In Methyl-IT, the signal detection step requires for the knowledge of the probability distribution of the methylation noise, which is the null hypothesis used to discriminate the signal from the noise. Both, signal and noise, are expressed in terms of an information divergence of methylation levels, which (currently in Methyl-IT) could be the Hellinger divergence or the total variation distance. As shown in reference (1), on statistical physics basis, the probability distribution of the noise is a member of the generalized gamma distribution. In particular, if the methylation changes on the DNA probability distribution.
Function 'gofReport' search for the best fitted model between the set of models requested by the user. Two goodness-of-fit (GoF) criteria are applied to select the best fitted model: Akaike's information criterion (AIC) and the correlation coefficient of cross-validations for the nonlinear regressions (R.Cross.val) (2). These criteria evaluate different information inferred from the models. AIC deals with the trade-off between the goodness of fit of the model and the complexity of the model, while R.Cross.val provides information on the prediction power/performance of the model when confronted with external dataset.
Although the numerical algorithms to accomplish the nonlinear fit are not
perfect, in general, the model with the lowest AIC must have the highest
R.Cross.val. If the model with the lowest AIC has not the highest
R.Cross.val, then further analyzes are required. These analyzes could include
the visualization of the graphics for the density distribution, evaluation of
whether the parameter values can be meaningful or not, etc. Nevertheless, the
best model will, in general, lead to the identification of the greater amount
of potential DMPs and DMPs, as well as, the highest classification accuracy
estimated with functions estimateCutPoint and
evaluateDIMPclass. In the worse scenario, these observations
can ultimately lead to a post-hoc decision on which the best model is.
gofReport(
HD,
model = c("Weibull2P", "Weibull3P", "Gamma2P", "Gamma3P", "GGamma3P", "GGamma4P"),
column = 9,
absolute = FALSE,
output = c("best.model", "all"),
alt_models = FALSE,
r.cv = FALSE,
npoints = NULL,
min.scale = 1e-05,
min.mu = 0.001,
mu.rel.err = 0.89,
num.cores = 1L,
verbose = FALSE,
...
)
# S3 method for ProbDistrList
print(x, ...)An 'InfDiv' object returned by function
estimateDivergence.
A character vector naming the models to fit. Default is
model = c('Weibull2P', 'Weibull3P', 'Gamma2P', 'Gamma3P'). See
nonlinearFitDist for more options.
An integer number denoting the index of the GRanges column where
the information divergence is given. Default column = 9, which is the
column where the Hellinger divergence values are reported (by default) by
function estimateDivergence.
Logic (default, FALSE). Total variation (TV, the difference of methylation levels) is normally an output in the downstream MethylIT analysis. If 'absolute = TRUE', then TV is transformed into |TV|, which is an information divergence that can be fitted as well.
If output == 'all', the table with the GoF statistics is returned in a list together with the best fitted model and the corresponding statistics. Default is 'best.model'
logical(1). If TRUE, then the best model based on highest R.Cross.val is returned for those samples where the model(s) with lowest AIC has not the highest R.Cross.val.
logical(1). Whether to select the best model based on the highest R.Cross.val (2) (see details section).
number of points to be used in the fit. Default is NULL.
A number. The nonlinear fit of GGamma family distributions sometimes yields weird estimation of the scale parameter with values < 1e-5. Based on our experience, scale parameter values lower than min.scale = 1e-5 are probably meaningless. The result of numerical errors, the nonlinear algorithm are not perfect.
Numbers. Models with location parameter \(\mu\) sometimes reports \(\mu\) values close to zero, which generally corresponds to a non statistically significant model coefficient/parameter. This detail is not detected by the AIC or the cross-validations correlation coefficient. In this case the best model is the model without location parameter, where \(\mu = 0\). If \(\mu < min.mu\) and \(|E(\mu) - min(idiv)|/min(idiv) > mu.rel.err\), where \(E(\mu)\) is the estimated value and \(min(idiv)\) is the minimum observed value of the information divergence provided, then the issue is reported in the column named 'model'. It is expected that, at least \(mu.rel.err <= 0.89\). That is, that its expected that the difference between \(E(\mu)\) and \(min(idiv)\) is not greater than the 89%.
The number of cores to use in the nonlinear fit step, i.e.
at most how many child processes will be run simultaneously (see
bplapply function from BiocParallel package).
If TRUE, prints the function log to stdout
Further arguments to pass to nonlinearFitDist.
An object from 'ProbDistrList' class
If 'output = 'best.model'', a character vector with the name of the
best fitted model for each sample and model statistics, which can be used the
next step to get the potential DMPs with function
getPotentialDIMP. Otherwise, it will return a list with the
table carrying the GoF values and the previously mentioned data.
The best model selection is based on the lowest AIC. However, if \(alt_models = TRUE\) and \(r.cv = FALSE\), then the returned list will contain two sublists named 'alt_models' and 'alt_nlms' with the best model selected based on the highest R.Cross.val. If \(r.cv = TRUE\), best model selection is based on the highest R.Cross.val and sublist named 'alt_models' will carry the model selected based on the lowest AIC. This sublist is returned only if at least there is one sample where the model with the lowest AIC has not the highest R.Cross.val. This report is useful to analyze any conflict between models. For example, some times the best model selected based on AIC has a R.Cross.val = 0.99945, while the highest R.Cross.val is 0.99950. In such a situation the model with lowest AIC is still fine. However, some times the model with the best AIC has some meaningless parameter value. For example, scale = 0.0000000001 in 'GGamma3P' or in 'Weibull2P' models. This last situation can result from the numerical algorithm used in the parameter estimation. The numerical algorithms for nonlinear fit estimation are not perfect!
For distribution models that include a location parameter ('Weibull3P', 'Gamma3P', and 'GGamma4P') there is an additional important constraint, which cannot be evaluated with 'AIC' or 'R.Cross.val': \(\mu > 0\). That is, the information divergences are strictly positive magnitudes, therefore, any best model in terms of AIC with values \(\mu < 0\) are meaningless and rejected.
R. Sanchez and S. A. Mackenzie, “Information Thermodynamics of Cytosine DNA Methylation,” PLoS One, vol. 11, no. 3, p. e0150427, Mar. 2016.
Stevens JP. Applied Multivariate Statistics for the Social Sciences. Fifth Edit. Routledge Academic; 2009.
## Loading information divergence dataset
data(HD)
## Subsetting object HD (for the sake of runnig a faster example)
hd <- lapply(HD, function(x) x[seq_len(100)], keep.attr = TRUE)
## The GoF report
dt <- gofReport(hd)
#>
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#>
#> *** Creating report ...
#> w2p_AIC w2p_R.Cross.val w3p_AIC w3p_R.Cross.val g2p_AIC g2p_R.Cross.val
#> C1 -444.0595 0.9964743 Inf 0 -441.8100 0.9961760
#> C2 -405.1271 0.9945160 Inf 0 -390.5324 0.9937742
#> C3 -491.5704 0.9977646 Inf 0 -511.8578 0.9981487
#> T1 -479.9246 0.9973971 Inf 0 -448.9375 0.9967083
#> T2 -470.1729 0.9969760 Inf 0 -473.6995 0.9972818
#> T3 -531.2909 0.9984178 Inf 0 -504.9352 0.9978628
#> g3p_AIC g3p_R.Cross.val gg3p_AIC gg3p_R.Cross.val gg4p_AIC gg4p_R.Cross.val
#> C1 Inf 0 -442.4498 0.9962049 Inf 0
#> C2 Inf 0 -434.9769 0.9965627 Inf 0
#> C3 Inf 0 -515.8301 0.9983450 Inf 0
#> T1 Inf 0 -486.5861 0.9976635 Inf 0
#> T2 Inf 0 -472.8684 0.9971326 Inf 0
#> T3 Inf 0 -525.0245 0.9982592 Inf 0
#> bestModel
#> C1 w2p
#> C2 gg3p
#> C3 gg3p
#> T1 gg3p
#> T2 g2p
#> T3 w2p
#>
## Report the model where AIC and R.Cross.val are in conflict.
dt <- gofReport(HD = hd, output = "all", alt_models = TRUE)
#>
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#>
#> *** Creating report ...
#> Warning:
#> The best fitted model for sample(s) C1 require(s) for further analysis.
#> The model with the lowest AIC must have the highest R.Cross.val