# 6.5.12. Hotelling’s T²¶

The final quantity from a PCA model that we need to consider is called Hotelling’s $$T^2$$ value. Some PCA models will have many components, $$A$$, so an initial screening of these components using score scatterplots will require reviewing $$A(A-1)/2$$ scatterplots. The $$T^2$$ value for the $$i^\text{th}$$ observation is defined as:

$T^2 = \sum_{a=1}^{a=A}{\left(\dfrac{t_{i,a}}{s_a}\right)^2}$

where the $$s_a^2$$ values are constants, and are the variances of each component. The easiest interpretation is that $$T^2$$ is a scalar number that summarizes all the score values. Some other properties regarding $$T^2$$:

• It is a positive number, greater than or equal to zero.

• It is the distance from the center of the (hyper)plane to the projection of the observation onto the (hyper)plane.

• An observation that projects onto the model’s center (usually the observation where every value is at the mean), has $$T^2 = 0$$.

• The $$T^2$$ statistic is distributed according to the $$F$$-distribution and is calculated by the multivariate software package being used. For example, we can calculate the 95% confidence limit for $$T^2$$, below which we expect, under normal conditions, to locate 95% of the observations. • It is useful to consider the case when $$A=2$$, and fix the $$T^2$$ value at its 95% limit, for example, call that $$T^2_{A=2, \alpha=0.95}$$. Using the definition for $$T^2$$:

$T^2_{A=2, \alpha=0.95} = \dfrac{t^2_{1}}{s^2_1} + \dfrac{t^2_{2}}{s^2_2}$

On a scatterplot of $$t_1$$ vs $$t_2$$ for all observations, this would be the equation of an ellipse, centered at the origin. You will often see this ellipse shown on $$t_i$$ vs $$t_j$$ scatterplots of the scores. Points inside this elliptical region are within the 95% confidence limit for $$T^2$$.

• The same principle holds for $$A>2$$, except the ellipse is called a hyper-ellipse (think of a rugby-ball shaped object for $$A=3$$). The general interpretation is that if a point is within this ellipse, then it is also below the $$T^2$$ limit, if $$T^2$$ were to be plotted on a line. 