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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

- 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)

- 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

- 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?

- Formula
- = the predicted value of Y (Darts)
- X = Dow value

- “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

- Slope
- Intercept

- b = 11.13/5.43 = 2.04
- a = 14.52 - 2.04*5.95 = 2.37
- See SPSS printout on next slide

R2, Percentage of Variance

Error of prediction

Is regression

Significant?

Intercept

Slope

- The values we obtained are shown on printout.
- The intercept is labeled “constant.”
- Slope is labeled by name of predictor variable.

- 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

- Residual variance
- The variability of predicted values

- Standard error of estimate
- The standard deviation of predicted values

- A common measure of the accuracy of our predictions
- We want it to be as small as possible.

- Define Sum of Squares

- Predicting one dependent variable from multiple predictor variables
- Example with Product Advisor Data
- Multiple correlation
- Regression equation
- Predictions

- 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?

R2, Percentage of Variance

Regression is significant

Importance of each variable

Is contribution significant?

- 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

- A separate coefficient for each variable
- These are slopes

- An intercept (here called b0 instead of a)