Determination of Effectiveness of the Square Root of N Plus One Rule in Lot Acceptance Sampling Using an Operating Characteristic Curve

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1 Determinion of Effectiveness of the Square Root of N Plus One Rule in Lot Acceptance Sampling Using an Opering Characteristic Curve J. Muralimanohar* and K. Jaianand Department of Laborory Management & Quality Assurance, LifeCell Internional Pvt. Ltd, Chennai, India Abstract Sampling is an inevitable means of testing in quality inspections. However sampling is always associed with sampling risks th inspectors have to control. This paper aims to reveal the reliability and efficacy of the commonly used formula, Square root of N plus one, in lot acceptance sampling. An Opering Characteristic Curve is exploited to test the validity of the sampling plan and it offers a further dimension to unveil the unreliability of it. The probability of accepting a defective lot is increased substantially from larger to smaller lot sizes with the deployment of the N + 1 rule as a sampling plan. Using a sampling plan blindly from recognized sources and pursuing a traditional practice does not ensure th the sampling plan is stistically valid. Regardless of the source of the sampling plan and how it is indexed, it is the actual acceptable quality level (AQL) and lot tolerance percent defective (LTPD) of the sampling plan th describes its protection and determines validity. Copyright 2011 John Wiley & Sons, Ltd. Key Words: square root of N plus one; opering characteristic curve; lot acceptance sampling; acceptable quality level; percent defective Introduction Bulk product characteristics are routinely tested to determine whether the product meets in-house QA specificions. Inspection for acceptance *Correspondence to: J. Muralimanohar, LifeCell Internional Pvt. ltd - Laborory Management & Quality Assurance, Chennai, India muralimanoharj@ ymail.com purposes is carried out many stages in manufacturing. Much of this acceptance inspection is necessarily on a sampling basis. Sampling inspection is likely to be better than 100% inspection because of the influence of inspection figue in 100% inspection and substantially reduces inspection costs. A sampling plan indices the number of units of product from each lot or bch which are to be inspected Copyright 2011 John Wiley & Sons, Ltd. Qual Assur J 2011; 14,

2 34 J. Muralimanohar and K. Jaianand (sample size or series of sample sizes) and the criteria for determining the acceptability of the lot or bch (acceptance and rejection numbers). Acceptance Sampling is used to make decisions on accepting or rejecting a lot or bch of product th has already been produced or purchased. A sample of product is used to determine acceptability. It is most often used to evalue products th are received from outside sources and where it is not possible to implement stistical process control. Pharmaceutical companies deploy the simple rule, square root of N plus one, for determining sample sizes to be inspected from the given populion as most quality inspectors are trained to use it as a rule-of-thumb model. The square root of the lot size plus one is taken to determine the number of units to inspect. Each unit is tested individually. The lot on zero defectives is accepted, or, in the case of continuous measurements, the lot is accepted if the average falls within given specificions. Origining in the 1920s as a sampling scheme for agricultural regulory inspectors, the square root (Sqrt) of the lot size (N) + 1 was semi formalized in an unpublished report by the Associion of Official Agricultural Chemists (now Associion of Analytical Chemists) in 1927 [1]. A simple method for choosing a sample size from a populion is through wh quality engineers refer to as the square root of N plus one sampling rule. This rule is apparently not stistically motived nor is it mentioned by sampling theorists, practitioners, or reviewers of the field. Despite the lack of theoretical support, this rule has been adopted by many federal regulory agencies. Sqrt(N) sampling is recommended in FDAs Investigions Operions Manual, which stes,... a general rule is to collect samples from the square root of the number of cases or shipping containers but not less than 12 or more than 36 subs in duplice [2]. Sqrt(N) + 1 sampling is recommended in the FDA document, CBER 03/01/92 Draft Points to Consider in the Manufacture of In Vitro Monoclonal Antibody. According to FDA s docket no. 91N-0466, section V3,... another protocol for testing representive lots (e.g., square root n + 1/yr, where n equals the number of lots of product produced per year) may also be found to be sisfactory [3]. The uncertainties and fallacy in using this sampling plan was well illustred by Hewa Saranadasa in 2003[4]. Lynn Torbek in 2009 suggested Sqrt (N) + 1 is a stistically correct and valid sampling plan and can be used with the same care and caution as ANSI Z1.4 General Level I would be used [5]. A sampling plan is best understood through the Opering Characteristic (OC) Curve. The behavior of a sampling plan is described by the sampling plan s OC Curve. OC Curves are powerful tools in the field of QC, as they display the discriminory power of a sampling plan. It refers to a graph of tributes of a sampling plan considered during project management which depicts the percent of lots or bches which are expected to be acceptable under the specified sampling plan and for a specified process quality. The OC Curve is usually summarized by two points on the curve: the Acceptable Quality Level or AQL and the Lot Tolerance Percent Defective or LTPD. The Acceptable Quality Level (AQL) is generally defined as the percent defectives th the plan will accept 95% of the time. Lots th are or better than the AQL will be accepted 95% of the time. If the lot fails, we can say with 95% confidence th the lot quality level is worse than the AQL. Likewise, we can say th a lot the AQL th is acceptable has a 5% chance of being rejected. The Lot Tolerance Percent Defective (LTPD) is generally defined as percent defective th the plan will reject 90% of the time. We can say th a lot or worse than the LTPD will be rejected 90% of the time. If the lot passes, we can say with 90% confidence th the lot quality is better than the LTPD. Plot for testing N + 1 sampling rule using Opering Characteristic Curve The plan is to employ an OC Curve to find the efficacy and consistency of Square root of N plus one plan in lot acceptance sampling.

3 Efficacy of Sqrt (N) + 1 rule in Lot Acceptance Sampling 35 Populions of different sizes (N 1,N 2...N n ) are used to examine whether this plan can produce reliable acceptance quotients with irrespective of lot sizes. Sampling sizes (n 1, n 2...n n ) are calculed as the square root of lot sizes plus one respectively. The OC Curve is a graph th shows the probability of acceptance of a lot through all possible levels of percent defective. The percent defective of any given quantity of units of product is one hundred times the number of defective units of product contained therein divided by the total number of units of the product, i.e.: Number of defectives Percent Defective ¼ Number of units inspected 100 Expected average fraction defectives (m) present in the sample(s) chosen is determined as the percent defective of the sample size. Then the distribution of the acceptance, a, in a random sample of n items is approximely Poisson with parameters n and m, where m is the fraction of defectives in sample. The probability of acceptance is thus calculed as the summion of terms of poisson s exponential binomial limit. Probability of acceptance is the probability th m, the number of defectives, is less than or equal to C (Acceptance limit). The OC Curve is creed by plotting the percent defective versus the mching probabilities of acceptance. The probability of acceptance is based on the number of samples to be evalued and the quantity of rejects th are to be allowed. The curve becomes more sensitive, having a lower percent defective for the same probability of acceptance as the sample quantity increases. Results Lot sizes, from 100 to 1000, are considered to obtain different acceptance sampling plans with same sampling size of n +1 to compare their performance over a range of possible quality protection levels. The respective sample sizes (n) for each lot size are determined as mentioned in Table 1. The expected fraction defectives (m) in the samples to be retrieved to determine the quality of the inspecting lots is calculed with respect to their corresponding percent defectives as shown in Table 1. The corresponding probabilities of acceptance, P (a), for the estimed average fraction defectives (m) in samples are obtained using Poisson distribution law are shown in Table 2. The probability of finding a defective sample from derived samples against a lot size and a known percent defective is calculed as Probability of acceptance (Pa). Probability of acceptance of lot with size ranging from 100 to 1000, with sampling plan of square root of N plus one and acceptance number of 0 is plotted on the Y-axis against percent defectives (1% - 10%) on the X-axis. Table 1. Expected average fraction defectives in the samples different percent defectives S. No Lot size (N) Sample Size Sqrt of (N) + 1 (n) 1% 2% Expected average fraction defectives in sample(μ) 3% 4% 5% 6% 7% 8% 9% 10%

4 36 J. Muralimanohar and K. Jaianand Table 2. Probability of accepance calculed using poisson distribution Probability of acceptance (Pa) S. No Sample Size (n) Acceptance limit (C) 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% The opering characteristic curves indice the percentage of lots or bches which may be expected to be accepted under the same sampling plan of Square root of N plus one for different lot sizes. Discussion The predominant reason for having a sampling based quality inspection is to reduce the figue th occurs with 100% inspection and to effectively minimize the cost of inspections. Such sampling plans should ensure consistency, good productivity and provide enough confidence irrespective of the size of the lot produced or purchased. Square root of N plus one provides a simple mhemical way to enumere the number of items to be inspected for quality. However, there is some controversy over whether this sampling plan is consistent or stistically valid. An OC Curve has been used to show how effective the sampling plan is. OC Curves will show th deploying this plan in lot acceptance sampling may enhance potential sampling risks. Figure 1 illustres just how wrong this idea is. This figure compares the OC Curves of ten sampling acceptance plans, all of which involve a N + 1 sample size and an acceptance number of zero. The difference in the quality protections provided by these plans are obvious and impressive. For instance, if the existence of 4% defectives in submitted lots is assumed, the opering characteristic curves reveal different acceptance levels for different lot sizes with the same sampling plan of N + 1. With lot size of 100, acceptance number of 0, the probability of acceptance is identified as i.e., a lot of size 100 with defectives of 4% is accepted 63.8% of Figure 1. Showing variance between Opering characteristic curves of different lot sizes with the same sampling plan of N +1.

5 Efficacy of Sqrt (N) + 1 rule in Lot Acceptance Sampling 37 the time. In contrast, the probability of acceptance is identified as with lot size of 1000 and acceptance number of 0, i.e., a lot of size 1000 with defectives of 4% is accepted 27.3% of the time only. This clearly shows th, higher the size of the lot the higher is the chance of finding the defectives. Square root of N plus one sampling plan lessen the chances of rejecting a defective lot of smaller size. Obviously, a producer making product 4% percent defective would have a strong motive for trying to have its product inspected in lots of smaller size rher than in lots of larger size. Thus, Square root of N plus one sampling plan is proved stistically not valid as it gives irrespective acceptance rios with change in lot size. Acceptance sampling plans providing a high probability of detecting a failure should be selected for inspections. Sampling plans obtained from recognized sources do not ensure stistical validity. Standards like ANSI Z1.4 contain a wide variety of sampling plans as they try to accommode a wide variety of industries and products and so they cannot be reliable for all industries type. For instance, one of the sampling plans in ANSI Z1.4 requires 2 samples and accepts on 30 defects. This sampling plan would not be valid for the inspection of critical defects on a medical device. If AQL is the nonconformities per 100 items and if the product has only one specificion, sampling plans in ANSI Z1.4 with AQL greer than 100 are practically not applicable and this holds another evidence for its generic nure [6]. But, regardless of the source of the sampling plan and how it is indexed, it is the actual AQL and LTPD of the sampling plan th describes its protection and th determines whether it is valid [7]. To select a stistically valid sampling plan, aspects like past performance, criticality of the product and potential failure modes must be considered. A sampling plan must be selected in such a way th its opering characteristics meet the defined directories such as AQL and LTPD. AQL and LTPD can be indexed based on failure mode and effect analysis (FMEA) report. References 1. Blanck, F.C. (1927). Report of the Committee on Sampling, J. Assoc. Official Agricultural Chemists, 10, FDA, Investigions Operions Manual, Subchapter 4.3: Collection Technique, section Random Sampling. Obtained through the Internet: htm# , [Last accessed March 15, 2010]. 3. FDA, CBER. Draft Points to Consider in the Manufacture of In Vitro Monoclonal Antibodies, Docket 91N-0466, Mar Saranadasa, H. (2003). The Square Root of N Plus One Sampling Rule: How Much Confidence Do We Have? Pharmaceutical Technology, 27 (5), Torbeck L. D. (2009). Square Root of (N) + 1 Sampling Plan Is the square root of (N) + 1 a stistically valid scheme? Pharmaceutical Technology, 33, (10), ANSI/ASQ Z1.4 (2008). Sampling procedures and Tables for inspection by tributes. 7. Taylor W. A. (1997). Selecting Stistically Valid Sampling Plans, Quality Engineering, 10 (2),