12.1: When to Use the ANOVA
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The Analysis of Variance is an umbrella term for a host of analyses. The first use of ANOVA is to compare the means of three or more groups. The common application is to determine if race is associated with mean scores on a dependent variable. We compare White, Black, Hispanic, and Asian populations on a dependent variable, such as depression.
Note that using ANOVA to compare three or more groups is only one way we use ANOVA. ANOVA has many uses, but for the purpose of simplicity, we will start with using ANOVA to compare three or more groups.
12.1.1: Hypotheses
Null hypothesis: The three or more groups have the same mean score
Alternative hypothesis: one group has a higher/lower mean score than the other two or more groups.
12.1.2: Design Set Up
The independent variable consists of three or more groups. The independent variable is a nominal or categorical variable, which is the same term. There can be three or more groups for that nominal/categorical variable. Examples are for the variable race, which we use: White, Black, Hispanic, and Asian. For the variable treatment, we use the new treatment, control, and usual care treatment. For the variable first responders, we use firefighters, paramedics, and police officers.
You can use a continuous variable and divide it into three or more groups. You could take scores on a depression scale and divide them into non-clinical depression, sub-clinical depression, and clinical depression. You could take scores on an alcoholism scale and divide them into social drinking, binge drinking, and heavy drinking. Essentially, you are creating ordinal variables, which, as you recall, are rankings. You could ask yourself if there is a better way to analyze ordinal ranking variables, because dividing the continuous variable into three ranked groups does not seem to fit the ANOVA framework. You would be right, and yes, you could use a rank Spearman correlation. In general, using the F-test for this situation would not dramatically change the results if you used the Spearman rank correlation. Using both tests will give you unique information, but they likely will give you the same result: whether the relationship between the IV and DV is significant or not. For now, using ANOVA to analyze an independent variable that is scaled as an ordinal variable would be sufficient.
The dependent variable is continuous. Recall that continuous variables are the ordinal, interval, and ratio variables. There must be a low to high level for these continuous variables. Similar to the t-test, examples are student ratings of teacher effectiveness, level of income, number of sessions attended, ratings of helpfulness of therapy sessions, and number of alcoholic drinks consumed.
Note that you can use an ordinal variable as your outcome variable, but interpreting the ordinal variable is difficult. It is difficult to use an ANOVA to state that there are changes in ordinal ranking when ordinal ranks are based on an all-or-nothing premise. You are in first place, or you are not in first place. There is no such interpretation where you can say that you have a higher average first-place ranking compared to the second-place ranking. It does not make a lot of intuitive sense. You are better off using logistic regression.