5.1.2: Two-Factor Factorial - Greenhouse Example (Minitab)
- Page ID
- 33635
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For Minitab, we also need to convert the data to a stacked format (Lesson 4 2 way Stacked Dataset). Once we do this, we will need to use a different set of commands to generate the ANOVA. We use...
Stat > ANOVA > General Linear Model > Fit General Linear Model
and get the following dialog box:
Click on Model…, hold down the shift key and highlight both factors. Then click on the Add box to add the interaction to the model.
These commands will produce the ANOVA results below which are similar to the output generated by SAS (shown in the previous section).
Analysis of Variance
| Source | DF | Adj SS | Adj MS | F-value | P-value |
|---|---|---|---|---|---|
| fert | 3 | 745.44 | 248.479 | 73.10 | 0.000 |
| species | 1 | 236.74 | 236.741 | 69.65 | 0.000 |
| fert*species | 3 | 50.58 | 16.861 | 4.96 | 0.005 |
| Error | 40 | 135.97 | 3.399 | ||
| Total | 47 | 1168.73 |
Following the ANOVA run, you can generate the mean comparisons by
Stat > ANOVA > General Linear Model > Comparisons
Then specify the fert*species interaction term for the comparisons by checking the box.
Then choose Graphs to get the following dialog box, where "Interval plot for difference of means" should be checked.
The outputs are shown below.
Grouping Information Using the Tukey Method and 95% Confidence
| fert | species | N | Mean | Grouping |
|---|---|---|---|---|
| f3 | SppB | 6 | 37.0667 | A |
| f1 | SppB | 6 | 31.6167 | B |
| f2 | SppB | 6 | 30.0500 | B |
| f3 | SppA | 6 | 29.2000 | B C |
| f1 | SppA | 6 | 28.6000 | B C |
| f2 | SppA | 6 | 25.8667 | C D |
| control | SppB | 6 | 23.7000 | D E |
| control | SppA | 6 | 21.0000 | E |
Means that do not share a letter are significantly different.
