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Math Details

Both statistics, both layouts, both phases, the limits, and the decomposition.

Throughout: \(p\) is the number of variable columns, \(n\) the subgroup size, and \(m\) the number of subgroups (or of rows, when \(n = 1\)).

T², the joint mean

Subgrouped, \(n \ge 2\)

Each subgroup contributes its mean vector \(\bar{\mathbf{x}}_k\), and the plotted statistic is that vector's squared distance from the centre, measured in the metric the covariance defines:

\[ T^2_k = n\,(\bar{\mathbf{x}}_k - \bar{\bar{\mathbf{x}}})' \mathbf{S}^{-1} (\bar{\mathbf{x}}_k - \bar{\bar{\mathbf{x}}}) \]

\(\bar{\bar{\mathbf{x}}}\) is the average of the subgroup mean vectors, and \(\mathbf{S}\) is the averaged within-subgroup sample covariance: each diagonal entry is the mean of the subgroup variances for that variable, and each off-diagonal entry the mean of the subgroup covariances for that pair. Both are taken over the estimation subgroups only.

The subgroup size is not required to exceed the variable count. A design with more variables than observations per subgroup is accepted, because \(\mathbf{S}\) pools across subgroups.

Individuals, \(n = 1\)

Each row is its own point, and the covariance comes from successive differences rather than from deviations about the mean:

\[ \mathbf{S} = \frac{1}{2(m-1)} \sum_{i=1}^{m-1} \mathbf{v}_i \mathbf{v}_i' , \qquad \mathbf{v}_i = \mathbf{x}_{i+1} - \mathbf{x}_i \]
\[ T^2_i = (\mathbf{x}_i - \bar{\mathbf{x}})' \mathbf{S}^{-1} (\mathbf{x}_i - \bar{\mathbf{x}}) \]

This estimator depends on the order of the rows, which the ordinary covariance does not. That is deliberate: consecutive differences see the variation between neighbouring observations and are not inflated by a mean that drifts across the run.

The T² limits

The lower limit is 0 on every T² chart. The statistic is a squared distance and cannot be negative.

The upper limit's form depends on the phase and the layout. In all four cases the centre line is the same scale factor evaluated at the 0.5 quantile of the same reference distribution, so the centre line is the median of what an in-control process would produce, not the mean of the plotted points.

Subgrouped

\[ \text{UCL} = \frac{p\,(m-1)(n-1)}{mn - m - p + 1}\; F_{\alpha}\!\left(p,\; mn - m - p + 1\right) \qquad \text{Phase I} \]
\[ \text{UCL} = \frac{p\,(m+1)(n-1)}{mn - m - p + 1}\; F_{\alpha}\!\left(p,\; mn - m - p + 1\right) \qquad \text{Phase II} \]

The two differ only in \(m-1\) against \(m+1\). In Phase I the parameters came from the same data the chart is judging, and the limit is narrower for it; in Phase II they came from elsewhere, and the extra term is the price of the plotted point being new.

In Phase II, \(m\) is the sample count behind the historical covariance that you typed, not the number of subgroups on the chart.

Individuals

Phase I uses a Beta distribution rather than an F, which is the form for a statistic whose covariance was estimated from the very rows being tested:

\[ \text{UCL} = \frac{(m-1)^2}{m}\; \text{Beta}_{\alpha}\!\left(\frac{p}{2},\; \frac{Q-p-1}{2}\right), \qquad Q = \frac{2(m-1)^2}{3m-4} \]
\[ \text{UCL} = \frac{p\,(m+1)(m-1)}{m\,(m-p)}\; F_{\alpha}\!\left(p,\; m-p\right) \qquad \text{Phase II} \]

The Phase II individuals limit needs \(m > p\), which is why the dialog refuses a historical sample count at or below the variable count on an individuals chart.

The default alpha

\(\alpha\) defaults to \(2\,\Phi(-3) = 0.0026997960632601918\), the exact two-sided three-sigma false-alarm rate, so an untouched T² chart is set as tightly as the three-sigma Shewhart charts. The dialog rounds it to 0.0027 in its own annotation.

\[ \varphi(z) = \frac{1}{\sqrt{2\pi}}\,e^{-z^2/2} \qquad \Phi(z) = \int_{-\infty}^{z}\varphi(u)\,du \qquad z_P = \Phi^{-1}(P) \]

Quantum XL evaluates \(\Phi\) and \(\Phi^{-1}\) numerically: the C# analysis tools use the Math.NET Numerics implementation, and the C++ engine uses the Boost.Math implementation.

Full details: Standard Normal Distribution.

Generalized Variance, the joint spread

Subgrouped

\[ \text{plotted}_k = |\mathbf{S}_k| \]

the determinant of subgroup \(k\)'s own sample covariance. The centre is \(|\bar{\mathbf{S}}|\), the determinant of the averaged estimation covariance, and the limits are three-sigma moment limits:

\[ \text{UCL} = \frac{|\bar{\mathbf{S}}|}{b_1}\left(b_1 + 3\sqrt{b_2}\right), \qquad \text{LCL} = \max\left(0,\; \frac{|\bar{\mathbf{S}}|}{b_1}\left(b_1 - 3\sqrt{b_2}\right)\right) \]
\[ b_1 = \frac{\prod_{i=1}^{p}(n-i)}{(n-1)^p}, \qquad b_2 = \frac{\prod_{i=1}^{p}(n-i)}{(n-1)^{2p}}\left[\prod_{j=1}^{p}(n-j+2) - \prod_{j=1}^{p}(n-j)\right] \]

There is no alpha in either limit. \(b_1\) and \(b_2\) are the mean and the variance of the determinant of a sample covariance matrix, up to the scale \(|\Sigma|\), so these are moment limits placed three standard deviations either side, exactly as an R chart's are. That is why the dialog disables alpha on a Generalized Variance chart.

The lower limit is floored at 0, and it is often 0 in practice: a determinant cannot be negative, and \(b_1 - 3\sqrt{b_2}\) goes negative for small \(n\).

The factors require \(n > p\). Every \((n-i)\) factor has to be positive, so a Generalized Variance chart needs more observations per subgroup than variables. This is the one place where the two panels differ in what they need: T² has no such requirement.

With a historical covariance

The plotted points do not change. They are still \(|\mathbf{S}_k|\), computed from your data. Only the centre swaps to \(|\Sigma_0|\), and the same \(b_1\) and \(b_2\) structure is placed around it.

The centre is \(|\Sigma_0|\) directly, not \(b_1|\Sigma_0|\).

Individuals

A determinant needs a covariance matrix, and a single row has none, so the individuals panel plots a different quantity: the sample standard deviation of that row's standardized values, with divisor \(k_i - 1\) where \(k_i\) is the count of available cells in row \(i\). Its limits use the \(c_4\) constant at each point's own sample size:

\[ \hat{\sigma} = \frac{\sum_i c_4(k_i)\, s_i \,/\, \left(1 - c_4(k_i)^2\right)}{\sum_i c_4(k_i)^2 \,/\, \left(1 - c_4(k_i)^2\right)} \]
\[ \text{CL}_i = c_4(k_i)\,\hat{\sigma}, \qquad \text{UCL}_i,\ \text{LCL}_i = \text{CL}_i \pm 3\,\hat{\sigma}\sqrt{1 - c_4(k_i)^2} \]

with the lower limit floored at 0. The limits vary from point to point whenever the rows have different numbers of available cells, which is the same behaviour a subgroup-size-varying chart has.

A row with fewer than two available cells has no point and no limits.

\[ c_4(n) = \sqrt{\frac{2}{n - 1}}\; \frac{\Gamma\!\left(\frac{n}{2}\right)}{\Gamma\!\left(\frac{n - 1}{2}\right)} \]

\(c_4(n)\) is the expected value of the sample standard deviation of \(n\) independent standard normal observations, so \(s/c_4(n)\) is an unbiased estimate of \(\sigma\). \(\Gamma\) is the gamma function. Quantum XL evaluates the formula directly (a log-gamma form is used for large \(n\); the value approaches \(1\) as \(n\) grows and is taken as \(1\) for extremely large \(n\)).

Full details: Control Chart Constants.

Missing data

Missing cells are handled rather than refused, and the rules differ by panel and layout.

Subgrouped

Case T² Generalized Variance
An empty subgroup, every cell missing nothing plots, and the per-point centre and limits are missing the same
A partial subgroup, some cells missing the point still plots, using each column's mean over its own available cells and the nominal \(n\) multiplier. The lines stay at their nominal values the determinant of the pairwise-available covariance: deviations from those column-wise means, each pair divided by \(n_{jl} - 1\), and a pair with fewer than two available observations contributing 0. It plots only when that determinant is positive, and its lines are missing when it does not plot

Estimation over partial subgroups. The grand mean averages the subgroup column-wise means, and \(\bar{\mathbf{S}}\) averages entry by entry, each entry over only those subgroups where its pair had at least two observations. The \(m\) in the Phase I limit counts every included non-empty subgroup, so a partial subgroup counts.

Individuals

  • Column means are column-wise over the included rows' available cells, so a partial row's surviving cells still take part.
  • The successive-differences covariance keeps the whole-row chain: only pairs of consecutive complete included rows contribute a difference.
  • The Beta limit's \(m\) counts every included row with at least one available cell.
  • T² plots only complete rows, but its lines are populated at every point, which is unlike the subgrouped layout.

The decomposition

A T² point above the limit says the group moved. The decomposition says which variables moved it.

It is computed for out-of-control points only, and only on the T² panel. For each variable \(j\):

\[ \text{DECO}_j = T^2 - T^2_{(-j)} \]

where \(T^2_{(-j)}\) is the same statistic computed with variable \(j\) removed: the deviation vector and the covariance matrix both lose row and column \(j\), and

\[ T^2_{(-j)} = n\,\mathbf{d}_{(-j)}' \left(\mathbf{S}_{(-j)}\right)^{-1} \mathbf{d}_{(-j)} \]

The inverse of the smaller matrix is computed afresh. Deleting a row and a column from \(\mathbf{S}^{-1}\) does not give \(\left(\mathbf{S}_{(-j)}\right)^{-1}\), and using it would give the wrong answer.

The contributions do not sum to \(T^2\), and that is not an error to look for. Each one is the amount variable \(j\) adds given every other variable, so with correlated variables the parts overlap and their total can exceed or fall short of the whole.

A variable whose reduced covariance matrix is singular gets no contribution. The cell is left empty and the chart is unaffected: the point had already plotted. With \(p = 1\) there is no decomposition at all.

The P-Value column

The conditional term follows the univariate version of the chart's own limit, with no dependence on \(p\). For the subgrouped layout the reference distribution is

\[ \frac{m^{*} \pm 1}{m^{*}}\; F\!\left(1,\; m^{*}(n-1)\right) \]

taking \(m^{*}-1\) in Phase I and \(m^{*}+1\) in Phase II, the same substitution the chart limit makes. The denominator degrees of freedom are \(m^{*}(n-1)\), the pooled within-subgroup count, which is the chart's own denominator with \(p\) set to 1, and the leading \(p\) drops out of the scale.

\(m^{*}\) is the count of plotted subgroups, not the estimation count. Omitting a subgroup, whether through First N subgroups: or by excluding it from the limit calculation, takes it out of the estimate only: it stays plotted and it stays counted in this sample size. With nothing omitted the two counts are equal and the distinction does not arise.

The listing threshold is 0.05. A contribution is listed when its p-value falls at or below it.

Out-of-control tests

One test runs on this chart: a point beyond a control limit. It applies to both panels and it cannot be switched off. The other eight tests the Shewhart charts offer are not run here, and the dialog offers no control for them.

References

  1. Hotelling, H. (1947). "Multivariate Quality Control." In C. Eisenhart, M. W. Hastay, and W. A. Wallis, eds., Techniques of Statistical Analysis. McGraw-Hill, New York, pp. 111-184.
  2. Holmes, D. S., and Mergen, A. E. (1993). "Improving the Performance of the T-Square Control Chart." Quality Engineering, 5(4), 619-625.
  3. Tracy, N. D., Young, J. C., and Mason, R. L. (1992). "Multivariate Control Charts for Individual Observations." Journal of Quality Technology, 24(2), 88-95.
  4. Mason, R. L., Tracy, N. D., and Young, J. C. (1995). "Decomposition of T-Square for Multivariate Control Chart Interpretation." Journal of Quality Technology, 27(2), 99-108.
  5. Montgomery, D. C. (2013). Introduction to Statistical Quality Control (7th ed.). Wiley.
  6. NIST/SEMATECH e-Handbook of Statistical Methods. https://www.itl.nist.gov/div898/handbook/

See Also