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First, it is essential to identify the correct phenotypic staging threshold recognizing that the signature only exists in a subset of tumors that exceed that particular stage.
Figure 1 shows the values of the cumulative hypergeometric probability for the COL11A1 gene using the TCGA ovarian cancer data set and the staging threshold for the definition of the binary phenotype set between IIIb and IIIc.
A multi-cancer gene expression signature involving a large set of genes, several among them being EMT markers coordinately overexpressed only in a subset of malignant samples that have exceeded a particular staging threshold specific to each cancer type was recently identified [ 5].
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where is the stage threshold, is the weak classifier's weight, and is the total number of weak classifiers (features).
The likelihood ratio test for the first stage threshold estimate is: F 1 = S 0 - S 1 ( γ ̂ 1 ) σ ̂ 2 (6).
Assuming the first threshold is given, the optimal second-stage threshold estimate is found by minimizing the residual sum of squares for Equation 7. The likelihood ratio test of one versus two thresholds is be based on the statistic: F 2 = S 1 ( γ ̂ 1 ) - S 2 ( γ ̂ 2 ) σ ̂ 2 (8).
The power was calculated based on the probability of passing both the 1st-stage p-value threshold (<5×10−5) and the 2nd-stage threshold (<0.0008 or 0.05/63 by Bonferroni correction) given the effect size.
Likewise, extending the first-stage threshold would have resulted in more false positives being considered in the second stage and higher risks of overparameterization.
With respect to P-value threshold selection at each stage, a lenient first-stage threshold was used to capture most true positive associations at the expense of some false positives.
Our two-stage threshold selection procedure is similar to the procedure that controls both statistical significance and biological significance in [ 86, 88] and we demonstrated that our results were not sensitive to different thresholds of FDR and PER (see Additional file 1, Text S11).
Nonetheless, the strength of association based on all available data (P=8.8 × 10−5) remains significant after a conservative multiple testing correction that accounts for the number of stages of data accumulation and also for the total number of SNPs that could have been followed through these stages (threshold P≤4.6 × 10−4).
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com