The Not-So-Geeky World of Statistics

Transcription

1 FEBRUARY 3 5, 2015 / THE HILTON NEW YORK The Not-So-Geeky World of Statistics Chris Emerson Chris Sweet (a/k/a Chris 2 )

2 2 Who We Are Chris Sweet JPMorgan Chase VP, Outside Counsel & Engagement Management Chris Emerson Bryan Cave Director, Practice Economics

3 3 Backdrop Problem Statement GOAL: How long will it take to resolve these cases, and how much will it cost? Litigation Cases in Some State 206 Open Cases from Closed Cases pre Projected New Cases in July 2014 Dec attorneys available from various law firms

4 4 What to Expect Turning quantitative data into information Shape of the data Center of the data Spread of the data 3-Point Estimating Linear Programming in Excel Case study on understanding effort and expense in portfolios Going beyond the basics

5 5 Turning Quantitative Data Into Information Shape of the data (Distribution) Center of the data (Average) Spread of the data (Variability)

6 6 Example: Matter Hours Data For similar types of matters

7 FREQUENCY 7 Hours Data: Histogram A histogram is a graphical representation of the frequency in which ranges of measures occur MATTER HOURS

8 8 Histogram: How To

9 9 Shape of the Data Symmetric bell shaped other symmetric shapes Asymmetric skewed to the right skewed to the left Bimodal

10 10 Symmetric Distributions Bell-Shaped

11 11 Symmetric Distributions Mound-Shaped

12 12 Symmetric Distributions Uniform

13 13 Asymmetric Distributions Skewed to the Left

14 14 Asymmetric Distributions Skewed to the Right

15 15 Gamma Distribution Skewed to the Right

16 16 Bimodal

17 17 Shape Takeaways Helps you understand where your matters fall Could show how likely / unlikely certain circumstances are Bet the farm litigation Protracted rounds of negotiation Potentially useful for evaluating confidence levels Potentially useful for probability estimation

18 18 Outliers (The Tail) Extreme values, far from the rest of the data. May occur naturally May occur due to error in recording May occur due to error in measuring Observational unit may be fundamentally different

19 Example: Number of Timekeepers on Your Matters Number of Timekeepers

20 20 Center of the Data (Average) Mean Median Mode

21 Average or Mean ( ) 21 X Traditional measure of center Sum the values and divide by the number of values Example: Mean of the matter hours is Common Applications Average hours on a matter Average duration of a matter Highly influenced by extreme values (esp. outliers)

22 22 Median (M) A measure of the data s center, resistant to excess variation and outliers At least half of the ordered values are less than or equal to the median value At least half of the ordered values are greater than or equal to the median value Example: The median of the matter is hours is 165. Common Applications More useful than mean when looking at hours, duration, etc. in skewed data (remember from shape of data) In a normal distribution, the mean and median are roughly identical

23 23 Mode Measured as most frequently occurring number in a data set. It will show in a histogram is the highest column (most frequency) It can reflect the most common outcome Example: The mode of the matter hours is 180. Common Applications Mode is not commonly used other than to identify the most common outcome

24 24 Spread of the Data (Variability) If all values are the same, then they all equal the mean. There is no spread (variability). Variability exists when some values are different from (above or below) the mean. We will consider the following measures of spread: variance, standard deviation, and inter-quartile range

25 25 Variance and Standard Deviation When variability exists, each data value has an associated deviation from the mean: x xi What is a typical deviation from the mean? (standard deviation) Small values of this typical deviation indicate small spread in the data Large values of this typical deviation indicate large spread in the data

26 26 Standard Deviation typical deviation from the mean

27 27 Quartiles Three numbers that divide the ordered data into four equal sized groups. Q 1 has 25% of the data below it. Q 2 has 50% of the data below it. (Median) Q 3 has 75% of the data below it. Chapter 12

28 28 Five-Number Summary minimum = 100 Q 1 = Q 2 or M = 165 Q 3 = 185 maximum = 260 Inter-quartile Range (IQR) = Q 3 Q 1 = 57.5 IQR gives spread of middle 50% of the data ( middle 50% of data ranges from Q 1 to Q 3 )

29 29 Visual of 5-Number Summary min Q 1 M Q 3 max Matter Hours Boxplot of Matter Hours Data

30 30 Choosing a Summary Outliers affect the values of the mean and standard deviation. The five-number summary should be used to describe center and spread for skewed distributions, or when outliers are present. Use the mean and standard deviation for reasonably symmetric distributions that are free of outliers.

31 31 With the Mean and Standard Deviation of the Normal Distribution We Can Determine: What proportion of matters fall into any range of values At what percentile a given matter falls, if you know its value What value corresponds to a given percentile Because the area under a distribution equals 100%

32 32 PERT 3-Point Estimating Program Evaluation and Review Technique Pessimistic: 16 Hours Most likely: 8 Hours Optimistic: 4 Hours Expected Time Pessimistic + 4 Most Likely + Optimistic = 8.67 Hours Estimating technique that uses a weighted average of 3 numbers to come up with an estimate.

33 33 PERT Standard Deviation Measures the volatility of the estimate Std. Deviation Pessimistic Optimistic = 2 68% Confidence: /- 2 = Hours 95% Confidence: /- 4 = Hours 99.7% Confidence: /- 6 = Hours The larger the std. deviation, the less confidence you have in your estimate.

34 34 Linear Programming Linear programming (also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming.

35 35

36 The TBA Airlines Problem TBA Airlines is a small regional company that specializes in short flights in small airplanes. The company has been doing well and has decided to expand its operations. The basic issue facing management is whether to purchase more small airplanes to add some new short flights, or start moving into the national market by purchasing some large airplanes, or both. Question: How many airplanes of each type should be purchased to maximize their total net annual profit?

37 Data for the TBA Airlines Problem Small Airplane Large Airplane Net annual profit per airplane $1 million $5 million Capital Available Purchase cost per airplane $5 million $50 million $100 million Maximum purchase quantity 2

38 Linear Programming Formulation Let S = Number of small airplanes to purchase L = Number of large airplanes to purchase Maximize Profit = S + 5L ($millions) subject to Capital Available: 5S + 50L 100 ($millions) Max Small Planes: S 2 and S 0, L 0 S, L are integers.

39 Spreadsheet Model Small Airplane Large Airplane Unit Profit ($millions) 1 5 Capital Capital Capital Per Unit Produced Spent Available Capital ($millions) <= 100 Total Profit Small Airplane Large Airplane ($millions) Units Produced <= Maximum Small Airplanes 2 Capital Spent: =SUMPRODUCT(CapitalPerUnitProduced,UnitsProduced) Total Profit: =SUMPRODUCT(UnitProfit,UnitsProduced)

40 40 Excel s Solver

41 Spreadsheet Model Small Airplane Large Airplane Unit Profit ($millions) 1 5 Capital Capital Capital Per Unit Produced Spent Available Capital ($millions) <= 100 Total Profit Small Airplane Large Airplane ($millions) Units Produced <= Maximum Small Airplanes 2 Capital Spent: =SUMPRODUCT(CapitalPerUnitProduced,UnitsProduced) Total Profit: =SUMPRODUCT(UnitProfit,UnitsProduced)

42 42 Case Study: Predicting Portfolio Litigation Reserve Balances Case Study using Statistics, PERT, & Linear Programming

43 43 Backdrop Problem Statement GOAL: How long will it take to resolve these cases, and how much will it cost? Litigation Cases in Some State 206 Open Cases from Closed Cases pre Projected New Cases in July 2014 Dec attorneys available from various law firms

44 44 Parameters Inbound cases Case life Hours Needed per month Hours available per month Attorney Rate Attorney Efficiency Level

45 45 Outside Counsel Parameters 9 Choices with different efficiency levels Efficiency Levels Rate Rate for lawyers 3% anticipated annual rate increase

46 46 Solution Methods Gamma Probability-based Predictions PERT 3-Point Estimating Linear programming model

47 Frequency 47 Inbound Cases June Case Arrival by Month Months ARRIVAL PERCENTAGES BY MONTH Oct 9% Nov 7% Dec 9% Jan 6% Feb 7% Mar 11% Sep 8% Apr 8% Aug 8% Jul 7% Jun 9% May 11%

48 48 Inbound Case Projection, July

49 FREQUENCY CUMULATIVE FREQUENCY Histogram & Gamma Fit 9% 8% 7% 6% 5% 4% 3% 2% 1% 0% MONTHS OF LIFE This distribution tells us the probability of a given matter closing within a certain amount of time. Frequency Gamma PDF Cumulative % Gamma CDF 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0%

50 FREQUENCY Cases in Quarters 2011 Histogram & Gamma Fit 25% 100% 90% 20% 80% 70% 15% 60% 50% 10% 40% 30% 5% 20% 0% MONTHS OF LIFE Frequency Gamma PDF Cumulative % Gamma CDF 10% 0%

51 51 Probability-Based Prediction Model Gamma Distribution Bound on the left at zero Right skewed Excel does most of this for you! Probability Density Function Excel: gamma.dist function Formula for Gamma Mean μ[x] = α β Cumulative Density Function Excel: gamma.dist function Formula for Gamma Variance Var x = α β 2

52 52 How the Predictions Work Actual Gamma Distribution Month Case Count Cumulative % % PDF CDF Cases Closing % 0.96% 0.254% 0.08% % 1.28% 1.037% 0.69% % 1.60% 2.123% 2.25% % 1.28% 3.277% 4.96% % 3.19% 4.332% 8.77% % 4.15% 5.193% 13.56% % 6.39% 5.817% 19.08% % 6.07% 6.202% 25.11% % 7.67% 6.367% 31.41% % 7.03% 6.345% 37.78% % 5.75% 6.172% 44.05% % 5.43% 5.885% 50.09% % 6.07% 5.518% 55.79% total cases in 2011 LEGALTECH 313 NEW total YORK cases / in FEBRUARY

53 53 Estimating Effort (PERT) Based upon sample closed case population of 858 cases.

54 HOURS 54 Boxplot of Monthly Hours First Month Second Month Third Month Fourth Month

55 55 Estimating Effort (PERT) Based upon sample closed case population of 858 cases. Expected Value = Optimistic + 4 Most Likely + Pessimistic 6

56 56 Calculating Production Hours Needed

57 57 Monthly Hours Needed

58 58 Objective Function Minimize Z = Decision Variable: X am = binary decision variable; whether an attorney was chosen to work a particular month for attorney a, a=1, a=2, 9 Parameters: 9 a=1 y=1 m=1 m = a month between July 2014 and December 2020 for month m, m=1, m=2, 78 C = rate per attorney in a particular year 7 Y = year between 2014 and 2020 for year y, y=1, y=2, 7 H m = hours available for an attorney to work during a particular month 78 X am C ay H m

59 59 Constraints Total hours performed each month must meet monthly production hour requirement: 9 a=1 X am E a H m P m A terminated attorney will not be rehired: X am 1 X am 0

60 60 Linear Programming Model opensolver.org

61 61 So what s the answer? We should be down to one half-time lawyer by 12/2017 We have will have spent $5.8M to: Finish the 206 open cases Finish 284 of the new cases Have 22 cases left remaining (the tail) The remainder should complete within 6 12 months The Heisenberg Uncertainty Principle Applies Here

62 Process Improvement Enables visibility into process improvement By improving upon the number of hours worked by month against the historical baseline By resolving / concluding matters faster than the baseline

63 63 Questions For a copy of this PPT or the excel template, cemerson@bryancave.com

64 64 Additional Resources Anderson, David, Dennis Sweeney, Thomas Williams. Modern Business Statistics with Microsoft Office Excel, Fourth Edition. Mason: Cengage Learning, Document. Bishop, Christopher, and Michael Tipping. "Bayesian Regression and Classification." n.d. Microsoft Research. Web. 14 July Carrier, Richard. "Bayesian Calculator." Richard Carrier, Ph.D. Web. 14 July COIN OR. "Welcome to the Cbc home page." 1 July COIN OR. Web. 23 July "Does my data come from a gamma or beta distribution?" 11 May Cross Validated. Web. 16 July "Gamma distribution." 27 April Math Wiki. Web. 16 July Jacobs, Robert F. and Richard B. Chase. Operations and Supply Chain Management, Fourteenth Edition. New York: McGraw-Hill/Irwin, Document. "OpenSolver for Excel." 3 Jul OpenSolver.org. Web. 23 July Pasik-Duncan, Bozenna. "Example Distribution Generator." University of Kansas Mathematics Department. Web. 16 July Plymouth, University of. "Chapter 20 Probability." n.d. Centre for Innovation in Mathematics Teaching. Web. 15 July Quantitative Skills. "Distributions: Simple Interactive Statistical Analysis." n.d. Quantitative Skills. Web. 17 July Ricci, Vito. "Fitting Distributions with R." February The Comprehensive R Archive Network. Web. 16 July Stracener, Jerrell T. Ph.D. "Gamma Beta Distributions." SMU Lyle School of Engineering. Web. 17 July Technology, National Institute of Standards and. " Beta Distribution." n.d. Information Technology Laboratory. Web. 16 July "The Excel BETA.DIST Function." n.d. ExcelFunctions.net. Web. 16 July Wilmott Electronic Media Limited. "Gamma Distribution in Excel." 22 August Wilmott. Web. 16 July Wolfram Research. "Gamma Distribution." n.d. Wolfram MathWorld. Web. 16 July "Probability Density Function." n.d. Wolfram MathWorld. Web. 16 July Zaiontz, Charles. "Gamma Distribution." n.d. Real Statistics Using Excel. Web. 16 July 2014.

65 65 Going Beyond the Basics Correlation, Regression, Root Cause Analysis