SOCIOLOGY 005
Lecture 6
Correlation and Regression
By doing things the easy way, we were quickly able to determine whether
or not the variation in Y is significantly related to X
We were also able to quickly calculate R
2
and Pearson’s r
But the Shortcuts we took are not without cost
•
These only work with bivariate regression (converting r to R
2
)
•
We don’t have a clear idea about what error really is
•
Our measure of model fit told us nothing about the regression line
itself in terms of inference and confidence intervals
We must now look at error and the form of a regression equation in more
detail
Correlation and Regression
If there was an a strong linear law in which only education affected
number of crimes committed, our previous equation would reflect it.
Y
=
a
+
bX
Predicted Values and Error
ˆ
Y
=
a
+
bX
Unfortunately, Education is but one factor which might be related to
crime and as such we are likely to commit error trying to predict
Y using
X.
This being the case, the real value for Y is not predicted and instead we
typically note that the predicted value for Y is just that 
A
Predicted Value
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Although we adjusted the equation to reflect that Y is a predicted
value, we are still interest in the actual value for Y
We can compare the predicted value of
Y to the actual value of
Y in
order to better understand the relationship between X and Y
This allows us to calculate the amount of error we made when
predicting Y
e
i
=
Y
i
!
ˆ
Y
i
Predicted Values and Error
Since we are now taking error into account, we can rewrite our
formula so the actual value of Y is predicted
Y
=
a
+
bX
+
e
This new equation is the formula for the linear regression model
Predicted Values and Error
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 Spring '10
 dunn
 Sociology, Regression Analysis, SSERROR

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