Explore the World of Software: Engineering & Science

home *** CD-ROM | disk | FTP | other *** search

/ Explore the World of Soft…e: Engineering & Science / Explore_the_World_of_Software_Engineering_and_Science_HRS_Software_1998.iso / programs / statistc / cqnt9311.txt < prev next >

Wrap

Text File | 1997-09-22 | 110KB | 1,949 lines

Introduction to CapQuant's Capability Quantification Methodology . . 1 Usage of the CapQuant Pop-Up Menus . . . . . . . . . . . . . . . . . 3 Preserving Time-Series Order and Forming Moving Statistics . . . . .18 REGRET FUNCTIONS: Quantify Impact of Deviations from Target. . . . .22 Equivalent Nonconformance and Expectancy . . . . . . . . . . . . . .27 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30 Appendix: Technical Descriptions of the Poissonization Process . . .33 CapQuant Software Update History . . . . . . . . . . . . . . . . . .38 ┌─────────────────┐ │ INTRODUCTION │ └─────────────────┘ CapQuant.EXE is a software system for IBM-compatible (MS-DOS) Personal Computers that implements the statistical methodology described in my 1991 and 1993 manuscripts "Regret Indices and the Quantification of Process Capability" and "Cumulative Capability Curves," respectively. Here, "regret" can be almost any surrogate measure of how cost-of- poor-quality (from a customer and/or regulatory viewpoint) increases as product deviation from an intended "target" value increases. CapQuant.EXE constructs Cumulative Capability (CC) curves and displays them in an "interactive" graphics mode that allows you to read off numerical values by moving an on-screen cursor. CC curves quantify process "yields" (conformance fractions) for a entire class of intervals. When regret is a "2-to-1" transformation of the original process characteristic (so that the only information loss is the numerical sign of the deviation from target), CC yields correspond to ALL intervals with equal, maximum regret at their two end-points. Specifically, when the regret function is symmetric-about-the-target CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 2 of 39 as well as 2-to-1, CC yields are those for ALL symmetric-about-the- target intervals. CC curves provide a highly-intuitive and widely-applicable capability quantification methodology that characterizes performance on a cost- unitless "index" scale. One ends up with a CC curve for each product or process studied. In summary, ┌───────────────────────────────────────────────────────────────────┐ │ CC curves not only... serve as benchmarks for current capability │ │ as well as for quality improvement efforts │ │ │ │ but also... enable and, in fact, encourage comparisons │ │ across diverse processes. │ └───────────────────────────────────────────────────────────────────┘ Any type of quantitative (variables) data measurement can be used as a basis for CC quantification via a three-stage analysis process: (i) Convert each observed numeric value into a REGRET, which is a measure of long-range economic impact or customer/regulatory dissatisfaction that results from product deviations from an intended TARGET value (or values.) CapQuant allows the user to pick a regret function from any of 10 general families and to specify its target value. (ii) Smooth and Standardize the observed distribution of historical regret values (via POISSONIZATION) to form a REGRET INDEX. CapQuant determines an EQUIVALENT NONCONFORMANCE value for each regret and the corresponding EQUIVALENT EXPECTANCY, the expected regret within historical capability levels. The corresponding INDEX is simply the ratio of observed to expected regret. An index value of 0 is ideal, while an index value of 1 is the historical quality standard (or capability threshold.) Customer and/or regulatory dissatisfaction increases as the index value increases. Deciding on exactly how much of available historical data to use in defining current process capability can be troublesome, especially if you are using a regret function (like quadratic) which is highly sensitive to outliers. Anyway, I recommend that you "start out" by examining all available history that you feel are still relevant to current operations. (iii) Display the CC curve, which is the Empirical Distribution Function (EDF) of observed Regret Indices. CapQuant draws CC curves in interactive, medium-resolution (CGA, 320x200 pixel) mode. CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 3 of 39 CapQuant can plot the cumulative distribution function for process history consisting of as many as 1200 observations. Both the observed (raw) and Poisson smoothed forms are displayed. CapQuant also allows direct graphical comparisons between the EDF of at most 400 of your "newest" sample results (most recent production) with your historical CC curve. CapQuant provides all of the computational/graphical tools you needed to implement specific analyses of the above type. CapQuant creates an ASCII (text) output file of extremely detailed results that, following minor manual editing, provides an ideal data input file to statistical and/or presentation graphics systems, such as SAS. But only a fraction of CapQuant's outputs (those which are displayed on your computer screen as well as written to the ASCII output file) are needed to characterize the (historical) capability of a process. ┌─────────────────────────────┐ │ *** USAGE of CAPQUANT *** │ └─────────────────────────────┘ The CapQuant software system is invoked from MS-DOS by entering its 8-character name, CAPQUANT, at your DOS prompt... Prompt> capquant CapQuant does not expect or accept any "command-line" arguments. Instead, CapQuant will prompt you to provide all information it needs to perform analyses. And each such prompt usually displays a [default value] in square brackets that you may accept by simply pressing the ENTER key. WARNING ONE: Aberrant behavior WILL result if you attempt to execute CapQuant at any time that your system has less than 375kBytes of available random access memory (RAM). CapQuant is actually a much larger program than it might appear to be. Yes, its .EXE module occupies only 85kBytes. But, before compression with LZEXE 0.91 (c) Fabrice BELLARD, CapQuant.EXE was 251kBytes in size, and this does NOT include the space CapQuant needs to read data and store intermediate results. WARNING TWO: CapQuant assumes that your personal computer has a COLOR monitor that is capable of making (at least) CGA graphical displays. CapQuant uses color in its text-mode windows/menus. Screens with text written in black on a green or cyan background will appear to be BLANK if you do not have a color monitor. And if your system is incapable of graphics, your system may crash when CapQuant attempts to draw P-P probability plots and Cumulative Capability Curves. CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 4 of 39 ┌────────────────────────────────────────────────────┐ │ CapQuant Input/Output File Naming Conventions... │ └────────────────────────────────────────────────────┘ The ASCII (text) files used by CapQuant for data input and output consist of a filename (of at most eight characters) specified by the user followed by a period (.) and one of four specific 3-character filename EXTensions. CONTAMIN.HST ...Batch input of Historical Data (at most 1200 obs.) CONTAMIN.NEW ...Batch input of New Data (at most 400 observations) CONTAMIN.OUT ...CapQuant Output Save File. CONTAMIN.XXX ...CapQuant Output of Trial REGret Parameter Settings. CONTAMIN.REG ...Edited .XXX File for Automatic REGret Calculations. ┌──────────────────────────────────────┐ │ Example CapQuant Walk-Through... │ └──────────────────────────────────────┘ The CapQuant start-up screen will appear as follows... ╔═════════════════════ Capability Quantification ══════════════════╗ ║ ║ ║ CAPQUANT.EXE...Version 9311 ║ ║ ║ ║ A Quality Assurance Training Tool: ║ ║ Statistics Committee of the QA Section of the PMA ║ ║ ║ ║ ╔═══════════════════════════════════════════════════════════════╗ ║ ║ ║ Will Process HISTORY be Input via K = Keyboard ? ...or ║ ║ ║ ║ B = Batch File ? ║ ║ ║ ║ ║ ║ ║ ║ Press the K or B key now --> ║ ║ ║ ╚═══════════════════════════════════════════════════════════════╝ ║ ╚═══════════════════════════════════════════════════════════════════╝ If you select Keyboard input, you will be prompted to type in all historical data values using your keyboard. But, in this case, CapQuant will also be creating a "Batch Input" file that you could use in future CapQuant runs to repeat and/or modify your initial analyses. On the other hand, if you are familiar with a "screen oriented" personal computer (ASCII) text editor, you might prefer to start by creating your own "Batch Input" file. This strategy can give you more flexibility in data entry and validation than is possible inside of CapQuant. Simply mimic the format of an example CapQuant Batch Input file. For example, the Batch Input file could consist for the following 7 records (rows), each starting in column 1: 1 ...Number of Data Values per Observation THICKNES ...Name of Variable ONE 1.09 2 13 CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 5 of 39 4.7 3 q where the letter Q at the start of the last record signals the "end of data entry." Once you respond to the initial CapQuant prompt by pressing either the K or the B key, the "Window" will clear itself, and you will either (i) be prompted to specify a MS-DOS filename, consisting of at most 8 characters, for capture of Keyboard input or (ii) select an existing Batch Input (.HST) file from the bounce-bar menu... ┌───────────────────── Capability Quantification ─────────────────────┐ │ │ │ Batch Input of Process History Selected... │ │ │ │ At colon Prompts : ...simply press ENTER to get the [default]. │ │ │ │ ╔File Names╗ │ │ ║ CONTAMIN ║ Arrow Keys: Move Highlight Bar Up / Down. │ │ ║█NORMAL███║ │ │ ║ DISSOLUT ║ Return Key: Selects the Highlighted File. │ │ ╚══════════╝ │ │ Escape Key: Abandon BATCH CapQuant Input. │ │ │ └─────────────────────────────────────────────────────────────────────┘ Whether input comes from your Keyboard or from a Batch file, the CapQuant Output File can have the same filename (of at most 8 characters) as does the Input File. But the 3-character MS-DOS filename EXTension for CapQuant Output Files will always be ".OUT". On the other hand, although the [default] output filename will always be to use the same filename as the input file, there are circumstances when you will probably want to us a DIFFERENT filename for your output. After all, if you always use the same output filename, that file will get over-written and all previously saved results will be lost. Below, we illustrate changing the output filename to NORMDIFF... ╔══════════════════ Capability Quantification ═══════════════════╗ ║ ║ ║ Batch File Input Selected... ║ ║ ║ ║ At colon Prompts : ...simply press ENTER to get the [default]. ║ ║ -------------------------------------------------------------- ║ ║ ║ ║ The Batch Input file is to be: normal.hst ║ ║ ║ ║ Specify filename for Detailed Output [normal.out] : normdiff <<<<<< ║ The CAPQUANT Output Save file is to be: normdiff.out ║ ║ ║ ║ Press Q to QUIT now...Other Key to Continue... ║ ╚═════════════════════════════════════════════════════════════════╝ CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 6 of 39 A [Black on Green] window for Parameter Specification appears next... ╔═══════════════════ CapQuant Parameters ══════════════════╗ ║ ║ ║ Does data consist of Single Values or Pairs (1 or 2) : ║ ║ Number of Data Values per Observation = 1. ║ ║ ║ ║ Should "Moving" statistics be calculated ? ║ ║ 0 => No moving statistics... ║ ║ 1 => Form first differences... ║ ║ 2 => Exp.Wgt.Mov.Average... ║ ║ 3 => Form Deviations from EWMA... ║ ║ ║ ║ WARNING: Options 1, 2, & 3 can only be used when ║ ║ data are input in time-series order. ║ ║ ...And the first observation will be lost ║ ║ unless a Start-Up Value is specified. ║ ║ ║ ║ What is your choice ? [0] : <<<< Here we accept the DEFAULT by ║ Moving Statistics Option = 0. simply pressing the ENTER key. ║ ║ ║ What is the NAME of Variable ONE ? [variable1] : ║ ║ First Variable Name: THICKNES ║ ║ ║ ║ Historical Data, in any sequence, are to be entered ║ ║ in the Next Window... Press Q to QUIT now... ║ ╚═════════════════════════════════════════════════════════════════╝ We will discuss the Moving Statistics options (1,2,3 above) in the section of this documentation that starts on page 11. Anyway, another good reason to change the name of the Output File on the previous screen would be because you want to explore several "smoothing" and/or "differencing" transformations of the data from one Input File. A [Blue on Cyan] Data Entry window now appears. If you are using Keyboard input, CapQuant will continue prompting for either single values or data pairs until you ENTER the letter Q to signal an end to data entry. The number of observations entered will then be displayed... ╔═════════════════════ Data Entry Window ════════════════════╗ ║ ║ ║ List Historical Data in Output File? [y|n] : n ║ ║ ║ ║ Number of Historical Observations = 500 ║ ║ ║ ║ Data Entry Complete... ║ ║ Press Q to QUIT now...Other Key to Continue... ║ ╚═════════════════════════════════════════════════════════════════╝ Small windows will now Pop-Up within the [Blue on Cyan] Large Window. You first selection from the left-hand "Process Capability MENU" should always be "R". (...Unless you wish to "eXit" without making any calculations!) CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 7 of 39 You first selection from the middle "REGRET Analysis MENU" should always be "S" or "A"...unless you wish to Return without making any calculations! After "S", you choose a Regret-Function-Family from the right-hand "REGRET Function MENU"... ╔═════════════════════════════╗ ║ Process Capability MENU ║ ║ ║ ╔═════════════════════════╗ ║ R = Regret Analysis ║ ║ REGRET Function MENU ║ ║ ╔═══════════════════════════║ ║ ║ D = Disp║ REGRET Analysis MENU ║ 1...GOAL-POSTS ║ ║ ║ ║ 2...QUADRATIC ║ ║ C = CC C║ S = Specify Regret Functio║ 3...ABSVALUE ║ ║ ║ ║ 4...1-SIDED QUADRATIC ║ ║ M = Moni║ I = Regret INDEX Computati║ 5...BILINEAR ║ ║ ║ ║ 6...END-POINT QUADRATIC ║ ║ X = eXit║ L = Lack-of-Fit to a Poiss║ 7...LOGISTIC ║ ║ ║ ║ 8...INVERTED NORMAL ║ ║ Choice --> R║ C = Composite Lack-of-Fit ║ 9...RANGEMAX ║ ║ ║ ║ 0...NONCONFORM/EXPTANCY ║ ╚═════════════║ A = Automatic Regret Calcs║ A...Abandon/Quit ║ ║ ║ ║ ║ R = Return to Main Menu ║ ║ ║ ║ Choice --> ║ ║ Choice --> S ║ ║ ║ ╚═════════════════════════╝ ╚══════════════════════════════╝ The regret functions which can be defined in CapQuant belong to one of ten general families. 1) attributes regret, which is zero inside an interval and one outside that same interval, 2) quadratic regret, where the regret is the square of the deviation of the measurement from its target value, 3) linear regret, in which the economic impact of a value is directly proportional to that value, 4) one-sided regret, which is quadratic on one side of a target and zero on the other side of that target value, 5) opportunity regret, in which the economic impact of a demand for a perishable product is measured relative to the inventory on hand, 6) end-point quadratic regret, in which the minimum and maximum observed values in samples of fixed size are monitored, and regret is the sum of the quadratic regret at the two end-points, CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 8 of 39 7) bounded max regret, in which regret follows a double- logit pattern that is V-shaped in its center (at the target) but flat like attribute regret in its tails, 8) inverse gaussian regret, in which regret looks like an upside-down bell-shaped curve (quadratic at the target but flat like attributes regret in its tails), 9) range (linear) regret, in which the minimum and maximum observed values in samples of fixed size are monitored, and regret is the difference, maximum minus minimum, and 0) an unspecified pattern called "nonconformance-expectancy." In this case, the data are assumed to have been "Poisson"ized outside of CapQuant, but you can still use CapQuant to plot Cumulative Capability Curves and/or to test Lack-of-Fit. See the section of this CapQuant documentation, starting on page 10, that discusses REGRET FUNCTIONS for detailed information on these ten families. The numerical example used here in the documentation is based upon... Regret Type: QUADRATC TARGET = Value corresponding to ZERO REGRET = 20.00 Once you have specified a functional form for your Regret Function and have selected values for its Parameters (TARGET, PARM1, PARM2), the "REGRET Analysis MENU" will re-appear. One's second selection from this menu is usually the second item, "I"... CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 9 of 39 ╔══════════════════════════════╗ ║ REGRET Analysis MENU ║ ║ S = Specify Regret Function ║ ║ I = Regret INDEX Poisson Fit ║ ║ C = Composite Poisson Fit ║ ║ Q = Q-Q Plot for Gamma Fit ║ ║ A = Automatic Regret Calcs ║ ║ R = Return to Main Menu ║ ║ Choice --> I ║ ╚══════════════════════════════╝ ╔═════════════════ Regret Calculations ════════════════╗ ║ ║ ║ Observations (Periods) = 500 ║ ║ Regret Type = 2 ║ ║ ║ ║ Observed Average (Expected) Regret = 34.962 ║ ║ Observed Variance of Regret = 2354.166 ║ ║ Corresponding Equivalent Expectancy = 0.5192 ║ ║ ║ ║ Enter Objective Expected Regret [ 1.000] : 35 ║ ║ using... 35.000 ║ ║ ║ ║ Enter Objective Variance [2354.321] : 2500 ║ ║ using... 2500.000 ║ ║ ║ ║ Equivalent Expectancy per Observation = 0.4900 ║ ╚═════════════════════════════════════════════════════════╝ As in the above example, the "Objective Expected Regret" value is usually taken to be a convenient, "rounded" value (35, here) that is fairly close to the observed, sample value (34.9). The initial default value for Expected Regret will always be [ 1.000]. But, if you select item "I" on the "REGRET Analysis MENU" more than once, the subsequent default value will be the value you entered in the previous iteration. The default value [2354.321] for the "Objective Variance," is the numerical value that (given an objective expected regret of 35 rather than the observed 34.9) would equate the objective "Equivalent Expectancy per Observation" to the observed value of 0.5192. This default value will not, necessarily, also equate the mean and variance of the rescaled regret distribution...unless the objective and observed expected regret values are exactly equal. In the above example, a convenient, "rounded" value (2500) is also entered for the "Objective Variance," and the resulting "Equivalent Expectancy per Observation" is computed by CapQuant to be 0.49. Here... EXPREGRET objective = 35.0 ...observed = 34.9621 VARREGRET objective = 2500.0 ...observed = 2354.3206 You should try out several choices for ER = Objective Expected Regret and VR = Objective Variance (35 and 50, above.) For example, you CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 10 of 39 should change ER and/or VR when you are unhappy with the resulting "Equivalent Expectancy per Observation" (0.49 above ) or when the resulting Kolmogorov-Smirnov Lack-of-Fit statistic is significant...as it is here! ╔════════════════Poisson Lack-of-Fit Test═════════════════╗ ║ ║ ║ Kolmogorov-Smirnov Statistic = 0.56263 ║ ║ 5% Critical Value = 0.06 ║ ║ 1% Critical Value = 0.07 ║ ║ Lack-of-Fit is Highly Significant, ║ ║ 1% Critical Level. ║ ║ ║ ║ Press a Key to View the P-P Plot... ║ ║ ║ ║ Index, ObsFreq, CumFreq, and Poisson Fit... ║ ║ 0.00 0.05000 0.05000 0.61263 ║ ║ 0.01 0.00000 0.05000 0.61263 ║ ║ 0.02 0.12000 0.17000 0.61263 ║ ╚═════════════════════════════════════════════════════════╝ See the section of this documentation, starting on page 17, that discusses Equivalent Nonconformances and Expectancy for detailed information on this CRITICAL phase of the analysis. Statistically significant Poisson Lack-of-Fit can even suggest the need to change the very form of your regret function ( item "S" ). NOTE1: Always re-run item "I" after each change you make to the regret function via item "S". And never select items "C" or "Q" except after item "I". NOTE2: All "Equivalent Nonconformance" values were rounded to the nearest full integer value for the purpose of the above Lack-of-Fit test because Poisson random variables take on only integer values. But this rounding really represents an unnecessary loss-of-precision in most quality monitoring applications. Thus CapQuant does NOT round nonconformance numbers to integers in most of its calculations. In fact, CapQuant lists nonconformance/expectancy results to 2 decimal places in the ASCII output file. The highly significant lack-of-fit in the above example is probably not "practically important." It resulted, primarily, because the equivalent expectancy was "small" (0.49 => a step-size larger than 2 units on the index scale) and our sample of data is "large" (500.) To illustrate this, choose item "C" and examine composites of 6 consecutive regret values... CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 11 of 39 ╔══════════════════════════════╗ ║ REGRET Analysis MENU ║ ║ S = Specify Regret Function ║ ║ I = Regret INDEX Poisson Fit ║ ║ C = Composite Poisson Fit ║ ║ Q = Q-Q Plot for Gamma Fit ║ ║ A = Automatic Regret Calcs ║ ║ R = Return to Main Menu ║ ║ Choice --> C ║ ╚══════════════════════════════╝ ╔═════════════Composite Poisson Lack-of-Fit═══════════════╗ ║ ║ ║ Composite Regret Calculations... ║ ║ Number of Original Observations = 500 ║ ║ Regret Type = 2 ║ ║ Equivalent Expectancy (Single Obs.) = 0.4900 ║ ║ ║ ║ How many consecutive Regret values should be Added ║ ║ to form Composite Regrets ? (2 to 10) [4] : 6 ║ ║ ║ ║ Number of Composite Observations = 83 ║ ║ Composite Equivalent Expectancy = 2.9400 ║ ║ Composite Poisson Lack-of-Fit... ║ ║ ║ ║ Kolmogorov-Smirnov Statistic = 0.13451 ║ ║ 5% Critical Value = 0.15 ║ ║ 1% Critical Value = 0.18 ║ ║ Composite distribution is not significantly ║ ║ different from the fitted Poisson. ║ ║ ║ ║ Press a Key to View the Composite P-P Plot... ║ ╚═════════════════════════════════════════════════════════╝ In other words, lack-of-fit to a Poisson distribution declines steadily for this dataset as 2, 3, 4, or 5 regrets are "composited"; all lack-of-fit becomes statistically insignificant when 6 or more are "composited." The hallmark of continuous improvement is then this: the EE=ER*ER/VR resulting from individual measurements made on the process will continually drop. And yet, statistically significant lack-of-fit in discrete Poisson approximations is common in cases where EE is small (0.5 or less) and data are plentiful. After all, the Poisson integer "step-size" then balloons to 2 units or greater when re-expressed on the regret index scale. The rounded and un-rounded regret index values (corresponding to integer or non-integer ENs) can thus be quite different numerically. CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 12 of 39 The fourth item ("Q") on the "REGRET Analysis MENU" provides a way to smooth a sample of observed regret indices with a continuous gamma distribution fit only to a specified number of the smallest regret order statistics... =============== ================ ╔══════════════════════════════╗ ║ REGRET Analysis MENU ║ ║ S = Specify Regret Function ║ ║ I = Regret INDEX Poisson Fit ║ ║ C = Composite Poisson Fit ║ ║ Q = Q-Q Plot for Gamma Fit ║ ║ A = Automatic Regret Calcs ║ ║ R = Return to Main Menu ║ ║ Choice --> Q ║ ╚══════════════════════════════╝ ┌───────────────── Gamma Q-Q Probability Plots ────────────────┐ │ │ │ Gamma Lack-of-Fit... │ │ Initial Equivalent Expectancy = 0.4900 │ │ Sorting Observed Regret Values... │ │ │ │ How many of the smallest Regret Order Statistics should used │ │ to estimate Gamma Parameters ? [125 to 500] : 200 │ │ │ │ Number of Iterations = 5 │ │ Final Slope = 1.0081 │ │ Final Equivalent Expectancy = 0.5426 │ │ │ │ Press a Key to View the Q-Q Probability Plot... │ │ │ └──────────────────────────────────────────────────────────────┘ Poisson or gamma lack-of-fit that occurs only in the extreme right-hand tail of an EN sample is easily tolerated. The essential feature of a "successful" smoothing for a regret index distribution is that it provides a good representation over the range that represents relatively good performance, I < or = 1. In my experience at least, processes in doubtful states of statistical control tend to be unstable primarily in their right-hand regret tail. Q-Q probability plotting methods yield an estimate of scale as well as shape of the best fitting gamma distribution. Thus, while preliminary EN rescaling of regret is not really necessary, EN rescaling is still recommended because the estimated gamma scale parameter will then be approximately 1, at least when all of the available data are used in the analysis. For gamma Q-Q probability plotting, Chambers, Cleveland, Kleiner and Tukey(1983), Chapter 6, recommend use of the Wilson-Hilferty(1931) normal approximation for the cube root of gamma variables and the Hastings(1955) rational function approximation for normal quantiles. Then it is straight-forward to estimate the slope, ß, of a zero-intercept regression of the k smallest observed EN regrets onto CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 13 of 39 their corresponding approximate gamma EE quantiles. At iteration i+1, the trial value for EE(i+1) would be, say, one third of the way from EE(i) to [EE(i) times the i-th ß estimate.] Iterative fitting would then halt when either [1] the i+1-th zero-intercept regression fails to show increased multiple correlation between the k smallest observed EN(i+1) regrets and their fitted gamma EE(i+1) quantiles, or [2] ß converges to 1.0. Let us suppose that you now select item "R" on the REGRET MENU followed by item "D" on the Main MENU... ╔═════════════════════════════╗ ║ Process Capability MENU ║ ║ R = Regret analysis ║ ║ D = Display CC curves ║ <--- 2 nd Selection = D ║ C = CC Confidence Limits║ ║ M = Monitor capability ║ ║ X = eXit CAPQUANT ║════════════════╗ ║ Choice --> C ║ysis MENU ║ ╚═════════════════════════════╝oss Function ║ ║ . . . . . . . . . . . . . . ║ R = 1 st Selection ---> ║ R = Return to Main Menu ║ ║ Choice --> R ║ ╚═════════════════════════════╝ ╔═══════════════════ Process Capability Plots ═══════════════════╗ ║ Number of Observations = 500 ║ ║ Show CC Curve as Stair-Steps ? [Y|n] :n <---Alternative ║ ║ Show Poisson or Gamma Fitted CDF ? [P|g] :g <---Alternative ║ ╚════════════════════════════════════════════════════════════════╝ While viewing the CGA graphics display of the Observed and Fitted Cumulative Capability Functions, you may always press the SpaceBar or Function Key F1 to view the following HELP Screen... ┌─────────────────────────────────────────┐ │ This HELP screen .... Space Bar or F1. │ │ │ │ ERASE PLOT......press ENTER or ESCape. │ │ Note: 2nd KeyPress sometimes needed. │ │ │ │ On-Screen Cursor & VALUES ... press V. │ │ Move Right/Left with ArrowKeys by 0.01 │ │ or... Tab/BackSpace by 0.25 │ │ │ │ Screen PRINT/SAVE Keys: First Press S │ │ and then press a Second Special Key... │ │ L or F10 dumps to HP LaserJet/DeskJet │ │ I or F9 dumps to IBM Graphics Matrix │ │ E or F8 dumps to Epson FX, JX, or LQ │ │ O or F7 dumps to Okidata Microline │ │ S saves screen to disk in PCX format. │ │ │ │ CapQuant, ver9311.......Press any key. │ └─────────────────────────────────────────┘ CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 14 of 39 One iteration through the three-stages of quality monitoring calculations is now complete. You now have the options of either pressing "R" to return to the REGRET Menu for further calculations, "C" to calculate and display Confidence Limits on the CC curve, "M" to Monitor Capability using NEW observed process results, or "X" to exit CapQuant and return to DOS. (Except possibly in "training" exercises, there will usually be no need to simply repeat Cumulative Capability display, item "D", without first changing either the regret function or the objective regret mean or variance.) Let's choose item "C" next... ╔═════════════════════════════╗ ║ Process Capability MENU ║ ║ R = Regret analysis ║ ║ D = Display CC curves ║ ║ C = CC Confidence Limits║ ║ M = Monitor capability ║ ║ X = eXit CAPQUANT ║ ║ Choice --> C ║ ╚═════════════════════════════╝ ╔═════════════════════ CC Confidence Limits ═════════════════════╗ ║ ║ ║ Number of Observations = 500 ║ ║ ║ ║ On the graphics screen, press F1 or Space Bar for HELP. ║ ║ Press the V key to interactively View numerical Values. ║ ║ Press a Key to View the CC curve and Confidence Limits... ║ ║ ║ ╚════════════════════════════════════════════════════════════════╝ Nair and Freeny(1993) point out that asymptotic confidence bands, constructed from (nonparametric) empirical distribution functions, are of the form CC(I) +/- K * sqrt[ CC(I) * ( 1 - CC(I) ) / N ] where K is a constant that depends upon the desired confidence level (one minus alpha) and N is the total number of regret indices defining the CC curve. Exact confidence intervals that are valid pointwise (for a given index, I) can be constructed from the fact that N * CC(I) follows a Binomial distribution. When a Normal approximation to this Binomial is used, the constant K above can be taken to be the (one minus alpha/two) quantile of the standard Normal distribution. Simultaneous (rather than pointwise) confidence bands of the above form can be constructed from the variance-weighted Kolmogorov-Smirnov statistic, but the constant K will then be sensitive to the index range covered, L <= CC(I) <= U. In fact, K approaches infinity as L approaches 0 and U approaches 1. Stephens(1986) provides tables of asymptotic K values (large N), a few of which are: CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 15 of 39 ┌──────────────────────────┐ │ alpha significance level │ ┌────────────────────────┼──────────────────────────┤ │ index range covered │ 0.01 0.05 0.10 │ ├────────────────────────┼──────────────────────────┤ │ L=0.01, U=0.99 │ 3.81 3.31 3.08 │ │ L=0.05, U=0.95 │ 3.68 3.16 2.91 │ │ L=0.10, U=0.90 │ 3.59 3.06 2.79 │ └────────────────────────┴──────────────────────────┘ Nair and Freeny(1993) recommend the choice L=0.05 and U=0.95 as yielding a relatively wide index range without producing excessively wide confidence limits. Thus CapQuant uses K = 2.91 to display asymptotic 95% confidence lower and upper simultaneous limits on CC(), so that the region between these lower and upper limits provides an central 90% confidence band about CC(). When N is small (less that 25, say), this asymptotic band tends to be too narrow. The exact critical values of Niederhausen(1981) could be used in these cases. Remember that, while viewing the CGA graphics display of the CC curve confidence limits, you may simply press the SpaceBar or Function Key F1 to get HELP on the KEYs used for on-screen annotation. CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 16 of 39 Next, try item "M" from the main menu... ╔═════════════════════════════╗ ║ Process Capability MENU ║ ║ R = Regret analysis ║ ║ D = Display CC curves ║ ║ C = CC Confidence Limits║ ║ M = Monitor capability ║ ║ X = eXit CAPQUANT ║ ║ Choice --> M ║ ╚═════════════════════════════╝ ╔═════════════════════ Capability Monitoring ════════════════════╗ ║ ║ ║ Display Historical Capability in R = Raw ...or ║ ║ S = Smoothed Form ? ║ ║ ║ ║ Press the R or S key now --> R ║ ║ ║ ║ Will NEW Process Data be Input via K = Keyboard ? ...or ║ ║ B = Batch File ? ║ ║ ║ ║ Press the K or B key now --> B ║ ║ ║ ║ Batch Input of New Data Selected... ║ ║ ║ ║ Specify filename for New Data Batch Input [NORMAL] : ║ ║ ║ ║ The New Data Batch Input file is to be: NORMAL.new ║ ║ Pause at each New Data Value [Y|n] : ║ ╚════════════════════════════════════════════════════════════════╝ NOTE: Because the NORMAL.NEW file contains 200 observations, you should probably either respond 'N' to the final prompt above or else press-and-hold-down the ENTER key when the Capability Monitoring graphics screen appears... CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 17 of 39 ╔═══════════════════ CapQuant Regret Function ═══════════════════╗ ║ ║ ║ Regret Type: QUADRATC ║ ║ ║ ║ Variables Measurement --> ║ ║ * | * ║ ║ * | * ║ ║ * | * ║ ║ ** | ** ║ ║ ** | ** ║ ║ ** | *** ║ ║ *** | *** ║ ║ **** | **** ║ ║ Regret = 0 ------------------*******--------------------- ║ ║ | ║ ║ Target ║ ║ ║ ║ REGRET = ( Value - Target )^2 ║ ║ ║ ║ TARGET = Value corresponding to ZERO REGRET = 20.0000 ║ ║ Press a Key to Continue... ║ ╚════════════════════════════════════════════════════════════════╝ ╔═════════════════════ Capability Monitoring ════════════════════╗ ║ ║ ║ Show Cumulative Distributions as Stair-Steps ? [Y|n] :n ║ ║ ║ ║ Between data-entry prompts, press F1 or Space Bar for HELP. ║ ║ Press the V key to interactively View numerical Values. ║ ║ ║ ║ Press a Key to Start Capability Monitoring... ║ ╚════════════════════════════════════════════════════════════════╝ The 300 "new" process observations stored in the NORMAL.NEW datafile start out looking good. Unfortunately, they end up showing that performance is actually deteriorating somewhat! Try it and SEE! CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 18 of 39 When you do eXit from CapQuant, you screen will display... ╔═════════════════════════════╗ ║ Process Capability MENU ║ ║ R = Regret Analysis ║ ║ D = Display CC Curves ║ ║ C = CC Confidence Limits║ ║ M = Monitor Capability ║ ║ X = eXit CAPQUANT ║ ║ Choice --> X ║ ╚═════════════════════════════╝ REMINDER(S): CAPQUANT created a Regret Calcs file... NORMAL.xxx You may edit this file and change its filename extension to .REG to use Automatic Regret Calculations in future CAPQUANT runs with Batch History Input from MORMAL.hst. CAPQUANT created an Output File named... NORMAL.out Use the DOS invocation... TYPE NORMAL.out | MORE to review detailed computational results from ┌───────────────────────────────┐ │ Preserving Time-Series Order │ │ & Forming "Moving" Statistics │ └───────────────────────────────┘ The primary statistical methodology implemented in CapQuant involves study of the "marginal distribution" of regret indices...i.e. the distribution which results from discarding all information about the time-series order in which values were actually observed. Therefore, CapQuant displays screen messages such as... ┌─────────────────────────────────────────────────────────────────┐ │ Historical Data, in any sequence, are to be entered │ │ in the Next Window... Press Q to QUIT now... │ └─────────────────────────────────────────────────────────────────┘ On the other hand, there usually are very good reasons for keeping your data stored in their proper time-series order in History (.HST) and MostRecent (.NEW) Batch Input files. For example, CapQuant can be used to convert almost any kind of variables data into Equivalent Expectancies and Equivalent Defects (EEs and EDs). These statistics will be recorded in the same order in CapQuant .OUT files that they were input from .HST and .NEW files. So their proper time-series order is maintained if they are input in the desired order. In fact, it is quite simple and straight-forward for you (using your favorite personal computer editor or word processor) CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 19 of 39 to read-in a CapQuant .OUT file and create a file suitable for input to my Quality Trend Monitoring software system, QMPT.EXE (from the QMPTC.EXE archive.) Consider the following two segments of the NORMAL.OUT file, each of which contains 6 lines of data... ┌─────────────────────────────────────────────────────┐ │ Process Capability Calculations... │ │ exptc noncf var1 var2 index │ │ 0.49 1.69 9.00 0.00 3.46 │ │ 0.49 0.13 23.00 0.00 0.26 │ │ 0.49 2.37 33.00 0.00 4.83 │ │ 0.49 0.06 22.00 0.00 0.11 │ │ 0.49 0.22 16.00 0.00 0.46 │ │ 0.49 0.90 28.00 0.00 1.83 │ ├─────────────────────────────────────────────────────┤ │ Capability Monitoring Calculations... │ │ exptc noncf var1 var2 index │ │ 0.49 2.02 8.00 0.00 4.11 │ │ 0.49 2.02 8.00 0.00 4.11 │ │ 0.49 1.13 11.00 0.00 2.31 │ │ 0.49 0.00 20.00 0.00 0.00 │ │ 0.49 0.00 20.00 0.00 0.00 │ │ 0.49 0.35 25.00 0.00 0.71 │ └─────────────────────────────────────────────────────┘ CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 20 of 39 Using these segments to create input files for QMPT.EXE is a simple matter of adding 11 appropriate lines at the top of each file... ┌─────────────────────────────────────────────────────┐ │ │ │ I <-- I-plots │ │ 1989 <-- Year │ │ 6 <-- 6 Periods to Plot │ 1 2 3 4 5 6 <-- Labels for Periods │ Quadratic Regret Example <-- Header 1 │ │ Historical Normal Data <-- Header 2 │ │ Target at Thickness = 20 <-- Footer 1 │ │ Exp.Regret=35, Variance=2500 <-- Footer 2 │ │ 6 <-- QMP Window Width │ 1 <-- E-factor │ │ 0 <-- No Previous History │ 0.49 1.69 9.00 0.00 3.46 │ │ 0.49 0.13 23.00 0.00 0.26 │ │ 0.49 2.37 33.00 0.00 4.83 │ │ 0.49 0.06 22.00 0.00 0.11 │ │ 0.49 0.22 16.00 0.00 0.46 │ │ 0.49 0.90 28.00 0.00 1.83 │ │ │ ├─────────────────────────────────────────────────────┤ │ │ │ F <-- Forecast Plots │ │ 1991 <-- Year │ │ 6 <-- 6 Periods to Plot │ 1 2 3 4 5 6 <-- Labels for Periods │ Quadratic Regret Example <-- Header 1 │ │ Most Recent Data <-- Header 2 │ │ Target at Thickness = 20 <-- Footer 1 │ │ Exp.Regret=35, Variance=2500 <-- Footer 2 │ │ 6 <-- QMP Window Width │ 1 <-- E-factor │ │ 0 <-- No Previous History │ 0.49 2.02 8.00 0.00 4.11 │ │ 0.49 2.02 8.00 0.00 4.11 │ │ 0.49 1.13 11.00 0.00 2.31 │ │ 0.49 0.00 20.00 0.00 0.00 │ │ 0.49 0.00 20.00 0.00 0.00 │ │ 0.49 0.35 25.00 0.00 0.71 │ │ │ └─────────────────────────────────────────────────────┘ NOTE: QMPT.EXE actually reads only the FIRST 2 VALUES off of each of the lines transferred from the CapQuant Output file to the QMPT Input file. You could delete the last three numbers from each of these lines using your Editor/WordProcessor, but that is not really necessary. CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 21 of 39 The "Moving" Statistics options offered by CapQuant can only be used when you enter data in their proper time-series sequence... ┌───────────────────── CapQuant Parameters ────────────────────┐ │ │ │ Should "Moving" statistics be calculated ? │ │ 0 => No moving statistics... │ │ 1 => Form first differences... │ │ 2 => Exp.Wgt.Mov.Average... │ │ 3 => Form Deviations from EWMA... │ │ │ └─────────────────────────────────────────────────────────────────┘ These options are NOT available when pairs of data-values are entered. But, if single values are entered (again, in correct time-series order!), the values actually input can be (optionally) shifted into storage space reserved for Variable2, and a TRANSFORMATION of the original data can be retained as Variable1. The transformations offered by CapQuant involve first computing some sort of "Moving Average" for Variable1 over time and, then, either replacing Variable1 with this Moving Average or else replacing Variable1 by its deviation from that Moving Average. Option 1: First Differences... ========= Moving Average[T] = Previous Observation in the Series = var1[T-1] Transformation = Current Deviation from Moving Average = var1[T] - var1[T-1] Option 2: Exponentially Weighted Moving Average... ========= Short-Term-Memory Weight = Proportion between 0.1 and 0.9 EWMA[T] = Weight * var1[T-1] + (1.0-Weight) * EWMA[T-1] Transformation = Moving Average = EWMA[T] Option 3: EWMA Deviations... ========= Short-Term Memory Weight and EWMA[T] same as in Option 2. Transformation = Current Deviation from Moving Average = var1[T] - EWMA[T] NOTE ONE: Option 3 with Weight=1.0 would be the same as Option 1. NOTE TWO: Regret functions with Target Value at ZERO are frequently appropriate with Options 1 and 3 because these Options re-express Variable1 as a Deviation from its Moving Average. CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 22 of 39 ┌───────────────────── CapQuant Parameters ────────────────────┐ │ │ │ How much Short-Term-Memory should the EWMA have ? │ │ 0.1 => MINimum Weight on Most Recent Observation... │ │ 0.9 => MAXimum Weight on Most Recent Observation... │ │ │ │ NOTE: 1.0 => EWMA is the Most Recent Observation... │ │ │ │ How much weight ? [0.20] : │ │ Short-Term-Memory Weight = 0.20. │ └─────────────────────────────────────────────────────────────────┘ As the Short-Term-Memory Weight decreases (down from its maximum allowed value of 0.9) toward its minimum at 0.1, considerable SMOOTHING of the Variable1 data series can result. In other words, the EWMA series can have much smaller variance that the original series. ┌───────────────────── CapQuant Parameters ────────────────────┐ │ │ │ Is there a Start-Up Value for the First History Observation ? │ │ -999.0 => No Start-Up Available; Start at 2nd Observation. │ │ OTHER => Start at 1st Observation with this Start-Up Value.│ │ │ │ What Start-Up Value ? [-999.0] : │ │ Start-Up Value = -999.0000. │ │ │ └─────────────────────────────────────────────────────────────────┘ NOTE: The final value of the Moving Average for the Historical data from the NORMAL.HST file is automatically used as the Start-Up Value for Capability Monitoring with the data from the NORMAL.NEW file. ┌─────────────────────────┐ │ REGRET FUNCTIONS │ └─────────────────────────┘ The methodology employed in CapQuant requires a general "rule" for transforming every possible value for a variable (or a pair) into a corresponding REGRET value, a surrogate measure of Cost-of-Poor- Quality. In other words, we need to specify a "functional relationship." The variable may assume negative values, but the corresponding regret must be zero or positive. This usually means that the regret function is "non-linear" and that more than one value for the variable can yield the same value for the regret...i.e. regret may be a non-invertible function of its input(s). All that is really necessary is that the regret function provide some sort of common-sense "proxy" for the economic impact of deviations from intended target value(s). In CapQuant, you may select a Regret Function from any of ten general families: GOALPOST QUADRATC ABSVALUE ONESIDED BILINEAR ENDPOINT LOGISTIC INVTNORM RANGEMAX NONCEXPT CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 23 of 39 Each of these Regret Functions can have as many as three Parameters: TARGET (the value of the variable resulting to Zero-Regret), PARM1, and PARM2. REGRET Type: GOALPOST Description: Economic Impact is either 0 (negligible) or 1 (complete). A variable measurement outside of an acceptable range indicates a nonconformance (defect.) Variables Measurement --> | | Regret = 1 ************| |************* | | | | | | Regret = 0 ------------*********************------------ | | Lower Upper Bound Bound Parameters: No single TARGET value is specified. Regret is zero over the interval from PARM1 = Lower Bound to PARM2 = Upper Bound. REGRET Type: QUADRATC (Taguchi's "Loss-to-Society") Description: Economic Impact is assumed to be proportional to the SQUARE of the deviation of the measured value from its (zero nonconformance) TARGET value. * Variables Measurement --> * | * | * * | * ** | ** ** | ** *** | *** *** | *** Regret = 0 ------------------*******--------------------- | TARGET REGRET = ( Value - TARGET )^2 Parameters: TARGET = Value corresponding to ZERO REGRET. No PARM1 or PARM2 values need be specified. ------------------------------------------------------------------- CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 24 of 39 REGRET Type: ABSVALUE (...symmetric BILINEAR ) Description: Economic Impact is assumed to be directly proportional to the measured value. | Variables Measurement --> *** | *** *** | *** *** | *** *** | *** *** | *** *** | *** *** | *** *** | *** *** | *** Regret = 0 -------------------***------------------------------ | TARGET = 0 Parameters: No TARGET, PARM1 or PARM2 values need be specified. Restrictions: Variable measurements must be non-negative. TARGET is implicitly ZERO. REGRET Type: ONESIDED Description: Economic Impact is ZERO on one side of the TARGET and proportional to the SQUARE of the deviation from TARGET on the other side. Variables Measurement --> * | | * | PARM1 = +1 * | ** | Zero Regret on ** | Right Side. *** | *** | Regret = 0 ------------------********************************** | TARGET Parameters: PARM1 is a LEFT(-1) or RIGHT(+1) indicator. PARM1 = -1 indicates zero regret left of TARGET. No PARM2 value need be specified. ----------------------------------------------------------------- CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 25 of 39 REGRET Type: BILINEAR (...asymmetric ABSVALUE) Description: Two variables measurements are required, a DEMAND level & an INVENTORY level for a perishable product. Economic Impact is proportional to production cost per unit when inventory > demand, but there is a Profit (Opportunity) Loss per unit when demand > inventory. **** Demand --> **** ** **** ** **** ** **** | ** **** | ** **** | ** **** |** Regret = 0 ------------------------****----------------------- | Inventory Level Parameters: PARM1 = Regret per Unit of Excess Inventory. PARM2 = Regret per Unit of Inventory Shortage. No TARGET value need be specified. REGRET Type: ENDPOINT (quadratic) Description: Two variables measurements are required, the MINIMUM and MAXIMUM observed values in a random sample of fixed size (say, 10 measurements.) Economic Impact is assumed to be the SUM of the QUADRATC regrets at the min. and max. values. *---------------| * |* | * | ** | ** | ** | ** | *** |-------*** | *** | ***| Regret = 0 -----+-----------*********---+---------------- MIN | MAX | TARGET | REGRET = ( MIN - TARGET )^2 + ( MAX - TARGET )^2 Parameters: TARGET = Value corresponding to ZERO REGRET. No PARM1 or PARM2 values need be specified. --------------------------------------------------------------------- CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 26 of 39 REGRET Type: LOGISTIC Description: Economic Impact follows a double-logit pattern that is V-shaped at the target but flat like GOALPOST in the tails. Measurement--> Regret = 1 ------------|---------|---------|------------ * | | | * * | | | * * | | | * |* | *| | * | * | | * | * | | * | * | | * | * | | *|* | Regret = 0 ----------------------*---------------------- | Target | Target-PARM1 Target+PARM1 REGRET = ( DEVIATION ) / ( PARM1 + DEVIATION ) Parameters: TARGET = Value corresponding to ZERO REGRET. PARM1 = DEVIATION from TARGET with regret = half MAX. REGRET Type: INVTNORM Description: Economic Impact follows an inverted bell-shaped pattern that is quadratic at the target but flat like GOALPOST in the tails. Measurement--> Regret = 1 ------------|---------|---------|------------ * | | | * * | | | * * | | | * *| | |* |* | *| | * | * | | * | * | | * | * | | * | * | Regret = 0 ---------------------***--------------------- | Target | Target-PARM1 Target+PARM1 REGRET = 1.0 - 0.5 to the power [square(DEVIATION/PARM1)] Parameters: TARGET = Value corresponding to ZERO REGRET. PARM1 = DEVIATION from TARGET with regret = half MAX. --------------------------------------------------------------------- CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 27 of 39 REGRET Type: RANGEMAX (...linear in Max - Min ) Description: Two variables measurements are required, the MINIMUM and MAXIMUM observed values in a random sample of fixed size (say, 10 measurements.) Economic Impact is assumed to be proportional to the RANGE from the min. to the max. values. | Range = ( Max. - Min. ) --> | *** | *** | *** | *** | *** | *** | *** | *** | *** Regret = 0 ---------***------------------------------ | TARGET = 0 Parameters: No TARGET, PARM1 or PARM2 values need be specified. Restrictions: Range must be non-negative; TARGET is implicitly ZERO. REGRET Type: NONCEXPT Description: Two variables measurements are required, the EQUIVALENT NONCONFORMANCES and the EQUIVALENT EXPECTANCY for each observation (production lot, period, etc.) With input data of this type, no Regret Function is actually specified. Similarly, no Regret Mean or Variance are needed to Poissonize the regret. Parameters: No TARGET, PARM1 or PARM2 values need be specified. --------------------------------------------------------------------- Equivalent Nonconformance & Expectancy The process of converting Regret into "Equivalent Nonconformances" and "Equivalent Expectancy" is called "Poissonization"... REGRET = Long Range Economic Impact and/or Customer/Regulatory Dissatisfaction associated with Deviations from intended Target Value(s). EXPREGRET = Expected Value of REGRET within the Historical Distribution of Process Capability. CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 28 of 39 VARREGRET = Variance of REGRET within the Historical Distribution of Process Capability. EN = Equivalent Nonconformances = REGRET * EXPREGRET / VARREGRET. EE = Equivalent Expectancy = EXPREGRET * EXPREGRET / VARREGRET. Note, in particular, that this conversion process involves nothing more than a simple "Change-of-Scale." REGRET is multiplied by the EXPREGRET / VARREGRET ratio to compute Equivalent Nonconformances, EN. Multiplication by this constant value assures that the mean and variance of EN are equal (to EE = Equivalent Expectancy.) This equality of mean and variance is an essential characteristic of Poisson distributions. The two key parameters needed to establish a successful application of variables data to regret quantification are thus the expected value (EXPREGRET) and the variance of the regret (VARREGRET) at standard quality. Many different shapes for regret functions might be appropriate for any particular variable in any particular production process. The exact form of the regret function chosen is sometimes not terribly important. As a result, even after the regret function and all of its describing parameters (TARGET, PARM1, and PARM2) have been specified, the critical final phase of customizing a CPQ application still remains...choice of numerical values for EXPREGRET and VARREGRET. While it might be possible to specify the distribution of regret at standard quality using only technical assumptions and theoretical calculations, the more standard approach is an empirical one in which actual data on the operation of the process are converted to regrets, and the statistical distribution of those sample regrets is then studied. If one has data on only a dozen or so lots, it could be misleading to use the observed sample values of the mean and variance of regret as EXPREGRET and VARREGRET, respectively, because small sample estimates are imprecise. On the other hand, even if one had data for hundreds of lots, it could still be misleading to use the observed sample mean and variance of regret to establish CapQuant parameter values because even large sample results can be biased. There could be several different causes of bias. The data at hand may not represent the results of the process operating at (or even near) an industry-wide "standard" quality level. The history being used might, in fact, reflect an operation in which quality is either well above or well below standard. Modern, new manufacturing equipment (using, say, robotics) or a change in ingredient formulations may have been introduced without changing the quality standard that was barely within the capability of the old technology. If you aren't an expert on the operation of the manufacturing process in question, consult with an expert. Develop opinions about the quality level of the process as reflected in its history. Ask CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 29 of 39 yourself, "How did I react to these quality monitoring results when I first saw them?" (Or, "How would I have reacted if I had been the manager in charge at that time?") Were you ever concerned that the process might be out of "statistical control?" Did you ever become alarmed because rejection rates jumped at final inspection or because in-process yield dropped? Might you have reacted quite differently if the data had been only slightly different? Or, would your reactions have been basically the same even if the data had shifted even more than it did in level and/or were even more highly variable from lot-to-lot? A common situation is that all the observed history reflects data taken from the process when it was operating well within its current process capability. In other words, the variable could have deviated much more from its target without management having been alarmed at all! This means that the values for EXPREGRET and VARREGRET that should be specified to CapQuant will indeed differ from what is observed in the history. General rules-of-thumb for this well- within-capability situation are... * the EXPREGRET parameter should usually be chosen to be rather close to the observed value of the sample mean regret if the history that is available consists of 50 or more observations. * the VARREGRET parameter, on the other hand, might be selected to be 10 or even 100 times larger than the observed regret variance of the available data. (Note: This corresponds to a "Standard Error" value between 3 and 10 times larger than that actually observed.) There are two primary reasons for using a rather large VARREGRET value. Again, the history data may be very well behaved, and management would not have been alarmed even if the deviations of measurements from the target had been much larger than observed. The second reason for using a large variance is that variables data are very rich in information. (Variables data are usually much more informative than nonconformance data.) ┌──────────────────────────────────────────────────────────────┐ │ NOTE: CapQuant illustrates accumulation of results across │ │ variables, across products or over time to form either │ │ COMPOSITE statistics or reporting-period summaries. Due to │ │ "Poissonization", composite Equivalent Nonconformances and │ │ Expectancies result simply by SUMMING their components over │ │ all relevant variables, periods, or processes. │ └──────────────────────────────────────────────────────────────┘ CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 30 of 39 The CapQuant philosophy is to fine-tune parameter values by making repeated "iterations" as follows: (i) The general form and specific parameter values of the Regret Function are set using item "S" on the Regret Menu. (ii) Using item "I" on the Regret Menu, the sample mean and variance of regret are displayed and you are prompted to enter values for EXPREGRET and VARREGRET. As explained above, you may wish to consider using EXPREGRET and VARREGRET values that are not very close, numerically, to the values in your sample. (iii) Goodness-of-Fit of sample results to a Poisson distribution (with "intensity" EXPREGRET*EXPREGRET/VARREGRET) can then be checked using a Kolmogorov-Smirnov test (item "L" on the Regret Menu.) Statistically significant Lack-of-Fit is NOT fatal. On the other hand, insignificance of distributional differences is always a welcome indication that your choices are supported by the available data. [Of course, you could return to step (ii) now to see if different values of EXPREGRET and VARREGRET reduce Lack-of-Fit.] (iv) Plot a detailed Cumulative Capability Curve using item "C" on the Main MENU. Study these results, using the on-screen cursor activated by the "V" key, and determine whether your results are consistent with the opinions of local "experts." (v) If you are unhappy with the outcome of step (iv), you should either (1st choice): return to step (ii) and try out different values of EXPREGRET and VARREGRET or (2nd choice): return to step (i) and try different regret function parameters or even a different regret function family. Once you are happy with the outcome of step (iv), the modeling process is complete. REFERENCES AN INTRODUCTION TO THE QUALITY MEASUREMENT PLAN. (1980.) Western Electric Company, Inc., Quality Assurance, September 1980, 16 page booklet. CHAMBERS, J. M., CLEVELAND, W. S., KLEINER, B., and TUKEY, P. A. (1983). Graphical Methods for Data Analysis, Monterey, CA: Wadsworth. CONOVER, W. J. (1980). Practical Nonparametric Statistics, 2nd. Ed., Wiley [Kolmogorov-Smirnov table and approximation, page 462.] CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 31 of 39 GUNTER, Bert. (1991a,b,c,d,e). "Process Capability Studies:"...parts 1 through 5. Statistics Corner, Quality Progress, Volume 24 (February, April, June, August, and October Issues, 1991.) HASTINGS, C., Jr. (1955). Approximations for Digital Computers. Princeton, NJ: Princeton University Press. HOADLEY, Bruce. (1978). "Equivalent Defects: A Unified Approach to Attributes Quality Rating." Bell Telephone Laboartories, Quality Assurance Center, Technical Memorandum, November 4. HOADLEY, Bruce. (1981). "The Quality Measurement Plan, QMP." Bell System Technical Journal, 60, 215-273. HOADLEY, Bruce. (1986). "QUALITY MEASUREMENT PLAN(QMP)." Encyclopedia of Statistical Sciences. (Kotz, Johnson and Read, Editors) Volume 7, pages 393-398. New York, John Wiley. INTRODUCTION TO QUALITY TREND CHARTS. Bell Communications Research, Inc. (Bellcore), January 1986, 18 page booklet. JOHNSON, N. L. and KOTZ, S. (1969). Distributions in Statistics: Discrete Distributions. [Chapter 4: Poisson Distribution] New York, NY: John Wiley. JOHNSON, N. L. and KOTZ, S. (1970). Distributions in Statistics: Continuous Univariate Distributions - 1. [Chapter 17: Gamma Distribution.] New York, NY: John Wiley. LINK, P. A., and RIPACKI, B. R., "Quality Assurance Approach to Fiber Optic Cable," International Wire and Cable Symposium Proceedings, 1985. MONTGOMERY, D. (1985). Introduction to Statistical Quality Control. New York, NY: John Wiley. NAIR, V. N. and FREENY, A. E. (1993). "Methods for Assessing Distributional Assumptions in One and Two Sample Problems." Probabilistic and Statistical Methods in the Physical Sciences. (J. Stanford and S. Vardeman, editors.) San Diego, CA: Academic Press. NIEDERHAUSEN, H. (1981). "Tables and Significant Points for the Variance Weighted Kolmogorov-Smirnov Statistic." Technical Report 298, Department of Statistics, Stanford University. OBENCHAIN, Robert. (1991). "Regret Indices and the Quantification of Process Capability." Indianapolis, IN: Lilly Research Laboratories. (29 pages.) OBENCHAIN, Robert. (1993). "Cumulative Capability Curves." Indianapolis, IN: Lilly Research Laboratories. (49 pages.) CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 32 of 39 QUALITY MEASUREMENT PLAN (QMP). Bell Communications Research Technical Reference, TR-TSY-000438. Issue 1, April 1987. SPIRING, Fred. (1991). "The Inverse Normal Loss Function." Unpublished manuscript, University of Manitoba. STEPHENS, M. A. (1986). "Tests based upon EDF Statistics." Goodness- of-Fit Techniques, Chapter 4. (R. B. D'Agostino and M. A. Stephens, editors.) New York, NY: Marcel Dekker. WILSON, E. B. and HILFERTY, M. M. (1931). "The Distribution of Chi- Square." Proceedings of the National Academy of Sciences 17, 684-688. --------------------------------------------------------------------- CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 33 of 39 APPENDIX ONE Poissonization: Conversion of Economic Impact Measures into Equivalent Nonconformances and Equivalent Expectancy. Section A.1 Nonconformance Expectancy Attributes Data Definition: Nonconformance Expectancy is the product of the Sample Size times the Quality Standard (expressed in expected nonconformances per unit.) Example: When the quality standard is 0.004 nonconformances per unit, one needs to inspect 250 units in order to expect to see 1 nonconformance when quality is at the standard level. Information Content: Under a Binomial model, the "moments" of the sampling distribution would be given by the familiar expressions: mean = n*p and variance = n*p*(1-p). QMP (the Quality Measurement Plan) is based upon the Poisson (maximum variance) model with mean = variance = n*p, which is the nonconformance expectancy. Expectancy is the primary measure of "information content" used in QMP and CPQ for sampling inspection data. [See Section A.3, below, for details on information content.] Variables Data Definition: Equivalent nonconformances and equivalent expectancy can be defined by "Poissonizing" any continuous measure of regret, R. [See Section A.2, below, for details on poissonization.] Reporting Experience with use of QMP at both Bell Communications Period: Research (Bellcore) and at AT&T Bell Laboratories suggests the following "Rule-of-Thumb" for aggregating sampling inspection data over "reporting periods" (usually either the 12 calendar months or 8-per-year): ┌─────────────────────────────────────────────────────────────┐ │ An effective quality monitoring program requires sufficient │ │ sampling inspection to produce a nonconformance expectancy │ │ of at least 2 nonconformances per reporting period. │ └─────────────────────────────────────────────────────────────┘ CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 34 of 39 If production is so low or nonconformances are so rare that one would have to inspect more than 20% of total production to achieve a nonconformance expectancy of at least 2 nonconformances per reporting period, then "quantitative" quality monitoring is probably impractical. (After all, QMP is the "most powerful" statistical method currently available; if QMP can't get the monitoring job done, what method can?) Section A.2 Equivalent Nonconformances & Expectancy Definition: Equivalent nonconformances and expectancy can be defined by "Poissonizing" a continuous measure of regret, R. For example, if E is the expected value of R and V is the variance of R, then the number of "equivalent nonconformances" is said to be R*E/V, which is usually not an integer value. But the mean and variance of R*E/V are both equal to the "equivalent expectancy" of E*E/V...and equality of mean and variance is an intrinsic property of all Poisson distributions. Example: Suppose a measurement produces a continuous variable, Z, that has (approximately) a "normal" distribution with mean = target value = T and variance = D, say, when quality is "at standard." Consider the "Taguchi- like" measure of regret given by the sum-of-squares of N deviations of independent Z-values from the target, T, all divided by D. Then normal distribution theory implies that E(R) = N and V(R) = 2N. It follows that the number of "equivalent nonconformances" is R/2, and the "equivalent expectancy" is N/2. (Note that the sample index value = nonconformances/expectancy is R/N; the average squared deviation from target, scaled by dividing by D.) "Power" of Variables Data: Variables data can be incredibly "rich" in sample information about quality compared with attributes data. The above "normal theory" example shows that a single continuous observation can yield a nonconformance expectancy of one-half! And the "uniform" distribution would yield an even higher nonconformance expectancy of .66 per observation with the above "Taguchi-like" R measure! Do-It-Right from the START: Quality engineers should consult a professional statistician if there is any doubt in their minds about how to implement the above basic concepts. One CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 35 of 39 needs to utilize a realistic "Regret Function," which might need to be one-sided (rather than symmetric about the target) or linear or saturating (rather than always quadratically increasing.) The calculation of the mean and variance of the regret depends upon the statistical distribution of test results when the manufacturing process is within its historical "process capability." Theoretical calculations can be complicated or require numerical integration; empirical studies of historical results are generally more straight-forward and more satisfactory. Section A.3 Expectancy and Information Contents of This Section: Two fundamental concepts are established by the highly technical arguments of this section. ...The information content in a set of regret measurements is directly proportional to the Poissonized "nonconformance expectancy" of the regrets at standard process capability. ...The sample regret index (ratio of observed to expected nonconformances) is the minimum variance, unbiased estimate of the true index. Minimum Variance Bound: If a sample statistic, tau, is an unbiased estimate of a parameter, theta, of the statistical distribution of the observed data vector, x, then the so-called "Cramer-Rao- Aitken-Silverstone Inequality" states that: ┌──────────────────────────────────────────────────┐ │ the product of the variance of tau times the │ │ --------------- │ │ Fisherian Information in the data must be >= 1, │ │ --------- ----------- │ └──────────────────────────────────────────────────┘ where this "information" is defined to be the expected value of the square of the derivative of the log- likelihood function of the data with respect to the parameter theta. Furthermore, equality holds if and only if the derivative of the log-likelihood with respect to theta is of the form: CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 36 of 39 A(theta) * (tau-theta), in which case, tau is the minimum variance estimate of theta, and A(theta) is the Fisherian Information. Poisson Example: If the observed numbers of nonconformances, { x(1),..., x(n) }, are stochastically independent and each follows a Poisson distribution with intensity parameter lambda(i), then the total number of observed nonconformances is also Poisson distributed with intensity parameter equal to the sum of the individual lambdas. This total intensity can be re-expressed as: lambda(1)+...+lambda(n) = (nonconformance expectancy) * theta where theta is the true regret index (total lambda divided by the "standard" nonconformance intensity) and expectancy is n times the standard intensity. As a result the log-likelihood function is [ x(1)+...+x(n) ] * log(theta) - expectancy * theta + terms that do not depend on theta. The theta-derivative of the log-likelihood is thus: ┌─────────────────────────────────────────────┐ │ expectancy x(1)+...+x(n) │ │ ---------- ( ------------- - theta ) │ │ theta expectancy │ └─────────────────────────────────────────────┘ Thus two basic facts are established by the C-R-A-S inequality... ...the sample index (observed / expected nonconformance ratio) is the minimum variance estimate of the true regret index, theta, and ...the (Fisherian) information content of the data is A(theta) = expectancy / theta, which is directly proportional to the nonconformance expectancy at standard process capability. Section A.4 Scale Invariance Definition: Equivalent nonconformances and expectancy are "scale invariant" quantities in the following sense: multiplying the regret by any strictly positive, constant value does not change the implied equivalent CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 37 of 39 nonconformances and expectancy. One implication of this is that it doesn't make any difference whether regret is expressed in terms of dollars or pennies or pounds sterling. Proof: Suppose that a regret measure, R, is multiplied by any constant value, K. Then the expected value of K times R would be K times E, the original expected value of R. And the variance of K times R would be K-squared times V, the original variance of R. As a result, multiplicative factors of K-squared cancel from both the numerators and the denominators of the ratios, R*E/V and E*E/V, that define equivalent nonconformances and expectancy. Specific Implications for the Ten Regret Families: GOALPOST.....Max Impact is unitless; call it ONE or LOGISTIC use any convenient value. INVTNORM LINEREGRET...The slope of the regret line is unimportant. RANGREGRET QUADRATC.....Regrets that are squared deviations from ENDPOINT target values can be measured in any ONESIDED convenient units. BILINEAR.....Either the Inventory Shortage or the Excess Inventory Regret Slope parameter can be set equal to 1 if the second Slope is expressed as a MULTIPLE of the first. CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 38 of 39 Section A.5 Summary ┌───────────────────────────────────────────────────────────────────┐ │ │ │ Knowledge of the statistical distribution of Regret when Quality │ │ is at a STANDARD LEVEL (that is within the Present Capability │ │ of the Manufacturing Process) is needed for Poissonization. │ │ │ │ The two statistical parameters which must be specified are: │ │ EXPREGRET = Expected Regret at Standard Quality, │ │ VARREGRET = Regret Variance at Standard Quality. │ │ │ │ Poissonization is accomplished by multiplying Regret by │ │ EXPREGRET and then dividing by VARREGRET... │ │ │ │ EQUIVALENT DEFECTS = REGRET * EXPREGRET / VARREGRET. │ │ │ │ Equivalent nonconformances have a statistical distribution that │ │ is like the Poisson in the sense that the mean and the variance │ │ of this distribution are EQUAL. This common value is... │ │ │ │ EQUIVALENT EXPECTANCY = EXPREGRET^2 / VARREGRET. │ │ │ └───────────────────────────────────────────────────────────────────┘ CapQuant Software Update History: ================================= Version 9009 ...Beta Test Version Version 9010 ...First Upload to CompuServe ...Fix "bugs" in User-Interface of Beta Test Version 9101 ...Update for Submission to Technometrics ...Pop-Up Batch-Input-File Selection Menu ...StreamLine of User-Interface / Default-Settings ...Second Upload to CompuServe Version 9107 ...Update & Submission to Journal of Quality Technology ...Add Inverse-Gaussian Regret Function Option ...Automatic Regret Calcs: Add Screen Pause & Rewinds Version 9110 ...Change titles on plots to "Cumulative Capability..." Version 9201 ...Improve staircase "smoothing" algorithm. Previous versions connected the points at the top/front edge of each step; present version reduces perception bias by connecting the mid-points of the vertical-rise portions. In the diagram below, the points numbered (2) are thus connected instead of those labeled (1): CapQuant [Version 9311] . . . . . . . . . . . . . . . Page 39 of 39 (1)─── │ (2) │ (1)─────────┘ │ (2) │ (1)──────────┘ │ When a Poisson distribution is fitted to an empirical cumulative capability curve, the horizontal spacing of the fitted steps will be exactly uniform. When equivalent expectancy is small, the fitted steps can be quite LARGE. With the old algorithm, one might get the false impression that the fitted capability is always "higher" than the observed capability. Version 9202 ...List/Save only those numerical values from cumulative distributions at which observed/fitted values CHANGE. ...Reduce CapQuant maximum capacity from 2,500 history values to 1,200; this makes CapQuant will much less of a "memory hog." ...Add capability to form COMPOSITES of 2 or more observed regret values. CapQuant now shows how accumulation of expectancy reduces Poisson step-size & can greatly reduce Kolmogorov-Smirnov lack-of-fit. ...Add capability to form 1st Differences, Exp.Wgt.Mov. Averages (EWMAs), or Deviations from an EWMA when data are input in time-series order. Version 9302 ...Add calculation and display of CC curve confidence limits (lower 95% & upper 95% limits => central 90% interval.) ...Add calculation and display of Gamma Q-Q plots for estimation of Equivalent Expectancy. ...Combine items "I" and "L" on the REGRET Analysis MENU so that Kolmogorov-Smirnov lack-of-fit can be computed for several different ER and VR combinations before making the P-P plot for the Poisson approximation. ...Display variances rather than standard deviations on "Poissonization" screens. ...The ATTRIBUT regret function was renamed to GOALPOST. The BOUNDMAX regret function was renamed to LOGISTIC. The INVGAUSS regret function was renamed to INVTNORM. Version 9310 ...Add display of Summary Statistics (mean, std.dev., variance, min & max) and a 20-line Histogram of historical data (Var1 and, if present, Var2.) Version 9311 ...Provide option to display standardized Gamma fit (instead of Poisson fit) to the empirical CC curve.