I have been reading about Hotelling's $T^2$ control charts and I'm unsure on how to deal with the case where the mean observations come from unequal-sized subgroups.

Consider $m$ observations $\mathbf{X}_1,\mathbf{X}_2, \ldots,\mathbf{X}_m $ from a multivariate process that has $p$ variables being monitored. Thus, let's assume that $$\mathbf{X}_i \sim \mathcal{N}_p\left(\boldsymbol{\mu},\Sigma\right)$$ for $i = 1,\ldots,m$. Since $\boldsymbol{\mu}$ and $\Sigma$ are usually unknown, they will be estimated by $$\mathbf{\bar{\bar{x}}}=(1 / m) \sum_{i=1}^m \bar{\mathbf{x}}_i \qquad \text{and} \qquad S =(1 / m) \sum_{i=1}^m S_i,$$ where $S_i$ is the usual covariance matrix for observations in group $i$.Now, for $T^2$ control charts, the statistic $$T_i^2 = n (\bar{\mathbf{x}}_i - \mathbf{\bar{\bar{x}}})^T S^{-1}(\bar{\mathbf{x}}_i - \mathbf{\bar{\bar{x}}})$$ is typically used for the case where there are $m$ subgroups and each one consists of $n$ observations. This theoretical framework has been studied already (see, e.g., Alt, F. and Smith, N. (1988)). In this case, $$T_i^2 \sim \frac{p(m-1)(n-1)}{(m n-m-p+1)}F_{p,mn-m-p+1}.$$ So, the upper control limit is set as the upper quantile of this distribution (with a desired $\alpha)$. That is, $$UCL = \frac{p(m-1)(n-1)}{(m n-m-p+1)}F_{p,mn-m-p+1,\alpha}.$$ However, what happens if one has a process where subgroups are not of the same size? That is, what if the subgroups have sizes $n_1, n_2, \ldots, n_m$, respectively? I have considered using a statistic $$T_i^2 = n_i (\bar{\mathbf{x}}_i - \mathbf{\bar{\bar{x}}})^T S^{-1}(\bar{\mathbf{x}}_i - \mathbf{\bar{\bar{x}}}),$$ as it seems analogous to standardizing in a univariate setting, but then the upper control limit requires one $n$ value. Something seems off. Any help or pointing in the right direction will be appreciated!

  • $\begingroup$ By unequal size of the subgroups, do you mean you have missing observation but only for some $i$ of X? Can you upload an example data-set? $\endgroup$
    – Kozolovska
    Commented Mar 29 at 6:30
  • $\begingroup$ @Kozolovska By unequal size I mean that not all of the subgroups contain the same number of observations - this could be due to missing observations or simply due to the nature of a process. For very similar values of subgroup sizes, one could average them and use $\overline{n}$ in the statistics mentioned in the question. However, for more "wildly" varying sample sizes, I would expect the idea of an average sample size not to work well in practice, thus the need for an alternative. I will try to upload a reproducible example a bit later. $\endgroup$
    – Bergson
    Commented Mar 29 at 13:39
  • $\begingroup$ Why not using pooled covariance matrix? see for example here: stats.stackexchange.com/a/400036/144600 $\endgroup$
    – Spätzle
    Commented Mar 31 at 8:05
  • $\begingroup$ @Spätzle Using the pooled covariance matrix is indeed a good way to estimate the underlying covariance. However, my question is in terms of the $T^2$ chart, i.e. to know the distribution of $T_i^2 = n (\bar{\mathbf{x}}_i - \mathbf{\bar{\bar{x}}})^T S^{-1}(\bar{\mathbf{x}}_i - \mathbf{\bar{\bar{x}}}) $ and an appropriate control limit. The distribution of the statistics depend on $n$, and is problematic when they are not equal. $\endgroup$
    – Bergson
    Commented Mar 31 at 10:36
  • $\begingroup$ See the first page here, might clarify: ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/… $\endgroup$
    – Spätzle
    Commented Mar 31 at 11:18


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