Chapter 7 Part 1

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Chapter 7 -Part 1

• Correlation

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Correlation Topics

• Correlational research what is it and how do
  you do co-relational research?
• The three questions
• Is it a linear or curvilinear correlation?
• Is it a positive or negative relationship?
• How strong is the relationship?
• Solving these questions with t scores and r, the
  estimated correlation coefficient derived from
  the tx and ty scores of individuals in a random
  sample.

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Correlational research how to start.

• To begin a correlational study, we select a
  population or, far more frequently, select a
  random sample from a population.
• We then obtain two scores from each individual,
  one score on each of two variables. These are
  usually variables that we think might be related
  to each other for interesting reasons). We call
  one variable X and the other Y.

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Rho and, its estimate, r.

• Since we use samples most of the time, for the
  most part, we will use the formulae and symbols
  for estimating the correlation in the population
  from a sample.)
• The actual correlation in the population is
  called rho or Pearsons rho.
• Your best estimate or rho, derived from a random
  sample, is called r or Pearsons r.
• Pearson invented the technique.

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Comparing tX tY scores to compute r

• We translate the raw scores on the X variable to
  t scores (called tX scores) and raw scores on the
  Y variable to tY scores.
• So each individual has a pair of scores, a tX
  score and a tY score.
• You determine how similar or different the tX and
  tY scores in the pairs are, on the average, by
  subtracting tY from tX, then squaring, summing,
  and (kind of) averaging the tX and tY differences.

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The estimated correlation coefficient, Pearsons r

• With a simple formula, you transform the average
  squared differences between the tX tY scores to
  Pearsons correlation coefficient, r
• Pearsons r indicates (with a single number),
  both the direction and strength of the
  relationship between the two variables in your
  sample.
• r also estimates the correlation in the
  population from which the sample was drawn
• In Ch. 8, you will learn when you can use r that
  way.

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Going from pairs of raw scores to r Linearity -
A preliminary question.

• Once you have scores on two variables, you
• ask, Is this a linear or curvilinear
  relationship?
• If you mistake a curvilinear relationship for a
  linear one and then use the correlation to
  predict values of Y from values of X, you can
  wind up predicting that the average 70 year old
  will be 13 feet tall!
• We dont like making that kind of mistake.
• So you have to watch out for curvilinearity.

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Linearity vs. Curvilinearity

• In a linear relationship, as scores on one
• variable go from low to high, scores on the
• other variable either generally increase or
• generally decrease.
• In a curvilinear relationship, as scores on one
• variable go from low to high, scores on the
• other variable change directions. They can go
• 1.)down and then up, 2.) up and then down, 3.)
  up and down and then up again, 4.) up or down
  then flat, 5.) and so on.

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Curvilinearity An example

• New furniture can be fairly expensive.
  Alternatively, it is hard to get very much for
  used furniture, unless it is very old. At some
  point such furniture, if in reasonably good
  condition, becomes a set of antiques and can be
  worth a good deal of money.
• Thus the value of furniture goes from high to
  low, then, when enough time has passed from low
  to very high.
• GRAPH that relationship. See how the line changes
  direction.

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Examples of linear relationships.

• For example, think of the relationship of the
  size of a pleasure boat (X) and its cost (Y).
• As one variable (boat size) increases, scores
• on the other variable (cost) also increase.
• Another example of a linear relationship the
  relationship between the size of a car and the
  number of miles per gallon it gets.
• In general, as cars get gradually larger (X),
  they tend to get fewer miles per gallon (Y).

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A curvilinear relationship

• In a curvilinear relationship, as scores on the X
  variable go gradually from low to high, the Y
  variable changes direction.
• For example, think of the relationship between
  age (X) and height (Y).
• As age increases from 0-14 or so, height
  increases also.
• But then people stop growing. As age increases,
  height stays the same.
• Thus the Y variable, height, changes direction.
  It goes from gradually rising to flat.
• If you graph age and height, the best fitting
  line is a curved line.

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Correlation Characteristics Which line best
shows the relationship between age (X) and height
(Y)
Linear vs Curvilinear
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Another non-linear relationship shortstops and
linemen great shortstops may be too small to be
great football lineman.
Football potential Terrible Average Average Very
Good Excellent Good Poor
Is this a linear relationship?
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Plot the dots!

• To check whether a relationship is linear, make a
  graph and place the scores on it.
• Thats what I mean by Plot the dots.
• If you really want to know what is going on with
  data, Plot the dots!
• Here is a graph for the baseball skills and
  football potential data.

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When you plot the dots, 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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After you know a correlation is linear, there are
other two questions Direction and Strength of a
correlation. But first, a definition of high and
low scores.

• Definition of high and low scores
• High scores are scores above the mean. They are
  represented by positive t scores.
• Low scores are scores below the mean of each
  variable.
• They are represented by negative t scores.

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Positive relationships

• In a positive relationship, as X scores gradually
  increase, Y scores tend 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 positive correlation, X and Y scores tend to
  be on the same side of their respective means.
  Scores below the mean on X are paired with scores
  below the mean on Y and scores above the mean on
  X tend to be paired with scores above the mean on
  Y.
• As a result, the tX and tY scores tend to be
  similar and the difference between them (tX tY)
  tends to be small.
• Since (tX tY) is small, the squared difference
  between them, (tX tY)2 also tends to be small

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In a positive correlation, the tX and tY scores
are relatively __________, so the difference and
the squared difference between the t scores in
each pair tends to be ________.
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In a positive correlation, the tX and tY scores
are relatively similar, so the difference and the
squared difference between the t scores in each
pair tends to be small (or, to put it another
way, close to zero).
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Graphing a positive relationship.

• In a positive correlation high scores on X tend
  to go with high scores on Y. On a graph, as the
  line runs from left to right, scores increase on
  the X axis. At the same time, Y scores also
  generally get higher. So, the line will tend to
  rise as it runs.
• Remember from math, slope equals how far a line
  rises on the Y axis for each unit it moves from
  left to right or runs along the X axis.
• If a line rises from left to right, rise is
  positive. Run is always positive. So a positive
  rise divided by an (always) positive run results
  in a positive slope. (Thats why we call it a
  positive correlation.)

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Positive vs Negative scatterplot
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Graphic display of a strong POSITIVE correlation.
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Negative relationships

• In a negative relationship, as X scores gradually
  increase, Y scores tend to decrease. Example
  The larger a car is, the fewer miles it tends to
  get for each gallon of gas. As size goes up,
  miles per gallon tends to go down.
• In a negative correlation, X and Y scores tend to
  be on opposite sides of their respective means.
• As a result, the tX and tY scores tend to be
  dissimilar and the difference between them (tX
  tY) tends to be large.
• Since (tX tY) is large, the squared difference
  between them, (tX tY)2 also tends to be large.

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Graphing a negative relationship

• In a negative correlation, high scores on X tend
  to go with low scores on Y. On a graph, as the
  line runs from left to right, scores increase on
  the X axis. At the same time, Y scores get lower.
  So, the line will tend to fall as it runs.
• Remember from math, slope equals how far a line
  rises on the Y axis for each unit it moves from
  left to right or runs along the X axis.
• If a line falls from left to right, rise is
  negative. Run is always positive. So a negative
  rise divided by an (always) positive run results
  in a negative slope. (Thats why we call it a
  negative correlation.)

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Positive vs Negative scatterplot
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Summary

• When t scores are consistently more similar than
  different, we have a positive correlation. On a
  graph the dots will rise from your left to your
  right. So, a best fitting line will have a
  positive slope.
• When t scores are consistently more different
  than similar, we have a negative correlation. On
  a graph the dots will fall from your left to your
  right. So, a best fitting line will have a
  negative slope.

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Positive vs Negative scatterplot
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How strong is the relationship between the tX and
tY scores?

• Here the question is about the consistency with
  which tX and tY scores are either similar or
  dissimilar.

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t scores sign and size

• There are two aspects to the consistency of the
  relationship between tX and tY scores.
• First, are the t scores consistently of the same
  sign (positive correlation) or opposite signs
  (negative correlation).
• If they are almost always one way or the other,
  you have at least a moderately strong
  relationship.
• On the other hand, if you sometimes see t scores
  on the same side of the mean and sometimes on
  opposite sides, you have a relatively weak
  correlation.

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t scores sign and size

• If there is a consistent pattern of same signed t
  scores (positive correlation) or a consistent
  pattern of opposite signed t scores (negative
  correlation), then whether the tX and tY scores
  are about the same distance from the mean comes
  into play.
• The large majority of t scores (close to 90),
  usually range from 1.50 to 1.50
• Given a consistent positive or negative
  correlation, the more similar in size the t
  scores, the stronger the correlation. This is
  especially true at the extremes (t lt-1.5 or t
  gt1.5)

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Positive correlations

• Perfect tX and tY scores are all the same sign
  and are identical in size.
• Strong tX and tY scores are almost all the same
  sign and are fairly similar in size.
• Moderate tX and tY scores are predominately the
  same sign. This is especially true for pairs in
  which one of the values is one or more standard
  deviations from the mean. Size may be fairly
  dissimilar.
• Weak tX and tY scores are a little more often
  the same sign than opposite in sign. Nothing can
  be said about size.

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Negative correlations

• Perfect tX and tY scores are all of the opposite
  sign and are identical in size.
• Strong tX and tY scores are almost all of
  opposite sign and are fairly similar in size.
• Moderate tX and tY scores are predominately
  opposite in sign. This is especially true for
  pairs in which one of the values is one or more
  standard deviations from the mean. Size may be
  fairly dissimilar.
• Weak tX and tY scores are a little more often of
  opposite signs than the same in sign. Nothing can
  be said about size.

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Unrelated (independent) variables

• When the size and sign of the tX scores bears no
  relationship to the size and sign of the tY
  scores, the variables are unrelated.
• We also can call the variables independent of
  or orthogonal to each other. The three terms,
  unrelated, independent and orthogonal are
  synonymous in this context.

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Graphing it on t axes The strength of a
relationship tells us approximately how the dots
representing pairs of t scores will fall around a
best fitting line.

• Perfect - scores fall exactly on a straight line
  whose slope will be 1.00 or 1.00.
• Strong - most scores fall near the line whose
  slope will be close to .750 or -.750.
• Moderate - some are near the line, some not. The
  slope of the line will be close to .500 or -.500.

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Graphing it on t axes The strength of a
relationship tells us approximately how the dots
representing pairs of t scores will fall around a
best fitting line.

• Weak some scores fall fairly close to the line,
  but others fall quite far from it. The slope of
  the line will be close to .250 or -.250
• Independent - the scores are not close to the
  line and form a circular or square pattern. The
  best fitting line will be the X axis, a line with
  a slope of 0.000.

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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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Computing the correlation coefficient.
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Comparing apples to oranges? Use Z or 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 (the
  variables may be 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 ZX and ZY scores if you are
  studying a population or tX and tY scores if you
  have a sample.
• ZX and ZY scores (or 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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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 Z or 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 BOTH the
  consistency and direction of a correlation with a
  single number

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rho the population parameter

• Pearsons rho (?) is the parameter that
  characterizes the strength and direction of a
  linear relationship (and only a linear
  relationship) between two variables. To compute
  rho, you must have the entire population. Then
  you can compute sigma, mu, Z scores and rho.
• The formula rho 1 -(1/2 ?(ZX - ZY)2 / (NP))
  where NP is the number of pairs of Z scores in
  the population
• In English The correlation coefficient equals 1
  minus half the average squared distance between
  the pairs of Z scores.

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Pearsons rho

• When you have a perfect positive correlation, the
  Z scores will be identical in size and sign. So
  the average squared distance will be zero and
• rho 1.000-1/2(0.000) 1.000

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Pearsons rho

• When you have a perfect negative correlation, the
  Z scores will be identical in size and opposite
  in sign. It can be proven algebraically that the
  average squared distance in that case will be
  4.000
• rho 1.000-1/2(4.000) -1.000

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Pearsons rho

• When you have two totally independent variables,
  the average squared distance will be 2.000
  (halfway between 0.000 and 4.000).
• rho 1.000-1/2(2.000) 0.000

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Pearsons Correlation Coefficient

• Thus, rho varies from -1.000 (perfect negative
  correlation to 0.000 (independent variables) to
  1.000 (perfect positive correlation).
• 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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Estimating rho with r

• Computing rho involves finding the actual average
  squared distance between the ZX and ZY scores in
  the whole population.
• In computing r, we are estimating rho.

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The formula for r

• Pearsons r is a least squares, unbiased estimate
  of rho, based on the relationships found between
  tX and tY scores in a random sample.
• r 1 - (1/2 ?(tX - tY)2 / (nP - 1)) where nP-1
  equals one less than the number of pairs of t
  scores in the sample.
• In English Pearsons r equals 1.000 minus half
  the estimated average squared difference between
  the Z scores in the population based on squared
  differences between the t scores in the sample.

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Look at those formulae again.

• ?(ZX - ZY)2 / (NP) is the average squared
  distance between the Z scores.
• The rest of the formula, simply transforms the
  average squared distance between the Z scores
  into a variable that goes from 1.000 to 1.000.
• rho 1 -(1/2 ?(ZX - ZY)2 / (NP)) where NP is the
  number of pairs of Z scores in the population

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Look at those formulae again.

• . ?(tX - tY)2 / (nP - 1)) is a least squared,
  unbiased estimate of the average squared
  difference between the Z scores in the population
  based on the differences between the tX and tY
  scores in a random sample.
• The rest of the formula, simply transforms the
  estimated average squared distance between the Z
  scores into a variable that goes from 1.000 to
  1.000.
• r 1 - (1/2 ?(tX - tY)2 / (nP - 1)) where nP-1
  equals one less than the number of pairs of t
  scores in the sample.
• REMEMBER, t scores are estimated Z scores

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Thus, r, the least squared, unbiased estimate of
rho, is basically an estimate of the average
squared difference between the ZX and ZY scores
in the population transformed into a variable
that goes from -1.00 to 1.00.
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Similarities of r and rho

• r and rho vary from -1.000 to 1.000.
• For both r and rho, a negative value indicates a
  negative relationship a positive value indicates
  a positive relationship.
• Values of r or rho 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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Since we almost always are studying random
samples, not populations, we almost always
compute Pearsons r, not Pearsons rho.
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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 between the t scores by (nP -
  1). ?(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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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
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By the way - True graphs.

• Ch.7 has true graphs, displays in which each dot
  stands for a score on two (in this case) or more
  (in more advanced cases) variables.
• In Ch. 1 through Ch. 6, most of the figures
  represented the frequency of scores on a single
  variable.
• Formally, displays of frequencies are figures,
  but they are not graphs.

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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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End ch. 7, part 1 slides here.