Correlations

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Correlations

• Bernardo Aguilar-Gonzalez

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Describing Relationships

• Positive relationship high values are paired
  with high values, low with low.
• Negative relationship high values are paired
  with low values, low with high.
• No relationship no regularity appears between
  pairs of scores in two distributions.

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Scatterplots

• One variable is measured on the x-axis, the
  other on the y-axis.
• Positive relationship a cluster of dots sloping
  upward from the lower left to the upper right.
• Negative relationship a cluster of dots sloping
  down from upper left to lower right.
• No relationship no apparent slope.

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Strength of Relationship

• The more closely the dots approximate a straight
  line, the stronger the relationship.
• A perfect relationship forms a straight line.
• Dots forming a line reflect a linear
  relationship.
• Dots forming a curved or bent line reflect a
  curvilinear relationship.

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Correlation Coefficient

• Pearsons r a measure of how well a straight
  line describes the cluster of dots in a plot.
• Ranges from -1 to 1.
• The sign indicates a positive or negative
  relationship.
• The value of r indicates strength of
  relationship.
• Pearsons r is independent of units of measure.

1900, I suggested this, when studying natural
selecion, remember?
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Interpreting Pearsons r

• The value of r needed to assert a strong
  relationship depends on
• The size of n
• What is being measured.
• Pearsons r is NOT the percent or proportion of a
  perfect relationship.
• Correlation is not causation.
• Experimentation is used to confirm a suspected
  causal relationship.

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Other Correlation Coefficients

• Spearmans rho (r) based on ranks rather than
  values.
• Used with ordinal data (qualitative data that can
  be ordered least to most).
• Point biserial correlation -- correlations
  between quantitative data and two coded
  categories.
• Cramers phi correlation between two ordered
  qualitative categories.

In 1904! Im Spearman, remember?
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Chapter 14 of the text

• Do the problem on the correlation between the
  physical and intellectual effects of exercising
  in Ch. 14 of the book

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Procedure and Output
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Example 1

• Do Exercise 1 in the handout
• Hypothesis membership growth in large city
  churches in positively correlated with distance
  from the central business district.

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Results
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Example 2

• Do Exercise 18 in the handout
• Hypothesis There is a positive correlation
  between the ranking of counties by their schools
  by two different consulting agencies.

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Results
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SPSS allows to do a scatter plot too
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The GSS

• The GSS (General Social Survey) is an almost
  annual See Note 1, "omnibus," personal
  interview survey of U.S. households conducted by
  the National Opinion Research Center (NORC) with
  James A. Davis, Tom W. Smith, and Peter V.
  Marsden as principal investigators (PIs). The
  first survey took place in 1972 and since then
  more than 38,000 respondents have answered over
  3,260 different questions. The special features
  of the GSS follow from its unique origin as the
  first, perhaps only, social science data set
  designed to be analyzed by "users," rather than
  the PIs and project staff.
• The mission of the GSS is to make timely,
  high-quality, scientifically relevant data
  available to the social science research
  community.
• Key features of the GSS are its broad coverage,
  its use of replication, its cross-national
  perspective, and its attention to data quality.

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Example 3

• How does education influence the types of
  occupations that people enter?  One way to think
  about occupations is in terms of  occupational
  prestige. Load the data set gss00a.sav. Your
  data set includes a variable, PRESTG80, in which
  a prestige score was assigned to respondents
  occupations, where higher numbers indicate
  greater prestige.  (To get more information about
  how the occupational prestige scale was
  constructed, you can go to http//www.csub.edu/ssr
  ic-trd/SPSS/xtras.html) Lets hypothesize that
  as education increases, the level of prestige of
  ones occupation also increases.  To test this
  hypothesis, click on "Analyze," "Correlate," and
  "Bivariate."  The following dialog box shown will
  appear on your screen. Click on EDUC, and then
  click the arrow to move it into the box.  Do the
  same with PRESTG80.

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Variable Code
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• The most widely used bivariate test is the
  Pearson correlation.  It is intended to be used
  when both variables are measured at either the
  interval or ratio level, and each variable is
  normally distributed.  However, sometimes we do
  violate these assumptions. If you do a histogram
  of both EDUC, chapter 4, and PRESTG80, you will
  notice that neither is actually normally
  distributed.  Furthermore, if you noted that
  PRESTG80 is really an ordinal measure, not an
  interval one, you would be correct. 

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• Nevertheless, most analysts would use the Pearson
  correlation because the variables are close to
  being normally distributed, the ordinal variable
  has many ranks, and because the Pearson
  correlation is the one they are used to.  SPSS
  includes another correlation test, Spearmans
  rho, that is designed to analyze variables that
  are not normally distributed, or are ranked, as
  is PRESTG80.  We will conduct both tests to see
  if our hypothesis is supported, and also to see
  how much the results differ depending on the test
  used in other words, whether those who use the
  Pearson correlation on these types of variables
  are seriously off base.

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In the dialog box, the box next to Pearson is
already checked, as this is the default.  Click
in the box next to Spearman.  Your dialog box
should now look like the following figure  Click
OK to run the tests.

• Your output screen will show two tables  one for
  the Pearson correlation, and one for the
  Spearmans rho.  The results of the Pearsons
  correlation, which is called a correlation
  matrix, should look like the following one

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Notice that the Pearson coefficient for the
relationship between education and occupational
prestige is .520, and it is positive.  This tells
us that, just as we predicted, as education
increases, occupational prestige increases.  But
should we consider the relationship strong?  At
.520, the coefficient is only about half as large
as is possible.  It should not surprise us,
however, that the relationship is not perfect
(a coefficient of 1).  Education appears to be an
important predictor of occupational prestige, but
no doubt you can think of other reasons why
people might enter a particular occupation. For
example, someone with a college degree may decide
that they really wanted to be a cheese-maker,
which has an occupational prestige score of only
29, while a high-school dropout may one day
become an owner of a bowling alley, which has a
prestige score of 44.  Given the variety of
factors that may influence ones occupational
choice, a coefficient of .520 suggests that the
relationship between education and occupational
prestige is actually quite strong.
The correlation matrix also gives the probability
of being wrong if we assume that the relationship
we find in our sample accurately reflects the
relationship between education and occupational
prestige that exists in the total population from
which the sample was drawn (labeled as Sig.
(2-tailed)).  The probability value is .000
(remember that the value is rounded to three
digits), which is well below the conventional
threshold of p lt .05.  Thus, our hypothesis is
supported.  There is a relationship (the
coefficient is not 0), it is in the predicted
direction (positive), and we can generalize the
results to the population (p lt .05).
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Recall that we had some concerns about using the
Pearson coefficient, given that  PRESTG80 is
measured as an ordinal variable.  The following
figure shows the results using Spearmans rho. 
Notice that the coefficient, .523, is nearly
identical to coefficient obtained using the
Pearson correlation.  What do you conclude?
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Example 4

• the size of breeding pairs of penguins was
  measured to see if there was correlation between
  the sizes of the two sexes.
• Calculate both the Parametric and the non
  parametric correlation between the sizes and
  sexes.

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Data Procedure
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Parametric Result
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Non Parametric
What happened?