
How do you test the covariation between
two continuous variables?

Most typically:

One independent variable

and:

One dependent variable
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Types of Relationships
Scatterplot Diagrams
Y
Y
X
X
No Apparent Relationship Between X and Y
Perfect Positive Relationship Between X and Y
Y
Y
X
Perfect Negative Relationship Between X and Y
X
Parabolic Relationship Between X and Y
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Types of Relationships
Scatterplot Diagrams
Y
Y
X
General Negative Relationship Between X and Y
X
General Positive Relationship Between X and Y
Y
Y
X
Negative Curvilinear Relationship Between X and Y
X
No Apparent Relationship Between X and Y
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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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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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
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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Least-Square Estimation Procedure
• 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.
Y
X
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Liquor
Consumption
# Churches
Correlations
EXAM_1
EXAM_2
REVIEW_1
Pears on Correlation
Sig. (1-tailed)
N
Pears on Correlation
Sig. (1-tailed)
N
Pears on Correlation
Sig. (1-tailed)
N
EXAM_1
1.000
.
42
.334*
.015
42
.226
.075
42
EXAM_2
REVIEW_1
.334*
.226
.015
.075
42
42
1.000
.438**
.
.002
42
42
.438**
1.000
.002
.
42
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*. Correlation is s ignificant at the 0.05 level (1-tailed).
**. Correlation is s ignificant 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
Unstandardized
Coefficients
Model
1
(Constant)
EXAM_1
B
51.344
.291
Std. Error
10.047
.130
Standardized
Coefficients
Beta
.334
t
5.110
2.242
Sig.
.000
.031
a. Dependent Variable: EXAM_2
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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.
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Too
complicated
by hand!
Ouch!

Relationship between 1 dependent & 2 or
more independent variables is a linear
function
Population
Y-intercept
Population
slopes
Random
error
Yi   0   1X 1i   2 X 2i   k X ki   i
Dependent
(response)
variable
Independent
(explanatory)
variables
Bivariate 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
SS yy  SSE
Explained variation
SSE
R 

 1
Total variation
SS yy
SS yy
2

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
Coefficientsa
Model
1
(Constant)
EXAM_1
EXAM_2
REVIEW_1
Unstandardized
Coefficients
Std.
B
Error
56.730 75.172
1.626
.791
8.E-02
.983
.949
.656
Standardized
Coefficients
Beta
.319
.013
.235
t
.755
2.056
.080
1.446
Sig.
.455
.047
.937
.156
a. Dependent Variable: TOTAL
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