Equivalent statistics for a one-sample t-test

Recent insights into problems with common statistical practice in psychology have motivated scientists to consider alternatives to the traditional frequentist approach that compares p-values to a significance criterion. While these alternatives have worthwhile attributes, Francis (Behavior Research Methods, 40, 1524-1538, 2017) showed that many proposed test statistics for the situation of a two-sample t-test are based on precisely the same information in a given data set; and for a given sample size, one can convert from any statistic to the others. Here, we show that the same relationship holds for the equivalent of a one-sample t-test. We derive the relationships and provide an on-line app that performs the computations. A key conclusion of this analysis is that many types of tests are based on the same information, so the choice of which approach to use should reflect the intent of the scientist and the appropriateness of the corresponding inferential framework for that intent.