W13 - MANOVA & LDA Flashcards
(13 cards)
Define a structured data matrix
n objects (rows) from k levels of a categorical predictor variable and p response variables (columns)
What is a Linear Discriminant Analysis (LDA)
Define linear combination of variables (y1 etc ) that has the
biggest difference in mean between levels of X
Define MANOVA
multivariate extension of linear model
Multiple continuous response variables
At least one categorical predictor variable
What are the steps in a multivariate data study
- MANOVA: test H0
if you reject H0, then: - LDA find the linear combination of the variables (i.e., the vector) that maximize the difference between groups.
What is MANOVA, statistically
MANOVA statistically – difference among groups in their
multivariate mean (multivariate centroid)
How are the SS & SCP partitioned in MANOVA
Between two levels:
among group (hypothesis) – group’s mean deviating from grand mean
* Within group (error or residual) – objects deviating from their group mean
What are the df’s in MANOVA
Degrees of freedom:
* k =number of groups
* n = number of objects
what is a MANOVA f-ratio
ratio of among group MS matrix to within group (error, or residual)
MS matrix
What do we do if our groups don’t differ in mean
H0 = no difference in multivariate centroid between groups.
= the AMONG group variation IS NOT greater than the
WITHIN group variation (H ≤ E; MS hypothesis is not greater
than MS error)
= eigenVALUES of F -> length
What do we do if our groups differ in mean
How do group means differ?
what is linear combination of variables that
describes the greatest distance between the
group centroids?
= eigenVECTORS of F -> direction
What are the 4 ways to test MANOVA
- Wilk’s lambda (Most used)
Accounts for difference in mean between groups in >1 dimension (orthogonal variable combination) – useful when you have more than 2 levels of predictor
* In the range from 0 ≤ Λ ≤ 1; smaller is more significant - Pillai’s trace
- Lawley-Hotelling trace
- Roy’s greatest root
Considers only the first eigenvalue.
* If only one dimension (s = 1): most powerful test
How does LDA function
Uses the eigenvectors of E-1H to describe the axis in multivariate space along which our groups differ the most.
structured covariance matrix - eigenvectors are directions in multivariate space that maximize the ratio of between-group and within-group variances (E-1H).
