Hotelling T² Statistic for Phase I

Calculate the Hotelling T² statistic for Phase I using estimated mean vector and covariance matrix. For individual observations (\(n = 1\)): \(T^2 = x_i' S^{-1} x_i\). For subgroups (\(n > 1\)): \(T^2_i = n (\bar{x}_i - \bar{\bar{x}})' S^{-1} (\bar{x}_i - \bar{\bar{x}})\).

Usage

T2.1(estat, m, n)

Arguments

estat: A list returned by stats with three components: [[1]] grand mean vector, [[2]] pooled covariance matrix, [[3]] matrix of subgroup means (\(m \times p\)).
m: Number of subgroups (Phase I sample size).
n: Subgroup size. Use \(n = 1\) for individual observations, \(n > 1\) for subgroups.

Value

A numeric vector of length \(m\) with the T² statistics.

Details

Before using this function it is necessary to execute stats (that calculates the auxiliary statistics involved in the T² formula) and the function data.1 (or other way to supply the data).

Phase convention

Phase I --- uses parameters estimated from the same data being plotted. The T² statistics are not independent.

Errors

Stop if \(n < 1\).

Decision rule

A Phase I point signals when T² exceeds the UCL from cchart.T2.1.

References

Montgomery, D.C., (2009). "Introduction to Statistical Quality Control". Chapter 11. Wiley.

Author

Daniela R. Recchia, Emanuel P. Barbosa

Examples


mu <- c(5.682, 88.22)
Sigma <- miscTools::symMatrix(c(3.770, -5.495, 13.53), 2)
#Example with individual observations
datum <- data.1(50, 1, mu, Sigma)
estat <- stats(datum, 50, 1, 2) 
T2.1(estat, 50, 1)
#>  [1]  1.63010843  4.92020396  1.54140603  2.96122927  0.24859337  0.52825846
#>  [7]  6.37793912  4.47023938  0.28213725  1.34780483  0.23775930  1.75896711
#> [13]  2.80333336  0.77451939  3.70185703  0.67761718  2.60881542  0.07459600
#> [19]  3.26686800  1.06436244  0.63787579  0.15962278  8.03795869  0.23803994
#> [25]  4.58825929  0.67745231  0.05583719  0.20899034 10.95558897  0.12548359
#> [31]  1.48228606  1.05986251  2.63978937  0.87332724  3.77130072  1.80004590
#> [37]  1.41978289  1.58868763  0.31655744  0.06265461  4.70580564  0.67197986
#> [43]  1.30294367  5.74965846  0.39175984  1.16881545  0.85410895  2.95273285
#> [49]  0.84829113  1.47748803
#Example with sub group observations
datum <- data.1(20, 10, mu, Sigma)
estat <- stats(datum, 20, 10, 2) 
T2.1(estat, 20, 10)
#>  [1] 3.2842282 1.5617464 1.1382521 2.9967570 4.1172577 1.2393318 1.2037962
#>  [8] 0.8167576 2.4744763 0.6787133 1.1616673 2.5645355 2.3503479 0.3277436
#> [15] 1.0340342 0.3318650 2.8054783 1.3435299 6.1620795 1.4261519