Combining
correlational analysis with linear regression
 Here is the simple
"scattergram" that plotted how the states voted for
president in 1980 and 1984.
correlation, r = .90


 The entries in
twodimensional space stand for the number of states that
voted about the same for Reagan in each
election.
 The correlation in vote
for Reagan between the two years is very high, meaning
that one could fairly well predict a state's vote in 1984
(Y) with knowledge of the state's vote in 1980
(X)
  IF one knew
the formula for the underlying "regression
line."

Determining the "regression line" that underlies the
correlation coefficient:
 The plot of
reagan84 with reagan80 suggests that a
"linear relationship" exists.
 One can imagine a
straight line through the swarm of points.
 The formula for such
a line is
 Where:
=
the predicted value of the dependent variable,
Y_{i}
 a = a
constant, the point at which the line
crosses the Y axis when X = 0
 b = a
coefficient representing the "slope" of
the line
 X_{i
}= the observed value of the
independent variable for the ith
case
 In fact, a line can be
drawn that constitutes the "best fit" in the sense that
it minimizes the squared deviations of observed Ys
from any alternative line.
 Such a criterion for
drawing a line is referred to as ordinary least
squares (OLS).
 More formally:
 Where: =
sum of squared errors in
prediction
 The square root
of these mean squared deviations is the standard error
of estimate.
Computing the OLS
(Ordinary Least Squares) regression line (these
values are automatically computed within SPSS):
 The slope of the line,
b, is computed by this basic formula:
 In words, this is
equivalent to
 It is also
equivalent to
 The formula for,
a, the intercept is
 Note that if there is
no slope (i.e., an increase in X produces no
increase in Y), b=0
 Thus the second term
on the right would also be 0
 and the intercept,
a, would be equal to the mean of the
dependent variable, Y.
 Thus, the "slope" in
the scatterplot would be a straight line from right to
left, drawn at the mean of Y.

Computing the regression coefficient,
b_{yx} (variable Y regressed on
X):
 Conceptually, the
regression coefficient is the ratio of the
covariation between both variables to the
variation of the independent variable.
 The regression
coefficient byx is an unstandardized coefficient,
which means that it is calculated for the "raw" or
unstandardized data.
 It represents the
slope of the regression linethe amount of
change in Y due to a change of 1 unit of
X.
 Calculating b
using crossproducts and standard deviations: for
variable Y regressed on X,
Here is the line and the regression equation superimposed
on the scatterplot:
How to
produce a scatterplot with the regression line in SPSS
10
Example: Consider the
regression equation predicting to REAGAN84 from REAGAN80, as
calculated by the scatterplot output:

= a + b X_{i}
[REAGAN84
= 14.86 +
(.88)REAGAN80 ]
 The
Correlation output (from a previous run) shows
 CrossProd
Dev of REAGAN84 = 3554.4369 [Thus,
SS_{yx} = 3554]
 Variance
of REAGAN80 = 80.95; N = 50 [Thus,
SS_{x} = 4047 (variance x N)]
 Calculating
the regression coefficient, b:
 The
coefficient .88 in the regression equation
means that the percentage of a state's vote for
Reagan in 1984 (Y_{i}) increased by
.88 for each percentage point that the state voted for
Reagan in 1980
(X_{i}).
 If the
slope is only .88, how did Reagan win more votes in 1984
than in 1980?
 Note
the value of the intercept, which is
14.86.
 Overall,
Reagan ran almost 15 percentage points better in every
state in 1984 than he did in 1980.
What
is the relationship between b ( the slope) and
r (the correlation coefficient)?

 That is,
if the two variables being correlated have
equal standard deviations (s_{y} =
s_{x})
 Then
b=r, for r would be multiplied by
1 (1/1=1)
 The
implication of all this is
 the
value of the slope, b, always differs
from the correlation coefficient, r,
 to
the extent that the two variables being
correlated, X and Y,
 vary
in their standard deviations, (s_{y}
and s_{x})
 Therefore,
the value of b (the slope) does not necessarily
indicate the value of r (the correlation).
 Indeed,
if the two variables, X and Y, vary
greatly in their standard deviations,
(s_{y} and
s_{x}),
 it is
possible to encounter a very small slope (e.g.,
b=.001) and a high correlation (e.g.,
r=.60)
Equivalent methods for calculating
b_{yx}
 Using
covariance of XY and variance of X:
 Or,
using r and standard deviations of the x and y
variables as described in the section above,
 For
REAGAN84 regressed on
REAGAN80:
How can we interpret the b coefficient?
 These
coefficients refer to the slopes of the regression
lines.
 b
coefficients are interpreted as the amount of
change in the dependent variable (Y) that
is associated with a change in one unit of the
independent variable (X).
 All
b coefficients are unstandardized, which
means that the magnitude of their values is relative
to the means and standard deviations of the
independent and dependent variables in the
equation.
 This
means that the slopes can be interpreted
directly in terms of the raw values of X and
Y,
 whether
the values are
 percents
(as in the regression of REAGAN84 on REAGAN80),
or
 dollars
(as in the regression of tax paid to income
earned); or
 mixed
scales (e.g., battlefield casualties on
tonnage of bombs dropped).
 In
the case of REAGAN84 regressed on REAGAN80, for
example, b=.878 can be interpreted in terms of a
state's voting percentage for Reagan in 1984 and in
1980.
 Because
the value of a b coefficient depends on the scaling of
the raw data, which is somewhat arbitrary (should time
be measured in years, months, days?), b
coefficients cannot be easily compared within a
regression equation.
 Later,
we will consider another type of regression
coefficient, beta,which is standardized
such that it adjusts for the different means and
variances of the variables being
related.
Note that there is "another" regression line for any two
correlated variables, X and Y.
 The productmoment
correlation, r, is symmetrical
 the correlation is
the same whether either X or Y is regarded as
independent or dependent variables.
 But regression analysis
is asymmetrical:
 when the dependent
and independent variables are switched, a different
formula defining the least squares line for X
regressed on Y.
 See Schmidt, pp.
192193, for calculating this "other"
line.
There is another way to measure prediction "error" in
units of measurement for Y:
 The standard error of
the estimate is the standard deviation of
observed values, Y, around predicted
values, Y.
 It is discussed in
the handout from Schmidt on pp. 191192.
 The
STD
ERR OF EST
(not computed
in the scatterplot statistics) is
3.87
for
REAGAN84.
 The standard error of
the estimate is less frequently used in statistical
analysis than the coefficient of determination,
r^{2}
Comments on the effect of the pattern of plots on the
regression line and the value of the correlation
coefficient
 Regression and
correlation analysis is most appropriate when the plot is
linear and homoscedastic.
 Linear regression
analysis underestimates a curvilinear plot between
variables:

 A
homoscedastic plot occurs when the variances of
observed Y values are equal regardless of the X values.
 When the plot is
heteroscadestic, the accuracy of predictions from
X to Y depends on the value of X:

HOMOSCEDASTIC

HETEROSCEDASTIC

 Note also that outliers  such as
Washington, D.C.can affect the relationshipacting to
either lower or raise it.
Example: Go here to see
the effect of dropping Washington, D.C. from the
analysis.
 Whether one does or does not exclude a case from the
analysis rests with the analyst.
 In this instance, the researcher might exclude
Washington, D.C., which is 100% urban, because it is not
really a "state."
