Lecture Eleven Probability Models

Outline Bayesian Probability Duration Models

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

Joint and Marginal Probabilities

Filling In Our Facts

Using Conditional Probability Pr(+ H)= Pr(+/H)*Pr(H)= 0.02*0.999=.01998 Pr(+ S) = Pr(+/S)*Pr(S) = 0.99*0.001=.00099

Filling In Our Facts

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(+) = 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

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

Duration of Post-War Economic Expansions in Months

Estimated Survivor Function for Ten Post-War Expansions

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

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

So l = 0.022

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] =

Model Building Reference: Ch 20

20.2 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.

Polynomial Models with One Predictor Variable y = b0 + b1x1+ b2x2 +…+ bpxp + e y = b0 + b1x + b2x2 + …+bpxp + e

Polynomial Models with One Predictor Variable First order model (p = 1) y = b0 + b1x + e Second order model (p=2) y = b0 + b1x + b2x2 + e b2 < 0 b2 > 0

Polynomial Models with One Predictor Variable Third order model (p = 3) y = b0 + b1x + b2x2 + e b3x3 + e b3 < 0 b3 > 0

Polynomial Models with Two Predictor Variables First order model y = b0 + b1x1 + e b2x2 + e b1 < 0 x1 x2 x1 x2 y b2 > 0 b2 < 0

20.3 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 the temperature was below 50o 0 if the temperature was 50o or more 1 if a first condition out of two is met 0 if a second condition out of two is met 1 if data were collected before 1980 0 if data were collected after 1980 1 if a degree earned is in Finance 0 if a degree earned is not in Finance I=

Nominal 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 colors Note: Color is a nominal variable.

Nominal Independent Variables; Example: Auction Car Price (II) Example 18.2 - revised (Xm18-02b) 1 if the color is white 0 if the color is not white I1 = 1 if the color is silver 0 if the color is not silver I2 = The category “Other colors” is defined by: I1 = 0; I2 = 0

How Many Indicator Variables? 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.

Nominal Independent Variables; Example: Auction Car Price Solution the proposed model is y = b0 + b1(Odometer) + b2I1 + b3I2 + e The data White car Other color Silver color

Example: Auction Car Price The Regression Equation From Excel (Xm18-02b) we get the regression equation PRICE = 16701-.0555(Odometer)+90.48(I-1)+295.48(I-2) Odometer Price The equation for a silver color car. 16996.48 - .0555(Odometer) Price = 16701 - .0555(Odometer) + 90.48(0) + 295.48(1) The equation for a white color car. 16791.48 - .0555(Odometer) 16701 - .0555(Odometer) Price = 16701 - .0555(Odometer) + 90.48(1) + 295.48(0) Price = 16701 - .0555(Odometer) + 45.2(0) + 148(0) The equation for an “other color” car.

Example: Auction Car Price The Regression Equation From Excel we get the regression equation PRICE = 16701-.0555(Odometer)+90.48(I-1)+295.48(I-2) For one additional mile the auction price decreases by 5.55 cents. 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 $295.48 more than a car of the “Other color” category.

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. Xm18-02b There is sufficient evidence to infer that a silver color car sells for a larger price than a car of the “other color” category.

Nominal Independent Variables; Example: MBA Program Admission (MBA II) 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. Note: The undergraduate degree is nominal data.

Nominal Independent Variables; Example: MBA Program Admission (II) 1 if B.A. 0 otherwise I1 = 1 if B.B.A 0 otherwise I2 = 1 if B.Sc. or B.Eng. 0 otherwise I3 = The category “Other group” is defined by: I1 = 0; I2 = 0; I3 = 0

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

20.4 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.

Human Resources Management: Pay-Equity Solution Construct the following multiple regression model: y = b0 + b1Education + b2Experience + b3Gender + e Note the nature of the variables: Education – Interval Experience – Interval Gender – Nominal (Gender = 1 if male; =0 otherwise).

Human Resources Management: Pay-Equity Solution – Continued (Xm20-03) 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.

Human Resources Management: Pay-Equity Solution – Continued (Xm20-03) 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