6. Process or Product Monitoring and Control
6.5. Tutorials
6.5.4. Elements of Multivariate Analysis

6.5.4.3.

Hotelling's T squared

Hotelling's $T^2$ distribution

A multivariate method that is the multivariate counterpart of Student's $t$ and which also forms the basis for certain multivariate control charts is based on Hotelling's $T^2$ distribution, which was introduced by Hotelling (1947).

Univariate $t$-test for mean

Recall, from Section 1.3.5.2, $$t=\frac{\overline{x}-\mu }{s/\sqrt{n}}$$ has a $t$ distribution provided that $X$ is normally distributed, and can be used as long as $X$ doesn't differ greatly from a normal distribution. If we wanted to test the hypothesis that $\mu ={\mu }_0$, we would then have $$t=\frac{\overline{x}-{\mu }_0}{s/\sqrt{n}}$$ so that $$\begin{array}{rcl}{t}^2& =& \frac{(\overline{x}-{\mu }_0{)}^2}{s^2/n}\\ & & \\ & =& n(\overline{x}-{\mu }_0)(s^2{)}^{-1}(\overline{x}-{\mu }_0).\end{array}$$

Generalize to $p$ variables

When $T^2$ is generalized to $p$ variables it becomes $$T^2=n(\overline{\mathbf{x}}-{\mu }_0){\mathbf{S}}^{-1}(\overline{\mathbf{x}}-{\mu }_0),$$ with $$\overline{\mathbf{x}}=\left[\begin{array}{c}{\overline{x}}_1\\ {\overline{x}}_2\\ ⋮\\ {\overline{x}}_p\end{array}\right]{\mu }_0=\left[\begin{array}{c}{\mu }_1^0\\ {\mu }_2^0\\ ⋮\\ {\mu }_p^0\end{array}\right].$$ ${\mathbf{S}}^{-1}$ is the inverse of the sample variance-covariance matrix, $\mathbf{S}$, and $n$ is the sample size upon which each ${\overline{x}}_i,i=1,2,\dots ,p$, is based. (The diagonal elements of $\mathbf{S}$ are the variances and the off-diagonal elements are the covariances for the $p$ variables. This is discussed further in Section 6.5.4.3.1.)

Distribution of $T^2$

It is well known that when $\mu ={\mu }_0$ $$T^2\sim \frac{p(n-1)}{n-p}{F}_{(p,n-p)},$$ with $F_{(p,n-p)}$ representing the F distribution with $p$ degrees of freedom for the numerator and $n-p$ for the denominator. Thus, if $\mu$ were specified to be ${\mu }_0$, this could be tested by taking a single $p$-variate sample of size $n$, then computing $T^2$ and comparing it with $$\frac{p(n-1)}{n-p}{F}_{\alpha (p,n-p)}$$ for a suitably chosen $\alpha$.

Result does not apply directly to multivariate Shewhart-type charts

Although this result applies to hypothesis testing, it does not apply directly to multivariate Shewhart-type charts (for which there is no ${\mu }_0$, although the result might be used as an approximation when a large sample is used and data are in subgroups, with the upper control limit (UCL) of a chart based on the approximation.

Three-sigma limits from univariate control chart

When a univariate control chart is used for Phase I (analysis of historical data), and subsequently for Phase II (real-time process monitoring), the general form of the control limits is the same for each phase, although this need not be the case. Specifically, three-sigma limits are used in the univariate case, which skirts the relevant distribution theory for each Phase.

Selection of different control limit forms for each Phase

Three-sigma units are generally not used with multivariate charts, however, which makes the selection of different control limit forms for each Phase (based on the relevant distribution theory), a natural choice.

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