1 Lecture Eleven Probability Models

2 Outline Bayesian Probability Duration Models

3 Bayesian 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 100

4 Joint and Marginal Probabilities

5 Filling In Our Facts

Using Conditional Probability Pr(+ H)= Pr(+/H)*Pr(H)= 0.02*0.999= Pr(+ S) = Pr(+/S)*Pr(S) = 0.99*0.001=.00099

7 Filling In Our Facts

8 By Sum and By Difference

False Positive Paradox Probability of Being Sick If You Test + Pr(S/+) ? From Conditional Probability: Pr(S/+) = Pr(S +)/Pr(+) = / Pr(S/+) =

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* *.999] Pr(S/+) = /[ ] Pr(S/+) = / =

11 Duration Models Exploratory (Graphical) Estimates – Kaplan-Meier Functional Form Estimates –Exponential Distribution

12 Duration of Post-War Economic Expansions in Months

13 Housing starts: Fred, St. Louis Fed

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17 Estimated Survivor Function for Ten Post-War Expansions

18 Kaplan-Meyer Estimate of Survivor Function Survivor Function = (# at risk - # ending)/# at risk

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Exponential 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

22 So 

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

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] =

25 Model Building Reference: Ch 20 (Ch. 18, 8 th Ed.)

Polynomial Models There are models where the independent variables (x i ) may appear as functions of a smaller number of predictor variables. Polynomial models are one such example.

27 y =  0 +  1 x 1 +  2 x 2 +…+  p x p +  y =  0 +  1 x +  2 x 2 + …+  p x p +  Polynomial Models with One Predictor Variable

y  0  1 x   First order model (p = 1) y =  0 +  1 x +  2 x 2 +   2 < 0  2 > 0 Second order model (p=2) Polynomial Models with One Predictor Variable

y =  0 +  1 x +  2 x 2 +   3 x 3 +   3 < 0  3 > 0 Third order model (p = 3) Polynomial Models with One Predictor Variable

First order model y =  0 +  1 x 1 +  Polynomial Models with Two Predictor Variables x1x1 x2x2 y  2 x 2 +   1 < 0  1 > 0 x1x1 x2x2 y  2 > 0  2 < 0

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 met 0 if a second condition out of two is met I= 1 if data were collected before if data were collected after if the temperature was below 50 o 0 if the temperature was 50 o or more 1 if a degree earned is in Finance 0 if a degree earned is not in Finance

32 Nominal Independent Variables; Example: Auction Car Price (II) Example revised (Xm18-02a)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 colors Note: Color is a nominal variable.

33 Example revised (Xm18-02b)Xm18-02b I 1 = 1 if the color is white 0 if the color is not white I 2 = 1 if the color is silver 0 if the color is not silver The category “Other colors” is defined by: I 1 = 0; I 2 = 0 Nominal Independent Variables; Example: Auction Car Price (II)

34 Note: To represent the situation of three possible colors we need only two indicator variables. Conclusion: To represent a nominal variable with m possible categories, we must create m-1 indicator variables. How Many Indicator Variables?

35 Solution –the proposed model is y =  0 +  1 (Odometer) +  2 I 1 +  3 I 2 +  –The data White car Other color Silver color Nominal Independent Variables; Example: Auction Car Price

36 Odometer Price Price = (Odometer) (0) (1) Price = (Odometer) (1) (0) Price = (Odometer) (0) + 148(0) (Odometer) (Odometer) (Odometer) The equation for an “other color” car. The equation for a white color car. The equation for a silver color car. From Excel (Xm18-02b) we get the regression equationXm18-02b PRICE = (Odometer)+90.48(I-1) (I-2) Example: Auction Car Price The Regression Equation

37 From Excel we get the regression equation PRICE = (Odometer)+90.48(I-1) (I-2) A white car sells, on the average, for $90.48 more than a car of the “Other color” category A silver color car sells, on the average, for $ more than a car of the “Other color” category. For one additional mile the auction price decreases by 5.55 cents. Example: Auction Car Price The Regression Equation

There is insufficient evidence to infer that a white color car and a car of “other color” sell for a different auction price. There is sufficient evidence to infer that a silver color car sells for a larger price than a car of the “other color” category. Xm18-02b Example: Auction Car Price The Regression Equation

39 Recall: The Dean wanted to evaluate applications for the MBA program by predicting future performance of the applicants. The following three predictors were suggested: –Undergraduate GPA –GMAT score –Years of work experience It is now believed that the type of undergraduate degree should be included in the model. Nominal Independent Variables; Example: MBA Program Admission (MBA II)MBA II Note: The undergraduate degree is nominal data.

40 Nominal Independent Variables; Example: MBA Program Admission (II) I 1 = 1 if B.A. 0 otherwise I 2 = 1 if B.B.A 0 otherwise The category “Other group” is defined by: I 1 = 0; I 2 = 0; I 3 = 0 I 3 = 1 if B.Sc. or B.Eng. 0 otherwise

41 Nominal Independent Variables; Example: MBA Program Admission (II) MBA-II

Applications in Human Resources Management: Pay-Equity Pay-equity can be handled in two different forms: –Equal pay for equal work –Equal pay for work of equal value. Regression analysis is extensively employed in cases of equal pay for equal work.

43 Solution –Construct the following multiple regression model: y =  0 +  1 Education +  2 Experience +  3 Gender +  –Note the nature of the variables: Education – Interval Experience – Interval Gender – Nominal (Gender = 1 if male; =0 otherwise). Human Resources Management: Pay-Equity

44 Solution – Continued (Xm20-03)Xm20-03 Human Resources Management: Pay-Equity Analysis and Interpretation The model fits the data quite well. The model is very useful. Experience is a variable strongly related to salary. There is no evidence of sex discrimination.

45 Solution – Continued (Xm20-03)Xm20-03 Human Resources Management: Pay-Equity Analysis and Interpretation Further studying the data we find: Average experience (years) for women is 12. Average experience (years) for men is 17 Average salary for female manager is $76,189 Average salary for male manager is $97,832