Part IVSignificantly DifferentUsing Inferential Statistics

Chapter 15

Using Linear Regression

Predicting Who’ll Win the Super Bowl

What you will learn in Chapter 15

How prediction works and how it can be used in the social and behavioral sciences

How and why linear regression works predicting one variable from another

How to judge the accuracy of predictions

The usefulness of multiple regression

What is Prediction All About?

Correlations can be used as a basis for the prediction of the value of one variable from the value of another Correlation can be determined by using a set

of previously collected data (such as data on variables X and Y)

calculate how correlated these variables are with one another

use that correlation and the knowledge of X to predict Y with a new set of data

Remember…

The greater the strength of the relationship between two variables (higher the absolute value of the correlation coefficient) the more accurate the predictive relationship

Why??? The more two variables share in common

(shared variance) the more you know about one variable from the other.

The Logic of Prediction

Prediction is an activity that computes future outcomes from present ones What if you wanted to predict college GPA

based on high school GPA?

Scatter Plot

Regression Line

Regression line – reflects our best guess as to what score on the Y variable would be predicted by the X variable. Also known as the “line of best fit.”

Prediction of Y given X = 3.0

Error in Prediction

Prediction is rarely perfect…

Drawing the World’s Best Line

Linear Regression Formula Y=bX + a

Y = dependent variable the predicted score or criterion

X = independent variable the score being used as the predictor

b = the slope direction of the line

a = the intercept point at which the line crosses the y-axis

Hasbro

Slope & Intercept

Slope – calculating b

Intercept – calculating a

Number of Complaints (y) by Reindeer Age (x)

Complaints by Reindeer Age: Intermediate Calculations

SS Reg, SS Error, R2, and Correlation

Chapter 6 16

Now You Try!!

Participant Hours/Week Video Games College GPA

1 3 3.8

2 15 2.1

3 22 2.5

4 30 0.6

5 11 3.1

6 25 1.9

7 6 3.9

8 12 3.8

9 17 1.7

Printout: Slope Int, SS Reg, SS Error

and R2

College GPA by SAT scores

Slope 0.003478 -1.07148Intercept

0.000832 0.957866

Rsquare 0.686069 0.445998

F 17.48335 8dfsSS Regression 3.477686 1.591314

SS Residual

Severity of Injuries by # hrs per week strength

training;

Slope -0.12507 6.847277Intercept

Stand Error 0.045864 1.004246

R2 0.209854 2.181672

7.436476 28SS Regression 35.39532 133.2713

SS Residual

Using the Computer

SPSS and Linear Regression

SPSS Output

What does it all mean?

SPSS Scatterplot

The More Predictors the Better? Multiple Regression

Multiple Regression Formula Y = bX1 + bX2 + a

Y = the value of the predicted score X1 = the value of the first independent variable

X2 = the value of the second independent variable

b = the regression weight for each variable

The BIG Rule…

When using multiple predictors keep in mind... Your independent variables (X1,, X2 ,, X3 , etc.)

should be related to the dependent variable (Y)…they should have something in common

However…the independent variables should not be related to each other…they should be “uncorrelated” so that they provide a “unique” contribution to the variance in the outcome of interest.

Glossary Terms to Know

Regression line Line of best fit

Error in prediction Standard error of the estimate

Criterion Independent variable

Predictor Dependent variable

Y prime Multiple Regression