Chapter 6: Correlational Research

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Chapter 6 Correlational Research

• Examine whether variables are related to one
  another (whether they vary together).
• Correlation coefficient statistic indicating how
  well two variables are related to one another
  (how well they vary together) in a linear
  fashion.
• Must obtain a score on each variable for each
  participant.
• Pearson correlation coefficient (r) most common.
  Values range from -1.00 to 1.00
• The direction of the relationship is indicated by
  the sign of the correlation coefficient.

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• Positive correlation indicates a direct, linear,
  positive relationship (as one variable increases
  the other variable also increases).
• Negative correlation indicates a direct, linear,
  negative relationship (as one variable increases
  the other variable decreases)
• Magnitude of the correlation the numerical value
  (ignoring the sign) which expresses the strength
  of the relation
• Correlation of .33, indicates that the variables
  are not a strongly related as variables with a
  correlation of .65
• The stronger the correlation the more tightly the
  data cluster around the mean

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• Two variables may be related in a curvilinear
  fashion.
• The correlation will be 0 but the variables may
  still be related in a non-linear way.

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• Coefficient of determination represents the
  proportion of the variance in one variable (x)
  that is accounted for by the other variable (y).
  
• r2 (square the correlation coefficient).
• If the correlation between two variables (x and
  y) is 0.3. Then 0.3 squared 0.09, or 9 is the
  variance in x is accounted for y
• Proportion of variance in x that is systemic
  variance shared with y.

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• Practice correlation calculation
• In this study, 12 participants were given as
  much time as they needed to memorize a poem. When
  they thought they had memorized the poem, the
  participants recited it, and the number of errors
  they made were counted. Calculate the correlation
  between the amount of time participants worked on
  memorizing the poem and the number of errors they
  made.

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• Practice correlation calculation
• x and y represent the variables of interest.
• ?xy means you multiply each participants x and y
  score and then sum all the products across
  participants
• (?x)(?y) means that you sum all the participants
  x scores, sum all the y scores, and then multiply
  these two sums together.

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• Statistical significance of r
• exists when the correlation coefficient has a
  very low chance of being 0 in the population.
• Statistically significant means the chance that
  our correlation is truly 0 in the population is
  very low (usually less than .05). Meaning there
  is a 5 probability that our result is not really
  significant but happened by chance.
• Statistical significance can be influenced by
• sample size the larger the sample size the more
  likely you are to conclude that a correlation is
  statistically significant.

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• The magnitude of the correlation the larger the
  more confident you are in concluding that the
  correlation is statistically significant
• P value the level of significance you set before
  you calculate the correlation.
• Most common is .05
• Some researchers are more conservative and use
  .01 meaning there is only a 1 probability the
  correlation could be found significant even if it
  really is not significant (or due to chance).
• With a P value of .01 you must have a larger
  correlation than with a P value of .05 for it to
  be significant.

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• Factors that distort correlation coefficients
• 1) Restricted range the size of the correlation
  may be reduced by a restriction of the range in
  the variables being correlated.
• A restricted range occurs when most participants
  have similar scores (less variability).
• This can occur when you are correlating scores
  that are either either high or low on one
  variable.
• E.g. If you correlate SAT scores of people who
  get into college with their college GPA, you may
  be dealing with a restricted range because
  usually those with higher SAT scores get in to
  college.
• Must ensure you have a broad range of scores.

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• 2) Outliers
• Outliers that are far off the correlation line
  (high on x but lower on y) tend to deflate the
  value of r.
• Outliers that are on the correlation line but to
  the extreme on both x and y tend to inflate the
  value of r.

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• 3) Reliability of measures the less reliable
  the measures the lower the correlation
  coefficients.
• Correlation and Casualty you can not infer that
  one variable causes the other in a correlation.
• The variables may be related a correlation
  between obesity and depression (more obese people
  are more depressed) does NOT mean that obesity
  causes depression, or that depression causes
  people to become obese.
• Experimental studies must be conducted to infer
  causality in which there must be
• Covariation changes in the value of one variable
  are associated with changes in the value of
  another variable

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• Directionality the presumed cause must precede
  the effect in time. Very difficult to do in
  correlational research.
• Elimination of extraneous variables eliminate
  all other factors that may influence the
  relationship between the two variables.
• Two variables may be correlated only because they
  are actually correlated with a third variable.
• E.g. There is a correlation between eating ice
  cream and drowning. But these variable are only
  correlated because they are both correlated with
  a third variable called summer (heat). People eat
  more ice cream in the summer (when it is hotter)
  and people drown more in the summer (swim more
  when it is hotter).

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• Partial Correlation The correlation between two
  variables after the influence of the third
  variable is statistically removed.
• E.g. Correlation between viewing violent TV and
  childhood aggression (children who watch more
  violent TV are more aggressive in their play)
• But, parent discipline style may also be related
  to childhood aggression. More harsh and mean
  parents may have more aggressive children.
• So with a partial correlation we can determine
  the correlation between violent TV viewing (x)
  and childhood aggression (y) once we
  statistically remove the influence of parents
  discipline style (z).

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Aggression (y)
Parental Discipline (z)
Violent TV (x)
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• If the correlation between x and y is still
  significant after removing z
• we can conclude that x and y are correlated even
  after we account for parent discipline style (z)
• and the relationship between x and y is unlikely
  due to parent discipline style (z).

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Aggression (y)
Parental Discipline (z)
Violent TV (x)
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• If the correlation between x and y is no longer
  significant after you remove z
• then we conclude that the previous observed
  correlation between x and y was likely due to
  another variable parent discipline style (z).
• Sometimes after removing another variable (z) the
  correlation between x and y is smaller but still
  significant, which means that z did have an
  influence, but x and y are still related.

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Aggression (y)
Parental Discipline (z)
Violent TV (x)
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• Other indices of correlation
• Spearman rank-order correlation correlation
  between two variables when one or both of the
  variables is on an ordinal scale (the numbers
  reflect rank ordering).
• E.g. Correlation between teachers ranking of the
  best to worst students (ordinal scale) and the
  students IQ scores (interval scale).

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• Point biserial correlation used when one
  variable is dichotomous
• Gender is dichotomous (male or female). To
  correlate gender with spatial memory you would
  assign all males a 1 and all females a 2.
• If you get a significant positive correlation
  that would mean that females tend to score higher
  on spatial memory than males. A significant
  negative correlation would mean that males score
  higher.
• Phi coefficient used when both variables being
  correlated are dichotomous (e.g., gender,
  handedness, yes/no answer)

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• Group Task Single People Attract Crime
• Statistics show that people who are not married
  are three to four times more likely to be victims
  of violent crime as people who are currently
  married. The number of violent crimes per 1,000
  people age 12 years or older are shown in the
  following list. Clearly, marital status
  correlates with victimization.
• Marital Status Violent Crimes per 1,000
  people
• Married 13
• Widowed 8
• Divorced or separated 42
• Never married 51

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• 1. Speculate regarding possible explanations of
  this relationship. Suggest at least three reasons
  that marital status and victimization may be
  linked.
• 2. Consider how you would conduct a
  correlational study to test each of your
  explanations. You will probably want to design
  studies that allow you to partial out variables
  that may mediate the relationship between marital
  status and victimization.

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• Class Discussion
• 1. Imagine you predicted a moderate correlation
  between peoples scores on a measure of anxiety
  and the degree to which they report having
  insomnia. You administered measures of anxiety
  and insomnia to a sample of 30 participants, and
  obtained a correlation of .28. Because this
  correlation is not statistically significant (the
  critical value is .30), you must treat it as if
  it were zero. Yet you still think that anxiety
  and insomnia are correlated. If you were going to
  conduct the study again, what could you do to
  provide a more powerful test of your hypothesis?

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• 2. Imagine you obtained a point biserial
  correlation of .35 between gender and
  punctuality, showing that men arrived later to
  class than women. You think that this correlation
  might be due to the fact that more women wear
  watches, so you calculate the partial correlation
  between gender and punctuality while removing the
  influence of watch-wearing. The resulting
  correlation was .35
• Interpret the partial correlation.
• What if the correlation was .10 and no longer
  significant?
• What if the correlation was .25 and still
  significant?

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• 3. Following the rash of school shootings that
  occurred in the late 1990s, some individuals
  suggested that violent video games were making
  children and adolescents more aggressive. Imagine
  that you obtained a sample of 150 15-years-old
  males and correlated their level of
  aggressiveness with the amount of time per week
  that they played violent video games. The
  correlation coefficient was .56 (highly
  significant). Does this finding support the idea
  that playing violent video games increases
  aggression?