Withdrawal time using 6 min as the threshold is not a strong predictor of the likelihood of finding a polyp during colonoscopy and should not be used as a quality indicator.

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Background Machine learning techniques are increasingly being applied in health research. It is not clear how useful these approaches are for modeling continuous outcomes. Child quality of life is associated with parental socioeconomic status and physical activity and may be associated with aerobic fitness and strength. It is unclear whether diet or academic performance is associated with quality of life. Objective The purpose of this study was to compare the predictive performance of machine learning techniques with that of linear regression in examining the extent to which continuous outcomes (physical activity, aerobic fitness, muscular strength, diet, and parental education) are predictive of academic performance and quality of life and whether academic performance and quality of life are associated. Methods We modeled data from children attending 9 schools in a quasi-experimental study. We split data randomly into training and validation sets. Curvilinear, nonlinear, and heteroscedastic variables were simulated to examine the performance of machine learning techniques compared to that of linear models, with and without imputation. Results We included data for 1711 children. Regression models explained 24% of academic performance variance in the real complete-case validation set, and up to 15% in quality of life. While machine learning techniques explained high proportions of variance in training sets, in validation, machine learning techniques explained approximately 0% of academic performance and 3% to 8% of quality of life. With imputation, machine learning techniques improved to 15% for academic performance. Machine learning outperformed regression for simulated nonlinear and heteroscedastic variables. The best predictors of academic performance in adjusted models were the child’s mother having a master-level education (P<.001; β=1.98, 95% CI 0.25 to 3.71), increased television and computer use (P=.03; β=1.19, 95% CI 0.25 to 3.71), and dichotomized self-reported exercise (P=.001; β=2.47, 95% CI 1.08 to 3.87). For quality of life, self-reported exercise (P<.001; β=1.09, 95% CI 0.53 to 1.66) and increased television and computer use (P=.002; β=−0.95, 95% CI −1.55 to −0.36) were the best predictors. Adjusted academic performance was associated with quality of life (P=.02; β=0.12, 95% CI 0.02 to 0.22). Conclusions Linear regression was less prone to overfitting and outperformed commonly used machine learning techniques. Imputation improved the performance of machine learning, but not sufficiently to outperform regression. Machine learning techniques outperformed linear regression for modeling nonlinear and heteroscedastic relationships and may be of use in such cases. Regression with splines performed almost as well in nonlinear modeling. Lifestyle variables, including physical exercise, television and computer use, and parental education are predictive of academic performance or quality of life. Academic performance is associated with quality of life after adjusting for lifestyle variables and may offer another promising intervention target to improve quality of life in children.

Machine learning (ML) approaches are increasingly being used in health research. It is not clear how useful these approaches are for modelling continuous health outcomes. Child quality of life (QoL) is associated with parental socioeconomic status and child activity levels, and may be associated with aerobic fitness and strength. It is not clear whether diet, or academic performance (AP) is associated with QoL.

Consider the equationwhere x, y ∈ N and k ∈ Z. It is easy to see that (*) has infinitely many solutions where k = 0 (let x = t 2 and y = t 3 where t is a natural number). It turns out that (*) only has finitely many solutions in x and y when k is a given integer different from 0. Moreover, it is hard to find solutions of (*) where k is small compared to x and y. Hall's Conjecture states that there exists a constant C such that for any solution of (*) where k = 0, we have C √ x < |k|. For more on Hall's Conjecture, see [1] and [3].Hall's Conjecture is neither proved nor disproved. To shed some light upon the conjecture, researchers has tried to find solutions of (*) where 0 < |k| < √ x. We will refer to such solutions as good examples of Hall's Conjecture, and we will say that (x, y, k) is a good triplet when x, y ∈ N and 0This paper is a preliminary report on our search for new good examples of Hall's Conjecture. We present a new algorithm that will detect all good examples