PS 225Lecture 20

Linear Regression Equation and Prediction

Adding Regression Line

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?

Examples of Dependence

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

Cost WeightPrice

Linear Relationship

s

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

Using SPSS to Determine a Linear Relationship Is there a relationship?

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’

bxay

SPSS Output for A Regression Line

Y = -18331.2 + 3909.907*x

X = Education Level

Y = Current Salary

Interpreting the Constant

Only has meaning if:

• Data present to validate

• Can naturally occur

Interpreting the Coefficient

Change in dependent variable for each unit change in the independent variable

2-Step Hypothesis Process

Test Overall Linear Relationship Test Contribution of Each Component

Similar to 2-Way ANOVA

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 significance<.05 conclude relationship Otherwise, stop analysis

Step 2: Component Tests

Is the component significant? Intercept Coefficient

Ho: Not Significant Ha: Significant

If significance<.05 conclude significant Otherwise, eliminate from analysis and

recreate model

Line of Best Fit Regression line that minimizes the

distance to data points SPSS calculations

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

Deviations

Sum of Squares

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

Accuracy of Predictions

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

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

SPSS AssignmentLast 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.