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Machine Learning - Dimensionality Reduction - Feature Extraction & Selection## An R package implementation of multifactor dimensionality reduction | BioData Mining | Full Text

One strategy for addressing missing heritability in genome-wide association study is gene-gene interaction analysis, which, unlike a single gene approach, involves high-dimensionality. The multifactor dimensionality reduction method MDR has been widely applied to reduce multi-levels of genotypes into high or low risk groups.

The Cox-MDR method has been proposed to detect gene-gene interactions associated with the survival phenotype by using the martingale residuals from a Cox model.

However, this method requires a cross-validation procedure to find the best SNP pair among all possible pairs and the permutation procedure should be followed for the significance of gene-gene interactions. Recently, the unified model based multifactor dimensionality reduction method UM-MDR has been proposed multifactor dimensionality reduction google unify the significance testing with the MDR algorithm within the regression model framework, in which neither cross-validation nor permutation testing are needed.

We also applied Cox UM-MDR to a dataset of leukemia patients and detected gene-gene interactions with regard to multifactor dimensionality reduction google survival time. The simulation results are shown to demonstrate the utility of the proposed method, which achieves at least the same power as Cox-MDR in most scenarios, and outperforms Cox-MDR when some SNPs having only marginal effects might mask the detection of the causal epistasis.

Many statistical methods in genome-wide association studies GWAS have been developed to identify susceptibility genes by considering a single SNP at a time. However, multifactor dimensionality reduction google effective sizes of the loci identified via GWAS are relatively small, and multifactor dimensionality reduction google individual loci may not be useful in assessing risk in personal genetics, as pointed out by Moore and Williams [ 2 ] and Manolio [ 3 ].

Furthermore, only a small proportion of heritability has been explained, leading to the missing heritability problem [ 4 ]. In order to overcome the missing heritability, the single-locus approach has been moved into gene-gene interaction analysis because complex diseases might be associated with multiple genes and their interactions [ 3 ].

However, the study multifactor dimensionality reduction google gene-gene interactions in GWAS involves the challenge of higher-order dimensionality, which Ritchie et al. MDR reduces multi-dimensional genotypes into one-dimensional binary attributes, in which multi-level genotypes of SNPs are classified into either high or low risk groups, using a ratio of cases and controls. The MDR mechanism can apply higher-order interactions such as two-way, three-way and so forth because all combinations of multi-way interactions can be reduced to either high or low risk groups using the appropriate classification rules.

Many modifications and extensions to MDR have been published by generalizing the classification rules and phenotypes, including the use of odds ratios [ 8 ], log-linear models [ 9 ], a generalized multifactor dimensionality reduction method GMDR for generalized linear models [ 10 ], methods for imbalanced data [ 11 ], model-based multifactor dimensionality reduction methods MB-MDR [ 12 ] and quantitative multifactor dimensionality reduction QMDR for the continuous response variables [ 13 ].

On the other hand, for a prospective cohort study, the MDR concept has been also extended to investigate those gene-gene interactions associated with the survival time. These methods extend the MDR algorithm to the survival time by using alternative classification rules, which are more applicable to survival data.

Whereas Surv-MDR is nonparametric and no covariate effect can be adjusted for [ 17 ]. However, the MDR algorithm requires cross-validation to identify the best multi-locus model multifactor dimensionality reduction google all possible combinations of SNPs and further implements computationally intensive permutation testing to check the significance of the selected multi-locus model.

A variety of classification rules has been proposed but the intensive computational procedure for cross-validation and permutation testing should be implemented as done in the original MDR method. Recently, the UM-MDR method has been proposed to address this issue by unifying the significance test with the MDR algorithm using regression model [ 17 ]. UM-MDR provides the significance test for the multi-locus model by introducing an indicator variable for the high risk after bamburasta adobe. It also allows a variety of classification rules and phenotypes.

We compared it with Cox-MDR by simulation studies. We also applied the proposed method to a real dataset of Korean leukemia patients and concluded with a discussion. As described in Multifactor dimensionality reduction google et al.

To this, they proposed a two-step unified model based MDR approach, in which multi-genetic levels were classified into high and low risk groups and an indicator variable for high risk group was defined in the first step, and then the significance of multi-locus model was achieved in the regression model with an indicator variable as well as adjusting covariates in the second step. The key idea of UM-MDR is to unify the algorithm of MDR and the significance testing of multi-locus model by using an indicator variable for high risk group.

By testing the null hypothesis of H 0: We can estimate the non-centrality parameter for each multi-locus model or pool all the statistics and then estimate the common non-centrality parameter for all multi-loci models as mentioned in [ 18 ]. We considered two disease-causal SNPs among 10 unlinked diallelic loci with the assumption of Hardy-Weinberg equilibrium and linkage equilibrium.

For the covariate adjustment, we consider only the one covariate which is associated with the survival time but has no interactions with any SNPs. We generated simulation datasets from different penetrance functions [ 11 ], which define a probabilistic relationship between the high or low risk status of groups and SNPs. We then considered 14 different combinations of two different minor allele frequencies of 0. Let f ik be an element from the i th row and the k th column of a penetrance function.

We generated patients from each of 70 penetrance models quicktime for lion os create one simulated dataset and repeated this procedure times.

We simulated the survival time from a Cox model specified as follows:. Here x is an indicator variable with value 1 for the high-risk group and 0 for the low-risk group.

In addition, the baseline hazard function follows a Weibull distribution with a shape parameter of 5 and a scale parameter of 2, the censoring time being generated from a uniform distribution, U 0, c depending on the censoring fractions which have four different censoring fractions of 0. For the power comparison, simulated datasets for each of the 70 models were generated including two disease-causal SNPs. The power of Cox UM-MDR is defined as the percentage of times that the corrected after Bonferroni correction p -value for testing the significance of the indicator variable S is less than or equal to the nominal size, called PBonf, as referred in [ 18 ].

On the other hand, the power of Cox-MDR is defined as the percentage of times that Cox-MDR correctly chooses the two disease-causal SNPs as the best model out of each set of datasets for each model. For a fair comparison, the alternative power of Cox UM-MDR is defined similarly as that of Cox-MDR, being the percentage of times that the causal model is ranked first by the corrected p -value, called PRankmultifactor dimensionality reduction google referred in [ 18 ].

As mentioned in the previous section, we considered two different scenarios, with and without the marginal SNP effects. This real dataset of 97 AML patients who had been followed-up which have age, sex and genetic information of SNPs.

At the end of the study, multifactor dimensionality reduction google were 40 deaths and 57 patients still alive. We considered two adjusting covariates, age and sex, in detecting gene-gene interaction associated with the survival time. Venn diagram for the number of SNP pairs identified by the four models.

Among the 68 SNP pairs, only 16 pairs provided statistically significant interaction effects with a p -value less than 0. The gene-gene interaction effect can be described in various terms, for example, using a semi-parametric model like a Cox model or a nonparametric approach like Cox UM-MDR and so forth. The more important point is that the interaction effect detected by the statistical method should be interpreted from a biological point of view. However, it fm10 tactics itunes not easy to connect the statistical significance directly to the biological findings.

Age, Sex, rs, rs a. Furthermore, we applied the proposed method to a real dataset of Korean leukemia patients and compared the results with those of Cox-MDR. In addition, the multi-locus model identified by Cox UM-MDR improves the power in detecting the high risk group by a log-rank test. The real data of Korean leukemia patients is not available. SYL wrote the manuscript. All authors read the paper and approved the final manuscript. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Unified Cox model based multifactor dimensionality reduction method for gene-gene interaction analysis of the survival phenotype. BioData Mining Abstract Background One strategy for addressing missing heritability in genome-wide association study is gene-gene interaction analysis, which, unlike a single gene approach, involves high-dimensionality.

Survival time Cox model Multifactor dimensionality reduction method Gene-gene interaction Unified model based method. In the cheloo operatiunea cur pansat necenzurat zippy adi step of Cox UM-MDR, we classify the multi-level genotypes into high or low risk groups by using the martingale residual of a Cox model with only the baseline hazard function.

We then define an indicator variable, Staking 1 for the high-risk group and 0 for the low-risk group. In the second step, we fit a Cox model given as follows: For the power comparison, we consider two different scenarios for the multifactor dimensionality reduction google study.

First, we conducted the power comparison when there is no marginal effect of SNPs. Under this scenario, the survival times are generated from the Cox model as follows: Under this scenario, the survival times are generated from multifactor dimensionality reduction google given Cox model as follows: Simulation results We first considered whether the type I error is controlled under the null hypothesis. For type I error, the simulation data sets were iteratively generated times under the null hypothesis multifactor dimensionality reduction google no genetic effect model across 5 different MAFs and 4 different censoring fractions.

The raw type I error was calculated without adjusting the non-centrality of the asymptotic chi-square distribution while the corrected type I error was calculated by adjusting the non-centrality.

As shown in Table 1the raw type Multifactor dimensionality reduction google error is not controlled and increases as the minor allele frequency increases while the corrected type I error is well-controlled regardless of MAF.

Here, we tried 5 times permutation for estimating the non-centrality because the number of permutations did not affect the test statistics, W. In addition, Fig. Next, we fit multifactor dimensionality reduction google following Cox model: If this null hypothesis is rejected, it implies that there is a significant gene-gene interaction associated with the survival time.

Since there are 5 models available for each combination of MAF and heritability, a total of 70 different powers are plotted consecutively on the x-axis, in which 14 different points represents the heritability within each MAF. The power increases as the heritability increases but decreases as the censoring fraction increases. Next, we fitted the following Cox model: We found that 21 SNPs had a significant marginal effect on the survival time.

The Venn diagram in Fig. As shown in the Venn diagrams, pairs, pairs, pairs and pairs are detected by models 123 and 4respectively. More SNP pairs are detected when the PC effect is unadjusted rather than adjusted in the model, for example,vs. However, the adjusting effect of PC is not critical when the main effect of SNPs is adjusted because the number of SNP pairs decreases from to As shown in Figs. For these 68 multi-locus we investigate whether the interaction effect of the corresponding SNP pairs was statistically significant or not by testing the interaction coefficient multifactor dimensionality reduction google the Cox model given as follows: Since the Cox model is commonly used to explain the association between risk factors and survival time, we compared the power of both Cox UM-MDR and Cox-MDR by significance testing for the interaction effects in a Cox model.

Although Cox-MDR selects the two pairs of rs, rs and rs, rs as the best, the interaction effect of these pairs is not found to be significant in the Cox regression model. This is multifactor dimensionality reduction google of drawbacks of Cox-MDR method, in which it cannot be guaranteed that the multifactor dimensionality reduction google SNP pairs are statistically significant without permutation testing.

To this end, we fitted a Cox model given in 1 with the attributed SNP pairs and calculated a risk score from the fitted model. We then classified all subjects into high and low risk groups based on the median risk score and tested the equivalence of the survival curves of these multifactor dimensionality reduction google groups by a log-rank test.

We found significant log-rank test results with very low p -values for all 68 SNP pairs. As shown in Fig. Furthermore, we investigated the effect of SNPs on the significant separation of these two survival curves by comparing the change of the log-rank test statistics. For all multifactor dimensionality reduction google SNP pairs showing the significant interaction effects, which are attributed by Cox UM-MDR, the power of the log-rank test is always greater than that under the model only with age and sex data not given here.

It would be said that the multi-locus model identified by Cox UM-MDR performs better in detecting the high risk group. Availability of data and materials The real data of Korean leukemia patients is not available.

One strategy for addressing missing heritability in genome-wide association study is gene-gene interaction analysis, which, unlike a single gene approach, involves high-dimensionality. The multifactor dimensionality reduction method MDR has been widely applied to reduce multi-levels of genotypes into high or low risk groups. The Cox-MDR method has been proposed to detect gene-gene interactions associated with the survival phenotype by multifactor dimensionality reduction google the martingale residuals from a Cox model.

However, this method requires a cross-validation procedure to find the best SNP pair among all possible pairs and the permutation procedure should be followed for the significance of gene-gene interactions. Recently, the unified model based multifactor dimensionality reduction method UM-MDR has been proposed to unify the significance testing with the MDR algorithm within the regression model framework, in which neither cross-validation nor permutation testing are needed.

We also applied Cox UM-MDR to a dataset of leukemia patients and detected gene-gene interactions with regard to the survival time. The simulation results are shown to demonstrate the utility of the proposed method, which achieves at least the same power as Multifactor dimensionality reduction google in most scenarios, and outperforms Cox-MDR when some SNPs having only marginal effects might mask the detection of the causal epistasis.

Many statistical methods in genome-wide darkorbit game studies GWAS have been developed to identify susceptibility genes by considering a single SNP at a time. However, the effective sizes of the loci identified via GWAS are relatively small, and these individual loci may not be useful in multifactor dimensionality reduction google risk in multifactor dimensionality reduction google genetics, as pointed out by Moore and Williams [ 2 ] and Manolio [ 3 ].

Furthermore, only a small proportion of heritability has been explained, leading to the missing heritability problem [ 4 ]. In order to overcome the missing heritability, the single-locus approach has been moved into gene-gene interaction analysis because multifactor dimensionality reduction google diseases might be associated with multiple genes and their interactions [ 3 ].

However, the study of gene-gene interactions in GWAS involves the challenge of higher-order dimensionality, which Ritchie et al. MDR reduces multi-dimensional genotypes into one-dimensional binary attributes, in which multi-level genotypes of SNPs are classified into either high or low risk groups, using a ratio of cases and controls. The MDR mechanism can apply higher-order interactions such as two-way, three-way and so forth because all combinations of multi-way interactions can be reduced to either high or low risk groups using the appropriate classification rules.

Many modifications and extensions to MDR have been published by generalizing the classification rules and phenotypes, including the use of odds ratios [ 8 ], log-linear models [ 9 ], a generalized multifactor dimensionality reduction method GMDR for generalized linear models [ 10 ], methods for imbalanced data [ 11 ], model-based multifactor dimensionality reduction methods MB-MDR [ 12 ] and quantitative multifactor dimensionality reduction QMDR for the multifactor dimensionality reduction google response variables [ 13 ].

On the other hand, for a prospective cohort study, the MDR concept has been also extended to investigate those gene-gene interactions associated with the survival time. These methods extend the MDR algorithm to the survival time by using alternative classification rules, which are more applicable to survival data.

Whereas Surv-MDR is nonparametric and no covariate effect can be adjusted for [ 17 ]. However, the MDR algorithm requires cross-validation to identify the best multi-locus model among all possible combinations of SNPs and further implements computationally intensive permutation testing to check the significance of the selected multi-locus model.

A variety of multifactor dimensionality reduction google rules has been proposed but the intensive computational procedure for cross-validation and permutation testing should be implemented as done in the original MDR method. Recently, the UM-MDR method has been proposed to address this issue by unifying the significance test with the MDR algorithm using regression model [ 17 ].

UM-MDR provides the significance test for the multi-locus model by introducing an indicator variable for the high risk after classification. It also allows a variety of classification rules and phenotypes. We compared it with Cox-MDR by simulation studies. We also applied the proposed method to a real dataset of Korean leukemia patients and multifactor dimensionality reduction google with a discussion. As described in Yu et al. To this, they proposed a two-step unified model based MDR approach, in which multi-genetic levels were classified into high and low risk groups multifactor dimensionality reduction google an indicator variable for high risk group was defined in the first step, and then the significance of multi-locus model was achieved in the regression model with an indicator variable as well as adjusting covariates in the second step.

The key idea of UM-MDR is to unify the algorithm multifactor dimensionality reduction google MDR and the significance testing of multi-locus model by using an indicator variable for high risk group.

By testing the null hypothesis of H 0: We can estimate the non-centrality parameter for each multi-locus model or pool all the statistics and then estimate the common non-centrality parameter for all multi-loci models as mentioned in [ 18 ]. We considered two disease-causal SNPs among 10 unlinked diallelic loci with the assumption of Hardy-Weinberg equilibrium and linkage equilibrium. For the covariate adjustment, we consider only the one covariate which is associated multifactor dimensionality reduction google the survival time but has no interactions with any SNPs.

Multifactor dimensionality reduction google generated simulation datasets from different penetrance functions [ 11 ], which define a probabilistic relationship between the high or low risk status of groups and SNPs. We then considered 14 different combinations of two different minor allele frequencies of 0. Let f ik be an element from the i th row and the k th column of a penetrance function. We generated patients from each of 70 penetrance models to create one simulated dataset and repeated this procedure times.

We simulated the survival time from a Cox model specified as follows:. Here x is an indicator variable with value 1 for the high-risk group and 0 for the low-risk group. In addition, the baseline hazard function follows a Weibull distribution with a shape parameter of 5 and a scale parameter of 2, multifactor dimensionality reduction google censoring time being generated from a uniform distribution, U 0, c depending on the censoring fractions which have four different censoring fractions of 0.

For the power comparison, simulated datasets for each of the 70 models were generated including two disease-causal SNPs. The power of Cox UM-MDR is defined as the percentage of times that the corrected after Bonferroni correction p -value for testing the significance of the indicator variable S is less than or equal to the nominal size, called PBonf, as referred in [ 18 ]. On the other hand, the power of Cox-MDR is defined as the percentage of times that Cox-MDR correctly chooses the two disease-causal SNPs as the best model out of each set of datasets for each model.

For a fair comparison, the alternative power of Cox UM-MDR is defined similarly as that of Cox-MDR, being the percentage of times that the causal model is ranked first by the corrected p -value, called PRankas referred in [ 18 ]. As mentioned in the previous section, we considered two different scenarios, with and without the marginal SNP effects.

This real dataset of 97 AML patients who had been followed-up which have age, sex multifactor dimensionality reduction google genetic information of SNPs.

At the end of the study, there were 40 deaths and 57 patients khiladi 786 3gp alive. We considered two adjusting covariates, age and sex, in detecting gene-gene interaction associated with the survival time. Venn diagram for multifactor dimensionality reduction google number of SNP pairs identified by the four multifactor dimensionality reduction google. Among the 68 SNP pairs, only 16 pairs provided statistically significant interaction effects multifactor dimensionality reduction google a p -value less than 0.

The gene-gene interaction effect can be described in various terms, for example, using a semi-parametric model like a Cox model or a nonparametric approach like Cox UM-MDR and so forth. The more important point is that the interaction effect detected by the statistical method should be interpreted from a biological multifactor dimensionality reduction google of view.

However, it is not easy to connect the multifactor dimensionality reduction google significance directly to the biological findings.

Age, Sex, rs, rs a. Furthermore, we applied the proposed method to a real dataset of Korean leukemia patients and compared the results with those of Cox-MDR. In addition, the multi-locus model identified by Cox UM-MDR improves the power in detecting the high risk group by a log-rank test. The real data of Korean leukemia patients is not available.

SYL wrote the manuscript. All authors read the paper and approved the final manuscript. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Unified Cox model based multifactor dimensionality reduction method for gene-gene interaction analysis of the survival phenotype. BioData Mining Abstract Background One strategy for addressing missing heritability in genome-wide association study is gene-gene interaction analysis, which, unlike a single gene approach, involves high-dimensionality.

Survival time Cox model Multifactor dimensionality reduction method Gene-gene interaction Unified model based method. In the first step of Cox UM-MDR, we classify the multi-level genotypes into high or low risk groups by using the martingale residual of a Unspeakable ace of base mp3 model with only the baseline hazard function.

We then define an indicator variable, Staking 1 for the high-risk group and 0 for the low-risk group. In the second step, we fit a Cox model given as follows: For the power comparison, we consider two different scenarios for the simulation study. First, we conducted the power comparison when there is no marginal effect of SNPs. Under this scenario, the survival times are generated from the Cox model as follows: Under this scenario, the survival times are generated from the given Cox model as follows: Simulation results We first considered whether the type I error is controlled under the null hypothesis.

For type I error, the simulation data sets were iteratively multifactor dimensionality reduction google times under the null hypothesis of no genetic effect model across 5 different MAFs and 4 different censoring fractions. The raw type I error was calculated without adjusting the non-centrality of the asymptotic chi-square distribution while the corrected type I error was calculated by adjusting the non-centrality.

As shown in Table 1the raw type I error is not controlled and increases as ca bogat e greu yahoo minor allele frequency increases while the corrected type I error is well-controlled regardless of MAF. Here, we tried 5 times permutation for estimating the non-centrality because the number of permutations did not affect the test statistics, W. In addition, Fig. Next, we fit the following Cox model: If this null hypothesis is rejected, it implies that there is a significant gene-gene interaction associated with the survival time.

Since there are 5 models available for each combination of MAF and heritability, a total of 70 different powers are plotted consecutively on the x-axis, in which 14 different points represents the heritability within each MAF. The power increases as the heritability increases but decreases as the censoring fraction increases. Next, we fitted the following Cox multifactor dimensionality reduction google We found that 21 SNPs had a significant marginal effect on the survival time.

The Venn diagram in Fig. As shown in the Venn diagrams, pairs, pairs, pairs and pairs are detected by models 123 and 4respectively. More SNP pairs are detected when the PC effect is unadjusted rather than adjusted in the model, for example,vs. However, the adjusting effect of PC is not critical when the main effect of SNPs is adjusted because the number of SNP pairs decreases from to As shown in Figs. For these 68 multi-locus we investigate whether the interaction effect of the corresponding SNP pairs was statistically significant or not by testing the interaction coefficient under the Cox model given as follows: Since the Cox model is commonly used to explain the association between risk factors and survival time, we compared the power of both Cox UM-MDR and Cox-MDR by significance testing for the interaction effects in a Cox model.

Although Cox-MDR selects the two pairs of rs, rs and rs, rs as the best, the interaction effect of these pairs is not found to be significant in the Cox regression model. This is one of drawbacks of Cox-MDR method, in which it cannot be guaranteed that the best SNP pairs are statistically significant without permutation testing. To this end, we fitted multifactor dimensionality reduction google Cox model given in 1 with the attributed SNP pairs and calculated a risk score from the fitted model.

We then classified all subjects into high and low risk groups based on the median risk score and tested the equivalence of the survival curves of these two groups by a log-rank test. We found significant log-rank test results with very low p -values for all 68 SNP pairs. As shown in Fig. Furthermore, we investigated the effect of SNPs on the significant separation of these two survival curves by comparing the change of the log-rank test statistics.

For all 16 SNP pairs showing the significant interaction effects, which are attributed by Cox UM-MDR, the power of the log-rank test is always greater than that under the model only with age and sex data not given here. It would be said that the multi-locus model identified by Cox UM-MDR performs better in detecting the high risk group. Availability of data and materials The real data of Korean leukemia patients is not available.

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