02-20-2024

Examples adopted from the following worK:

Pedhazur, E. J. (1997). *Multiple regression in behavioral
research: Explanation and prediction* (3rd ed.). Wadsworth.

From the chapter: “Canonical and Discriminant Analysis, and Multivariate Analysis of Variance” (pp. 924-981).

Pedhazur (1997) has the toy case where:

a researcher wishes to study how Conservatives (A

_{1}), Republicans (A_{2}), and Democrats (A_{3}) differ in their expectations regarding government spending on social welfare programs (Y_{1}) and on defense (Y_{2}) (p. 947).

And he presents the following table:

```
| --------------------------------------------------------------------------------------------------------- |
| A_1 | A_2 | A_3 |
| --------------------------------- | --------------------------------------------------------------------- |
| Y_1 | Y_2 | Y_1 | Y_2 | Y_1 | Y_2 |
| --------------- | --------------- | --------------- | --------------- | --------------- | --------------- |
| 3 | 7 | 4 | 5 | 5 | 5 |
| 4 | 7 | 4 | 6 | 6 | 5 |
| 5 | 8 | 5 | 7 | 6 | 6 |
| 5 | 9 | 6 | 7 | 7 | 7 |
| 6 | 10 | 6 | 8 | 7 | 8 |
| --------------- | --------------- | --------------- | --------------- | --------------- | --------------- |
```

I had some trouble figuring out how to organize this data for a MANOVA analysis, and then after some searching, I found help via set of class slides on MANOVA and R here: http://faculty.smu.edu/kyler/courses/7314/manova.pdf (see also: http://faculty.smu.edu/kyler/). Although it might work with other structures, the data structure I used looks like this:

group | y1 | y2 | |
---|---|---|---|

1 | 1 | 3 | 7 |

2 | 1 | 4 | 7 |

3 | 1 | 5 | 8 |

4 | 1 | 5 | 9 |

5 | 1 | 6 | 10 |

6 | 2 | 4 | 5 |

7 | 2 | 4 | 6 |

8 | 2 | 5 | 7 |

9 | 2 | 6 | 7 |

10 | 2 | 6 | 8 |

11 | 3 | 5 | 5 |

12 | 3 | 6 | 5 |

13 | 3 | 6 | 6 |

14 | 3 | 7 | 7 |

15 | 3 | 7 | 8 |

The code to create this from Pedhazur’s toy sample is:

```
manova.test <- data.frame(group = as.factor(rep(1:3, c(5,5,5))),
y1 = c(3,4,5,5,6,4,4,5,6,6,5,6,6,7,7),
y2 = c(7,7,8,9,10,5,6,7,7,8,5,5,6,7,8))
```

Pedhazur calculates Wilks’ Lambda. To do that in R, we build the model and then call the summary:

```
fit.1 <- manova(cbind(y1,y2 ~ group, manova.test)
summary.manova(fit.1, test = "Wilks")
```

And the results, which include the **F** statistic,
agree with Pedhazur.

Df | Wilks | approx F | num Df | den Df | Pr(>F) | |
---|---|---|---|---|---|---|

group | 2 | 0.096541 | 12.201 | 4 | 22 | 0.000 |

Residuals | 12 |

Pedhazur writes:

The differences among the three groups on the two dependent variables, when these are analyzed simultaneously, are statistically significant (p. 955).

Note: Pedhazur follows this with univariate analyses, which show that not all differences are statistically significant. Will add this later.