PS 225 Lecture 20

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PS 225Lecture 20

• Linear Regression Equation and Prediction

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Adding Regression Line
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Dependence

• What if two variables are correlated?
• What if the mean of a variable is dependent on
  the value of another variable?
• Is it dependent?
• How much is it dependent?
• How can we express the dependence algebraically?

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Examples of Dependence
Linear Relationships

• The distance traveled at a given speed
• x
• The cost of a bag of bulk mixed nuts with a given
  price per pound
• 
  x

Distance
Speed
Time
Weight
Cost
Price
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Types of Relationships

• Deterministic Relationship
• One variable totally determines the value of
  another variable with perfect accuracy
• Algebraic linear relationship
• Previous examples
• Variable
• One variable affects the value of another
  variable with some element of variability
• Example Height and weight

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Using SPSS to Determine a Linear Relationship

• Is there a relationship?

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Linear Regression Form of a Line

• Algebraic Form of Line
• A is the y-intercept
• B is the slope
• Linear Regression Meaning of the Line
• A is the constant
• B is a coefficient

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SPSS Output for A Regression Line
Y -18331.2 3909.907x X Education Level Y
Current Salary
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Interpreting the Constant

• Only has meaning if
• Data present to validate
• Can naturally occur

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Interpreting the Coefficient
Change in dependent variable for each unit change
in the independent variable
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2-Step Hypothesis Process

• Test Overall Linear Relationship
• Test Contribution of Each Component

Similar to 2-Way ANOVA
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Step 1 Overall Test

• Is there a linear relationship?
• Ho Means are the same at all values of x (No
  relationship)
• Ha There is a linear relationship between x and
  y
• If significancelt.05 conclude relationship
• Otherwise, stop analysis

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Step 2 Component Tests

• Is the component significant?
• Intercept
• Coefficient
• Ho Not Significant
• Ha Significant
• If significancelt.05 conclude significant
• Otherwise, eliminate from analysis and recreate
  model

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Line of Best Fit

• Regression line that minimizes the distance to
  data points
• SPSS calculations

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Sum of Squares

• Sum of squared differences for each data point
• Regression- Difference between overall mean and
  regression line
• Residual- Difference Between the regression line
  and data points
• Regression lines minimize the residual sum of
  squares

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Deviations
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Sum of Squares
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Predicting Values from a Linear Regression

• Write equation for the regression line
• Plug in independent variable
• Gain a prediction for the dependent variable
• The relationship between the values of the
  independent variable and the prediction are
  deterministic

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Accuracy of Predictions

• The BEST guess
• Probably not exact due to variability
• Correct on average

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Quality of Prediction

• Predicted values must be within the range of the
  data
• Relationship must be linear over the entire range
  of the data
• Line must not depend too strongly on one point

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SPSS Assignment

• Last class we answered the following questions
• Does the number of years of education an
  individual has affect the hours of television a
  person watches?
• Does age affect the hours of television a person
  watches?
• This class Use SPSS to find the regression
  equation that best represents each relationship.
• Write the full regression equation.
• Make a prediction for yourself with each
  regression equation
• How different is each prediction from the number
  of hours you watch? If the equation under
  predicts, report your answer as a negative
  number. If it over predicts report your answer as
  a positive number. Add your prediction error to
  the class data.