Chapter 8: Bivariate Regression and Correlation

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Chapter 8 Bivariate Regression and Correlation

• Overview
• The Scatter Diagram
• Two Examples Education Prestige
• Correlation Coefficient
• Bivariate Linear Regression Line
• SPSS Output
• Interpretation
• Covariance

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Overview
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Overview
Independent Variables
Nominal Interval
Considers how a change in a variable affects a
discrete outcome
Lambda
Dependent Variable
Interval Nominal
Considers the difference between the mean of one
group on a variable with another group
Considers the degree to which a change in one
variable results in a change in another
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Overview
You already know how to deal with two nominal
variables
Independent Variables
Nominal Interval
Considers how a change in a variable affects a
discrete outcome
Lambda
Dependent Variable
Interval Nominal
Considers the degree to which a change in one
variable results in a change in another
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Overview
You already know how to deal with two nominal
variables
Independent Variables
Nominal Interval
Considers how a change in a variable affects a
discrete outcome
Lambda
Dependent Variable
Interval Nominal
Regression Correlation
Confidence Intervals T-Test
We will deal with this later in the course
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Overview
You already know how to deal with two nominal
variables
Independent Variables
Nominal Interval
Logistic Regression
Lambda
Dependent Variable
Interval Nominal
Regression Correlation
Confidence Intervals T-Test
We will deal with this later in the course
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General Examples
Does a change in one variable significantly
affect another variable? Do two scores tend to
co-vary positively (high on one score high on the
other, low on one, low on the other)? Do two
scores tend to co-vary negatively (high on one
score low on the other low on one, hi on the
other)?
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Specific Examples
Does getting older significantly influence a
persons political views? Does marital
satisfaction increase with length of
marriage? How does an additional year of
education affect ones earnings?
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Scatter Diagrams

• Scatter Diagram (scatterplot)a visual method
  used to display a relationship between two
  interval-ratio variables.
• Typically, the independent variable is placed on
  the X-axis (horizontal axis), while the dependent
  variable is placed on the Y-axis (vertical axis.)

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Scatter Diagram Example

• The data

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Scatter Diagram Example
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A Scatter Diagram Example of a Negative
Relationship
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Linear Relationships

• Linear relationship A relationship between two
  interval-ratio variables in which the
  observations displayed in a scatter diagram can
  be approximated with a straight line.
• Deterministic (perfect) linear relationship A
  relationship between two interval-ratio variables
  in which all the observations (the dots) fall
  along a straight line. The line provides a
  predicted value of Y (the vertical axis) for any
  value of X (the horizontal axis.

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Graph the data below and examine the relationship
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The Seniority-Salary Relationship
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Example Education Prestige
Does education predict occupational prestige? If
so, then the higher the respondents level of
education, as measured by number of years of
schooling, the greater the prestige of the
respondents occupation. Take a careful look at
the scatter diagram on the next slide and see if
you think that there exists a relationship
between these two variables
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Scatterplot of Prestige by Education
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Example Education Prestige

• The scatter diagram data can be represented by a
  straight line, therefore there does exist a
  relationship between these two variables.
• In addition, since occupational prestige becomes
  higher, as years of education increases, we can
  say also that the relationship is a positive one.

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Take your best guess?
If you know nothing else about a person, except
that he or she lives in United States and I asked
you to his or her age, what would you guess?

• The mean age for U.S. residents.
• Now if I tell you that this person owns a
  skateboard, would you change your guess? (Of
  course!)
• With quantitative analyses we are generally
  trying to predict or take our best guess at value
  of the dependent variable. One way to assess the
  relationship between two variables is to consider
  the degree to which the extra information of the
  second variable makes your guess better. If
  someone owns a skateboard, that is likely to
  indicate to us that s/he is younger and we may be
  able to guess closer to the actual value.

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Take your best guess?

• Similar to the example of age and the skateboard,
  we can take a much better guess at someones
  occupational prestige, if we have information
  about her/his years or level of education.

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Equation for a Straight Line
Y a bX where a intercept b slope Y
dependent variable X independent variable
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Bivariate Linear Regression Equation

• Y a bX
• Y-intercept (a)The point where the regression
  line crosses the Y-axis, or the value of Y when
  X0.
• Slope (b)The change in variable Y (the dependent
  variable) with a unit change in X (the
  independent variable.)

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SPSS Regression Output (GSS)Education Prestige
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SPSS Regression Output (GSS)Education Prestige
Now lets interpret the SPSS output...
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The Regression Equation
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The Regression Equation
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Interpreting the regression equation

• If a respondent had zero years of schooling, this
  model predicts that his occupational prestige
  score would be 6.120 points.
• For each additional year of education, our model
  predicts a 2.762 point increase in occupational
  prestige.

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Ordinary Least Squares

• Least-squares line (best fitting line) A line
  where the errors sum of squares, or e2, is at a
  minimum.
• Least-squares method The technique that
  produces the least squares line.

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Estimating the slope b

• The bivariate regression coefficient or the slope
  of the regression line can be obtained from the
  observed X and Y scores.

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Covariance and Variance
Covariance Variance of X Covariance of
X and Ya measure of how X and Y vary together.
Covariance will be close to zero when X and Y are
unrelated. It will be greater than zero when the
relationship is positive and less than zero when
the relationship is negative. Variance of Xwe
have talked a lot about variance in the dependent
variable. This is simply the variance for the
independent variable
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Estimating the Intercept
The regression line always goes through the point
corresponding to the mean of both X and Y, by
definition. So we utilize this information to
solve for a
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Back to the original scatterplot
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A Representative Line
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Other Representative Lines
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Calculating the Regression Equation
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Calculating the Regression Equation
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The Least Squares Line!
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Summary Properties of the Regression Line

• Represents the predicted values for Y for any and
  all values of X.
• Always goes through the point corresponding to
  the mean of both X and Y.
• It is the best fitting line in that it minimizes
  the sum of the squared deviations.
• Has a slope that can be positive or negative
  null hypothesis is that the slope is zero.

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Coefficient of Determination

• Coefficient of Determination (r2) A PRE measure
  reflecting the proportional reduction of error
  that results from using the linear regression
  model. It reflects the proportion of the total
  variation in the dependent variable, Y, explained
  by the independent variable, X.

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Coefficient of Determination
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Coefficient of Determination
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The Correlation Coefficient

• Pearsons Correlation Coefficient (r) The
  square root of r2. It is a measure of
  association between two interval-ratio variables.
• Symmetrical measureNo specification of
  independent or dependent variables.
• Ranges from 1.0 to 1.0. The sign (?) indicates
  direction. The closer the number is to ?1.0 the
  stronger the association between X and Y.

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The Correlation Coefficient
r 0 means that there is no association between
the two variables.
r 0
Y
X
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The Correlation Coefficient
r 0 means that there is no association between
the two variables. r 1 means a perfect
positive correlation.
r 1
Y
X
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The Correlation Coefficient
r 0 means that there is no association between
the two variables. r 1 means a perfect
positive correlation. r 1 means a perfect
negative correlation.
Y
r 1
X