USTPADPDD601

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UST/PAD/PDD601 Applied Quantitative Reasoning
Lecture 8. Multiple Regression Analysis
Sugie Lee, Ph.D. Assistant Professor Urban
Planning, Design and Development Program Levin
College of Urban Affairs Cleveland State
University
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Multiple Regression

• Extends the concept of simple regression
• Includes one criterion (dependent) variable and
  several predictor (independent) variables
• Needs Interval-ratio dependent variable and two
  or more independent variables (Independent
  variables can be either dichotomous or interval
  level)
• Evaluates direct effect of a single independent
  variable controlling for the other independent
  variables
• Reduces our errors of prediction using many
  predictor variables instead of just one predictor
  variable

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Multiple Regression Equation
Population
Estimate

• Examples
• Salaryf( education, gender, experience, etc)
• College grade point averagef (high school grade
  point average, aptitude test scores, household
  income, entrance test scores, etc)
• Housing sales ()f (square feet, lot size, of
  bedrooms, of bathrooms, year built, etc school
  quality, transportation accessibility, etc tax
  policy, public services, etc)
• Commuting timef( , ,
  , , )
• Poverty ratef( , ,
  , , ,.)

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Multiple Regression Equation (cont.)

• is the expected value of Y
• are the independent variables
• is the y-intercept (when all
  independent variables are zero)
• are the regression coefficients (the
  expected change in Y from an one-unit increase in
  , holding all the other Xs constant)

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Multiple Regression Equation (Cont.)
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Multiple Regression Analysis (Example)
SPSS data opm91.sav
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Multiple Regression Analysis (Example)

• The first regression coefficient suggests that,
  on average, male employees earn 7,983 more than
  female employees, holding education constant
• The second regression coefficient suggests that,
  on average, employees earns 3,076 more than
  people one grade below them, holding gender
  constant
• Y-intercept is the salary of female employee
  with 0 of education

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Multiple Regression Analysis (Example)

• The coefficients of the standardized predictor
  variables are referred to as beta coefficients or
  beta weights
• The second standardized coefficients .458 has a
  more important contribution to the dependent
  variable than the gender variable
• The beta regression coefficients can inform us
  only of the relative importance of the various
  predictor variables, not the absolute
  contributions
• The beta regression coefficients will be
  changed if we add other independent variables

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Multiple Regression Analysis (Example)

• .373 indicates that 37.3 of the
  variance of the salary is predictable by two
  independent variablesgender and education.
• If the independent variable is uncorrelated with
  each other, the multiple is the summation
  of individual of each independent variable.
• If the independent variable is correlated with
  each other, the sum of the individual will
  be greater than , since most of the
  independent variables are duplicating the
  predictive power contained in another independent
  variable

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Multiple Regression Analysis (Example)

• The adjusted adjusts for a bias in
  R-square when the model has a small sample size
  and many predictors (independent variables)

Increase in the sample size will make a small
adjustment
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Multiple Regression Analysis (Example)
SSR
SSE
SST
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Importance of the Predictor Variables

• The multiple correlation coefficients R tells
  us the correlation between the weighted sum of
  the predictor (independent) variables and the
  criterion (dependent) variable
• The squared multiple correlation coefficient
  tells us what proportion of the variance of
  the criterion variable is accounted for by all
  the predictor variables combined for example, a
  multiple of .7 indicates that 70 of the
  variance of the dependent variable is accounted
  by a given set of independent variables

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Multiple Regression Dummy and Interval Variables

• Reference group female with no education
• Y-intercept(-21,690) is the expected income of
  female with no education
• The coefficient (13,550) on male is the expected
  difference in income between male and female
  holding education constant. That is, the expected
  income of male is 13,550 higher than that of
  female of same education status
• The coefficient (3,350) on educ is the increase
  in the expected income with an one-year increase
  in education holding gender constant. That is,
  each additional year of education would raise the
  income by 3,350 holding gender constant.

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Multiple Regression Dummy and Interval Variables
(cont.)
If male1(male), If male0(female),
Income
male
female
education
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Multiple Regression Dummy Variables

• Reference group white population
• Y-intercept(41.91) is the expected working hours
  of the White
• The coefficient (.33) on Asian is the expected
  difference in working hours between the Asian and
  the White. That is, the Asian are .33 hours more
  working than the white
• The coefficient (-.94) on oth_minority is the
  expected difference in working hours between the
  other minority and the white. That is, other
  minorities are .94 hours less working than the
  white.

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Multiple Regression Interaction Variable

• Reference group female with no education
• Y-intercept(-12,050) is the expected income of
  female with no education
• The coefficient (-4,410) on male is the expected
  difference in income between male and female
  holding education zero (no education). That is,
  the expected income of male is 4,410 lower than
  that of female holding education zero
• The coefficient (2,640) on educ is the increase
  in the expected income with an one-year increase
  in education holding male zero (female). In other
  words, an additional year of education of female
  would raise the income by 2,640
• The coefficient (1,310) on maleeduc
  (interaction variable) is the additional income
  advantage being male for an additional year of
  education.

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Multiple Regression Interaction Variable (cont.)
If male1(male), If male0(female),
Income
male
female
education
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Multiple Regression Polynomial Model

• The relationship between dependent and
  independent variables are curvilinear rather than
  linear. Adding the squared term is the most
  common way to model a curvilinear relationship.
• Y-intercept(-29,660) is the expected income of
  someone with zero age
• We cannot interpret the coefficients as the
  effect of increase in age while holding age
  squared constant.

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Multiple Regression Polynomial Model (cont.)
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Multicolinearity

• Multicolinearity means co-dependence among
  independent variables
• When multicolinearity is severe, it leads to
  unreasonable coefficient estimates and large
  standard errors
• The common detecting method of multicolinearity
  is the variance inflation factor (VIF) or
  tolerance (1/VIF) a rule of thumb of
  multicolinearity VIFgt10.
• When you detect multicolinearity among your
  independent variables, you will have to drop some
  variables

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Autoregression

• Predict values on the dependent variable based
  on values of the same dependent variable obtained
  earlier in time
• Example Predict the price of Stock A on a given
  day based on its price the previous day
• Useful in identifying dependencies among data
  collected sequentially which we may wish to
  extract before submitting the data to further
  analysis
• Useful for projecting time series data such as
  crime rates, fertility rates, etc.

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Multiple Regression Modeling Process in Planning
and Policy Analysis

• Research Background
• Hypothesis
• Data Preparation and Variable Selection
• Descriptive Analysis
• Multiple Regression Analysis
• Interpretation
• Limitations
• Policy Implications

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Multiple Regression Modeling Process Example

• Research Background
• The purpose of this research is to investigate
  the economic impact of publicly owned street
  trees on residential property values involving
  case studies of Clevelands neighborhoods. By
  quantifying the positive economic impact of trees
  on property values, it should encourage planners
  and decision makers to provide adequate program
  funding to maintain a healthy and vigorous tree
  resource.
• Hypothesis
• The citys street trees has a positive economic
  impact on single family property values at the
  neighborhood level controlling for neighborhood
  characteristics and housing characteristics

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Multiple Regression Modeling Process Example
(cont.)

• Data Preparation and Variable Selection

Figure 1. Example of Case Study Blocks, Belden
Avenue, City of Cleveland With Street Trees
(A) and Without Street Trees (B)
Figure 2. Sold Houses (1994-2005)
Multiple Regression Model y (property sales
value) ß0 ßx housing characteristics ßy
neighborhood impacts ßz street tree
Housing characteristics s.f in living space,
of bedrooms, of bathrooms, year built, and year
sold Neighborhood characteristics Street
trees dummy variable of with tree and without
tree
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Multiple Regression Modeling Process Example
(cont.)

• Analysis (Descriptive Analysis and Multiple
  Regression)

Data tree_regression.sav on the class
website Dependent variables convamt (sale
price) Independent variables livatot, bedrooms,
baths, ryrbuilt, transyr, tree
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Multiple Regression Modeling Process Example
(cont.)

• Output, Interpretation, limitations, and Policy
  Implications