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- Classification and Regression Trees
- Article Info.
- Genome-wide prediction using Bayesian additive regression trees

*Nikos P. E-mail this Article Article.*

The goal of genome-wide prediction GWP is to predict phenotypes based on marker genotypes, often obtained through single nucleotide polymorphism SNP chips. The major problem with GWP is high-dimensional data from many thousands of SNPs scored on several thousands of individuals. A large number of methods have been developed for GWP, which are mostly parametric methods that assume statistical linearity and only additive genetic effects. The Bayesian additive regression trees BART method was recently proposed and is based on the sum of nonparametric regression trees with the priors being used to regularize the parameters. Each regression tree is based on a recursive binary partitioning of the predictor space that approximates an unknown function, which will automatically model nonlinearities within SNPs dominance and interactions between SNPs epistasis.

## Classification and Regression Trees

The goal of genome-wide prediction GWP is to predict phenotypes based on marker genotypes, often obtained through single nucleotide polymorphism SNP chips. The major problem with GWP is high-dimensional data from many thousands of SNPs scored on several thousands of individuals.

A large number of methods have been developed for GWP, which are mostly parametric methods that assume statistical linearity and only additive genetic effects. The Bayesian additive regression trees BART method was recently proposed and is based on the sum of nonparametric regression trees with the priors being used to regularize the parameters.

Each regression tree is based on a recursive binary partitioning of the predictor space that approximates an unknown function, which will automatically model nonlinearities within SNPs dominance and interactions between SNPs epistasis.

In evaluations using real data on pigs, the prediction error was smaller with BART than with the other methods. BART was shown to be an accurate method for GWP, in which the regression trees guarantee a very sparse representation of additive and complex non-additive genetic effects.

Moreover, the Markov chain Monte Carlo algorithm with Bayesian back-fitting provides a computationally efficient procedure that is suitable for high-dimensional genomic data.

The online version of this article doi The concept of genome-wide prediction GWP was introduced by Meuwissen et al. In order to identify SNPs that affect the phenotype of interest, state of the art genome-wide marker data comprise several thousands, sometimes millions of SNPs.

Other problems with big genome-wide datasets include spurious random correlations, incidental endogeneity, noise accumulation, and measurement error [ 3 ]. Two popular statistical approaches to overcome some of these challenges are regularized regression and variable selection [ 4 ]. Several studies have evaluated the predictive abilities of different statistical and machine learning methods in genome-wide selection situations e.

Howard et al. They found that the parametric methods predicted phenotypic values less accurately when the underlying genetic architecture was entirely based on epistasis, whereas the parametric methods resulted in only slightly better predictions than nonparametric methods when the underlying genetic architecture was additive.

However, they did not evaluate any regression tree method. A regression tree consists of three components: a tree structure with internal nodes, decision rules and a set of terminal nodes also denoted leaves. Each observation moves down a tree according to the binary decision rules contained at each internal node until it reaches a terminal node. The terminal nodes are parameterized such that each observation that is contained within a terminal node is assigned the same value.

Tree size determines the complexity of the model and it needs to be tuned to reach the optimum size. Regression trees yield a flexible model that allows for nonlinearities and interaction effects in the unknown regression function, but single trees have problems with high variance, lack of smoothness and difficulty to capture additive structure [ 2 ].

The random forest RF method [ 9 ] is a collection of many trees, often hundreds to thousands, where the trees are constructed from nonparametric bootstrap samples of the original data. RF belongs to the category of randomized independent regression trees, where trees are grown independently and predictions are averaged to reduce variance. Instead of finding the best split rule at a tree node by using all the predictor variables, RF selects at each node of each tree a random subset of variables that are used as candidates to find the best split rule for the node.

The idea behind this is to de-correlate trees so that the average over the forest ensemble will have a lower variance. Thus, for RF choices need to be made on the number of bootstrap samples and the number of sub-samples of predictors for the decision rules.

RF can also select and rank variables through different variable importance measures, which make it an important tool for genomic data analysis and bioinformatics research [ 10 , 11 ]. Chipman et al. Unfortunately, this means that the chains tend to get stuck in locally-optimal regions of the tree-space.

As an alternative, Chipman et al. BART belongs to the family of approaches based on additive regression trees, where each consecutive tree fits the residuals that are not explained by the remaining trees. Over-fitting is controlled by three prior distributions that result in simpler tree structures and less extreme estimates at the leaves. Empirical studies have frequently shown that BART outperforms alternative prediction methods [ 14 ]. We used simulated data as well as real pig data to compare methods.

A decision tree contains three parts, i. The nodes with zero children are denoted leaves or terminal nodes, and are located at the bottom of the tree. When a decision tree is applied to a regression problem, it is usually referred to as a regression tree [ 17 ]. A regression tree for two SNPs and one phenotype, the genetic interpretation and response surface are described in Additional file 1. Single regression trees are easy to construct and still relatively flexible, but there are some limitations.

First, regression trees tend to have a high variance because of the binary splits and because the errors in the higher nodes are propagated downwards. Hence, a small change in the data may result in a very different tree structure, i. Second, the terminal node surface is not smooth. This is a minor problem for SNP predictors that only have three possible values. However, it can be challenging in situations where other continuous predictors are included in the model. Third, the binary splits will favor a non-additive structure see [ 2 ] for further details.

In order to address the problems described above, Breiman [ 9 ] proposed the random forests RF methodology. The main idea of RF is to fit regression trees to bootstrap samples of the original data, and then average the result.

The trees are often grown until a minimum node size is reached and each tree is likely to have different split points and tree structures. One of the key improvements in RF is the reduction in variance obtained by reducing the correlation between bootstrapped trees. The BART model is defined as:.

The number of trees M also needs to be set. Although it would be possible to estimate the optimal number of trees by assigning a hyper-prior to this number, Chipman et al.

An alternative is to choose the hyperparameters based on cross-validation. For each MCMC iteration, the Gibbs sampler draws successively from the following conditional distributions:. This algorithm is known as Bayesian backfitting [ 18 ]. In order to draw the trees in 7 , a Metropolis—Hastings step is needed. The algorithm proposes new trees based on four possible changes of the current tree. Since the main goal of genomic prediction is to predict the future phenotypes based on available genotype and phenotype data, the full dataset was divided into training and test datasets.

For the simulated QTLMAS dataset, the individuals of generations 1—4 were used as training data and the individuals of generation 5 were used as test data.

This strategy corresponds to the two-generation cross-validation approach [ 19 ]. The real dataset of Cleveland et al.

This approach is an example of repeated random sub-sampling validation [ 19 ]. Hence, the first 25, iterations were excluded as burn-in, and the remaining iterations were thinned to a final sample of It is possible to obtain different variable importance measures VIMP. In the RF approach, there are several measures of variable importance. One common approach for regression trees is to calculate the decrease in prediction accuracy from the OOB data.

The difference between the two are then averaged over all trees, and normalized by the standard deviation of the differences [ 2 ].

The variable showing the largest decrease in prediction accuracy is the most important variable. The result is often displayed in a variable importance plot of the top ranked variables, or in Manhattan type scatter plots of all variables. BART uses a different approach where the selected variables are those that appear most often in the fitted sum-of-trees models of the MCMC chains.

It should be noted that this approach depends on M and irrelevant predictors can get mixed with relevant predictors when M is very large [ 13 ]. The simulated pedigree was founded by 20 individuals i. The pedigree structure was created by assuming that each female mates with only one male mostly with males from their own generation and gives birth to approximately 30 progeny. The biallelic SNP data was simulated using a neutral coalescent model. The continuous quantitative trait used in this study was determined by 37 quantitative trait loci QTL , including nine known genes and 28 random genes.

The known genes were selected based on their high level of polymorphism and high linkage disequilibrium LD with SNPs. They were selected if the absolute value of their additive effect was less than 2, i. The two epistatic pairs of QTL were located on chromosomes 1 and 2, respectively, and determined by four controlled additive QTL with an additional epistatic effect of 4 for the lowest homozygous pairs.

The imprinting effect was equal to 3. The narrow-sense heritability h 2 was equal to 0. A final set of SNPs was available. In order to also evaluate if BART can detect various forms of dominance and epistasis, a second simulated dataset was created based on the QTLMAS data by adding effects at different loci on chromosome 5: 1 SNP was a dominant locus by setting a value of 5 and 5. Finally, the values of these new SNPs were summed to the original y -values.

Cleveland et al. Missing genotypes were imputed using a probability score which results in non-integer values. SNPs with both known and unknown positions were included and imputed, but the map order was randomized and SNP identity was recoded. Genotyped animals had phenotypes for five purebred traits phenotypes from a single nucleus line , with heritabilities ranging from 0. For this study, we chose the trait that had a heritability of 0.

This phenotype was corrected for environmental factors and rescaled by correcting for the overall mean. Individuals with missing phenotype data were removed and a final number of individuals was used.

The second best MSPE The lowest MSPE Hence, RF can be considered to perform considerably worse than all other methods in terms of prediction error when the majority of the genetic effects are additive. The lowest MSPE obtained with each method is highlighted in italics. These results show that BART can detect complicated non-additive genetic effects and accommodate these in the predictions of phenotypes. The regression coefficient and variable importance plots in Fig.

The epistatic locus on chromosome 1 was also detected by all methods, but not the epistatic locus on chromosome 2. Neither of the imprinting effects were detected.

## Article Info.

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition. Introduction to Tree Classification. Right Sized Trees and Honest Estimates. Splitting Rules. Strengthening and Interpreting. Medical Diagnosis and Prognosis. Mass Spectra Classification.

PDF | Fifty years have passed since the publication of the first regression tree Classiﬁcation And Regression Trees (CART) (Breiman et al., ) was instrumental in Chipman, H.A., George, E.I. & McCulloch, R.E. ().

## Genome-wide prediction using Bayesian additive regression trees

Decision tree learning is one of the predictive modelling approaches used in statistics , data mining and machine learning. It uses a decision tree as a predictive model to go from observations about an item represented in the branches to conclusions about the item's target value represented in the leaves. Tree models where the target variable can take a discrete set of values are called classification trees ; in these tree structures, leaves represent class labels and branches represent conjunctions of features that lead to those class labels.

Background : Audience segmentation strategies are of increasing interest to public health professionals who wish to identify easily defined, mutually exclusive population subgroups whose members share similar characteristics that help determine participation in a health-related behavior as a basis for targeted interventions. However, it is not commonly used in public health. This is a preview of subscription content, access via your institution. Pacific Grove, CA: Wadsworth, Google Scholar.

*Classification and regression tree CART models are tree-based exploratory data analysis methods which have been shown to be very useful in identifying and estimating complex hierarchical relationships in ecological and medical contexts. In this paper, a Bayesian CART model is described and applied to the problem of modelling the cryptosporidiosis infection in Queensland, Australia.*

One approach to learning classification rules from examples is to build decision trees. That paper considered a number of different measures and experimentally examined their behavior on four domains. The main conclusion was that a random splitting rule does not significantly decrease classificational accuracy. This note suggests an alternative experimental method and presents additional results on further domains. Our results indicate that random splitting leads to increased error.

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PDF | Classification and regression trees are machine-learning methods for [5] L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone. CRC Press, [6] K.-Y. [10] H. A. Chipman, E. I. George, and R. E. McCulloch.

It can be considered a Bayesian version of machine learning tree ensemble methods where the individual trees are the base learners.

Sanjib Saha and Debashis Nandi.