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Transcript
```CORRELATION AND
REGRESSION
Topic #13
An Example
Recall sentence #11 in Problem Set #3A (and #9):
If you want to get ahead, stay in school.
Underlying this nagging parental advice is the following claimed empirical
relationship:
+
LEVEL OF EDUCATION =====> LEVEL OF SUCCESS IN LIFE
Suppose we collect data through by means of a survey asking respondents
(say a representative sample of the population aged 35-55) to report the
number of years of formal EDUCATION they completed and also their
current INCOME (as an indicator of SUCCESS). We then analyze the
association between the two interval variables in this reformulated
hypothesis.
+
LEVEL OF EDUCATION =========> LEVEL OF INCOME
(# of years reported)
(\$000 per year)
Since these are both interval continuous variables, we analyze their association
by means of a scattergram.
Having collected such data in two rather different societies
A and B, we produce these two scattergrams.
An Example (cont.)
• Note that the two scattergrams are drawn with the same
horizontal and vertical scales.
– With respect to the vertical axis in scattergram A, this violates
the usual guideline that scales be set to minimize “white space”
in the scattergram.
– But here we want to facilitate comparison between the two
charts.
• Both scattergrams show a clear positive association
between the two variables, i.e., the plotted points in both
form an upward-sloping pattern running from Low – Low
to High – High.
• At the same time there are obvious differences between
the two scattergrams (and thus between the relationships between INCOME and EDUCATION in societies A
and B).
Questions For Discussion
• In which society, A or B, is the hypothesis most powerfully confirmed?
• In which society, A or B, is their a greater incentive for people to stay in
school?
• Which society, A or B, does the U.S. more closely resemble?
• How might we characterize the difference between societies A and B?
An Example (cont.)
• We can visually compare and contrast the nature of the
associations between the two variables in the two
scattergrams by drawing a number of vertical strips in
each scattergram (as we did in the earlier Pearson
scattergram of SON’S HEIGHT by FATHER’S HEIGHT).
– Points that lie within each vertical strip represent
respondents who have (just about) the same value on
the independent (horizontal) variable of EDUCATION.
• Within each strip, we can estimate (by “eyeball
methods”) the average magnitude of the dependent
(vertical) variable INCOME and put a mark at the
appropriate level.
Average Income for Selected Levels of Education
We can connect these marks to form a line of averages
that is apparently (close to being) a straight line.
An Example (cont.)
• Now we can assess two distinct characteristics of the
relationships between EDUCATION and INCOME in
scattergrams A and B.
– How much the does the average level of INCOME change
among people with different levels of education?
– How much dispersion of INCOME there is among people with
the same level of EDUCATION?
An Example (cont.)
• In both scattergams, the line of averages is upwardsloping, indicating a clear apparent positive effect on
EDUCATION on INCOME.
– But in the scattergram for society A, the upward slope
of the line of averages is fairly shallow.
• The line of averages indicates that average INCOME
EDUCATION.
– On the other hand, in the scattergram for society B,
the upward slope of the line of averages is much
steeper.
• The graph in Figure 1B indicates that average INCOME
EDUCATION.
– In this sense, EDUCATION is on average more
“rewarding” in society B than A.
An Example (cont.)
• There is another difference between the two
scattergrams.
– In scattergram A, there is almost no dispersion within each
vertical strip (and almost no dispersion around the line of
averages as a whole).
– In scattergram B, there is a lot of dispersion within each vertical
strip (and around the line of averages as a whole).
• We can put this point in substantive language.
– In society A, while additional years of EDUCATION produce
rewards in terms of INCOME that are modest (as we saw
before), these modest rewards are essentially certain.
– In society B, while additional years of EDUCATION produce on
average much more substantial rewards in terms of INCOME (as
we saw before), these large expected rewards are highly
uncertain and are indeed realized only on average.
• For example, in scattergram B (but not A), we can find many pairs
of cases such that one case has (much) higher EDUCATION but
the other case has (much) higher INCOME.
An Example (cont.)
• This means that in society B, while EDUCATION has a
big impact on EDUCATION, there are evidently other
(independent) variables (maybe family wealth, ambition,
career choice, athletic or other talent, just plain luck, etc.)
that also have major effects on LEVEL OF INCOME.
• In contrast, in society A it appears that LEVEL OF
EDUCATION (almost) wholly determines LEVEL OF
INCOME and that essentially nothing else matters.
– Another difference between the two societies is that, while both
societies have similar distributions of EDUCATION, their
INCOME distributions are quite different.
• A is quite egalitarian with respect to INCOME, which ranges only
egalitarian with respect to INCOME, which ranges from under to
\$10,000 to at least \$100,000 — and possibly higher.)
• In summary, in society A the INCOME rewards of
EDUCATION are modest but essentially certain, while in
society B the INCOME rewards of EDUCATION are
substantial on average but quite uncertain in individual
cases.
Two Kinds of “Strength of Association”
• This example illustrates that, given bivariate data for
interval variables, “strength of association” between
them can mean either of two quite different things:
– the independent variable has a very reliable or certain
association with the dependent variable, as is true for society A
but not B, or
– the independent variable has on average a big impact on the
dependent variable, as is true for society B but not A.
• There are two bivariate summary statistics that capture
these two different kinds of strength of association
between interval variables:
– the first second is called the regression coefficient, customarily
designated b, which is the slope of the line of averages; and
– the second is called the correlation coefficient, customarily
designated r , which is (more or less) determined by the
magnitude of dispersion in each vertical strip.
• In scattergrams A and B, A has the greater correlation
coefficient and B has the greater regression coefficient.
Review: The Equation of a Line
• Having drawn (at least by “eyeball methods”) the line of averages in
a scattergram, it is convenient to write an equation for the line.
• You should recall from high-school algebra that, given any graph
with a horizontal axis X and a vertical axis Y, any straight line drawn
on the graph has an equation of the form (using the symbols you
probably used in high school)
y = mx + b,
where
– m is the slope of the line expressed as Δ y / Δ x (“change in y”
divided by “change in x” or “rise over run”), and
– b is the y-intercept (i.e., the value of y when x = 0).
• Evidently to further torment students, in college statistics the symbol
b is used in place of m to represent the slope of the line and the
symbol a is used in place of b to represent the intercept (and a is
customarily placed in front of the bx term), so the equation for a
straight line is usually written as
y= a+bx.
Slope and Intercept in Scattergrams A and B
Equation of a Line (cont.)
• The equation for the line of averages in scattergram A appears to be
approximately
Y
= a
+
b ×
x
AVERAGE INCOME = \$40,000 + \$1000 × EDUCATION .
• The equation for the line of averages in Figure B appears to be
approximately
AVERAGE INCOME = \$10,000 + \$4000 × EDUCATION .
• Given such an equation (or formula), we can take any value for the
independent variable EDUCATION, plug it into the appropriate
formula above, and calculate or “predict” the corresponding average
or “expected” value of the dependent variable INCOME.
– In society A, such a prediction is likely to be quite reliable,
because there is very little dispersion in any vertical strip and the
association/correlation between the two variables is almost
perfect
– In society B, this prediction will be much less reliable or “fuzzier,”
because there is considerable dispersion in every vertical strip
and the association/correlation between the two variables is
much less than perfect.
Before we used eyeball methods to draw the line of averages in the SON’S
HEIGHT by FATHER’S HEIGHT scattergram. Let’s write its equation in the form
SON’S HEIGHT = a + b × FATHER’S HEIGHT .
Equation for Son’s Height
• What is the equation for the line of averages?
– SH = 63 + 0.5 × FH
– SH = 63 + 0.5 × 74 = 63 + 37 = 100 (8' 4")
• Clearly something is wrong:
– 63" is not the true intercept, i.e., it is
• not average SH when FH = 0,
• but rather average SH when FH = 58".
– My height is 16" greater than 58", so my son’s expected height is
• 63 + 0.5 × 16 = 72"
• To read the true y-intercept a on the chart, we need to
extend the FH scale down to FH = 0 to see where the
line of averages intersects the true SH axis.
Determining the “Line of Averages”
[Regression Line] and the Degree of
Association [Correlation] Numerically
• Clearly statisticians are not satisfied with “eyeballing” a
scattergram and
– visually estimating the slope and intercept of the line averages,
or (especially)
– guessing at the degree of association between the DEPENDENT
and INDEPENDENT variables.
• Before we can make exact numerical calculations, we
must have an exact definition of the “line of averages.”
– This will lead to precise formula for the slope, intercept, and
association/correlation.
Example: Worksheet in Topic #13
Correlation and Regression (cont.)
Consider the scattergram
to the right, which
displays the bivariate
numerical data for x
and y in the Sample
Problem presented on
p. 9 of Handout #13.
The “vertical strips” kind
of argument simply
will not work, because
most strips include no
data points and only
one strip (for x ≈ 5)
includes more than
one point.
Using this simple
example, we will now
proceed a little more
formally.
Correlation and Regression (cont.)
• Suppose we were to ask what horizontal line (i.e., a line
for which the slope b = 0, so its equation is simply y = a)
would “best fit” (come as close as possible to) the plotted
points.
• In order to answer this question, we must have a specific
criterion for “best fitting.”
• The one statisticians use is called the least squares
criterion.
– For each horizontal line, we calculate the sum (or mean) of
squared deviations of the y-observations from a line.
– We’re looking for the horizontal line that minimizes this
sum/mean.
• Lets try the horizontal line y = 6:
– The sum of squared deviations = 138,
– so the mean squared deviation is 23.
• Can we do better that this?
Correlation and Regression (cont.)
• That is, we want to find the horizontal line such that the
squared vertical deviations from the line to each point
are (in total or on average) as small as possible.
– You might remember (or should be able to guess) what line this
is — it is the line y = mean of y , or in this case y = 5.
– Recall from Handout #6, p. 4, point (e) that “the sum (or average)
of the squared deviations from the mean is less than the sum of
the squared deviations from any other value of the variable.”
– You should also recall that we have a special name for “the
average squared deviation from the mean (of y)” – namely, the
variance (of y)
• This (or its positive square root, i.e., the standard deviation of
y) is the standard univariate measure of dispersion in y.
Correlation and Regression (cont.)
• So the line y = 5 is the best fitting horizontal line by the
least squares criterion.
• Now suppose that, in finding the best fitting line by the
least squares criterion, we are no longer restricted to
horizontal lines but can tip the line up or down, so it has
a non-zero slope and its height varies with the
independent variable x.
• More particularly, suppose that we pivot such straight
lines on the point that is right in the middle of the all data
plotted points,
– specifically the point that represents the mean of x
and the mean of y (in this case, x = 4, y = 5),
– until it has a slope that best fits all the plotted points
by the same least squares criterion.
• In our example, we can clearly improve the fit by tipping
the line counter-clockwise.
Tipping the Horizontal Line Can Improve the
Fit by the Least Squares Criterion
Finding the Best Fitting Line
• The question is how far we must tip the line to get the
best fit (by the least squares criterion).
– If we tip it too far, the fit becomes worse.
• When we have found this best fitting (by the least
squares criterion) line, we have found what is thereby
defined as the regression line.
• Fortunately, we don’t have to find this best fitting line by
trial and error methods.
– There are formulas for finding slope of the regression line (i.e.,
the regression coefficient) and intercept from the numerical data.
– There is a related formula for finding the correlation coefficient
from the numerical data, which tells us how well this “best fitting”
regression line fits the data points.
Association Between X and Y?
• Mean Squared Deviation from regression line equals
7.9375.
• The degree of assiciation would be related to how big
the MSD is compared with MSD around the best fitting
horizontal line, which equals 22.
• Degree of association = 1 – 7.9375
22
= 1 - .3608 = 0.6392
• Note that this number must be between 0 and 1 (but
cannot be negative).
Correlation and Regression (cont.)
• One statistical theorem tells us that the regression line
goes through the point corresponding to the mean of x
and the mean of y.
• Another theorem gives us formulas to calculate the slope
and intercept of the regression line, given the numerical
data.
• A third theorem
– gives us the formula to calculate the correlation coefficient, and
also
– tells us that the squared correlation coefficient r 2 measures how
much we can improve the goodness of fit (by the least squares
criterion) when we move from the best fitting horizontal line to
the best fitting tipped line.
How to Calculate the Regression and
Correlation Coefficients
• Let’s call the independent
variable X and the
dependent variable Y.
(This usage is standard.)
• Set up a worksheet like
the one that follows (also
p. 9 of Handout #13).
How to Calculate Coefficients (cont.)
Compute the following univariate statistics: (1) the mean of X, (2) the
mean of Y, (3) the variance of X, (4) the variance of Y, (5) the SD of
X, and (6) the SD of Y.
How to Calculate Coefficients (cont.)
Now we make bivariate calculations for each case by
multiplying its deviation from the mean with respect to X
by its deviation from the mean with respect to Y.
– This is called the crossproduct of the deviations.
How to Calculate Coefficients (cont.)
• Notice that a crossproduct in a given case:
– is positive if, in that case, the X and Y values both deviate in the same
direction (i.e., both upwards or both downwards) from their respective
means; and
– it is negative if, in that case, the X and Y values deviate in opposite
directions (i.e., one upwards and one downwards) from their respective
means.
– In either event, the [absolute] crossproduct is large if both [absolute]
deviations are large and small if either deviation is small.
– The crossproduct is zero if either deviation is zero.
• Thus:
– (a) if there is a positive relationship between the two variables,
most crossproducts are positive and many are large;
– (b) if there is a negative relationship between the two variables,
most crossproducts are negative and many are large; and
– (c) if the two variables are unrelated, the crossproducts are
positive in about half the cases and negative in about half the
cases.
• So, the average of all crossproducts is indicative of the
association between variables.
How to Calculate Coefficients (cont.)
• Add up the crossproducts over all cases.
• The sum (and thus the average) of crossproducts is
– positive in the event of (a) above,
– negative in the event of (b) above, and
– (close to) zero in the event of (c) above.
• Divide the sum of the crossproducts by the number of cases to get
the average (mean) crossproduct. This average is called the
covariance of X and Y, and its formula is the bivariate parallel to the
univariate variance (of X or Y).
How to Calculate Coefficients (cont.)
• Notice the following:
– If the association between X and Y is positive, their covariance is
positive (because most crossproducts are positive).
– If the association between X and Y is negative, their covariance
is negative (because most crossproducts are negative).
– If there is [almost] no association between X and Y , their
covariance is [approximately] zero (because positive and
negative crossproducts [approximately] cancel out when added
up).
• Thus the covariance does measure the association
between the two variables.
How to Calculate Coefficients (cont.)
• However, the covariance is not a (fully) valid measure of
the association, because the magnitude of the average
(positive or negative) crossproduct depends on not only
– how closely the two variables are associated, but also
– on the magnitude of their dispersions (as indicated by
their standard deviations or variances).
– So, for example:
• two very closely (and positively) associated
variables have a positive but small covariance if
they both have small standard deviations;
• two not so closely (but still positively) associated
variables have a larger covariance if their standard
deviations are sufficiently larger.
Multiplying all Values by 10 does not Change
the Degree of Association between the Two
Variables X and Y
But Doing This Increases the
Covariance of X and Y 100-Fold
The Correlation Coefficient
• The correlation coefficient is a measure of association
only, which (like other measures of association) is
standardized so that it always falls in the range from -1
to +1.
– This is accomplished by dividing the covariance of X
and Y by the standard deviation of X and also by the
standard deviation of X.
– This is equivalent to calculating the covariance of X
and Y if the X and Y data is converted into standard
scores.
• The degree of correlation in a scattergram can be
observed by looking only at the plotted points, the
regression line, and the orientation of the horizontal and
vertical axes.
– The units of measurement on each axis make no
difference and do not have to be shown.
The Correlation Coefficient (cont.)
• Divide the covariance of X and Y by the standard
deviation of X and also by the standard deviation of Y.
• This gives the correlation coefficient r, which measures
the degree (or “reliability” or “completeness”) and
direction of association between two interval variables X
and Y.
Correlation coefficient = r =
Cov (X,Y)
SD(X) × SD(Y)
• If you calculate r to be greater than +1 or less than -1,
– The correlation coefficient is the one measure of association for which
you are expected to know how to use the calculating formula (above).
• It is a good idea first to construct a scattergram of the
data on X and Y and then to check whether your
calculated correlation coefficient looks plausible in light
of the scattergram.
Calculating the Correlation Coefficient
Properties of the Correlation Coefficient
• The correlation coefficient formula is based on a ratio of “X-units ×
Y-units” (i.e., their respective deviations from the mean) divided by
“X-units × Y-units” (i.e., their respective SDs).
– This means that all units of measurement cancel out and the correlation
coefficient a pure number that is not expressed in any units.
– For example, suppose the correlation between the height (measured in
inches) and weight (measured in pounds) of students in the class is r =
+.45.
• This is not r = +.45 inches or r = +.45 pounds, or r = +.45 pounds
per inch, etc.,
• rather it is just r = +.45.
– Moreover, the magnitude of the correlation coefficient is independent of
the units used to measure the two variables.
• If we measured students’ heights in feet, meters, etc. (instead of
inches) and/or measured their weights in ounces, kilograms, etc.
(instead of pounds), and then calculated the correlation coefficient
based on the new numbers, it would be just the same as before,
i.e., r = +.45.
• In addition, if you check the formula above, you can see that r is
symmetric, i.e., it is unchanged if we interchange X and Y.
– Thus, the correlation between two variables is the same regardless of
which variable is considered to be independent and which dependent.
Calculating the Regression Coefficient
• To compute the regression coefficient b, divide the covariance of X
and Y by the variance of the independent variable X.
Cov(X,Y)
Regression coefficient = b = ----------Var (X)
Properties of the Regression Coefficient
• So, from one point of view, the regression coefficient answers this
question:
– How big is the covariance of the IND and DEP variables compared with
the variance of the IND variable?
• But (as we have already seen) from a more practical point of view,
the regression coefficient (i.e., the slope of the regression line [or
line of averages]) answers this question:
How many units, on average, does the dependent variable increase
when the independent variable increases by one unit?
– If the dependent variable decreases (has a “negative increase”)
when the independent variable increases, the regression
coefficient is negative.
• Thus the magnitude of the regression coefficient [unlike the
correlation coefficient] depends on
– which variable is considered to be independent and which
dependent, and also
– the units in which each variable is measured.
Properties of the Regression
Coefficient (cont.)
• The regression coefficient is a ratio of “X-units × Y-units”
(i.e., their respective deviations from the mean) divided
by “X-units × X-units” (i.e., the variance of X).
• Thus the regression coefficient is expressed in units of
the dependent variable per unit of the independent
variable; that is,
– it is a rate,
• like “miles per hour” [rate of speed] or
• “miles per gallon” [rate of fuel efficiency], and so
– its numerical value changes as these units change.
Properties of the Regression
Coefficient (cont.)
• Remember that we informally calculated regression coefficients (by
eyeball methods) in the INCOME by EDUCATION example.
– In society A, the regression coefficient was about +\$1000 (of
INCOME [DEP]) per YEAR (of EDUCATION [IND]).
– In society B, the regression coefficient was about +\$4000 (of
INCOME [DEP]) per YEAR (of EDUCATION [IND]).
• Likewise we informally calculated the regression coefficient in the
Pearson (SON’S HEIGHT by FATHER’S HEIGHT)
– It was about +0.5 INCHES (of SH [DEP]) per INCH (of FH
[IND]).
– In this (rather special) case, both variables are “in the same
currency” and measured in the same units (INCHES), so
– if we change the unit of measurement for both variables to
FEET, CENTIMETERS, etc.), the magnitude of the regression
coefficient does not change.
Properties of the Regression Coefficient (cont.)
• But suppose we measure the height in inches and
weight in pounds of all students in the class, suppose
we treat height as the independent variable, and
suppose the regression coefficient is b = +6.
– This means +6 pounds (dependent variable units) of weight per
inch (independent variable units) of height.
• That is, if we lined students up in order from shortest to tallest, we
would observe that their weights increase, on average, by about 6
pounds for each additional inch of height.
– This also means that students’ weights increase, on average, by
pound) for each additional inch of height.
• So if weight were measured in kilograms instead of pounds, the
regression coefficient would be b = +6 x.45 = +2.7 kilograms per
inch.
• Likewise if we measured weight in pounds but height in feet, the
regression coefficient would be about b = +6 × 12 = 72 pounds
per foot.
• Remember that the magnitude of the correlation
coefficient is unchanged by such changes in units.
Properties of the Regression Coefficient (cont.)
• If we took weight to be the independent variable and
height the dependent variable,
– the magnitude of the regression coefficient would clearly change,
because
– we would divide Cov(X,Y) by Var(Y), instead of by Cov(X)
• This make sense because the regression coefficient is
– If we lined students up in order of their weights and observed
how their heights varied with weight, the regression coefficient
would be telling us how much students’ heights increase, on
average, in some unit of height (inch, meter, etc.) as their
weights increase by one unit (pound, kilogram, or whatever), so
it would be yet a different number, e.g., perhaps about b = +0.1
inches per pound.
• Remember that the magnitude of the correlation
coefficient is unchanged by reversing the roles of the
independent and dependent variables.
Specifying the Regression Line
• To specify the regression line of the relationship between
independent variable X and dependent variable Y, we
need to know, in addition to the slope of the regression
line (i.e., the regression coefficient b), how high or low the
line with that slope lies in the scattergram.
• Actually, we already know enough to draw the regression
line into the scattergram precisely.
– The regression line is the line that passes through the point that
equals the mean of x and the mean of y and that has a slope b.
• But to write the equation of the regression line, we need
to know the intercept a, i.e., where the regression line
passes through the vertical axis representing values of
the dependent variable Y when the vertical Y axis
intersects the horizontal X axis at the point x = 0.
• This intercept a is equal to the mean of Y minus b times
the mean of X.
Finding the Intercept
Finding the Intercept
Properties of the Intercept
• The magnitude of the intercept is expressed in Y-units.
• The value of the intercept answers this (sometimes quite
artificial or even absurd) question:
What is the average (or expected or predicted) value
of the dependent variable
when the independent variable X is zero?
• Using a and b together, we can answer this question:
what is the expected or predicted value of Y when X is
any specified value. The expected or predicted value of
Y, given that X is some particular value x, is given by the
regression equation (i.e., the equation of the regression
line):
y = a + b × x.
Relationship between Calculations for
Correlation and Regression Coefficients
• Note from the formulas that the sign (i.e., “+” or “-”) of b
and r are both determined by the sign of cov(X,Y), from
which it follows that
– b and r themselves always have the same sign, and
– if one is zero, the other is also zero.
• Notice also that the regression and correlation coefficients
are equal in the event the independent and dependent
variables have the same dispersion, i.e., same SD.
– For example, in the SON’S HEIGHT by FATHER’S HEIGHT
scattergram, by eyeball methods we can determine the regression
– It is also apparent, both from common expectations and
examination of the scattergram, that the dispersions (SDs) of
SON’s HEIGHT and FATHER’s HEIGHT are just about the same,
so we also know that the correlation coefficient is also about +0.5.
• In general, b = r x SD(Y)
SD(X)
and
r = b x SD(X)
SD(Y)
Other Formulas
• You will find other formulas for the correlation and regression
coefficients in textbooks.
– For a horrendous looking example, see Weisberg et al., near top of p.
305 [and below].
– Such formulas are mathematically equivalent to (i.e., give the same
answers as) the formulas given here.
– They are actually easier to work with if you are processing many cases
or programming a calculator or computer, because they require you (or
the computer) to pass through the data only once. (They involve only X
and Y values, not X and Y deviations from the mean)
• But the formulas presented here make more intuitive sense and are
easy enough to use in the simple sorts of problems that you will be
presented with in problems sets and tests.
The Coefficient of Determination r2
• For various reasons, the squared correlation coefficient
r ² is often reported.
– To compute r ², just multiply the correlation coefficient
by itself.
– This results in a number that
• (i) always lies between 0 and 1 (i.e., is never negative and so
does not indicate the direction of association), and
• (ii) is closer to 0 than r is for all intermediate values of r — for
example, (± 0.45)² = + 0.2025.
• The statistic r ² is sometimes called the coefficient of
determination.
• There are three reasons for using this statistic.
The Coefficient of Determination r2
(1) A scattergram with r about +0.5 does not appear to be “halfway between” a
scattergram with r = +1 and one with r = 0 — it looks closer to the one with
zero association. The scattergram that looks “halfway in between” perfect
and zero association has a correlation of about r = +0.7 (and r2 = 0.5).
The Coefficient of Determination r2
(2) r 2 has a specific interpretation that r itself lacks.
– Recall that the line y = 5 (= mean of y) is the horizontal line that
best fits the plotted points by the least squares criterion.
– Recall also that the quantity “average squared deviation from the
line y = 5” has a special and familiar name
• It is the variance of the dependent variable Y (the square root of
which is the SD of Y.)
The Coefficient of Determination r2
• When we tip the line, we can almost always improve the fit at least a
bit.
• The regression line is the tipped line with the best fit according to the
least squares criterion.
• But even this line usually fits the points far from perfectly.
– For each plotted point, there is some vertical distance (positive if the point
lies above the line, negative if it lies below) between the (best fitting)
regression line and the point.
– This vertical distance the called the residual for that case.
– These residuals are the errors in prediction that are “left over” after we
use the regression line to predict the value of the dependent variable in
each case.
• The ratio of the average squared residuals divided by the variance of
Y can be characterized as the proportion of the variance in Y that is
not “predicted” or “explained” by the regression equation that has X
as the independent variable.
• Therefore 1 minus this ratio can be characterized as the proportion of
the variance in Y that is “predicted” or “explained by” the regression
equation.
• It turns out that the latter proportion is exactly equal to r2.
The Coefficient of Determination r2
(3) Finally, serious regression analysis is almost always
multiple (multivariate) regression, where the effects of
multiple independent (and/or “control”) variables on a
single dependent variable are analyzed.
– In this case, we want some summary measure of the
overall extent to which the set of all independent
variables “explains” variation in the dependent
variable, regardless of whether individual independent
variables have positive or negative effects (i.e.,
regardless of whether bivariate correlations are
positive or negative).
– The coefficient of determination r 2 provides this
measure.
Look “Underneath” the Correlation
Correlation Reflects Linear
Association Only.
• Suppose that you calculate
a correlation coefficient and
find that r = 0 (so b = 0
also).
conclusion that there is no
association of any kind
between the variables.
• The zero coefficient tells
you that there is no linear
(straight-line) association
between the variables.
• But there may be a very
clear curvilinear (curvedline) association between
them.
Look “Underneath” the Correlation (cont.)
A Univariate Outlier May
Create a Correlation.
• Suppose that you calculate
a correlation coefficient and
find that r . +0.9.
conclusion that there is a
strong and reliable association between the variables.
• In the adjacent scattergram, a single univariate
outlier produces the high
correlation by itself.
• You should at least double
this case.
Look “Underneath” the Correlation (cont.)
A Bivariate Outlier May Greatly
Attenuate a Correlation.
• Clerical errors (or deviant
cases) can attenuate, as well
as enhance, apparent correlation.
• In the adjacent scattergram, a
single bivariate outlier
reduces what is otherwise an
almost perfect association
between the variables to a
more modest level.
• Note that the deviant case is
not an outlier in this univariate
sense.
– Its value on each variable
separately is unexceptional.
– What is exceptional is the
combination of values on
the two variables that it
exhibits.
Applied Regression (and
Correlation) Analysis
• Regression (especially multiple regression) analysis is
now very commonly used in political science research.
• But perhaps its application is most intuitive in practical
situations in which researchers literally want to make
predictions about future cases based on analysis of past
cases.
• Here are two examples.
Predicting College GPAs.
• A college Admissions Office has recorded the combined SAT scores
of all of its incoming students over a number of years.
• It has also recorded the first-year college GPAs of the same students.
• The Admissions Office can therefore calculate the regression
coefficient b, the intercept a, and the correlation coefficient r for the
data they have collected in the past. It can then use the regression
equation
PREDICTED COLLEGE GPA = a + b × SAT SCORE
to predict the potential college GPAs of the next crop of applicants on
the basis of their SAT scores (and use these predictions in making
•
Even better, it can collect and record more extensive data and use a
multiple regression equation such as:
GPA = a + b1 × SATV + b2× SATM + b3 × HSGPA + b4 × # AP + . . .
Restricting the Values of the Independent Variable
Attenuates Correlation
Predicting Presidential Elections Months
• A number of political scientists have devised predictive
models that you may have heard about during the past
Presidential election year.
– Much like the hypothetical college Admissions Office, these
political scientists have assembled data concerning the past 15
or so Presidential elections, in particular:
• the percent of the popular vote won by the candidate of the
incumbent party (that controlled the White House going into the
election) [the dependent variable of interest], and
• data on a number of independent variables whose values become
available well in advance of the election, typically including:
– one or more indicators of the state of the economy (growth rate,
unemployment rate, etc.) usually as of the end of the second
quarter (June 30) of the election year;
– some poll measure of the incumbent President’s approval
rating as of about June 30.
– Using this data, they calculate the coefficients for the regression
equation.
Predicting Presidential Elections (cont.)
• You constructed two bivariate scattergrams along these
lines in PS #11:
– INCUMBENT VOTE by GDP
– INCUMBENT VOTE by PRESIDENTIAL APPROVAL
– Remember that:
• GDP was real Gross Domestic Product (economic) growth
over the Fall, Winter, and Spring quarters preceding the
election (e.g., from October 1, 2003 through June 30, 2004);
• PRES APPROVAL was the incumbent President’s approval
rating in the first Gallup Poll taken after June 30 of the
election year.
• Let’s look at each of these more closely, and find (by
either eyeball or calculation) the regression equation.
Predicting Presidential Elections (cont.)
Predicting Presidential Elections (cont.)
Predicting Presidential Elections (cont.)
• In July 2008, GDP was about +1%, so we could
predict:
McCain POP VOTE = 47.7% + 1.12 x 1%
= 47.7 + 1.12% = 48.8%
This would be a pretty fuzzy prediction because
the [calculated] correlation coefficient is only
about r = +0.5 (r 2 = +.25)
The Regression Equation
• Putting this all together, we now have the
equation for the regression line in the example:
Predicting Presidential Elections (cont.)
Predicting Presidential Elections (cont.)
• We can make still sharper predictions by using both
independent variables simultaneously to predict the
value of the dependent variable.
• Here is the multiple (vs. bivariate) regression equation
(coefficients calculated by SPSS):
INC VOTE = 35.8 + .49 x GDP + .30 x POP (with r2 = .71)
• It might seems surprising the apparent impact of each
independent variable (its regression coefficient) is
smaller in this multivariate analysis.
– This is because the two independent variables are themselves
correlated (r = +.4)
“Out-of-Sample” Predictions
An Relevant Example
How Much Is an Additional Year of
Education Really Worth?
How Much Is an Additional Year of
Education Really Worth? (cont.)
How Much Is an Additional Year of
Education Really Worth? (cont.)
How Much Is an Additional Year of
Education Really Worth? (cont.)
Test 2
• If you answered (C) to M-C Q#9, one point