Correlation

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Correlation

• Advanced Research Methods in Psychology
• - lecture -
• Matthew Rockloff

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When to use correlation

• 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.)

• 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.
• 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 ?

• Both tests can be valuable, but answer completely
  different questions.
• 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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Example 5.1

• 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.)
Xi Anxiety Yi Depression Yi Depression Zxi Zyi
65 42 42 1.5 0.5 0.75
55 46 46 0.5 1.5 0.75
50 40 40 0 0 0
45 34 34 -0.5 -1.5 0.75
35 38 38 -1.5 -0.5 0.75
50 40 40 0 0
10 4 4 1 1
df n 2 3 .60
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Example 5.1 (cont.)

• 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.)

• 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.)

• 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.)

• To accomplish this goal, multiply the 2 sets of
  z-scores.
• Summing this final column and dividing it by the
  number of observations (n5) 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 n5 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 gt .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

• 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.)

• Next, we enter the data into the SPSS data view

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Example 5.1 Using SPSS (cont.)

• 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.)

• The results appear in the SPSS output viewer

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Example 5.1 Using SPSS (cont.)

• 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.)

• In other words, it doesnt matter which variable
  comes first in either the hand calculations, or
  when entered as syntax in SPSS.
• 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.)

• 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
Thus concludes

• Advanced Research Methods in Psychology
• Week 4 lecture
• Matthew Rockloff