8.5: Interpreting a Statistical Test Result
- Page ID
- 50675
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Statistical inference, or determining if a result is significant, is based on two pieces of information—first, the statistical test result, and second, the p value. For example, you will read the following numbers: t-test = 2.45. ANOVA f-test = 4.34. correlation r = .58. Chi-square χ2 = 10.21. Higher numbers indicate significant test results.
Statistical test results are essentially ratios. The ratio is the numerator over the denominator. The numerator represents the variability we want to see, the pattern we want to observe. These patterns are usually one of two types – group A is higher/lower than group B, or increases in variable X are related to increases/decreases in variable Y. The numerator is the pattern we want to see, and that is the true variance. The denominator is the error variance or random error. Observations can happen on a random basis. If so, then these observations are random and are part of the denominator. We always want to beat error or have true variability that is greater than error. When the true variance is greater than the error variance, the numerator is higher than the denominator. Then we have a large number, hence a large stat value and a significant result.
For comparing groups, usually, any test (t-test, ANOVA) results above 1.96 are significant, indicating that there are differences between groups. You could think that 1.96 as the true variance is twice as much as the error variance, which is good. For determining associations between variables, usually results above .20, ideally .30, indicates there are associations between variables. Anytime the statistical test result is around 0, it is not significant; there are no differences between groups and no associations between variables.
Statistical inference is based on a second piece of information, the p value. For example, t-test = 2.45, p = .03. ANOVA f-test = 4.34, p = .02. correlation r = .58, p < .05. Chi-square χ2 = 10.21, p < .01. Lower p values indicate significant test results. For comparing groups, a p value less than .05 is usually significant, indicating differences between groups. For determining associations between variables, usually, a p value less than .05 is significant, indicating there are associations between variables. Anytime the p value is above .05, it is not significant and there are no differences between groups and no associations between variables.
The value of the test results and the p value are inversely related. As the test result value goes up, the p value goes down. This situation is what you want. Do you look at the test result and the p value, or one or the other? Basically, you want to attend to the p value. The statistical tests can be difficult to interpret using hard, fast numbers. Statistical test values can be fickle, meaning they can change based on several factors, such as the sample size. You can de-emphasize the test result by using the p value.