As an expert in statistical analysis, I can provide you with a thorough explanation regarding the concept of negative F values in the context of statistical tests, particularly the F-test. The F-test is a statistical test that is used to compare the variances of two or more groups. It is commonly applied in the analysis of variance (ANOVA) and regression analysis. The F-test is based on the F-distribution, which is a continuous probability distribution that arises when two independent chi-squared variables are added together. The F-distribution is defined by two parameters: the degrees of freedom for the numerator (df1) and the degrees of freedom for the denominator (df2). When we talk about F values, we are referring to the ratio of variances. The F-test statistic is calculated as the variance between groups divided by the variance within groups. This ratio is used to determine whether the variances of the groups are significantly different from each other. Now, let's address the question of whether an F value can be negative. In theory, the F statistic is calculated as a ratio, and ratios can take on negative values. However, in the context of statistical tests, F values are not expected to be negative because variances are always non-negative quantities. Variance measures the dispersion of data points around the mean, and it is mathematically defined as the average of the squared differences from the mean. Since the square of any real number is non-negative, the variance, and by extension the F statistic, should also be non-negative. The statement you provided, "Thus, any -statistic will always be non-negative," seems to contain a typographical error. It likely meant to say that any F statistic will always be non-negative. This is because the F statistic is derived from variances, which are inherently non-negative. If you were to calculate an F statistic and obtained a negative value, it would suggest an error in the calculation process. There are a couple of scenarios mentioned in the reference material that could theoretically lead to an F statistic that is undefined or zero. The first scenario is when all conditional means are identical. In this case, there is no variation to explain, and the F statistic would be zero, indicating that there is no evidence of a difference in variances between the groups. The second scenario is when all data points exactly equal the conditional means. This is an extremely unlikely event, and it would also result in an F statistic that is undefined or zero. In practice, obtaining a negative F value is highly improbable. If it were to occur, it would typically be due to computational errors, incorrect data inputs, or misunderstandings in the statistical methodology. It is important for statisticians and researchers to carefully check their calculations and assumptions to ensure the accuracy of their results. In conclusion, while mathematically a ratio can be negative, in the context of F statistics used in statistical tests, F values are expected to be non-negative. Negative F values are not a standard outcome of correctly conducted statistical tests and should prompt a review of the methodology and calculations. read more >>

Thus, any -statistic will always be non-negative. For a given sample, it is possible to get if all conditional means are identical, or undefined if all data exactly equal the conditional means, but these are extremely unlikely to happen in practice even if the null hypothesis is completely true.May 6, 2014read more >>