Lecture ElevenOutlineBayesian ProbabilityJoint and Marginal ProbabilitiesFilling In Our FactsUsing Conditional ProbabilitySlide 7By Sum and By DifferenceFalse Positive ParadoxBayesian Probability By FormulaDuration ModelsDuration of Post-War Economic Expansions in MonthsPowerPoint PresentationEstimated Survivor Function for Ten Post-War ExpansionsKaplan-Meyer Estimate of Survivor FunctionSlide 16Slide 17Exponential DistributionSlide 19Exponential Distribution (Cont.)Exponential Distribution(Cont.)Model Building20.2 Polynomial ModelsPolynomial Models with One Predictor VariableSlide 25Slide 26Polynomial Models with Two Predictor Variables20.3 Nominal Independent VariablesNominal Independent Variables; Example: Auction Car Price (II)Slide 30How Many Indicator Variables?Nominal Independent Variables; Example: Auction Car PriceExample: Auction Car Price The Regression EquationSlide 34Slide 35Nominal Independent Variables; Example: MBA Program Admission (MBA II)Nominal Independent Variables; Example: MBA Program Admission (II)Slide 3820.4 Applications in Human Resources Management: Pay-EquityHuman Resources Management: Pay-EquitySlide 41Slide 421Lecture ElevenProbability Models2Outline•Bayesian Probability•Duration Models3Bayesian Probability•Facts•Incidence of the disease in the population is one in a thousand•The probability of testing positive if you have the disease is 99 out of 100•The probability of testing positive if you do not have the disease is 2 in a 1004Joint and Marginal ProbabilitiesSick: S Healthy: HTest +Pr(+ S) Pr(+H)Pr(+)Test-Pr(-S) Pr(-H)Pr(-)Pr(S) Pr(H)5Filling In Our FactsSick: S Healthy: HTest +Test -Pr(s) =0.001Pr(H) =0.999Using Conditional Probability•Pr(+ H)= Pr(+/H)*Pr(H)= 0.02*0.999=.01998•Pr(+ S) = Pr(+/S)*Pr(S) = 0.99*0.001=.000997Filling In Our FactsSick: S Healthy: HTest +Pr(+S)=0.00099Pr(+H)=0.01998Test -Pr(s) =0.001Pr(H) =0.9998By Sum and By DifferenceSick: S Healthy: HTest +Pr(+S)=0.00099Pr(+H)=0.00198Pr(+)=0.02097Test -Pr(-S)=0.00901Pr(-H)=0.88802Pr(s) =0.001Pr(H) =0.999False Positive Paradox•Probability of Being Sick If You Test +•Pr(S/+) ?•From Conditional Probability:•Pr(S/+) = Pr(S +)/Pr(+) = 0.00099/0.02097•Pr(S/+) = 0.0472Bayesian Probability By Formula•Pr(S/+) = Pr(S +)/Pr(+) = PR(+/S)*Pr(S)/Pr(+)•Where PR(+) = PR(+/S)*PR(S) + PR(+/H)*PR(H)•And Using our facts;Pr(S/+) = 0.99*(0.001)/[0.99*.001 + 0.02*.999]•Pr(S/+) = 0.00099/[0.00099+0.01998]•Pr(S/+) = 0.00099/0.02097 = 0.047211Duration Models•Exploratory (Graphical) Estimates– Kaplan-Meier•Functional Form Estimates–Exponential Distribution12Duration of Post-War Economic Expansions in Months13Trough Peak DurationOct. 1945 Nov. 1948 37Oct. 1949 July 1953 45May 1954 August 1957 39April 1958 April 1960 24Feb. 1961 Dec. 1969 106Nov. 1970 Nov. 1973 36March 1975 January 1980 58July 1980 July 1981 12Nov. 1982 July 1990 92March 1991 March 2000 12014Estimated Survivor Function for Ten Post-War Expansions15Kaplan-Meyer Estimate of Survivor Function•Survivor Function = (# at risk - # ending)/# at risk16Duration # Ending # At Risk Survivor0 0 10 112 1 10 0.924 1 9 0.836 1 8 0.737 1 7 0.639 1 6 0.545 1 5 0.458 1 4 0.392 1 3 0.2106 1 2 0.1120 1 1 017Figure 2: Estimated Survivor Function for Post-War Expansions00.20.40.60.811.20 20 40 60 80 100 120 140Duration in MonthsSurvivor FunctionExponential Distribution•Density: f(t) = exp[ - t], 0 t •Cumulative Distribution Function F(t)•F(t) = •F(t) = - exp[- u] •F(t) = -1 {exp[- t] - exp[0]} •F(t) = 1 - exp[- t]•Survivor Function, S(t) = 1- F(t) = exp[- t]•Taking logarithms, lnS(t) = - t f (u)du 0t exp[ u]du0t0t19Postwar Expansionsy = -0.0217x + 0.1799R2 = 0.9533-2.5-2-1.5-1-0.500.50 20 40 60 80 100 120Duration (Months)Ln Survivor FunctionSo  Exponential Distribution (Cont.)•Mean = 1/ =•Memoryless feature: •Duration conditional on surviving until t = :•DURC( ) = = + 1/ •Expected remaining duration = duration conditional on surviving until time , i.e DURC, minus•Or 1/ , which is equal to the overall mean, so the distribution is memorylesst * f (t)dtt * f (t)dt / S()Exponential Distribution(Cont.)•Hazard rate or function, h(t) is the probability of failure conditional on survival until that time, and is the ratio of the density function to the survivor function. It is a constant for the exponential.•h(t) = f(t)/S(t) = exp[- t] /exp[- t] = 22Model Building•Reference: Ch 202320.2 Polynomial Models•There are models where the independent variables (xi) may appear as functions of a smaller number of predictor variables.•Polynomial models are one such example.24y = 0 + 1x1+ 2x2 +…+ pxp +  y = 0 + 1x + 2x2 + …+pxp +  Polynomial Models with One Predictor Variabley01x•First order model (p = 1)y = 0 + 1x + 2x2 + 2 < 02 > 0•Second order model (p=2) Polynomial Models with One Predictor Variabley = 0 + 1x + 2x2 + 3x3 + 3 < 0 3 > 0•Third order model (p = 3) Polynomial Models with One Predictor Variable•First order modely = 0 + 1x1 +  Polynomial Models with Two Predictor Variablesx1x2y2x2 + 1 < 01 > 0x1x2y2 > 02 < 02820.3 Nominal Independent Variables•In many real-life situations one or more independent variables are nominal.•Including nominal variables in a regression analysis model is done via indicator variables.•An indicator variable (I) can assume one out of two values, “zero” or “one”.1 if a first condition out of two is met0 if a second condition out of two is metI=1 if data were collected before 19800 if data were collected after 19801 if the temperature was below 50o0 if the temperature was 50o or more1 if a degree earned is in Finance0 if a degree earned is not in Finance29Nominal Independent Variables; Example: Auction Car Price (II)•Example 18.2 - revised (Xm18-02a)–Recall: A car dealer wants to predict the auction price of a car.–The dealer believes now that odometer reading and the car color are variables that affect a car’s price.–Three color categories are considered:•White•Silver•Other colorsNote: Color is a nominal variable.30•Example