Correlation Coefficients

1
Correlation Coefficients

• Expressing the strength and direction of
  relationships
• Between variables

2
The Correlation Coefficient

• A correlation coefficient is a statistic that
  indicates the strength and direction of a
  relationship between two variables.
• Pearsons r is a frequently used correlation
  coefficient.
• Possible range or strength of r
• 0.00 to 1.00
• No correlation r 0.00
• A perfect correlation r 1.00 or -1.00
• The larger the decimal, the stronger the
  relationship.

3
What does it mean when variables are perfectly
correlated?

• If height and weight were perfectly correlated
  such as (r 1.00) in a group of 20 people
• The person with the highest weight would also be
  the tallest person,
• The person with the second-highest weight would
  be the second-tallest person, and
• The person with the lowest weight would be the
  shortest person.

4
Strength/Magnitude What values of r are felt to
reflect strong, moderate, or weak
relationships?
5
Direction of the relationship Positive
Correlations

• A sign indicates a positive or direct
  relationship. r .34
• In a positive correlation, there is a direct
  relationship between the two variables
• This means, as variable A increases, variable B
  increases
• Examples
• Height and weight
• Hours of TV watching and weight among children.
• Length of years smoking and incidence of lung
  cancer.

6
Negative Correlations

• A minus sign indicates an indirect or negative
  (inverse) relationship. r -.72
• This means, as variable A increases, variable B
  decreases.
• age and acuity of eye sight
• Hours of TV watching and physical activity in
  children.
• As the temperature drops, the numbers of layers
  of peoples clothing rises

7
Prediction and Correlation

• Once a correlation between variables has been
  established, prediction becomes possible.
• If we know the correlation between A and B,
  knowing the value of variable A will allow us to
  predict the value of variable B for a given
  individual.
• Example There is a correlation between college
  freshman GPAs and the SAT. Therefore knowing
  someones SAT will help to predict their freshman
  GPA. (Thats the value of the SAT.)

8
Are Aggression and TV Viewing Correlated?
First, data are collected. In this example,
researchers might collect data on two variables
how many hours of TV children watch each day and
the number of fights they have been in at school
this year.
9
A Scatterplot
Hours of television viewing are measured along
the horizontal axis, the X variable, and the
number of fights is measured along the vertical
axis, the Y variable. Each dot on the graph,
then, represents a pairing of two scores from a
particular child TV watching and fights. Or, in
mathematical terms, XY.
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Researchers then use a statistical correlation
calculation to see how the two variables (X and
Y) are related to one another. The correlation
procedure fits a line through all of the XY pairs
of scores. This line is called a regression line.
The regression line allows researchers to use
information from one variable to predict the
expected score on the other. In this example, we
can predict that the more hours of TV a child
watches everyday, the more fights he or she is
likely to get into.
11
Zero correlation means that two variables are not
related. One cannot be used to predict the other.
Examples of unrelated variables include number of
paper clips in a desk drawer and height of the
desks legs, or IQ and gender.
12
A negative correlation is one that varies
inversely, or in opposite directions. As one
variable increases, we can predict that the other
will decrease. If the researchers in our
fictional TV watching study found that increased
hours of TV watching were associated with lower
grade-point averages, that would be a negative
correlation.
13
Nonlinear Relationships

• Nonlinear relationships exist, and cannot be
  expressed by the statistic r.
• Example the relationship between psychological
  arousal and test performance
• Research has shown that in many situations,
  people perform best at moderate levels of
  arousal. At very low and high levels, performance
  declines.
• The scatterplot will show a U-shaped curve.

14
Multivariate Designs

• Correlation Matrix
• A grid that displays the correlations between
  each variable and every other variable in the
  study.
• This allows the reader to see patterns within the
  data.

15
Table 2. Correlations among scores on measures
of shame, guilt, self-esteem, and depression.
16
Multivariate Techniques

• Factor Analysis
• Extracts descriptive factors from a large set of
  correlations.
• Multiple Regression
• Summarizes relationships among multiple variables
• Analysis of Variance (ANOVA)