The authors thank the Editor and the reviewers for their helpful suggestions, comments and encouragement, which helped in improving the final version of the paper.
Purpose – The purpose of this paper is to examine extreme value charts and analyse means based on half logistic distribution.
Design/methodology/approach – Variable control charts with subgroup observations based on the extreme values at each subgroup are constructed without specially going to any subgroup statistic. The control chart constants depend on the probability model of the extreme order statistic of each subgroup and the size of the subgroup. Accordingly the proposed chart is normal as extreme value chart. As a by-product the technique of analysis of means for a skewed population is exemplated through half logistic distribution and extreme value control charts. The results are illustrated by examples on live data.
Findings – H.L.D is found to be better test for the data of the three examples, ANOM gave a larger (complete) homogeneity of data than those of Ott.
Research limitations/implications – Supposing arithmetic means of k subgroups of size “n” each drawn from a half logistic model. If these subgroup means are used to develop control charts to assess whether the population from which these subgroups are drawn is operating with admissible quality variations. Depending on the basic population model, we may use the control chart constants developed by the authors or the popular Shewart constants given in any SQC text book. Generally the authors say that the process is in control if all the subgroup means fall within the control limits. Otherwise it is said that the process lacks control.
Originality/value – Half logistic distribution is a better model, exhibiting significant linear relation between sample and population quantiles.
Research paper
Analysis of means; Half logistic distribution; Q-Q plot; Logistic data processing; Statistics.
International Journal of Quality & Reliability Management
29
5
2012
501-511
Emerald Group Publishing Limited
0265-671X
f(x)=probability density function (pdf)
F(x)=cumulative distribution function(cdf)
HLD=half logistic distribution
σ=scale parameter
ANOM=analysis of means
SQC=statistical quality control
Normal distribution is the most commonly cited example of a probability model for any data to apply a statistical technique. The reason is perhaps the most powerful result of central limit theorem. However, there may be some exceptions where we may have to go to some probability models other than normal distribution. With this backdrop, one of the first modifications for normal distribution that is suggested is its folded version at its median. Such a distribution is named as half normal distribution. The advantage of this folding mechanism is that the resulting distribution eliminates negative values of the random variable from the population leading to a more practical model. In a similar manner the well known logistic distribution can also be folded at its median giving rise to half logistic distribution. In view of a number of similarities between normal and logistic distributions one can expect the same trend between half normal and half logistic distributions. Therefore, half logistic distribution can be an alternative to half normal distribution. Balakrishnan (1985) seems to be the person to have thought of half logistic distribution. The probability density function (pdf) of a half logistic distribution with scale parameter σ is given by: Equation 1 Its cumulative distribution function (cdf) is: Equation 2 When σ=1, the above equations are called standard pdf and cdf. In order to construct a control chart using the extreme observations of a subgroup drawn from the production process with the quality variate following half logistic distribution we need the percentiles of extreme order statistics in samples from half logistic distribution. Specifically, the test statistic on extreme value control chart is the original sample vector X=(x1, x2 … xn) from the ongoing production. In this chart all the individual sample observations are plotted into control chart without calculating any statistic out of them. A corrective action is taken after one or either of the extreme values namely x(1) (sample minimum) and x(n) (sample maximum) of the sample, respectively, fall below or above two specified lines (limits). Because of this, the chart is called extreme value control chart.
The Shewart control chart is a common tool of statistical quality control for many practitioners. When these charts indicate the presence of an assignable cause, an adjustment of the process is made if the remedy is known. Otherwise the suspected presence of assignable cause is regarded to be an indication of heterogeneity of the subgroup statistic for which the control chart is developed. For instance if the statistic is sample mean, this leads to heterogeneity of process mean indicating departures from target mean. Such an analysis is generally carried out with the help of means to divide a collection of a given number of subgroup means into categories such that means within a category are homogenous and those between categories are heterogeneous and the procedure is called analysis of means (ANOM) by Ott (1967).
For using the ANOM technique the concept of the control chart for means is viewed in a different way – grouping of plotted means to fall within the control limits or some outside the control limits. For the homogeneity of all the means, it is necessary that all the means should fall within the control limits. If (1?α) is taken as the confidence coefficient we should have the probability of all the subgroup means to fall within the control limits is (1?α). Assuming independence of subgroups the above probability statement becomes nth power of the probability of a subgroup mean to fall within the limits., i.e. in the sampling distribution of xˉ the confidence interval for xˉ to lie between two specified limits should be equal to (1?α)1/n. The same principle is adopted in the rest of this paper through half logistic distribution. Because this paper aims at exploring ANOM using control limits of extreme value statistics we have considered only the control chart aspects but not any recently developed ANOM tables or techniques. However, a detailed literature about ANOM is available in Rao (2005) and some related works in this direction are: Ramig (1983) computed decision lines of ANOM based on proportions, Bakir (1994) developed a non-parametric procedure called the analysis of means by using ranks (ANOMR) for testing the equality of several population means, Bernard and Wludyka (2001) developed robust ANOM-type randomization tests for variances in balanced and unbalanced designs, Wludyka et al. (2001) presented power curves for ANOM procedure and offered guidance regarding the choice of sample size and interpretation of the results based on power considerations, Wardsworth et al. (1986) and Montgomery (2001) presented some of the ANOM techniques with illustrative examples in their books, Nelson and Dudewicz (2002) developed a heteroscedastic analysis of means (HANOM) procedure for testing the equality of several means when the population variances are not equal, Rao and Pran Kumar (2002) developed another ANOM type graphical procedure for testing the equality of several correlation coefficients, Farnum (2004) developed new formulas to calculate the ANOM constants using commonly available mathematical processors. This approach makes the ANOM constants easily accessible, portable and unrestricted with regard to the choice of significance level, sample size and the number of populations, Guirguis and Tobias (2004) presented the procedure to develop ANOM for a given distribution, and the references there in. The rest of the paper is organized as follows.
The basic exposure to extreme value control charts is given in Section 2. ANOM applied to half logistic distribution using extreme value control charts of HLD is given in Section 3 followed by numerical examples.
Summary and conclusions are given in Section 4.
The given sample observations are assumed to follow half logistic model. The control lines are determined by the theory of extreme order statistics based on half logistic model. The control lines are to be determined in such a way that an arbitrarily chosen xi of X=(x1, x2 … xn) lies with probability (1?α)1/n within the limits. This can be formulated as a probability inequality in the following way.
P{x(1)≤L}=α/2 and P{x(n)≥U}=α/2. The theory of order statistics say that the cdf of the least and highest order statistics in a sample of size n from any continuous population are [F(x)]n and 1?[1?F(x)]n, respectively, where F(x) is the cdf of the population. If 1?α is desired at 0.9973 then α would be 0.0027. Taking F(x) as the CDF of a standard half logistic model we can get solutions of the two equations 1?[1?F(x)]n=0.00135 and F(x)n=0.99865, which in turn can be used to develop the control limits of extreme value chart. The solutions of the above two equations for n=2 (1) 10 are given in Table I denoted as Z(1)0.00135 and Z(n)0.99865.
The values of Table I indicate the following probability statement: Equation 3 Equation 4 Taking xˉ?loge n as an unbiased estimate of σ, the above expression becomes: Equation 5 where D3 *=Z(1)(0.00135)/ln?4 and D4 *=Z(n)(0.99865)/ln?4. Thus, D3 *, D4 * would constitute the control chart constants for the extreme value charts. These are given in Table II for n=2 (1) 10.
These constants can be used to decide the in-control status of a given sample of size n=2 (1) 10 directly by plotting the minimum and maximum of the sample on the graph against the serial number of the subgroup. The control lines would be two horizontal lines at D3 *xˉ?and?D4 *xˉ. If both minimum and maximum of the sample – the extreme values are between the control limits, the process is judged to be in control otherwise it falls out of control. Another important application of these constants is that they can be used to the popular ANOM procedure, when the process variate follows half logistic distribution. This is explained in the next section.
Suppose xˉ 1,xˉ 2,…xˉ k are arithmetic means of k subgroups of size “n” each drawn from a half logistic model. If these subgroup means are used to develop control charts to assess whether the population from which these subgroups are drawn is operating with admissible quality variations. Depending on the basic population model, we may use the control chart constants developed by us or the popular Shewart constants given in any SQC text book. Generally we say that the process is in control if all the subgroup means fall within the control limits. Otherwise we say the process lacks control. If α is the level of significance of the above decisions we can have the following probability statements: Equation 6 Using the notion of independent subgroups equation (3.1) becomes: Equation 7 With equi-tailed probability for each subgroup mean, we can find two constants say L* and U* such that: Equation 8 In the case of normal population L* and U* satisfy U*=?L*. For the skewed populations like HLD we have to calculate L*, U* separately from the sampling distribution of xˉ i . Accordingly these depend on the subgroup size “n” and the number of subgroups “k”. The percentiles of sampling distribution of xˉ in samples from half logistic distribution are worked out through Monte-Carlo simulation and are given in the Table III. We make use of its percentiles in equation (3.3) for specified “n” and k to get L* and U* for α=0.01 and 0.05. These are given in Tables IV and V.
A control chart for averages giving “In Control” conclusion indicates that all the subgroup means though vary among themselves are homogenous in some sense. This is exactly the null hypothesis in an analysis of variance technique. Hence the constants of Tables IV and V can be used as an alternative to analysis of variance technique. For a normal population one can use the tables of Ott (1967). For a HLD our tables can be used. We therefore present below some examples for which the goodness of fit of half logistic model is assessed with Q-Q plot technique (strength of linearity between observed and theoretical quantiles of a model) and tested the homogeneity of means involved in each case.
Consider the following data of 20 observations on manufacture of metal products that suspect variations in iron content of raw material supplied by five suppliers. Five ingots were randomly selected from each of the five suppliers. Table VI contains the data for the iron determinations on each ingots in percent by weight (Wadsworth et al., 1986).
Three bands of batteries are under study. It is suspected that the life (in weeks) of the three brands is different. Five batteries of each brand are tested with the following results (Satyaprasad, 1987) (Table VII).
Four catalysts that may affect the concentration of one component in a three-component liquid mixture are being investigated. The following concentrations are obtained (Table VIII). Test whether the four catalysts have the same affect on the concentration at 5 per cent level of significance (Satyaprasad, 1987).
The goodness-of-fit of data in these three examples as revealed by Q-Q plot (correlation coefficient) are summarized in Table IX, which shows that half logistic distribution is a better model, exhibiting significant linear relation between sample and population quantiles.
Treating these observations in the data as a single sample, we have calculated the decision limits for the HLD population, Normal population and have given them in the Tables X and XI, respectively.
The decision lines using normal distribution, ANOM tables of Ott (1967) yield that the number of homogenous means for each data set are 3, 2, 2, respectively. And those away from homogeneity are 2, 1, 2, respectively. On the other hand when the ANOM tables of our model (HLD) are used for the same data sets we get the number of homogenous means to be 5, 3, 4, respectively, without exhibiting deviation of any mean from homogeneity. Thus, usage of normal model resulted in homogeneity for some means and deviation for some other means, indicating a possible rejection of those means. This decision is valid if normal distribution is a good fit to the data. As a comparison, we have already established by Q-Q plot that HLD is a better model than Normal as supported by the Q-Q plot correlation coefficient of each data set with normal as well as HLD separately. Therefore, we have assumed that more error is likely to be associated with decision process of normal distribution. Therefore, all the means to be homogenous with the help of HLD (Table XI) is a better decision than some means to be away from homogeneity using normal, ANOM procedure.
The ANOM chart differs from Shewart control chart because it is a formal test of hypothesis, where in the latter is used to distinguish between common causes and special causes of variation. Both approaches complement each other in trying to evaluate the efficiency of the data. We have calculated the percentiles of sampling distribution of xˉ in samples from half logistic distribution with ANOM technique. If the population is normal one can use the tables prepared by Ott (1967) else if the population is HLD the values proposed in Tables IV and V can be used.
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Srinivasa Rao Boyapati can be contacted at: boyapatisrinu@yahoo.com
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