Lecture Eleven

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Lecture Eleven

• Probability Models

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Outline

• Bayesian Probability
• Duration Models

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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

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Joint and Marginal Probabilities
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Filling In Our Facts
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Using Conditional Probability

• Pr( H) Pr(/H)Pr(H) 0.020.999.01998
• Pr( S) Pr(/S)Pr(S) 0.990.001.00099

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Filling In Our Facts
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By Sum and By Difference
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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

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Bayesian Probability By Formula

• Pr(S/) Pr(S H)/Pr() PR(/S)Pr(S)/Pr()
• Where PR() PR(/S)PR(S) PR(/H)PR(H)
• And Using our factsPr(S/) 0.99(0.001)/0.99
  .001 0.02.999
• Pr(S/) 0.00099/0.000990.01998
• Pr(S/) 0.00099/0.02097 0.0472

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Duration Models

• Exponential Distribution

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Duration of Post-War Economic Expansions in Months
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Estimated Survivor Function for Ten Post-War
Expansions
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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 - exp0
• F(t) 1 - exp- t
• Survivor Function, S(t) 1- F(t) exp- t
• Taking logarithms, lnS(t) - t

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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

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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

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Model Building

• Reference Ch 20

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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.

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Polynomial Models with One Predictor Variable

• y b0 b1x1 b2x2 bpxp e
• y b0 b1x b2x2 bpxp e

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Polynomial Models with One Predictor Variable

• First order model (p 1)

y b0 b1x e

• Second order model (p2)

y b0 b1x e
b2x2 e
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Polynomial Models with One Predictor Variable

• Third order model (p 3)

y b0 b1x b2x2 e
b3x3 e
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Polynomial Models with Two Predictor Variables
y
b1 gt 0

• First order modely b0 b1x1 e

b2x2 e
b1 lt 0
x1
x2
b2 gt 0
b2 lt 0
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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 a degree earned is in Finance 0 if a
degree earned is not in Finance
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
I
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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 cars
  price.
• Three color categories are considered
• White
• Silver
• Other colors

Note Color is a nominal variable.
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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
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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.

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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
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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)
The equation for a silver color car.
Price 16701 - .0555(Odometer) 90.48(0)
295.48(1)
The equation for a white color car.
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.
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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.
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Example Auction Car Price The Regression
Equation
Xm18-02b
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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.
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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
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Nominal Independent Variables Example MBA
Program Admission (II)
MBA-II
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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.

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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).

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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.

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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