METAS DE LA INTERPRETACIÓN 1 Concepción subjetiva

Next, we compared the performance of the estimation methods for each of the models by applying them to the nucleosomal array data. All analyses were run in the statistical software R (http://www.r-project.org/). The MCMC procedure was initialized at values generated from a N(0,1) distribution for log(λ), log(µ), log(θ) log(β),ν00 and ν10, while

the variance parameters were initialized to 1. Multiple starting points were observed to make no significant difference in the results. Convergence was attained for all the models within 1000-5000 iterations. About 10000 iterations were used for each model after burn-in, for inference. The posterior estimates of the parameters were calculated from the mean of the lagged samples after burn-in, taken to be the first 10% of the iterations. We present below a subset of the analyses that indicate most strongly the power of the new approach. Table 3.1 gives the numerical summary of the parameter estimates obtained from all three models. The parameters corresponding to the second principal component variable was the only covariate that turned out to be significant at a 5% level in either state, in both models M1 and M2. On analyzing the data set with Model M3, MCMC convergence appeared to be very slow, hence we did not use this for further analysis. We used the same dataset consisting of 12760 probes for all the three methods. For this dataset,as we had mentioned earlier, the number of non missing probes varied. But the hierarchical model was able to take into account this discrepancy by being able to integrate out all the probe specific parameters at each probe level. We also ran the algorithm on a reduced data set where we had three non missing observations at each probe. The parameters obtained from there are very similar to the that obtained for the original data set, except for the values ofσ0 and σ1. These two values are further

separated out ( .4 and .6 respectively) for the reduced data set. This indicates that the pattern of missig data is a little different between the two hidden states.

In the base model, M0, the parameter estimates for log(λ) and log(µ) implied that if a particular probe was covered by a nucleosome free region, the probability of remaining in the state is very high, approximately about 0.98. These probability estimates support the belief that nucleosomes are separated by long nucleosome-free sequences. The nucleosomal states predicted also appear to be quite long; which is natural in this case as the low resolution of the data is motivated towards enrichment of nucleosome-free potential

regulatory regions, and do not allow for detection of short nucleosome free linker regions (<10 bp) between adjacent nucleosomes, which are absorbed into the nucleosomal state. In the transition model, M1, the estimates for the emission means and variances were exactly equal to that obtained in M0. The intercept estimates of the transition

parameters matched quite well with the transition rates obtained from model M0. The second principal component was significant both in the the nucleosomal and NFR

category (Table 3.1). The opposing signs of these estimates imply that the AT oligomers associated with this covariate help in continuation of the NFR subsequences and are detrimental to nucleosome formation. The estimates of other variables were not strictly negligible, but their absolute values were comparably lower than that of the second principal component variable. Recomputing the weights of the actual dinucleotide counts with respect to this variable, we see the following oligomers play an important role in the differentiation between nucleosomal and NFR regions. (All these oligomers have a weight of greater than .1): A,T, AA, TT, AT,TA, AAT,ATA,ATT,TTA,TAT,TAA,TTTTA,TTTAT. The maximum weights were seen to be given to AT and TA simultaneously. The mono-nucleotides, the tetra-nucleotides and dinucleotides AA and TT had the lowest weights. The tri-nucleotides occupied the intermediate ranks. Interestingly, all

statistically significant oligomers (of the AT combination) having the same number of nucleotides in their configuration, shared very similar weights.

In the emission model, M2, the estimates for the transition probabilities and variances were very similar to that obtained in the model M0. Again, the second principal

component of the oligomer counts turned out to be the only significant variable. However unlike the transition model, this covariate was significant only in the NFR state.

We refitted the emission and transition models using only those 14 oligomer counts which were associated with the second principal component. The parameter estimates are given in tables 3.2 and 3.3

The starred values indicate that the estimates were less than .005.

AA and TT dinucleotides turned out to be the most significant variable for the emission model. As was in the case, when the principal components were fitted, none of the parameters corresponding to nucleosomes appeared to be significant. In fact many of them have starred values (less than .05) indicating that their effects were almost

negligible. This suggests that the intensity data within the nucleosomal segments is more or less uniformly distributed, however in the nucleosome free regions, the intensity mean is a variable function of the the AA and TT dinucleotide counts. It is interesting to observe that the effect of the other oligomer combinations, which were a part of the second principal component, vanishes when we refit the model with the oligomer counts. This implies that the effect of the second principal component was borne primarily by the counts of these two dinucleotide categories. The results from the transition model

reiterates the sole importance of the AA and TT dinucleotides in influencing the state lengths. Here as in the PCA fitted transition model, these two features were important both for the nucleosomal and the NFR states. The other features though not individually significant, the combination may be significant in determining the states

Model comparison using the BIC.One important question is which of the models M0, M1 or M2 is most appropriate for the data. One possible criterion for model choice is the Bayes factor, which however would be difficult to compute analytically here due to the complexity of the model. A simpler alternative is the Bayesian Information Criterion (BIC) which can under many circumstances be considered an approximation to the Bayes Factor. It is straightforward to compute the BIC under the different models by the formulaBIC =−2 log( ˆL) +klog(n), where ˆLis the modal posterior likelihood,k is the number of parameters, andnis the number of data points (the log-likelihood is computed through a forward algorithm). The BIC for the three models are 17154.27 (M0) 14386.92 (M1) and 13337.19 (M2), showing that the incorporation of the sequence features was an essential part in determining the structural classification. The emission model, M2, turned out to have the best fit for this data set, indicating that local sequence features indeed influence nucleosome formation; however, the sequence does not exhibit as strong an effect in determining the lengths of the state of neighboring regions.

3.5.2 Comparison with known NFR regions from UCSC genome