Complete the following steps to interpret general MANOVA. Key output includes the p-value, the coefficients, R^{2}, and the residual plots.

- Step 1: Test the equality of means from all the responses
- Step 2: Determine which response means have the largest differences for each factor
- Step 3: Assess the differences between group means
- Step 4: Assess the univariate results to examine individual responses
- Step 5: Determine whether your model meets the assumptions of the analysis

To simultaneously test the equality of means from all the responses, compare the p-values in the MANOVA test tables for each term to your significance level. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that an association exists when there is no actual association.

- P-value ≤ α: The differences between the means are statistically significant
- If the p-value is less than or equal to the significance level, you can conclude that the differences between the means are statistically significant.
- P-value > α: The differences between the means are not statistically significant
- If the p-value is greater than the significance level, you cannot conclude that the differences between the means are statistically significant. You may want to refit the model without the term.

If a model term is statistically significant, the interpretation depends on the type of term. The interpretations are as follows:

- If a main effect is significant, the level means for the factor are significantly different from each other across all responses in your model.
- If an interaction term is significant, the effects of each factor are different at each level of the other factors across all responses in your model. For this reason, you should not analyze the individual effects of terms involved in significant higher-order interactions.

Test Statistic | DF | ||||
---|---|---|---|---|---|

Criterion | F | Num | Denom | P | |

Wilks' | 0.63099 | 16.082 | 2 | 55 | 0.000 |

Lawley-Hotelling | 0.58482 | 16.082 | 2 | 55 | 0.000 |

Pillai's | 0.36901 | 16.082 | 2 | 55 | 0.000 |

Roy's | 0.58482 |

Test Statistic | DF | ||||
---|---|---|---|---|---|

Criterion | F | Num | Denom | P | |

Wilks' | 0.89178 | 1.621 | 4 | 110 | 0.174 |

Lawley-Hotelling | 0.11972 | 1.616 | 4 | 108 | 0.175 |

Pillai's | 0.10967 | 1.625 | 4 | 112 | 0.173 |

Roy's | 0.10400 |

Test Statistic | DF | ||||
---|---|---|---|---|---|

Criterion | F | Num | Denom | P | |

Wilks' | 0.85826 | 2.184 | 4 | 110 | 0.075 |

Lawley-Hotelling | 0.16439 | 2.219 | 4 | 108 | 0.072 |

Pillai's | 0.14239 | 2.146 | 4 | 112 | 0.080 |

Roy's | 0.15966 |

The p-values for the production method are statistically significant at the 0.10 significance level. The p-values for the manufacturing plant are not significant at the 0.10 significance level for any of the tests. The p-values for the interaction between plant and method are statistically significant at the 0.10 significance level. Because the interaction is statistically significant, the effect of the method depends on the plant.

Use the eigen analysis to assess how the response means differ between the levels of the different model terms. You should focus on the eigenvectors that correspond to high eigenvalues. To display the eigen analysis, go to Eigen analysis under Display of Results.

and selectEigenvalue | 0.5848 | 0.00000 |
---|---|---|

Proportion | 1.0000 | 0.00000 |

Cumulative | 1.0000 | 1.00000 |

Eigenvector | 1 | 2 |
---|---|---|

Usability Rating | 0.144062 | -0.07870 |

Quality Rating | -0.003968 | 0.13976 |

In these results, the first eigenvalue for method (0.5848) is greater than the second eigenvalue (0.00000). Therefore, you should put higher importance on the first eigenvector. The first eigenvector for method is 0.144062, -0.003968. The highest absolute value within this vector is for the usability rating. This suggests that the means for usability have the largest difference between the factor levels for method. This information is helpful for assessing the means table.

Use the Means table to understand the statistically significant differences between the factor levels in your data. The mean of each group provides an estimate of each population mean. Look for differences between group means for terms that are statistically significant.

For main effects, the table displays the groups within each factor and their means. For interaction effects, the table displays all possible combinations of the groups. If an interaction term is statistically significant, do not interpret the main effects without considering the interaction effects.

To display the means, go to Univariate analysis of variance, and enter the terms in Display least squares means corresponding to the terms.

, selectUsability Rating | Quality Rating | |||
---|---|---|---|---|

Mean | SE Mean | Mean | SE Mean | |

Method | ||||

Method 1 | 4.819 | 0.165 | 5.242 | 0.193 |

Method 2 | 6.212 | 0.179 | 6.026 | 0.211 |

Plant | ||||

Plant A | 5.708 | 0.192 | 5.833 | 0.226 |

Plant B | 5.493 | 0.232 | 5.914 | 0.273 |

Plant C | 5.345 | 0.206 | 5.155 | 0.242 |

Method*Plant | ||||

Method 1 Plant A | 4.667 | 0.272 | 5.417 | 0.319 |

Method 1 Plant B | 4.700 | 0.298 | 5.400 | 0.350 |

Method 1 Plant C | 5.091 | 0.284 | 4.909 | 0.334 |

Method 2 Plant A | 6.750 | 0.272 | 6.250 | 0.319 |

Method 2 Plant B | 6.286 | 0.356 | 6.429 | 0.418 |

Method 2 Plant C | 5.600 | 0.298 | 5.400 | 0.350 |

In these results, the Means table shows how the mean usability and quality ratings varies by method, plant, and the method*plant interaction. Method and the interaction term are statistically significant at the 0.10 level. The table shows that method 1 and method 2 are associated with mean usability ratings of 4.819 and 6.212 respectively. The difference between these means is larger than the difference between the corresponding means for quality rating. This confirms the interpretation of the eigen analysis.

However, because the Method*Plant interaction term is also statistically significant, do not interpret the main effects without considering the interaction effects. For example, the table for the interaction term shows that with method 1, plant C is associated with the highest usability rating and the lowest quality rating. However, with method 2, plant A is associated with the highest usability rating and a quality rating that is nearly equal to the highest quality rating.

When you perform General MANOVA, you can choose to calculate the univariate statistics to examine the individual responses. The univariate results can provide a more intuitive understanding of the relationships in your data. However, the univariate results can differ from the multivariate results.

To display the univariate results, go to Univariate analysis of variance under Display of Results.

and selectTo determine whether the association between the response and each term in the model is statistically significant, compare the p-value for the term to your significance level to assess the null hypothesis. The null hypothesis is that there is no association between the term and the response. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that an association exists when there is no actual association.

- P-value ≤ α: The association is statistically significant
- If the p-value is less than or equal to the significance level, you can conclude that there is a statistically significant association between the response variable and the term.
- P-value > α: The association is not statistically significant
- If the p-value is greater than the significance level, you cannot conclude that there is a statistically significant association between the response variable and the term. You may want to refit the model without the term.

If a model term is statistically significant, the interpretation depends on the type of term. The interpretations are as follows:

- If a categorical factor is significant, you can conclude that not all the level means are equal.
- If an interaction term is significant, the relationship between a factor and the response depends on the other factors in the term. In this case, you should not interpret the main effects without considering the interaction effect.
- If a covariate is statistically significant, you can conclude that changes in the value of the covariate are associated with changes in the mean response value.
- If a polynomial term is significant, you can conclude that the data contain curvature.

Source | DF | Seq SS | Adj SS | Adj MS | F | P |
---|---|---|---|---|---|---|

Method | 1 | 31.264 | 29.074 | 29.0738 | 32.72 | 0.000 |

Plant | 2 | 1.366 | 1.499 | 0.7495 | 0.84 | 0.436 |

Method*Plant | 2 | 7.099 | 7.099 | 3.5494 | 3.99 | 0.024 |

Error | 56 | 49.754 | 49.754 | 0.8885 | ||

Total | 61 | 89.484 |

Source | DF | Seq SS | Adj SS | Adj MS | F | P |
---|---|---|---|---|---|---|

Method | 1 | 8.8587 | 9.2196 | 9.2196 | 7.53 | 0.008 |

Plant | 2 | 6.7632 | 7.0572 | 3.5286 | 2.88 | 0.064 |

Method*Plant | 2 | 0.7074 | 0.7074 | 0.3537 | 0.29 | 0.750 |

Error | 56 | 68.5900 | 68.5900 | 1.2248 | ||

Total | 61 | 84.9194 |

In these results, the p-value for the main effect of method and the method*plant interaction effect are statistically significant at the 0.10 level in the model for usability rating. The main effects of both method and plant are statistically significant in the model for quality rating. You can conclude that changes in these variables are associated with changes in the response variables.

Use the residual plots to help you determine whether the model is adequate and meets the assumptions of the analysis. If the assumptions are not met, the model may not fit the data well and you should use caution when you interpret the results.

When you perform General MANOVA, Minitab displays residual plots for all response variables that are in your model. You must determine whether the residual plots for all response variables indicate that the model meets the assumptions.

For more information on how to handle patterns in the residual plots, go to Residual plots for General MANOVA and click the name of the residual plot in the list at the top of the page.

Use the residuals versus fits plot to verify the assumption that the residuals are randomly distributed and have constant variance. Ideally, the points should fall randomly on both sides of 0, with no recognizable patterns in the points.

The patterns in the following table may indicate that the model does not meet the model assumptions.

Pattern | What the pattern may indicate |
---|---|

Fanning or uneven spreading of residuals across fitted values | Nonconstant variance |

Curvilinear | A missing higher-order term |

A point that is far away from zero | An outlier |

A point that is far away from the other points in the x-direction | An influential point |

In this residuals versus fits plot, the data appear to be randomly distributed about zero. There is no evidence that the value of the residual depends on the fitted value.

Use the residuals versus order plot to verify the assumption that the residuals are independent from one another. Independent residuals show no trends or patterns when displayed in time order. Patterns in the points may indicate that residuals near each other may be correlated, and thus, not independent. Ideally, the residuals on the plot should fall randomly around the center line:

If you see a pattern, investigate the cause. The following types of patterns may indicate that the residuals are dependent.

In this residuals versus order plot, the residuals appear to fall randomly around the centerline. There is no evidence that the residuals are not independent.

Use the normal probability plot of the residuals to verify the assumption that the residuals are normally distributed. The normal probability plot of the residuals should approximately follow a straight line.

The patterns in the following table may indicate that the model does not meet the model assumptions.

Pattern | What the pattern may indicate |
---|---|

Not a straight line | Nonnormality |

A point that is far away from the line | An outlier |

Changing slope | An unidentified variable |

In this normal probability plot, the points generally follow a straight line. There is no evidence of nonnormality, outliers, or unidentified variables.