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One-way multivariate analysis of variance

`d = manova1(X,group)`

d = manova1(X,group,alpha)

[d,p] = manova1(...)

[d,p,stats] = manova1(...)

`d = manova1(X,group)`

performs
a one-way Multivariate Analysis of Variance (MANOVA) for comparing
the multivariate means of the columns of `X`

, grouped
by `group`

. `X`

is an *m*-by-*n* matrix
of data values, and each row is a vector of measurements on *n* variables
for a single observation. `group`

is a grouping variable
defined as a categorical variable, vector, character array, or cell
array of character vectors. Two observations are in the same group
if they have the same value in the `group`

array.
The observations in each group represent a sample from a population.

The function returns `d`

, an estimate of the
dimension of the space containing the group means. `manova1`

tests
the null hypothesis that the means of each group are the same *n*-dimensional
multivariate vector, and that any difference observed in the sample `X`

is
due to random chance. If `d`

= `0`

, there is no evidence to reject that
hypothesis. If `d`

= `1`

,
then you can reject the null hypothesis at the 5% level, but you cannot
reject the hypothesis that the multivariate means lie on the same
line. Similarly, if `d`

= `2`

the
multivariate means may lie on the same plane in *n*-dimensional
space, but not on the same line.

`d = manova1(X,group,alpha)`

gives
control of the significance level, `alpha`

. The return
value `d`

will be the smallest dimension having `p`

> `alpha`

, where `p`

is
a *p*-value for testing whether the means lie in
a space of that dimension.

`[d,p] = manova1(...)`

also
returns a `p`

, a vector of *p*-values
for testing whether the means lie in a space of dimension 0, 1, and
so on. The largest possible dimension is either the dimension of the
space, or one less than the number of groups. There is one element
of `p`

for each dimension up to, but not including,
the largest.

If the *i*th *p*-value is
near zero, this casts doubt on the hypothesis that the group means
lie on a space of *i*-1 dimensions. The choice
of a critical *p*-value to determine whether the
result is judged statistically significant is left to the researcher
and is specified by the value of the input argument `alpha`

.
It is common to declare a result significant if the *p*-value
is less than 0.05 or 0.01.

`[d,p,stats] = manova1(...)`

also
returns `stats`

, a structure containing additional
MANOVA results. The structure contains the following fields.

Field | Contents |
---|---|

`W` | Within-groups sum of squares and cross-products matrix |

`B` | Between-groups sum of squares and cross-products matrix |

`T` | Total sum of squares and cross-products matrix |

`dfW` | Degrees of freedom for |

`dfB` | Degrees of freedom for |

`dfT` | Degrees of freedom for |

`lambda` | Vector of values of Wilk's lambda test statistic for testing whether the means have dimension 0, 1, etc. |

`chisq` | Transformation of |

`chisqdf` | Degrees of freedom for |

`eigenval` | Eigenvalues of |

`eigenvec` | Eigenvectors of `C` ,
and they are scaled so the within-group variance of the canonical
variables is 1 |

`canon` | Canonical variables |

`mdist` | A vector of Mahalanobis distances from each point to the mean of its group |

`gmdist` | A matrix of Mahalanobis distances between each pair of group means |

The canonical variables `C`

are linear combinations
of the original variables, chosen to maximize the separation between
groups. Specifically, `C(:,1)`

is the linear combination
of the `X`

columns that has the maximum separation
between groups. This means that among all possible linear combinations,
it is the one with the most significant *F* statistic
in a one-way analysis of variance. `C(:,2)`

has
the maximum separation subject to it being orthogonal to `C(:,1)`

,
and so on.

You may find it useful to use the outputs from `manova1`

along
with other functions to supplement your analysis. For example, you
may want to start with a grouped scatter plot matrix of the original
variables using `gplotmatrix`

. You can use `gscatter`

to
visualize the group separation using the first two canonical variables.
You can use `manovacluster`

to
graph a dendrogram showing the clusters among the group means.

The MANOVA test makes the following assumptions about the data
in `X`

:

The populations for each group are normally distributed.

The variance-covariance matrix is the same for each population.

All observations are mutually independent.

you can use `manova1`

to determine whether
there are differences in the averages of four car characteristics,
among groups defined by the country where the cars were made.

load carbig [d,p] = manova1([MPG Acceleration Weight Displacement],... Origin) d = 3 p = 0 0.0000 0.0075 0.1934

There are four dimensions in the input matrix, so the group
means must lie in a four-dimensional space. `manova1`

shows
that you cannot reject the hypothesis that the means lie in a 3-D
subspace.

[1] Krzanowski, W. J. *Principles
of Multivariate Analysis: A User's Perspective*. New York:
Oxford University Press, 1988.

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