Correlation

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Correlation

• Howell Ch 9

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Correlational Research

• In correlational research, the variables are not
  manipulated by the researcher.
• The two variables are simply measured on the same
  subject
• Value of variable A
• Value of variable B
• Once the variables have been measured on several
  (many) subjects, the data are plotted on a graph
  called a scatterplot

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Scatterplots

• Each data point represents a single subjects
  pair of scores on the two variables

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Creating a Scatterplot Raw Data
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Creating a Scatterplot
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Correlations

• A correlation indicates a relationship between
  two or more variables
• As the value of one variable changes, the value
  of the other variable tends to change in a
  consistent way
• In other words, the 2 variables covary (change
  together in a consistent way).
• Correlations can be negative or positive
• They range from -1.00 to 1.00

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Positive Correlations

• The values of the two variables change in the
  same direction.
• Ex) As time spent studying increases, scores on
  exams increase also.

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Positive Correlations

• Ex) A recent study found that cholesterol levels
  were positively correlated with increased
  probability of developing Alzheimers Disease
  (AD)

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Positive Correlations

• If there is a positive correlation, then low
  scores on one variable will tend to be associated
  with low scores on the second variable, and high
  scores will be associated with high scores.

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Negative Correlations

• The values of the two variables change in
  opposite directions.
• Ex) As the hours spent watching TV increases,
  test scores decrease

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Negative Correlations

• Ex) As age increases, working memory capacity and
  duration decrease.

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Negative Correlations

• If there is a negative correlation, then low
  scores on one variable will tend to be associated
  with high scores on the second variable, and high
  scores will be associated with low scores.

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Zero Correlation

• When values on Variable 1 and Variable 2 are NOT
  related to each other.

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Zero Correlation

• If there is a zero correlation, then low and high
  scores on one variable are not consistently
  associated with either high or low scores on the
  second variable.

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Identifying Relationships

• As college students take more credit hours, will
  they be able to read more books or fewer books
  for pleasure?
• What kind of relationship is this? (i.e. positive
  or negative?)
• Who will tend to want more children Someone who
  grew up in a large family or someone who grew up
  in a small family?
• What kind of relationship?
• As students take more credit hours, how many
  children will they want to have?
• What is the relationship here?

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Correlations

• A Correlation Coefficient is a statistic that
  measures the strength of the relationship.
• Its value is between 1.0 and 1.0
• Large negative or positive values indicate a
  strong relationship.
• Values near zero indicate little or no
  relationship.
• Pearsons r is the most widely used correlation
  coefficient

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Interpreting r

• The size of r is a measure of effect size
• How important is the relationship between the two
  variables?
• Here are Cohens general guidelines
• Small effect - r .10
• Medium effect - r .30
• Large effect - r .50

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Example Correlations
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Interpreting r Statistical Significance

• Significance is determined by the p value
• If p lt .05 then you can say that the correlation
  is statistically significant
• Two things determine the statistical significance
  of r
• The absolute value of r
• The number of pairs of scores (N)
• A small value of r may be statistically
  significant (plt.05) if the number of scores in
  the sample is large

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Reporting r APA Style

• Single correlations are usually reported in the
  text
• The correlation between shoe size and facial
  expressivity was significant (r .51, p lt .01).
• If you have a lot of correlations, you can also
  use a Table.
• The correlations between emotional reactions to
  infidelity and each of the outcome measures are
  summarized in Table 10.

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Reporting r APA Style
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Correlations

• Pearsons r is appropriate for data that uses
  interval or ratio scales (both variables).
• It can also be used for some ordinal data (Likert
  scales)
• The relationship between the 2 variables must be
  linear for r to be used.
• If the relationship between the variables is
  curvilinear, then r is not an appropriate
  statistic and should not be used.

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Linear Relationships

• In a linear relationship as the X scores
  increase, the Y scores tend to change in only one
  direction
• In a positive linear relationship, as the scores
  on the X variable increase, the scores on the Y
  variable also tend to increase
• In a negative linear relationship, as the scores
  on the X variable increase, the scores on the Y
  variable tend to decrease

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Nonlinear Relationships

• In a nonlinear, or curvilinear, relationship, as
  the X scores change, the Y scores do not tend to
  only increase or only decrease At some point,
  the Y scores change their direction of change.

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Linear vs. Curvilinear Relationships

• So how do you know if the relationship between
  your variables is linear?
• Look at the scatter plot

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Limitations of Correlational Research

• Correlations are extremely sensitive to outliers
  and small sample sizes
• Correlations DO NOT tell us whether one variable
  causes the other.
• 3rd Variable problem
• Spurious correlations
• Directionality Problem
• Selection Bias
• Restriction of Range
• In order to say that one variable causes changes
  in another variable, you have to do an experiment

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Correlation and Causation
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Selection Bias Restriction of Range

• The size of the correlation between 2 variables
  may be artificially reduced if there is a
  restriction of range in the variables being
  correlated
• Restriction of Range occurs when most
  participants have similar scores on one the
  variables being correlated

r .80
r .30
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Correlation and Causation
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Correlation and Causation

• Dont live together if you want to stay married
• A nationwide study of over 2,000 couples found
  that couples who lived together before getting
  married were 2.3 times as likely to get divorced
  as couples who had not lived together.
• Does living together before marriage cause
  divorce?
• How else can you explain this relationships

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Correlation and Causation

• Small colleges drive students to drink
• Parents around the country are withdrawing their
  children from small colleges. Their action comes
  after the release of a survey last week that
  found that students attending small colleges
  (less than 2000 students) consumed an average of
  7.2 alcoholic beverages a week. By comparison,
  those attending large schools (more than 20,000
  students) consumed an average of 4.5 alcoholic
  drinks. Parents speculated that the pressures of
  the small college environment were pushing their
  children to drink.
• Does attending a small college cause students to
  drink?
• How else can you explain this relationships

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Causal vs Non-Causal Language

• Which statement is appropriate to describe a
  correlational relationship?
• Sexual lyrics prompt teens to have sex
• Listening to sexual lyrics is associated with
  teen sex
• Memory retention enhanced by sleep
• People who sleep more, remember more!
• Kids who take music lessons have bigger brains
• Music lessons improve kids brain development

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Benefits of Correlational Research

• Some variables cannot be manipulated
  experimentally
• You cant change a persons height or age
• Others?
• Correlations between two variables can be used to
  make predictions
• Colleges predict academic success based on high
  school GPA and ACT scores

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Interpreting Correlations

• Several years ago, a large-scale study was
  conducted in Taiwan to determine what factors
  were related to the use of contraceptives. Social
  scientists administered lots of questionnaires to
  a random sample of people and found that the best
  single predictor of contraceptive use was the
  number of electrical appliances in the home.
• Conclusion To curb population growth and STDs,
  buy people more blenders, or at least make them
  more affordable.

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Interpreting Correlations

• If we gather data across all 50 state in the US,
  we find that there is essentially no correlation
  between the amount of money spent per capita on
  education and the states average SAT score.
• Conclusion There is no sense in increasing
  spending on education, because states that spend
  more money arent doing any better than states
  that spend less.

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Interpreting Correlations

• According to FBI crime statistics, the number of
  churches in a city is positively correlated with
  the number of sex crimes in the city.
• Conclusion Burn all churches to decrease the
  rates of sex crimes.

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Interpreting Correlations

• Parents often discourage their children from
  beginning to shave because shaving makes hair
  grow back faster and thicker, so starting earlier
  means having to shave more often.
• Conclusion Lets use shaving to combat baldness.
  Shave it or lose it!

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Illusory Correlations

• The perception of a relationship where none
  exists