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2 mean and variance) from the data or obtain them from prior experience. Such a process yields a distribution hypothesis, also called the null hypothesis (or H 0 ), with several parts that must be jointly true. The negation of the assumed distribution (or its parameters) is the alternative hypothesis (also called H 1 ). We test the assumed (hypothesized) distribution using the data set. Finally, H 0 is rejected whenever any one of the several elements composing the null hypothesis H 0 is not supported by the data. In the distance tests, when the assumed distribution is correct, the theoretical (assumed) CDF (denoted F 0 ) closely follows the empirical, step function CDF (denoted F n ). This is conceptually illustrated in Figure 1. There, the data are given as an ordered sample {X 1 < X 2 <... < X n } and the assumed (H 0 ) theoretical distribution has a CDF, F 0 (x). Then we obtain the corresponding GoF test statistic values. Finally, we compare the theoretical and empirical results. If they agree (probabilistically) then the data supports the assumed distribution. If they do not, the distribution assumption is rejected. z = x µ σ Empirical CDF: F n Step Theoretical CDF (Normal) Compare both and assess the agreement Figure 1. Distance Goodness of Fit Test Conceptual Approach There is, however, an important caveat to make. Theoretically, distance tests require the knowledge of the assumed distribution parameters. These are seldom known in practice. Therefore, adaptive procedures are used to circumvent this problem, when implementing GoF tests in the real world (e.g., see [4], Chapter 7). This is a major drawback of the KS GoF test, which otherwise is very powerful and general. In the examples we develop here some of these adaptive procedures will be used. Under a Normal Assumption, F 0 is normal (mu, sigma). Diff Fitting a Normal Distribution Using the Kolmogorov-Simirnov GoF Test The KS GoF test is widely used in practice. It is very versatile (any continuous distribution can be fit with it), works well with small samples and is included in statistical software packages. In this section we develop an example using KS for testing Normality. In the next section we develop another one for testing the Weibull assumption. Finally, we develop a counter example, where the assumption fails the test. First, we use real data from Section of MIL-HDBK-17 (1E), Composite Materials. The data set (Table 1) contains six tensile strengths, drawn from the same population. Table 1. Data for the KS GoF Test We obtain the descriptive statistics (Table 2) yielding point estimations of the assumed Normal distribution unknown parameters: sample mean and standard deviation. Table 2. Descriptive Statistics of the Data in Table 1 Variable N Mean Median Data Set There are several computational methods for the KS [3, 4]. Here, we follow the KS test statistic version in [4] and a welldefined scheme. First, sort the data. Then establish the assumed distribution (null hypothesis) and estimate its parameters. Then, obtain both the theoretical (assumed CDF) distribution (F 0 ) as well as the empirical (F n ) at each data point. Since KS is a distance test, we need to find the maximum distance F 0 - F n between the theoretical and empirical distributions. Its two basic functions are defined as: F0 (Xi ) = P0 (X Xi ) = CDF(Xi ); That is F 0 (X i ) is the assumed cumulative distribution function evaluated at Xi and F n (X i ) is the empirical distribution function obtained by the proportion of the data smaller than X i in the data set size n. # of X's X i F (X ) i n i = = ; i = 1,..., n n n Then, define: D+ = F n - F 0 and D- = F 0 - F n-1 for every data point X i. The KS statistic is: D = Maximum of all D+ and D- (> 0); for i = 1,, n (1) The KS logic is as follows: if the maximum departure between the assumed (CDF) and empirical distributions is small, then the assumed CDF will likely be correct. But if this discrepancy is large then the assumed F 0 is likely not the underlying data distribution. As an example, we calculate the values for the smallest data point (294.2) in Table 1. F0 (294.2) = Pµ= , σ = 14.85(294.2) = 2

3 Normal = Which is the assumed normal probability evaluated at Fn (294.2) = = = 0.167; Fn-1 (294.2) = = = 0 n 6 n 6 Which is the empirical probability at D + (294.2) = F n (294.2) - F 0 (294.2) = = D - (294.2) = F 0 (294.2) - F n-1 (294.2) = = The resulting values are shown in Table 3. The two columns of KSINDEX denote F n and F n-1, respectively. Column NormProb denotes the evaluation of the data using the assumed (Normal) distribution F 0 and the last two columns are statistics D+ and D. The result of the KS GoF test statistic (1) for this data set and assumption is: D = Maximum (D+, D-) = KS tables of critical values (CV) can be found in several texts [3, 4]. These CV are valid when the distribution parameters are known. When the parameters are estimated from the data, these CV are only approximate. One adaptive procedure uses a CV for Table 3. Intermediate Values for the KS GoF Test for Normality Row DataSet F 0 F n F n-1 D+=Fn-Fo D-=Fo-Fn Max: an error α', four times the a for which we are testing. Critical values are given in Table 4. Table 4. KS CV for α = 0.05 and α' = 0.2 and n = 4(1)10 Size n CV (0.20) CV (0.05) For n>10, use the approximation C.V. (0.05) = 1.36/ n C.V. (0.20) = 1.07/ n Approximation Error < 0.02 The KS statistic value (D = 0.155) is less than the corresponding table value (0.521) for n = 6 and an error α = If the true Normal parameters were used, the results for a would allow us to assume the null hypothesis (H 0 ) that the distribution of the population is Normal (315.8; 14.9). Since the two Normal parameters were estimated from the data, we must use an adaptive test procedure. We test at level α = 0.05 but compare the KS statistic with the CV at level α' = 4*α = 4*0.05 = 0.2. Hence, the adaptive KS CV for this case would be 0.411, which is still larger than the KS statistic value of There are software versions of the KS GoF test available. We present, in Figure 2, the Minitab graphical version. Verify how D+, D- and D (KS Statistics) coincide with the hand calculations in Table 3. In addition, the Minitab version of the KS test provides the approximate p-value = 0.15 > α = 0.05 (also leading us to assume Normality of the data). Finally, the KS graph from the Minitab program shows a linear trend. This fact lends additional support to the Normality assumption. Probability Average: StDev: N: 6 Kolmogorov Smirnov 310 Data Figure 2. Computer (Minitab) Version of the KS GoF Test for Normality Hence, since all results coincide, we can assume that the data comes from a Normal (µ = 315.8; σ = 14.9) population. The stepwise KS GoF test procedure is in Table Kolmogorov-Smirnov/Normality Test D+: D-: D: Approximate P-Value>0.15 3

4 Table 5. Step-by-Step Summary of the KS GoF Test for Establish the (H 0 ) assumed distribution: Normal (µ; σ) Estimate the Normal parameters: µ = 315.8; σ = 14.9 Sort the data in ascending order (Col. 1, Table 3) Obtain the Theoretical distribution (Col. 2) Obtain the Empirical distributions (Cols. 3 & 4) Obtain D+ and D- (Cols. 5 & 6) Obtain the KS Statistic: D = Maximum (D+, D-) = Obtain the KS table critical value CV(α') = (Adaptive) Since KS < CV assume the data come from a Normal (µ = 315.8; σ = 14.9) When available, use the computer software and the p-value Normality Finally, if we want to fit a Lognormal distribution, we first take the logarithm of the data and then implement the above KS GoF procedure on these transformed data. For, if the original data is Lognormal then its logarithm is Normally distributed and we can use KS to test for Lognormality. Fitting a Weibull Distribution Using the Kolmogorov-Simirnov GoF Test We now develop an example of testing for the Weibull assumption, using the data in Table 6. These data consist of six measurements, drawn from the same Weibull (α = 10; β = 2) population. In our example, however, parameters are unknown and estimated. Table 6. Data for Testing the Weibull Assumption We obtain the descriptive statistics (Table 7) and, using graphical methods in [11], also the point estimations of the assumed Weibull parameters: shape β = 1.3 and scale α = 8.7. Using the KS statistic (1) and following the procedures already discussed in the previous section, we obtain the intermediate KS results shown in Table 8. Table 7. Descriptive Statistics Variable N Mean Median StDev Min Max Q1 Q2 DataSet Table 8. Intermediate Values for the KS GoF Test for Weibull Row DataSet F 0 F n F n-1 D+=Fn-Fo D-=Fo-Fn Max: The step-by-step procedure (Table 9) shows how the KS GoF test statistic (1) value (0.23) is smaller than the KS table critical values (0.52 or 0.411) for α = 0.05 and a sample of size n = 6. Based on these results, we do not reject the hypothesis that the population from where these data were obtained, is distributed Weibull (α = 8.7; β = 1.3). Again, recall that the KS test is theoretically used with the true (but usually unknown) distribution parameters and not their estimations. Therefore, when implementing KS with parameters estimated from the data, use an adaptive test procedure (e.g., α' = 4 times Error α), as we have done in the present case. Finally, assume we want to assess the data for the Exponential (or any other continuous distribution) assumption. In such case, we obtain the Exponential mean (or any other appropriate distribution parameter) and apply the KS procedure, described in this section for the Weibull distribution. But now, we use instead the Exponential (or other pertinent probabilities) instead of the Weibull, in Column 2 of Table 8. Table 9. Step-by-Step Summary of the KS GoF Test for the Weibull Establish the (Null Hypothesis) assumed distribution: Weibull (α; β). Estimate the Weibull parameters: α = 8.7; β = 1.3. Sort the data in ascending order (Col. 1, Table 8). Obtain the Theoretical distribution (Col. 2). Obtain the Empirical distributions (Cols. 3 and 4). Obtain D+ and D- (Cols. 5 and 6). Obtain the KS statistic: D = Maximum (D+, D-) = Obtain the KS table critical value(s) = (or 0.411; Adaptive). Since D < CV do not reject the Weibull (α = 8.7; β = 1.3). Software for Weibull version of KS is not commonly available. 4

5 A Counter Example As an example of the use of KS where the assumed distribution is not true, we now refit the data (recall that these are Normal data) in Table 1 for the Weibull. We again estimate, using the graphical methods given in [11], the two unknown Weibull parameters: shape = 8 and scale = 350. Again, we use formula (1) and the KS procedures of the previous section and obtain the intermediate results shown in Table 10. Table 10. Intermediate Values for the KS GoF Test for the Weibull Row DataSet F 0 F n F n-1 D+=Fn-Fo D-=Fo-Fn Max: The KS GoF test statistic (1) value (0.463) is now three times higher than before. It is larger than the KS test CV for α' = 0.2 (0.411) and smaller than the CV for α = 0.05 (0.52), both for n = 6. Given that both Weibull parameters were estimated from the sample, the test error α is different from the one stated and we have to use an adaptive procedure (i.e., compare with 0.411). Based on these adaptive KS results, we reject the hypothesis that the population from where these data were obtained, is distributed Weibull (α = 350; β = 8). The entire KS GoF test procedure is summarized in Table 11. Table 11. Step-by-Step Summary of the KS GoF Test for the Weibull Establish the (Null Hypothesis) assumed distribution: Weibull (α; β). Estimate the Weibull parameters: α = 350; β = 8. Sort the data in ascending order (Col. 1, Table 10). Obtain the Theoretical distribution (Col. 2). Obtain the Empirical distributions (Cols. 3 and 4). Obtain D+ and D- (Cols. 5 and 6). Obtain the KS statistic: D = Maximum (D+, D-) = Obtain the KS table critical value CV(α') = (Adaptive). Since KS > CV reject that data comes from Weibull (α = 350; β = 8). Software for Weibull version of KS is not commonly available. Summary In this START sheet we have discussed the important problem of (GoF) assessment of statistical distributions, especially for small samples, via the Kolmogorov Smirnov GoF test. Alternatively, one can also implement the Anderson Darling test [10]. These tests can also be used with large samples. We have provided several numerical and graphical examples for testing the Normal, Lognormal, Exponential and Weibull distributions, relevant in reliability and maintainability studies (Lognormal is a special case of the Normal). We have also discussed some related theoretical and practical issues, providing several references to background information and further readings. The large sample GoF problem is often better dealt with via the Chi-Square test [9]. For, it does not require knowledge of the distribution parameters -something that both, AD and KS tests do require and affects their power. On the other hand, the Chi Square GoF test requires that the number of data points be large enough for the test statistic to converge to its underlying Chi- Square distribution -something that neither AD nor KS require. Due to their complexity, the Chi Square and the Anderson Darling GoF tests are treated in more detail in separate papers [9, 10]. Bibliography 1. Practical Statistical Tools for Reliability Engineers, Coppola, A., RAC, A Practical Guide to Statistical Analysis of Material Property Data, Romeu, J.L. and C. Grethlein, AMPTIAC, An Introduction to Probability Theory and Mathematical Statistics, Rohatgi, V.K., Wiley, NY, Methods for Statistical Analysis of Reliability and Life Data, Mann, N., R. Schafer, and N. Singpurwalla, John Wiley, NY, Statistical Confidence, Romeu, J.L., RAC START, Volume 9, Number 4, STAT_CO.pdf. 6. Statistical Assumptions of an Exponential Distribution, Romeu, J.L., RAC START, Volume 8, Number 2, 7. Empirical Assessment of Normal and Lognormal Distribution Assumptions, Romeu, J.L., RAC START, Volume 9, Number 6, NLDIST.pdf. 8. Empirical Assessment of Weibull Distribution, Romeu, J.L., RAC START, Volume 10, Number 3, alionscience.com/pdf/weibull.pdf. 5

6 9. The Chi-Square: A Large-Sample Goodness of Fit Test, Romeu, J.L., RAC START, Volume 10, Number 4, Anderson-Darling: A Goodness of Fit Test for Small Samples Assumptions, Romeu, J.L., RAC START, Volume 10, Number Weibull Analysis, Dodson, B., ASQ Press, WI, About the Author Dr. Jorge Luis Romeu has over thirty years of statistical and operations research experience in consulting, research, and teaching. He was a consultant for the petrochemical, construction, and agricultural industries. Dr. Romeu has also worked in statistical and simulation modeling and in data analysis of software and hardware reliability, software engineering and ecological problems. Dr. Romeu has taught undergraduate and graduate statistics, operations research, and computer science in several American and foreign universities. He teaches short, intensive professional training courses. He is currently an Adjunct Professor of Statistics and Operations Research for Syracuse University and a Practicing Faculty of that school's Institute for Manufacturing Enterprises. Dr. Romeu is a Chartered Statistician Fellow of the Royal Statistical Society, Full Member of the Operations Research Society of America, and Fellow of the Institute of Statisticians. Romeu is a senior technical advisor for reliability and advanced information technology research with Alion Science and Technology. Since joining Alion in 1998, Romeu has provided consulting for several statistical and operations research projects. He has written a State of the Art Report on Statistical Analysis of Materials Data, designed and taught a three-day intensive statistics course for practicing engineers, and written a series of articles on statistics and data analysis for the AMPTIAC Newsletter and RAC Journal. Other START Sheets Available Many Selected Topics in Assurance Related Technologies (START) sheets have been published on subjects of interest in reliability, maintainability, quality, and supportability. START sheets are available on-line in their entirety at < alionscience.com/rac/jsp/start/startsheet.jsp>. For further information on RAC START Sheets contact the: Reliability Analysis Center 201 Mill Street Rome, NY Toll Free: (888) RAC-USER Fax: (315) or visit our web site at: < About the Reliability Analysis Center The Reliability Analysis Center is a world-wide focal point for efforts to improve the reliability, maintainability, supportability and quality of manufactured components and systems. To this end, RAC collects, analyzes, archives in computerized databases, and publishes data concerning the quality and reliability of equipments and systems, as well as the microcircuit, discrete semiconductor, electronics, and electromechanical and mechanical components that comprise them. RAC also evaluates and publishes information on engineering techniques and methods. Information is distributed through data compilations, application guides, data products and programs on computer media, public and private training courses, and consulting services. Alion, and its predecessor company IIT Research Institute, have operated the RAC continuously since its creation in

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### START Selected Topics in Assurance

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