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Least-Squares Regression• Linear relationships
between two variables are defined using a regression line• A mathematical expression
(equation)• Simplest type of
relationship – straight line
Least-Squares Regression• How do we find the best straight
line that relates two variables?• Take a ruler and try to fit it?• How many answers would that
give?
• The most common practice is to find the Least-Squares Line – How?• Find a line that passes through
points such that there is a minimum distance between each value of the response variable and the line.
• These distances are squared and added up for all points in the sample
• For the least-squares line, that sum is smaller than it would be for any other line
Least-Squares RegressionEquation of any straight line
• Response variables on vertical axis – Y• Explanatory variable on horizontal axis
–X• The equation that relates these two
Y=a + bXWhere a and b are numbers
• a: intercept – point where line crosses the y axis, when X=0
• b: slope – how much an increase there is in variable Y when variable X increases by 1 unit• Positive slope• Negative slope
Least-Squares Regression
• Regression line:• Y = a + bX
Where b is the slope calculated by
And the intercept “a”
Where and represent the coordinates of a point on the line
x
y
SSrb
xbya
yx
Least-Squares Regression• It is possible to find a
regression line if we know the slope and one point through which the line passes.
• Example: What is the equation of the least-squares line that has the following characteristics?
•
4and3,5.0,5,20 yxrSS xy
Least-Squares Regression
• When developing a regression equation, it matters which is the explanatory and which is the response variable.
• When calculating the equation we consider distances from the response variable to the line.
• Reversing the roles will in turn produce different equations.
Correlation coefficient – r• r describes the strength of the
relationshipIn regression• r2 = fraction of variation in
values of Y that is explained by X
Example:• The correlation between IQ
and GPA was found to be • a: r=0.50 • b: r=0.99
• What percent of the observed variation in the students’ GPAs can be explained by IQ alone?