2. Modeling the methylation signal

Here, the methylation signal is expressed in terms of Hellinger divergence of methylation levels. Here, the signal distribution is modelled by a Weibull probability distribution model. Basically, the model could be a member of the generalized gamma distribution family. For example, it could be gamma distribution, Weibull, or log-normal. To describe the signal, we may prefer a model with a cross-validations: R.Cross.val > 0.95. Cross-validations for the nonlinear regressions are performed in each methylome as described in (Stevens 2009). The non-linear fit is performed through the function nonlinearFitDist.

The above statistical description of the dataset (evidently) suggests that there two main groups: control and treatments, while treatment group would split into two subgroups of samples. In the current case, to search for a good cutpoint, we do not need to use all the samples. The critical value $H_{\alpha=0.05}=33.51223$ suggests that any optimal cutpoint for the subset of samples T1 to T5 will be optimal for the samples T6 to T10 as well.

Below, we are letting the PCA+LDA model classifier to take the decision on whether a differentially methylated cytosine position is a treatment DMP. To do it, Methyl-IT function getPotentialDIMP is used to get methylation signal probabilities of the oberved $H$ values for all cytosine site (alpha = 1), in accordance with the 2-parameter Weibull distribution model. Next, this information will be used to identify DMPs using Methyl-IT function estimateCutPoint. Cytosine positions with $H$ values above the cutpoint are considered DMPs. Finally, a PCA + QDA model classifier will be fitted to classify DMPs into two classes: DMPs from control and those from treatment. Here, we fundamentally rely on a relatively strong $tv.cut \ge 0.34$ and on the signal probability distribution (nlms.wb) model.

dmps.wb <- getPotentialDIMP(LR = divs[1:10],
                             nlms = nlms.wb[1:10],  div.col = 9L, 
                             tv.cut = 0.34, tv.col = 7, alpha = 1, 
                             dist.name = "Weibull2P")
cut.wb = estimateCutPoint(LR = dmps.wb, simple = FALSE,
                            column = c(hdiv = TRUE, TV = TRUE, 
                                       wprob = TRUE, pos = TRUE),
                            classifier1 = "pca.lda", 
                             classifier2 = "pca.qda", tv.cut = 0.34,
                            control.names = control.nam, 
                            treatment.names = treatment.nam1,
                            post.cut = 0.5, cut.values = seq(15, 38, 1),
                            clas.perf = TRUE, prop = 0.6,
                            center = FALSE, scale = FALSE,
                            n.pc = 4, div.col = 9L, stat = 0)
cut.wb

## Cutpoint estimation with 'pca.lda' classifier 
## Cutpoint search performed using model posterior probabilities 
## 
## Posterior probability used to get the cutpoint = 0.5 
## Cytosine sites with treatment PostProbCut >= 0.5 have a 
## divergence value >= 3.121796 
## 
## Optimized statistic: Accuracy = 1 
## Cutpoint = 37.003 
## 
## Model classifier 'pca.qda' 
## 
## The accessible objects in the output list are: 
##                    Length Class           Mode     
## cutpoint           1      -none-          numeric  
## testSetPerformance 6      confusionMatrix list     
## testSetModel.FDR   1      -none-          numeric  
## model              2      pcaQDA          list     
## modelConfMatrix    6      confusionMatrix list     
## initModel          1      -none-          character
## postProbCut        1      -none-          numeric  
## postCut            1      -none-          numeric  
## classifier         1      -none-          character
## statistic          1      -none-          character
## optStatVal         1      -none-          numeric

The cutpoint is higher from what is expected from the higher treatment empirical critical value and DMPs are found for $H$ values: $H^{TT_{Emp}}_{\alpha=0.05}=33.98<37≤H$. The model performance in the whole dataset is:

# Model performance in in the whole dataset
cut.wb$modelConfMatrix

## Confusion Matrix and Statistics
## 
##           Reference
## Prediction   CT   TT
##         CT 4897    0
##         TT    2 9685
##                                      
##                Accuracy : 0.9999     
##                  95% CI : (0.9995, 1)
##     No Information Rate : 0.6641     
##     P-Value [Acc > NIR] : <2e-16     
##                                      
##                   Kappa : 0.9997     
##  Mcnemar's Test P-Value : 0.4795     
##                                      
##             Sensitivity : 1.0000     
##             Specificity : 0.9996     
##          Pos Pred Value : 0.9998     
##          Neg Pred Value : 1.0000     
##              Prevalence : 0.6641     
##          Detection Rate : 0.6641     
##    Detection Prevalence : 0.6642     
##       Balanced Accuracy : 0.9998     
##                                      
##        'Positive' Class : TT         
## 

# The False discovery rate
cut.wb$testSetModel.FDR

## [1] 0