3.6: One-Way ANOVA Greenhouse Example in Minitab
- Page ID
- 33439
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Step 1: Import the data
The data (Lesson 1 Data) can be copied and pasted from a word processor into a worksheet in Minitab:
Step 2: Run the ANOVA
To run the ANOVA, we use the sequence of tool-bar tabs: Stat > ANOVA > One-way…
You then get the pop-up box seen below. Be sure to select from the drop-down in the upper right, "Response data are in a separate column for each factor level":
Then we double-click from the left-hand list of factor levels to the input box labeled "Responses", and then click on the box labeled Comparisons.
We check the box for Tukey and then exit by clicking on OK. To generate the Diagnostics, we then click on the box for Graphs and select the "Three in one" option:
You can now "back out" by clicking on OK in each nested panel.
Step 3: Results
Now in the Session Window, we see the ANOVA table along with the results of the Tukey Mean Comparison:
One-Way ANOVA: Control, F1, F2, F3
Method
Null Hypothesis: All means are equal
Alternative Hypothesis: Not all means are equal
Significance Level: \(\alpha=0.05\)
Equal variances were assumed for the analysis.
Factor Information
| Factor | Levels | Values |
|---|---|---|
| Factor | 4 | Control, F1, F2, F3 |
Analysis of Variance
| Source | DF | Adj SS | Adj MS | F-Value | P-Value |
|---|---|---|---|---|---|
| Factor | 3 | 251.44 | 83.813 | 27.46 | 0.000 |
| Error | 20 | 61.03 | 3.052 | ||
| Total | 23 | 312.47 |
(Extracted from the output that follows from above.)
Grouping Information Using Tukey Method
| N | Mean | Grouping | |||
|---|---|---|---|---|---|
| F3 | 6 | 29.200 | A | ||
| F1 | 6 | 28.600 | A | B | |
| F2 | 6 | 25.867 | B | ||
| Control | 6 | 21.000 | C | ||
Means that do not share a letter are significantly different.
As can be seen, Minitab provides a difference in means plot, which can be conveniently used to identify the significantly different means by following the rule: if the confidence interval does not cross the vertical zero line, then the difference between the two associated means is statistically significant.
The diagnostic (residual) plots, as we asked for them, are in one figure:
Note that the Normal Probability plot is reversed (i.e, the axes are switched) compared to the SAS output. Assessing straight line adherence is the same, and the residual analysis provided is comparable to SAS output.