SPSS Chapter 8 Correlation and Linear Regression

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SPSS Chapter 8Correlation and Linear Regression
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Review Measures of Association

• The most important thing is to answer the
  question How strong is the relationship between
  the independent and the dependent variables?
• This distinction is important because it is
  possible that a change in the independent
  variable can cause only slight changes in the
  dependent variable.  
• The calculated value of most measures of
  association ranges from 0 to 1 or 1. A value of
  zero indicates no relationship, a value of 1
  indicates perfect positive relationship, and a
  value of 1 indicates perfect negative
  relationship. The sign of the measure indicates
  the direction of the relationship

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Review PRE Measures (nominal or ordinal data)

• Prediction based How much better can we predict
  the DV by knowing the IV than by not knowing the
  IV?
• General rules about PRE Relationships
• Less than 0.1 Weak
• Greater than 0.1 but less than 0.2 Moderate
• Greater than 0.2 but less than 0.3 Moderately
  strong
• Greater than 0.3 strong  

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Pearsons r

• Relationship between 2 interval level variables
• Not PRE measure
• -1 (perfect negative) to 1 (perfect positive)
• Used to get overall picture of relationships
  between variables

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Regression

• More specialized than correlation
• Regression Coefficient
• Estimates the effect of an IV on a DV
• PRE Measure of association
• How completely the IV explains the DV
• Variables
• DV Interval Level
• IV nominal, ordinal or interval
• Used to model causal relationships between one or
  more IVs and a DV

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Correlation Example

• Interest Gender composition of State
  Legislatures
• Using Descriptives finds 6 to 40
• Explanations
• of college grads
• Christian residents

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Correlation Matrix
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Scatterplot

• Graphic plot of all cases across 2 variables
• Horizontal Axis IV (christad)
• Vertical Axis DV (womleg)
• Interactive Scatterplot tailored to report BASIC
  regression estimates
• Can also use scatter but does not report
  regression estimates

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Scatterplot Regression Estimates

• Y a b (X)
• Y Dependent Variable
• X Independent Variable
• a Y Intercept
• Estimated value of Y when X 0
• b Regression Coefficient
• Change in DV for each 1-unit change in IV
• R-Square Strength of Relationship
• proportion of the variation in DV explained by
  the IV

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Scatterplot Regression Estimates

• Y a b (X)
• Women Legislators 3.01 0.74 (college)
• For each 1-unit increase in the percentage of
  college graduates, there is a .74-unit increase
  in the percentage of female legislators.

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Example of scatterplot
This scatter plot represents age vs. size of a
plant.  As the plant ages, its size tends to
increase.  If it seems that the points follow a
linear pattern then we say that there is a high
linear correlation, while if it seems that the
data do not follow a linear pattern, we say that
there is no linear correlation.  If the data
somewhat follow a linear path, then we say that
there is a moderate linear correlation
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Scatterplot Regression Estimates

• Estimate of women legislators for a state
• Start with intercept (3.01)
• Add 0.74 for each percentage of states pop with
  a college degree
• If 10 Y 3.01 .74 10 10.4

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Regression

• y a bx
• Bivariate
• One IV to predict a DV
• Multiple
• More than one IV to predict a DV

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Regression Example

• Factors related to motor vehicle deaths
• Population density
• Sparsly populated
• Drive More
• Drive Faster
• Fewer People to witness/help
• Fewer Hospitals
• Younger drivers more likely to get in accidents

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Hypotheses

• In comparing states, those with lower population
  density will have higher number of car crash
  fatalities than more populated states
• In comparing states, those with more people under
  20 will have higher number of car crash
  fatalities than states with fewer people under 20
  
• Variables
• Carfatal motor vehicle deaths per 100k
  residents
• Density state population per square mile
• Under20 of population 19 years and younger

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Regression

• Y a b (X)
• MV Fatalities 20.278 0.014density
• Alaska 1 per sq mile
• 20 per 100,000
• New Jersey 1,000 per sq mile
• 6 per 100,000
• b for each addition person per square mile, MV
  Fatality drops by .014.

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Regression Null Hypothesis

• In the population no relationship exists between
  the IV and the DV
• In the population the true regression coefficient
  is equal to 0
• Further, the regression coefficient of -.014 was
  occurred by chance.

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Regression Interpretation

• Coefficients
• Constant Y Intercept
• Pop Per Sq Mile IV
• B values for constant and IV
• T value from analysis (get p-value from
  t-table)
• Generally look for t (absolute value) above 2
  significant
• Sig P-values for this test
• Ignore Constant (for most applications)
• Probability of obtaining the results if null is
  correct
• If over .05 observed results occur too frequently
  by chance

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Regression Interpretation

• Model Summary
• R-Square
• Overall measure of how well IV explains DV
• Multiply by 100 represents of variance
  explained
• Of all variation among states in MV fatality
  rates 38.3 is explained by population density

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R-Squared

• ? b (standard error of b)

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Regression Example What do we know

• Population density significant negative effect on
  MV Fatality
• Percentage of Younger people has significant
  positive effect on MV Fatality
• Correlation between density and of younger
  people

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Regression Example What do we know

• Population density significant negative effect on
  MV Fatality
• Percentage of Younger people has significant
  positive effect on MV Fatality
• Small correlation between density and of
  younger people (-.139)
• Density up of Young Down
• Perhaps states with higher percentages of young
  people have higher fatality rates because they
  have lower population densities, not younger
  drivers.
• Relationship between under20 and carfatal may be
  spurious

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Multiple Regression

• Designed to estimate the partial effect of an IV
  on DV, controlling for effect of other IVs
• Y a b(X1) b(X2)

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• MV Fatalities 7.535 -.012(density) 0.428
  (under20)