ACTA IMEKO  December 2014, Volume 3, Number 4, 38 – 45  www.imeko.org  ACTA IMEKO | www.imeko.org  December 2014 | Volume 3 | Number 4 | 38  A method for assessing multivariate measurement systems  Michele Scagliarini  Department of Statistical Sciences, University of Bologna, Via Belle Arti 41, 40126 Bologna, Italy  Section: RESEARCH PAPER   Keywords: Covariance Matrices; Eigenvalues; Gauge Study; MANOVA; Wishart Distribution  Citation: Michele Scagliarini, A method for assessing multivariate measurement systems, Acta IMEKO, vol. 3, no. 4, article 8, December 2014, identifier:  IMEKO‐ACTA‐03 (2014)‐04‐08  Editor: Paolo Carbone, University of Perugia   Received December 16 th , 2013; In final form December 16 th , 2013; Published December 2014  Copyright: © 2014 IMEKO. This is an open‐access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits  unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited  Funding: This work was supported by grant from University of Bologna  Corresponding author: Michele Scagliarini, e‐mail: michele.scagliarini@unibo.it  1. INTRODUCTION In a manufacturing environment, critical decisions about process and product quality depend on the quality of the measurement systems. Measurement systems analysis (MSA) is a set of statistical techniques used to quantify the uncertainty of the measurement instruments. [1] and [2] provided a review of gauge repeatability and reproducibility (R&R) methods for assessing the precision of measurement systems. In the case of univariate measurement systems, several MSA-approval metrics are commonly used. For an overview on this topic we suggest [3] and [4]. In current manufacturing industry processes are often characterized by many important characteristics. Accordingly, [5] proposed multivariate extensions of three commonly used MSA-approval criteria using the volume of constant-density contours to characterize the variability of the measurement system. These multivariate MSA-metrics require a multivariate analysis of variance (MANOVA) for estimating the covariance matrices for one factor and two- factor gauge studies. In order to ensure constant flows of reliable data, manufacturers should periodically assess their measurement systems and the costs involved in maintaining well performing measurement systems are normally relevant. This issue motivates the present work. Multivariate measurement systems analysis is usually performed by designing suitable gauge (R&R) experiments ignoring available data generated by the measurement system while used for inspection or process control. In recent literature, the use of these measurements from regular use of the instrument has been suggested for univariate MSA studies (see e.g, [6]). Here we propose the following approach. In the initial set up, after the multivariate measurement instrument is assessed as adequate, its performances are assumed as benchmark. Therefore, using the data from the regular activity of the instrument, the periodic assessments of the measurement device are performed by comparing the present precision with the benchmark through a statistical test. Since the proposed method does not require a multivariate gauge study, our proposal can be a useful tool for reducing the costs of multivariate MSA carried out with a certain frequency. Here is the outline of the paper. The next section introduces the multivariate measurement error model, ABSTRACT Multivariate measurement systems analysis is usually performed by designing suitable gauge R&R experiments ignoring available data  generated by the measurement system while used for inspection or process control. This article proposes an approach that, using the  data  that  are  routinely  available  from  the  regular  activity  of  the  instrument,  offers  the  possibility  of  assessing  multivariate  measurement systems without the necessity of performing a multivariate gauge study. It can be carried out more frequently than a  multivariate  gauge  R&R  experiment,  since  it  can  be  implemented  at  almost  no  additional  cost.  Therefore  the  synergic  use  of  the  proposed approach and the traditional multivariate gauge R&R studies can be a useful strategy for  improving the overall quality of  multivariate measurement systems and is effective for reducing the costs of a multivariate MSA performed with a certain frequency.  ACTA IMEKO | www.imeko.org  December 2014 | Volume 3 | Number 4 | 39  describes the multivariate MSA-approval criteria proposed in recent literature and explains the multivariate analysis of variance (MANOVA) method for estimating the covariance matrices of interest. Section 3 develops the test for assessing the multivariate measurement instruments. Section 4 studies the performances of the proposed method. Finally, the last section contains a discussion and the conclusions. 2. MULTIVARIATE MEASUREMENT SYSTEM ANALYSIS  Let  1 2, ,..., mX X X X (1) represent the vector of m quality characteristics with mean vector µ and covariance matrix  positive definite. We assume that the multivariate process data are from a multivariate normal distribution. Let us denote with  1 2, ,...,  mLSL LSL LSLLSL , (2)  1 2, ,...,  mUSL USL USLUSL , (3) and  1 2, ,...,  mT T TT (4) the m-vectors values of the lower specification limits, upper specification limits and target values, respectively. MSA methodology assumes the model  Y X e (5) where Y is the vector of the observable quality characteristics, which is usually obtained from some physical measurements, X is the true quality characteristics vector and e is the multivariate measurement error vector. It is assumed that e  ,0 eN Σ (6) with c positive definite and that X and e are independent. As a result, Y  , yN μ Σ (7) where  y eΣ Σ Σ . Let us denote with i, ei and yi (i=1,2,…,m) the eigenvalues of , e and y, respectively. In the multivariate framework, [5] developed multivariate versions of three univariate gauge-approval criteria. The author proposed the following statistics. The multivariate precision-to-tolerance ratio, which is defined as the m-th root of the ratio of (1-)100% volume of the multivariate error distribution and the volume of the tolerance region. This ratio, according to the specification of the tolerance region, simplifies to   1/ 2 /2 , 1 1 1 1 2 m m m m ei i m m i I P T TOL m                           (8) when a hypercube-shaped tolerance region is used, and 1/ 2 , 12 2 m m m ei im i P T TOL             (9) for the case of a hyperellipsoid-shaped tolerance region. In the above equations (·) is the gamma function, TOLi=USLi-LSLi and 2,m is the 100(1-)-th percentile of the 2 distribution with m degrees of freedom with (1-) usually fixed at 0.99. Therefore the P/T1m and P/T2m criteria compare the multivariate instrument variability, computed on the base of the constant-density contour ellipsoid, with the multivariate tolerance region (hypercube or hyperellipsoid). The multivariate percent R&R ratio, which is defined taking the m-th root of the (1-)100% volumes of the gauge error distribution and measured-values distribution. The statistic in question simplifies in 1/ 1 % & 100 m m ei m i yi R R             (10) and expresses the relative widths of the multivariate distributions of the error e and the measured values Y. The third multivariate approval-metric is the multivariate-signal-to-noise ratio which compares the (1- )100% volume of the gauge-error distribution. This statistic is 1/ 1 2 m m i m i ei SNR            . (11) The author in [5] also gave the guidelines for gauge acceptance. Approval values for P/T1m and P/T2m range from 0 to 0.3, %R&Rm should be ≤30%, while based on SNRm a measurement system is adequate when SNRm≥5. 2.1. Multivariate MSA in practice  The covariance matrices , e and y are usually unknown, for this reason [5] also proposes a multivariate analysis of variance (MANOVA) method of estimating the covariance matrices for one-factor and two-factor gauge studies. According to the adopted notation, Y′=[Y1,Y2,...,Ym] represents the measured values or data generated by the gauge. Let us consider a random-effects MANOVA model [7] where the factors in question are all m-dimensional vectors. Let us denote the factors as ip (i=1,2,…,p) for part, jo (j=1,2,…,o) for operator, ijpo for part-operator interaction. The error term is denoted by ijkε , where k=1,2,…,r indicates the repeated reading of the same part by the same operator. Therefore ijkY is an m-vector containing the k-th reading, by operator j, of the i-th part for the m quality characteristics. In the two-factor gauge study the MANOVA model is:    ijk i j jk ijkY p o po ε (12) where the random components ip , jo j, ijpo and ijkε are mutually independent with distributions ip   ,μ PN Σ , jo   ,0 ON Σ , ijpo   ,0 PON Σ and ijkε   , 0N Σ , respectively. Within this framework, ACTA IMEKO | www.imeko.org  December 2014 | Volume 3 | Number 4 | 40    .. ... .. ... 11       p i i i or p MSP Y Y Y Y (13) is the mean-square for the part matrix, where .. 1 1 1     o r i ijk j kor Y Y and ... 1 1 1 1      p o r ijk i j kpor Y Y . The mean square for the operator matrix is   . . ... . . ... 11       o j j j pr o MSO Y Y Y Y (14) where . . 1 1 1     p r j ijk i kpr Y Y . The mean square for the part- operator interaction matrix is given by      . .. . . ... . .. . . ... 1 1 1 1             p o ij i j ij i j i j r o p MSPO Y Y Y Y Y Y Y Y (15) and the MSE matrix is     ... ...1 1 1 1 1         p o r ijk ijk i j kpo r MSE Y Y Y Y . (16) The covariance matrices are estimated using the expected mean squares. The parts covariance matrix is estimated by ˆ P or MSP MSPO Σ , (17) the operator factor covariance matrix is estimated by ˆ O pr   MSO MSPO Σ , (18) the part-operator interaction covariance matrix is estimated by ˆ PO r MSPO MSE Σ (19) and the covariance matrix of the error terms ijkε is estimated by ˆ  Σ MSE . (20) By adopting the gauge R&R notation, P corresponds to ,the covariance matrix of the quality characteristic. Repeatability and reproducibility are given by Σ and O POΣ Σ , respectively. The sum of repeatability and reproducibility gives the gauge measurement error covariance matrix     e O POΣ Σ Σ Σ . (21) Therefore, the estimators of the covariance matrices of interest are: ˆ ˆ PΣ Σ , (22)  ˆ ˆ ˆ ˆ  e O POΣ Σ Σ Σ , (23) and ˆ ˆ ˆ y eΣ Σ Σ . (24) 3. A TEST FOR MULTIVARIATE MEASUREMENT SYSTEMS  The multivariate MSA-approval criteria described in the previous section are based on constant-density contours of the multivariate normal distribution. A change in the precision of a measurement instrument will cause a change in the corresponding ellipsoid of constant density. Therefore, since we are interested in the detection of a worsening in the measurement instrument precision, we will focus on significant reduction of the ellipsoid coverage. Let us suppose that at the beginning of the manufacturing activity, which for notation purpose we denote as time T=0, a multivariate MSA is performed and that the measurement instrument is assessed as adequate. We denote with e0 the precision of the measurement instrument, with0 the covariance matrix of the true quality characteristic and with y0=0+e0 the covariance matrix of the measurements, at time T=0. Assuming the multivariate normality an ellipsoid of constant density is defined by     10 0:    yU CY Y μ Σ Y μ . (25) If we assume C=2,m, then U0 is the boundary of the multivariate region in which 100 (1-)% process fall. After the initial set up, the measurement device is usually used for inspection or process control generating a lot of data at no additional costs. Let us consider a time interval in which the instrument has been routinely used. At time T=t the measurement instrument is characterized by a precision et. Usually, the process variability is monitored by a suitable control chart thus if no out of control signals occur we can assume the stability of the process i.e. 0=t. Under these assumptions the variability of Y is 0 yt etΣ Σ Σ . (26) In this framework, differences in the variability of the observed measures are only caused by changes in the precision since yt=y0 if and only if et=e0. At time T=t, the 0UY define the quadratic form             1 1 0 . t yt et Q           Y Y μ Σ Y μ Y μ Σ Σ Y μ (27)  tQ Y under the normality assumption is distributed as a 2 with m degrees of freedom and has several useful properties.  tQ Y is not constant,   tQ CY with equality only when 0yt yΣ Σ , i.e. 0et eΣ Σ . The minimum and the maximum values of  tQ Y can be determined analytically. Using results from [8] we find     max maximum eigenvalue of  tQ CY Γ (28)     min minimum eigenvalue of  tQ CY Γ (29) where 10  y ytΓ Σ Σ . Therefore, if at time T=t the measurement instrument is worse than at time T=0, then the 0UY define ellipsoids with coverage ranging from    2min Pr min m tP Q Y (30) to ACTA IMEKO | www.imeko.org  December 2014 | Volume 3 | Number 4 | 41     2max Pr max m tP Q Y . (31) The difference between (1 ) , the ellipsoid coverage at time T=0, and minP (or maxP ), the ellipsoid coverage at time T=t, quantifies the possible worsening in the instrument precision at instant t. A test for detecting the decreasing of the coverage can be derived as shown below. Let us consider the null hypothesis that the instrument precision at instant T=t is equal to the precision at instant T=0 0 0H : et eΣ Σ (32) If 0H holds, then 0yt yΣ Σ , Γ I ,   tQ CY and therefore the ellipsoid coverage is min 1  P . Let the alternative hypothesis be 1 0H : is positive definiteet eΣ Σ (33) If 1H holds, then   tQ CY hence the ellipsoid coverage is smaller than 1  : min 1  P . Let 1 2 ,...,       m (34) be the eigenvalues of the matrix Γ . Under hypothesis 1H we have 1 1m    (35) and the (upper) limit 1 is reached only under 0H . Let 1 be the largest eigenvalue of the matrix 1 1 0    yt yΩ Γ Σ Σ . (36) Then, when 1H holds 1 1 1m      (37) with 1 1  only under 0H . Let S be the sample covariance matrix of a random sample of size n from Y at time t. If 0H holds, then 1/2 1/2 0 0( 1)   y yn Σ SΣ  ( , 1)W nI (38) where ( , 1)W nI denotes a Wishart distribution with parameters I and 1n  . From [8] we have that matrix 1/ 2 1/ 2 0 0   y yΣ SΣ and matrix 1 0  yΣ S , have the same eigenvalues. Furthermore, also matrices 10  yΣ S and 1 0  ySΣ have the same eigenvalues. Therefore, if we denote with 1̂ the largest eigenvalue of the matrix 10  ySΣ , then 0H is not rejected if and only if 1 1 ˆ( 1)n u  , where 1u is the upper percentage point of the largest characteristic root of a Wishart matrix. The advantage of this method is that the measurement instrument is assessed by comparing its performance instant t with those at instant 0, without the necessity of performing a multivariate gauge study (MANOVA): the sample covariance matrix S can be estimated using the data available by the routine use of the measurement device at no additional costs. 4. CASE STUDIES  In this Section we discuss the ability of the test for detecting worsening in the measurement instrument performances. Before entering in the details of the case studies it is useful to spend a few words reminding that the multivariate MSA-metrics are designed thinking at different ways for assessing the measurement precision. The 1mP T and 2mP T criteria compare the multivariate instrument variability with the multivariate tolerance region (hypercube or hyperellipsoid). The remaining metrics do not involve the tolerances: % & mR R expresses the relative widths of the multivariate distributions of the errors e and the measured values Y; mSNR compares the width of the multivariate distribution of the true quality characteristics X with the corresponding volume of the multivariate errors e. Since the test in question does not involve the tolerance regions, we expect a test behaviour similar to those of % & mR R and mSNR . For this reason we shall compare the outcomes of the test only with the MSA-metrics % & mR R and mSNR . We consider as the situation at time T=0 (the benchmark) the case discussed by [5], then we examine a variety of worsening-precision scenarios at time T=t. For each of the proposed scenarios, we compute the multivariate MSA-approval metrics presented in Section 2 and we design suitable simulation experiments for studying the performances of the proposed test. Let us therefore consider the case discussed by [5] where the data come from an automotive body panel gauge-study involving m=4 quality characteristics, with p=5 parts, o=2 operators and r=3 repeated measurements (see Table 1 in [5] for the original data). Using a two-factor MANOVA method the matrices estimates are 0.01811 0.01600 0.02180 0.00763 0.01600 0.25163 0.15732 0.35463ˆ 0.02180 0.15732 0.20856 0.39249 0.00763 0.35463 0.39249 0.98631                Σ 0.00094 0.00168 0.00141 0.00189 0.00168 0.00632 0.00475 0.00702ˆ 0.00141 0.00475 0.00486 0.00581 0.00189 0.00702 0.00581 0.00852              eΣ and 0.01905 0.01768 0.02321 0.00574 0.01768 0.25795 0.16207 0.36165ˆ 0.02321 0.16207 0.21342 0.39830 0.00574 0.36165 0.38830 0.99483                yΣ The eigenvalues of the covariance matrices Σ̂ , ˆ eΣ and ˆ yΣ are reported in Table 1. Using equations (10) and (11) we obtain % & 12.26061mR R and 11.30385mSNR respectively. The results show that the measurement instrument is assessed as acceptable by both the multivariate ACTA IMEKO | www.imeko.org  December 2014 | Volume 3 | Number 4 | 42  criteria, therefore it can be used in the manufacturing process and for our purposes we can assume 0ˆ e eΣ Σ , 0 ˆ Σ Σ and 0ˆ y yΣ Σ . Now we examine several scenarios where a realistic worsening of the measurement instrument after a period of use is considered. We base this discussion on the spectral decomposition of 0eΣ 0 0 0 0 e e e eΣ U D U (39) where 0 01 02 0( , ,..., )e e e e mU u u u is the matrix of eigenvectors with columns 0e iu (i=1,2,…,m), and  0 01 02 0, ,...,  e e e e mdiagD is the diagonal matrix of the eigenvalues. The diagonal matrix 0eD is the covariance matrix of the latent independent factors that represent the primary independent sources of variability introduced by the instrument at time T=0. The instrument after a period of use is characterised by a measurement error covariance matrix: et et et etΣ U D U (40) where 1 2( , ,..., )et et et etmU u u u with columns etiu (i=1,2,…,m) and  1 2, ,...,  et et et etmdiagD . Many alternative cases for etΣ worse than 0eΣ can be considered, however etΣ cannot be attained by changing the elements of 0eΣ arbitrarily. It is realistic to examine for etΣ cases where changes in the variability are due to changes in variance of the latent independent factors. In other words, the factors that cause the variability in the instrument at instant t remain the same as for instant 0, but with larger variance. This means that cases for etΣ with practical meaning should be those involving the eigenvalues, 0eit ei  for (i=1,2,…,m), but maintaining unchanged the eigenvectors ( 0 e etU U ).A change in the eigenvectors can be interpreted as a the presence of serious problems in the instrument such that the independent sources of variability become dependent. It is worth noting that this concept has been used also by other authors. For instance [9] used this definition of plausible changes in a process capability analysis framework. In what follows, we consider three cases for etΣ where the eigenvectors remain unchanged. Furthermore, for the sake of completeness, we will also consider the case of a change in the eigenvectors. 4.1. Case 1  Now we examine the case where the eigenvalues of 0eΣ are increased by the same additive term  : 0  et eD D I . (41) Note that this case is equivalent to add the diagonal matrix  Ι directly to 0eΣ 0  et eΣ Σ I , (42) since from the spectral decomposition of 0eΣ we get the expression  0 0 0 0 0 0 0 0 0             e e e e e e e e e et U Σ I U U Σ U U U D I D (43) Thus, the eigenvectors remain unchanged, 0 e etU U , and the eigenvalues can be expressed as 0eti e i    for i=1,2,…,m. Within this case we consider scenarios where the worsening term  ranges from 0 to 0.006 with a step of 0.0001. Therefore, for each value of  : a) we compute the multivariate MSA approval criteria % & mR R and mSNR and the results are shown in Figure 1; b) using the R- software we generate 104 samples (n=50,75,100,150) from Y   0,  etN μ Σ Σ where 0  et eΣ Σ I . For each sample we compute the statistic   1ˆ1n  , where 1̂ is the largest eigenvalue of the matrix 10  ySΣ and S is the sample covariance matrix estimated from the sample. Therefore, we evaluate the power of the test computing, for each value of  and n, the proportion of rejections of 0H . Note that, fixed 0.05  , we used the function qWishartMax of the R- package RMTstat [10] for computing the critical values of the test. The simulation results are summarized in Figure 2 where, for each sample size, the rejection rates of hypothesis 0H as a function of  are reported. Examining the results we note that mSNR assesses the instrument as unacceptable for  ≥ 0.0033, while the test concludes, with a power greater than 80%, that instrument at time T=t is worse than the instrument at time T=0 for  ≥ 0.00325, when the sample size is n=75, and for  ≥ 0.00415 when n=50. Therefore, pointing out that % & mR R evaluates the instrument as inadequate for  ≥ 0.0059, we can conclude that in this case, for moderate sample sizes (n=50 and n=75), the performances of the test are among the outcomes of metrics mSNR and % & mR R . 4.2. Case 2  Next, we consider the case where the eigenvalues at time T=t are proportional to those at time T=0 0et eD D . (44) Table 1. Eigenvalues of the estimated covariance matrices. Σ̂ 1 2 3 4 1.29428 0.11185 0.05438 0.00410 ˆ eΣ 1e 2e 3e 4e 0.01908 0.00081 0.00050 0.00025 ˆ yΣ 1y 2y  3y  4y  1.311189 0.11392 0.05557 0.00457 ACTA IMEKO | www.imeko.org  December 2014 | Volume 3 | Number 4 | 43  Note that (44) is equivalent to consider the measurement error covariance matrix of instrument at instant t to be proportional to the instrument precision at instant 0 0et eΣ Σ (45) since we can write    0 1 0 0 0 0 0     e e e e e e e etU D U U D U Σ Σ . (46) Also in this case, the eigenvectors do not change, 0 e etU U , and the eigenvalues are expressed as 0eti e i  , for i=1,2,…,m. We perform our study by considering scenarios where the worsening factor  ranges from 1 to 10 with a step of 0.1. For each value of  we proceed as for case 1: a) we compute the multivariate MSA approval metrics, the results are displayed in Figure 3; b) we generate for each sample size (n=50, 75, 100, 125), 104 samples from Y   0,  etN μ Σ Σ where 0et eΣ Σ . For each sample we compute the test statistic   1ˆ1n  and, fixed as before an -level of 5%, we evaluate the test power by computing the proportion of the 104 replications where 0H is rejected in favour of 1H . The rejection rates for hypothesis 0H as a function of  are shown in Figure 4. In this case % & mR R assesses the measurement instrument as inadequate for  ≥7.4. The test concludes that the instrument at instant t is worse than the instrument m at instant 0, with a power greater than 80%, when δ≥7.2, δ≥6.2, and δ≥5.6 for n=75, n=100 and n=125 respectively. Therefore, considering that using SNR the instrument is unacceptable for  ≥5.2, we conclude that for sample sizes ranging from n=75 to n=125 the results of the test are among the outcomes of metrics mSNR and % & mR R . 4.3. Case 3  Since in the previous cases we examined cases where all the eigenvalues simultaneously change, now we discuss the case where only several eigenvalues change their values. Let us consider the scenario where three eigenvalues are increased by a factor  ranging from 1 to 20 with a step of 0.25:  01 02 03 04, , ,   et e e e ediagD (47) and the covariance matrix etΣ is given by 0 0et e et eΣ U D U . For each value of  we have followed the same procedure as for Cases 1 and 2: the results are summarized in Figures 5 and 6. The pattern of the results is similar to that observed for Case 1. mSNR assesses the instrument as unacceptable for  ≥9 and % & mR R for  ≥16, while the test concludes, that the instrument at time T=t is worse than the instrument at time 0, with a power greater than 80%, when  ≥8 and  ≥10 for n=75 and n=50 respectively. Figure 4. H0 rejection rates versus    for Case 2.  Figure 2. H0 rejection rates versus    for Case 1.  Figure 3. Multivariate MSA‐approval criteria as a function of    for Case 2.  Figure 1. Multivariate MSA‐approval criteria as a function of    for Case 1.  ACTA IMEKO | www.imeko.org  December 2014 | Volume 3 | Number 4 | 44  4.4. Case 4  Finally, let us examine the situation where the variations involve also the eigenvectors, which can be interpreted as the presence of a serious problem in the measurement instrument. We examine the case where the first two diagonal elements of etΣ increase, while the other matrix elements are equal to the corresponding elements of 0eΣ : 0.00094 0.00168 0.00141 0.00189 0.00168 0.00632 0.00475 0.00702 0.00141 0.00475 0.00486 0.00581 0.00189 0.00702 0.00581 0.00852               etΣ (48) In the analysis the term  ranges from 1 to 8 with a step of 0.1. Figure 7 shows the multivariate MSA metrics computed for each values of  . The Monte Carlo experiment has been performed as in the previous cases and the results are summarized in Figure 8. The results show that the test tends to be more sensitive to the increasing of  than % & mR R and mSNR . For instance, for n=50 the power of the test is greater than 80% for  ≥6, while mSNR evaluates the instrument as inadequate for  ≥6.2. Note that in this case, although the multivariate metric % & mR R detects the worsening in the measurement instrument, it assesses the instrument as acceptable for all the values considered of  . Summarizing, in the cases examined we aimed to study the performances of the test in realistic worsening scenarios of the measurement instrument after a period of use. The results show that the test provides outcomes with an appreciable level of agreement with the issues of % & mR R and mSNR . 5. DISCUSSION AND CONCLUSIONS As any activity involving personnel, materials, tools and equipment, MSA usually requires a non-negligible financial support. Furthermore, the fact that these systems measure more than a single quality characteristic and that periodic assessments of measurement system performance are often required engages manufacturers in important challenges. In this work, we have proposed a method which can be an additional tool for assessing the statistical properties of a multivariate measurement system. The method makes use of the data that are routinely available from the regular activity of the instrument and offers the possibility of assessing multivariate measurement systems without the necessity of performing a multivariate gauge study (MANOVA). Since the illustrated strategy can be implemented at almost no additional costs it may carried out more frequently than a MANOVA gauge study. Therefore, the synergic use of the proposed approach and Figure 5. Multivariate MSA‐approval criteria as a function of    for Case 3.  Figure 7. Multivariate MSA‐approval criteria as a function of    for Case 4.  Figure 8. H0 rejection rates versus    for Case 4.  Figure 6. H0 rejection rates versus    for Case 3.  ACTA IMEKO | www.imeko.org  December 2014 | Volume 3 | Number 4 | 45  the traditional multivariate gauge R&R studies can be: effective for reducing the costs of a multivariate MSA performed with a certain frequency; a useful strategy for improving the overall quality of multivariate measurement systems. ACKNOWLEDGEMENT  The author would like to thank the two anonymous reviewers for their helpful comments. REFERENCES  [1] R.K. Burdick, C.M. Borror, D.C. Montgomery, A review of methods for measurements systems capability analysis, J. Qual. Technol. 35 (2003) pp. 342-354. [2] R.K. Burdick, C.M. Borror, D.C. 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