Evidence From Marginally Significant t Statistics

Figures & data

Fig. 1 Maximum integrated likelihood ratio for one-sided t-test yielding p = 0.05.

Fig. 2 ILRs for one-sided tests. The black curve represents the integrated likelihood ratio for one-sided t-tests yielding p = 0.05 under the alternative hypothesis specified in (10). The red curve represents the “average” ILR for a one-sided t-test yielding p = 0.05. The red curve was obtained by replacing the marginal density of the t statistic under the alternative hypothesis by its expectation. The blue curve represents the ILR obtained under the alternative hypothesis corresponding to $a=\overline{X}$ and $g=1/n$ in (1).

Fig. 3 ILRs for two-sided tests. The black curve depicts the maximum ILR for a two-sided t-test yielding p = 0.05. The alternative hypothesis underlying this curve assumes that $\mu =T_{0.025}^{\nu }s/\sqrt{n}$, the value of the sample mean that produces a two-sided p-value of 0.05. The green curve represents the ILR for a two-sided t-test yielding p = 0.05 obtained by setting $a=\pm \overline{X}$, each with probability one-half, and g = 0. The blue curve was obtained similarly, except that $g=1/n$ to account for variation in the sample mean. The purple curve was obtained by taking $a=\overline{X}$ and $g=1/n$ in (1). The red curve represents the “average” ILR for two-sided t-tests yielding p = 0.05. The marginal likelihood for this curve was obtained by replacing the marginal density of the data under the alternative hypothesis with its expected value at the true value of μ.