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1/15/2016Marketing Research2 How do you test the covariation between two continuous variables? Most typically: One independent variable and: One dependent variable
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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
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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.
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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)
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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
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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
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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.
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Too complicated by hand! Ouch!
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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
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Bivariate model
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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 ^ ^ ^
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Proportion of Variation in Y ‘Explained’ by All X Variables Taken Together
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If you add a variable to the model ◦ How will that affect the R-squared value for the model?
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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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