An Approach To ANOM Chart. Muhammad Riaz

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1 An Approach To ANOM Chart Muhammad Riaz Department of tatistics, Quaid-i-Azam University, Islamabad, Pakistan Abstract The study proposes a scheme for the structure of Analysis of Means (ANOM) chart assuming normality. A comparison of proposed scheme is made with the well known existing range and standard deviation based schemes for the design structure of ANOM chart. The effect of departure from normality is also examined on the three schemes for the design structure of ANOM chart. It is observed that all the three schemes result into same state of statistical significance in case of normal data. In case of departure from normality the proposed scheme shows the most robust behavior among three schemes under consideration. Key words: Normality; Decision Limits; Power Curves; Robustness. Introduction Analysis of Variance (ANOVA) is generally used for analyzing single and multi factor experimental designs. Analysis of Means (ANOM) can be used as an alternative to ANOVA when the effects of factors under study are fixed. ANOM is a graphical technique initially developed for testing equality of several means. Ott (967) introduced ANOM procedure for controlling group of treatment means following Halperin (955). Later it has been used for other purposes like testing of several variances, correlations coefficients, proportions etc. An ANOM chart is conceptually similar to a hewhart type control chart. It portrays decision lines for testing purposes. Any subgroup mean lying outside these

2 decision lines is declared to be significantly different from target/overall mean value about which decision limits are constructed. For comparing individual subgroup means with overall mean Ott (967) introduced a range based scheme for ANOM chart with the following structure: CL = x LDL x H σˆ = ( α ) x = + ˆ ( α ) σ x UDL x H () where x is average of an appropriate number, say k, of subgroup means, CL is abbreviation for Center Line, LDL and UDL are abbreviations for Lower Decision ˆ σ R Limit and Upper Decision Limit respectively, ˆ σ x =, ˆ σ = * n d, R is average of k subgroup ranges and * d is a factor for estimating σ from R depending on the number of subgroups k and subgroup size n. The tables of critical values H α, for xn x x x H =max, ˆ σ ˆ x σx, at α =.5 and. have been developed by Ott (967) for some combinations of k and v (the degrees of freedom associated with the estimate of variability). Following Ott (967), heesley (98) developed range based structure for ANOM chart based on simplified factors A α as: CL = x LDL = x Aα R UDL = x + Aα R () H where A α α = * and other terms as defined in (). d n Nelson (98) used a scheme for decision limits of ANOM chart as:

3 CL = x LDL x h ( k ) / kn = ( α ) = + ( α ) ( ) / UDL x h k kn (3) where is the pooled estimate of variance, x is mean of k subgroup averages and h( α ) represents the exact critical points obtained by Nelson (98) which depends on k and v (degrees of freedom in ). Nelson (98) provided critical values for few combinations of k and v. Nelson (983) and Nelson (993) provided more detailed tables (as function of number of subgroups k and degrees of freedom associated with the estimate of variability ) of critical values to be used for ANOM. Nelson and Dudewicz () considered the ANOM procedure for heteroscedastic situations. They constructed power curves for ANOM charts for heteroscedastic data which enable an experimenter in designing a study to detect differences among different subgroup means when any two of them differ by a given amount. A Proposed cheme for ANOM Chart Muhammad and Riaz (6) proposed and developed the design structure of a variability control chart namely chart. Their proposed control chart is based on statistic which is basically a probability weighted moments estimate of σ. For an i.i.d subgroup X, X,..., X n of size n from normal distribution, the statistic is defined as: n π i.5 = X X n n () i () i i= = n π n i= (i n ) X ( i ), (4) 3

4 ' where () i X s represent the ordered observations and ( ) empirical distribution function F ( ) n i.5 n, i =,,..., n is an x. Muhammad and Riaz (6) used a relationship between and σ as Q= / σ in their study. They obtained coefficients, r and quantile points of Q as function of n in their study. The r 3 quantities, r and quantile points of Q are provided in Appendix Tables A- and r 3 A- as function of n. Based on these quantities they developed the control structure for σ and proposed an estimator of σ as: ˆ σ = / r. (5) This study proposes a based scheme for the structure of ANOM chart following Ott (967), L.. Nelson (974), heesley (98), P. R. Nelson (98), L.. Nelson (983) and P. R. Nelson (993). The proposed scheme for the design structure of ANOM chart is given as: CL = x h k LDL x, UDL ( α, m, k ) = r nk = x + h( α, m, k ) k r nk (6) where n : subgroup size, k : number of subgroups, x : mean of k subgroup means, m : degrees of freedom associated with the estimate of variability, h( α, mk, ): the critical points derived by Nelson (983) which are function of k and m, 4

5 : mean of an appropriate number, say k, of subgroup s, r : factor for unbiased estimation of σ from 3 Numerical Computations-imulations s, provided in Appendix Table A-. In this section numerical calculations are carried out for the design structure of ANOM chart using proposed scheme and the existing range and standard based schemes. A hypothetical data set (provided in Appendix Table A-3) consisting of twenty subgroups each of size ten is used in this study. Each subgroup is assumed to follow an independent normal distribution. The subgroup means of hypothetical data set are given in the following Table. Table : Means of the Data et ubgroup # Means ubgroup # Means For the hypothetical data set, 8 is the estimate of σ when all the subgroups validate the state of control with respect to σ. Using Monte Carlo simulations technique random subgroups, each of size ten, are generated from normal distribution with mean 6. (the mean of all twenty subgroup means given in Table ) and standard deviation 8. Then for the 5

6 structures given in (), (3) and (6), the R, and based estimates of σ are computed using these simulated, subgroups. The same is done times and the averages of the results, along with their respective standard errors, are provided in the following Table. Table : ummary tatistics of tandard Deviation Based on imulations from N (6., 8.) Estimate R based based based Mean 8.36 (.9) 8.69 (.5) 8.48 (.8) Based on the estimates of σ provided in Table the structures of ANOM chart, for the hypothetical data set given in Appendix Table A-3, are constructed using all the three schemes given in (), (3) and (6) based on subgroups from N (6.,8.). Different significance levels have been used for constructing these charts and one set (using α =.5 ) of ANOM charts, based on the three schemes, is provided in the following Figures (a-c). Fig. (a) : R Based tructure of ANOM Chart for N (6., 8.) ubgroup Mean Fig. : R Chart Based tructure of X Chart when Process follows N (5.8,.3833) ample Mean 5 5 ubgroup Number ample Number UCL=9.668 CL=5.8 LCL=.955 UDL= CL=6. LDL=

7 Fig. (b) : Based tructure of ANOM Chart for N (6., 8.) 5 ubgroup Mean 4 3 UDL = CL =6. LDL = ubgroup Number Fig. (c) : Based tructure of ANOM Chart for N (6., 8.) 5 ubgroup Mean 4 3 UDL = CL =6. LDL = ubgroup Number In the above figures a * marked represents that the individual subgroup mean is not consistent with the true/overall mean. It is observed that in a normally distributed environment if the dispersion level of all the subgroups (i.e.σ ) remains stable then the three schemes under consideration produce almost same control structure(i.e. whether some mean is consistent with the overall/target mean level or not) for ANOM chart as obvious from Figures (a-c). 7

8 Consequently all the three schemes possess same power efficiency for changes in subgroup means. 4 Effect of Non-Normality A fundamental assumption for the development of ANOM chart is that the underlying distribution of the data under consideration should be normal. In this section, effect of deviation from normality on the structures of ANOM chart based on the three schemes under discussion is examined using, simulated random subgroups of sizes from comparable, t and exponential distributions t (comparable means that the simulated subgroups are transformed such that they have same mean and standard deviation as that of subgroups from N (6., 8.)). Then for the structures given in (), (3) and (6) the R, and based estimates of σ are computed using these simulated, subgroups. The same is done times and the averages of the results, along with their respective standard errors, are provided in the following Table 3, 4 and 5 respectively. Table 3: ummary tatistics of tandard Deviation Based on imulations from Comparable t Estimate R based based based Mean 8.83 (.39) 8.94 (.8) (.) 8

9 Table 4: ummary tatistics of tandard Deviation Based on imulations from Comparable t Estimate R based based based Mean 8.86 (.9) 8.97 (.) (.) Table 5: ummary tatistics of tandard Deviation Based on imulations from Comparable Exponential Distribution Estimate R based based based Mean (.) (.4) (.) Based on the estimates of σ provided in Table 3 the structures of ANOM chart, for the hypothetical data set given in Appendix Table A-3, are constructed using all the three schemes given in (), (3) and (6) based on subgroups from comparable t. Different significance levels have been used for constructing these charts and one set (using α =.5 ) of ANOM charts, based on the three schemes, is provided in the following Figures (a-c). 9

10 Fig. (a): R Based tructure of ANOM Chart for Comparable t 5 ubgroup Mean 4 3 UDL =33.96 CL=6. LDL =8.374 ubgroup Number Fig. (b): Based tructure of ANOM Chart for Comparable t 5 ubgroup Mean 4 3 UDL = X=6. LDL =8.387 ubgroup Number Fig. (c): Based tructure of ANOM Chart for Comparable t 5 ubgroup Mean 4 3 UDL =33.76 CL=6. LDL = ubgroup Number

11 Based on the estimates of σ provided in Table 5 the structures of ANOM chart, for the hypothetical data set given in Appendix Table A-3, are constructed using all the three schemes given in (), (3) and (6) based on subgroups from comparable exponential distribution. Different significance levels have been used for constructing these charts and one set (using α =.5 ) of ANOM charts, based on the three schemes, is provided in the following Figures 3(a-c). Fig. 3(a): R Based tructure of ANOM Chart for Comparable Exponential Dist. 5 ubgroup Mean 4 3 UDL = CL=6. UDL =7.67 ubgroup Number Fig. 3(b): Based tructure of ANOM Chart for Comparable Exponential Dist. 5 ubgroup Mean 4 3 UDL = CL=6. UDL =7.757 ubgroup Number

12 Fig. 3(c): Based tructure of ANOM Chart for Comparable Exponential Dist. 5 ubgroup Mean 4 3 UDL = CL=6. UDL =7.736 ubgroup Number It is examined that in case of deviation from normality, based scheme for the structure of ANOM chart is least disturbed from the original structure (i.e. the structure based on normal data as provided in Figures (a-c) as compared to R and based schemes for the design structure of ANOM chart as obvious from the above Figures (a-c) and 3(a-c). It is interesting to examine Figures (a-c) for subgroup # 5, of the hypothetical data set given in Appendix Table A-3, which is actually inconsistent with the data. In case of deviation from normality the R and based schemes of ANOM chart are showing subgroup # 5 to be consistent with the data as obvious from Figures (a) and (b), while based scheme for ANOM chart is showing it to be inconsistent with the data as obvious from Figure (c). Thus based scheme for the structure of ANOM chart is reasonably effective even in case of deviation from normality as it shows the most robust behavior against non-normality among three schemes under study.

13 5 Power Curves Using parameters of the hypothetical data set given in Appendix Table A-3 and the simulations carried out in ection 3 from N (6., 8.), power curves of ANOM chart are constructed using the three structures under consideration. Later the same hypothetical data set and simulations made in ection 4 from comparable t and exponential distribution are used and power curves of ANOM chart using the three schemes under consideration are constructed. Using different significance levels, power curves of ANOM chart are constructed for all the three schemes given in (), (3) and (6) based on subgroups from normal, comparable t and exponential distributions. For constructing power curves, the inconsistencies in data are considered in terms of kσ, where k is a constant which helps specifying the least amount by which any two of the subgroup means differ in terms of common population standard deviation σ. For α =.5 the original power curves (the curves based on normal distribution) and the affected power curves (the curves based on comparable t and exponential distributions) of ANOM chart using the three schemes are produced in the following Figure 4(a-c). 3

14 Fig. 4(a): Power Curves of ANOM Chart using R Based cheme. N E T Power.5. k 3 Fig. 4(b): Power Curves of ANOM Chart using Based cheme. N E T Power.5. 3 k 4

15 Fig. 4(c): Power Curves of ANOM Chart using Based cheme. N E T Power.5. 3 k Figures 4(a-c) provide a comparison of the power curves of ANOM chart using R, and based schemes. The symbol N represents the situation when subgroups are simulated from normal distribution, and E and T represent the situations when subgroups are simulated from comparable exponential and t distributions respectively. It is observed that discriminatory power of chart based scheme of ANOM chart is least influenced by departure from normality among the three schemes under study as obvious from the Figures 4(a-c). Thus the proposed based scheme for the structure of ANOM chart enjoys the most robust behavior against non-normality among three schemes under consideration. 5

16 6 Conclusion In a normally distributed environment, the proposed based scheme for ANOM chart provides an equally powerful design structure as of the well known existing R and based schemes for ANOM chart. In case of departure from normality, the proposed scheme gets an advantage over the existing R and based schemes for ANOM chart in the sense that it possesses the most robust behavior against non-normality. Appendix TABLE A- n r r

17 TABLE A- n Q. Q.5 Q. Q.5 Q. Q. Q.5 Q.5 Q.75 Q.8 Q.9 Q.95 Q.99 Q.995 Q

18 Table A-3 A Hypothetical Data et ubgroup # ubgroup Values

19 References Halperin, M., Greenhouse,. W., Cornfield, J. and Zalokar, J. (955). Tables of percentage points for the studentized maximum absolute deviate in normal samples. Journal of American tatistical Association 5, Muhammad, F. and Riaz, M. (6). Probability Weighted Moments approach to Quality Control Charts. Economic Quality Control, (), Nelson, L.. (974). Factors for the Analysis of Means. Journal of Quality Technology, 6, Nelson, L.. (983). Exact critical values for use with the analysis of means. Journal of Quality Technology, 5, Nelson, P. R. (98). Exact critical points for the analysis of means. Communications in tatistics, Theory and Methods,, Nelson, P. R. (993). Additional uses for the Analysis of Means and Extended Tables of Critical Values. Technometrics, 35 (), 6-7. Nelson, P. R, and Dudewicz, E. J. (). Exact Analysis of Means with unequal variances. Technometrics, 44(), 5-6. Ott, E. R. (967). Analysis of Means: A graphical procedure. Industrial Quality Control., 4, -9. heesley, J. H., (98). implified factors for analysis of means when the standard deviation is estimated from the range. Journal of Quality Technology., 3, Muhammad Riaz, Department of tatistics, Quaid-i-Azam University, Islamabad, Pakistan, riaz76qau@yahoo.com. 9