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Multivariate Analyses |
The variable is distributed
as a bivariate normal variate with mean 0 and covariance
,and it is independent of S.
The confidence ellipse for
is based on
Hotelling's T2 statistic:
A confidence ellipse for prediction is a confidence region
for predicting a new observation in the population.
It also approximates a region containing
a specified percentage of the population.
Consider Z as a bivariate random
variable for a new observation.
The variable is distributed
as a bivariate normal variate with mean 0 and
covariance
,and it is independent of S.
A confidence ellipse
for prediction is then given by the equation
The family of ellipses generated by different F critical values has a common center (the sample mean) and common major and minor axes. The ellipses graphically indicate the correlation between two variables. When the variable axes are standardized (by dividing the variables by their respective standard deviations), the ratio of the two axis lengths (in Euclidean distances) reflects the magnitude of the correlation between the two variables. A ratio of 1 between the major and minor axes corresponds to a circular confidence contour and indicates that the variables are uncorrelated. A larger value of the ratio indicates a larger positive or negative correlation between the variables.
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