4. Análisis de la normativa de EP Petroecuador
4.3 Normativa Interna de EP Petroecuador
4.3.2 Normativa de Gestión de EP Petroecuador
To assess the sensitivity of inferences on treatment effect improvements (r1−r3 and r2−r3) to the informative priors on the sensitivity parameters (˜τ1,
˜
τ2 and ˜τ3), we consider a set of point-mass priors for each ˜τx along the [0, 1]
grid. Figure 4.4 summarizes how inferences on r1− r3 and r2− r3 change for
different choices of ˜τ1, ˜τ2 and ˜τ3. Considering a significance level of 0.05. The
sensitivity analysis corroborates our conclusion that there is no evidence that the test drug has better performance than placebo. For all the choices of ˜τ1
and ˜τ3, the posterior probability of r1− r3 < 0 does not reach the 0.95 cutoff.
On the other hand, the sensitivity analysis shows that there is some evidence that the active drug is superior than placebo. For all the combinations of ˜τ2
and ˜τ3, the posterior probability of r2− r3 < 0 is greater than 0.84. For most
favorable values of ˜τ2 and ˜τ3, the posterior probability of r2− r3 < 0 is greater
than 0.95, although it only occurs when ˜τ2 is substantially smaller than ˜τ3.
In summary, for all the choices of ˜τx, we do not reach substantially different
results, which improves our confidence on the previous conclusions. Finally, inferences on r1− r3 and r2− r3 under the uniform prior ˜τx ∼ Unif(0, 1) are
roughly the same as inferences under the point-mass prior ˜τx = 0.5, which
means using the uniform prior does not induce much more uncertainty in this scenario.
0.0 0.2 0.4 0.6 0.8 1.0
˜
τ
1 0.0 0.2 0.4 0.6 0.8 1.0˜
τ
3 0.240 0.320 0.400 0.480 0.560 0.640 0.0 0.2 0.4 0.6 0.8 1.0˜
τ
2 0.0 0.2 0.4 0.6 0.8 1.0˜
τ
3 0.860 0.880 0.900 0.920 0.940 0.960 1.8 1.2 0.6 0.0 0.6 1.2 1.8 2.4 3.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0Figure 4.4: Contour plots showing inferences on treatment effect improvements r1−r3(left) and r2−r3(right) for different choices of the sensitivity parameters
along the [0, 1] grid. The colors represent posterior means of rx − r3, where
a deeper color indicates more improvement compared to placebo. The black lines show posterior probabilities of rx− r3 < 0.
4.7
Discussion
In this work, we have developed a nonparametric Bayesian approach to monotone missing data with non-ignorable missingness in the presence of auxiliary covariates. Under the extrapolation factorization, we flexibly model the observed data distribution and specify the extrapolation distribution using identifying restrictions. We have shown the inclusion of auxiliary covariates in the model could in general improve the accuracy of inferences and reduce the extent of sensitivity analysis. We have also shown more accurate inferences can be obtained by using the proposed nonparametric Bayesian approach com- pared to using more restrictive parametric approaches.
pattern and auxiliary covariates, we have assumed the effect of past responses on current response is linear (Equation (4.3) case j ≥ 2). To make the model more flexible, we could include past responses in the index set of the stochastic process a(·), i.e. to model Yj | ¯Yj−1, S, V , π = a( ¯Yj−1, V , j, S) + εjs. This
way the model could account for possible nonlinearity and interactions in the past responses. However, this modeling approach is complicated by the fact that for different time j, the dimension of past responses ¯Yj−1 is different,
so we leave it as an extension of this work. A possible compromise could be including only lag-1 response in a(·).
The computation complexity of the Gaussian process is cubic in the number of data points. The problem is manageable in our application since the schizophrenia clinical trial dataset only contains 204 subjects. When a much larger number of subjects is considered, several methods have been proposed to tackle the computational bottleneck of the GP (see Banerjee et al., 2008, 2013, Hensman et al., 2013, Datta et al., 2016). To identify the extrapolation distribution under NFD, we assume a location shift. Alternatively, we can consider exponential tilting (Rotnitzky et al., 1998, Birmingham et al., 2003). A possible extension of our work is to consider continuous time dropout. The Gaussian process is naturally suitable for the continuous case. Another extension would be more flexible incorporation of auxiliary covariates beyond the mean. Another possible future direction is to extend our method to non- monotone missing data without imposing the partial ignorability assumption. In the setting of binary outcomes, our method can be extended by using a
Chapter 5
Future Work
In the preceding chapters, we have developed nonparametric Bayesian models for biomedical data analysis. In particular, we have presented a nov- el feature allocation model for tumor subclone reconstruction using mutation pair data, a treed feature allocation model for tumor subclone phylogeny recon- struction using mutation pair data, and a nonparametric Bayesian approach to monotone missing data with auxiliary covariates in longitudinal studies. We have shown how inferences under our proposed models compare favorably with inferences under existing methods, and significantly outperform inferences un- der existing methods in certain scenarios.
As we have mentioned in the discussion section in each chapter, there are several directions for future works. For the tumor heterogeneity prob- lem, the proposed models can be extended for data where a local haplotype segment consists of more than two SNVs. We can accommodate n-tuples in- stead of pairs of SNVs by increasing the number of categorical values (Q) that the entries of Z can take. The current model measures tumor heterogeneity with single nucleotide variants (SNVs) data in copy number neutral regions. We have discussed possibilities to incorporate copy number variants (CNVs)
and have specific research plans to formally incorporate CNVs into the model and software. Also, structural variants (SVs) such as deletion, duplication, inversion, translocation and other large genome rearrangement provide more information for characterizing tumor heterogeneity. Utilizing information from SVs is another direction of characterizing tumor heterogeneity. The major mo- tivation of tumor subclone reconstruction is application to precision medicine. The reconstructed tumor subclones can be used as basis for adaptive Bayesian clinical trial design. Finally, we plan to develop computation efficient algo- rithms to handle large numbers (e.g. millions) of SNVs.
For the missing data problem, our model focuses on monotone missing data with discrete time dropout. A possible extension of this work is to con- sider continuous time dropout. The Gaussian process is suitable for this case. Another possible future direction is to extend our method to non-monotone missing data without imposing the partial ignorability assumption. In the setting of binary outcomes, our method can be extended by using a probit link. For large numbers of subjects, existing methods (Banerjee et al., 2008, 2013, Hensman et al., 2013, Datta et al., 2016) that tackle the computational bottleneck of the Gaussian process can be employed in our application.