Download 2. The One-Sample t-Test

Correlation
Advanced Research Methods in Psychology
- lecture -
Matthew Rockloff
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When to use correlation
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Correlation is a technique that summarizes the
relationship between 2 paired variables.
The technique gives one number, the
correlation coefficient, that expresses whether:
 “higher” numbers in one variable tend to be
paired with “higher” numbers in the other
variable (a positive correlation),
 or “higher” numbers in one variable are
associated with “lower” numbers in the
second variable (a negative correlation).
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When to use correlation (cont.)
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Because the paired t-test and correlation
use the same type of data (i.e., paired
numbers), it is easy to confuse the two
techniques.
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The paired t-test is used to test for
differences in the mean values of each
variable, while correlation shows
associations between the pairs of values.
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Paired t-test OR correlation ?
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Both tests can be valuable, but answer
completely different questions.
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The important point to remember is that
correlation describes whether the
individual values within each pair tend to
move in the same direction (a positive
correlation) or opposite directions (a
negative correlation).
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Depressed Duck
Example 5.1
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In the next example, we correlate
two scores taken from the same persons.
We want to see if clinical measures of
Anxiety and Depression are related.
Is an anxious person also likely to be
depressed and vise versa?
The Anxiety and Depression test scores
for 5 randomly selected Psychiatric hospital
patients are illustrated in columns 1 and 2 on
the next slide
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Example 5.1 (cont.)
Z xi 
i  
Sx
Z yi 
Yi  Y
Sy
Xi:
Anxiety
Yi:
Depression
65
42
1.5
0.5
0.75
55
46
0.5
1.5
0.75
50
40
0
0
45
34
-0.5
-1.5
0.75
35
38
-1.5
-0.5
0.75
40
0
0
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1
1
  50
S x  10
Y 
Sy 
rxy
Column 1
Column 2
Zxi Zyi
0
Z Z


x
y
n
df = n – 2 = 3
 .60
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Example 5.1 (cont.)
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Calculating a correlation coefficient requires
a relatively simple transformation of both
sets of values.
In order to compare these two sets of
values, the Anxiety and Depression scores
must first be measured on comparable
scales.
The average of Anxiety scores is 50, and the
average of Depression scores is 40. To
begin our comparison, we must eliminate
these mean differences.
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Example 5.1 (cont.)
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A simple way to do this is to subtract
the average for each set.
This will leave each set of values with
a mean of “0.”
The next way in which we need to
make these values comparable is to
make the variance, and likewise
standard deviation of the two sets the
same.
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Example 5.1 (cont.)
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This is easily accomplished by dividing by
the standard deviation of each set.
The 2 sets of transformed scores for Anxiety
and Depression both have a mean of “0”
and a standard deviation of “1.”

These are so-called z-scores,
or standard normal deviates.
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Example 5.1 (cont.)

A correlation coefficient summarizes
whether the scores …
 move in the same direction,
= positive correlation,
 move in the opposite direction,
= negative correlation,
 or are not linearly related
= zero correlation.
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Example 5.1 (cont.)
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To accomplish this goal,
multiply the 2 sets of z-scores.
Summing this final column and dividing it by the
number of observations (n=5) yields the correlation
coefficient (=.60).
Since we expected that Anxiety and Depression would
be positively correlated, this is a 1-tailed test.
In some Statistics textbooks you can find a “Table of
Critical Values for Pearson Correlation.”
The critical correlation for n=5 is r = .805.
Since our calculated value is less than the critical
value, we cannot conclude that this correlation is
significant.
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Example 5.1 - Conclusion
There was a non-significant
positive correlation between
Anxiety and Depression scores,
r(3) = .60, p > .05, ns.

Notice that there is no need to include
“mean” values, because unlike previous
techniques, the correlation coefficient does
not answer a question regarding the means
of each variable, but rather the association
between 2 variables.
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Example 5.1 Using SPSS
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First, we must setup each variable in the
SPSS variable view. Although not strictly
necessary, we add a variable for “personid.”
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Example 5.1 Using SPSS (cont.)
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Next, we enter the data into the SPSS data view:
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Example 5.1 Using SPSS (cont.)
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The syntax for a correlation is as follows:
correlation Variable1 Variable2.
In our example, the following syntax is entered:
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Example 5.1 Using SPSS (cont.)
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The results appear in the SPSS output viewer:
Row 1
Row 2
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Example 5.1 Using SPSS (cont.)
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The correlation syntax allows 2 or more
variables to be entered in one command.
The SPSS output shows all pairs of
correlations in a matrix format.
As is shown in the output, on the previous
slide, it is possible to correlate both Anxiety
with Depression (row 1), and separately,
Depression with Anxiety (row 2).
Both answers, however, are the same.
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Example 5.1 Using SPSS (cont.)
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In other words, it doesn’t matter which
variable comes first in either the hand
calculations, or when entered as
syntax in SPSS.
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In our example, the answer for the
correlation coefficient is always:
r = .60.
Notice that SPSS calls the correlation
coefficient the “pearson correlation.”
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Example 5.1 Using SPSS (cont.)
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Unlike the tables in the back of a
statistics textbook, which give a
“critical value” for the correlation
coefficient, SPSS provides a p-value,
or significance, associated with the
correlation (i.e., p = .285).
As usual, when this p-value
is below “.05” we can declare a
significant correlation between
our 2 paired variables.
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Example 5.1 Using SPSS (Conclusion)

In our example, however, the
correlation was not significant,
thus we conclude:
There was a non-significant
positive correlation between
Anxiety and Depression scores,
r(3) = .60, p = .29, ns.
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Correlation
Advanced Research Methods in Psychology
Week 4 lecture
Matthew Rockloff
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