The first set of output includes descriptions of the ANOVA run. These are somewhat trivial here, but will become important as we work with more complicated situations.

Model Information | |
---|---|

Data Set | WORK.GREENHOUSE |

Dependent Variable | Height |

Covariance Structure | Diagonal |

Estimation Method | Type 3 |

Residual Variance Method | Factor |

Fixed Effects SE Method | Model-Based |

Degrees of Freedom Method | Residual |

Class Level Information | ||
---|---|---|

Class | Levels | Values |

fert | 4 | Control F1 F2 F3 |

Dimensions | |
---|---|

Covariance Parameters | 1 |

Columns in X | 5 |

Columns in Z | 0 |

Subjects | 0 |

Max Obs Per Subject | 24 |

The main output we are interested in is the **Type 3 Analysis of Variance**. It does not include the Total SS, but other than that we get the same ANOVA table that we developed earlier.

Type 3 Analysis of Variance | ||||||||
---|---|---|---|---|---|---|---|---|

Sources | DF | Sum of Squares | Mean Square | Expected Mean Square | Error Term | Error DF | F Value | Pr > F |

fert | 3 | 251.440000 | 83.813333 | Var(Residual)+Q(fert) | MS(Residual) | 20 | 27.46 | <.0001 |

Residual | 20 | 61.033333 | 3.051667 | Var(Residual) |

Covariance Parameter Estimates | |
---|---|

Cov Parm | Estimate |

Residual | 3.0517 |

Fit Statistics | |
---|---|

-2 Res Log Likelihood | 86.2 |

AIC (smaller is better) | 88.2 |

AICC (smaller is better) | 88.5 |

BIC (smaller is better) | 89.2 |

Type 3 Tests of Fixed Effects | ||||
---|---|---|---|---|

Effect | Num DF | Den DF | F Value | Pr > F |

fert | 3 | 20 | 27.46 | <.0001 |

The next output that we should discuss is that labeled as “Type 3 Tests of Fixed Effects”. This is the standard output we would get if we weren’t specifying “method = type 3”. This is the method SAS uses to calculate the ANOVA (if this is confusing don't worry, we haven't talked about this yet). These are the *F*_{calculated} and *p*-value for the test of any variables that are specified in the model statement. Note that the *F *and *p*-values are identical to that which we see in the full ANOVA table.

#### Diagnostic Plots

The Mixed procedure produces the following diagnostic plots:

We included the LSmeans statement in the PLM procedure and so we get these:

fert Least Squares Means | ||||||||
---|---|---|---|---|---|---|---|---|

fert | Estimate | Standard Error | DF | t Value | Pr > |t| | Alpha | Lower | Upper |

Control | 21.0000 | 0.7132 | 20 | 29.45 | <.0001 | 0.05 | 19.5124 | 22.4876 |

F1 | 28.6000 | 0.7132 | 20 | 40.10 | <.0001 | 0.05 | 27.1124 | 30.0876 |

F2 | 25.8667 | 0.7132 | 20 | 36.27 | <.0001 | 0.05 | 24.3790 |
27.3543 |

F3 | 29.2000 | 0.7132 | 20 | 40.94 | <.001 | 0.05 | 27.7124 | 30.6876 |

In the "Least Squares Means" table note that the *t*-value and *P* > *t* are testing Null hypotheses that the estimate = 0. (They aren't usually very important). The Lower and Upper values are the 95% confidence limits for the estimates. You may also note that there is a standard error listed after the estimates of the Least Squares Means. Here in this example we see only one value here for all the treatment levels. The reason is that these standard error are based on the residual variance estimate (MSE). The LS means will be the same as the original arithmetic means that we got in the Summary procedure because we have equal sample sizes.

With unequal sample sizes or if there is a covariate present, the LSmeans can differ from the original sample means. Plots of final results should be made with the LSmeans as shown below.. When we make plots of these, we are constrained by the software to only use error bars showing the confidence limits.

The results of the Tukey test appear in the "Difference of Least Squares Means". Notice this is different than the previous table because this table is testing each pairwise comparison. For example, the first row compares the control to the F1. We interpret this output as we would any other confidence interval for two means. The confidence interval does not contain zero, so we reject the null and conclude the difference is not zero.

Differences of fert Least Squares Means Adjustment for Multiple Comparisons: Tukey |
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fert | _fert | Estimate | Standard Error | DF | t Value | Pr > |t| | Adj P | Alpha | Lower | Upper | Adj Lower | Adj Upper |

Control | F1 | -7.6000 | 1.0086 | 20 | -7.54 | <.0001 | <.0001 | 0.05 | -9.7038 | -5.4962 | -10.4229 | -4.7771 |

Control | F2 | -4.8667 | 1.0086 | 20 | -4.83 | 0.0001 | 0.0006 | 0.05 | -6.9705 | -2.7628 | -7.6896 | -2.0438 |

Control | F3 | -8.2000 | 1.0086 | 20 | -8.13 | <.0001 | <.0001 | 0.05 | -10.3038 | -6.0962 | -11.0229 | -5.3771 |

F1 | F2 | 2.7333 | 1.0086 | 20 | 2.71 | 0.0135 | 0.0599 | 0.05 | 0.6295 | 4.8372 | -0.08957 | 5.5562 |

F1 | F3 | -0.6000 | 1.0086 | 20 | -0.59 | 0.5586 | 0.9324 | 0.05 | -2.7038 | 1.5038 | -3.4229 | 2.2229 |

F2 | F3 | -3.3333 | 1.0086 | 20 | -3.30 | 0.0035 | .0171 | 0.05 | -5.4372 | -1.2295 | -6.1562 | -0.5104 |

Tukey Grouping for fert Least Squares Means (Alpha=0.05) |
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---|---|---|---|

LS means with the same letter are not significantly different | |||

fert | Estimate | ||

F3 | 29.2000 | A | |

A | |||

F1 | 28.6000 | B | A |

B | |||

F2 | 25.8667 | B | |

Control | 21.000 | C |

The lettering here can be used now to label the LSmeans, using the Graphics Editor. Although this may seem tedious, Minitab will not produce the means plot using the LSmeans. Minitab makes a means plot, but but only from the raw data (using the original sample means).