Power 14 Goodness of Fit

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Power 14Goodness of Fit Contingency Tables
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II. Goodness of Fit Chi Square

• Rolling a Fair Die
• The Multinomial Distribution
• Experiment 600 Tosses

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The Expected Frequencies
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The Expected Frequencies Empirical Frequencies
Empirical Frequency
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Hypothesis Test

• Null H0 Distribution is Multinomial
• Statistic (Oi - Ei)2/Ei, observed minus
  expected squared divided by expected
• Set Type I Error _at_ 5 for example
• Distribution of Statistic is Chi Square

One Throw, side one comes up multinomial
distribution
P(n1 1, n2 0, n3 0, n4 0, n5 0, n6 0) n!/
P(n1 1, n2 0, n3 0, n4 0, n5 0, n6 0)
1!/1!0!0!0!0!0!(1/6)1(1/6)0 (1/6)0 (1/6)0 (1/6)0
(1/6)0
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Chi Square x2 ? (Oi - Ei)2 6.15
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5
11.07
Chi Square Density for 5 degrees of freedom
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Contingency Table Analysis

• Tests for Association Vs. Independence For
  Qualitative Variables

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Does Consumer Knowledge Affect Purchases? Frost
Free Refrigerators Use More Electricity
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Marginal Counts
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Marginal Distributions, f(x) f(y)
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Joint Disribution Under Independence f(x,y)
f(x)f(y)
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Expected Cell Frequencies Under Independence
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Observed Cell Counts
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Contribution to Chi Square (observed-Expected)2/E
xpected
Upper Left Cell (314-324)2/324 100/324 0.31
Chi Sqare 0.31 0.93 0.46 1.39
3.09 (m-1)(n-1) 111 degrees of freedom
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5
5.02
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Conclusion

• No association between consumer knowledge about
  electricity use and consumer choice of a
  frost-free refrigerator

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Using Goodness of Fit to Choose Between Competing
Probability Models

• Men on base when a home run is hit

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Men on base when a home run is hit
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Conjecture

• Distribution is binomial

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Average of men on base
Sum of products np 0.2980.2500.081 0.63
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Using the binomialkmen on base, n of trials

• P(k0) 3!/0!3! (0.21)0(0.79)3 0.493
• P(k1) 3!/1!2! (0.21)1(0.79)2 0.393
• P(k2) 3!/2!1! (0.21)2(0.79)1 0.105
• P(k3) 3!/3!0! (0.21)3(0.79)0 0.009

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Assuming the binomial

• The probability of zero men on base is 0.493
• the total number of observations is 765
• so the expected number of observations for zero
  men on base is 0.493765377.1

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Goodness of Fit
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Chi Square, 3 degrees of freedom
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7.81
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Conjecture Poisson where mnp 0.63

• P(k3) 1- P(k2)-P(k1)-P(k0)
• P(k0) e-m mk /k! e-0.63 (0.63)0/0! 0.5326
• P(k1) e-m mk /k! e-0.63 (0.63)1/1! 0.3355
• P(k2) e-m mk /k! e-0.63 (0.63)2/2! 0.1057

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Average of men on base
Sum of products np 0.2980.2500.081 0.63
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Conjecture Poisson where mnp 0.63

• P(k3) 1- P(k2)-P(k1)-P(k0)
• P(k0) e-m mk /k! e-0.63 (0.63)0/0! 0.5326
• P(k1) e-m mk /k! e-0.63 (0.63)1/1! 0.3355
• P(k2) e-m mk /k! e-0.63 (0.63)2/2! 0.1057

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Goodness of Fit
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Chi Square, 3 degrees of freedom
5
7.81
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Likelihood Functions

• Review OLS Likelihood
• Proceed in a similar fashion for the probit

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Likelihood function

• The joint density of the estimated residuals can
  be written as
• If the sample of observations on the dependent
  variable, y, and the independent variable, x, is
  random, then the observations are independent of
  one another. If the errors are also identically
  distributed, f, i.e. i.i.d, then

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Likelihood function

• Continued If i.i.d., then
• If the residuals are normally distributed
• This is one of the assumptions of linear
  regression errors are i.i.d normal
• then the joint distribution or likelihood
  function, L, can be written as

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Likelihood function

• and taking natural logarithms of both sides,
  where the logarithm is a monotonically increasing
  function so that if lnL is maximized, so is L

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Log-Likelihood

• Taking the derivative of lnL with respect to
  either a-hat or b-hat yields the same estimators
  for the parameters a and b as with ordinary least
  squares, except now we know the errors are
  normally distributed.

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Probit

• Example expenditures on lottery as a of
  household income
• lotteryi a bincomei ei
• if lotteryi gt0, i.e. a bincomei ei gt0,
  then Berni , the yes-no indicator variable is
  equal to one and ei gt- a - bincomei
• this determines a threshold for observation i in
  the distribution of the error ei
• assume

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i
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i
Area above the threshold is the probability of
playing the lottery for observation i, Pyes
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i
Area above the threshold is the probability of
playing the lottery for observation i, Pyes
Pno for observation i
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Probit

• Likelihood function for the observed sample
• Log likelihood

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i
Area above the threshold is the probability of
playing the lottery for observation i, Pyes
Pno for observation i
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Probit

• Substituting these expressions for Pno and Pyes
  in the ln Likelihood function gives the complete
  expression.

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Probit

• Likelihood function for the observed sample
• Log likelihood

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Outline

• I. Projects
• II. Goodness of Fit Chi Square
• III.Contingency Tables

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Part I Projects

• Teams
• Assignments
• Presentations
• Data Sources
• Grades

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Team One

• Project choice
• Data Retrieval
• Statistical Analysis
• PowerPoint Presentation
• Executive Summary
• Technical Appendix
• Graphics (Excel, Eviews, other)

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Assignments

• 1. Project choice Markus Ansmann
• 2. Data Retrieval Theodore Ehlert
• 3. Statistical Analysis David Sheehan
• 4. PowerPoint Presentation Qun Luo
• 5. Executive Summary Steven Comstock
• 6. Technical Appendix Alan Weinberg
• 7. Graphics Gregory Adams

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PowerPoint Presentations Member 4

• 1. Introduction Members 1 ,2 , 3
• What
• Why
• How
• 2. Executive Summary Member 5
• 3. Exploratory Data Analysis Members 3, 7
• 4. Descriptive Statistics Member 3, 7
• 5. Statistical Analysis Member 3
• 6. Conclusions Members 3 5
• 7. Technical Appendix Table of Contents, Member
  6

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Executive Summary and Technical Appendix
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Grades
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Data Sources

• FRED Federal Reserve Bank of St. Louis,
  http//research.stlouisfed.org/fred/
• Business/Fiscal
• Index of Consumer Sentiment, Monthly (195211)
• Light Weight Vehicle Sales, Auto and Light Truck,
  Monthly (1976.01)
• Economagic, http//www.economagic.com/
• U S Dept. of Commerce, http//www.commerce.gov/
• Population
• Economic Analysis, http//www.bea.gov/

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Data Sources (Cont. )

• Bureau of Labor Statistics, http//stats.bls.gov/
• California Dept of Finance, http//www.dof.ca.gov/