Why and How We Should Join the Shift From Significance Testing to Estimation

A paradigm shift away from null hypothesis significance testing seems in progress. Based on simulations, we illustrate some of the underlying motivations. First, P-values vary strongly from study to study, hence dichotomous inference using significance thresholds is usually unjustified. Second, statistically significant results have overestimated effect sizes, a bias declining with increasing statistical power. Third, statistically non-significant results have underestimated effect sizes, and this bias gets stronger with higher statistical power. Fourth, the tested statistical hypotheses generally lack biological justification and are often uninformative. Despite these problems, a screen of 48 papers from the 2020 volume of the Journal of Evolutionary Biology exemplifies that significance testing is still used almost universally in evolutionary biology. All screened studies tested the default null hypothesis of zero effect with the default significance threshold of p = 0.05, none presented a pre-planned alternative hypothesis, and none calculated statistical power and the probability of ‘false negatives’ (beta error). The papers reported 49 significance tests on average. Of 41 papers that contained verbal descriptions of a ‘statistically non-significant’ result, 26 (63%) falsely claimed the absence of an effect. We conclude that our studies in ecology and evolutionary biology are mostly exploratory and descriptive. We should thus shift from claiming to “test” specific hypotheses statistically to describing and discussing many hypotheses (effect sizes) that are most compatible with our data, given our statistical model. We already have the means for doing so, because we routinely present compatibility (“confidence”) intervals covering these hypotheses.

ON A NEW GENERALIZED BETA FUNCTION DEFINED BY THE GENERALIZED WRIGHT FUNCTION AND ITS APPLICATIONS

Various extensions of classical gamma, beta, Gauss hypergeometric and confluent hypergeometric functions have been proposed recently by many researchers. In this paper, we further generalized extended beta function with some of its properties such as symmetric properties, summation formulas, integral representations, connection with some other special functions such as classical beta, error, Mittag – Leffler, incomplete gamma, hypergeometric, classical Wright, Fox – Wright, Fox H and Meijer G – functions. Furthermore, the generalized beta function is used to generalize classical and other extended Gauss hypergeometric, confluent hypergeometric, Appell’s and Lauricella’s functions.

Significance Testing Needs a Taxonomy

Accurate measurement and a cutoff probability with inferential statistics are not wholly compatible. Fisher understood this when he developed the F test to deal with measurement variability and to make judgments on manipulations that may be worth further study. Neyman and Pearson focused on modeled distributions whose parameters were highly determined and concluded that inferential judgments following an F test could be made with accuracy because the distribution parameters were determined. Neyman and Pearson’s approach in the application of statistical analyses using alpha and beta error rates has played a dominant role guiding inferential judgments, appropriately in highly determined situations and inappropriately in scientific exploration. Fisher tried to explain the different situations, but, in part due to some obscure wording, generated a long standing dispute that currently has left the importance of Fisher’s p < .05 criteria not fully understood and a general endorsement of the Neyman and Pearson error rate approach. Problems were compounded with power calculations based on effect sizes following significant results entering into exploratory science. To understand in a practical sense when each approach should be used, a dimension reflecting varying levels of certainty or knowledge of population distributions is presented. The dimension provides a taxonomy of statistical situations and appropriate approaches by delineating four zones that represent how well the underlying population of interest is defined ranging from exploratory situations to highly determined populations.