Regression
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Regression. The basic problem Regression and Correlation Accuracy of prediction in regression Hypothesis testing Regression with multiple predictors. The Basic Problem. How do we predict one variable from another? How does one variable change as the other changes?

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Regression

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Regression

  • The basic problem

  • Regression and Correlation

  • Accuracy of prediction in regression

  • Hypothesis testing

  • Regression with multiple predictors


The Basic Problem

  • How do we predict one variable from another?

  • How does one variable change as the other changes?

  • Cause and effect (can only be inferred if it makes theoretical sense)


An Example

  • Effect of Dow Jones Performance on Darts performance (to what degree can Dow Jones predict Dart performance)


The Data


Relationship can be represented by line of best fit


Why use regression?

  • We may want to make a prediction.

  • More likely, we want to understand the relationship.

    • How fast does Darts rise with one unit rise in Dow Jones?


Regression Line

  • Formula

    • = the predicted value of Y (Darts)

    • X = Dow value


Regression Coefficients

  • “Coefficients” are a and b

  • b = slope (also called rate of change)

    • Change in predicted Y for one unit change in X

  • a = intercept

    • value of when X = 0


Calculation

  • Slope

  • Intercept


For Our Data

  • b = 11.13/5.43 = 2.04

  • a = 14.52 - 2.04*5.95 = 2.37

  • See SPSS printout on next slide


SPSS Printout for one Predictor

R2, Percentage of Variance


SPSS printout cont.

Error of prediction

Is regression

Significant?

Intercept

Slope


Note:

  • The values we obtained are shown on printout.

  • The intercept is labeled “constant.”

  • Slope is labeled by name of predictor variable.


Making a Prediction

  • Suppose that we want to predict Darts score for a new Dow Score of 200

  • We predict that Darts will be at 23.65 when Dow is at 25

  • Check with data: what is real value of Darts when Dow is 25


Prediction

Residual


Errors of Prediction

  • Residual variance

    • The variability of predicted values

  • Standard error of estimate

    • The standard deviation of predicted values


Standard Error of Estimate

  • A common measure of the accuracy of our predictions

    • We want it to be as small as possible.


r 2 as % Predictable Variability

  • Define Sum of Squares


Major Points

  • Predicting one dependent variable from multiple predictor variables

  • Example with Product Advisor Data

  • Multiple correlation

  • Regression equation

  • Predictions


The Problem

  • In the product advisor study, we asked participants to rate the system on a number of aspects: e.g, usefulness, ease of use, trust, kind of product information, number of ratings etc.

  • Lets think of overall usefulness as our dependent variable. Which of the above factors can predict overall usefulness?

  • What percentage variance do they explain in the usefulness overall?

  • What factors play the more important role?


Product Advisor Data


Correlational Matrix


Regression Results(using simple linear regression using method “enter”

R2, Percentage of Variance


Regression is significant

Importance of each variable

Is contribution significant?


Regression Coefficients

  • Slopes and an intercept.

  • Each variable adjusted for all others in the model.

  • Just an extension of slope and intercept in simple regression

  • SPSS output on next slide


Regression Equation

  • A separate coefficient for each variable

    • These are slopes

  • An intercept (here called b0 instead of a)


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