1/15/2016Marketing Research2  How do you test the covariation between two continuous variables?  Most typically:  One independent variable  and:  One dependent variable

3 No Apparent Relationship Between X and Y X Y Perfect Positive Relationship Between X and Y X Y Perfect Negative Relationship Between X and Y Parabolic Relationship Between X and Y X Y Types of Relationships Scatterplot Diagrams

4 General Positive Relationship Between X and Y X Y No Apparent Relationship Between X and Y X Y Y X Negative Curvilinear Relationship Between X and Y General Negative Relationship Between X and Y X Y Types of Relationships Scatterplot Diagrams

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1/15/2016Marketing Research8  examines the strength of the relationship between two continuous variables  range: ◦ between -1 (perfect inverse relationship), ◦ through 0 (no relationship at all) ◦ to +1 (perfect positive relationship)

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15 Correlation Assessing Measures of Association Measure of Association using interval or ratio data. Measure of Association using ordinal or rank order data.

1/15/2016Marketing Research16  How do you test the covariation between  one continuous independent variable ◦ (e.g., age, income)  and:  one continuous dependent variable ◦ (e.g., cost of automobile purchased)

17 Used to fit data for X and Y Enables estimation of non-plotted data points Results in a straight line that fits the actual observations (plotted dots) better than any other line that could be fit to the observations. Least-Square Estimation Procedure

Liquor Consumption # Churches

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1/15/2016Marketing Research20  Y i = B 0 + B 1 X 1 + e i ◦ Y is the dependent variable (estimated outcome) ◦ B0 is the value of Y when X = 0 (the Y intercept) ◦ B1 is the rate at which Y changes for every unit change in X (the slope) ◦ and e is the error in the model

1/15/2016Marketing Research21  Test statistic:  H0: B 1 = 0  Ha: B 1 does not = 0

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1/15/2016Marketing Research25  Yi = B 0 + B 1 X 1 +B 2 X 2 + B 3 X 3 +e ◦ where each of the Betas estimate the effect of one independent variable.  This allows the regression to "control" for each of the other factors simultaneously ◦ e.g., control for exercise, eating habits, AND fish consumption on heart attacks.

Too complicated by hand! Ouch!

 Relationship between 1 dependent & 2 or more independent variables is a linear function Dependent (response) variable Independent (explanatory) variables Population slopes Population Y-intercept Random error

Bivariate model

 1.Slope (  k ) ◦ Estimated Y Changes by  k for Each 1 Unit Increase in X k Holding All Other Variables Constant  2.Y-Intercept (  0 ) ◦ Average Value of Y When X k = 0 ^ ^ ^

 Proportion of Variation in Y ‘Explained’ by All X Variables Taken Together

 If you add a variable to the model ◦ How will that affect the R-squared value for the model?

 R 2 Never Decreases When New X Variable Is Added to Model ◦ Only Y Values Determine SS yy ◦ Disadvantage When Comparing Models  Solution: Adjusted R 2 ◦ Each additional variable reduces adjusted R 2, unless SSE goes up enough to compensate

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