PSY 360

1
PSY 360

• Correlation
• Regression

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Correlation and Regression

• Both examine linear (straight line) relationships
• Both work with a pair of scores, one on each of
  two variables, X and Y
• Correlation
• Defined as the degree of linear relationship
  between X and Y
• Is measured/described by the statistic r
• Regression
• Describes the form or function of the linear
  relationship between X Y
• Is concerned with the prediction of Y from X
• Forms a prediction equation to predict Y from X

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Why do we care?

• Knowing the values of the one most representative
  score (central tendency) and a measure of spread
  or dispersion (variability) is critical for
  DESCRIBING the characteristics of a distribution
• However, sometimes we are interested in the
  relationship between variables or to be more
  precise, how the value of one variable changes
  when the value of another variable changes.
  Correlation and regression help us understand
  these relationships

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Correlation

• The aspect of the data that we
• want to describe/measure is
• the degree of linear relationship
• between X and Y
• The statistic r describes/measures the degree of
  linear relationship between X and Y
• r?zXzY/N, the average product of z scores for X
  and Y
• Works with two variables, X and Y
• -1relationships
• Measures only the degree of linear relationship
• r2proportion of variability in Y that is
  explained by X
• r is undefined if X or Y has zero spread

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Correlation -1

The sign of r shows the type of linear
relationship between X and Y. We can use the
definitional

formula for r and these scatterplots to see
positive, negative and zero relationships
r 1
r 0
r -1
Y Y
Y Y
Y Y



-

-
-

-
X X
X X
X X
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Correlation -1
7
Correlation -1
8
Correlation -1
9
Correlation -1
10
Correlation -1
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Interpreting Correlation Coefficients
Size of the correlation coefficient General
Interpretation .8 to 1.0 very strong
relationship .6 to .8 strong relationship
.4 to .6 moderate relationship .2 to
.4 weak relationship .0 to .2 weak or
no relationship
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Interpreting Correlation Coefficients
Size of the correlation coefficient General
Interpretation .8 to 1.0 very strong
relationship .6 to .8 strong relationship
.4 to .6 moderate relationship .2 to
.4 weak relationship .0 to .2 weak or
no relationship
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Interpreting Correlation Coefficients
Size of the correlation coefficient General
Interpretation .8 to 1.0 very strong
relationship .6 to .8 strong relationship
.4 to .6 moderate relationship .2 to
.4 weak relationship .0 to .2 weak or
no relationship
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Interpreting Correlation Coefficients
Size of the correlation coefficient General
Interpretation .8 to 1.0 very strong
relationship .6 to .8 strong relationship
.4 to .6 moderate relationship .2 to
.4 weak relationship .0 to .2 weak or
no relationship
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Interpreting Correlation Coefficients
Size of the correlation coefficient General
Interpretation .8 to 1.0 very strong
relationship .6 to .8 strong relationship
.4 to .6 moderate relationship .2 to
.4 weak relationship .0 to .2 weak or
no relationship
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Correlation Linear

• If there is a curvilinear
• relationship between X and Y,
• then r will not detect it. The value of r
  will be zero if there is no linear relationship
  between X and Y.

r 0
r 0
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Correlation r2

• r2proportion of variability in Y
• that is explained by X.
• If r.5, r2.25, so the proportion of
• variability in Y that is explained by X is .25
  (as a percentage, this shows 25 explained by X,
  75 unexplained).
• Scatterplots
• r.5, r2.25 r.7, r2.49 r.9,
  r2.81
• Venn Diagrams r2 is represented by the
  proportion of overlap.
  
• Y X Y X Y X

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Correlation Undefined

• If there is no spread in X or Y, then r is
  undefined. Note that any z is undefined if the
  standard deviation is zero, and r?zXzY/N.

sY0
sX0



Y
Y
r is undefined
r is undefined


X
X
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Correlation

• Example of correlation
• Murder rates and ice cream sales are positively
  correlated
• As murder rates increase, ice cream sales also
  increase
• Why?
• CORRELATION DOES NOT MEAN CAUSATION
• Murders may cause increases in ice cream sales
• Ice cream sales may cause more murders
• Some other variable may cause both murders and
  ice cream sales

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Correlation

• Things to remember
• Correlations can range from -1 to 1
• The absolute value of the correlation coefficient
  reflects the strength of the correlation. So a
  correlation of -.7 is stronger than a correlation
  of .5
• Do not assign a value judgment to the sign of the
  correlation. Many students assume that negative
  correlations are bad and positive correlations
  are good. This is not true!
• Population correlation coefficient, ? (rho)
• Impact on r
• Restriction of range
• Extreme scores (outliers)

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Regression

• Not only can we compute the degree to which two
  variables are related (correlation coefficient),
  but we can use these correlations as the basis
  for predicting the value of one variable from the
  value of another
• Prediction is an activity that computes future
  outcomes from present ones. When we want to
  predict one variable from another, we need to
  first compute the correlation between the two
  variables

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Regression
Total High School GPA and First-Year College GPA
are Correlated
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Regression
r .68 for these two variables
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Regression

• Regression is concerned with
• forming a prediction equation to
• predict Y from X
• Uses the formula for a straight line, YbXa
• Y is the predicted Y score on the criterion
  variable
• b is the slope, b?Y/ ? Xrise/run
• X is a score on the predictor variable
• a is the Y-intercept, where the line crosses the
  Y axis, the value of Y when X0
• Example if b.695, a.739, and X3.5,
• then Y .695(3.5).739 3.17

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Regression

Regression line of Y on X
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Regression

Prediction of Y, given X 3.5
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Regression

• Linear only
• Generalize only for X values in your sample
• Actual observed Y is different from Y by an
  amount called error, e, that is, YYe
• Error in regression is eY-Y
• Many different potential regression lines

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Regression Best Line

• There are many different potential regression
  lines, but only one best-fitting line

• The statistics b and a are computed so as to
  minimize the sum of squared errors,
• ?e2?(Y-Y)2 is a minimum which is called the
  Least Squares Criterion.
• This means that it minimizes the distance between
  each individual point and the regression line

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Regression

• Error in prediction the distance between
  each individual data point and the
  regression line (a direct
  reflection of the correlation
  between two variables)

Error in prediction
X 3.3, Y 3.7
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Regression sy.x

• Standard error of estimate is a
• statistic that measures/describes
• spread of errors or Y scores
• in regression.
• syx is the standard deviation of errors in
  regression
• syx ??e2/(N-2) ??(Y-Y)2/(N-2).
• syx ?(N-1)/(N-2)(sy)?(1-r2)
• As r2 increases, syx decreases. For example, if
  N100 and sy4
• r2 syx
• .2 3.94
• .4 3.68
• .6 3.22
• .8 2.41
• .9 1.75

syx is the standard deviation of Y around the
regression line Y
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Regression Partitioning

• Partitioning total variability
• Total Explained Not Explained
• This is true for proportion of spread and amount
  of spread.
• Proportion 1 r2 (1-r2)
• Amount s2y r2s2y (1-r2)s2y
• Formulas Total Expl.
  Not Expl.
• Proportion
• Amount

1
r2
1-r2
s2y
r2.s2y
(1-r2)s2y

• Example Total Expl.
  Not Expl.
• r.7, s2y150, Proportion
• Amount

1
.49
.51
150
73.5
76.5
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• Quiz 1 Quiz 2
• 9.8 10.4
• 8.6 9.9
• 10.0 10.4
• 7.0 7.8
• 9.1 8.5
• 8.1 9.4
• 7.8 9.4
• 6.1 5.4
• 4.8 5.0
• 7.6 8.0
• 7.3 9.5
• 3.8 1.5
• 7.4 3.5
• 5.8 4.2
• 8.8 7.4
• 8.8 5.5
• 6.0 7.9
• 6.1 6.4

Compute the correlation between Quiz 1 and Quiz 2
scores using SPSS. Run a regression model
predicting Quiz 2 scores using Quiz 1 scores.
What is the slope? What is the y-intercept? Based
on the regression results, estimate the Quiz 2
score of someone who earned a 9.0 on Quiz 1.