Whole-genome Regression (WGR) Analysis
A detailed description of a set of statistical models and algorithms used in whole-genome regression (WGR) analysis for genetic studies. WGR is a method utilized in quantitative genetics and genomics to predict traits or phenotypes using genetic information.
Model Description:
The whole-genome regression model is given by the equation:
y = mu + Xb + u + e
- y: Response variable (phenotypic trait).
- mu: Intercept term.
- X: Genotypic matrix (genetic markers).
- b: Regression coefficients or allele effects.
- u: Polygenic term (kernel-based if applicable).
- e: Residual term.
Methods and Algorithms:
The model can be used with various methods and algorithms for parameter estimation, offering a comprehensive toolkit for conducting whole-genome regression analyses and genetic prediction studies in quantitative genetics and genomics research.
Bayesian Approach in Genetic Prediction
The Bayesian approach is a powerful statistical framework used to predict traits or phenotypes using genetic information in genetics and genomics research.
Model Specification:
In the Bayesian framework, a probabilistic model is specified to describe the relationship between genetic markers and the phenotypic trait to be predicted. This model includes assumptions about the distribution of genetic effects, the contribution of each genetic marker to the trait, and the associated uncertainty.
Prior Distributions:
Bayesian analysis involves specifying prior distributions for the model parameters, reflecting prior beliefs or knowledge about the parameters before observing the data. For instance, prior distributions for regression coefficients may convey expectations about the effect sizes of different genetic markers.
Likelihood Function:
The likelihood function quantifies how well the model's predictions match the actual observations. In genetic prediction, it captures the relationship between genetic markers and the trait, considering both genetic effects and residual variability.
Bayes' Theorem and Posterior Distribution:
Using Bayes' theorem, the prior distributions and the likelihood function are combined to obtain the posterior distribution of the model parameters. The posterior distribution represents updated beliefs about the parameters after taking the data into account.
Model Description:
The whole-genome regression model is given by the equation:
y = mu + Xb + u + e
- y: Response variable (phenotypic trait).
- mu: Intercept term.
- X: Genotypic matrix (genetic markers).
- b: Regression coefficients or allele effects.
- u: Polygenic term (kernel-based if applicable).
- e: Residual term.
Methods and Algorithms:
The model can be used with various methods and algorithms for parameter estimation, including:
- BayesA: Incorporates a variable selection step using a Bayesian approach.
- BayesB: Similar to BayesA but allows for a proportion (pi) of markers to have no effect.
- BayesC: Another method with variable selection using Bayesian techniques.
- BayesL (Bayesian LASSO): Applies LASSO (L1 regularization).
The predictive models were validated considering the following statistical measures:
$$ RMSE= { \sqrt{\sum_{i=1}^{n}{\left(Predicted\left(y_{i}\right) - observed\left(y_{i}\right)\right)^{2} \over n}}}..............(2) $$
To check the predictive accuracy of each model Relative mean absolute percentage error (RMAPE) was used. The RMAPE is widely used to validate forecast accuracy, which provides an indication of the average size of prediction error expressed as a percentage of the relevant observed value irrespective of whether that prediction error is positive or negative.
$$ RMAPE= \frac{1}{n}{ \sum_{i=1}^{n}|\left(observed\left(y_{i}\right) - Predicted\left(y_{i}\right)\right)| \over observed\left(y_{i}\right)}\times100 .....(2)$$
$$ Model\space Accuracy = 100 - RMAPE ..............................(3) $$
Once the model has been trained. We can't assume, it can work well on data that it hasn't seen before. In other words, it can not be certain that the model will have the desired accuracy and variance in the environment of production. It needs some form of confirmation that the predictions our model is sending out are accurate. For that, our model needs to be validated. The process of determining whether the numerical results which quantify hypothesized relationships between variables are appropriate as data descriptions is known as validation. To evaluate the performance of any machine learning model we need to test it on some unseen data. Based on the models performance on unseen data we can say weather our model is Under-fitting/Over-fitting/Well generalized. Cross-validation (CV) is one of the authentication technique used to test the effectiveness of the build model, and is also a re-sampling procedure used to evaluate that model. We need to hold a sample / portion of the data that is not used to train the model to perform a CV. A 10-fold cross-validation method was applied to verify the accuracy of the models, and accuracy was tested based on the RMSE value.
The Bayesian Approach to Genetic Prediction
The Bayesian approach to genetic prediction offers several advantages in the fields of quantitative genetics, genomic selection, and association studies.
- Incorporation of Prior Information: Bayesian methods allow the incorporation of prior information, enabling the utilization of existing knowledge about the genetic architecture of traits. This facilitates more informed and personalized predictions.
- Quantification of Uncertainty: Bayesian approaches provide a probabilistic framework, allowing for the quantification of uncertainty in genetic predictions. This is particularly valuable in scenarios with limited or noisy data, as the model can account for the inherent uncertainty in genetic information.
- Handling Complex Models: Bayesian methods are well-suited to handle complex models in genetic studies. Whether applied to genomic selection or association studies, Bayesian approaches can effectively capture intricate relationships within the data, leading to more accurate predictions and meaningful insights into the genetic basis of complex traits.
The Bayesian approach is widely employed to predict traits, identify genetic markers, and enhance our understanding of the genetic factors influencing complex traits.
