Developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 91 Cost Risk Analysis How to adjust your estimate for historical cost growth.

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 92 Unit Index Unit I – Cost Estimating Unit II – Cost Analysis Techniques Unit III – Analytical Methods 6.Basic Data Analysis Principles 7.Learning Curves 8.Regression Analysis 9.Cost Risk Analysis 10.Probability and Statistics Unit IV – Specialized Costing Unit V – Management Applications

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 95 Overview Risk is a significant part of cost estimation and is used to allow for cost growth due to anticipatable and un-anticipatable causes There are several approaches to risk estimation Incorrect treatment of risk, while better than ignoring it, creates a false sense of security Risk is perhaps best understood through a detailed examination of an example method

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 96 Definitions Cost Growth: –Increase in cost of a system from inception to completion Cost Risk: –Predicted Cost Growth. In other words: Cost Growth = actuals Cost Risk = projections In other words: Cost Growth = actuals Cost Risk = projections

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 97 Types of Risk Cost Growth = Cost Estimating Growth + Sked/Tech Growth + Requirements Growth + Threat Growth Cost Risk = Cost Estimating Risk + Sked/Technical Risk + Requirements Risk + Threat Risk –Cost Estimating Risk: Risk due to cost estimating errors, and the statistical uncertainty in the estimate –Schedule/Technical Risk: Risk due to inability to conquer problems posed by the intended design in the current CARD or System Specifications –Requirements Risk: Risk resulting from an as-yet-unseen design shift from the current CARD or System Specifications arising due to shortfalls in the documents Due to the inability of the intended design to perform the (unchanged) intended mission We didn’t understand the solution –Threat Risk: Risk due to an unrevealed threat; e.g. shift from the current STAR or threat assessment The problem changed Often implicit or omitted 12

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 98 Basic Flow of the Risk Process Structure & Execution Includes the organization, the mathematical assumptions, and how the model runs Structure & Execution Includes the organization, the mathematical assumptions, and how the model runs InputsOutputs From the cost analyst and technical experts The CARD Expert rating/scoring Point Estimate To the decision maker and the cost analyst Means Standard Deviations Risk by CWBS Inputs and outputs, although outside the purview of the risk analyst, are determined by the structure and execution of the risk model

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 99 Engineers Work in physical materials, with –Physics-based responses –Physical connections Typically examine or discuss a specific outcome –System Parameters –Designs Typically seek to know: –Given this solution, what will go wrong? –Are my design margins enough? Engineers’ and Cost Analysts’ View of Risk Cost Analysts Work in dollars and parameters, with –Statistical relationships –Correlation Typically examine or discuss a general outcome set –Probability distribution –Statistical parameters such as mean and standard deviation Typically seek to know: – Given this relationship, what is the range of possibilities? – Are my cost margins enough?

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 911 General Model Architecture – Coverage & Partition Cost Estimating Schedule / Technical Requirements Threat – Assigning Cost to Risk CERs Direct Assessment of Distribution Parameters Factors Rates – Below-the-Line Yes No – Distribution Normal Log Normal Triangular Beta Bernoulli – Correlation Functional Relational Injected None Scoring Organization Interval w/ objective criteria Interval Ordinal None Monte Carlo Method of Moments Deterministic Probability Model Cross Checks Means CVs Inputs Tip: Higher is better except in Cross Checks Historical Domain Experts Conceptual Dollar Basis Inputs Compu- tation Structure Execution

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 912 Inputs – Scoring Interval with objective criteria – Set scoring based on objective criteria, and for which the distance (interval) between scores has meaning. (Note: the below example is also Ratio, because it passes through the origin.) A schedule slip of 1 week gets a score of 1, a slip of 2 weeks gets a score of 2, a slip of 4 weeks gets a 4, a slip of 5 weeks gets a score of 5, etc. The difference between a score of 1 and 2 is as big as a difference between score of 4 and 5 A scale is interval if it acts interval under examination* “Nominal, ordinal, interval, and ratio typologies are misleading,” P.F. Velleman and L. Wilkinson, The American Statistician, 1993, 47(1), 65-72 8

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 913 Inputs – Scoring Interval –Set scoring for which the distance (interval) between scores has meaning Low risk is assigned a 1, medium risk is assigned a 5, and a high risk is assigned a 10 Note that it is not immediately clear that the scale is interval, but it is surely not subject to objective criteria. Ordinal –Score is relative to the measurement e.g., difficulty in achieving schedule is high, medium, or low None

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 914 Inputs – Dollar Basis Historical –Actual costs of similar programs or components of programs are used to predict costs Domain Experts –Persons with expertise regarding similar programs or program components assess the cost based on their experience Conceptual –An arbitrary impact is assigned Any scale without a historical basis or expert assessment is conceptual

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 915 Org – Coverage & Partition How the four types of risk are covered and partitioned –Cost Estimating –Schedule/Technical –Requirements –Threat These risk types may be covered implicitly or explicitly in any combination.

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 916 Org – Assigning Cost to Risk Risk CERs: Equations are developed that reflect the relationship between an interval risk score and the cost impact of the risk (this might also be termed a Risk Estimating Relationship (RER)) –These equations amount to the same thing as CERs used in the cost estimate –e.g., Risk Amount = 0.12 * Risk Score Direct Assessment of Distribution Parameters: Costs are captured in shifts of parameters of the risk, e.g., shifted end points for triangulars, shifted end points or means for betas, etc. –Note: Scoring is completely eliminated from this mapping method –e.g., triangles assessed by domain experts 9

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 917 Org – Assigning Cost to Risk Factors: Fractions or percents are used in conjunction with the scores and the cost of the component or program –e.g., a score of 2 increases the cost of the component by 8% –Antenna Risk Score = 2 –Cost of Antenna = $4090K –Risk Amount = 0.08 * 4090K = $327.2K Rates: Predetermined costs are associated with the scores –e.g., a score of 2 has a cost of $100K –Antenna Risk Score = 2 –Cost of Antenna = $4090K –Risk Amount = $100K Warning: Rates are independent of the element’s cost.

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 918 Org – Below-the-Line Below-the-Line Elements –Elements that are driven by hardware, software, and the like –Below-the-Line Elements include: Systems Engineering/Program Management (SE/PM) System Test and Evaluation (ST&E) –Not all models account for this cost growth –Functional Correlation is another approach to address the risk in these elements 9

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 919 Probability Model – Distribution Normal –Best behavior, most iconic –Theoretically (although not practically) allows negative costs, which spook some users –Symmetric, needs mean shift to reflect propensity for positive growth Lognormal –A natural result in non-linear CERs –Indistinguishable from Normal at CVs below 25% –Skewed 10 4

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 920 Probability Model – Distribution Triangular –Most common –Easy to use, easy to understand –Modes, medians do not add –Skewed Beta –Rare now, but formerly popular –Solves negative cost and duration issues –Many parameters – simplifications like PERT Beta are possible –Skewed Bernoulli – Probability is only assigned to two possible outcomes, success and failure (p and 1-p) – Simplest of all discrete distributions – Mean = p – Variance = p*(1-p) 10

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 921 Probability Model – Correlation Functional: Arises between source and derivative variables as a result of functional dependency. The lines of the Monte Carlo are cell-referenced wherever relationships are known. –CERs are entered as equations –Cell references are left in the spreadsheet –When the Monte Carlo runs, input variables fluctuate, and outputs of CERs reflect this Correlation is a measure of the relation between two or more variables/WBS elements An Overview of Correlation and Functional Dependencies in Cost Risk and Uncertainty Analysis, R. L. Coleman and S. S. Gupta, DoDCAS, 1994 3

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 922 Functional Correlation Old: No Functional Correlation; Simulation run with WBS items entered as values New: Simulation run with functional dependencies entered as they are in cost model Correlated Not Correlated Note shift of mean, and increased variability

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 923 Probability Model – Correlation Relational: Introduces the geometry of correlation and provides a substantial improvement over injected correlations, and fills a gap in FC –Relational Correlation provides insight into the tilt of the data, i.e. the regression line, and the variance around the regression line Relational Correlation: What to do when Functional Correlation is Impossible, R. L. Coleman, J. R. Summerville, M. E. Dameron, C. L. Pullen, S. S. Gupta, ISPA/SCEA Joint International Conference,2001

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 924 Probability Model - Correlation Injected: Imposed by setting the correlation directly between variables without having a functional relationship. None: No relationship exists among the variables. The lines of the Monte Carlo are self contained.

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 925 Shortcomings of Injected Correlation Correlations are very hard to estimate No check of the functional implications of the correlations is done –This is troublesome because of the regression line that arises when we insert a correlation. –Simply injecting arbitrary correlations of 0.2 - 0.3 to achieve dispersion is unsatisfactory as well. Unless the injected correlations are among elements that are actually correlated If correlations are actually known, no harm is done.

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 926 Execution – Computation Monte Carlo: A widely accepted method, used on a broad range of risk assessments for many years. It produces cost distributions. The cost distributions give decision makers insight into the range of possible costs and their associated probabilities. Method of Moments: The mean and standard deviation of lower-level WBS lines are known, and are rolled up assuming independence to provide higher-level distributions. –Only provides an analysis of distribution at a top level –Easy to calculate –Negated by the rapid advances in microcomputer technology –Only works for independent elements, unless covariances are allowed for, which is difficult. Deterministic: Only point values are used. No shifts or other probabilistic effects are taken into account. 10

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 927 Risk Assessment Techniques Add a Risk Factor/Percentage (Minutes) –Low accuracy, no intervals Bottom Line Monte Carlo/Bottom Line Range/Method of Moments (Hours) –Moderate accuracy, provides intervals Historically based Detailed Monte Carlo (Months of non-recurring work, but recurring in days) –Time consuming non-recurring work, but with recurring implementation being easier, accurate if done right. Provides intervals. Expert Opinion-Based Probability and Consequence (Pf*Cf) or Expert Opinion-Based Detailed Monte Carlo (Months) –Time consuming with no gains in recurring effort, but accurate if done right. Provides intervals. Detailed Network and Risk Assessment (Month) –Time consuming with no gains in recurring effort, but accurate if done right. Provides intervals.

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 928 Execution – Cross Checks Means: The mean cost growth factor for WBS items can be compared to history as a way to cross check results CVs: The CV of the cost growth factors for WBS items can be compared to history as a way to cross check results Inputs: Checks are performed on inputs or other parameters to see if historical values are in line with program assumptions –Example: Historical risk scores can be compared to program risk scores to see if risk assessors are being realistic, and to see if the underlying database is representative of the program. 11

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 930 Intro to SARs – Sample A SAR report is submitted for each year of a program’s Acquisition cycle. The most recent SAR is used to determine cost growth To calculate the CGF, adjust the current estimate for quantity changes, then divide by the baseline estimate 12 Sample Program: XXX, December 31, 19XX

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 931 NAVAIR Cost Growth Study: A Cohorted Study of The Effects of Era, Size, Acquisition Phase, Phase Correlation and Cost Drivers, R. L. Coleman, J. R. Summerville, M. E. Dameron, C. L. Pullen, D. M. Snead, DoDCAS, 2001 and ISPA/SCEA International Conference, 2001 Contract Data Hard to use – problems with changing baselines, lack of reasons for variances, and access to data Preliminary comparative analysis suggests Contract Data mimics patterns in SAR data –Shape of distribution –Trends in tolerance for cost growth K-S tests find no statistically significant difference between Contract data and SAR data for programs <$1B in RDT&E –Failed to reject the null hypothesis of identical distributions Descriptive statistics indicate amount of Contract Data growth and dispersion is more extreme than previously found in SAR studies SAR data remains the best choice for analysis and predictive modeling

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 932 Contract Data Exploratory Analysis Contract Data blends well: Continues trend that tolerance for growth increases as program size decreases SAR Data Contract Data CGF vs IPE-Contract and SAR (RDT&E) ZOOM IN with common Scale

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 933 Common Problems Most historically-based methods rely on SARs –Adjusting for quantity – important to remove quantity changes from cost growth –Beginning points – the richest data source is found by beginning with EMD –Cohorting must be introduced to avoid distortions EVM data is also potentially useable, but re- baselined programs are a severe complication. “Applicability” and “currency” are the most common criticisms 15

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 934 Applicability and Currency Applicability: “Why did you include that in your database?” –Virtually all studies of risk have failed to find a difference among platforms (some exceptions) –If there is no discoverable platform effect, more data is better Currency: “But your data is so old!” –Previous studies have found that post-1986 data is preferable –Data accumulation is expensive

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 936 Example Model Architecture – Coverage & Partition Cost Estimating Schedule / Technical Requirements Threat – Assigning Cost to Risk CERs Direct Assessment of Distribution Parameters Factors Rates – Below-the-Line Yes No – Distribution Normal Log Normal Triangular Beta Bernoulli – Correlation Functional Relational Injected None Scoring Organization Interval w/ objective criteria Interval Ordinal None Monte Carlo Method of Moments Deterministic Probability Model Cross Checks Means CVs Inputs Historical Domain Experts Conceptual Dollar Basis Inputs Compu- tation Structure Execution Tip: Higher is better except in cross checks 13

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 937 Assessment Approach Develop a cost estimating risk distribution for each CWBS element Develop a schedule/technical risk distribution for each WBS entry for: –Hardware –Software –Note that “Below-the-line” WBS elements get risk from Above-the-line WBS elements via Functional Correlation Combine these risk distributions and the point estimate using a Monte Carlo simulation

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 938 Example Model in Blocks Functional Correlation Standard Errors & SEEs Mapping Risk Scoring Monte Carlo Risk Report CARD IPE Cost Estimating Risk Sked/Tech Risk Cost Risk Analysis of the Ballistic Missile Defense (BMD) System, An Overview of New Initiatives Included in the BMDO Risk Methodology, R. L. Coleman, J. R. Summerville, D. M. Snead, S. S. Gupta, G. E. Hartigan, N. L. St. Louis, DoDCAS, 1998 (Outstanding Contributed Paper), and ISPA/SCEA International Conference, 1998

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 939 Cost Estimating Risk Assessment Consists of a standard deviation and a bias associated with the costing methodologies –Standard deviation comes from the CERs and factors –Bias is a correction for underestimating 14

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 940 Sked/Tech Risk Assessment Technical risk is decomposed into categories and each category into sub categories –Hardware sub categories: Technology Advancement, Engineering Development, Reliability, Producibility, Alternative Item and Schedule –Software sub categories: Technology Approach, Design Engineering, Coding, Integrated Software, Testing, Alternatives, and Schedule

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 942 Calculating Sked/Tech Risk Endpoints Technical experts score each of the categories from 0 (no risk) to 10 (high risk) Each category is weighted depending on the relevancy of the category –Weights are allowed, but rarely used Weighted average risk scores are mapped to a cost growth distribution –This distribution is based on a database of cost growth factors of major weapon systems collected from SARs. These programs range from those which experienced tremendous cost growth due to technical problems to those which were well managed and under budget.

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 943 Sked/Tech Score Mapping Maximum possible cost growth Average cost growth Minimum possible cost growth MIN=0.77 AVG=1.17 MAX=1.58 MIN=0.61 AVG=1.28 MAX=1.96 MIN=0.46 AVG=1.40 MAX=2.34

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 944 Sked/Tech Risk Distribution Bars are the frequency of occurrence of each risk score Model 1357910 This is the composite PDF for all SARs S/T Risk Score These are the PDFs for 3 risk scores above. More risk has higher mode, wider base, all are symmetric. MIN=0.77 AVG=1.17 MAX=1.58 MIN=0.61 AVG=1.28 MAX=1.96 MIN=0.46 AVG=1.40 MAX=2.34

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 945 Cost Growth Database 2 4 15 28 20 8 9 4 7 11 3 5 -30%-20%-10%0%10%20%30% 40% 50%60% 70%80%90%100%400% Cost Decrease Cost Increase No Change Risk appears skewed, perhaps Triangular or Lognormal This distribution, found in databases, is the result of a blending of a family of distributions as shown on the previous slide.

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 947 Example Cost Estimate with Risk – R&D 0 50 100 150 Initial Point Estimate Add Cost Estimating Risk Add Sched/Tech Risk S/T Risk CE Risk Init Pt Est $ 8.7% 25% 33.7% 6 7 Note: These are means – there is an associated confidence interval, not portrayed.

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 948 Summary –Why include risk? Risk adjusts the cost estimate so that it more closely represents what historical data and experts know to be true … it predicts cost growth –How to treat risk? We have seen an overview of the many different options in terms of inputs, the structure of the risk model, and how to execute the risk model The choices are varied, but it is important that the model “fits together” and that it predicts well. –Closing thought: Always include cross checks to support the accuracy of the model and the specific results for a program The model may seem right, but will it (did it) predict accurate results?

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 949 Risk Resources – Books Against the Gods: The Remarkable Story of Risk, Peter L. Bernstein, August 31, 1998, John Wiley & Sons Living Dangerously! Navigating the Risks of Everyday Life, John F. Ross, 1999, Perseus Publishing Probability Methods for Cost Uncertainty Analysis: A Systems Engineering Perspective, Paul Garvey, 2000, Marcel Dekker Introduction to Simulation and Risk Analysis, James R. Evan, David Louis Olson, James R. Evans, 1998, Prentice Hall Risk Analysis: A Quantitative Guide, David Vose, 2000, John Wiley & Sons

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 950 Risk Resources – Web Decisioneering –Makers of Crystal Ball for Monte Carlo simulation –http://www.decisioneering.comhttp://www.decisioneering.com Palisade –Makers of @Risk for Monte Carlo simulation –http://www.palisade.comhttp://www.palisade.com

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 951 Risk Resources – Papers Approximating the Probability Distribution of Total System Cost, Paul Garvey, DoDCAS 1999 Why Cost Analysts should use Pearson Correlation, rather than Rank Correlation, Paul Garvey, DoDCAS 1999 Why Correlation Matters in Cost Estimating, Stephen Book, DoDCAS 1999 General-Error Regression in Deriving Cost-Estimating Relationships, Stephen A. Book and Mr. Philip H. Young, DoDCAS 1998 Specifying Probability Distributions From Partial Information on their Ranges of Values, Paul R. Garvey, DoDCAS 1998 Don't Sum EVM WBS Element Estimates at Completion, Stephen Book, Joint ISPA/SCEA 2001 Only Numbers in the Interval –1.0000 to +0.9314… Can Be Values of the Correlation Between Oppositely-Skewed Right-Triangular Distributions, Stephen Book, Joint ISPA/SCEA 1999

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 952 Risk Resources – Papers An Overview of Correlation and Functional Dependencies in Cost Risk and Uncertainty Analysis, R. L. Coleman, S. S. Gupta, DoDCAS, 1994 Weapon System Cost Growth As a Function of Maturity, K. J. Allison, R. L. Coleman, DoDCAS 1996 Cost Risk Estimates Incorporating Functional Correlation, Acquisition Phase Relationships, and Realized Risk, R. L. Coleman, S. S. Gupta, J. R. Summerville, G. E. Hartigan, SCEA National Conference, 1997 Cost Risk Analysis of the Ballistic Missile Defense (BMD) System, An Overview of New Initiatives Included in the BMDO Risk Methodology, R. L. Coleman, J. R. Summerville, D. M. Snead, S. S. Gupta, G. E. Hartigan, N. L. St. Louis, DoDCAS, 1998 (Outstanding Contributed Paper) and ISPA/SCEA International Conference, 1998

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 953 Risk Resources – Papers Risk Analysis of a Major Government Information Production System, Expert-Opinion-Based Software Cost Risk Analysis Methodology, N. L. St. Louis, F. K. Blackburn, R. L. Coleman, DoDCAS, 1998 (Outstanding Contributed Paper), and ISPA/SCEA International Conference, 1998 (Overall Best Paper Award) Analysis and Implementation of Cost Estimating Risk in the Ballistic Missile Defense Organization (BMDO) Risk Model, A Study of Distribution, J. R. Summerville, H. F. Chelson, R. L. Coleman, D. M. Snead, Joint ISPA/SCEA International Conference 1999 Risk in Cost Estimating - General Introduction & The BMDO Approach, R. L. Coleman, J. R. Summerville, M. DuBois, B. Myers, DoDCAS, 2000 Cost Risk in Operations and Support Estimates, J. R. Summerville, R. L. Coleman, M. E. Dameron, SCEA National Conference, 2000

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 954 Risk Resources – Papers Cost Risk in a System of Systems, R.L. Coleman, J.R. Summerville, V. Reisenleiter, D. M. Snead, M. E. Dameron, J. A. Mentecki, L. M. Naef, SCEA National Conference, 2000 NAVAIR Cost Growth Study: A Cohorted Study of the Effects of Era, Size, Acquisition Phase, Phase Correlation and Cost Drivers, R. L. Coleman, J. R. Summerville, M. E. Dameron, C. L. Pullen, D. M. Snead, ISPA/SCEA Joint International Conference, 2001 Probability Distributions of Work Breakdown Structures,, R. L. Coleman, J. R. Summerville, M. E. Dameron, N. L. St. Louis, ISPA/SCEA Joint International Conference, 2001 Relational Correlation: What to do when Functional Correlation is Impossible, R. L. Coleman, J. R. Summerville, M. E. Dameron, C. L. Pullen, S. S. Gupta, ISPA/SCEA Joint International Conference,2001 The Relationship Between Cost Growth and Schedule Growth, R. L. Coleman, J. R. Summerville, DoDCAS, 2002 The Manual for Intelligence Community CAIG Independent Cost Risk Estimates, R. L. Coleman, J. R. Summerville, S. S. Gupta, DoDCAS, 2002

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 956 Geometry of Bivariate Normal Random Variables ( μ x, μ y) σx σy ρ=.75 μxμx σy σx μyμy x y ρ=0 The dispersion and axis tilt of the “data cloud” is a function of correlation: less correlation, more dispersion about the axis more correlation, more axis tilt tilt

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 957 This line is with perfect correlation … The slope that would be true if ρ = 1 Implications for Regression Line (μ x, μ y) σx σy 2σx ρ=.75 b= μ y- ρσy / σx * μ x b μxμx σy σx μyμy x y 2σy This line has correlation added y = ρ(σy / σx) (x- μ x) + μ y

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 958 Intercept b varies with ρ, σx, σy, μx, and μy Geometry of Regression Line (μ x, μ y) σx σy 2σx ρ=1 ρ=-1 Dispersion varies with ρ b= μ y- ρσy / σx * μ x b μxμx σy σx Slope m varies with ρ, σx, σy μyμy x y ρ=02σy Range of intercepts Range of slopes y = ρ(σy / σx) (x- μ x) + μ y The regression line of y on x depends on their means, their standard deviations and their correlation

developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 959 r 2 is the percent reduction between these two variances: σ y 2 and σ y|x 2 or σ x 2 and σ x|y 2 Geometry of r squared σx σy b μxμx σx μyμy x y σy|x r 2 = 0.75 r 2 = 0 Variance of y|x = (1- ρ 2 )* σ y 2

Developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 91 Cost Risk Analysis How to adjust your estimate for historical cost growth.

Similar presentations

Presentation on theme: "Developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 91 Cost Risk Analysis How to adjust your estimate for historical cost growth."— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 91 Cost Risk Analysis How to adjust your estimate for historical cost growth.

Similar presentations

Presentation on theme: "Developed by© 2002 SCEA. All rights reserved. CostPROF Unit III - Module 91 Cost Risk Analysis How to adjust your estimate for historical cost growth."— Presentation transcript:

Similar presentations

About project

Feedback