Chapter 7 Part 1

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Chapter 7 -Part 1

• Correlation

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Correlation Topics

• Co-relationship between two variables.
• Linear vs Curvilinear relationships
• Positive vs Negative relationships
• Strength of relationship

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Mythical relationship between Baseball and
Football performance
Football skill predicts baseball skill.
There is a strong relationship.
Baseball skill Very good Very poor Good Terrible P
oor Average Excellent
Football skill Very good Very poor Good Terrible P
oor Average Excellent
Al Ben Chuck David Ed Frank George
Baseball skill predicts football skill.
Is this a linear relationship?
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First we must arrange the scores in order
Football skill Terrible Very Poor Poor Average Goo
d Very Good Excellent
Baseball skill Terrible Very Poor Poor Average Goo
d Very Good Excellent
David Ben Ed Frank Chuck Al George
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Then we plot the scores
Football Skill
George
Baseball Skill
This is definitely a linear relationship!
David
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Lets get more abstract?
Football Skill
Y
X
Baseball Skill
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Linear or nonlinear? Lets look at another set of
values.
Football skill Terrible Average Average Very
Good Excellent Good Poor
Is this a linear relationship?
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Is this linear?
Football Skill
Chuck
Frank
Al
Baseball Skill
Ben
Ed
George
NO! It is best described by a curved line. It is
a curvilinear relationship!
David
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Positive vs Negative relationships

• In a positive relationship, as one value
  increases the other value tends to increase as
  well. Example The longer a sailboat is, the
  more it tends to cost. As length goes up, price
  tends to go up.
• In a negative relationship, as one value
  increases, the other value decreases.Example
  The older a sailboat is, the less it tends to
  cost. As years go up, price tends to go down.

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Positive vs Negative scatterplot
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Correlation Characteristics
Linear vs Curvilinear
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The strength of a relationship tells us
approximately how the dots will fall around a
best fitting line.

• Perfect - scores fall exactly on a straight
  line.
• Strong - most scores fall near the line.
• Moderate - some are near the line, some not.
• Weak lots of scores fall close to the line, but
  many fall quite far from it.
• Independent - the scores are not close to the
  line and form a circular or square pattern

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Strength of a relationship
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Strength of a relationship
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Strength of a relationship
Moderate
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Strength of a relationship
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What is this relationship?
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What is this?
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What is this?
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What is this?
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Comparing apples to oranges? Use t scores!

• You can use correlation to look for the
  relationship between ANY two values that you can
  measure of a single subject.
• However, there may not be any relationship
  (independent).
• A correlation tells us if scores are consistently
  similar on two measures, consistently different
  from each other, or have no real pattern

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Comparing apples to oranges? Use t scores!

• To compare scores on two different variables, you
  transform them into tX and tY scores.
• tX and tY scores can be directly compared to each
  other to see whether they are consistently
  similar, consistently quite different, or show no
  consistent pattern of similarity or difference

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Similar tX and tY scores positive correlation.
dissimilar negative correlation. No pattern
independence.

• When t scores are consistently more similar than
  different, we have a positive correlation.
• When t scores are consistently more different
  than similar, we have a negative correlation.
• When t scores show no consistent pattern of
  similarity or difference, we have independence.

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Comparing variables

• Anxiety symptoms, e.g., heartbeat, with number of
  hours driving to class.
• Hat size with drawing ability.
• Math ability with verbal ability.
• Number of children with IQ.
• Turn them all into t scores

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Pearsons Correlation Coefficient

• coefficient - noun, a number that serves as a
  measure of some property.
• The correlation coefficient indexes the
  consistency and direction of a correlation
• Pearsons rho (?) is the parameter that
  characterizes the strength and direction of a
  linear relationship (and only a linear
  relationship) between two population variables.
• Pearsons r is a least squares, unbiased estimate
  of rho.

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Pearsons Correlation Coefficient

• r and rho vary from -1.000 to 1.000.
• A negative value indicates a negative
  relationship a positive value indicates a
  positive relationship.
• Values of r close to 1.000 or -1.000 indicate a
  strong (consistent) relationship values close
  to 0.000 indicate a weak (inconsistent) or
  independent relationship.

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r, strength and direction
Perfect, positive 1.00 Strong, positive
.75 Moderate, positive .50 Weak, positive
.25 Independent .00 Weak, negative -
.25 Moderate, negative - .50 Strong, negative
- .75 Perfect, negative -1.00
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Calculating Pearsons r

• Select a random sample from a population obtain
  scores on two variables, which we will call X and
  Y.
• Convert all the scores into t scores.

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Calculating Pearsons r

• First, subtract the tY score from the tX score in
  each pair.
• Then square all of the differences and add them
  up, that is, ?(tX - tY)2.

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Calculating Pearsons r

• Estimate the average squared distance between ZX
  and ZY by dividing by the sum of squared
  differences by(nP - 1), that is, ?(tX - tY)2 /
  (nP - 1)
• To turn this estimate into Pearsons r, use the
  formula r 1 - (1/2 ?(tX - tY)2 / (nP - 1))

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Note seeming exception

• Usually we divide a sum of squared deviations
  around a mean by df to estimate the variance.
• Here the sum of squares is not around a mean and
  we are not estimating a variance.
• So you divide ?(tX - tY)2 by (nP - 1)
• nP - 1 is not df for corr regression (dfREG
  nP - 2)

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Example Calculate t scores for X
DATA 2 4 6 8 10
MSW 40.00/(5-1) 10
sX 3.16
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Calculate t scores for Y
DATA 9 11 10 12 13
MSW 10.00/(5-1) 2.50
sY 1.58
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Calculate r
tY -1.26 0.00 -0.63 0.63 1.26
tX -1.26 -0.63 0.00 0.63 1.26
tX - tY 0.00 -0.63 0.63 0.00 0.00
(tX - tY)2 0.00 0.40 0.40 0.00 0.00
This is a very strong, positive relationship.
? (tX - tY)2 / (nP - 1)0.200
r 1.000 - (1/2 (? (tX - tY)2 / (nP - 1)))
r 1.000 - (1/2 .200)
1 - .100 .900