Multivariate Analyses

Confidence Ellipses

SAS/INSIGHT software provides two types of confidence ellipses for pairs of analysis variables. One is a confidence ellipse for the population mean, and the other is a confidence ellipse for prediction. A confidence ellipse for the population mean is displayed with dashed lines, and a confidence ellipse for prediction is displayed with dotted lines. Using these confidence ellipses assumes that each pair of variables has a bivariate normal distribution. Let ${\overline{Z}}$ and S be the sample mean and the unbiased estimate of the covariance matrix of a random sample of size n from a bivariate normal distribution with mean $\mu$ and covariance matrix ${\Sigma}$ .

The variable ${\overline{Z}}-{\mu}$ is distributed as a bivariate normal variate with mean 0 and covariance $n^{-1}{\Sigma}$ ,and it is independent of S. The confidence ellipse for $\mu$ is based on Hotelling's T² statistic:

$T^2 = n {({\overline Z}-{\mu})'} S^{-1} ({\overline Z}-{\mu})$

A $100(1-\alpha)\%$ confidence ellipse for $\mu$ is defined by the equation

${({\overline Z}-{\mu})'} S^{-1} ({\overline Z}-{\mu}) = \frac{2(n-1)}{n(n-2)} F_{2 , n-2}(1-{\alpha})$

where $F_{2, n-2}(1-\alpha)$ is the $(1-\alpha)$ critical value of an F variate with degrees of freedom 2 and n-2.

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 $Z-{\overline{Z}}$ is distributed as a bivariate normal variate with mean 0 and covariance $(1 + 1/n){\Sigma}$ ,and it is independent of S.

A $100(1-\alpha)\%$ confidence ellipse for prediction is then given by the equation

${(Z-{\overline Z})'} S^{-1} (Z-{\overline Z}) = \frac{2(n+1)(n-1)}{n(n-2)} F_{2 , n-2}(1-\alpha)$

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.

Scatter Plot Confidence Ellipses

Canonical Discriminant Confidence Ellipses

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