Download Correlations and causality

Correlation
Association between 2 variables
Suppose we wished to graph the relationship between
foot length and height of 20 subjects.
In order to create the graph, which is called a
scatterplot or scattergram, we need the foot length
and height for each of our subjects.
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Height
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Foot Length
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1. Find 12 inches on the x-axis.
2. Find 70 inches on the y-axis.
3. Locate the intersection of 12 and 70.
4. Place a dot at the intersection of 12 and 70.
Assume our first subject had a 12
inch foot and was 70 inches tall.
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Height
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Foot Length
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5. Find 8 inches on the x-axis.
6. Find 62 inches on the y-axis.
7. Locate the intersection of 8 and 62.
8. Place a dot at the intersection of 8 and 62.
9. Continue to plot points for each pair of scores.
Assume that our second subject
had an 8 inch foot and was 62
inches tall.
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Notice how the scores cluster to form a pattern.
The more closely they cluster to a line that is drawn
through them, the stronger the linear relationship between
the two variables is (in this case foot length and height).
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If the points on the scatterplot
have an upward movement
from left to right, we say the
relationship between the
variables is positive.
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If the points on the
scatterplot have a
downward movement from
left to right, we say the
relationship between the
variables is negative.
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A positive relationship means that high scores on one
variable are associated with high scores on the other
variable
It also indicates that low scores on one variable
are associated with low scores on the other variable.
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A negative relationship means that high scores on one
variable are associated with low scores on the other variable.
It also indicates that low scores on one variable
are associated with high scores on the other variable.
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Not only do relationships have direction (positive and
negative), they also have strength (from 0.00 to 1.00 and
from 0.00 to –1.00).
The more closely the points cluster toward a straight line,
the stronger the relationship is.
A set of scores with r= –0.60 has the same strength as
a set of scores with r= 0.60 because both sets cluster
similarly.
For this procedure, we use Pearson’s r (also known as a
Pearson Product Moment Correlation Coefficient). This
statistical procedure can only be used when BOTH
variables are measured on a continuous scale and you
wish to measure a linear relationship.
NO
Pearson r
Linear Relationship
Curvilinear Relationship
Formula for correlations

( x  x )( y  y )  / n

r

SxS y
or
Covxy
SDx SDy
1  xi  x  yi  y 

r   
n  s x  s y 
Assumptions of the PMCC
1. The measures are approximately
normally distributed
2. The variance of the two measures is
similar (homoscedasticity) -- check with
scatterplot
3. The relationship is linear -- check with
scatterplot
4. The sample represents the population
5. The variables are measured on a interval
or ratio scale
Example
• We’ll use data from the class
questionnaire in 2005 to see if a
relationship exists between the number
of times per week respondents eat fast
food and their weight
• What’s your guess (hypothesis) about
how the results of this test will turn out?
.5? .8? ???
Example
• To get a correlation
coefficient:
• Slide the variables
over...
Example
• SPSS output
The red is our correlation coefficient. The blue is our
level of significance resulting from the test…what does
that mean?
Digression - Hypotheses
• Many research designs involve statistical tests
– involve accepting or rejecting a hypothesis
• Null (statistical) hypotheses assume no
relationship between two or more variables.
• Statistics are used to test null hypotheses
– E.g. We assume that there is no relationship
between weight and fast food consumption until we
find statistical evidence that there is
Probability
• Probability is the odds that a certain event will
occur
• In research, we deal with the odds that
patterns in data have emerged by chance vs.
they are representative of a real relationship
• Alpha (a) is the probability level (or
significance level) set, in advance, by the
researcher as the odds that something occurs
by chance
Probability
• Alpha levels (cont.)
– E.g. a = .05 means that there will be a 5%
chance that significant findings are due to
chance rather than a relationship in the data
– The lower the a the better, but…a level
must be set in advance
Probability
• Most statistical tests produce a p-value
that is then compared to the a-level to
accept or reject the null hypothesis
• E.g. Researcher sets significance level at
.05 a priori; test results show p = .02.
• Researcher can then reject the null
hypothesis and conclude the result was not
due to chance but to there being a real
relationship in the data
• How about p = .051, when a-level = .05?
Error
• Significance levels (e.g. a = .05) are set
in order to avoid error
– Type I error = rejection of the null hypothesis
when it was actually true
• Conclusion = relationship; there wasn’t one
(false positive) (= a)
– Type II error = acceptance of the null
hypothesis when it was actually false
• Conclusion = no relationship; there was one
Error – Truth Table
Null True
Null False
Accept

Type II error
Reject
Type I error

Back to Our Example
• Conclusion: No relationship exists between
weight and fast food consumption with this
group of respondents
Really?
• Conclusion: No relationship exists
between weight and fast food
consumption with this group of subjects
– Do you believe this? Can you critique it?
Construct validity? External validity?
– Thinking in this fashion will help you adopt
a critical stance when reading research
Another Example
• Now let’s see if a relationship exists
between weight and the number of
piercings a person has
– What’s your guess (hypothesis) about how
the results of this test will turn out?
– It’s fine to guess, but remember that our
null hypothesis is that no relationship
exists, until the data shows otherwise
Another Example (continued)
• What can we conclude from this test?
• Does this mean that  weight causes 
piercings, or vice versa, or what?
Correlations and causality
•
•
•
Correlations only describe the
relationship, they do not prove cause and
effect
Correlation is a necessary, but not
sufficient condition for determining
causality
There are Three Requirements to Infer a
Causal Relationship
Correlations and causality
 A statistically significant relationship
between the variables
 The causal variable occurred prior to the
other variable
 There are no other factors that could
account for the cause

Correlation studies do not meet the last
requirement and may not meet the second
requirement (go back to internal validity –
497)
Correlations and causality

If there is a relationship between weight
and # piercings it could be because




weight  # piercings
weight  # piercings
weight  some other factor  # piercings
Which do you think is most likely here?
Other Types of Correlations
• Other measures of correlation between
two variables:
– Point-biserial correlation=use when you
have a dichotomous variable
• The formula for computing a PBC is actually
just a mathematical simplification of the formula
used to compute Pearson’s r, so to compute a
PBC in SPSS, just compute r and the result is
the same
Other Types of Correlations
• Other measures of
correlation between two
variables: (cont.)
– Spearman rho
correlation; use with
ordinal (rank) data
• Computed in SPSS the
same way as Pearson’s
r…simply toggle the
Spearman button on the
Bivariate Correlations
window
Coefficient of Determination

Correlation Coefficient Squared
 Percentage of the variability among scores on
one variable that can be attributed to
differences in the scores on the other variable
 The coefficient of determination is useful
because it gives the proportion of the
variance of one variable that is predictable
from the other variable
 Next week we will discuss regression, which
builds upon correlation and utilizes this
coefficient of determination
Correlation in excel
Use the function
“correl”
The “arguments”
(components) of
the function are
the two arrays
Applets (see applets page)
•
http://www.stat.uiuc.edu/courses/stat100/java/GCApplet/GCAppletFrame.html
• http://www.stat.sc.edu/~west/applets/clicktest.html
• http://www.stat.sc.edu/~west/applets/rplot.html