### Correlation Coefficient

```Social Statistics: Correlation
coefficient
Whether the correlation is significant
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Once you know the correlation coefficient for your
sample, you might want to determine whether this
correlation occurred by chance.
Or does the relationship you found in your sample
really exist in the population or were your results a
fluke?
Or in the case of a t-test, did the difference between
the two means in your sample occurred by chance
and not really exist in your population.
Whether the correlation is significant
If you set your confidence level at 0.05
 Let’s assume that you collected your data with
100 different samples from the same
population and calculate correlation each time.
So, the maximum of 5 out of 100 samples
might show a relationship when there really
was no relationship (r=0)
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Correlation
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Any relationship should be assessed for its
significance as well as its strength
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Pearson correlation measures the strength of a
relationship between two continuous variables
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Significance is measured by t-test with p=0.05 (which tells
how unlikely a given correlation coefficient, r, will occur
given no relationship in the population)
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Correlation coefficient: r
Coefficient of determination: r2
The smaller the p-level, the more significant the relationship
The larger the correlation, the stronger the relationship
Classical model for testing significance
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You have a sample from a population
Whether you observed statistic for the sample is
likely to be observed given some assumption of the
corresponding population parameter.
Classical model for testing significance
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The classical model makes some assumptions about the
population parameter: Population parameters are expressed
as Greek letters, while corresponding sample statistics are
expressed in lower-case Roman letters:
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r = correlation between two variables in the sample
(rho) = correlation between the same two variables in the
population
A common assumption is that there is NO relationship
between X and Y in the population:  = 0.0
Under this common null hypothesis in correlational analysis: r
= 0.0
Classical model for testing significance
When the test is against the null hypothesis: r
xy = 0.0
 What is the likelihood of drawing a sample
with r xy =0.0?
 The sampling distribution of r is
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approximately normal (but bounded at -1.0 and
+1.0) when N is large
 and distributes t when N is small.
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T test for the significance of the
correlation coefficient
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The simplest formula for computing the
appropriate t value to test significance of a
correlation coefficient employs the t
distribution:
=
−2
1− 2
The degrees of freedom for entering the tdistribution is N - 2
Quality of Marriage
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Example
Quality of parent-child
relationship
T test for the significance of the
correlation coefficient

Step1: a statement of the null and research
hypotheses
Null hypothesis: there is no relationship between
the quality of the marriage and the quality of the
relationship between parents and children
 Research hypothesis: (two-tailed, nondirectional)
there is a relationship between the two variables

H0 :  xy  0
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H1 : rxy  0
Correlation coefficient
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CORREL() and PEARSON()
r=0.393
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T test for the significance of the
correlation coefficient
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Step2: setting the level of risk (or the level of
significance or Type I error) associated with the null
hypothesis
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0.05 or 0.01
What does it mean?
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Why not 0.0001?
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on any test of the null hypothesis, there is a 5% (1%) chance you
will reject it when the null is true when there is no group
difference at all.
So rigorous in your rejection of false null hypothesis that you may
miss a true one; such stringent Type I error rate allows for little
leeway
T test for the significance of the
correlation coefficient
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Step 3 and 4: select the appropriate test
statistics
The relationship between variables, and not the
difference between groups, is being examined.
 Only two variables are being used
 The appropriate test statistic to use is the t test
for the correlation coefficient
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=
13
−2
=2.22
1− 2
Types of t test
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T test for the significance of the
correlation coefficient
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Step5: determination of the value needed for rejection of the
null hypothesis using the appropriate table of critical values
for the particular statistic.
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From t table, the critical value=2.052 (two tailed, 0.05, df=27)
T=2.22
If obtained value>the critical value reject null hypothesis
If obtained value<the critical value accept null hypothesis
T test for the significance of the
correlation coefficient
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Step6: compare the obtained value with the
critical value
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T Distribution Critical Values Table (Critical value r table)
compute the correlation coefficient (r=0.393)
Compute df =n-2 (df=27)
obtained value: 0.393
critical value: 0.367
hrt.htm
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T test for the significance of the
correlation coefficient
Step 7 and 8: make decisions
 What could be your decision? And why, how
to interpret?
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obtained value: 0.393 > critical value: 0.349 (level
of significance: 0.05)
 Coefficient of determination is 0.154, indicating
that 15.4% of the variance is accounted for and
84.6% of the variance is not.
 There is a 5% chance that the two variables are
not related at all
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Causes and associations
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Two variables are related to each other
One causes another
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having a great marriage cannot ensure that the parent-child
relationship will be of a high quality as well;
The two variables maybe correlated because they
share some traits that might make a person a good
husband or wife and also a good parent;
It’s possible that someone can be a good husband or wife
but have a terrible relationship with his/her children.
A critique
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a correlation can be
taken as evidence for a
possible causal
relationship, but
cannot indicate what
the causal relationship,
if any, might be.
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These examples
indicate that the
correlation coefficient,
as a summary statistic,
cannot replace the
individual examination
of the data.
Exercise
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To investigate the effect of a new hay fever drug on driving skills, a
researcher studies 24 individuals with hay fever: 12 who have been
taking the drug and 12 who have not. All participants then entered a
simulator and were given a driving test which assigned a score to
each driver as summarized in the below figure.
Explain whether this drug has an effect or not?
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y
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0
```