12.5: Observed Significance Levels
- Page ID
- 65558
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)When another researcher is reading a paper based on a statistical analysis using a statistical hypothesis test, they may or may not be willing to take the same risk of a Type I error as the researcher who wrote the paper. Because of this, statisticians have found a method for reporting the results of a hypothesis test that allows each researcher to make their own determination as to whether they would reject the null hypothesis for the given data. The value that it used is called the observed significance level and more simply the \(p\)-value.
The observed significance level or \(p\)-value for a statistical hypothesis test is the smallest significance level for which the null hypothesis would be rejected for the observed data.
We will not discuss how a \(p\)-value is computed. We will focus on how it is used to determine if someone with a specified significance level would reject the null hypothesis when given a \(p\)-value. The usage of the \(p\)-value follows directly from the definition. Because a \(p\)-value is the smallest significance level for which the null hypothesis would be rejected given the observed data, the hypothesis would be rejected anytime the significance level is greater than the \(p\)-value. If the significance level is smaller than the \(p\)-value, the null hypothesis would not be rejected.
When looking at \(p\)-values, keep in mind that if a null hypothesis is rejected at a certain significance level, the null hypothesis would also be rejected for any significance level larger than that. For example, if a null hypothesis is rejected when \(\alpha=0.01\), the null hypothesis would also be rejected at \(\alpha=0.05\). This is why the \(p\)-value defined as is the smallest for which the null hypothesis would be rejected given the observed data.
For example, suppose a study reports that the \(p\)-value from a statistical test is \(p=0.02\). If a researcher wanted to use a significance level of \(\alpha=0.05\), they would reject the null hypothesis because the \(p\)-value is less than their specified significance level. On the other hand, if the researchers wanted to use the significance level \(\alpha=0.01\), they would fail to reject the null hypothesis because their significance level is less than the \(p\)-value.
Some research papers will not report an exact \(p\)-value but will give an upper bound on the \(p\)-value. For example, researchers may report that the \(p\)-value was less then 0.001, or \(p<0.001\). For these situations it usually suffices for the researcher to take the bound as the \(p\)-value and draw their conclusion accordingly. So, if a researcher sees a report that states \(p<0.001\) the smallest significance level they would reject for is 0.001. Then they would reject the null hypothesis with \(\alpha=0.01\) for example. Usually when \(p\)-values are reported in this way, the upper bound is small, so it usually does not make a difference in the conclusions of most researchers about the truth of the null hypothesis.
Another common technique is to show the value of a test statistic, which may not have much meaning for the reader, but annotate the value with one, two, or sometimes three asterisks. In most cases one asterisk implies that the result is significant when \(\alpha=0.05\), two asterisks when the result is significant at \(\alpha=0.01\) and three asterisks when the result is significant at \(\alpha=0.001\). For example, a research article may report that the test statistic as \(t=3.22∗∗\), which would mean that the test statistic is in the rejection region at both \(\alpha=0.05\) and \(\alpha=0.01\). The notation varies depending on the author, so the reader should look for an explanation of the notation somewhere in the research article.