Multivariate Statistics – Questions & Solutions
MANOVA ✔️Ans - - One variable for situations with several variables.
- Determines significant differences among treatments when compared across
all dependent variables
- Analyzes all the dependent variables simultaneously
- Single F-statistic
- Significant (due to a small Wilk's lambda) = samples are significantly
different across the dependent variables and warrants individual ANOVAs for
each variable
- Protects the statistician against an inflated overall Type I error by high n in
ANOVA
- Examines Colinearity (through the covariance matrix) among the variables,
which can point to a group of variables discriminating among the products,
thus guarding against Type II error
Correlation and Co-variance ✔️Ans - - From the original matrix of n
objects and p variables correlation / co-variance matrix.
- Co-variance between two variables X and Y = measure of their linear
association
- Correlation between two variables by dividing the co-variance with the
product of their respective standard deviations.
- Thus, Correlation is a unit-less, scaled co-variance measure.
- Correlation is the most useful measure of interdependence between
variables b/c correlations are directly comparable, regardless of the units by
which the variables are measured.
**Correlation = with variables measured in different units
**Co-variance = same measurement
, Factor Analysis Methods (FAM) ✔️Ans - - Describe data matrix in terms
of the sum of a few underlying latent factors (axes / dimensions / principal
components / factors/ main tendencies of variation)
- Factors constitute a simpler axis system, e.g., a space of 2-3 dimensions than
the p-dimensional space spanned by the original p variables in the data
matrix.
- Boil down a matrix of 20-30 sensory attributes to a couple of factors, without
loosing much of the information in those original variables.
- Two independent steps:
1. Rank reduction:
finding the number of important factors in the data matrix
- factor scores for the n objects
- factor loading for the p variables.
2. Factor interpretation:
finding the linear transformation of the obtained factors that is most suitable
for understanding the results (expansion/contraction, rotation, translation).
Principal Component Analysis (PCA)
Vectors ✔️Ans - - Plotted as bi-plots with the objects and/or the
variables.
- Locates the center of gravity for the data cloud. Then it searches for the
straight line (through this center) that accounts for as much as possible of the
variation in the data cloud
- Creates new variables that are linear combinations of the original variables
- Principal components, are mutually orthogonal (perpendicular) and account
for decreasing amounts of the variance
- Eigenvalues are associated with each PC. A PC with an eigenvalue > 1 usually
has some 'significance'= the Kaiser criterion.
MANOVA ✔️Ans - - One variable for situations with several variables.
- Determines significant differences among treatments when compared across
all dependent variables
- Analyzes all the dependent variables simultaneously
- Single F-statistic
- Significant (due to a small Wilk's lambda) = samples are significantly
different across the dependent variables and warrants individual ANOVAs for
each variable
- Protects the statistician against an inflated overall Type I error by high n in
ANOVA
- Examines Colinearity (through the covariance matrix) among the variables,
which can point to a group of variables discriminating among the products,
thus guarding against Type II error
Correlation and Co-variance ✔️Ans - - From the original matrix of n
objects and p variables correlation / co-variance matrix.
- Co-variance between two variables X and Y = measure of their linear
association
- Correlation between two variables by dividing the co-variance with the
product of their respective standard deviations.
- Thus, Correlation is a unit-less, scaled co-variance measure.
- Correlation is the most useful measure of interdependence between
variables b/c correlations are directly comparable, regardless of the units by
which the variables are measured.
**Correlation = with variables measured in different units
**Co-variance = same measurement
, Factor Analysis Methods (FAM) ✔️Ans - - Describe data matrix in terms
of the sum of a few underlying latent factors (axes / dimensions / principal
components / factors/ main tendencies of variation)
- Factors constitute a simpler axis system, e.g., a space of 2-3 dimensions than
the p-dimensional space spanned by the original p variables in the data
matrix.
- Boil down a matrix of 20-30 sensory attributes to a couple of factors, without
loosing much of the information in those original variables.
- Two independent steps:
1. Rank reduction:
finding the number of important factors in the data matrix
- factor scores for the n objects
- factor loading for the p variables.
2. Factor interpretation:
finding the linear transformation of the obtained factors that is most suitable
for understanding the results (expansion/contraction, rotation, translation).
Principal Component Analysis (PCA)
Vectors ✔️Ans - - Plotted as bi-plots with the objects and/or the
variables.
- Locates the center of gravity for the data cloud. Then it searches for the
straight line (through this center) that accounts for as much as possible of the
variation in the data cloud
- Creates new variables that are linear combinations of the original variables
- Principal components, are mutually orthogonal (perpendicular) and account
for decreasing amounts of the variance
- Eigenvalues are associated with each PC. A PC with an eigenvalue > 1 usually
has some 'significance'= the Kaiser criterion.
