSO&Permuted based maxT algorithm.

<p>Statistically significant genes associated with SNPs and/or CpGs selected by LASSO&Permuted based maxT algorithm.</p

Group of neoplastic cells showing CD 133 positive cytoplasmic expression.

<p>Group of neoplastic cells showing CD 133 positive cytoplasmic expression.</p

Example of a correlation plot for <i>MMP7</i> detected by the Global model using ENET but not using LASSO.

<p>The bar color represents the levels of correlation from 0 (no correlation) to 1 (perfect correlation) between SNPs and CpGs that were selected for the <i>MMP7</i> models. Three nets of correlated variables are the ones responsible that the gene is only selected by ENET and not by LASSO.</p