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Improvement of causal analysis using multivariate statistical process control

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Abstract

Statistical process control (SPC) is a conventional means of monitoring software processes and detecting related problems, where the causes of detected problems can be identified using causal analysis. Determining the actual causes of reported problems requires significant effort due to the large number of possible causes. This study presents an approach to detect problems and identify the causes of problems using multivariate SPC. This proposed method can be applied to monitor multiple measures of software process simultaneously. The measures which are detected as the major impacts to the out-of-control signals can be used to identify the causes where the partial least squares (PLS) and statistical hypothesis testing are utilized to validate the identified causes of problems in this study. The main advantage of the proposed approach is that the correlated indices can be monitored simultaneously to facilitate the causal analysis of a software process.

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Notes

  1. The range of variable X can be treated as a variable and expressed as W = w/σ w (standardized range), in which σ w denotes the standard deviation of W. The E(Wk) represents the k-th moments of W, and the value of E(Wk) depends on n and k, in which the n denotes the subgroup size. The d 2 is defined as the first moment (or expectation) of W. Therefore, the value of d 2 depends on n, and can be treated as a control chart constant for given n.

  2. The d 3 is defined as the standard deviation of the range and can be expressed as \(E[(\hbox{W}-\upeta_{\rm w})^{2}]\) , where \(\upeta_{{\rm w}}\) is the mean of W. The value of d 3 can be calculated using the second moment of W and d 2, as follows: \(d_{3}=E[(\hbox{W}-\upeta_{\rm w})^{2}]=E[\hbox{W}^{2}-2\hbox{W}\upeta_{{\rm w}}+\upeta_{{\rm w}}^{2}]=E(\hbox{W}^{2})-2E(\hbox{W})\upeta_{w}+\upeta_{w}^{2}= E(\hbox{W}^{2})-2\upeta_{{\rm w}}^{2}+\upeta_{{\rm w}}^{2}= E(\hbox{W}^{2})-\upeta_{{\rm w}}^{2}=E(\hbox{W}^{2})-d_{2}^{2}.\) The values of d 2 and d 3 are tabulated for different n.

  3. This is because \(\sigma_{\overline{\rm X}}=(1.6555-1.5189)/3=0.0455,\) and the z value of the sixth point is (1.67 − 1.5189)/0.0455≈3.32. Therefore, the probability of P(Z < 3.32) is about 0.9995.

  4. The \(\hbox{UCL}_{T^{2}}\) is calculated using α = 0.02, m = 18 and k = 2, and obtains B 0.99(1,7.5) = 0.4588. Therefore, the \(\hbox{UCL}_{T^{2}}\) is about 16.0556 × 0.4583 = 7.37 and \(\hbox{LCL}_{T^{2}}\) is about 0.02.

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Acknowledgements

This work is partially supported by the National Science Council of Taiwan, R.O.C., under grant NSC-92-2213-E-309-005, and partially sponsored by the Ministry of Economic Affairs of Taiwan, under grant 93-EC-17-A-02-S1-029.

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  1. Department of Computer Science and Information Engineering, National Cheng-Kung University, Taiwan No.1, Ta-Hsueh Road, Tainan, 701, Taiwan

    Ching-Pao Chang & Chih-Ping Chu

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Correspondence to Ching-Pao Chang.

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Chang, CP., Chu, CP. Improvement of causal analysis using multivariate statistical process control. Software Qual J 16, 377–409 (2008). https://doi.org/10.1007/s11219-007-9042-3

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