Regression and Prediction

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Regression and Prediction

• Minium, Clarke Coladarci, Chapter 8

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Regression and Prediction

• If there is some association (correlation)
  between two variables, this means that knowing a
  persons score on one variable (X) provides
  information about his or her score on the other
  variable (Y)
• So, we should be able to predict (with some level
  of accuracy) the Y score from knowledge of the X
  score
• Situations in which wed like to be able to do
  this
• university performance from high school
  performance
• job performance from questionnaire score
• childs depression score from parents depression
  score
• chance of going to jail from schoolyard behaviour

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Regression and Prediction

• Correlation vs Prediction
• The magnitude of r is related to how well we can
  predict the y-value from the x-value
• To understand this we need to consider a
  prediction line (what well later call a
  regression line)
• Remember the definition of a line?
• Y a b(X)
• b is the slope of the line and a is the intercept
• cost of gas
• e.g., cost of a fillup b(X) where X is the
  number of liters and b is the price per liter
• cost of a car repair
• a b(X) where X is time, b is the cost per hour
  and a is a flat rate that you pay just for
  bringing your car to the mechanic
• e.g., cost of repair 50 70 (X)

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Determining the Line of Best Fit

• The Regression Equation
• The slope
• The intercept

SAT GPA Mean 545.8 2.57 S 123.2 0.52 r .5
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Determining the Line of Best Fit

• The regression equation defines the line of best
  fit
• The line of best fit minimizes the sum of squared
  prediction errors
• This is called the least squares criterion, or
  the least squares solution

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Determining the Line of Best Fit

• An Example

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Regression and Sums of Squares

• The concept of Explained Variance
• The total variation in the Y scores is the sum of
  squared deviations from the mean (weve seen this
  before)
• The total variation can be divided into two parts
• The explained variation is the sum of squared
  deviations for the predicted Y scores (Y) from
  the mean
• The unexplained variation is the sum of squared
  deviations of the actual Y scores from the
  predicted Y scores (Y)
• The explained variation can be thought of as the
  variation in the Y scores explained by Ys
  association with X

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Regression and Sums of Squares

• The concept of Explained Variance
• Therefore, the total variation can be expressed
  as the sum of the explained and unexplained
  variation
• The proportion of explained variation is called
  the coefficient of determination and equals r2
• we wont try to prove this relationship, well
  just trust the statisticians who assure us that
  its true.

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The Regression Equation in terms of z Scores

• Weve seen before that the correlation
  coefficient can be calculated as follows
• It can also be calculated this way
• In words, convert the raw scores in X and raw
  scores in Y to z-scores (ZX and ZY) then compute
  their cross product.

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The Regression Equation in terms of z Scores

• The z-score regression formula is
• where ZY is the predicted value of Y expressed
  as a z score
• r is the correlation coefficient between X and Y
• ZX is the z score of X
• In other words, r is the slope of the regression
  equation when X and Y are transformed into z
  scores.
• For each standard deviation in X, Y changes by r
  standard deviations p. 140
• The smaller r is, the closer the predicted value
  of Y is to the mean.
• This is called regression toward the mean

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The Regression Equation in terms of z Scores

• Note that if we know the mean and standard
  deviation of X and Y, and we know r, we can
  calculate the predicted Y score as follows
• where