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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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05/03/23Marketing Research 2
How do you test the covariation between two continuous variables?
Most typically: One independent variable and: One dependent variable
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No Apparent Relationship Between X and Y
X
Y
Perfect Positive Relationship Between X and Y
X
Y
Y
X
Perfect Negative Relationship Between X and YParabolic Relationship Between X and Y
X
Y
Types of RelationshipsScatterplot Diagrams
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General Positive Relationship Between X and Y
X
Y
No Apparent Relationship Between X and Y
X
YY
X
Negative Curvilinear Relationship Between X and Y
General Negative Relationship Between X and Y
X
Y
Types of RelationshipsScatterplot Diagrams
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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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CorrelationAssessing Measures of Association
Measure of Association using interval or ratio data.
Measure of Association using ordinal or rank order data.
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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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X
Y
• 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 Liquor ConsumptionConsumption
# Churches# Churches
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Correlations
1.000 .334* .226. .015 .075
42 42 42.334* 1.000 .438**.015 . .002
42 42 42.226 .438** 1.000.075 .002 .
42 42 42
Pearson CorrelationSig. (1-tailed)NPearson CorrelationSig. (1-tailed)NPearson CorrelationSig. (1-tailed)N
EXAM_1
EXAM_2
REVIEW_1
EXAM_1 EXAM_2 REVIEW_1
Correlation is significant at the 0.05 level (1-tailed).*.
Correlation is significant at the 0.01 level (1-tailed).**.
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Yi = B0 + B1 X1 + ei
◦ 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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Test statistic:
H0: B1 = 0
Ha: B1 does not = 0
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Coefficientsa
51.344 10.047 5.110 .000.291 .130 .334 2.242 .031
(Constant)EXAM_1
Model1
B Std. Error
UnstandardizedCoefficients
Beta
StandardizedCoefficients
t Sig.
Dependent Variable: EXAM_2a.
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Yi = B0 + B1 X1 +B2 X2 + B3 X3 +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
Y X X Xi i i k ki i 0 1 1 2 2
Dependent Dependent (response) (response) variablevariable
Independent Independent (explanatory) (explanatory) variablesvariables
Population Population slopesslopes
Population Population Y-interceptY-intercept
Random Random errorerror
Bivariate modelBivariate model
1. Slope (k)
◦ Estimated Y Changes by k for Each 1 Unit Increase in Xk Holding All Other Variables
Constant
2. Y-Intercept (0)
◦ Average Value of Y When Xk = 0
^̂
^̂^̂
Proportion of Variation in Y ‘Explained’ by All X Variables Taken Together
yyyy
yy
SSSSE
SSSSESS
R
1 variationTotal
variationExplained2
If you add a variable to the model◦ How will that affect the R-squared value for the
model?
R2 Never Decreases When New X Variable Is Added to Model◦ Only Y Values Determine SSyy
◦ Disadvantage When Comparing Models Solution: Adjusted R2
◦ Each additional variable reduces adjusted R2, unless SSE goes up enough to compensate
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Coefficientsa
56.730 75.172 .755 .4551.626 .791 .319 2.056 .047
8.E-02 .983 .013 .080 .937.949 .656 .235 1.446 .156
(Constant)EXAM_1EXAM_2REVIEW_1
Model1
BStd.Error
UnstandardizedCoefficients
Beta
StandardizedCoefficients
t Sig.
Dependent Variable: TOTALa.
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