Linear Regression

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Linear Regression

• Shavelson Chapter 7

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S7-1 Linear RegressionS7-1. Be able to describe
in your own words what "linear regression" is.
(182)

• Fitting a straight line through a scatter plot (a
  joint distribution)
• Creating a best fit line for a number of
  observations

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S7-2S7-2. Know the two (mean and slope) things
necessary to develop a prediction rule
(predicting Y from X based on existing data)
(184,0). You should know how to calculate the
slope of a line (formula 7-1 and 7-2).

• Linear Equation Formula formula used to
  describe the relationship of two variables whose
  joint relationship forms a straight line (thus a
  perfect correlation!)

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S7-2,3 Linear Equation Formula S7-3. You should
know the general linear equation formula Y a
bX and what each of the parts of the formula
represent. That is, if I give you the formula,
you should be able to identify which part is the
slope, which the y intercept, etc. (186,3-187)
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S7-2 S7-2. Know the two (mean and slope) things
necessary to develop a prediction rule
(predicting Y from X based on existing data)
(184,0). You should know how to calculate the
slope of a line.(formula 7-1 and 7-2).

• Given y 22x
• Given x 4, what is the value of y?
• If x 20 what is the value of y?

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S7-2,3

• Systematic Relationship
• The change in one variable can be described as a
  function of another variable (thus, predict y
  from x)
• y x

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• y x2 y 2x
  y 22x

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S7-3 S7-3. You should know the general linear
equation formula Y a bX and what each of the
parts of the formula represent. That is, if I
give you the formula, you should be able to
identify which part is the slope, which the y
intercept, etc. (186,3-187)

• Slope the increase in y for every unit increase
  in x (or, the increase in y for a single unit
  increase in x)
• Change in y y2-y1
• Change in x x2-x1

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S7-2,3

• Linear Regression Equation

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S7-2,3

• Differences from the Linear equation
• Used for imperfect correlations
• Uses the mean of y instead of the y intercept
• Uses the calculated slope byx
• Since we are using the mean of y, also need
  deviation of x from its mean (to locate x on
  scatter plot)

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S7-4 S7-4. Be able to explain what the
"criterion of Least Squares" is. (187,1)

• Criterion of Least Squares
• The distance from all points in a scatter plot
  from the created line (regression line) is the
  least possible distance from that line (the
  squared deviations from the mean are the smallest
  obtainable by using the regression line)

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S7-5 S7-5. Given the Linear regression equation
(formula 7-7, p 189) be able to explain it in the
same manner as you did (will do?) the linear
equation formula (obj 2 above). Also, What does
the regression equation accomplish? (189,4)

• Linear Regression Formula
• Allows one to fit a straight line to a joint
  distribution (scatter plot)
• That line fitted adheres to the Least Squares
  Criterion
• That line (and the formula) can be used to
  predict a value for y for every value of x

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S7-5 S7-5. Given the Linear regression equation
(formula 7-7, p 189) be able to explain it in the
same manner as you did (will do?) the linear
equation formula (obj 2 above). Also, What does
the regression equation accomplish? (189,4)

• Problem when all the points in a joint
  distribution do not fall on a straight line, it
  is not clear how we determine the value of a(y
  intercept) and b (the slope).
• Solution Criterion of Least Squares estimates
  a and b so as to minimize the variability of
  points about the line fitted to the joint
  distribution.

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S7-5 1st Example

• Scatter Plot of test scores by hours studied
• Insert graph here

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S7-5 2nd Example

• Scatter Plot of test scores by hours studied

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S7-6 S7-6. You should be generally familiar
with what the by.x (slope of the regression line)
is (in terms of variance and covariance) (190
lecture)

• Boils down to the increase in y for each unit of
  x the slope!

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S7-7 S7-7. Know the assumption in using linear
regression (191,2) and how it is checked.

• Assumption when using the regression equation
• x and y are approximately linearly related
• How do you check this?

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S7-8 S7-8. Given the appropriate data, be able
to plot a regression line (184). Once the
regression line is drawn, be able to predict Y
from a given value of x. (195)

• Given the following
• The mean of x 2.5 (GPA scores)
• The mean of y 500 (GRE scores)
• Bxy 150 (the slope of the joint distribution)
• Draw a regression line!

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S7-9 S7-9. Know What the Standard Error Estimate
is (that is be able to describe it), and
generally how it is calculated (196-198). Also
read the section on "interpreting the Standard
Error of the estimate and be able to briefly
describe it as the author has (It is a good way
of understanding the Standard Error Estimate).
(198-200)

• Prediction Error the difference in the
  regression line predictions and the actual value.
• Standard Error of Estimate Similar to the
  standard deviation the SEE is the discrepancy
  in observed y scores and predicted y scores.
• (Big error lots of variability!)

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• If the actual y values are normally distruibuted
  about the predicted y value (the point on the
  regression line!)
• Insert Graph here

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S7-9

• Interpretation of the standard error of estimate
  as the average standard deviation of y scores
  about their predicted y scores

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S7-9 Standard Error of the Estimate

• Errors in predicting the exam scores from the
  belief scores in class 3 in the Majasan study