Download Chapter 7 - Lone Star College

Regression
Regression
 A regression line attempts to predict one variable
based on the relationship with another variable (its
correlation).
 The regression line is placed so that the error (distance
from the line to each data point) is minimized.
 The placement of the regression line minimizes the
total squared predictive error. (That way there are no
negative values.)
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Prediction
 What is predictive error?
 The amount of error associated with the placement of a
best fitting regression line.
 The placement of the regression line:


Minimizes the total predictive error, and
Minimizes the total squared predictive error
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Progress Check 7.1
20
15
10
5
1
5
10
15
20
a) Predict the approximate rate of inflation, given an unemployment rate of 5 percent.
b) Predict the approximate rate of inflation, given an unemployment rate of 15 percent.
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Estimating regression and relationship
 Witte web demonstration
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Least Squares Regression Equation
 Y = bX + a
 Where Y = the predicted value
 X = a known value
 b = slope of the line
 a = Y-intercept
 Ability to predict an outcome for a variable, given a
regression line and a value of a second paired variable.
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Calculating Least Squares Regression
Page 159
Calculate SSx, SSy, and r for the data.
2. Substitute numbers into the formula below
SSy
(r)
b=
SSx
3. Find the mean for X and mean for Y
4. Solve for a
1.
√
a = Y-(b)(X)
5. Solve for the predicted value
Y’ = (b)(X) + a
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Standard error of estimate
 This represents a special kind of standard deviation
that reflects the magnitude of predictive error.
 It is the difference between known values and
predicted values based on the regression equation.
 It is how much we over/under estimate a value based
on the regression equation, which is related to the
strength of the correlation.
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Calculation of standard error of estimate
Page 162
 Square root of the quantity of sum of squares for Y
times one minus r squared divided by n minus 2.
 (n-2 because 2 paired variable results in n-2 degrees of
freedom)
SSy(1-r2)
 Sy|x=
n-2
√
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Assumptions
 Use of regression equation requires that the
underlying relationship be linear.
 Use of the standard error of estimate assumes that
except for chance, the dots in the original scatterplot
will be dispersed equally about all segments of the
regression line. (homoscedasticity)
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Progress Check 7.2 Page 160-1
Educational Level (X)
Weekly Reading Time (Y)
X = 13
Y=8
SSx = 25
SSy = 50
R = .30
a) Determine the least squares equation for predicting
weekly reading time from educational level.
b) Faith’s education level is 15. What is her predicted
reading time?
c) Keegan’s educational level is 11. What is his predicted
reading time?
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Calculate Standard Error of Estimate
 Calculate the Standard Error of Estimate using the
data in 7.2 on page 160
Educational Level (X)
Weekly Reading Time (Y)
X = 13
Y=8
SSx = 25
SSy = 50
R = .30
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Correlation, Prediction, Error
X
Y
5
4
4
6
1
4
2
3
4
5
2
4
X2
Y2
X*Y
X - mean number of cases of influenza in a month among employees
Y - mean number of bacteria x 1 million on a door knob on front door
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Regression toward the mean
 Extreme scores on multiple trials will tend toward the
mean.
 The regression fallacy: accepting that regression
toward the mean is real, rather than a chance effect.
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Regression toward the mean
 Tversky and Kahnemann
 Study of Israeli Air Force pilots in 1974
 Some trainees were praised after good landings, while
others were reprimanded after bad landings.
 On their next landings, praised trainees did more
poorly and reprimanded trainees did better.
 Conclusion:
 Praise hinders but a reprimand helps performance!
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