Correlation

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Correlation

• The relationship between X and Y

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Scatter Plot

• Graph of values of X and Y
• When look at scatterplot, see relationship
  between 2 variables

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Correlation Coefficient

• The size of the correlation indicates the
  strength of the relationship between X and Y
• 1.0 indicates a perfect relationship
• The closer the number is to 1.0, the stronger the
  relationship
• Positive or negative sign indicates the direction
  of the relationship

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Factors that affect size of correlation

• Linearity
• Homogeneity of group
• Size of the group

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Spearman rho (?)

• Correlation of variables measured using ordinal
  scale

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Simple Linear Regression

• The relationship between X and Y

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Conceptualization

• Take a scatterplot
• Fit a line to the points to minimize how much
  each point deviates from the line

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Fitting the Line

• Obviously, cannot draw a straight line among the
  points
• If we try to minimize each points deviance from
  the line, we are fitting a least squares line.

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Fitting a Line The Equation

• Y ß0 ß1X e
• Y predicted score
• ß1 slope (amount of change in Y that
  corresponds to a change of 1 unit in X)
• ß0 Y intercept (value of Y when X0)
• ei independent errors in distribution of Y at
  each level of X

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Slope (ß)

• Slope can be positive or negative
• Refers to direction (slant) of line
• Slope 0 means horizontal line

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Examples
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Determining the Regression Line

• Ordinary Least Squares
• Minimize the squared deviations of the actual
  value (Y) from the predicted value (Y)

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Simple Regression
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Purpose

• Predicting score on Y variable based on knowledge
  of score on X variable.
• Examines relationship among 2 or more variables
• Variables may be either continuous or categorical

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Example

• Random sample of 11 children
• Can we predict IQ from birthorder (Zajonc)?

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Simple Regression

• 1 Dependent Variable (Criterion)
• 1 Independent Variable (Predictor)
• Mathematically, doesnt matter which is which in
  case of simple regression

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Assumptions for Simple Regression

• Linear relationship between dependent variable
  (Y) and predictor (X)
• The errors are independent of each other
• The errors have constant variance
• The errors are normally distributed with a mean
  of 0

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Ways to Assess Violations of Assumptions

• Plot the residuals against predicted values
• Residuals are the deviations of each individual
  score from the regression line
• If violations are tenable, the scatterplot should
  not have any obvious pattern

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Testing Hypotheses

• H0 ß1 0
• X and Y are unrelated (slope of regression line
  is 0 and horizontal line)
• Simple Regression is nothing more than t-test
• Slope (b) represents correlation between X and Y

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Computer Examples
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Correlation between X Y when only one predictor
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Amount of variance in Y explained by predictors
(X)
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Predictors (X) significantly predict Y (F(1, 9)
21.74, p.001
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Y2.2 .98(X) e
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X is a statistically significant predictor of Y
(t4.66, p.001)
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Results

• Using simple regression analyses, X was a
  statistically significant predictor of Y (F (1,
  9) 21.74, p.001) explaining 67.5 of the
  variance in Y using the adjusted R squared
  statistic. The standardized beta coefficient for
  X was .84.

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Homework

• Hinkle p. 117-120 6 (find correlation with
  SPSS), 13 (find correlation with SPSS)
• p. 136 2b (use SPSS), 3b (use SPSS)
• Stangor p. 38 3