Aims: This study investigated the expression patterns of five crucial genes BRCA1, PALB2, CDK1, CDH1, and CHEK2 across breast cancer subtypes using Multivariate Analysis of Variance (MANOVA). Study Design: A quantitative research design based on MANOVA was employed using gene expression data from 1,139 breast cancer patients representing multiple subtypes. Place and Duration of Study: The study utilized secondary genomic data obtained from publicly available breast cancer databases and was conducted over a six-month period. Methodology: Preliminary diagnostic tests assessed normality, multicollinearity, and multivariate outliers. Shapiro–Wilk tests indicated approximate normality, VIF values below 5 confirmed the absence of multicollinearity, and Mahalanobis distance showed no significant outliers (12.68 < 20.515). Although Levene’s and Box’s M tests indicated heterogeneity of variances and covariance matrices (p < 0.001), Pillai’s trace was adopted due to its robustness. Univariate analyses and Scheffe post hoc tests were subsequently conducted. Results: MANOVA revealed a highly significant overall effect of breast cancer subtype on combined gene expression profiles (Pillai’s Trace = 0.525, F = 48.09, p < 0.001, observed power = 1.000). BRCA1, CDK1, CDH1, and CHEK2 significantly discriminated subtypes (p < 0.001), while PALB2 showed marginal significance (p = 0.003). Luminal A and HER2 demonstrated the strongest gene-specific effects, whereas Luminal B showed the least differentiation. Conclusion: The findings confirm distinct subtype-specific gene expression patterns and demonstrate the effectiveness of MANOVA for genomic studies, highlighting its relevance for precision oncology and targeted breast cancer management. Read more

Multicollinearity, a situation where explanatory variables in a regression model are highly correlated, remains a fundamental challenge in statistical analysis, particularly when dealing with complex datasets. When multicollinearity occurs, it becomes difficult to isolate the individual effects of predictor variables, often leading to inflated standard errors, unstable parameter estimates, and reduced statistical power. Numerous authors have contributed significantly to the body of literature surrounding the detection, consequences, and mitigation of multicollinearity across various regression models, and in recent years, innovative approaches have emerged that aim to address this issue directly. This section provides an overview of these advancements, highlighting the progress made in linear regression modeling and emphasizing the need for similar innovations in the context of simultaneous equation models (SEMs). Another significant challenge in estimating Linear Regression Model is the presence of heteroscedasticity, where the variance of the error terms is not constant across observations (Chen &amp; Wang, 2018). Heteroscedasticity violates one of the key assumptions of the OLS method, leading to inefficient and biased estimates of the regression parameters. To address this issue, heteroscedasticity-consistent estimators, such as Whites robust standard errors and the Generalized Method of Moments (GMM), have been developed (Pérez-Sánchez et al., 2021). These methods provide more reliable estimates in the presence of heteroscedasticity by accounting for the varying error variances. In SEMs, heteroscedasticity is particularly problematic because the error terms in different equations may have different variances, and these variances may be correlated with the explanatory variables. There is need to address this issue within SEMs framework Read more

In econometric models, a system of equations is designed to increase the efficiency of estimation. However, with an increase in data, even in large data sets, multicollinearity compromises the estimation efficiency of the models, particularly Seemingly Unrelated Regression (SUR) models. Shrinkage estimators are effective in overcoming the problem of multicollinearity but pose challenges of over shrinkage bias in large datasets, whereas ensemble learning improves prediction by aggregating multiple models. However, there is dearth of study in the combine application shrinkage estimators and ensemble learning methods. The objective of the study is to assess the performance of shrinkage and ensemble shrinkage-based estimators in the sense of alleviating the multicollinearity problem in SUR models under different sample size scenarios. The Monte Carlo method simulates the covariance matrix with collinearity level of 0.5, 0.6, 0.7, 0.8 and 0.9 among variables the error variance-covariance matrix with non-diagonal elements of 0.8. The estimators used in the study are Ridge, Lasso, Elastic Net, Boosting Ridge, Boosting Lasso, Boosting Elastic Net, Stacking Ridge, Stacking Lasso, and Stacking Elastic Net. Regularization parameters of 0.01 and 0.8, along with sample sizes of 30, 100, 1000, 5000, 10,000, 50,000, 100,000 and 200,000 were considered. Based on the results obtained in relation to the findings, it is clear that an improvement in SUR coefficients can be achieved by using ensemble shrinkage-based estimators instead of traditional shrinkage when dealing with a large sample size and high levels of multicollinearity. Of all the proposed estimators, Stacking Elastic Net performed best in all situations since it had the least value of RMSE. Read more

This study proposed and evaluated three novel robust estimators—Robust Shrinkage Generalized Method of Moments (RSGMM), Panel Adaptive Ridge GMM (PARGMM), and Heteroscedasticity-Autocorrelation-Robust Shrinkage GMM (HARSGMM)—for panel data models where classical assumptions are frequently violated. The estimators were designed to simultaneously address multicollinearity, heteroscedasticity, and autocorrelation, which commonly undermine the reliability of conventional estimators such as Ordinary Least Squares (OLS), Feasible Generalized Least Squares (FGLS), First Difference (FD), and Between Estimators (BTW). Using Monte Carlo simulations, the performance of all estimators were assessed across three scenarios of increasing violation severity and varying sample sizes. Performance metrics include bias, variance, mean squared error (MSE), and efficiency. Results revealed that HARSGMM and RSGMM consistently outperformed traditional estimators in terms of lower bias and MSE, particularly in settings with high assumption violations and larger samples. Even under baseline conditions with minimal violations, the proposed estimators maintained superior efficiency. These findings support the adoption of HARSGMM and RSGMM as more reliable alternatives for empirical researchers dealing with complex panel datasets. The study concluded with recommendations for broader application and integration of these robust techniques into econometric software and policy-oriented research. Read more

This study develops and evaluates robust estimation techniques for simultaneous equation models (SEMs) under conditions that violate the classical linear regression assumptions specifically multicollinearity, and heteroscedasticity. Building on limitations identified in conventional estimators such as Two-Stage Least Squares (2SLS), Three-Stage Least Squares (3SLS), and Full Information Maximum Likelihood (FIML), we propose five novel estimators: Adaptive Ridge IV (ARIV), Generalized Two-Stage Adaptive Elastic-Net (G2SAE), Elastic-Net IV (ENIV), Heteroscedasticity-Consistent Generalized Method of Moments (HCGMM), and Three-Stage Adaptive Elastic-Net (3SAEN). The performance of these estimators were assessed using extensive Monte Carlo simulations across varying degrees of multicollinearity, heteroscedasticity, and sample sizes (n = 30, 50, 100, 200), with 2,000 replications for each scenario. Evaluation metrics include Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and Bias. The results reveal that the proposed estimators consistently outperform traditional methods, especially under severe assumption violations. HCGMM emerges as the most robust and efficient estimator, exhibiting the lowest RMSE and bias across nearly all conditions, including small sample sizes. G2SAE and 3SAEN also demonstrate strong asymptotic properties and adaptability to complex data structures. In contrast, traditional estimators particularly 2SLS and 3SLS exhibit significant performance deterioration in the presence of heteroscedasticity and multicollinearity. A comparative analysis further highlights a trade-off between computational efficiency and estimation accuracy, with the proposed methods offering a favorable balance. These findings have practical implications for econometric modeling in applied research, particularly in fields where data irregularities are prevalent. The study underscores the need for methodological reform and adoption of robust estimation techniques to improve the reliability of policy-relevant empirical analysis. Read more

This study demonstrated the very essence of remedying the presence of heteroscedasticity, where it existed, in regression modelling. Two different hypothetical data, Data A (the Original) and Data B (the Original), were used in this study for the purpose of illustration. The normality, multicollinearity and autocorrelation assumptions were satisfied, but the Breusch-Pagan test and the White test established the existences of heteroscedasticity in the two datasets. The estimated multiple linear regression model for Data A (the Original) was statistically significant with an R-square value of 0.976, an AIC value of 332.5929, and an SBC value of 347.2533; and the one for Data B (the Original) was also statistically significant with an R-square value of 0.553, an AIC value of 69.89669, and an SBC value of 82.15499. The Log-transformation was applied on the variables in Data A (the Original) and Data B (the Original) to give rise to new sets of data, Data A (Now with Heteroscedasticity Remedied) and Data B (Now with Heteroscedasticity Remedied); which equally satisfied the normality, multicollinearity and autocorrelation assumptions, and also satisfied that there were no existences of heteroscedasticity in the two datasets. Now, the estimated multiple linear regression model for Data A (Now with Heteroscedasticity Remedied) was statistically significant with an R-square value of 0.986, an AIC value of -135.021, and an SBC value of -120.361; and the estimated model for Data B (Now with Heteroscedasticity Remedied) was statistically significant with an R-square value of 0.624, an AIC value of -32.0801, and an SBC value of -19.8218. From the points of view of the values of the R-square (0.986&gt;0.976 and 0.624&gt;0.553), AIC (-135.021&lt;332.5929 and -32.0801&lt;69.89669) and SBC (-120.361&lt;347.2533 and -19.8218&lt;82.15499), it was evident that the estimated regression models for Data A (Now with Heteroscedasticity Remedied) and Data B (Now with Heteroscedasticity Remedied) were, respectively, better models when compared to the regression models for Data A (the Original) and Data B (the Original). Read more

Machine Learning techniques such as Regression have been developed to investigate associations between risk factor and disease in multivariable analysis. However, multicollinearity amongst explanatory variables becomes a problem which makes interpretation more difficult and degrade the predictability of the model. This study compared Bridge and Elastic Net regressions in handling multicollinearity in multivariable analysis. Wisconsin Diagnostic Breast Cancer data was used for comparison for model fit and in handling multicollinearity between the regression techniques. Comparison were made using MSE, RMSE, R^2, VIF, AIC and BIC for efficiency. Scatter plots was employed to show fitted regression models. The results from the study show that, the Bridge regression performed better in solving the problem of multicollinearity with VIF value of 1.182296 when 𝛾 = 2 compared to Elastic Net regression with a VIF value of 1.204298 respectively. In comparison for best model fit, Bridge regression with 𝛾 = 0.5 performed better with MSE of 11.58667, AIC value of 258.9855 and BIC of 277.2217 respectively. Consequently, we can conclude that both the Bridge and Elastic Net Regressions can be used in handling multicollinearity problems that exist in multivariable regression analysis. Information on machine learning such as this, can help those in the medical fields to improve diagnosis, narrow clinical trials and biopsy to proffer effective treatment. Read more

Ridge regression as a solution to multicollinearity depends on the value of k, the ridge biasing constant. Since no optimum value can be found for k, as k is generally bounded between 0 and 1, i.e. 0≤ k ≤1 and it varies from one application to another. This has posed a major limitation of ridge regression in that ordinary inference procedures are not applicable and exact distributional properties are not known; and the choice of the biasing constant, k, is a judgmental one. This work examined the effect of ridge biasing constant, k, on different sample sizes using data combination from gamma and exponential distributions when multicollinearity exists. The sample sizes of 140, 100, 80, 50, 30, 20 and 10 and ridge constants, k=0.01, 0.02, . . . ., 0.1 respectively were used in the study. The Anderson Darling Test was used to check for the distribution of the data which were found to follow gamma and exponential distributions. The findings lay credence to how the ridge regression drastically remedies the effect of multicollinearity among independent variables. The study also revealed that the VIF consistently decreased as the ridge constant increased. While the ridge regression has a slight effect on the R-squared, sample sizes were found not to have any significant change or pattern on the VIFs. Since the study has shown that the VIF reduced drastically as the ridge constant increases, it is recommended to use a VIF that reduces multicollinearity to an acceptable minimum while maximizing the R-squared. This study recommends a ridge constant of 0.1 as all multicollinearity issues have been remedied at more than 90% if not completely. The study recommends using a large sample size to help stabilize the R2 values while remedying multicollinearity. Read more