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

6. 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

7. Filling In Our Facts

8. By Sum and By Difference

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

10. 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

14. Estimated Survivor Function for Ten Post-War Expansions

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

18. Exponential Distribution
Density: f(t) = exp[ - t], 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

19. So l = 0.022

20. Exponential Distribution (Cont.)
Mean = 1/ =
Memoryless feature:
Duration conditional on surviving until t = :
DURC( ) = = /
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

21. 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] =

22. Model Building
Reference: Ch 20

23. 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.

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

25. 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

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

27. 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

28. 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=

29. Nominal Independent Variables; Example: Auction Car Price (II)
Example 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.

30. Nominal Independent Variables; Example: Auction Car Price (II)
Example 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

31. 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.

32. 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

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

34. Example: Auction Car Price The Regression Equation
From Excel we get the regression equation
PRICE = (Odometer)+90.48(I-1) (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 $ more than a car of the “Other color” category.

35. 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.

36. 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.

37. 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

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

39. 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.

40. 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).

41. 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.

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