
Correlation
◦ The linear association between two variables
 Strength of relationship based on how tightly points in
an X,Y scatterplot cluster about a straight line
◦ -1 to 1unitless
◦ Observations should be quantitative





No categorical variables even if recoded
evaluate a visual scatterplot
Independent samples
Correlation does not imply causality
Do not assume infinite ranges of linearity
 Ho: there is no linear relationship between the 2 variables
 Ha: there is a linear relationship between the 2 variables

Simple Linear Regression
◦ Examine relationship between one predictor
variable (independent) and a single quantitative
response variable (dependent)
◦ Produces regression equation used for prediction
◦ Normality, equal variances, independence
◦ Least Squares Principle
 Do not extrapolate
 Analyze residuals
 Ho: there is no slope, no linear relationship between the 2
variables
 Ha: there is a slope, linear relationship between the 2
variables

Positive correlation: Indicates that
the values on the two variables
being analyzed move in the same
direction. That is, as scores on
one variable go up, scores on the
other variable go up as well (on
average) & vice versa

Negative correlation: Indicates that
the values on the two variables
being analyzed move in opposite
directions. That is, as scores on
one variable go up, scores on the
other variable go down, and viceversa (on average)



Correlation coefficients range
in strength from -1.00 to
+1.00
The closer the correlation
coefficient is to either -1.00
or + 1.00, the stronger the
relationship is between the
two variables
Perfect positive correlation of
+1.00 reveals that for every
member of the sample or
population, a higher score on
one variable is related to
higher score on the other
variable


Perfect negative correlation of
–1.00 indicates that for every
member of the sample or
population, a higher score on
one variable is related to a
lower score on the other
variable
Perfect correlations are never
found in actual social science
research

Positive and negative
correlations are represented
by scattergrams

Scattergrams: Graphs that
indicate the scores of each
case in a sample
simultaneously on two
variables
r: the symbol for the sample
Pearson correlation coefficient
Positive Correlation
100
90
80
70
60
50
40
30
20
10
0
Score on Exam

Score on Exam
Negative Correlation
1
6
Hours Spent Studying
11
100
90
80
70
60
50
40
30
20
10
0
1
6
11
Hours Spent Studying
The scattergrams presented here represent very
strong positive and negative correlations
(r = 0.97 and r = -0.97 for the positive and
negative correlations, respectively)

No Correlation Between Hours Spent Studying and Exam
Scores

100
90
Scores on Exam
80
70
60
50
40
30
20
10
0
0
2
4
6
8
10
12
Hours Spent Studying
Scattergram representing virtually no correlation
between the number of hours spent studying and
the scores on the exam is presented
No discernable pattern between
the scores on the two variables
We learn it is virtually impossible
to predict an individual’s test
score simply by knowing how
many hours the person studied
for the exam



The first step in understanding
how Pearson correlation
coefficients are calculated is to
notice that we are concerned with
a sample’s scores on two
variables at the same time
The data shown are scores on two
variables: hours spent studying
and exam score. These data are
for a randomly selected sample of
five students.
To be used in a correlation
analysis, it is critical that the
scores on the two variables are
paired.
Data for Correlation Coefficient
Hours Spent
Studying
(X variable)
Exam Score
(Y variable)
Student 1
5
80
Student 2
6
85
Student 3
7
70
Student 4
8
90
Student 5
9
85

Each student’s score on the X
variable must be matched with
his or her own score on the Y
variable

Once this is done a person can
determine whether, on average,
hours spent studying is related
to exam scores

Definitional Formula for Pearson Correlation
Finding the Pearson correlation
coefficient is simple when
following these steps:
1.
2.
r
Find the z scores on each of the
two variables being examined
for each case in the sample
z
Multiply each individual's z score
on one variable with that
individual's z score on the
second variable (i.e., find a
cross-product)
3.
Sum those across all of the
individuals in the sample
4.
Divide by N
Pearson product-moment correlation
coefficient
x
a z score for variable X
zy
a paired z score for variable Y
N
the number of pairs of X and Y scores
r = Σ(zx zy)
Ν

You then have an average
standardized cross product. If
we had not standardized these
scores we would have produced
a covariance.

This formula requires that you standardize your
variables
•
Note: When you standardize a variable, you are
simply subtracting the mean from each score in your
sample and dividing by the standard deviation

What this does is provide a z score for each case
in the sample

Members of the sample with scores below the
mean will have negative z scores, whereas those
members of the sample with scores above the
mean will have positive z scores


Correlation coefficients such as the Pearson are very powerful
statistics. They allow us to determine whether, on average,
the values on one variable are associated with the values on a
second variable
People often confuse the concepts of correlation and
causation
• Correlation (co-relation) simply means that variation in the
scores on one variable correspond with variation in the
scores on a second variable
• Causation means that variation in the scores on one variable
cause or create variation in the scores on a second variable.
Correlation does not equal causation.


Simple Pearson correlations are
designed to examine linear relations
among variables. In other words,
they describe average straight
relations among variables
Not all relations between variables
are linear
As previously mentioned, people
often confuse the concepts of
correlation and causation
Example: There is a curvilinear
relationship between anxiety and
performance on a number of
academic and non-academic
behaviors as shown in the figure
below
•

We call this a curvilinear relationship
because what began as a positive
relationship (between performance
and anxiety) at lower levels of
anxiety, becomes a negative
relationship at higher levels of
anxiety
70
60
50
Performance

40
30
20
10
0
1
2
3
Anxiety
4
5


The problem of truncated range
is another common problem
that arises when examining
correlation coefficients. This
problem is encountered when
the scores on one or both of the
variables in the analysis do not
have much variance in the
distribution of scores, possibly
due to a ceiling or floor effect
The data from the table at right
show all of the students did well
on the test, whether they spend
many hours studying for it or
not
•
The weak correlation that will be
produced by the data in the table
may not reflect the true
relationship between how much
students study and how much they
learn because the test was too
easy. A ceiling effect may have
occurred, thereby truncating the
range of scores on the exam
Data for Studying-Exam Score Correlation
Hours Spent
Studying
(X variable)
Exam Score
(Y variable)
Student 1
0
95
Student 2
2
95
Student 3
4
100
Student 4
7
95
Student 5
10
100


Researchers test whether the
correlation coefficient is
statistically significant
To test whether a correlation
coefficient is statistically
significant, the researcher begins
with the null hypothesis that
there is absolutely no
relationship between the two
variables in the population, or
that the correlation coefficient in
the population equals zero

The alternative hypothesis is
that there is, in fact, a statistical
relationship between the two
variables in the population, and
that the population correlation
coefficient is not equal to zero.
So what we are testing here is
whether our correlation
coefficient is statistically
significantly different from 0

What we want to be able to do
with a measure of association,
like a correlation coefficient, is
be able to explain some of the
variance in the scores on one
variable with the scores on a
second variable. The coefficient
of determination tells us how
much of the variance in the
scores of one variable can be
understood, or explained, by the
scores on a second variable

One way to conceptualize
explained variance is to
understand that when two
variables are correlated with
each other, they share a certain
percentage of their variance

See next slide for visual


In this picture, the two squares
are not touching each other,
suggesting that all of the
variance in each variable is
independent of the other
variable. There is no overlap
The precise percentage of
shared, or explained, variance
can be determined by squaring
the correlation coefficient. This
squared correlation coefficient
is known as the coefficient of
determination
r = 0.00
r² = 0.00
r = 0.30
r² = 0.09
r = 0.55
r² = 0.30
All of these statistics are very
similar to the Pearson correlation
and each produces a correlation
coefficient that is similar to the
Pearson r
Phi:

Sometimes researchers want
to know whether two
dichotomous variables are
correlated. In this case, we
would calculate a phi
coefficient (F), which is
specialized version of the
Pearson r

For example, suppose you
wanted to know whether
gender (male, female) was
associated with whether one
smokes cigarettes or not
(smoker, non smoker)
•
In this case, with two
dichotomous variables, you
would calculate a phi
coefficient
•
Note: Readers familiar with
chi-square analysis will
notice that two dichotomous
variables can also be
analyzed using chi square
test (see Chapter 14)
Point Biserial:

When one of our variables is a
continuous variable
(i.e., measured on an interval or
ratio scale) and the other is a
dichotomous variable we need to
calculate a point-biserial
correlation coefficient

This coefficient is a specialized
version of the Pearson
correlation coefficient

For example, suppose you
wanted to know whether there is
a relationship between whether a
person owns a car (yes or no)
and their score on a written test
of traffic rule knowledge, such as
the tests one must pass to get a
driver’s license
•
In this example, we are
examining the relation between
one categorical variable with two
categories (whether one owns a
car) and one continuous variable
(one’s score on the driver’s test)
•
Therefore, the point-biserial
correlation is the appropriate
statistic in this instance
Spearman Rho:
 Sometimes data are recorded as
ranks. Because ranks are a form
of ordinal data, and the other
correlation coefficients discussed
so far involve either continuous
(interval, ratio) or dichotomous
variables, we need a different
type of statistic to calculate the
correlation between two variables
that use ranked data

The Spearman rho is a
specialized form of the Pearson r
that is appropriate for such data

For example, many schools use
students’ grade point averages
(a continuous scale) to rank
students (an ordinal scale)
•
In addition, students’ scores on
standardized achievement tests
can be ranked
•
To see whether a students’ rank
in their school is related to their
rank on the standardized test, a
Spearman rho coefficient can be
calculated.

The correlations on the diagonal show the correlation between a single variable
and itself. Because we always get a correlation of 1.00 when we correlate a
variable with itself, these correlations presented on the diagonal are meaningless.
That is why there is not a p value reported for them


The numbers in the parentheses, just below the
correlation coefficients, report the sample size.
There were 314 eleventh grade students in this
sample
From the correlation coefficient that is off the
diagonal, we can see that students’ grade point
average (Grade) was moderately correlated with
their scores on the test (r = 0.4291). This
correlation is statistically significant, with a
p value of less than 0.0001 (p < 0.0001)
SPSS Printout of Correlation
Analysis
Grade
Grade
Test Score
1.0000
( 314)
P= .
Test Score
0.4291
1.0000
( 314)
( 314)
P = 0.000
P= .

To gain a clearer understanding of the relationship between grade and test
scores, we can calculate a coefficient of determination. We do this by squaring
the correlation coefficient. When we square this correlation coefficient (0.4291
* 0.4291 = 0.1841), we see that grades explains a little bit more than 18% of
the variance in the test scores

Because of 80% percentage of unexplained
variance, we must conclude that teacherassigned grades reflect something substantially
different from, and more than, just scores on
tests.
SPSS Printout of
Correlation Analysis
Grades
Grades
Test score
1.0000
( 314)
P= .
Test score
0.4291
1.0000
( 314)
( 314)
P = 0.000
P= .
Same table as in previous slide

Allows researchers to examine:
• How variables are related to each other
• The strength of the relations
• Relative predictive power of several
independent variables on a dependent variable
• The unique contribution of one or more
independent variables when controlling for
one or more covariates

Simple Regression
• Simple regression analysis involves a
single independent, or predictor
variable and a single dependent, or
outcome variable

Multiple Regression
• Multiple regression involves models
that have two or more predictor
variables and a single dependent
variable


The dependent and independent variables need
to be measured on an interval or ratio scale
Dichotomous (i.e., categorical variables with two
categories) predictor variables can also be used
• There is a special form of regression analysis,
logit regression, that allows us to examine
dichotomous dependent variables



Regression analysis yields more information
The regression equation allows us to think about
the relation between the two variables of interest
in a more intuitive way, using the original scales
of measurement rather than converting to
standardized scores
Regression analysis yields a formula for
calculating the predicted value of one variable
when we know the actual value of the second
variable

Assumes the two variables are linearly related
• In other words, if the two variables are actually
related to each other, we assume that every
time there is an increase of a given size in value
on the X variable (called the predictor or
independent variable), there is a corresponding
increase (if there is a positive correlation) or
decrease (if there is a negative correlation) of a
specific size in the Y variable (called the
dependent, or outcome, or criterion variable)
Ŷ = bX + a
Ŷ
is the predicted value of the Y variable
b
is the unstandardized regression coefficient, or the slope
a
is the intercept (i.e., the point where the regression line
intercepts the Y axis. This is also the predicted value of
Y when X is zero)

Is there a relationship between the amount of education
people have and their monthly income?
Education Level (X)
in years
Monthly Income (Y)
in thousands
Case 1
6
$1
Case 2
8
$1.5
Case 3
11
$1
Case 4
12
$2
Case 5
12
$4
Case 6
13
$2.5
Case 7
14
$5
Case 8
16
$6
Case 9
16
$10
Case 10
21
$8
Mean
12.9
$4.1
Standard Deviation
4.25
$3.12
Correlation Coefficient
0.83
Scatterplot for education and income:

12
11
9
10
9
10
8
7
8
6
I
n
c
o
m
e
7
5
4
5
3
2
1
4
2
With the data provided in
the table, we can calculate
a regression. The
regression equation allows
us to do two things:
1)
find predicted values for
the Y variable for any
given value of the X
variable
2)
produce the regression
line
6
3
1
0
-1
-2

-3
-4
0
2
4
6
8
10
12
Education (in years)
14
16
18
20
22
The regression line is the
basis for linear regression
and can help us
understand how
regression works


OLS is the most commonly
used regression formula
It is based on an idea that
we have seen before: the
sum of squares

To do OLS: find the line of
least squares (i.e., the
straight line that produces
the smallest sum of
squared deviations from the
line)
Sum of Squares:
Σ (observed value – predicted value)2
b  r*
sy
sx
b
is the regression coefficient
r
is the correlation between the X and Y variables
sy
is the standard deviation of the Y variable
sx
is the standard deviation of the X variable
a  Y  bX
Y
is the average value of Y
X
is the average value of X
b
is the regression coefficient


The regression equation does not
calculate the actual value of Y. It
can only make predictions about
the value of Y. So error (e) is
bound to occur.
e = Y - Ŷ
• Error is the difference between the
actual, or observed, value of Y and
the predicted value of Y
OR
To calculate error, use one of two
equations:
e = Y - a + bX
Y
Ŷ
is the actual, or observed
value of Y
is the predicted value of Y

For the predicted value of Y:
Ŷ = bX + a

For the actual / observed value of
Y; takes into account error (e):
Y = bX + a + e

Example: Is there a relationship between the amount of
education people have and their monthly income?
Ŷ = -3.77 + .61X


For every unit of increase in X, there is a corresponding
predicted increase of 0.61 units in Y
OR
For every additional year of education, we would predict
an increase of 0.61 ($1,000), or $610, in monthly income

Example: What would we predict the monthly
income to be for a person with 9 years of formal
education?
Yˆ
Yˆ
Yˆ

= -3.77 + .61(9)
= -3.77 + 5.59
= 1.82
So we would predict that a person
with 9 years of education would make
$1,820 per month, plus or minus our
error in prediction (e)
Drawing the Regression Line

To do this we need to calculate two points
Yˆ
Yˆ
Yˆ
Yˆ
Yˆ
Yˆ
= -3.77 + .61(9)
= -3.77 + 5.59
= 1.82
= -3.77 + .61(25)
= -3.77 + 15.25
= 11.48
12
11
9
10
9
10
8
I
n
c
o
m
e
7
8
6
7
5
5
4
3
2
1
4
2
6
3
1
0
-1
-2
-3
-4
0
2
4
6
8
10
12
Education
14
16
18
20
22



The regression line does not
always accurately predict the
actual Y values
In some cases there is a little
error, and in other cases there
is a larger error
• Residuals = errors in
prediction
In some cases, our predicted
value is greater than our
observed value.
• Overpredicted = observed
values of Y at given values of
X that are below the
predicted values of Y.
Produces negative residuals.

Sometimes our predicted value
is less than our observed value
• Underpredicted = observed
values of Y at given values of
X that are above the
predicted values of Y.
Produces positive residuals.